Theory Spec_Monad

(*
 * Copyright (c) 2024 Apple Inc. All rights reserved.
 *
 * SPDX-License-Identifier: BSD-2-Clause
 *)

text_raw ‹\part{AutoCorres}›

chapter ‹Spec-Monad›

theory Spec_Monad
  imports
  "Basic_Runs_To_VCG"
  "HOL-Library.Complete_Partial_Order2"
  "HOL-Library.Monad_Syntax"
  "AutoCorres_Utils"
begin

section ‹‹rel_map› and ‹rel_project››

definition rel_map :: "('a ⇒ 'b) ⇒ 'a ⇒ 'b ⇒ bool" where
  "rel_map f r x = (x = f r)"

lemma rel_map_direct[simp]: "rel_map f a (f a)"
  by (simp add: rel_map_def)

abbreviation "rel_project ≡ rel_map"

lemmas rel_project_def = rel_map_def

lemma rel_project_id: "rel_project id = (=)" 
   "rel_project (λv. v) = (=)"
  by (auto simp add: rel_project_def fun_eq_iff)

lemma rel_project_unit: "rel_project (λ_. ()) x y = True"
  by (simp add: rel_project_def)

lemma rel_projectI: "y = prj x ⟹ rel_project prj x y"
  by (simp add: rel_project_def)

lemma rel_project_conv: "rel_project prj x y = (y = prj x)"
  by (simp add: rel_project_def)

section ‹Misc Theorems›

declare case_unit_Unity [simp] ― ‹without this rule simplifier seems loops in unexpected ways ›

lemma abs_const_unit: "(λ(v::unit). f) = (λ(). f)"
  by auto

lemma SUP_mono'': "(⋀x. x∈A ⟹ f x ≤ g x) ⟹ (⨆x∈A. f x) ≤ (⨆x∈A. g x :: _::complete_lattice)"
  by (rule SUP_mono) auto

lemma wf_nat_bound: "wf {(a, b). b < a ∧ b ≤ (n::nat)}"
  apply (rule wf_subset)
  apply (rule wf_measure[where f="λa. Suc n - a"])
  apply auto
  done

lemma (in complete_lattice) admissible_le:
  "ccpo.admissible Inf (≤) (λx. (y ≤ x))"
  by (simp add: ccpo.admissibleI local.Inf_greatest)

lemma mono_const: "mono (λx. c)"
  by (simp add: mono_def)

lemma mono_lam: "(⋀a. mono (λx. F x a)) ⟹ mono F"
  by (simp add: mono_def le_fun_def)

lemma mono_app: "mono (λx. x a)"
  by (simp add: mono_def le_fun_def)

lemma all_cong_map:
  assumes f_f': "⋀y. f (f' y) = y" and P_Q: "⋀x. P x ⟷ Q (f x)"
  shows "(∀x. P x) ⟷ (∀y. Q y)"
  using assms by metis

lemma rel_set_refl: "(⋀x. x ∈ A ⟹ R x x) ⟹ rel_set R A A"
  by (auto simp: rel_set_def)

lemma rel_set_converse_iff: "rel_set R X Y ⟷ rel_set R¯¯ Y X"
  by (auto simp add: rel_set_def)

lemma rel_set_weaken:
  "(⋀x y. x ∈ A ⟹ y ∈ B ⟹ P x y ⟹ Q x y) ⟹ rel_set P A B ⟹ rel_set Q A B"
  by (force simp: rel_set_def)

lemma sim_set_refl: "(⋀x. x ∈ X ⟹ R x x) ⟹ sim_set R X X"
  by (auto simp: sim_set_def)

section ‹Galois Connections›

lemma mono_of_Sup_cont:
  fixes f :: "'a::complete_lattice ⇒ 'b::complete_lattice"
  assumes cont: "⋀X. f (Sup X) = (SUP x∈X. f x)"
  assumes xy: "x ≤ y"
  shows "f x ≤ f y"
proof -
  have "f x ≤ sup (f x) (f y)" by (rule sup_ge1)
  also have "… = (SUP x∈{x, y}. f x)" by simp
  also have "… = f (Sup {x, y})" by (rule cont[symmetric])
  finally show ?thesis using xy by (simp add: sup_absorb2)
qed

lemma gc_of_Sup_cont:
  fixes f :: "'a::complete_lattice ⇒ 'b::complete_lattice"
  assumes cont: "⋀X. f (Sup X) = (SUP x∈X. f x)"
  shows "f x ≤ y ⟷ x ≤ Sup {x. f x ≤ y}"
proof safe
  assume "f x ≤ y" then show "x ≤ Sup {x. f x ≤ y}"
    by (intro Sup_upper) simp
next
  assume "x ≤ Sup {x. f x ≤ y}"
  then have "f x ≤ f (Sup {x. f x ≤ y})"
    by (rule mono_of_Sup_cont[OF cont])
  also have "… = (SUP x∈{x. f x ≤ y}. f x)" by (rule cont)
  also have "… ≤ y" by (rule Sup_least) auto
  finally show "f x ≤ y" .
qed

lemma mono_of_Inf_cont:
  fixes f :: "'a::complete_lattice ⇒ 'b::complete_lattice"
  assumes cont: "⋀X. f (Inf X) = (INF x∈X. f x)"
  assumes xy: "x ≤ y"
  shows "f x ≤ f y"
proof -
  have "f x = f (Inf {x, y})" using xy by (simp add: inf_absorb1)
  also have "… = (INF x∈{x, y}. f x)" by (rule cont)
  also have "… = inf (f x) (f y)" by simp
  also have "… ≤ f y" by (rule inf_le2)
  finally show ?thesis .
qed

lemma gc_of_Inf_cont:
  fixes f :: "'a::complete_lattice ⇒ 'b::complete_lattice"
  assumes cont: "⋀X. f (Inf X) = (INF x∈X. f x)"
  shows "Inf {y. x ≤ f y} ≤ y ⟷ x ≤ f y"
proof safe
  assume "x ≤ f y" then show "Inf {y. x ≤ f y} ≤ y"
    by (intro Inf_lower) simp
next
  assume *: "Inf {y. x ≤ f y} ≤ y"
  have "x ≤ (INF y∈{y. x ≤ f y}. f y)" by (rule Inf_greatest) auto
  also have "… = f (Inf {y. x ≤ f y})" by (rule cont[symmetric])
  also have "… ≤ f y"
    using * by (rule mono_of_Inf_cont[OF cont])
  finally show "x ≤ f y" .
qed

lemma gfp_fusion:
  assumes f_g: "⋀x y. g x ≤ y ⟷ x ≤ f y"
  assumes a: "mono a"
  assumes b: "mono b"
  assumes *: "⋀x. f (a x) = b (f x)"
  shows "f (gfp a) = gfp b"
  apply (intro antisym gfp_upperbound)
  subgoal
    apply (subst *[symmetric])
    apply (subst gfp_fixpoint[OF a])
    ..
  apply (rule f_g[THEN iffD1])
  apply (rule gfp_upperbound)
  apply (rule f_g[THEN iffD2])
  apply (subst *)
  apply (subst gfp_fixpoint[OF b, symmetric])
  apply (rule monoD[OF b])
  apply (rule f_g[THEN iffD1])
  apply (rule order_refl)
  done

lemma lfp_fusion:
  assumes f_g: "⋀x y. g x ≤ y ⟷ x ≤ f y"
  assumes a: "mono a"
  assumes b: "mono b"
  assumes *: "⋀x. g (a x) = b (g x)"
  shows "g (lfp a) = lfp b"
  apply (intro antisym lfp_lowerbound)
  subgoal
    apply (rule f_g[THEN iffD2])
    apply (rule lfp_lowerbound)
    apply (rule f_g[THEN iffD1])
    apply (subst *)
    apply (subst (2) lfp_fixpoint[OF b, symmetric])
    apply (rule monoD[OF b])
    apply (rule f_g[THEN iffD2])
    apply (rule order_refl)
    done
  subgoal
    apply (subst *[symmetric])
    apply (subst lfp_fixpoint[OF a])
    ..
  done

section ‹‹post_state› type›

datatype 'r post_state = Failure | Success "'r set"

text ‹
\<^const>‹Failure› is supposed to model things like undefined behaviour in C. We usually have
to show the absence of \<^const>‹Failure› for all possible executions of the program. Moreover,
it is used to model the 'result' of a non terminating computation.
›

lemma split_post_state:
  "x = (case x of Success X ⇒ Success X | Failure ⇒ Failure)"
  for x::"'a post_state"
  by (cases x) auto

instantiation post_state :: (type) order
begin

inductive less_eq_post_state :: "'a post_state ⇒ 'a post_state ⇒ bool" where
  Failure_le[simp, intro]: "less_eq_post_state p Failure"
| Success_le_Success[intro]: "r ⊆ q ⟹ less_eq_post_state (Success r) (Success q)"

definition less_post_state :: "'a post_state ⇒ 'a post_state ⇒ bool" where
  "less_post_state p q ⟷ p ≤ q ∧ ¬ q ≤ p"

instance
proof
  fix p q r :: "'a post_state"
  show "p ≤ p" by (cases p) auto
  show "p ≤ q ⟹ q ≤ r ⟹ p ≤ r" by (blast elim: less_eq_post_state.cases)
  show "p ≤ q ⟹ q ≤ p ⟹ p = q" by (blast elim: less_eq_post_state.cases)
qed (fact less_post_state_def)
end

lemma Success_le_Success_iff[simp]: "Success r ≤ Success q ⟷ r ⊆ q"
  by (auto elim: less_eq_post_state.cases)

lemma Failure_le_iff[simp]: "Failure ≤ q ⟷ q = Failure"
  by (auto elim: less_eq_post_state.cases)

instantiation post_state :: (type) complete_lattice
begin

definition top_post_state :: "'a post_state" where
  "top_post_state = Failure"

definition bot_post_state :: "'a post_state" where
  "bot_post_state = Success {}"

definition inf_post_state :: "'a post_state ⇒ 'a post_state ⇒ 'a post_state" where
  "inf_post_state =
    (λFailure ⇒ id | Success res1 ⇒
      (λFailure ⇒ Success res1 | Success res2 ⇒ Success (res1 ∩ res2)))"

definition sup_post_state :: "'a post_state ⇒ 'a post_state ⇒ 'a post_state" where
  "sup_post_state =
    (λFailure ⇒ (λ_. Failure) | Success res1 ⇒
      (λFailure ⇒ Failure | Success res2 ⇒ Success (res1 ∪ res2)))"

definition Inf_post_state :: "'a post_state set ⇒ 'a post_state" where
  "Inf_post_state s = (if Success -` s = {} then Failure else Success (⋂ (Success -` s)))"

definition Sup_post_state :: "'a post_state set ⇒ 'a post_state" where
  "Sup_post_state s = (if Failure ∈ s then Failure else Success (⋃ (Success -` s)))"

instance
proof
  fix x y z :: "'a post_state" and A :: "'a post_state set"
  show "inf x y ≤ y" by (simp add: inf_post_state_def split: post_state.split)
  show "inf x y ≤ x" by (simp add: inf_post_state_def split: post_state.split)
  show "x ≤ y ⟹ x ≤ z ⟹ x ≤ inf y z"
    by (auto simp add: inf_post_state_def elim!: less_eq_post_state.cases)
  show "x ≤ sup x y" by (simp add: sup_post_state_def split: post_state.split)
  show "y ≤ sup x y" by (simp add: sup_post_state_def split: post_state.split)
  show "x ≤ y ⟹ z ≤ y ⟹ sup x z ≤ y"
    by (auto simp add: sup_post_state_def elim!: less_eq_post_state.cases)
  show "x ∈ A ⟹ Inf A ≤ x" by (cases x) (auto simp add: Inf_post_state_def)
  show "(⋀x. x ∈ A ⟹ z ≤ x) ⟹ z ≤ Inf A" by (cases z) (force simp add: Inf_post_state_def)+
  show "x ∈ A ⟹ x ≤ Sup A" by (cases x) (auto simp add: Sup_post_state_def)
  show "(⋀x. x ∈ A ⟹ x ≤ z) ⟹ Sup A ≤ z" by (cases z) (force simp add: Sup_post_state_def)+
qed (simp_all add: top_post_state_def Inf_post_state_def bot_post_state_def Sup_post_state_def)
end

primrec holds_post_state :: "('a ⇒ bool) ⇒ 'a post_state ⇒ bool" where
  "holds_post_state P Failure ⟷ False"
| "holds_post_state P (Success X) ⟷ (∀x∈X. P x)"

primrec holds_partial_post_state :: "('a ⇒ bool) ⇒ 'a post_state ⇒ bool" where
  "holds_partial_post_state P Failure ⟷ True"
| "holds_partial_post_state P (Success X) ⟷ (∀x∈X. P x)"

inductive
  sim_post_state :: "('a ⇒ 'b ⇒ bool) ⇒ 'a post_state ⇒ 'b post_state ⇒ bool"
  for R
where
  [simp]: "⋀p. sim_post_state R p Failure"
| "⋀A B. sim_set R A B ⟹ sim_post_state R (Success A) (Success B)"

inductive
  rel_post_state :: "('a ⇒ 'b ⇒ bool) ⇒ 'a post_state ⇒ 'b post_state ⇒ bool"
  for R
where
  [simp]: "rel_post_state R Failure Failure"
| "⋀A B. rel_set R A B ⟹ rel_post_state R (Success A) (Success B)"

primrec lift_post_state :: "('a ⇒ 'b ⇒ bool) ⇒ 'b post_state ⇒ 'a post_state" where
  "lift_post_state R Failure = Failure"
| "lift_post_state R (Success Y) = Success {x. ∃y∈Y. R x y}"

primrec unlift_post_state :: "('a ⇒ 'b ⇒ bool) ⇒ 'a post_state ⇒ 'b post_state" where
  "unlift_post_state R Failure = Failure"
| "unlift_post_state R (Success X) = Success {y. ∀x. R x y ⟶ x ∈ X}"

primrec map_post_state :: "('a ⇒ 'b) ⇒ 'a post_state ⇒ 'b post_state" where
  "map_post_state f Failure  = Failure"
| "map_post_state f (Success r) = Success (f ` r)"

primrec vmap_post_state :: "('a ⇒ 'b) ⇒ 'b post_state ⇒ 'a post_state" where
  "vmap_post_state f Failure  = Failure"
| "vmap_post_state f (Success r) = Success (f -` r)"

definition pure_post_state :: "'a ⇒ 'a post_state" where
  "pure_post_state v = Success {v}"

primrec bind_post_state :: "'a post_state ⇒ ('a ⇒ 'b post_state) ⇒ 'b post_state" where
  "bind_post_state Failure p = Failure"
| "bind_post_state (Success res) p = ⨆ (p ` res)"

subsection ‹Order Properties›

lemma top_ne_bot[simp]: "(⊥:: _ post_state) ≠ ⊤"
  and bot_ne_top[simp]: "(⊤:: _ post_state) ≠ ⊥"
  by (auto simp: top_post_state_def bot_post_state_def)

lemma Success_ne_top[simp]:
  "Success X ≠ ⊤" "⊤ ≠ Success X"
  by (auto simp: top_post_state_def)

lemma Success_eq_bot_iff[simp]:
  "Success X = ⊥ ⟷ X = {}" "⊥ = Success X ⟷ X = {}"
  by (auto simp: bot_post_state_def)

lemma Sup_Success: "(⨆x∈X. Success (f x)) = Success (⋃x∈X. f x)"
  by (auto simp: Sup_post_state_def)

lemma Sup_Success_pair: "(⨆(x, y)∈X. Success (f x y)) = Success (⋃(x, y)∈X. f x y)"
  by (simp add: split_beta' Sup_Success)

subsection ‹‹holds_post_state››

lemma holds_post_state_iff:
  "holds_post_state P p ⟷ (∃X. p = Success X ∧ (∀x∈X. P x))"
  by (cases p) auto

lemma holds_post_state_weaken:
  "holds_post_state P p ⟹ (⋀x. P x ⟹ Q x) ⟹ holds_post_state Q p"
  by (cases p; auto)

lemma holds_post_state_combine:
  "holds_post_state P p ⟹ holds_post_state Q p ⟹
    (⋀x. P x ⟹ Q x ⟹ R x) ⟹ holds_post_state R p"
  by (cases p; auto)

lemma holds_post_state_Ball:
  "Y ≠ {} ∨ p ≠ Failure ⟹
    holds_post_state (λx. ∀y∈Y. P x y) p ⟷ (∀y∈Y. holds_post_state (λx. P x y) p)"
  by (cases p; auto)

lemma holds_post_state_All:
  "holds_post_state (λx. ∀y. P x y) p ⟷ (∀y. holds_post_state (λx. P x y) p)"
  by (cases p; auto)

lemma holds_post_state_BexI:
  "y ∈ Y ⟹ holds_post_state (λx. P x y) p ⟹ holds_post_state (λx. ∃y∈Y. P x y) p"
  by (cases p; auto)

lemma holds_post_state_conj:
  "holds_post_state (λx. P x ∧ Q x) p ⟷ holds_post_state P p ∧ holds_post_state Q p"
  by (cases p; auto)

lemma sim_post_state_iff:
  "sim_post_state R p q ⟷
    (∀P. holds_post_state (λy. ∀x. R x y ⟶ P x) q ⟶ holds_post_state P p)"
  apply (cases p; cases q; simp add: sim_set_def sim_post_state.simps)
  apply safe
  apply force
  apply force
  subgoal premises prems for A B
    using prems(3)[THEN spec, of "λa. ∃b∈B. R a b"] prems(4)
    by auto
  done

lemma post_state_le_iff:
  "p ≤ q ⟷ (∀P. holds_post_state P q ⟶ holds_post_state P p)"
  by (cases p; cases q; force simp add: subset_eq Ball_def)

lemma post_state_eq_iff:
  "p = q ⟷ (∀P. holds_post_state P p ⟷ holds_post_state P q)"
  by (simp add: order_eq_iff post_state_le_iff)

lemma holds_top_post_state[simp]: "¬ holds_post_state P ⊤"
  by (simp add: top_post_state_def)

lemma holds_bot_post_state[simp]: "holds_post_state P ⊥"
  by (simp add: bot_post_state_def)



lemma holds_Sup_post_state[simp]: "holds_post_state P (Sup F) ⟷ (∀f∈F. holds_post_state P f)"
  by (subst (2) split_post_state)
    (auto simp: Sup_post_state_def split: post_state.splits)

lemma holds_post_state_gfp:
  "holds_post_state P (gfp f) ⟷ (∀p. p ≤ f p ⟶ holds_post_state P p)"
  by (simp add: gfp_def)

lemma holds_post_state_gfp_apply:
  "holds_post_state P (gfp f x) ⟷ (∀p. p ≤ f p ⟶ holds_post_state P (p x))"
  by (simp add: gfp_def)

lemma holds_lift_post_state[simp]:
  "holds_post_state P (lift_post_state R x) ⟷ holds_post_state (λy. ∀x. R x y ⟶ P x) x"
  by (cases x) auto

lemma holds_map_post_state[simp]:
  "holds_post_state P (map_post_state f x) ⟷ holds_post_state (λx. P (f x)) x"
  by (cases x) auto

lemma holds_vmap_post_state[simp]:
  "holds_post_state P (vmap_post_state f x) ⟷ holds_post_state (λx. ∀y. f y = x ⟶ P y) x"
  by (cases x) auto

lemma holds_pure_post_state[simp]: "holds_post_state P (pure_post_state x) ⟷ P x"
  by (simp add: pure_post_state_def)

lemma holds_bind_post_state[simp]:
  "holds_post_state P (bind_post_state f g) ⟷ holds_post_state (λx. holds_post_state P (g x)) f"
  by (cases f) auto

lemma holds_post_state_False: "holds_post_state (λx. False) f ⟷ f = ⊥"
  by (cases f) (auto simp: bot_post_state_def)

subsection ‹‹holds_post_state_partial››

lemma holds_partial_post_state_of_holds:
  "holds_post_state P p ⟹ holds_partial_post_state P p"
  by (cases p; simp)

lemma holds_partial_post_state_iff:
  "holds_partial_post_state P p ⟷ (∀X. p = Success X ⟶ (∀x∈X. P x))"
  by (cases p) auto

lemma holds_partial_post_state_True[simp]: "holds_partial_post_state (λx. True) p"
  by (cases p; simp)

lemma holds_partial_post_state_weaken:
  "holds_partial_post_state P p ⟹ (⋀x. P x ⟹ Q x) ⟹ holds_partial_post_state Q p"
  by (cases p; auto)

lemma holds_partial_post_state_Ball:
  "holds_partial_post_state (λx. ∀y∈Y. P x y) p ⟷ (∀y∈Y. holds_partial_post_state (λx. P x y) p)"
  by (cases p; auto)

lemma holds_partial_post_state_All:
  "holds_partial_post_state (λx. ∀y. P x y) p ⟷ (∀y. holds_partial_post_state (λx. P x y) p)"
  by (cases p; auto)

lemma holds_partial_post_state_conj:
  "holds_partial_post_state (λx. P x ∧ Q x) p ⟷
    holds_partial_post_state P p ∧ holds_partial_post_state Q p"
  by (cases p; auto)

lemma holds_partial_top_post_state[simp]: "holds_partial_post_state P ⊤"
  by (simp add: top_post_state_def)

lemma holds_partial_bot_post_state[simp]: "holds_partial_post_state P ⊥"
  by (simp add: bot_post_state_def)

lemma holds_partial_pure_post_state[simp]: "holds_partial_post_state P (pure_post_state x) ⟷ P x"
  by (simp add: pure_post_state_def)

lemma holds_partial_Sup_post_stateI:
  "(⋀x. x ∈ X ⟹ holds_partial_post_state P x) ⟹ holds_partial_post_state P (Sup X)"
  by (force simp: Sup_post_state_def)

lemma holds_partial_bind_post_stateI:
  "holds_partial_post_state (λx. holds_partial_post_state P (g x)) f ⟹
    holds_partial_post_state P (bind_post_state f g)"
  by (cases f) (auto intro: holds_partial_Sup_post_stateI)

lemma holds_partial_map_post_state[simp]:
  "holds_partial_post_state P (map_post_state f x) ⟷ holds_partial_post_state (λx. P (f x)) x"
  by (cases x) auto

lemma holds_partial_vmap_post_state[simp]:
  "holds_partial_post_state P (vmap_post_state f x) ⟷
    holds_partial_post_state (λx. ∀y. f y = x ⟶ P y) x"
  by (cases x) auto

lemma holds_partial_bind_post_state:
  "holds_partial_post_state (λx. holds_partial_post_state P (g x)) f ⟹
    holds_partial_post_state P (bind_post_state f g)"
  by (cases f) (auto intro: holds_partial_Sup_post_stateI)

subsection ‹‹sim_post_state››

lemma sim_post_state_eq_iff_le: "sim_post_state (=) p q ⟷ p ≤ q"
  by (simp add: post_state_le_iff sim_post_state_iff)

lemma sim_post_state_Success_Success_iff[simp]:
  "sim_post_state R (Success r) (Success q) ⟷ (∀a∈r. ∃b∈q. R a b)"
  by (auto simp add: sim_set_def elim: sim_post_state.cases intro: sim_post_state.intros)

lemma sim_post_state_Success2:
  "sim_post_state R f (Success q) ⟷ holds_post_state (λa. ∃b∈q. R a b) f"
  by (cases f; simp add: sim_post_state.simps sim_set_def)

lemma sim_post_state_Failure1[simp]: "sim_post_state R Failure q ⟷ q = Failure"
  by (auto elim: sim_post_state.cases intro: sim_post_state.intros)

lemma sim_post_state_top2[simp, intro]: "sim_post_state R p ⊤"
  by (simp add: top_post_state_def)

lemma sim_post_state_top1[simp]: "sim_post_state R ⊤ q ⟷ q = ⊤"
  using sim_post_state_Failure1 by (metis top_post_state_def)

lemma sim_post_state_bot2[simp, intro]: "sim_post_state R p ⊥ ⟷ p = ⊥"
  by (cases p; simp add: bot_post_state_def)

lemma sim_post_state_bot1[simp, intro]: "sim_post_state R ⊥ q"
  by (cases q; simp add: bot_post_state_def)

lemma sim_post_state_le1: "sim_post_state R f' g ⟹ f ≤ f' ⟹ sim_post_state R f g"
  by (simp add: post_state_le_iff sim_post_state_iff)

lemma sim_post_state_le2: "sim_post_state R f g ⟹ g ≤ g' ⟹ sim_post_state R f g'"
  by (simp add: post_state_le_iff sim_post_state_iff)

lemma sim_post_state_Sup1:
  "sim_post_state R (Sup A) f ⟷ (∀a∈A. sim_post_state R a f)"
  by (auto simp add: sim_post_state_iff)

lemma sim_post_state_Sup2:
  "a ∈ A ⟹ sim_post_state R f a ⟹ sim_post_state R f (Sup A)"
  by (auto simp add: sim_post_state_iff)

lemma sim_post_state_Sup:
  "∀a∈A. ∃b∈B. sim_post_state R a b ⟹ sim_post_state R (Sup A) (Sup B)"
  by (auto simp: sim_post_state_Sup1 intro: sim_post_state_Sup2)

lemma sim_post_state_weaken:
  "sim_post_state R f g ⟹ (⋀x y. R x y ⟹ Q x y) ⟹ sim_post_state Q f g"
  by (cases f; cases g; force)

lemma sim_post_state_trans:
  "sim_post_state R f g ⟹ sim_post_state Q g h ⟹ (⋀x y z. R x y ⟹ Q y z ⟹ S x z) ⟹
    sim_post_state S f h"
  by (fastforce elim!: sim_post_state.cases intro: sim_post_state.intros simp: sim_set_def)

lemma sim_post_state_refl': "holds_partial_post_state (λx. R x x) f ⟹ sim_post_state R f f"
  by (cases f; auto simp: rel_set_def)

lemma sim_post_state_refl: "(⋀x. R x x) ⟹ sim_post_state R f f"
  by (simp add: sim_post_state_refl')

lemma sim_post_state_pure_post_state2:
  "sim_post_state F f (pure_post_state x) ⟷ holds_post_state (λy. F y x) f"
  by (cases f; simp add: pure_post_state_def)

subsection ‹‹rel_post_state››

lemma rel_post_state_top[simp, intro!]: "rel_post_state R ⊤ ⊤"
  by (auto simp: top_post_state_def)

lemma rel_post_state_top_iff[simp]:
  "rel_post_state R ⊤ p ⟷ p = ⊤"
  "rel_post_state R p ⊤ ⟷ p = ⊤"
  by (auto simp: top_post_state_def rel_post_state.simps)

lemma rel_post_state_Success_iff[simp]:
  "rel_post_state R (Success A) (Success B) ⟷ rel_set R A B"
  by (auto elim: rel_post_state.cases intro: rel_post_state.intros)

lemma rel_post_state_bot[simp, intro!]: "rel_post_state R ⊥ ⊥"
  by (auto simp: bot_post_state_def Lifting_Set.empty_transfer)

lemma rel_post_state_eq_sim_post_state:
  "rel_post_state R p q ⟷ sim_post_state R p q ∧ sim_post_state R¯¯ q p"
  by (auto simp: rel_set_def sim_set_def elim!: sim_post_state.cases rel_post_state.cases)

lemma rel_post_state_weaken:
  "rel_post_state R f g ⟹ (⋀x y. R x y ⟹ Q x y) ⟹ rel_post_state Q f g"
  by (auto intro: sim_post_state_weaken simp: rel_post_state_eq_sim_post_state)

lemma rel_post_state_eq[relator_eq]: "rel_post_state (=) = (=)"
  by (simp add: rel_post_state_eq_sim_post_state fun_eq_iff sim_post_state_eq_iff_le order_eq_iff)

lemma rel_post_state_mono[relator_mono]:
  "A ≤ B ⟹ rel_post_state A ≤ rel_post_state B"
  using rel_set_mono[of A B]
  by (auto simp add: le_fun_def intro!: rel_post_state.intros elim!: rel_post_state.cases)

lemma rel_post_state_refl': "holds_partial_post_state (λx. R x x) f ⟹ rel_post_state R f f"
  by (cases f; auto simp: rel_set_def)

lemma rel_post_state_refl: "(⋀x. R x x) ⟹ rel_post_state R f f"
  by (simp add: rel_post_state_refl')

subsection ‹‹lift_post_state››

lemma lift_post_state_top: "lift_post_state R ⊤ = ⊤"
  by (auto simp: top_post_state_def)

lemma lift_post_state_unlift_post_state: "lift_post_state R p ≤ q ⟷ p ≤ unlift_post_state R q"
  by (cases p; cases q; auto simp add: subset_eq)

lemma lift_post_state_eq_Sup: "lift_post_state R p = Sup {q. sim_post_state R q p}"
  unfolding post_state_eq_iff holds_Sup_post_state
  using holds_lift_post_state sim_post_state_iff
  by blast

lemma le_lift_post_state_iff: "q ≤ lift_post_state R p ⟷ sim_post_state R q p"
  by (simp add: post_state_le_iff sim_post_state_iff)

lemma lift_post_state_eq[simp]: "lift_post_state (=) p = p"
  by (simp add: post_state_eq_iff)

lemma lift_post_state_comp:
  "lift_post_state R (lift_post_state Q p) = lift_post_state (R OO Q) p"
  by (cases p) auto

lemma sim_post_state_lift:
  "sim_post_state Q q (lift_post_state R p) ⟷ sim_post_state (Q OO R) q p"
  unfolding le_lift_post_state_iff[symmetric] lift_post_state_comp ..

lemma lift_post_state_Sup: "lift_post_state R (Sup F) = (SUP f∈F. lift_post_state R f)"
  by (simp add: post_state_eq_iff)

lemma lift_post_state_mono: "p ≤ q ⟹ lift_post_state R p ≤ lift_post_state R q"
  by (simp add: post_state_le_iff)

subsection ‹‹map_post_state››

lemma map_post_state_top: "map_post_state f ⊤ = ⊤"
  by (auto simp: top_post_state_def)

lemma mono_map_post_state: "s1 ≤ s2 ⟹ map_post_state f s1  ≤ map_post_state f s2"
  by (simp add: post_state_le_iff)

lemma map_post_state_eq_lift_post_state: "map_post_state f p = lift_post_state (λa b. a = f b) p"
  by (cases p) auto

lemma map_post_state_Sup: "map_post_state f (Sup X) = (SUP x∈X. map_post_state f x)"
  by (simp add: post_state_eq_iff)

lemma map_post_state_comp: "map_post_state f (map_post_state g p) = map_post_state (f ∘ g) p"
  by (simp add: post_state_eq_iff)

lemma map_post_state_id[simp]: "map_post_state id p = p"
  by (simp add: post_state_eq_iff)

lemma map_post_state_pure[simp]: "map_post_state f (pure_post_state x) = pure_post_state (f x)"
  by (simp add: pure_post_state_def)

lemma sim_post_state_map_post_state1:
  "sim_post_state R (map_post_state f p) q ⟷ sim_post_state (λx y. R (f x) y) p q"
  by (cases p; cases q; simp)

lemma sim_post_state_map_post_state2:
  "sim_post_state R p (map_post_state f q) ⟷ sim_post_state (λx y. R x (f y)) p q"
  unfolding map_post_state_eq_lift_post_state sim_post_state_lift by (simp add: OO_def[abs_def])

lemma map_post_state_eq_top[simp]: "map_post_state f p = ⊤ ⟷ p = ⊤"
  by (cases p; simp add: top_post_state_def)

lemma map_post_state_eq_bot[simp]: "map_post_state f p = ⊥ ⟷ p = ⊥"
  by (cases p; simp add: bot_post_state_def)

subsection ‹‹vmap_post_state››

lemma vmap_post_state_top: "vmap_post_state f ⊤ = ⊤"
  by (auto simp: top_post_state_def)

lemma vmap_post_state_Sup:
  "vmap_post_state f (Sup X) = (SUP x∈X. vmap_post_state f x)"
  by (simp add: post_state_eq_iff)

lemma vmap_post_state_le_iff: "(vmap_post_state f p ≤ q) = (p ≤ ⨆ {p. vmap_post_state f p ≤ q})"
  using vmap_post_state_Sup by (rule gc_of_Sup_cont)

lemma vmap_post_state_eq_lift_post_state: "vmap_post_state f p = lift_post_state (λa b. f a = b) p"
  by (cases p) auto

lemma vmap_post_state_comp: "vmap_post_state f (vmap_post_state g p) = vmap_post_state (g ∘ f) p"
  apply (simp add: post_state_eq_iff)
  apply (intro allI arg_cong[where f="λP. holds_post_state P p"])
  apply auto
  done

lemma vmap_post_state_id: "vmap_post_state id p = p"
  by (simp add: post_state_eq_iff)

lemma sim_post_state_vmap_post_state2:
  "sim_post_state R p (vmap_post_state f q) ⟷
    sim_post_state (λx y. ∃y'. f y' = y ∧ R x y') p q"
  by (cases p; cases q; auto) blast+

lemma vmap_post_state_le_iff_le_map_post_state:
  "map_post_state f p ≤ q ⟷ p ≤ vmap_post_state f q"
  by (simp flip: sim_post_state_eq_iff_le
           add: sim_post_state_map_post_state1 sim_post_state_vmap_post_state2)

subsection ‹‹pure_post_state››

lemma pure_post_state_Failure[simp]: "pure_post_state v ≠ Failure"
  by (simp add: pure_post_state_def)

lemma pure_post_state_top[simp]: "pure_post_state v ≠ ⊤"
  by (simp add: pure_post_state_def top_post_state_def)

lemma pure_post_state_bot[simp]: "pure_post_state v ≠ ⊥"
  by (simp add: pure_post_state_def bot_post_state_def)

lemma pure_post_state_inj[simp]: "pure_post_state v = pure_post_state w ⟷ v = w"
  by (simp add: pure_post_state_def)

lemma sim_pure_post_state_iff[simp]:
  "sim_post_state R (pure_post_state a) (pure_post_state b) ⟷ R a b"
  by (simp add: pure_post_state_def)

lemma rel_pure_post_state_iff[simp]:
  "rel_post_state R (pure_post_state a) (pure_post_state b) ⟷ R a b"
  by (simp add: rel_post_state_eq_sim_post_state)

lemma pure_post_state_le[simp]: "pure_post_state v ≤ pure_post_state w ⟷ v = w"
  by (simp add: pure_post_state_def)

lemma Success_eq_pure_post_state[simp]: "Success X = pure_post_state x ⟷ X = {x}"
  by (auto simp add: pure_post_state_def)

lemma pure_post_state_eq_Success[simp]: "pure_post_state x = Success X ⟷ X = {x}"
  by (auto simp add: pure_post_state_def)

lemma pure_post_state_le_Success[simp]: "pure_post_state x ≤ Success X ⟷ x ∈ X"
  by (auto simp add: pure_post_state_def)

lemma Success_le_pure_post_state[simp]: "Success X ≤ pure_post_state x ⟷ X ⊆ {x}"
  by (auto simp add: pure_post_state_def)

subsection ‹‹bind_post_state››

lemma bind_post_state_top: "bind_post_state ⊤ g = ⊤"
  by (auto simp: top_post_state_def)

lemma bind_post_state_Sup1[simp]:
  "bind_post_state (Sup F) g = (SUP f∈F. bind_post_state f g)"
  by (simp add: post_state_eq_iff)

lemma bind_post_state_Sup2[simp]:
  "G ≠ {} ∨ f ≠ Failure ⟹ bind_post_state f (Sup G) = (SUP g∈G. bind_post_state f g)"
  by (simp add: post_state_eq_iff holds_post_state_Ball)

lemma bind_post_state_sup1[simp]:
  "bind_post_state (sup f1 f2) g = sup (bind_post_state f1 g) (bind_post_state f2 g)"
  using bind_post_state_Sup1[of "{f1, f2}" g] by simp

lemma bind_post_state_sup2[simp]:
  "bind_post_state f (sup g1 g2) = sup (bind_post_state f g1) (bind_post_state f g2)"
  using bind_post_state_Sup2[of "{g1, g2}" f] by simp

lemma bind_post_state_top1[simp]: "bind_post_state ⊤ f = ⊤"
  by (simp add: post_state_eq_iff)

lemma bind_post_state_bot1[simp]: "bind_post_state ⊥ f = ⊥"
  by (simp add: post_state_eq_iff)

lemma bind_post_state_eq_top:
  "bind_post_state f g = ⊤ ⟷ ¬ holds_post_state (λx. g x ≠ ⊤) f"
  by (cases f) (force simp add: Sup_post_state_def simp flip: top_post_state_def)+

lemma bind_post_state_eq_bot:
  "bind_post_state f g = ⊥ ⟷ holds_post_state (λx. g x = ⊥) f"
  by (cases f) (auto simp flip: top_post_state_def)

lemma lift_post_state_bind_post_state:
  "lift_post_state R (bind_post_state x g) = bind_post_state x (λv. lift_post_state R (g v))"
  by (simp add: post_state_eq_iff)

lemma vmap_post_state_bind_post_state:
  "vmap_post_state f (bind_post_state p g) = bind_post_state p (λv. vmap_post_state f (g v))"
  by (simp add: post_state_eq_iff)

lemma map_post_state_bind_post_state:
  "map_post_state f (bind_post_state x g) = bind_post_state x (λv. map_post_state f (g v))"
  by (simp add: post_state_eq_iff)

lemma bind_post_state_pure_post_state1[simp]:
  "bind_post_state (pure_post_state v) f = f v"
  by (simp add: post_state_eq_iff)

lemma bind_post_state_pure_post_state2:
  "bind_post_state p (λx. pure_post_state (f x)) = map_post_state f p"
  by (simp add: post_state_eq_iff)

lemma bind_post_state_map_post_state:
  "bind_post_state (map_post_state f p) g = bind_post_state p (λx. g (f x))"
  by (simp add: post_state_eq_iff)

lemma bind_post_state_assoc[simp]:
  "bind_post_state (bind_post_state f g) h =
    bind_post_state f (λv. bind_post_state (g v) h)"
  by (simp add: post_state_eq_iff)

lemma sim_bind_post_state':
  "sim_post_state (λx y. sim_post_state R (g x) (k y)) f h ⟹
    sim_post_state R (bind_post_state f g) (bind_post_state h k)"
  by (cases f; cases h; simp add: sim_post_state_Sup)

lemma sim_bind_post_state_left_iff:
  "sim_post_state R (bind_post_state f g) h ⟷
    h = Failure ∨ holds_post_state (λx. sim_post_state R (g x) h) f"
  by (cases f; cases h) (simp_all add: sim_post_state_Sup1)

lemma sim_bind_post_state_left:
  "holds_post_state (λx. sim_post_state R (g x) h) f ⟹
    sim_post_state R (bind_post_state f g) h"
  by (simp add: sim_bind_post_state_left_iff)

lemma sim_bind_post_state_right:
  "g ≠ ⊥ ⟹ holds_partial_post_state (λx. sim_post_state R f (h x)) g ⟹
    sim_post_state R f (bind_post_state g h) "
  by (cases g) (auto intro: sim_post_state_Sup2)

lemma sim_bind_post_state:
  "sim_post_state Q f h ⟹ (⋀x y. Q x y ⟹ sim_post_state R (g x) (k y)) ⟹
    sim_post_state R (bind_post_state f g) (bind_post_state h k)"
  by (rule sim_bind_post_state'[OF sim_post_state_weaken, of Q])

lemma rel_bind_post_state':
  "rel_post_state (λa b. rel_post_state R (g1 a) (g2 b)) f1 f2 ⟹
    rel_post_state R (bind_post_state f1 g1) (bind_post_state f2 g2)"
  by (auto simp: rel_post_state_eq_sim_post_state
           intro: sim_bind_post_state' sim_post_state_weaken)

lemma rel_bind_post_state:
  "rel_post_state Q f1 f2 ⟹ (⋀a b. Q a b ⟹ rel_post_state R (g1 a) (g2 b)) ⟹
    rel_post_state R (bind_post_state f1 g1) (bind_post_state f2 g2)"
  by (rule rel_bind_post_state'[OF rel_post_state_weaken, of Q])

lemma mono_bind_post_state: "f1 ≤ f2 ⟹ g1 ≤ g2 ⟹ bind_post_state f1 g1 ≤ bind_post_state f2 g2"
  by (simp flip: sim_post_state_eq_iff_le add: le_fun_def sim_bind_post_state)


lemma Sup_eq_Failure[simp]: "Sup X = Failure ⟷ Failure ∈ X"
  by (simp add: Sup_post_state_def)

lemma Failure_inf_iff: "(Failure = x ⊓ y) ⟷ (x = Failure ∧ y = Failure)"
  by (metis Failure_le_iff le_inf_iff)

lemma Success_vimage_singleton_cancel: "Success -` {Success X} = {X}"
  by auto

lemma Success_vimage_image_cancel: "Success -` (λx. Success (f x)) ` X = f ` X"
  by auto

lemma Success_image_comp: "(λx. Success (g x)) ` X = Success ` (g ` X)"
  by auto

section ‹‹exception_or_result› type›

instantiation option :: (type) default
begin

definition "default_option = None"

instance ..
end

lemma Some_ne_default[simp]:
  "Some x ≠ default" "default ≠ Some x"
  "default = None ⟷ True" "None = default ⟷ True"
  by (auto simp: default_option_def)

typedef (overloaded) ('a::default, 'b) exception_or_result =
  "Inl ` (UNIV - {default}) ∪ Inr ` UNIV :: ('a + 'b) set"
  by auto

setup_lifting type_definition_exception_or_result

context assumes "SORT_CONSTRAINT('e::default)"
begin

lift_definition Exception :: "'e ⇒ ('e, 'v) exception_or_result" is
  "λe. if e = default then Inr undefined else Inl e"
  by auto

lift_definition Result :: "'v ⇒ ('e, 'v) exception_or_result" is
  "λv. Inr v"
  by auto

end

lift_definition
  case_exception_or_result :: "('e::default ⇒ 'a) ⇒ ('v ⇒ 'a) ⇒ ('e, 'v) exception_or_result ⇒ 'a"
  is case_sum .

declare [[case_translation case_exception_or_result Exception Result]]

lemma Result_eq_Result[simp]: "Result a = Result b ⟷ a = b"
  by transfer simp

lemma Exception_eq_Exception[simp]: "Exception a = Exception b ⟷ a = b"
  by transfer simp

lemma Result_eq_Exception[simp]: "Result a = Exception e ⟷ (e = default ∧ a = undefined)"
  by transfer simp

lemma Exception_eq_Result[simp]: "Exception e = Result a ⟷ (e = default ∧ a = undefined)"
  by (metis Result_eq_Exception)

lemma exception_or_result_cases[case_names Exception Result, cases type: exception_or_result]:
  "(⋀e. e ≠ default ⟹ x = Exception e ⟹ P) ⟹ (⋀v. x = Result v ⟹ P) ⟹ P"
  by (transfer fixing: P) auto

lemma case_exception_or_result_Result[simp]:
  "(case Result v of Exception e ⇒ f e | Result w ⇒ g w) = g v"
  by transfer simp

lemma case_exception_or_result_Exception[simp]:
  "(case Exception e of Exception e ⇒ f e | Result w ⇒ g w) = (if e = default then g undefined else f e)"
  by transfer simp


text ‹
Caution: for split rules don't use syntax
\<^term>‹(case r of Exception e ⇒ f e | Result w ⇒ g w)›, as this introduces
non eta-contracted \<^term>‹λe. f e› and \<^term>‹λw. g w›, which don't work with the
splitter.
›
lemma exception_or_result_split:
  "P (case_exception_or_result f g r)  ⟷
    (∀e. e ≠ default ⟶ r = Exception e ⟶ P (f e)) ∧
    (∀v. r = Result v ⟶ P (g v))"
  by (cases r) simp_all

lemma exception_or_result_split_asm:
  "P (case_exception_or_result f g r) ⟷
  ¬ ((∃e. r = Exception e ∧ e ≠ default ∧ ¬ P (f e)) ∨
    (∃v. r = Result v ∧ ¬ P (g v)))"
  by (cases r) simp_all

lemmas exception_or_result_splits = exception_or_result_split exception_or_result_split_asm

lemma split_exception_or_result: 
  "r = (case r of Exception e ⇒ Exception e | Result v ⇒ Result v)"
  by (cases r) auto

lemma exception_or_result_nchotomy:
  "¬ ((∀e. e ≠ default ⟶ x ≠ Exception e) ∧ (∀v. x ≠ Result v))" by (cases x) auto

lemma val_split:
  fixes r::"(unit, 'a) exception_or_result"
  shows
  "P (case_exception_or_result f g r)  ⟷
    (∀v. r = Result v ⟶ P (g v))"
  by (cases r) simp_all

lemma val_split_asm:
  fixes r::"(unit, 'a) exception_or_result"
  shows "P (case_exception_or_result f g r) ⟷
    ¬ (∃v. r = Result v ∧ ¬ P (g v))"
  by (cases r) simp_all

lemmas val_splits[split] = val_split val_split_asm

instantiation exception_or_result :: ("{equal, default}", equal) equal begin

definition "equal_exception_or_result a b =
  (case a of
    Exception e ⇒ (case b of Exception e' ⇒ HOL.equal e e' | Result s ⇒ False)
  | Result r    ⇒ (case b of Exception e ⇒ False   | Result s ⇒ HOL.equal r s)
)"
instance proof qed (simp add: equal_exception_or_result_def equal_eq
    split: exception_or_result_splits)

end

definition is_Exception:: "('e::default, 'a) exception_or_result ⇒ bool" where
  "is_Exception x ≡ (case x of Exception e ⇒ e ≠ default | Result _ ⇒ False)"

definition is_Result:: "('e::default, 'a) exception_or_result ⇒ bool" where
  "is_Result x ≡ (case x of Exception e ⇒ e = default | Result _ ⇒ True)"

definition the_Exception::  "('e::default, 'a) exception_or_result ⇒ 'e" where
  "the_Exception x ≡ (case x of Exception e ⇒ e | Result _ ⇒ default)"

definition the_Result::  "('e::default, 'a) exception_or_result ⇒ 'a" where
  "the_Result x ≡ (case x of Exception e ⇒ undefined | Result v ⇒ v)"

lemma is_Exception_simps[simp]:
  "is_Exception (Exception e) = (e ≠ default)"
  "is_Exception (Result v) = False"
  by (auto simp add: is_Exception_def)

lemma is_Result_simps[simp]:
  "is_Result (Exception e) = (e = default)"
  "is_Result (Result v) = True"
  by (auto simp add: is_Result_def)

lemma the_Exception_simp[simp]:
  "the_Exception (Exception e) = e"
  by (auto simp add: the_Exception_def)

lemma the_Exception_Result:
  "the_Exception (Result v) = default"
  by (simp add: the_Exception_def)

syntax "_Res" :: "pttrn ⇒ pttrn" ("Res _")

definition undefined_unit::"unit ⇒ 'b" where "undefined_unit x ≡ undefined"

translations "λRes x. b" ⇌ "CONST case_exception_or_result (CONST undefined_unit) (λx. b)"

term "λRes x. f x"
term "λRes (x, y, z). f x y z"
term "λRes (x, y, z) s. P x y s z"


lifting_update exception_or_result.lifting
lifting_forget exception_or_result.lifting

inductive rel_exception_or_result:: "('e::default ⇒ 'f::default ⇒ bool) ⇒ ('a ⇒ 'b ⇒ bool) ⇒
  ('e, 'a) exception_or_result ⇒ ('f, 'b) exception_or_result ⇒ bool"
  where
Exception:
  "E e f ⟹ e ≠ default ⟹ f ≠ default ⟹
  rel_exception_or_result E R (Exception e) (Exception f)" |
Result:
  "R a b ⟹ rel_exception_or_result E R (Result a) (Result b)"

lemma All_exception_or_result_cases:
  "(∀x. P (x::(_, _) exception_or_result)) ⟷
    (∀err. err ≠ default ⟶ P (Exception err)) ∧  (∀v. P (Result v))"
  by (metis exception_or_result_cases)

lemma Ball_exception_or_result_cases:
  "(∀x∈s. P (x::(_, _) exception_or_result)) ⟷
    (∀err. err ≠ default ⟶ Exception err ∈ s ⟶ P (Exception err)) ∧
    (∀v. Result v ∈ s ⟶ P (Result v))"
  by (metis exception_or_result_cases)

lemma Bex_exception_or_result_cases:
  "(∃x∈s. P (x::(_, _) exception_or_result)) ⟷
    (∃err. err ≠ default ∧ Exception err ∈ s ∧ P (Exception err)) ∨
    (∃v. Result v ∈ s ∧ P (Result v))"
  by (metis exception_or_result_cases)

lemma Ex_exception_or_result_cases:
  "(∃x. P (x::(_, _) exception_or_result)) ⟷
    (∃err. err ≠ default ∧ P (Exception err)) ∨
    (∃v. P (Result v))"
  by (metis exception_or_result_cases)

lemma case_distrib_exception_or_result:
  "f (case x of Exception e ⇒ E e | Result v ⇒ R v) = (case x of Exception e ⇒ f (E e) |  Result v ⇒ f (R v))"
  by (auto split: exception_or_result_split)

lemma rel_exception_or_result_Results[simp]:
  "rel_exception_or_result E R (Result a) (Result b) ⟷ R a b"
  by (simp add: rel_exception_or_result.simps)

lemma rel_exception_or_result_Exception[simp]:
  "e ≠ default ⟹ f ≠ default ⟹
    rel_exception_or_result E R (Exception e) (Exception f) ⟷ E e f"
  by (simp add: rel_exception_or_result.simps)

lemma rel_exception_or_result_Result_Exception[simp]:
  "e ≠ default ⟹ ¬ rel_exception_or_result E R (Exception e) (Result b)"
  "f ≠ default ⟹ ¬ rel_exception_or_result E R (Result a) (Exception f)"
  by (auto simp add: rel_exception_or_result.simps)

lemma is_Exception_iff: "is_Exception x ⟷ (∃e. e ≠ default ∧ x = Exception e)"
  apply (cases x)
   apply (auto)
  done

lemma rel_exception_or_result_converse:
  "(rel_exception_or_result E R)¯¯ = rel_exception_or_result E¯¯ R¯¯"
  apply (rule ext)+
  apply (auto simp add: rel_exception_or_result.simps)
  done

lemma rel_eception_or_result_Result[simp]:
  "rel_exception_or_result E R (Result x) (Result y) = R x y"
  by (auto simp add: rel_exception_or_result.simps)

lemma rel_exception_or_result_eq_conv: "rel_exception_or_result (=) (=) = (=)"
  apply (rule ext)+
  by (auto simp add: rel_exception_or_result.simps)
    (meson exception_or_result_cases)

lemma rel_exception_or_result_sum_eq: "rel_exception_or_result (=) (=) = (=)"
  apply (rule ext)+
  apply (subst (1) split_exception_or_result)
  apply (auto simp add: rel_exception_or_result.simps split:  exception_or_result_splits)
  done

definition
  map_exception_or_result::
    "('e::default ⇒ 'f::default) ⇒ ('a ⇒ 'b) ⇒ ('e, 'a) exception_or_result ⇒
      ('f, 'b) exception_or_result"
where
  "map_exception_or_result E F x =
    (case x of Exception e ⇒ Exception (E e) | Result v ⇒ Result (F v))"

lemma map_exception_or_result_Exception[simp]:
  "e ≠ default ⟹ map_exception_or_result E F (Exception e) = Exception (E e)"
  by (simp add: map_exception_or_result_def)

lemma map_exception_or_result_Result[simp]:
  "map_exception_or_result E F (Result v) = Result (F v)"
  by (simp add: map_exception_or_result_def)

lemma map_exception_or_result_id: "map_exception_or_result id id x = x"
  by (cases x; simp)

lemma map_exception_or_result_comp:
  assumes E2: "⋀x. x ≠ default ⟹ E2 x ≠ default"
  shows "map_exception_or_result E1 F1 (map_exception_or_result E2 F2 x) =
    map_exception_or_result (E1 ∘ E2) (F1 ∘ F2) x"
  by (cases x; auto dest: E2)

lemma le_bind_post_state_exception_or_result_cases[case_names Exception Result]:
  assumes
    "holds_partial_post_state (λ(x, t). ∀e. e ≠ default ⟶ x = Exception e ⟶ X e t ≤ X' e t) x"
  assumes "holds_partial_post_state (λ(x, t). ∀v. x = Result v ⟶ V v t ≤ V' v t) x"
  shows "bind_post_state x (λ(r, t). case r of Exception e ⇒ X e t | Result v ⇒ V v t) ≤
    bind_post_state x (λ(r, t). case r of Exception e ⇒ X' e t| Result v ⇒ V' v t)"
  using assms
  by (cases x; force intro!: SUP_mono'' split: exception_or_result_splits prod.splits)

section ‹‹spec_monad› type›

typedef (overloaded) ('e::default, 'a, 's) spec_monad =
  "UNIV :: ('s ⇒ (('e::default, 'a) exception_or_result × 's) post_state) set"
  morphisms run Spec
  by (fact UNIV_witness)

lemma run_case_prod_distrib[simp]:
  "run (case x of (r, s) ⇒ f r s) t = (case x of (r, s) ⇒ run (f r s) t)"
  by (rule prod.case_distrib)

lemma image_case_prod_distrib[simp]:
  "(λ(r, t). f r t) ` (λv. (g v, h v)) ` R = (λv. (f (g v) (h v))) ` R "
  by auto

lemma run_Spec[simp]: "run (Spec p) s = p s"
  by (simp add: Spec_inverse)

setup_lifting type_definition_spec_monad

lemma spec_monad_ext: "(⋀s. run f s = run g s) ⟹ f = g"
  apply transfer
  apply auto
  done

lemma spec_monad_ext_iff: "f = g ⟷ (∀s. run f s = run g s)"
  using spec_monad_ext by auto

instantiation spec_monad :: (default, type, type) complete_lattice
begin

lift_definition
  less_eq_spec_monad :: "('e::default, 'r, 's) spec_monad ⇒ ('e, 'r, 's) spec_monad ⇒ bool"
  is "(≤)" .

lift_definition
  less_spec_monad :: "('e::default, 'r, 's) spec_monad ⇒ ('e, 'r, 's) spec_monad ⇒ bool"
  is "(<)" .

lift_definition bot_spec_monad :: "('e::default, 'r, 's) spec_monad" is "bot" .

lift_definition top_spec_monad :: "('e::default, 'r, 's) spec_monad" is "top" .

lift_definition Inf_spec_monad :: "('e::default, 'r, 's) spec_monad set ⇒ ('e, 'r, 's) spec_monad"
  is "Inf" .

lift_definition Sup_spec_monad :: "('e::default, 'r, 's) spec_monad set ⇒ ('e, 'r, 's) spec_monad"
  is "Sup" .

lift_definition
  sup_spec_monad ::
    "('e::default, 'r, 's) spec_monad ⇒ ('e, 'r, 's) spec_monad ⇒ ('e, 'r, 's) spec_monad"
  is "sup" .

lift_definition
  inf_spec_monad ::
    "('e::default, 'r, 's) spec_monad ⇒ ('e, 'r, 's) spec_monad ⇒ ('e, 'r, 's) spec_monad"
  is "inf" .

instance
  apply (standard; transfer)
  subgoal by (rule less_le_not_le)
  subgoal by (rule order_refl)
  subgoal by (rule order_trans)
  subgoal by (rule antisym)
  subgoal by (rule inf_le1)
  subgoal by (rule inf_le2)
  subgoal by (rule inf_greatest)
  subgoal by (rule sup_ge1)
  subgoal by (rule sup_ge2)
  subgoal by (rule sup_least)
  subgoal by (rule Inf_lower)
  subgoal by (rule Inf_greatest)
  subgoal by (rule Sup_upper)
  subgoal by (rule Sup_least)
  subgoal by (rule Inf_empty)
  subgoal by (rule Sup_empty)
  done
end

lift_definition fail :: "('e::default, 'a, 's) spec_monad"
  is "λs. Failure" .

lift_definition
  bind_exception_or_result :: 
    "('e::default, 'a, 's) spec_monad ⇒
      (('e, 'a) exception_or_result ⇒ ('f::default, 'b, 's) spec_monad) ⇒ ('f, 'b, 's) spec_monad"
is
  "λf h s. bind_post_state (f s) (λ(v, t). h v t)" .

lift_definition bind_handle ::
    "('e::default, 'a, 's) spec_monad ⇒
      ('a ⇒ ('f, 'b, 's) spec_monad) ⇒ ('e ⇒ ('f, 'b, 's) spec_monad) ⇒
      ('f::default, 'b, 's) spec_monad"
  is "λf g h s. bind_post_state (f s) (λ(r, t). case r of Exception e ⇒ h e t | Result v ⇒ g v t)" .

lift_definition yield :: "('e, 'a) exception_or_result ⇒ ('e::default, 'a, 's) spec_monad"
  is "λr s. pure_post_state (r, s)" .

abbreviation "return r ≡ yield (Result r)"

abbreviation "skip ≡ return ()"

abbreviation "throw_exception_or_result e ≡ yield (Exception e)"

lift_definition get_state :: "('e::default, 's, 's) spec_monad"
  is "λs. pure_post_state (Result s, s)" .

lift_definition set_state :: "'s ⇒ ('e::default, unit, 's) spec_monad"
  is "λt s. pure_post_state (Result (), t)" .

lift_definition map_value ::
    "(('e::default, 'a) exception_or_result ⇒ ('f::default, 'b) exception_or_result) ⇒
      ('e, 'a, 's) spec_monad ⇒ ('f, 'b, 's) spec_monad"
  is "λf g s. map_post_state (apfst f) (g s)" .

lift_definition vmap_value ::
    "(('e::default, 'a) exception_or_result ⇒ ('f::default, 'b) exception_or_result) ⇒
      ('f, 'b, 's) spec_monad ⇒ ('e, 'a, 's) spec_monad"
  is "λf g s. vmap_post_state (apfst f) (g s)" .

definition bind ::
    "('e::default, 'a, 's) spec_monad ⇒ ('a ⇒ ('e, 'b, 's) spec_monad) ⇒ ('e, 'b, 's) spec_monad"
where
  "bind f g = bind_handle f g throw_exception_or_result"

adhoc_overloading
  Monad_Syntax.bind bind

lift_definition lift_state ::
    "('s ⇒ 't ⇒ bool) ⇒ ('e::default, 'a, 't) spec_monad ⇒ ('e, 'a, 's) spec_monad"
  is "λR p s. lift_post_state (rel_prod (=) R) (SUP t∈{t. R s t}. p t)" .

definition exec_concrete ::
  "('s ⇒ 't) ⇒ ('e::default, 'a, 's) spec_monad ⇒ ('e, 'a, 't) spec_monad"
where
  "exec_concrete st = lift_state (λt s. t = st s)"

definition exec_abstract ::
  "('s ⇒ 't) ⇒ ('e::default, 'a, 't) spec_monad ⇒ ('e, 'a, 's) spec_monad"
where
  "exec_abstract st = lift_state (λs t. t = st s)"

definition select_value :: "('e, 'a) exception_or_result set ⇒ ('e::default, 'a, 's) spec_monad" where
  "select_value R = (SUP r∈R. yield r)"

definition select :: "'a set ⇒ ('e::default, 'a, 's) spec_monad" where
  "select S = (SUP a∈S. return a)"

definition unknown :: "('e::default, 'a, 's) spec_monad" where
  "unknown ≡ select UNIV"

definition gets :: "('s ⇒ 'a) ⇒ ('e::default, 'a, 's) spec_monad" where
  "gets f = bind get_state (λs. return (f s))"

definition assert_opt :: "'a option ⇒ ('e::default, 'a, 's) spec_monad" where
  "assert_opt v = (case v of None ⇒ fail | Some v ⇒ return v)"

definition gets_the :: "('s ⇒ 'a option) ⇒ ('e::default, 'a, 's) spec_monad" where
  "gets_the f = bind (gets f) assert_opt"

definition modify :: "('s ⇒ 's) ⇒ ('e::default, unit, 's) spec_monad" where
  "modify f = bind get_state (λs. set_state (f s))"

definition "assume" :: "bool ⇒ ('e::default, unit, 's) spec_monad" where
  "assume P = (if P then return () else bot)"

definition assert :: "bool ⇒ ('e::default, unit, 's) spec_monad" where
  "assert P = (if P then return () else top)"

definition assuming :: "('s ⇒ bool) ⇒ ('e::default, unit, 's) spec_monad" where
  "assuming P = do {s  ← get_state; assume (P s)}"

definition guard:: "('s ⇒ bool) ⇒  ('e::default, unit, 's) spec_monad" where
  "guard P = do {s ← get_state; assert (P s)}"

definition assume_result_and_state :: "('s ⇒ ('a × 's) set) ⇒ ('e::default, 'a, 's) spec_monad" where
  "assume_result_and_state f = do {s ← get_state; (v, t) ← select (f s); set_state t; return v}"

(* TODO: delete *)
lift_definition assume_outcome ::
    "('s ⇒ (('e, 'a) exception_or_result × 's) set) ⇒ ('e::default, 'a, 's) spec_monad"
  is "λf s. Success (f s)" .

definition assert_result_and_state ::
    "('s ⇒ ('a × 's) set) ⇒ ('e::default, 'a, 's) spec_monad"
where
  "assert_result_and_state f =
    do {s ← get_state; assert (f s ≠ {}); (v, t) ← select (f s); set_state t; return v}"

abbreviation state_select :: "('s × 's) set ⇒ ('e::default, unit, 's) spec_monad" where
  "state_select r ≡ assert_result_and_state (λs. {((), s'). (s, s') ∈ r})"

definition condition ::
    "('s ⇒ bool) ⇒ ('e::default, 'a, 's) spec_monad ⇒ ('e, 'a, 's) spec_monad ⇒
      ('e, 'a, 's) spec_monad"
where
  "condition C T F =
    bind get_state (λs. if C s then T else F)"

notation (output)
  condition  ("(condition (_)//  (_)//  (_))" [1000,1000,1000] 999)

definition "when" ::"bool ⇒ ('e::default, unit, 's) spec_monad ⇒ ('e, unit, 's) spec_monad"
  where "when c f ≡ condition (λ_. c) f skip"

abbreviation "unless" ::"bool ⇒ ('e::default, unit, 's) spec_monad ⇒ ('e, unit, 's) spec_monad"
  where "unless c ≡ when (¬c)"

definition on_exit' ::
    "('e::default, 'a, 's) spec_monad ⇒ ('e, unit, 's) spec_monad ⇒ ('e, 'a, 's) spec_monad"
where
  "on_exit' f c ≡
     bind_exception_or_result f (λr. do { c; yield r })"

definition on_exit ::
    "('e::default, 'a, 's) spec_monad ⇒ ('s × 's) set ⇒ ('e, 'a, 's) spec_monad"
where
  "on_exit f cleanup ≡ on_exit' f (state_select cleanup)"

abbreviation guard_on_exit ::
    "('e::default, 'a, 's) spec_monad ⇒ ('s ⇒ bool) ⇒ ('s × 's) set ⇒ ('e, 'a, 's) spec_monad"
where
  "guard_on_exit f grd cleanup ≡ on_exit' f (bind (guard grd) (λ_. state_select cleanup))"

abbreviation assume_on_exit ::
    "('e::default, 'a, 's) spec_monad ⇒ ('s ⇒ bool) ⇒ ('s × 's) set ⇒ ('e, 'a, 's) spec_monad"
where
  "assume_on_exit f grd cleanup ≡
    on_exit' f (bind (assuming grd) (λ_. state_select cleanup))"

lift_definition run_bind ::
    "('e::default, 'a, 't) spec_monad ⇒ 't ⇒
     (('e, 'a) exception_or_result ⇒ 't ⇒ ('f::default, 'b, 's) spec_monad) ⇒
     ('f::default, 'b, 's) spec_monad"
  is "λf t g s. bind_post_state (f t) (λ(r, t). g r t s)" .
    ― ‹\<^const>‹run_bind› might be a more canonical building block compared to
   \<^const>‹lift_state›. See subsection on \<^const>‹run_bind› below.›

type_synonym ('e, 'a, 's) predicate = "('e, 'a) exception_or_result ⇒ 's ⇒ bool"

lift_definition runs_to ::
    "('e::default, 'a, 's) spec_monad ⇒ 's ⇒ ('e, 'a, 's) predicate ⇒ bool"
    ("_/ ∙ _ ⦃ _ ⦄" [61, 1000, 0] 30) ― ‹syntax ‹_do_block› has 62›
  is "λf s Q. holds_post_state (λ(r, t). Q r t) (f s)" .

lift_definition runs_to_partial ::
    "('e::default, 'a, 's) spec_monad ⇒ 's ⇒ ('e, 'a, 's) predicate ⇒ bool"
    ("_/ ∙ _ ?⦃ _ ⦄" [61, 1000, 0] 30)
  is "λf s Q. holds_partial_post_state (λ(r, t). Q r t) (f s)" .

lift_definition refines ::
  "('e::default, 'a, 's) spec_monad ⇒ ('f::default, 'b, 't) spec_monad ⇒ 's ⇒ 't ⇒
    ((('e, 'a) exception_or_result × 's) ⇒ (('f, 'b) exception_or_result × 't) ⇒ bool) ⇒ bool"
  is "λf g s t R. sim_post_state R (f s) (g t)" .

lift_definition rel_spec ::
  "('e::default, 'a, 's) spec_monad ⇒ ('f::default, 'b, 't) spec_monad ⇒ 's ⇒ 't ⇒
     ((('e, 'a) exception_or_result × 's) ⇒ (('f, 'b) exception_or_result × 't) ⇒ bool)
   ⇒ bool"
is
  "λf g s t R. rel_post_state R (f s) (g t)" .

definition rel_spec_monad ::
    "('s ⇒ 't ⇒ bool) ⇒ (('e, 'a) exception_or_result ⇒ ('f, 'b) exception_or_result ⇒ bool) ⇒
      ('e::default, 'a, 's) spec_monad ⇒ ('f::default, 'b, 't) spec_monad ⇒ bool"
where
  "rel_spec_monad R Q f g =
    (∀s t. R s t ⟶ rel_post_state (rel_prod Q R) (run f s) (run g t))"

lift_definition always_progress :: "('e::default, 'a, 's) spec_monad ⇒ bool" is
  "λp. ∀s. p s ≠ bot" .

named_theorems run_spec_monad "Simplification rules to run a Spec monad"
named_theorems runs_to_iff "Equivalence theorems for runs to predicate"
named_theorems always_progress_intros "intro rules for always_progress predicate"

lemma runs_to_partialI: "(⋀r x. P r x) ⟹ f ∙ s ?⦃ P ⦄"
  by (cases "run f s") (auto simp: runs_to_partial.rep_eq)

lemma runs_to_partial_True: "f ∙ s ?⦃ λr s. True ⦄"
  by (simp add: runs_to_partialI)

lemma runs_to_partial_conj: 
  "f ∙ s ?⦃ P ⦄ ⟹ f ∙ s ?⦃ Q ⦄ ⟹ f ∙ s ?⦃ λr s. P r s ∧ Q r s ⦄"
  by (cases "run f s") (auto simp: runs_to_partial.rep_eq)

lemma refines_iff':
  "refines f g s t R ⟷ (∀P. g ∙ t ⦃ λr s. ∀p t. R (p, t) (r, s) ⟶ P p t ⦄ ⟶ f ∙ s ⦃ P ⦄)"
  apply transfer
  apply (simp add: sim_post_state_iff le_fun_def split_beta' all_comm[where 'a='a])
  apply (rule all_cong_map[where f'="λP (x, y). P x y" and f="λP x y. P (x, y)"])
  apply auto
  done

lemma refines_weaken:
  "refines f g s t R ⟹ (⋀r s q t. R (r, s) (q, t) ⟹ Q (r, s) (q, t)) ⟹ refines f g s t Q"
  by transfer (auto intro: sim_post_state_weaken)

lemma refines_mono:
  "(⋀r s q t. R (r, s) (q, t) ⟹ Q (r, s) (q, t)) ⟹ refines f g s t R ⟹ refines f g s t Q"
  by (rule refines_weaken)

lemma refines_refl: "(⋀r t. R (r, t) (r, t)) ⟹ refines f f s s R"
  by transfer (auto intro: sim_post_state_refl)

lemma refines_trans:
  "refines f g s t R ⟹ refines g h t u Q ⟹
    (⋀r s p t q u. R (r, s) (p, t) ⟹ Q (p, t) (q, u) ⟹ S (r, s) (q, u)) ⟹
    refines f h s u S"
  by transfer (auto intro: sim_post_state_trans)

lemma refines_trans': "refines f g s t1 R ⟹ refines g h t1 t2 Q ⟹ refines f h s t2 (R OO Q)"
  by (rule refines_trans; assumption?) auto

lemma refines_strengthen:
  "refines f g s t R ⟹ f ∙ s ?⦃ F ⦄ ⟹ g ∙ t ?⦃ G ⦄ ⟹
    (⋀x s y t. R (x, s) (y, t) ⟹ F x s ⟹ G y t ⟹ Q (x, s) (y, t)) ⟹
  refines f g s t Q"
  by transfer (fastforce elim!: sim_post_state.cases simp: sim_set_def split_beta')

lemma refines_strengthen1:
  "refines f g s t R ⟹ f ∙ s ?⦃ F ⦄ ⟹
    (⋀x s y t. R (x, s) (y, t) ⟹ F x s ⟹ Q (x, s) (y, t)) ⟹
  refines f g s t Q"
  by (rule refines_strengthen[OF _ _ runs_to_partial_True]; assumption?)

lemma refines_strengthen2:
  "refines f g s t R ⟹ g ∙ t ?⦃ G ⦄ ⟹
    (⋀x s y t. R (x, s) (y, t) ⟹ G y t ⟹ Q (x, s) (y, t)) ⟹
  refines f g s t Q"
  by (rule refines_strengthen[OF _ runs_to_partial_True]; assumption?)

lemma refines_cong_cases:
  assumes "⋀e s e' s'. e ≠ default ⟹ e' ≠ default ⟹
    R (Exception e, s) (Exception e', s') ⟷ Q (Exception e, s) (Exception e', s')"
  assumes "⋀e s x' s'. e ≠ default ⟹
    R (Exception e, s) (Result x', s') ⟷ Q (Exception e, s) (Result x', s')"
  assumes "⋀x s e' s'. e' ≠ default ⟹
    R (Result x, s) (Exception e', s') ⟷ Q (Result x, s) (Exception e', s')"
  assumes "⋀x s x' s'. R (Result x, s) (Result x', s') ⟷ Q (Result x, s) (Result x', s')"
  shows "refines f f' s s' R ⟷ refines f f' s s' Q"
  apply (clarsimp intro!: arg_cong[where f="refines f f' s s'"] simp: fun_eq_iff del: iffI)
  subgoal for v s v' s' by (cases v; cases v'; simp add: assms)
  done

lemma runs_to_partial_weaken[runs_to_vcg_weaken]:
  "f ∙ s ?⦃Q⦄ ⟹ (⋀r t. Q r t ⟹ Q' r t) ⟹ f ∙ s ?⦃Q'⦄"
  by transfer (auto intro: holds_partial_post_state_weaken)

lemma runs_to_weaken[runs_to_vcg_weaken]: "f ∙ s ⦃Q⦄ ⟹ (⋀r t. Q r t ⟹ Q' r t) ⟹ f ∙ s ⦃Q'⦄"
  by transfer (auto intro: holds_post_state_weaken)

lemma runs_to_partial_imp_runs_to_partial[mono]:
  "(⋀a s. P a s ⟶ Q a s) ⟹ f ∙ s ?⦃ P ⦄ ⟶ f ∙ s ?⦃ Q ⦄"
  using runs_to_partial_weaken[of f s P Q] by auto

lemma runs_to_imp_runs_to[mono]:
  "(⋀a s. P a s ⟶ Q a s) ⟹ f ∙ s ⦃ P ⦄ ⟶ f ∙ s ⦃ Q ⦄"
  using runs_to_weaken[of f s P Q] by auto

lemma refines_imp_refines[mono]:
  "(⋀a s. P a s ⟶ Q a s) ⟹ refines f g s t P ⟶ refines f g s t Q"
  using refines_weaken[of f g s t P Q] by auto

lemma runs_to_cong_cases:
  assumes "⋀e s. e ≠ default ⟹ P (Exception e) s ⟷ Q (Exception e) s"
  assumes "⋀x s. P (Result x) s ⟷ Q (Result x) s"
  shows "f ∙ s ⦃ P ⦄ ⟷ f ∙ s ⦃ Q ⦄"
  apply (clarsimp intro!: arg_cong[where f="runs_to _ _"] simp: fun_eq_iff del: iffI)
  subgoal for v s by (cases v; simp add: assms)
  done

lemma le_spec_monad_le_refines_iff: "f ≤ g ⟷ (∀s. refines f g s s (=))"
  by transfer (simp add: le_fun_def sim_post_state_eq_iff_le)

lemma spec_monad_le_iff: "f ≤ g ⟷ (∀P (s::'a). g ∙ s ⦃ P ⦄ ⟶ f ∙ s ⦃ P ⦄)"
  by (simp add: le_spec_monad_le_refines_iff refines_iff' all_comm[where 'a='a])

lemma spec_monad_eq_iff: "f = g ⟷ (∀P s. f ∙ s ⦃ P ⦄ ⟷ g ∙ s ⦃ P ⦄)"
  by (simp add: order_eq_iff spec_monad_le_iff)

lemma spec_monad_eqI: "(⋀P s. f ∙ s ⦃ P ⦄ ⟷ g ∙ s ⦃ P ⦄) ⟹ f = g"
  by (simp add: spec_monad_eq_iff)

lemma runs_to_Sup_iff: "(Sup X) ∙ s ⦃ P ⦄ ⟷ (∀x∈X. x ∙ s ⦃ P ⦄)"
  by transfer simp

lemma runs_to_lfp:
  assumes f: "mono f"
  assumes x_s: "P x s"
  assumes *: "⋀p. (⋀x s. P x s ⟹ p x ∙ s ⦃ R ⦄) ⟹ (⋀x s. P x s ⟹ f p x ∙ s ⦃ R ⦄)"
  shows "lfp f x ∙ s ⦃ R ⦄"
proof -
  have "∀x s. P x s ⟶ lfp f x ∙ s ⦃ R ⦄"
    apply (rule lfp_ordinal_induct[OF f])
    subgoal using * by blast
    subgoal by (simp add: runs_to_Sup_iff)
    done
  with x_s show ?thesis by auto
qed

lemma runs_to_partial_alt:
  "f ∙ s ?⦃ Q ⦄ ⟷ run f s = ⊤ ∨ f ∙ s ⦃ Q ⦄"
  by (cases "run f s"; simp add: runs_to.rep_eq runs_to_partial.rep_eq top_post_state_def)

lemma runs_to_of_runs_to_partial_runs_to':
  "f ∙ s ⦃ P ⦄ ⟹ f ∙ s ?⦃ Q ⦄ ⟹ f ∙ s ⦃ Q ⦄"
  by (auto simp: runs_to_partial_alt runs_to.rep_eq)

lemma runs_to_partial_of_runs_to:
  "f ∙ s ⦃ Q ⦄ ⟹ f ∙ s ?⦃ Q ⦄"
  by (auto simp: runs_to_partial_alt)

lemma runs_to_partial_tivial[simp]: "f ∙ s ?⦃ λ_ _. True ⦄"
  by (cases "run f s"; simp add: runs_to_partial.rep_eq)

lemma le_spec_monad_le_run_iff: "f ≤ g ⟷ (∀s. run f s ≤ run g s)"
  by (simp add: le_fun_def less_eq_spec_monad.rep_eq)

lemma refines_le_run_trans: "refines f g s s1 R ⟹ run g s1 ≤ run h s2 ⟹ refines f h s s2 R"
  by transfer (rule sim_post_state_le2)

lemma le_run_refines_trans: "run f s ≤ run g s1 ⟹ refines g h s1 s2 R ⟹ refines f h s s2 R"
  by transfer (rule sim_post_state_le1)

lemma refines_le_trans: "refines f g s t R ⟹ g ≤ h ⟹ refines f h s t R"
  by (auto simp: le_spec_monad_le_run_iff refines_le_run_trans)

lemma le_refines_trans: "f ≤ g ⟹ refines g h s t R ⟹ refines f h s t R"
  by (auto simp: le_spec_monad_le_run_iff intro: le_run_refines_trans)

lemma le_run_refines_iff: "run f s ≤ run g t ⟷ refines f g s t (=)"
  by transfer (simp add: sim_post_state_eq_iff_le)

lemma refines_top[simp]: "refines f ⊤ s t R"
  unfolding top_spec_monad.rep_eq refines.rep_eq by simp

lemma refines_Sup1: "refines (Sup F) g s s' R ⟷ (∀f∈F. refines f g s s' R)"
  by (simp add: refines.rep_eq Sup_spec_monad.rep_eq image_image sim_post_state_Sup1)

lemma monotone_le_iff_refines:
  "monotone R (≤) F ⟷ (∀x y. R x y ⟶ (∀s. refines (F x) (F y) s s (=)))"
  by (auto simp: monotone_def le_fun_def le_run_refines_iff[symmetric] le_spec_monad_le_run_iff)

lemma refines_iff_runs_to:
  "refines f g s t R ⟷ (∀P. g ∙ t ⦃ λr t'. ∀p s'. R (p, s') (r, t') ⟶ P p s' ⦄ ⟶ f ∙ s ⦃ P ⦄)"
  by transfer (auto simp: sim_post_state_iff split_beta')

lemma runs_to_refines_weaken':
  "refines f g s t R ⟹ g ∙ t ⦃ λr t'. ∀p s'. R (p, s') (r, t') ⟶ P p s' ⦄ ⟹ f ∙ s ⦃ P ⦄"
  by (simp add: refines_iff_runs_to)

lemma runs_to_refines_weaken: "refines f f' s t (=) ⟹ f' ∙ t ⦃ P ⦄ ⟹ f ∙ s ⦃ P ⦄"
  by (auto simp: runs_to_refines_weaken')

lemma refinesD_runs_to:
  assumes f_g: "refines f g s t R" and g: "g ∙ t ⦃ P ⦄"
  shows "f ∙ s ⦃ λr t. ∃r' t'. R (r, t) (r', t') ∧ P r' t' ⦄"
  by (rule runs_to_refines_weaken'[OF f_g]) (auto intro: runs_to_weaken[OF g])

lemma rel_spec_iff_refines:
  "rel_spec f g s t R ⟷ refines f g s t R ∧ refines g f t s (R¯¯)"
  unfolding rel_spec_def by transfer (simp add: rel_post_state_eq_sim_post_state)

lemma rel_spec_symm: "rel_spec f g s t R ⟷ rel_spec g f t s R¯¯"
  by (simp add: rel_spec_iff_refines ac_simps)

lemma rel_specD_refines1: "rel_spec f g s t R ⟹ refines f g s t R"
  by (simp add: rel_spec_iff_refines)

lemma rel_spec_mono: "P ≤ Q ⟹ rel_spec f g s t P ⟹ rel_spec f g s t Q"
  using rel_post_state_mono by (auto simp: rel_spec_def)

lemma rel_spec_eq: "rel_spec f g s t (=) ⟷ run f s = run g t"
  by (simp add: rel_spec_def rel_post_state_eq)

lemma rel_spec_eq_conv: "(∀s. rel_spec f g s s (=)) ⟷ f = g"
  using rel_spec_eq spec_monad_ext by metis

lemma rel_spec_eqD: "(⋀s. rel_spec f g s s (=)) ⟹ f = g"
  using rel_spec_eq_conv by metis

lemma rel_spec_refl': "f ∙ s ?⦃ λr s. R (r, s) (r, s)⦄ ⟹ rel_spec f f s s R"
  by transfer  (simp add: rel_post_state_refl' split_beta')

lemma rel_spec_refl: "(⋀r s. R (r, s) (r, s)) ⟹ rel_spec f f s s R"
  by (rule rel_spec_mono[OF _ rel_spec_eq[THEN iffD2]]) auto

lemma refines_same_runs_to_partialI:
  "f ∙ s ?⦃λr s'. R (r, s') (r, s')⦄ ⟹ refines f f s s R"
  by transfer
     (auto intro!: sim_post_state_refl' simp: split_beta')

lemma refines_same_runs_toI:
  "f ∙ s ⦃λr s'. R (r, s') (r, s')⦄ ⟹ refines f f s s R"
  by (rule refines_same_runs_to_partialI[OF runs_to_partial_of_runs_to])

lemma always_progress_case_prod[always_progress_intros]:
  "always_progress (f (fst p) (snd p)) ⟹ always_progress (case_prod f p)"
  by (simp add: split_beta)

lemma refines_runs_to_partial_fuse:
  "refines f f' s s' Q ⟹ f ∙ s ?⦃P⦄ ⟹
    refines f f' s s' (λ(r,t) (r',t'). Q (r,t) (r',t') ∧ P r t)"
  by transfer (auto elim!: sim_post_state.cases simp: sim_set_def split_beta')

lemma runs_to_refines:
  "f' ∙ s' ⦃ P ⦄ ⟹ refines f f' s s' Q ⟹ (⋀x s y t. P x s ⟹ Q (y, t) (x, s) ⟹ R y t) ⟹
    f ∙ s ⦃ R ⦄"
  by transfer (force elim!: sim_post_state.cases simp: sim_set_def split_beta')

lemma runs_to_partial_subst_Res: "f ∙ s ?⦃λr. P (the_Result r)⦄ ⟷ f ∙ s ?⦃λRes r. P r⦄"
  by (intro arg_cong[where f="λP. f ∙ s ?⦃ P ⦄"]) (auto simp: fun_eq_iff the_Result_def)

lemma runs_to_subst_Res: "f ∙ s ⦃λr. P (the_Result r)⦄ ⟷ f ∙ s ⦃λRes r. P r⦄"
  by (intro arg_cong[where f="λP. f ∙ s ⦃ P ⦄"]) (auto simp: fun_eq_iff the_Result_def)

lemma runs_to_Res[simp]: "f ∙ s ⦃λr t. ∀v. r = Result v ⟶ P v t⦄ ⟷ f ∙ s ⦃λRes v t. P v t⦄"
  by (intro arg_cong[where f="λP. f ∙ s ⦃ P ⦄"]) (auto simp: fun_eq_iff the_Result_def)

lemma runs_to_partial_Res[simp]:
  "f ∙ s ?⦃λr t. ∀v. r = Result v ⟶ P v t⦄ ⟷ f ∙ s ?⦃λRes v t. P v t⦄"
  by (intro arg_cong[where f="λP. f ∙ s ?⦃ P ⦄"]) (auto simp: fun_eq_iff the_Result_def)

lemma rel_spec_runs_to:
  assumes f: "f ∙ s ⦃ P ⦄" "always_progress f"
    and g: "g ∙ t ⦃ Q ⦄" "always_progress g"
    and P: "(⋀r s' p t'. P r s' ⟹ Q p t' ⟹ R (r, s') (p, t'))"
  shows "rel_spec f g s t R"
  using assms unfolding rel_spec_def always_progress.rep_eq runs_to.rep_eq
  by (cases "run f s"; cases "run g t"; simp add: rel_set_def split_beta')
     (metis all_not_in_conv bot_post_state_def prod.exhaust_sel)

lemma runs_to_res_independent_res: "f ∙ s ⦃λ_. P⦄ ⟷ f ∙ s ⦃λRes _. P⦄"
  by (rule runs_to_subst_Res)

lemma lift_state_spec_monad_eq[simp]: "lift_state (=) p = p"
  by transfer (simp add: prod.rel_eq fun_eq_iff rel_post_state_eq)

lemma rel_post_state_Sup:
  "rel_set (λx y. rel_post_state Q (f x) (g y)) X Y ⟹ rel_post_state Q (⨆x∈X. f x) (⨆x∈Y. g x)"
  by (force simp: rel_post_state_eq_sim_post_state rel_set_def intro!: sim_post_state_Sup)

lemma rel_set_Result_image_iff:
  "rel_set (rel_prod (λRes v1 Res v2. (P v1 v2)) R)
     ((λx. case x of (v, s) ⇒ (Result v, s)) ` Vals1)
     ((λx. case x of (v, s) ⇒ (Result v, s)) ` Vals2)
   ⟷
   rel_set (rel_prod P R) Vals1 Vals2"
  by (force simp add: rel_set_def)

lemma rel_post_state_converse_iff:
  "rel_post_state R X Y ⟷ rel_post_state R¯¯ Y X"
  by (metis conversep_conversep rel_post_state_eq_sim_post_state)

lemma runs_to_le_post_state_iff: "runs_to f s Q ⟷ run f s ≤ Success {(r, s). Q r s}"
  by (cases "run f s") (auto simp add: runs_to.rep_eq)

lemma runs_to_partial_runs_to_iff: "runs_to_partial f s Q ⟷ (run f s = Failure ∨ runs_to f s Q)"
  by (cases "run f s"; simp add: runs_to_partial.rep_eq runs_to.rep_eq)

lemma run_runs_to_extensionality:
  "run f s = run g s ⟷ (∀P. f ∙ s ⦃P⦄ ⟷ g ∙ s ⦃P⦄)"
  apply transfer
  apply (simp add: post_state_eq_iff)
  apply (rule all_cong_map[where f'="λP (x, y). P x y" and f="λP x y. P (x, y)"])
  apply auto
  done

lemma le_spec_monad_runI: "(⋀s. run f s ≤ run g s) ⟹ f ≤ g"
  by transfer (simp add: le_fun_def)

subsection ‹\<^const>‹rel_spec_monad››

lemma rel_spec_monad_iff_rel_spec:
  "rel_spec_monad R Q f g ⟷ (∀s t. R s t ⟶ rel_spec f g s t (rel_prod Q R))"
  unfolding rel_spec_monad_def rel_fun_def rel_spec.rep_eq ..

lemma rel_spec_monadI:
  "(⋀s t. R s t ⟹ rel_spec f g s t (rel_prod Q R)) ⟹ rel_spec_monad R Q f g"
  by (auto simp: rel_spec_monad_iff_rel_spec)

lemma rel_spec_monadD:
  "rel_spec_monad R Q f g ⟹ R s t ⟹ rel_spec f g s t (rel_prod Q R)"
  by (auto simp: rel_spec_monad_iff_rel_spec)

lemma rel_spec_monad_eq_conv: "rel_spec_monad (=) (=) = (=)"
  unfolding rel_spec_monad_def by transfer (simp add: fun_eq_iff prod.rel_eq rel_post_state_eq)

lemma rel_spec_monad_converse_iff:
  "rel_spec_monad R Q f g ⟷ rel_spec_monad R¯¯ Q¯¯ g f"
  using rel_post_state_converse_iff
  by (metis conversep.cases conversep.intros prod.rel_conversep rel_spec_monad_def)

lemma rel_spec_monad_iff_refines:
  "rel_spec_monad S R f g ⟷
    (∀s t. S s t ⟶ (refines f g s t (rel_prod R S) ∧ refines g f t s (rel_prod R¯¯ S¯¯)))"
  using rel_spec_iff_refines rel_spec_monad_iff_rel_spec
  by (metis prod.rel_conversep)

lemma rel_spec_monad_rel_exception_or_resultI:
  "rel_spec_monad R (rel_exception_or_result (=) (=)) f g ⟹ rel_spec_monad R (=) f g"
  by (simp add: rel_exception_or_result_sum_eq)

lemma runs_to_partial_runs_to_fuse: 
  assumes part: "f ∙ s ?⦃Q⦄"
  assumes tot: "f ∙ s ⦃P⦄"
  shows "f ∙ s ⦃λr t. Q r t ∧ P r t⦄"
  using assms
  apply (cases "run f s")
   apply (auto simp add: runs_to_def runs_to_partial_def)
  done

section ‹VCG basic setup›

lemma runs_to_cong_pred_only:
  "P = Q ⟹ (p ∙ s ⦃ P ⦄) ⟷ (p ∙ s ⦃ Q ⦄)"
  by simp

lemma runs_to_cong_state_only[runs_to_vcg_cong_state_only]:
  "s = t ⟹ (p ∙ s ⦃ Q ⦄) ⟷ (p ∙ t ⦃ Q ⦄)"
  by simp

lemma runs_to_partial_cong_state_only[runs_to_vcg_cong_state_only]:
  "s = t ⟹ (p ∙ s ?⦃ Q ⦄) ⟷ (p ∙ t ?⦃ Q ⦄)"
  by simp

lemma runs_to_cong_program_only[runs_to_vcg_cong_program_only]:
  "p = q ⟹ (p ∙ s ⦃ Q ⦄) ⟷ (q ∙ s ⦃ Q ⦄)"
  by simp

lemma runs_to_partial_cong_program_only[runs_to_vcg_cong_program_only]:
  "p = q ⟹ (p ∙ s ?⦃ Q ⦄) ⟷ (q ∙ s ?⦃ Q ⦄)"
  by simp

lemma runs_to_case_conv[simp]:
  "((case (a, b) of (x, y) ⇒ f x y) ∙ s ⦃ Q ⦄) ⟷ ((f a b) ∙ s ⦃ Q ⦄)"
  by simp

lemma runs_to_partial_case_conv[simp]:
  "((case (a, b) of (x, y) ⇒ f x y) ∙ s ?⦃ Q ⦄) ⟷ ((f a b) ∙ s ?⦃ Q ⦄)"
  by simp

lemma always_progress_prod_case[always_progress_intros]:
  "always_progress (f (fst p) (snd p)) ⟹ always_progress (case p of (x, y) ⇒ f x y)"
  by (auto simp add: always_progress_def split: prod.splits)

lemma runs_to_conj:
  "(f ∙ s ⦃ λr s. P r s ∧ Q r s⦄) ⟷ (f ∙ s ⦃ λr s. P r s ⦄) ∧ (f ∙ s ⦃ λr s. Q r s⦄)"
  by (cases "run f s") (auto simp: runs_to.rep_eq)

lemma runs_to_all:
  "(f ∙ s ⦃ λr s. ∀x. P x r s ⦄) ⟷ (∀x. f ∙ s ⦃ λr s. P x r s ⦄)"
  by (cases "run f s") (auto simp: runs_to.rep_eq)

lemma runs_to_imp_const:
  "(f ∙ s ⦃ λr s. P r s ⟶ Q ⦄) ⟷ (Q ∧ (f ∙ s ⦃λ_ _. True ⦄)) ∨ (¬ Q ∧ (f ∙ s ⦃ λr s. ¬ P r s⦄))"
  by (cases "run f s") (auto simp: runs_to.rep_eq)

subsection ‹‹res_monad› and ‹exn_monad› Types›

type_synonym 'a val = "(unit, 'a) exception_or_result"
type_synonym ('e, 'a) xval = "('e option, 'a) exception_or_result"
type_synonym ('a, 's) res_monad = "(unit, 'a , 's) spec_monad"
type_synonym ('e, 'a, 's) exn_monad = "('e option, 'a, 's) spec_monad"

definition Exn :: "'e ⇒ ('e, 'a) xval" where
  "Exn e = Exception (Some e)"

definition
  case_xval :: "('e ⇒ 'a) ⇒ ('v ⇒ 'a) ⇒ ('e, 'v) xval ⇒ 'a" where
  "case_xval f g x =
    (case x of
       Exception v ⇒ (case v of Some e ⇒ f e | None ⇒ undefined)
     | Result v ⇒ g v)"

declare [[case_translation case_xval Exn Result]]

inductive rel_xval:: "('e ⇒ 'f ⇒ bool) ⇒ ('a ⇒ 'b ⇒ bool) ⇒
  ('e, 'a) xval ⇒ ('f, 'b) xval ⇒ bool"
  where
Exn: "E e f ⟹ rel_xval E R (Exn e) (Exn f)" | 
Result: "R a b ⟹ rel_xval E R (Result a) (Result b)"

definition map_xval :: "('e ⇒ 'f) ⇒ ('a ⇒ 'b) ⇒ ('e, 'a) xval ⇒ ('f, 'b) xval" where
  "map_xval f g x ≡ case x of Exn e ⇒ Exn (f e) | Result v ⇒ Result (g v)"

lemma rel_xval_eq: "rel_xval (=) (=) = (=)"
  apply (rule ext)+
  subgoal for x y
    apply (cases x)
    apply (auto simp add: rel_xval.simps default_option_def Exn_def)
    done
  done

lemma rel_xval_rel_exception_or_result_conv: 
  "rel_xval E R = rel_exception_or_result (rel_map the OO E OO rel_map Some) R"
  apply (rule ext)+
  subgoal for x y
    apply (cases x)
    subgoal
      by (auto simp add: rel_exception_or_result.simps rel_xval.simps Exn_def default_option_def rel_map_def)
        (metis option.sel rel_map_direct relcomppI)+
    subgoal
      by (auto simp add: rel_exception_or_result.simps rel_xval.simps Exn_def default_option_def rel_map_def)
    done
  done

lemmas rel_xval_Exn = rel_xval.intros(1)
lemmas rel_xval_Result = rel_xval.intros(2)

lemma case_xval_simps[simp]:
  "case_xval f g (Exn v) = f v"
  "case_xval f g (Result e) = g e"
   by (auto simp add: case_xval_def Exn_def)

lemma case_exception_or_result_Exn[simp]:
  "case_exception_or_result f g (Exn x) = f (Some x)"
  by (simp add: Exn_def)

lemma xval_split:
  "P (case_xval f g r) ⟷
    (∀e. r = Exn e ⟶ P (f e)) ∧
    (∀v. r = Result v ⟶ P (g v))"
  by (auto simp add: case_xval_def Exn_def split: exception_or_result_splits option.splits)

lemma xval_split_asm:
  "P (case_xval f g r) ⟷
  ¬ ((∃e. r = Exn e ∧ ¬ P (f e)) ∨
    (∃v. r = Result v ∧ ¬ P (g v)))"
  by (auto simp add: case_xval_def Exn_def split: exception_or_result_splits option.splits)

lemmas xval_splits = xval_split xval_split_asm

lemma Exn_eq_Exn[simp]: "Exn x = Exn y ⟷ x = y"
  by (simp add: Exn_def)

lemma Exn_neq_Result[simp]: "Exn x = Result e ⟷ False"
  by (simp add: Exn_def)

lemma Result_neq_Exn[simp]: "Result e = Exn x ⟷ False"
  by (simp add: Exn_def)

lemma Exn_eq_Exception[simp]:
  "Exn x = Exception a ⟷ a = Some x"
  "Exception a = Exn x  ⟷ a = Some x"
  by (auto simp: Exn_def)

lemma map_exception_or_result_Exn[simp]:
  "(⋀x. f (Some x) ≠ None) ⟹ map_exception_or_result f g (Exn x) = Exn (the (f (Some x)))"
  by (simp add: Exn_def)

lemma case_xval_Exception_Some_simp[simp]: "(case Exception (Some y) of Exn e ⇒ f e | Result v ⇒ g v) = f y "
  by (metis Exn_def case_xval_simps(1))

lemma rel_xval_simps[simp]: 
  "rel_xval E R (Exn e) (Exn f) = E e f"
  "rel_xval E R (Result v) (Result w) = R v w"
  "rel_xval E R (Exn e) (Result w) = False"
  "rel_xval E R (Result v) (Exn f) = False"
  by (auto simp add: rel_xval.simps)

lemma map_xval_simps[simp]: 
  "map_xval f g (Exn e) = Exn (f e)"
  "map_xval f g (Result v) = Result (g v)"
  by (auto simp add: map_xval_def)

lemma map_xval_Exn: "map_xval f g x = Exn y ⟷ (∃e. x = Exn e ∧ y = f e)"
  by (auto simp add: map_xval_def split: xval_splits)

lemma map_xval_Result: "map_xval f g x = Result y ⟷ (∃v. x = Result v ∧ y = g v)"
  by (auto simp add: map_xval_def split: xval_splits)

lemma Result_unit_eq: "(x:: unit val) = Result ()"
  by (cases x) auto

simproc_setup Result_unit_eq ("x::(unit val)") = ‹
let
  fun is_Result_unit \<^Const>‹Result ‹@{typ "unit"}› ‹@{typ "unit"}› for \<^Const>‹Product_Type.Unity›› = true
    | is_Result_unit _ = false
  in
    K (K (fn ct =>
      if is_Result_unit (Thm.term_of ct) then NONE
      else SOME (mk_meta_eq @{thm Result_unit_eq})))
  end
›

lemma ex_val_Result1:
  "∃v1. (x::'v1 val) = Result v1"
  by (cases x) auto

lemma ex_val_Result2:
  "∃v1 v2. (x::('v1 * 'v2) val) = Result (v1, v2)"
  by (cases x) auto

lemma ex_val_Result3:
  "∃v1 v2 v3. (x::('v1 * 'v2 * 'v3) val) = Result (v1, v2, v3)"
  by (cases x) auto

lemma ex_val_Result4:
  "∃v1 v2 v3 v4. (x::('v1 * 'v2 * 'v3 * 'v4) val) = Result (v1, v2, v3, v4)"
  by (cases x) auto

lemma ex_val_Result5:
  "∃v1 v2 v3 v4 v5. (x::('v1 * 'v2 * 'v3 * 'v4 * 'v5) val) = Result (v1, v2, v3, v4, v5)"
  by (cases x) auto

lemma ex_val_Result6:
  "∃v1 v2 v3 v4 v5 v6. (x::('v1 * 'v2 * 'v3 * 'v4 * 'v5  * 'v6) val) = Result (v1, v2, v3, v4, v5, v6)"
  by (cases x) auto

lemma ex_val_Result7:
  "∃v1 v2 v3 v4 v5 v6 v7. (x::('v1 * 'v2 * 'v3 * 'v4 * 'v5  * 'v6 * 'v7) val) = Result (v1, v2, v3, v4, v5, v6, v7)"
  by (cases x) auto

lemma ex_val_Result8:
  "∃v1 v2 v3 v4 v5 v6 v7 v8. (x::('v1 * 'v2 * 'v3 * 'v4 * 'v5  * 'v6 * 'v7 * 'v8) val) = Result (v1, v2, v3, v4, v5, v6, v7, v8)"
  by (cases x) auto

lemma ex_val_Result9:
  "∃v1 v2 v3 v4 v5 v6 v7 v8 v9. (x::('v1 * 'v2 * 'v3 * 'v4 * 'v5  * 'v6 * 'v7 * 'v8 * 'v9) val) = Result (v1, v2, v3, v4, v5, v6, v7, v8, v9)"
  by (cases x) auto

lemma ex_val_Result10:
  "∃v1 v2 v3 v4 v5 v6 v7 v8 v9 v10. (x::('v1 * 'v2 * 'v3 * 'v4 * 'v5  * 'v6 * 'v7 * 'v8 * 'v9 * 'v10) val) = Result (v1, v2, v3, v4, v5, v6, v7, v8, v9, v10)"
  by (cases x) auto


lemma ex_val_Result11:
  "∃v1 v2 v3 v4 v5 v6 v7 v8 v9 v10 v11. (x::('v1 * 'v2 * 'v3 * 'v4 * 'v5  * 'v6 * 'v7 * 'v8 * 'v9 * 'v10 * 'v11) val) = Result (v1, v2, v3, v4, v5, v6, v7, v8, v9, v10, v11)"
  by (cases x) auto


lemmas ex_val_Result[simp] =
  ex_val_Result1
  ex_val_Result2
  ex_val_Result3
  ex_val_Result4
  ex_val_Result5
  ex_val_Result6
  ex_val_Result7
  ex_val_Result8
  ex_val_Result9
  ex_val_Result10
  ex_val_Result11

lemma all_val_imp_iff: "(∀v. (r::'a val) = Result v ⟶ P) ⟷ P"
  by (cases r) auto

definition map_exn :: "('e ⇒ 'f) ⇒ ('e, 'a) xval ⇒ ('f, 'a) xval" where
  "map_exn f x =
    (case x of
       Exn e ⇒ Exn (f e)
     | Result v ⇒ Result v)"

lemma map_exn_simps[simp]:
  "map_exn f (Exn e) = Exn (f e)"
  "map_exn f (Result v) = Result v"
  "map_exn f x = Result v ⟷ x = Result v"
  by (auto simp add: map_exn_def split: xval_splits)

lemma map_exn_id[simp]: "map_exn (λx. x) = (λx. x)"
  by (auto simp add: map_exn_def split: xval_splits)

definition unnest_exn :: "('e + 'a, 'a) xval ⇒ ('e, 'a) xval" where
  "unnest_exn x =
     (case x of
        Exn e ⇒ (case e of Inl l ⇒ Exn l | Inr r ⇒ Result r)
      | Result v ⇒ Result v)"

lemma unnest_exn_simps[simp]:
  "unnest_exn (Exn (Inl l)) = Exn l"
  "unnest_exn (Exn (Inr r)) = Result r"
  "unnest_exn (Result v) = Result v"
   by (auto simp add: unnest_exn_def split: xval_splits)

lemma unnest_exn_eq_simps[simp]:
  "unnest_exn (Result r) = Result r' ⟷ r = r'"
  "unnest_exn (Result e) ≠ Exn e"
  "unnest_exn (Exn e) = Result r ⟷ e = Inr r"
  "unnest_exn (Exn e) = Exn e' ⟷ e = Inl e'"
  by (auto simp: unnest_exn_def split: xval_splits sum.splits)

definition the_Exn :: "('e, 'a) xval ⇒ 'e" where
  "the_Exn x ≡ (case x of Exn e ⇒ e | Result _ ⇒ undefined)"

abbreviation the_Res :: "'a val ⇒ 'a" where "the_Res ≡ the_Result"

lemma is_Exception_val[simp]: "is_Exception (x::'a val) = False"
  by (auto simp add: is_Exception_def)

lemma is_Exception_Exn[simp]: "is_Exception (Exn x)"
  by (simp add: Exn_def)

lemma is_Result_val[simp]: "is_Result (v::'a val)"
  by (auto simp add: is_Result_def)

lemma the_Exception_Exn[simp]: "the_Exception (Exn e) = Some e"
  by (auto simp add: the_Exception_def)

lemma the_Exn_Exn[simp]:
  "the_Exn (Exn e) = e"
  "the_Exn (Exception (Some e)) = e"
   by (auto simp add: the_Exn_def case_xval_def)

lemma the_Result_simp[simp]:
  "the_Result (Result v) = v"
  by (auto simp add: the_Result_def)

lemma Result_the_Result_val[simp]: "Result (the_Result (x::'a val)) = x"
  by (auto simp add: the_Result_def)

lemma rel_exception_or_result_val_apply:
  fixes x::"'a val"
  fixes y ::"'b val"
  shows "rel_exception_or_result E R x y ⟷ rel_exception_or_result E' R x y"
  by (auto simp add: rel_exception_or_result.simps)

lemma rel_exception_or_result_val:
  shows "((rel_exception_or_result E R)::'a val ⇒ 'b val ⇒ bool) = rel_exception_or_result E' R"
  apply (rule ext)+
  apply (auto simp add: rel_exception_or_result.simps)
  done

lemma rel_exception_or_result_Res_val:
  "(λRes x Res y. R x y) = rel_exception_or_result (λ_ _. False) R"
  apply (rule ext)+
  apply auto
  done

definition unite:: "('a, 'a) xval ⇒ 'a val" where
  "unite x = (case x of Exn v ⇒ Result v | Result v ⇒ Result v)"

lemma unite_simps[simp]:
  "unite (Exn v) = Result v"
  "unite (Result v) = Result v"
  by (auto simp add: unite_def)

lemma val_exhaust: "(⋀v. (x::'a val) = Result v ⟹ P) ⟹ P"
  by (cases x) auto

lemma val_iff: "P (x::'a val) ⟷ (∃v. x = Result v ∧ P (Result v))"
  by (cases x) auto

lemma val_nchotomy: "∀(x::'a val). ∃v. x = Result v"
  by (meson val_exhaust)

lemma "(⋀x::'a val. PROP P x) ⟹ PROP P (Result (v::'a))"
  by (rule meta_spec)

lemma val_Result_the_Result_conv: "(x::'a val) = Result (the_Result x)"
  by (cases x) (auto simp add: the_Result_def)

lemma split_val_all[simp]: "(⋀x::'a val. PROP P x) ≡ (⋀v::'a. PROP P (Result v))"
proof
  fix v::'a
  assume "⋀x::'a val. PROP P x"
  then show "PROP P (Result v)" .
next
  fix x::"'a val"
  assume "⋀v::'a. PROP P (Result v)"
  hence "PROP P (Result (the_Result x))" .
  thus "PROP P x"
    apply (subst val_Result_the_Result_conv)
    apply (assumption)
    done
qed

lemma split_val_Ex[simp]: "(∃(x::'a val). P x) ⟷ (∃(v::'a). P (Result v))"
  by (auto)

lemma split_val_All[simp]: "(∀(x::'a val). P x) ⟷ (∀(v::'a). P (Result v))"
  by (auto)

definition to_xval :: "('e + 'a) ⇒ ('e, 'a) xval" where
 "to_xval x = (case x of Inl e ⇒ Exn e | Inr v ⇒ Result v)"

lemma to_xval_simps[simp]:
  "to_xval (Inl e) = Exn e"
  "to_xval (Inr v) = Result v"
   by (auto simp add: to_xval_def)

lemma to_xval_Result_iff[simp]: "to_xval x = Result v ⟷ x = Inr v"
  apply (cases x)
   apply (auto simp add: to_xval_def default_option_def)
  done

lemma to_xval_Exn_iff[simp]: 
 "to_xval x = Exn v ⟷
    (x  = Inl v)"
  apply (cases x)
   apply (auto simp add: to_xval_def default_option_def Exn_def)
  done

lemma to_xval_Exception_iff[simp]: 
 "to_xval x = Exception v ⟷
    ((v = None ∧ x = Inr undefined) ∨ (∃e. v = Some e ∧ x  = Inl e))"
  apply (cases x)
   apply (auto simp add: to_xval_def default_option_def Exn_def)
  done


definition from_xval :: "('e, 'a) xval ⇒ ('e + 'a)" where
 "from_xval x = (case x of Exn e ⇒ Inl e | Result v ⇒ Inr v)"

lemma from_xval_simps[simp]:
  "from_xval (Exn e) = Inl e"
  "from_xval (Result v) = Inr v"
  by (auto simp add: from_xval_def)

lemma from_xval_Inr_iff[simp]: "from_xval x = Inr v ⟷ x = Result v"
  apply (cases x)
   apply (auto simp add: from_xval_def split: xval_splits)
  done

lemma from_xval_Inl_iff[simp]: "from_xval x = Inl e ⟷ x = Exn e"
  apply (cases x)
   apply (auto simp add: from_xval_def split: xval_splits)
  done

lemma to_xval_from_xval[simp]: "to_xval (from_xval x) = x"
  apply (cases x)
   apply (auto simp add: Exn_def default_option_def)
  done

lemma from_xval_to_xval[simp]: "from_xval (to_xval x) = x"
  apply (cases x)
   apply (auto simp add: Exn_def default_option_def)
  done

lemma rel_map_to_xval_Exn_iff[simp]:  "rel_map to_xval y (Exn e) ⟷ y = Inl e"
  by (auto simp add: rel_map_def to_xval_def split: sum.splits)

lemma rel_map_to_xval_Inr_iff[simp]:  "rel_map to_xval (Inr r) x ⟷ x = Result r"
  by (auto simp add: rel_map_def)

lemma rel_map_to_xval_Inl_iff[simp]:  "rel_map to_xval (Inl l) x ⟷ x = Exn l"
  by (auto simp add: rel_map_def)

lemma rel_map_to_xval_Result_iff[simp]:  "rel_map to_xval y (Result r) ⟷ y = Inr r"
  by (auto simp add: rel_map_def to_xval_def split: sum.splits)

lemma rel_map_from_xval_Exn_iff[simp]:  "rel_map from_xval (Exn l) x ⟷ x = Inl l"
  by (auto simp add: rel_map_def)

lemma rel_map_from_xval_Inl_iff[simp]:  "rel_map from_xval y (Inl e) ⟷ y = Exn e"
  by (auto simp add: rel_map_def from_xval_def split: xval_splits)

lemma rel_map_from_xval_Result_iff[simp]:  "rel_map from_xval (Result r) x ⟷ x = Inr r"
  by (auto simp add: rel_map_def)

lemma rel_map_from_xval_Inr_iff[simp]:  "rel_map from_xval y (Inr r) ⟷ y = Result r"
  by (auto simp add: rel_map_def from_xval_def split: xval_splits)

section ‹‹res_monad› and ‹exn_monad› functions›

abbreviation "throw e ≡ yield (Exn e)"

definition "try" :: "('e + 'a, 'a, 's) exn_monad ⇒ ('e, 'a, 's) exn_monad" where
  "try = map_value unnest_exn"

definition "finally" :: "('a, 'a, 's) exn_monad ⇒ ('a, 's) res_monad" where
  "finally = map_value unite"

definition
  catch :: "('e, 'a, 's) exn_monad ⇒
            ('e ⇒ ( 'f::default, 'a, 's) spec_monad) ⇒
            ('f::default, 'a, 's) spec_monad" (infix "<catch>" 10)
where
  "f <catch> handler ≡ bind_handle f return (handler o the)"

abbreviation bind_finally ::
  "('e, 'a, 's) exn_monad ⇒ (('e, 'a) xval ⇒ ('b, 's) res_monad) ⇒ ('b, 's) res_monad"
where
  "bind_finally ≡ bind_exception_or_result"

definition ignoreE ::"('e, 'a, 's) exn_monad ⇒ ('a, 's) res_monad" where
  "ignoreE f = catch f (λ_. bot)"

definition liftE :: "('a, 's) res_monad ⇒ ('e, 'a, 's) exn_monad" where
  "liftE = map_value (map_exception_or_result (λx. undefined) id)"

definition check:: "'e ⇒ ('s ⇒ 'a ⇒ bool) ⇒ ('e, 'a, 's) exn_monad" where
  "check e p =
    condition (λs. ∃x. p s x)
    (do { s ← get_state; select {x. p s x} })
    (throw e)"

abbreviation check' :: "'e ⇒ ('s ⇒ bool) ⇒ ('e, unit, 's) exn_monad" where
  "check' e p ≡ check e (λs _. p s)"

section "Monad operations"

subsection ‹\<^const>‹top››

declare top_spec_monad.rep_eq[run_spec_monad, simp]

lemma always_progress_top[always_progress_intros]: "always_progress ⊤"
  by transfer simp

lemma runs_to_top[simp]: "⊤ ∙ s ⦃ Q ⦄ ⟷ False"
  by transfer simp

lemma runs_to_partial_top[simp]: "⊤ ∙ s ?⦃ Q ⦄ ⟷ True"
  by transfer simp

subsection ‹\<^const>‹bot››

declare bot_spec_monad.rep_eq[run_spec_monad, simp]

lemma always_progress_bot_iff[iff]: "always_progress ⊥ ⟷ False"
  by transfer simp

lemma runs_to_bot[simp]: "⊥ ∙ s ⦃ Q ⦄ ⟷ True"
  by transfer simp

lemma runs_to_partial_bot[simp]: "⊥ ∙ s ?⦃ Q ⦄ ⟷ True"
  by transfer simp

subsection ‹\<^const>‹fail››

lemma always_progress_fail[always_progress_intros]: "always_progress fail"
  by transfer (simp add: bot_post_state_def)

lemma run_fail[run_spec_monad, simp]: "run fail s = ⊤"
  by transfer (simp add: top_post_state_def)

lemma runs_to_fail[simp]: "fail ∙ s ⦃ R ⦄ ⟷ False"
  by transfer simp

lemma runs_to_partial_fail[simp]: "fail ∙ s ?⦃ R ⦄ ⟷ True"
  by transfer simp

lemma refines_fail[simp]: "refines f fail s t R"
  by (simp add: refines.rep_eq)

lemma rel_spec_fail[simp]: "rel_spec fail fail s t R"
  by (simp add: rel_spec_def)

subsection ‹\<^const>‹yield››

lemma run_yield[run_spec_monad, simp]: "run (yield r) s = pure_post_state (r, s)"
  by transfer simp

lemma always_progress_yield[always_progress_intros]: "always_progress (yield r)"
  by (simp add: always_progress_def)

lemma yield_inj[simp]: "yield x = yield y ⟷ x = y"
  by transfer (auto simp: fun_eq_iff)

lemma runs_to_yield_iff[simp]: "((yield r) ∙ s ⦃ Q ⦄) ⟷ Q r s"
  by transfer simp

lemma runs_to_yield[runs_to_vcg]: "Q r s ⟹ yield r ∙ s ⦃ Q ⦄"
  by simp

lemma runs_to_partial_yield_iff[simp]: "((yield r) ∙ s ?⦃ Q ⦄) ⟷ Q r s"
  by transfer simp

lemma runs_to_partial_yield[runs_to_vcg]: "Q r s ⟹ yield r ∙ s ?⦃ Q ⦄"
  by simp

lemma refines_yield_iff[simp]: "refines (yield r) (yield r') s s' R ⟷ R (r, s) (r', s')"
  by transfer simp

lemma refines_yield: "R (a, s) (b, t) ⟹ refines (yield a) (yield b) s t R"
  by simp

lemma refines_yield_right_iff:
  "refines f (yield e) s t R ⟷ (f ∙ s ⦃ λr s'. R (r, s') (e, t) ⦄)"
  by (auto simp: refines.rep_eq runs_to.rep_eq sim_post_state_pure_post_state2 split_beta')

lemma rel_spec_yield: "R (a, s) (b, t) ⟹ rel_spec (yield a) (yield b) s t R"
  by (simp add: rel_spec_def)

lemma rel_spec_yield_iff[simp]: "rel_spec (yield x) (yield y) s t Q ⟷ Q (x, s) (y, t)"
  by (simp add: rel_spec.rep_eq)

lemma rel_spec_monad_yield: "Q x y ⟹ rel_spec_monad R Q (yield x) (yield y)"
  by (auto simp add: rel_spec_monad_iff_rel_spec)

subsection ‹\<^const>‹throw_exception_or_result››

lemma throw_exception_or_result_bind[simp]:
  "e ≠ default ⟹ throw_exception_or_result e >>= f = throw_exception_or_result e"
  unfolding bind_def by transfer auto

subsection ‹\<^const>‹throw››

lemma throw_bind[simp]: "throw e >>= f = throw e"
  by (simp add: Exn_def)

subsection ‹\<^const>‹get_state››

lemma always_progress_get_state[always_progress_intros]: "always_progress get_state"
  by transfer simp

lemma run_get_state[run_spec_monad, simp]: "run get_state s = pure_post_state (Result s, s)"
  by transfer simp

lemma runs_to_get_state[runs_to_vcg]: "get_state ∙ s ⦃λr t. r = Result s ∧ t = s⦄"
  by transfer simp

lemma runs_to_get_state_iff[runs_to_iff]: "get_state ∙ s ⦃Q⦄ ⟷ (Q (Result s) s)"
  by transfer simp

lemma runs_to_partial_get_state[runs_to_vcg]: "get_state ∙ s ?⦃λr t. r = Result s ∧ t = s⦄"
  by transfer simp

lemma refines_get_state: "R (Result s, s) (Result t, t) ⟹ refines get_state get_state s t R"
  by transfer simp

lemma rel_spec_get_state: "R (Result s, s) (Result t, t) ⟹ rel_spec get_state get_state s t R"
  by (auto simp: rel_spec_iff_refines intro: refines_get_state)

subsection ‹\<^const>‹set_state››

lemma always_progress_set_state[always_progress_intros]: "always_progress (set_state t)"
  by transfer simp

lemma set_state_inj[simp]: "set_state x = set_state y ⟷ x = y"
  by transfer (simp add: fun_eq_iff)

lemma run_set_state[run_spec_monad, simp]: "run (set_state t) s = pure_post_state (Result (), t)"
  by transfer simp

lemma runs_to_set_state[runs_to_vcg]: "set_state t ∙ s ⦃λr t'. t' = t⦄"
  by transfer simp

lemma runs_to_set_state_iff[runs_to_iff]: "set_state t ∙ s ⦃Q⦄ ⟷ Q (Result ()) t"
  by transfer simp

lemma runs_to_partial_set_state[runs_to_vcg]: "set_state t ∙ s ?⦃λr t'. t' = t⦄"
  by transfer simp

lemma refines_set_state:
  "R (Result (), s') (Result (), t') ⟹ refines (set_state s') (set_state t') s t R"
  by transfer simp

lemma rel_spec_set_state:
  "R (Result (), s') (Result (), t') ⟹ rel_spec (set_state s') (set_state t') s t R"
  by (auto simp add: rel_spec_iff_refines intro: refines_set_state)

subsection ‹\<^const>‹select››

lemma always_progress_select[always_progress_intros]: "S ≠ {} ⟹ always_progress (select S)"
  unfolding select_def by transfer simp

lemma runs_to_select[runs_to_vcg]: "(⋀x. x ∈ S ⟹ Q (Result x) s) ⟹ select S ∙ s ⦃ Q ⦄"
  unfolding select_def by transfer simp

lemma runs_to_select_iff[runs_to_iff]: "select S ∙ s ⦃Q⦄ ⟷ (∀x∈S. Q (Result x) s)"
  unfolding select_def by transfer auto

lemma runs_to_partial_select[runs_to_vcg]: "(⋀x. x ∈ S ⟹ Q (Result x) s) ⟹ select S ∙ s ?⦃ Q ⦄"
  using runs_to_select by (rule runs_to_partial_of_runs_to)

lemma run_select[run_spec_monad, simp]: "run (select S) s = Success ((λv. (Result v, s)) ` S)"
  unfolding select_def by transfer (auto simp add: image_image pure_post_state_def Sup_Success)

lemma refines_select:
  "(⋀x. x ∈ P ⟹ ∃xa∈Q. R (Result x, s) (Result xa, t)) ⟹ refines (select P) (select Q) s t R"
  unfolding select_def by transfer (auto intro!: sim_post_state_Sup)

lemma rel_spec_select:
  "rel_set (λa b. R (Result a, s) (Result b, t)) P Q ⟹ rel_spec (select P) (select Q) s t R"
  by (auto simp add: rel_spec_def rel_set_def)

subsection ‹\<^const>‹unknown››

lemma runs_to_unknown[runs_to_vcg]: "(⋀x. Q (Result x) s) ⟹ unknown ∙ s ⦃ Q ⦄"
  by (simp add: unknown_def runs_to_select_iff)

lemma runs_to_unknown_iff[runs_to_iff]: "unknown ∙ s ⦃ Q ⦄ ⟷ (∀x. Q (Result x) s)"
  by (simp add: unknown_def runs_to_select_iff)

lemma runs_to_partial_unknown[runs_to_vcg]: "(⋀x. Q (Result x) s) ⟹ unknown ∙ s ?⦃ Q ⦄"
  using runs_to_unknown by (rule runs_to_partial_of_runs_to)

lemma run_unknown[run_spec_monad, simp]: "run unknown s = Success ((λv. (Result v, s)) ` UNIV)"
  unfolding unknown_def by simp

lemma always_progress_unknown[always_progress_intros]: "always_progress unknown"
  unfolding unknown_def by (simp add: always_progress_intros)

subsection ‹\<^const>‹lift_state››

lemma run_lift_state[run_spec_monad]:
  "run (lift_state R f) s = lift_post_state (rel_prod (=) R) (SUP t∈{t. R s t}. run f t)"
  by transfer standard

lemma runs_to_lift_state_iff[runs_to_iff]:
  "(lift_state R f) ∙ s ⦃ Q ⦄ ⟷ (∀s'. R s s' ⟶ f ∙ s' ⦃λr t'. ∀t. R t t' ⟶ Q r t⦄)"
  by (simp add: runs_to.rep_eq lift_state.rep_eq rel_prod_sel split_beta')

lemma runs_to_lift_state[runs_to_vcg]:
  "(⋀s'. R s s' ⟹ f ∙ s' ⦃λr t'. ∀t. R t t' ⟶ Q r t⦄) ⟹ lift_state R f ∙ s ⦃ Q ⦄"
  by (simp add: runs_to_lift_state_iff)

lemma runs_to_partial_lift_state[runs_to_vcg]:
  "(⋀s'. R s s' ⟹ f ∙ s' ?⦃λr t'.  ∀t. R t t' ⟶ Q r t⦄) ⟹ lift_state R f ∙ s ?⦃ Q ⦄"
  by (cases "∀s'. R s s' ⟶ run f s' ≠ Failure")
     (auto simp: runs_to_partial_alt run_lift_state lift_post_state_Sup image_image
                 image_iff runs_to_lift_state_iff top_post_state_def)

lemma mono_lift_state: "f ≤ f' ⟹ lift_state R f ≤ lift_state R f'"
  by transfer (auto simp: le_fun_def intro!: lift_post_state_mono SUP_mono)

subsection ‹const‹exec_concrete››

lemma run_exec_concrete[run_spec_monad]:
  "run (exec_concrete st f) s = map_post_state (apsnd st) (⨆ (run f ` st -` {s}))"
  by (auto simp add: exec_concrete_def lift_state_def map_post_state_eq_lift_post_state fun_eq_iff
           intro!: arg_cong2[where f=lift_post_state] SUP_cong)

lemma runs_to_exec_concrete_iff[runs_to_iff]:
  "exec_concrete st f ∙ s ⦃ Q ⦄ ⟷ (∀t. s = st t ⟶ f ∙ t ⦃λr t.  Q r (st t)⦄)"
  by (auto simp: runs_to.rep_eq run_exec_concrete split_beta')

lemma runs_to_exec_concrete[runs_to_vcg]:
  "(⋀t. s = st t ⟹ f ∙ t ⦃λr t.  Q r (st t)⦄) ⟹ exec_concrete st f ∙ s ⦃ Q ⦄"
  by (simp add: runs_to_exec_concrete_iff)

lemma runs_to_partial_exec_concrete[runs_to_vcg]:
  "(⋀t. s = st t ⟹ f ∙ t ?⦃λr t.  Q r (st t)⦄) ⟹ exec_concrete st f ∙ s ?⦃ Q ⦄"
  by (simp add: exec_concrete_def runs_to_partial_lift_state)

lemma mono_exec_concrete: "f ≤ f' ⟹ exec_concrete st f ≤ exec_concrete st f'"
  unfolding exec_concrete_def by (rule mono_lift_state)

lemma monotone_exec_concrete_le[partial_function_mono]:
  "monotone Q (≤) f ⟹ monotone Q (≤) (λf'. exec_concrete st (f f'))"
  using mono_exec_concrete
  by (fastforce simp add: monotone_def le_fun_def)

lemma monotone_exec_concrete_ge[partial_function_mono]:
  "monotone Q (≥) f ⟹ monotone Q (≥) (λf'. exec_concrete st (f f'))"
  using mono_exec_concrete
  by (fastforce simp add: monotone_def le_fun_def)

subsection ‹const‹exec_abstract››

lemma run_exec_abstract[run_spec_monad]:
  "run (exec_abstract st f) s = vmap_post_state (apsnd st) (run f (st s))"
  by (auto simp add: exec_abstract_def vmap_post_state_eq_lift_post_state lift_state_def fun_eq_iff
           intro!: arg_cong2[where f=lift_post_state])

lemma runs_to_exec_abstract_iff[runs_to_iff]:
  "exec_abstract st f ∙ s ⦃ Q ⦄ ⟷ f ∙ (st s) ⦃λr t. ∀s'. t = st s' ⟶ Q r s'⦄ "
  by (simp add: runs_to.rep_eq run_exec_abstract split_beta' prod_eq_iff eq_commute)

lemma runs_to_exec_abstract[runs_to_vcg]:
  "f ∙ (st s) ⦃λr t. ∀s'. t = st s' ⟶ Q r s'⦄ ⟹ exec_abstract st f ∙ s ⦃ Q ⦄"
  by (simp add: runs_to_exec_abstract_iff)

lemma runs_to_partial_exec_abstract[runs_to_vcg]:
  "f ∙ (st s) ?⦃λr t. ∀s'. t = st s' ⟶ Q r s'⦄ ⟹ exec_abstract st f ∙ s ?⦃ Q ⦄"
  by (simp add: runs_to_partial.rep_eq run_exec_abstract split_beta' prod_eq_iff eq_commute)

lemma mono_exec_abstract: "f ≤ f' ⟹ exec_abstract st f ≤ exec_abstract st f'"
  unfolding exec_abstract_def by (rule mono_lift_state)

lemma monotone_exec_abstract_le[partial_function_mono]:
  "monotone Q (≤) f ⟹ monotone Q (≤) (λf'. exec_abstract st (f f'))"
  by (simp add: monotone_def mono_exec_abstract)

lemma monotone_exec_abstract_ge[partial_function_mono]:
  "monotone Q (≥) f ⟹ monotone Q (≥) (λf'. exec_abstract st (f f'))"
  by (simp add: monotone_def mono_exec_abstract)

subsection ‹\<^const>‹bind_exception_or_result››

lemma runs_to_bind_exception_or_result_iff[runs_to_iff]:
  "bind_exception_or_result f g ∙ s ⦃ Q ⦄ ⟷ f ∙ s ⦃ λr t. g r ∙ t ⦃ Q ⦄⦄"
  by transfer (simp add: split_beta')

lemma runs_to_bind_exception_or_result[runs_to_vcg]:
  "f ∙ s ⦃ λr t. g r ∙ t ⦃ Q ⦄⦄ ⟹ bind_exception_or_result f g ∙ s ⦃ Q ⦄"
  by (auto simp: runs_to_bind_exception_or_result_iff)

lemma runs_to_partial_bind_exception_or_result[runs_to_vcg]:
  "f ∙ s ?⦃ λr t. g r ∙ t ?⦃ Q ⦄⦄ ⟹ bind_exception_or_result f g ∙ s ?⦃ Q ⦄"
  by transfer (simp add: split_beta' holds_partial_bind_post_state)

lemma refines_bind_exception_or_result:
  "refines f f' s s' (λ(r, t) (r', t'). refines (g r) (g' r') t t' R) ⟹
    refines (bind_exception_or_result f g) (bind_exception_or_result f' g') s s' R"
  by transfer (auto intro: sim_bind_post_state)

lemma refines_bind_exception_or_result':
  assumes f: "refines f f' s s' Q"
  assumes g: "⋀r t r' t'. Q (r, t) (r', t') ⟹ refines (g r) (g' r') t t' R"
  shows "refines (bind_exception_or_result f g) (bind_exception_or_result f' g') s s' R"
  by (auto intro: refines_bind_exception_or_result refines_mono[OF _ f] g)

lemma mono_bind_exception_or_result:
  "f ≤ f' ⟹ g ≤ g' ⟹ bind_exception_or_result f g ≤ bind_exception_or_result f' g'"
  unfolding le_fun_def
  by transfer (auto simp add: le_fun_def intro!: mono_bind_post_state)

lemma monotone_bind_exception_or_result_le[partial_function_mono]:
  "monotone R (≤) (λf'. f f') ⟹ (⋀v. monotone R (≤) (λf'. g f' v)) ⟹
    monotone R (≤) (λf'. bind_exception_or_result (f f') (g f'))"
  by (simp add: monotone_def) (metis le_fun_def mono_bind_exception_or_result)

lemma monotone_bind_exception_or_result_ge[partial_function_mono]:
  "monotone R (≥) (λf'. f f') ⟹ (⋀v. monotone R (≥) (λf'. g f' v)) ⟹
    monotone R (≥) (λf'. bind_exception_or_result (f f') (g f'))"
  by (simp add: monotone_def) (metis le_fun_def mono_bind_exception_or_result)

lemma run_bind_exception_or_result_cong:
  assumes *: "run f s = run f' s"
  assumes **: "f ∙ s ?⦃ λx s'. run (g x) s' = run (g' x) s' ⦄"
  shows "run (bind_exception_or_result f g) s = run (bind_exception_or_result f' g') s"
  using assms
  by (cases "run f' s")
     (auto simp: bind_exception_or_result_def runs_to_partial.rep_eq
           intro!: SUP_cong split: exception_or_result_splits)

subsection ‹\<^const>‹bind_handle››

lemma bind_handle_eq:
  "bind_handle f g h =
    bind_exception_or_result f (λr. case r of Exception e ⇒ h e | Result v ⇒ g v)"
  unfolding spec_monad_ext_iff bind_handle.rep_eq bind_exception_or_result.rep_eq
  by (intro allI arg_cong[where f="bind_post_state _"] ext)
     (auto split: exception_or_result_split)

lemma runs_to_bind_handle_iff[runs_to_iff]:
  "bind_handle f g h ∙ s ⦃ Q ⦄ ⟷ f ∙ s ⦃ λr t.
    (∀v. r = Result v ⟶ g v ∙ t ⦃ Q ⦄) ∧
    (∀e. r = Exception e ⟶ e ≠ default ⟶ h e ∙ t ⦃ Q ⦄)⦄"
  by transfer
     (auto intro!: arg_cong[where f="λp. holds_post_state p _"] simp: fun_eq_iff
           split: prod.splits exception_or_result_splits)

lemma runs_to_bind_handle[runs_to_vcg]:
  "f ∙ s ⦃ λr t. (∀v. r = Result v ⟶ g v ∙ t ⦃ Q ⦄) ∧
                 (∀e. r = Exception e ⟶ e ≠ default ⟶ h e ∙ t ⦃ Q ⦄)⦄ ⟹
  bind_handle f g h ∙ s ⦃ Q ⦄"
  by (auto simp: runs_to_bind_handle_iff)

lemma runs_to_bind_handle_exeption_monad[runs_to_vcg]:
  fixes f :: "('e, 'a, 's) exn_monad"
  assumes f:
    "f ∙ s ⦃ λr t. (∀v. r = Result v ⟶ (g v ∙ t ⦃ Q ⦄)) ∧
                   (∀e. r = Exception (Some e) ⟶ (h (Some e) ∙ t ⦃ Q ⦄))⦄"
  shows "bind_handle f g h ∙ s ⦃ Q ⦄"
  by (rule runs_to_bind_handle[OF runs_to_weaken[OF f]]) (auto simp: default_option_def)

lemma runs_to_bind_handle_res_monad[runs_to_vcg]:
  fixes f :: "('a, 's) res_monad"
  assumes f: "f ∙ s ⦃ λr t. (∀v. r = Result v ⟶ (g v ∙ t ⦃ Q ⦄))⦄"
  shows "bind_handle f g h ∙ s ⦃ Q ⦄"
  by (rule runs_to_bind_handle[OF runs_to_weaken[OF f]]) (auto simp: default_option_def)

lemma runs_to_partial_bind_handle[runs_to_vcg]:
  "f ∙ s ?⦃ λr t. (∀v. r = Result v ⟶ g v ∙ t ?⦃ Q ⦄) ∧
                 (∀e. r = Exception e ⟶ e ≠ default ⟶ h e ∙ t ?⦃ Q ⦄)⦄ ⟹
  bind_handle f g h ∙ s ?⦃ Q ⦄"
  by transfer
     (auto intro!: holds_partial_bind_post_stateI intro: holds_partial_post_state_weaken
           split: exception_or_result_splits prod.splits)

lemma runs_to_partial_bind_handle_exeption_monad[runs_to_vcg]:
  fixes f :: "('e, 'a, 's) exn_monad"
  assumes f: "f ∙ s ?⦃ λr t. (∀v. r = Result v ⟶ (g v ∙ t ?⦃ Q ⦄)) ∧
    (∀e. r = Exception (Some e) ⟶ (h (Some e) ∙ t ?⦃ Q ⦄))⦄"
  shows "bind_handle f g h ∙ s ?⦃ Q ⦄"
  by (rule runs_to_partial_bind_handle[OF runs_to_partial_weaken[OF f]])
     (auto simp: default_option_def)

lemma runs_to_partial_bind_handle_res_monad[runs_to_vcg]:
  fixes f :: "('a, 's) res_monad"
  assumes f: "f ∙ s ?⦃ λr t. (∀v. r = Result v ⟶ (g v ∙ t ?⦃ Q ⦄))⦄"
  shows "bind_handle f g h ∙ s ?⦃ Q ⦄"
  by (rule runs_to_partial_bind_handle[OF runs_to_partial_weaken[OF f]])
     (auto simp: default_option_def)

lemma mono_bind_handle:
  "f ≤ f' ⟹ g ≤ g' ⟹ h ≤ h' ⟹ bind_handle f g h ≤ bind_handle f' g' h'"
  unfolding le_fun_def
  by transfer
     (auto simp add: le_fun_def intro!: mono_bind_post_state split: exception_or_result_splits)

lemma monotone_bind_handle_le[partial_function_mono]:
  "monotone R (≤) (λf'. f f') ⟹ (⋀v. monotone R (≤) (λf'. g f' v)) ⟹
    (⋀e. monotone R (≤) (λf'. h f' e)) ⟹
    monotone R (≤) (λf'. bind_handle (f f') (g f') (h f'))"
  by (simp add: monotone_def) (metis le_fun_def mono_bind_handle)

lemma monotone_bind_handle_ge[partial_function_mono]:
  "monotone R (≥) (λf'. f f') ⟹ (⋀v. monotone R (≥) (λf'. g f' v)) ⟹
    (⋀e. monotone R (≥) (λf'. h f' e)) ⟹
    monotone R (≥) (λf'. bind_handle (f f') (g f') (h f'))"
  by (simp add: monotone_def) (metis le_fun_def mono_bind_handle)

lemma run_bind_handle[run_spec_monad]:
  "run (bind_handle f g h) s = bind_post_state (run f s)
    (λ(r, t). case r of Exception e ⇒ run (h e) t | Result v ⇒ run (g v) t)"
  by transfer simp

lemma always_progress_bind_handle[always_progress_intros]:
  "always_progress f ⟹ (⋀v. always_progress (g v)) ⟹ (⋀e. always_progress (h e))
  ⟹  always_progress (bind_handle f g h)"
  by (auto simp: always_progress.rep_eq run_bind_handle bind_post_state_eq_bot prod_eq_iff
                 exception_or_result_nchotomy holds_post_state_False
           split: exception_or_result_splits prod.splits)

lemma refines_bind_handle':
  "refines f f' s s' (λ(r, t) (r', t').
    (∀e e'. e ≠ default ⟶ e' ≠ default ⟶ r = Exception e ⟶ r' = Exception e' ⟶
      refines (h e) (h' e') t t' R) ∧
    (∀e x'. e ≠ default ⟶ r = Exception e ⟶ r' = Result x' ⟶
      refines (h e) (g' x') t t' R) ∧
    (∀x e'. e' ≠ default ⟶ r = Result x ⟶ r' = Exception e' ⟶
      refines (g x) (h' e') t t' R) ∧
    (∀x x'. r = Result x ⟶ r' = Result x' ⟶
      refines (g x) (g' x') t t' R)) ⟹
  refines (bind_handle f g h) (bind_handle f' g' h') s s' R"
  apply transfer
  apply (auto intro!: sim_bind_post_state')[1]
  apply (rule sim_post_state_weaken, assumption)
  apply (auto split: exception_or_result_splits prod.splits)
  done

lemma refines_bind_handle_bind_handle:
  assumes f: "refines f f' s s' Q"
  assumes ll: "⋀e e' t t'. Q (Exception e, t) (Exception e', t') ⟹
    e ≠ default ⟹ e' ≠ default ⟹
    refines (h e) (h' e') t t' R"
  assumes lr: "⋀e v' t t'. Q (Exception e, t) (Result v', t') ⟹ e ≠ default ⟹
    refines (h e) (g' v') t t' R"
  assumes rl: "⋀v e' t t'. Q (Result v, t) (Exception e', t') ⟹ e' ≠ default ⟹
    refines (g v) (h' e') t t' R"
  assumes rr: "⋀v v' t t'. Q (Result v, t) (Result v', t') ⟹
    refines (g v) (g' v') t t' R"
  shows "refines (bind_handle f g h) (bind_handle f' g' h') s s' R"
  apply (rule refines_bind_handle')
  apply (rule refines_weaken[OF f])
  apply (auto split: exception_or_result_splits prod.splits intro: ll lr rl rr)
  done

lemma refines_bind_handle_bind_handle_exn:
  assumes f: "refines f f' s s' Q"
  assumes ll: "⋀e e' t t'. Q (Exn e, t) (Exn e', t') ⟹
    refines (h (Some e)) (h' (Some e')) t t' R"
  assumes lr: "⋀e v' t t'. Q (Exn e, t) (Result v', t') ⟹
    refines (h (Some e)) (g' v') t t' R"
  assumes rl: "⋀v e' t t'. Q (Result v, t) (Exn e', t') ⟹
    refines (g v) (h' (Some e')) t t' R"
  assumes rr: "⋀v v' t t'. Q (Result v, t) (Result v', t') ⟹
    refines (g v) (g' v') t t' R"
  shows "refines (bind_handle f g h) (bind_handle f' g' h') s s' R"
  apply (rule refines_bind_handle')
  apply (rule refines_weaken[OF f])
  using ll lr rl rr
  apply (auto split: exception_or_result_splits prod.splits simp: Exn_def default_option_def)
  done

lemma bind_handle_return_spec_monad[simp]: "bind_handle (return v) g h = g v"
  by transfer simp

lemma bind_handle_throw_spec_monad[simp]:
  "v ≠ default ⟹ bind_handle (throw_exception_or_result v) g h = h v"
  by transfer simp

lemma bind_handle_bind_handle_spec_monad:
  "bind_handle (bind_handle f g1 h1) g2 h2 =
    bind_handle f
      (λv. bind_handle (g1 v) g2 h2)
      (λe. bind_handle (h1 e) g2 h2)"
  apply transfer
  apply (auto simp: fun_eq_iff intro!: arg_cong[where f="bind_post_state _"])[1]
  subgoal for g1 h1 g2 h2 r v by (cases r) simp_all
  done

lemma mono_bind_handle_spec_monad:
  "mono f ⟹ (⋀v. mono (λx. g x v)) ⟹ (⋀e. mono (λx. h x e)) ⟹
    mono (λx. bind_handle (f x) (g x) (h x))"
  unfolding mono_def
  apply transfer
  apply (auto simp: le_fun_def intro!: mono_bind_post_state)[1]
  subgoal for f g h x y r q by (cases r) simp_all
  done

lemma rel_spec_bind_handle:
  "rel_spec f f' s s' (λ(r, t) (r', t').
    (∀e e'. e ≠ default ⟶ e' ≠ default ⟶ r = Exception e ⟶ r' = Exception e' ⟶
      rel_spec (h e) (h' e') t t' R) ∧
    (∀e x'. e ≠ default ⟶ r = Exception e ⟶ r' = Result x' ⟶
      rel_spec (h e) (g' x') t t' R) ∧
    (∀x e'. e' ≠ default ⟶ r = Result x ⟶ r' = Exception e' ⟶
      rel_spec (g x) (h' e') t t' R) ∧
    (∀x x'. r = Result x ⟶ r' = Result x' ⟶
      rel_spec (g x) (g' x') t t' R)) ⟹
  rel_spec (bind_handle f g h) (bind_handle f' g' h') s s' R"
  by (auto simp: rel_spec_iff_refines intro!: refines_bind_handle' intro: refines_weaken)

lemma bind_finally_bind_handle_conv: "bind_finally f g = bind_handle f (λv. g (Result v)) (λe. g (Exception e))"
  apply (rule spec_monad_eqI)
  apply (auto simp add: runs_to_iff Exn_def [symmetric] default_option_def elim!: runs_to_weaken)[1]
  using exception_or_result_nchotomy by force

subsection ‹\<^const>‹bind››

lemma run_bind[run_spec_monad]: "run (bind f g) s =
    bind_post_state (run f s) (λ(r, t). case r of
      Exception e ⇒ pure_post_state (Exception e, t)
    | Result v ⇒ run (g v) t)"
  by (auto simp add: bind_def run_bind_handle )

lemma run_bind_eq_top_iff:
  "run (bind f g) s = ⊤ ⟷ ¬ (f ∙ s ⦃ λx s. ∀a. x = Result a ⟶ run (g a) s ≠ ⊤ ⦄)"
  by (simp add: run_bind bind_post_state_eq_top runs_to.rep_eq prod_eq_iff split_beta'
           split: exception_or_result_splits prod.splits)

lemma always_progress_bind[always_progress_intros]:
  "always_progress f ⟹ (⋀v. always_progress (g v)) ⟹ always_progress (bind f g)"
  by (simp add: always_progress_intros bind_def)

lemma run_bind_cong:
  assumes *: "run f s = run f' s"
  assumes **: "f ∙ s ?⦃ λx s'. ∀v. x = (Result v) ⟶ run (g v) s' = run (g' v) s' ⦄"
  shows "run (bind f g) s = run (bind f' g') s"
  using assms
  by (cases "run f' s")
     (auto simp: run_bind runs_to_partial.rep_eq
           intro!: SUP_cong split: exception_or_result_splits)

lemma runs_to_bind_iff[runs_to_iff]:
  "bind f g ∙ s ⦃ Q ⦄ ⟷ f ∙ s ⦃ λr t.
    (∀v. r = Result v ⟶ g v ∙ t ⦃ Q ⦄) ∧
    (∀e. r = Exception e ⟶ e ≠ default ⟶ Q (Exception e) t)⦄"
  by (simp add: bind_def runs_to_bind_handle_iff)

lemma runs_to_bind[runs_to_vcg]:
  "f ∙ s ⦃ λr t. (∀v. r = Result v ⟶ g v ∙ t ⦃ Q ⦄) ∧
                 (∀e. r = Exception e ⟶ e ≠ default ⟶ Q (Exception e) t)⦄ ⟹
      bind f g ∙ s ⦃ Q ⦄"
  by (simp add: runs_to_bind_iff)

lemma runs_to_bind_exception[runs_to_vcg]:
  fixes f :: "('e, 'a, 's) exn_monad"
  assumes [runs_to_vcg]: "f ∙ s ⦃ λr t. (∀v. r = Result v ⟶ g v ∙ t ⦃ Q ⦄) ∧
                 (∀e. r = Exn e ⟶ Q (Exn e) t)⦄"
  shows "bind f g ∙ s ⦃ Q ⦄"
  supply runs_to_bind[runs_to_vcg]
  by runs_to_vcg (auto simp: Exn_def default_option_def)

lemma runs_to_bind_res[runs_to_vcg]:
  fixes f :: "('a, 's) res_monad"
  assumes [runs_to_vcg]: "f ∙ s ⦃ λRes v t. g v ∙ t ⦃ Q ⦄⦄"
  shows "bind f g ∙ s ⦃ Q ⦄"
  supply runs_to_bind[runs_to_vcg]
  by runs_to_vcg

lemma runs_to_partial_bind[runs_to_vcg]:
  "f ∙ s ?⦃ λr t. (∀v. r = Result v ⟶ g v ∙ t ?⦃ Q ⦄) ∧
                 (∀e. r = Exception e ⟶ e ≠ default ⟶ Q (Exception e) t)⦄ ⟹
      bind f g ∙ s ?⦃ Q ⦄"
  apply (simp add: bind_def)
  apply (rule runs_to_partial_bind_handle)
  apply simp
  done

lemma runs_to_partial_bind_exeption_monad[runs_to_vcg]:
  fixes f :: "('e, 'a, 's) exn_monad"
  assumes [runs_to_vcg]: "f ∙ s ?⦃ λr t. (∀v. r = Result v ⟶ g v ∙ t ?⦃ Q ⦄) ∧
    (∀e. r = Exn e ⟶ Q (Exn e) t)⦄"
  shows "bind f g ∙ s ?⦃ Q ⦄"
  by runs_to_vcg (auto simp add: default_option_def Exn_def)

lemma runs_to_partial_bind_res_monad[runs_to_vcg]:
  fixes f :: "('a, 's) res_monad"
  assumes [runs_to_vcg]: "f ∙ s ?⦃ λRes v t. g v ∙ t ?⦃ Q ⦄⦄"
  shows "bind f g ∙ s ?⦃ Q ⦄"
  by runs_to_vcg

lemma bind_return[simp]: "(bind m return) = m"
  apply (clarsimp simp: spec_monad_eq_iff runs_to_bind_iff fun_eq_iff
                  intro!: runs_to_cong_pred_only)
  subgoal for P r t
    by (cases r) auto
  done

lemma bind_skip[simp]: "bind m (λx. skip) = m"
  using bind_return[of m] by simp

lemma return_bind[simp]: "bind (return x) f = f x"
  unfolding bind_def by transfer simp

lemma bind_assoc: "bind (bind f g) h = bind f (λx. bind (g x) h)"
  apply (rule spec_monad_ext)
  apply (auto simp: split_beta' run_bind intro!: arg_cong[where f="bind_post_state _"]
              split: exception_or_result_splits)
  done

lemma mono_bind: "f ≤ f' ⟹ g ≤ g' ⟹ bind f g ≤ bind f' g'"
  unfolding bind_def
  by (auto intro: mono_bind_handle)

lemma mono_bind_spec_monad:
  "mono f ⟹ (⋀v. mono (λx. g x v)) ⟹ mono (λx. bind (f x) (g x))"
  unfolding bind_def
  by (intro mono_bind_handle_spec_monad mono_const) auto

lemma monotone_bind_le[partial_function_mono]:
  "monotone R (≤) (λf'. f f') ⟹ (⋀v. monotone R (≤) (λf'. g f' v))
  ⟹ monotone R (≤) (λf'. bind (f f') (g f'))"
  unfolding bind_def
  by (auto intro: monotone_bind_handle_le)

lemma monotone_bind_ge[partial_function_mono]:
  "monotone R (≥) (λf'. f f') ⟹ (⋀v. monotone R (≥) (λf'. g f' v))
  ⟹ monotone R (≥) (λf'. bind (f f') (g f'))"
  unfolding bind_def
  by (auto intro: monotone_bind_handle_ge)

lemma refines_bind':
  assumes f: "refines f f' s s' (λ(x, t) (x', t').
    (∀e. e ≠ default ⟶ x = Exception e ⟶
      ((∀e'. e' ≠ default ⟶ x' = Exception e' ⟶ R (Exception e, t) (Exception e', t')) ∧
      (∀v'. x' = Result v' ⟶ refines (throw_exception_or_result e) (g' v') t t' R))) ∧
    (∀v. x = Result v ⟶
      ((∀e'. e' ≠ default ⟶ x' = Exception e' ⟶
        refines (g v) (throw_exception_or_result e') t t' R) ∧
      (∀v'. x' = Result v' ⟶ refines (g v) (g' v') t t' R))))"
  shows "refines (bind f g) (bind f' g') s s' R"
  unfolding bind_def
  by (rule refines_bind_handle'[OF refines_weaken, OF f]) auto

lemma refines_bind:
  assumes f: "refines f f' s s' Q"
  assumes ll: "⋀e e' t t'.
    Q (Exception e, t) (Exception e', t') ⟹ e ≠ default ⟹ e' ≠ default ⟹
    R (Exception e, t) (Exception e', t')"
  assumes lr: "⋀e v' t t'. Q (Exception e, t) (Result v', t') ⟹ e ≠ default ⟹
    refines (yield (Exception e)) (g' v') t t' R"
  assumes rl: "⋀v e' t t'. Q (Result v, t) (Exception e', t') ⟹ e' ≠ default ⟹
    refines (g v) (yield (Exception e')) t t' R"
  assumes rr: "⋀v v' t t'. Q (Result v, t) (Result v', t') ⟹
    refines (g v) (g' v') t t' R"
  shows "refines (bind f g) (bind f' g') s s' R"
  by (rule refines_bind'[OF refines_weaken[OF f]])
     (auto simp: ll lr rl rr)

lemma refines_bind_bind_exn:
  assumes f: "refines f f' s s' Q"
  assumes ll: "⋀e e' t t'. Q (Exn e, t) (Exn e', t') ⟹ R (Exn e, t) (Exn e', t')"
  assumes lr: "⋀e v' t t'. Q (Exn e, t) (Result v', t') ⟹ refines (throw e) (g' v') t t' R"
  assumes rl: "⋀v e' t t'. Q (Result v, t) (Exn e', t') ⟹ refines (g v) (throw e') t t' R"
  assumes rr: "⋀v v' t t'. Q (Result v, t) (Result v', t') ⟹ refines (g v) (g' v') t t' R"
  shows "refines (f ⤜ g) (f' ⤜ g') s s' R"
  using ll lr rl rr
  by (intro refines_bind[OF f])
     (auto simp: Exn_def default_option_def)

lemma refines_bind_res:
  assumes f: "refines f f' s s' (λ(Res r, t) (Res r', t'). refines (g r) (g' r') t t' R) "
  shows "refines ((bind f g)::('a,'s) res_monad) ((bind f' g')::('b, 't) res_monad) s s' R"
  by (rule refines_bind[OF f]) auto

lemma refines_bind_res':
  assumes f: "refines f f' s s' Q"
  assumes g: "⋀r t r' t'. Q (Result r, t) (Result r', t') ⟹ refines (g r) (g' r') t t' R"
  shows "refines ((bind f g)::('a,'s) res_monad) ((bind f' g')::('b, 't) res_monad) s s' R"
  by (auto intro!: refines_bind_res refines_weaken[OF f] g)

lemma refines_bind_bind_exn_wp: 
  assumes f: "refines f f' s s' (λ(r, t) (r',t'). 
     (case r of 
        Exn e ⇒ (case r' of Exn e' ⇒ R (Exn e, t) (Exn e', t') | Result v' ⇒ refines (throw e) (g' v') t t' R)
      | Result v ⇒ (case r' of Exn e' ⇒ refines (g v) (throw e') t t' R | Result v' ⇒ refines (g v) (g' v') t t' R)))"
   shows "refines (f ⤜ g) (f' ⤜ g') s s' R"
  apply (rule refines_bind')
  apply (rule refines_weaken [OF f])
  apply (auto simp add: Exn_def default_option_def split: xval_splits )
  done

lemma rel_spec_bind_res:
  "rel_spec f f' s s' (λ(Res r, t) (Res r', t'). rel_spec (g r) (g' r') t t' R) ⟹
    rel_spec ((bind f g)::('a,'s) res_monad) ((bind f' g')::('b, 't) res_monad) s s' R"
  unfolding rel_spec_iff_refines
  by (safe intro!: refines_bind_res; (rule refines_weaken, assumption, simp))

lemma rel_spec_bind_res':
  assumes f: "rel_spec f f' s s' Q"
  assumes g: "⋀r t r' t'. Q (Result r, t) (Result r', t') ⟹ rel_spec (g r) (g' r') t t' R"
  shows "rel_spec ((bind f g)::('a,'s) res_monad) ((bind f' g')::('b, 't) res_monad) s s' R"
  by (auto intro!: rel_spec_bind_res rel_spec_mono[OF _ f] g)

lemma rel_spec_bind_bind:
  assumes f: "rel_spec f f' s s' Q"
  assumes ll: "⋀e e' t t'.
    Q (Exception e, t) (Exception e', t') ⟹ e ≠ default ⟹ e' ≠ default ⟹
    R (Exception e, t) (Exception e', t')"
  assumes lr: "⋀e v' t t'. Q (Exception e, t) (Result v', t') ⟹ e ≠ default ⟹
    rel_spec (yield (Exception e)) (g' v') t t' R"
  assumes rl: "⋀v e' t t'. Q (Result v, t) (Exception e', t') ⟹ e' ≠ default ⟹
    rel_spec (g v) (yield (Exception e')) t t' R"
  assumes rr: "⋀v v' t t'. Q (Result v, t) (Result v', t') ⟹
    rel_spec (g v) (g' v') t t' R"
  shows "rel_spec (bind f g) (bind f' g') s s' R"
  using assms unfolding rel_spec_iff_refines
  apply (intro conjI)
  subgoal by (rule refines_bind[where Q=Q]) auto
  subgoal by (rule refines_bind[where Q="Q¯¯"]) auto
  done

lemma rel_spec_bind_exn:
  assumes f: "rel_spec f f' s s' Q"
  assumes ll: "⋀e e' t t'. Q (Exn e, t) (Exn e', t') ⟹ R (Exn e, t) (Exn e', t')"
  assumes lr: "⋀e v' t t'. Q (Exn e, t) (Result v', t') ⟹ rel_spec (throw e) (g' v') t t' R"
  assumes rl: "⋀v e' t t'. Q (Result v, t) (Exn e', t') ⟹ rel_spec (g v) (throw e') t t' R"
  assumes rr: "⋀v v' t t'. Q (Result v, t) (Result v', t') ⟹ rel_spec (g v) (g' v') t t' R"
  shows "rel_spec (bind f g) (bind f' g') s s' R"
  using ll lr rl rr
  by (intro rel_spec_bind_bind[OF f])
     (auto simp: Exn_def default_option_def)

lemma refines_bind_right:
  assumes f: "refines f f' s s' Q"
  assumes ll: "⋀e e' t t'.
    Q (Exception e, t) (Exception e', t') ⟹ e ≠ default ⟹ e' ≠ default ⟹
    R (Exception e, t) (Exception e', t')"
  assumes lr: "⋀e v' t t'. Q (Exception e, t) (Result v', t') ⟹ e ≠ default ⟹
    refines (yield (Exception e)) (g' v') t t' R"
  assumes rl: "⋀v e' t t'. Q (Result v, t) (Exception e', t') ⟹ e' ≠ default ⟹
    R (Result v, t) (Exception e', t')"
  assumes rr: "⋀v v' t t'. Q (Result v, t) (Result v', t') ⟹
    refines (return v) (g' v') t t' R"
  shows "refines f (bind f' g') s s' R"
proof -
  have "refines (bind f return) (bind f' g') s s' R"
    using ll lr rl rr by (intro refines_bind[OF f]) auto
  then show ?thesis by simp
qed

lemma refines_bind_left:
  assumes f: "refines f f' s s' Q"
  assumes ll: "⋀e e' t t'.
    Q (Exception e, t) (Exception e', t') ⟹ e ≠ default ⟹ e' ≠ default ⟹
    R (Exception e, t) (Exception e', t')"
  assumes lr: "⋀e v' t t'. Q (Exception e, t) (Result v', t') ⟹ e ≠ default ⟹
    R (Exception e, t) (Result v', t')"
  assumes rl: "⋀v e' t t'. Q (Result v, t) (Exception e', t') ⟹ e' ≠ default ⟹
    refines (g v) (yield (Exception e')) t t' R"
  assumes rr: "⋀v v' t t'. Q (Result v, t) (Result v', t') ⟹
    refines (g v) (return v') t t' R"
  shows "refines (bind f g) f' s s' R"
proof -
  have "refines (bind f g) (bind f' return) s s' R"
    using ll lr rl rr by (intro refines_bind[OF f]) auto
  then show ?thesis by simp
qed

lemma rel_spec_bind_right:
  assumes f: "rel_spec f f' s s' Q"
  assumes ll: "⋀e e' t t'.
    Q (Exception e, t) (Exception e', t') ⟹ e ≠ default ⟹ e' ≠ default ⟹
    R (Exception e, t) (Exception e', t')"
  assumes lr: "⋀e v' t t'. Q (Exception e, t) (Result v', t') ⟹ e ≠ default ⟹
    rel_spec (yield (Exception e)) (g' v') t t' R"
  assumes rl: "⋀v e' t t'. Q (Result v, t) (Exception e', t') ⟹ e' ≠ default ⟹
    R (Result v, t) (Exception e', t')"
  assumes rr: "⋀v v' t t'. Q (Result v, t) (Result v', t') ⟹
    rel_spec (return v) (g' v') t t' R"
  shows "rel_spec f (bind f' g') s s' R"
  using assms unfolding rel_spec_iff_refines
  apply (intro conjI)
  subgoal by (rule refines_bind_right[where Q=Q]) auto
  subgoal by (rule refines_bind_left[where Q ="Q¯¯"]) auto
  done

lemma refines_bind_handle_left':
  "f ∙ s ⦃ λv s'. (∀r. v = Result r ⟶ refines (g r) k s' t R) ∧
    (∀e. e ≠ default ⟶ v = Exception e ⟶ refines (h e) k s' t R) ⦄ ⟹
  refines (bind_handle f g h) k s t R"
  by (auto simp: runs_to.rep_eq refines.rep_eq run_bind_handle split_beta' ac_simps
           intro!: sim_bind_post_state_left split: prod.splits exception_or_result_splits)

lemma refines_bind_left_res:
  "f ∙ s ⦃ λRes r s'. refines (g r) h s' t R ⦄ ⟹ refines (f >>= g) h s t R"
  unfolding bind_def by (rule refines_bind_handle_left') simp

lemma refines_bind_left_exn:
  "f ∙ s ⦃ λr s'. (∀a. r = Result a ⟶ refines (g a) h s' t R) ∧
    (∀e. r = Exn e ⟶ refines (throw e) h s' t R) ⦄ ⟹
    refines (f >>= g) h s t R"
  unfolding bind_def
  by (rule refines_bind_handle_left')
     (auto simp add: Exn_def default_option_def imp_ex)

lemma runs_to_partial_bind1:
  "(⋀r s. (g r) ∙ s ?⦃P⦄) ⟹ ((f >>= g)::('a, 's) res_monad) ∙ s ?⦃P⦄"
  apply (rule runs_to_partial_bind)
  apply (rule runs_to_partial_weaken[OF runs_to_partial_True])
  apply auto
  done

lemma unknown_bind_const[simp]: "unknown >>= (λx. f) = f"
  by (rule spec_monad_ext) (simp add: run_bind)

lemma bind_cong_left:
  fixes f::"('e::default, 'a, 's) spec_monad"
  shows "(⋀r. g r = g' r) ⟹ (f >>= g) = (f >>= g')"
  by (rule spec_monad_ext) (simp add: run_bind)

lemma bind_cong_right:
  fixes f::"('e::default, 'a, 's) spec_monad"
  shows "f = f' ⟹ (f >>= g) = (f' >>= g)"
  by (rule spec_monad_ext) (simp add: run_bind)

lemma rel_spec_bind_res'':
  fixes f::"('a, 's) res_monad"
  shows "f ∙ s ?⦃ λRes r t. rel_spec (g r) (g' r) t t R⦄ ⟹ rel_spec (f >>= g) (f >>= g') s s R"
  by (intro rel_spec_bind_res rel_spec_refl') (auto simp: split_beta')

lemma rel_spec_monad_bind_rel_exception_or_result:
  assumes mn: "rel_spec_monad R (rel_exception_or_result E P) m n"
    and fg: "rel_fun P (rel_spec_monad R (rel_exception_or_result E Q)) f g"
  shows "rel_spec_monad R (rel_exception_or_result E Q) (m >>= f) (n >>= g)"
  by (intro rel_spec_monadI rel_spec_bind_bind[OF rel_spec_monadD[OF mn]])
     (simp_all add: fg[THEN rel_funD, THEN rel_spec_monadD])

lemma rel_spec_monad_bind_rel_exception_or_result':
  "(⋀x. rel_spec_monad (=) (rel_exception_or_result (=) Q) (f x) (g x)) ⟹
    rel_spec_monad (=) (rel_exception_or_result (=) Q) (m >>= f) (m >>= g)"
  apply (rule rel_spec_monad_bind_rel_exception_or_result[where P="(=)"])
  apply (auto simp add: rel_exception_or_result_eq_conv rel_spec_monad_eq_conv)
  done

lemma rel_spec_monad_bind:
  "rel_spec_monad R (λRes v1 Res v2. P v1 v2) m n ⟹ rel_fun P (rel_spec_monad R Q) f g ⟹
    rel_spec_monad R Q (m >>= f) (n >>= g)"
  apply (clarsimp simp add: rel_spec_monad_iff_rel_spec[abs_def] rel_fun_def
                  intro!: rel_spec_bind_res)
  subgoal premises prems for s t
    by (rule rel_spec_mono[OF _ prems(1)[rule_format, OF prems(3)]])
       (clarsimp simp: le_fun_def prems(2)[rule_format])
  done

lemma rel_spec_monad_bind_left:
  assumes mn: "rel_spec_monad R (λRes v1 Res v2. P v1 v2) m n"
    and fg: "rel_fun P (rel_spec_monad R Q) f return"
  shows "rel_spec_monad R Q (m >>= f) n"
proof -
  from mn fg
  have "rel_spec_monad R Q (m >>= f) (n >>= return)"
    by (rule rel_spec_monad_bind)
  then show ?thesis
    by (simp)
qed

lemma rel_spec_monad_bind':
  fixes m:: "('a, 's) res_monad"
  shows "(⋀x. rel_spec_monad (=) Q (f x) (g x)) ⟹ rel_spec_monad (=) Q (m >>= f) (m >>= g)"
  by (rule rel_spec_monad_bind[where R="(=)" and P="(=)"])
     (auto simp: rel_fun_def rel_spec_monad_iff_rel_spec intro!: rel_spec_refl)

lemma run_bind_cong_simple:
  "run f s = run f' s ⟹ (⋀v s. run (g v) s = run (g' v) s) ⟹
    run (bind f g) s = run (bind f' g') s"
  by (simp add: run_bind)

lemma run_bind_cong_simple_same_head:  "(⋀v s. run (g v) s = run (g' v) s) ⟹
  run (bind f g) s = run (bind f g') s"
  by (rule run_bind_cong_simple) auto

lemma refines_bind_same:
  "refines (f >>= g) (f >>= g') s s R" if "f ∙ s ⦃λRes y t. refines (g y) (g' y) t t R ⦄"
  apply (rule refines_bind_res)
  apply (rule refines_same_runs_toI)
  apply (rule runs_to_weaken[OF that])
  apply auto
  done

lemma refines_bind_handle_right_runs_to_partialI:
  "g ∙ t ?⦃ λr t'. (∀e. e ≠ default ⟶ r = Exception e ⟶ refines f (k e) s t' R) ∧
     (∀a. r = Result a ⟶ refines f (h a) s t' R) ⦄ ⟹ always_progress g ⟹
  refines f (bind_handle g h k) s t R"
  by transfer
     (auto simp: prod_eq_iff split_beta'
           intro!: sim_bind_post_state_right
           split: exception_or_result_splits prod.splits)

lemma refines_bind_right_runs_to_partialI:
  "g ∙ t ?⦃ λr t'. (∀e. e ≠ default ⟶ r = Exception e ⟶
      refines f (throw_exception_or_result e) s t' R) ∧
     (∀a. r = Result a ⟶ refines f (h a) s t' R) ⦄ ⟹ always_progress g ⟹
  refines f (bind g h) s t R"
  unfolding bind_def by (rule refines_bind_handle_right_runs_to_partialI)

lemma refines_bind_right_runs_toI:
  "g ∙ t ⦃ λRes r t'. refines f (h r) s t' R ⦄ ⟹ always_progress g ⟹
    refines f (g >>= h) s t R"
  unfolding bind_def
  by (rule refines_bind_handle_right_runs_to_partialI)
     (auto intro: runs_to_partial_of_runs_to)

lemma refines_bind_left_refine:
  "refines (f >>= g) h s t R"
  if "refines f f' s s (=)" "refines (f' >>= g) h s t R"
  by (rule refines_trans[OF refines_bind[OF that(1)] that(2), where R="(=)"])
     (auto simp add: refines_refl)

lemma refines_bind_right_single:
  assumes x: "pure_post_state (Result x, u) ≤ run g t" and h: "refines f (h x) s u R"
  shows "refines f (g ⤜ h) s t R"
  apply (rule refines_le_run_trans[OF h])
  apply (simp add: run_bind)
  apply (rule order_trans[OF _ mono_bind_post_state, OF _ x order_refl])
  apply simp
  done

lemma return_let_bind: " (return (let v = f' in (g' v))) = do {v <- return f'; return (g' v)}"
  apply (rule spec_monad_eqI)
  apply (auto simp add: runs_to_iff)
  done

lemma rel_spec_monad_rel_xval_bind:
  assumes f_f': "rel_spec_monad S (rel_xval L P) f f'"
  assumes Res_Res: "⋀v v'. P v v' ⟹ rel_spec_monad S (rel_xval L R) (g v) (g' v')"
  shows "rel_spec_monad S (rel_xval L R) (f ⤜ g) (f' ⤜ g')"
  apply (intro rel_spec_monadI rel_spec_bind_exn)
  apply (rule f_f'[THEN rel_spec_monadD], assumption)
  using Res_Res[THEN rel_spec_monadD]
  apply auto
  done

lemma rel_spec_monad_rel_xval_same_bind:
  assumes f_f': "rel_spec_monad S (rel_xval L R) f f'"
  assumes Res_Res: "⋀v v'. R v v' ⟹  rel_spec_monad S (rel_xval L R) (g v) (g' v')"
  shows "rel_spec_monad S (rel_xval L R) (f ⤜ g) (f' ⤜ g')"
  using assms by (rule rel_spec_monad_rel_xval_bind)

lemma rel_spec_monad_rel_xval_result_eq_bind:
  assumes f_f': "rel_spec_monad S (rel_xval L (=)) f f'"
  assumes Res_Res: "⋀v. rel_spec_monad S (rel_xval L (=)) (g v) (g' v)"
  shows "rel_spec_monad S (rel_xval L (=)) (f ⤜ g) (f' ⤜ g')"
  apply (rule rel_spec_monad_rel_xval_bind [OF f_f'])
  subgoal using Res_Res by auto
  done

lemma rel_spec_monad_fail: "rel_spec_monad Q R fail fail"
  by (auto simp add: rel_spec_monad_def rel_set_def)

lemma bind_fail[simp]: "fail ⤜ X = fail"
  apply (rule spec_monad_ext)
  apply (simp add: run_bind)
  done

subsection ‹\<^const>‹assert››

lemma assert_simps[simp]:
  "assert True = return ()"
  "assert False = top"
   by (auto simp add: assert_def)

lemma always_progress_assert[always_progress_intros]: "always_progress (assert P)"
  by (simp add: always_progress_def assert_def)

lemma run_assert[run_spec_monad]:
  "run (assert P) s = (if P then pure_post_state (Result (), s) else ⊤)"
  by (simp add: assert_def)

lemma runs_to_assert_iff[simp]: "assert P ∙ s ⦃ Q ⦄ ⟷ P ∧ Q (Result ()) s"
  by (simp add: runs_to.rep_eq run_assert)

lemma runs_to_assert[runs_to_vcg]: "P ⟹ Q (Result ()) s ⟹ assert P ∙ s ⦃ Q ⦄"
  by simp

lemma runs_to_partial_assert_iff[simp]: "assert P ∙ s ?⦃ Q ⦄ ⟷ (P ⟶ Q (Result ()) s)"
  by (simp add: runs_to_partial.rep_eq run_assert)

lemma refines_top_iff[simp]: "refines ⊤ g s t R ⟷ run g t = ⊤"
  by transfer auto

lemma refines_assert: 
  "refines (assert P) (assert Q) s t R ⟷ (Q ⟶ P ∧ R (Result (), s) (Result (), t))"
  by (simp add: assert_def)

subsection "\<^const>‹assume›"

lemma assume_simps[simp]:
  "assume True = return ()"
  "assume False = bot"
   by (auto simp add: assume_def)

lemma always_progress_assume[always_progress_intros]: "P ⟹ always_progress (assume P)"
  by (simp add: always_progress_def assume_def)

lemma run_assume[run_spec_monad]:
  "run (assume P) s = (if P then pure_post_state (Result (), s) else ⊥)"
  by (simp add: assume_def)

lemma run_assume_simps[run_spec_monad, simp]:
  "P ⟹ run (assume P) s = pure_post_state (Result (), s)"
  "¬ P ⟹ run (assume P) s = ⊥"
  by (simp_all add: run_assume)

lemma runs_to_assume_iff[simp]: "assume P ∙ s ⦃ Q ⦄ ⟷ (P ⟶ Q (Result ()) s)"
  by (simp add: runs_to.rep_eq run_assume)

lemma runs_to_partial_assume_iff[simp]: "assume P ∙ s ?⦃ Q ⦄ ⟷ (P ⟶ Q (Result ()) s)"
  by (simp add: runs_to_partial.rep_eq run_assume)

lemma runs_to_assume[runs_to_vcg]: "(P ⟹ Q (Result ()) s) ⟹ assume P ∙ s ⦃ Q ⦄"
  by simp

subsection ‹\<^const>‹assume_outcome››

lemma run_assume_outcome[run_spec_monad, simp]: "run (assume_outcome f) s = Success (f s)"
  apply transfer
  apply simp
  done

lemma always_progress_assume_outcome[always_progress_intros]:
  "(⋀s. f s ≠ {}) ⟹ always_progress (assume_outcome f)"
  by (auto simp add: always_progress_def bot_post_state_def)

lemma runs_to_assume_outcome[runs_to_vcg]:
  "(assume_outcome f) ∙ s ⦃λr t. (r, t) ∈ f s⦄"
  by (simp add: runs_to.rep_eq)

lemma runs_to_assume_outcome_iff[runs_to_iff]:
  "(assume_outcome f) ∙ s ⦃Q⦄ ⟷ (∀(r, t) ∈ f s. Q r t)"
  by (auto simp add: runs_to.rep_eq)

lemma runs_to_partial_assume_outcome[runs_to_vcg]:
  "(assume_outcome f) ∙ s ?⦃λr t. (r, t) ∈ f s⦄"
  by (simp add: runs_to_partial.rep_eq)

lemma assume_outcome_elementary:
  "assume_outcome f = do {s ← get_state; (r, t) ← select (f s); set_state t; yield r}"
  apply (rule spec_monad_ext)
  apply (simp add: run_bind Sup_Success pure_post_state_def)
  done

subsection ‹\<^const>‹assume_result_and_state››

lemma run_assume_result_and_state[run_spec_monad, simp]:
  "run (assume_result_and_state f) s = Success ((λ(v, t). (Result v, t)) `  f s)"
  by (auto simp add: assume_result_and_state_def run_bind Sup_Success_pair pure_post_state_def)

lemma always_progress_assume_result_and_state[always_progress_intros]:
  "(⋀s. f s ≠ {}) ⟹ always_progress (assume_result_and_state f)"
  by (simp add: always_progress_def)

lemma runs_to_assume_result_and_state[runs_to_vcg]:
  "assume_result_and_state f ∙ s ⦃λr t. ∃v. r = Result v ∧ (v, t) ∈ f s⦄"
  by (auto simp add: runs_to.rep_eq)

lemma runs_to_assume_result_and_state_iff[runs_to_iff]:
  "assume_result_and_state f ∙ s ⦃Q⦄ ⟷ (∀(v, t) ∈ f s. Q (Result v) t)"
  by (auto simp add: runs_to.rep_eq)

lemma runs_to_partial_assume_result_and_state[runs_to_vcg]:
  "assume_result_and_state f ∙ s ?⦃λr t. ∃v. r = Result v ∧ (v, t) ∈ f s⦄"
  by (auto simp add: runs_to_partial.rep_eq)

lemma refines_assume_result_and_state_right:
  "f ∙ s ⦃λr s'. ∃r' t'.  (r', t') ∈ g t ∧ R (r, s') (Result r', t') ⦄ ⟹
  refines f (assume_result_and_state g) s t R"
  by (simp add: refines.rep_eq runs_to.rep_eq sim_post_state_Success2 split_beta' Bex_def)

lemma refines_assume_result_and_state:
  "sim_set R (P s) (Q t) ⟹
    refines (assume_result_and_state P) (assume_result_and_state Q) s t
     (λ(Res v, s) (Res w, t). R (v, s) (w, t))"
  by (force simp: refines.rep_eq sim_set_def)

lemma refines_assume_result_and_state_iff:
  "refines (assume_result_and_state A) (assume_result_and_state B) s t Q ⟷ 
        sim_set (λ(v, s') (w, t'). Q (Result v, s') (Result w, t')) (A s) (B t) "
  apply (simp add: refines.rep_eq, intro iffI)
  subgoal
    by (fastforce simp add: sim_set_def)
  subgoal
    by (force simp add: sim_set_def)
  done

subsection ‹\<^const>‹gets››

lemma run_gets[run_spec_monad, simp]:"run (gets f) s = pure_post_state (Result (f s), s)"
  by (simp add: gets_def run_bind)

lemma always_progress_gets[always_progress_intros]: "always_progress (gets f)"
  by (simp add: always_progress_def)

lemma runs_to_gets[runs_to_vcg]: "gets f ∙ s ⦃λr t. r = Result (f s) ∧ t = s⦄"
  by (simp add: runs_to.rep_eq)

lemma runs_to_gets_iff[runs_to_iff]: "gets f ∙ s ⦃Q⦄ ⟷ Q (Result (f s)) s"
  by (simp add: runs_to.rep_eq)

lemma runs_to_partial_gets[runs_to_vcg]: "gets f ∙ s ?⦃λr t. r = Result (f s) ∧ t = s⦄"
  by (simp add: runs_to_partial.rep_eq)

lemma refines_gets: "R (Result (f s), s) (Result (g t), t) ⟹ refines (gets f) (gets g) s t R"
  by (auto simp add: refines.rep_eq)

lemma rel_spec_gets: "R (Result (f s), s) (Result (g t), t) ⟹ rel_spec (gets f) (gets g) s t R"
  by (auto simp add: rel_spec_iff_refines intro: refines_gets)

lemma runs_to_always_progress_to_gets:
  "(⋀s. f ∙ s ⦃λr t. t = s ∧ r = Result (v s)⦄) ⟹ always_progress f ⟹ f = gets v"
  apply (clarsimp simp: spec_monad_ext_iff runs_to.rep_eq always_progress.rep_eq)
  subgoal premises prems for s
    using prems[rule_format, of s]
    by (cases "run f s"; auto simp add: pure_post_state_def Ball_def)
  done

lemma gets_let_bind: "(gets (λs. let v = f' s in (g' v s))) = do {v <- gets f'; gets (g' v)}"
  apply (rule spec_monad_eqI)
  apply (auto simp add: runs_to_iff)
  done

subsection ‹\<^const>‹assert_result_and_state››

lemma run_assert_result_and_state[run_spec_monad]:
  "run (assert_result_and_state f) s =
    (if f s = {} then ⊤ else Success ((λ(v, t). (Result v, t)) `  f s))"
  by (auto simp add: assert_result_and_state_def run_bind pure_post_state_def Sup_Success_pair)

lemma always_progress_assert_result_and_state[always_progress_intros]:
  "always_progress (assert_result_and_state f)"
  by (auto simp add: always_progress_def run_assert_result_and_state)

lemma runs_to_assert_result_and_state_iff[runs_to_iff]:
  "assert_result_and_state f ∙ s ⦃Q⦄ ⟷ (f s ≠ {} ∧ (∀(v,t) ∈ f s. Q (Result v) t))"
  by (auto simp add: runs_to.rep_eq run_assert_result_and_state)

lemma runs_to_assert_result_and_state[runs_to_vcg]:
  "f s ≠ {} ⟹ assert_result_and_state f ∙ s ⦃λr t. ∃v. r = Result v ∧ (v, t) ∈ f s⦄"
  by (simp add: runs_to_assert_result_and_state_iff)

lemma runs_to_partial_assert_result_and_state[runs_to_vcg]:
  "assert_result_and_state f ∙ s ?⦃λr t. ∃v. r = Result v ∧ (v, t) ∈ f s⦄"
  by (auto simp add: runs_to_partial.rep_eq run_assert_result_and_state)

lemma runs_to_state_select[runs_to_vcg]:
  "∃t. (s, t) ∈ R ⟹ state_select R ∙ s ⦃ λr t. r = Result () ∧ (s, t) ∈ R ⦄"
  by (auto intro!: runs_to_assert_result_and_state[THEN runs_to_weaken])

lemma runs_to_partial_state_select[runs_to_vcg]:
  "state_select R ∙ s ?⦃ λr t. r = Result () ∧ (s, t) ∈ R ⦄"
  by (simp add: runs_to_partial.rep_eq run_assert_result_and_state)

lemma refines_assert_result_and_state:
  assumes sim: "(⋀r' s'. (r', s') ∈ f s ⟹ (∃v' t'. (v', t') ∈ g t ∧ R (Result r', s') (Result v', t')))" 
  assumes emp: "f s = {} ⟹ g t = {}"
  shows "refines (assert_result_and_state f) (assert_result_and_state g) s t R"
  using sim emp by (fastforce simp add: refines_iff_runs_to runs_to_iff)

lemma refines_state_select: 
  assumes sim: "(⋀s'. (s, s') ∈ f ⟹ (∃t'. (t, t') ∈ g ∧ R (Result (), s') (Result (), t')))" 
  assumes emp: "∄s'. (s, s') ∈ f ⟹ ∄t'. (t, t') ∈ g"
  shows "refines (state_select f) (state_select g) s t R"
  using sim emp by (intro refines_assert_result_and_state) auto

subsection "\<^const>‹assuming›"

lemma run_assuming[run_spec_monad]:
  "run (assuming g) s = (if g s then pure_post_state (Result (), s) else ⊥)"
  by (simp add: assuming_def run_bind)

lemma always_progress_assuming[always_progress_intros]: "always_progress (assuming g) ⟷ (∀s. g s)"
  by (auto simp add: always_progress_def run_assuming)

lemma runs_to_assuming[runs_to_vcg]: "assuming g ∙ s ⦃λr t. g s ∧ r = Result () ∧ t = s⦄"
  by (simp add: runs_to.rep_eq run_assuming)

lemma runs_to_assuming_iff[runs_to_iff]: "assuming g ∙ s ⦃Q⦄ ⟷ (g s ⟶ Q (Result ()) s)"
  by (auto simp add: runs_to.rep_eq run_assuming)

lemma runs_to_partial_assuming[runs_to_vcg]:
  "assuming g ∙ s ?⦃λr t. g s ∧ r = Result () ∧ t = s ∧ g s⦄"
  by (simp add: runs_to_partial.rep_eq run_assuming)

lemma assuming_state_assume:
  "assuming P = assume_result_and_state (λs. (if P s then {((), s)} else {}))"
  by (simp add: spec_monad_eq_iff runs_to_iff)

lemma assuming_True[simp]: "assuming (λs. True) = skip"
  by (simp add: spec_monad_eq_iff runs_to_iff)

lemma refines_assuming:
  "(P s ⟹ Q t) ⟹ (P s ⟹ Q t ⟹ R (Result (), s) (Result (), t)) ⟹
    refines (assuming P) (assuming Q) s t R"
  by (auto simp add: refines.rep_eq run_assuming)

lemma rel_spec_assuming:
  "(Q t ⟷ P s) ⟹ (P s ⟹ Q t ⟹ R (Result (), s) (Result (), t)) ⟹
    rel_spec (assuming P) (assuming Q) s t R"
  by (auto simp add: rel_spec_def run_assuming rel_set_def)

lemma refines_bind_assuming_right:
  "P t ⟹ (P t ⟹ refines f (g ()) s t R) ⟹ refines f (assuming P ⤜ g) s t R"
  by (simp add: refines.rep_eq run_assuming run_bind)

lemma refines_bind_assuming_left:
  "(P s ⟹ refines (f ()) g s t R) ⟹ refines (assuming P >>= f) g s t R"
  by (simp add: refines.rep_eq run_assuming run_bind)

subsection "\<^const>‹guard›"

lemma run_guard[run_spec_monad]:
  "run (guard g) s = (if g s then pure_post_state (Result (), s) else ⊤)"
  by (simp add: guard_def run_bind)

lemma always_progress_guard[always_progress_intros]: "always_progress (guard g)"
  by (auto simp add: always_progress_def run_guard)

lemma runs_to_guard[runs_to_vcg]: "g s ⟹ guard g ∙ s ⦃λr t. r = Result () ∧ t = s⦄"
  by (simp add: runs_to.rep_eq run_guard)

lemma runs_to_guard_iff[runs_to_iff]: "guard g ∙ s ⦃Q⦄ ⟷ (g s ∧ Q (Result ()) s)"
  by (auto simp add: runs_to.rep_eq run_guard)

lemma runs_to_partial_guard[runs_to_vcg]: "guard g ∙ s ?⦃λr t. r = Result () ∧ t = s ∧ g s⦄"
  by (simp add: runs_to_partial.rep_eq run_guard)

lemma refines_guard:
  "(Q t ⟹ P s) ⟹ (P s ⟹ Q t ⟹ R (Result (), s) (Result (), t)) ⟹
    refines (guard P) (guard Q) s t R"
  by (auto simp add: refines.rep_eq run_guard)

lemma rel_spec_guard:
  "(Q t ⟷ P s) ⟹ (P s ⟹ Q t ⟹ R (Result (), s) (Result (), t)) ⟹
    rel_spec (guard P) (guard Q) s t R"
  by (auto simp add: rel_spec_def run_guard rel_set_def)

lemma refines_bind_guard_right:
  "refines f (guard P ⤜ g) s t R" if "P t ⟹ refines f (g ()) s t R"
  using that
  by (auto simp: refines.rep_eq run_guard run_bind)

lemma guard_False_fail: "guard (λ_. False) = fail"
  by (simp add: spec_monad_ext run_guard)

lemma rel_spec_monad_bind_guard:
  shows "(⋀x. rel_spec_monad (=) Q (f x) (g x)) ⟹
    rel_spec_monad (=) Q (guard P >>= f) (guard P >>= g)"
  by (auto simp add: rel_spec_monad_def run_bind run_guard)

lemma runs_to_guard_bind_iff: "((guard P >>= f) ∙ s ⦃ Q ⦄) ⟷ P s ∧ ((f ()) ∙ s ⦃ Q ⦄)"
  by (simp add: runs_to_iff)

lemma refines_bind_guard_right_iff:
  "refines f (guard P ⤜ g) s t R ⟷ (P t ⟶ refines f (g ()) s t R)"
  by (auto simp: refines.rep_eq run_guard run_bind)

lemma refines_bind_guard_right_end:
  assumes f_g: "refines f g s t R"
  shows "refines f (do {res <- g; guard G; return res}) s t 
            (λ(r, s) (q, t). R (r, s) (q, t) ∧ 
                (case q of Exception e ⇒ True | Result _ ⇒ G t))"
  apply (subst bind_return[symmetric])
  apply (rule refines_bind[OF f_g])
  apply (auto simp: refines_bind_guard_right_iff)
  done

lemma refines_bind_guard_right_end':
  assumes f_g: "refines f g s t R"
  shows "refines f (do {res <- g; guard (G res); return res}) s t 
            (λ(r, s) (q, t). R (r, s) (q, t) ∧ 
                (case q of Exception e ⇒ True | Result v ⇒ G v t))"
  apply (subst bind_return[symmetric])
  apply (rule refines_bind[OF f_g])
  apply (auto simp: refines_bind_guard_right_iff)
  done

subsection "\<^const>‹assert_opt›"

lemma run_assert_opt[run_spec_monad, simp]:
  "run (assert_opt x) s = (case x of Some v ⇒ pure_post_state (Result v, s) | None ⇒ ⊤)"
  by (simp add: assert_opt_def run_bind split: option.splits)

lemma always_progress_assert_opt[always_progress_intros]: "always_progress (assert_opt x)"
  by (simp add: always_progress_intros assert_opt_def split: option.splits)

lemma runs_to_assert_opt[runs_to_vcg]: "(∃v. x = Some v ∧ Q (Result v) s) ⟹ assert_opt x ∙ s ⦃Q⦄"
  by (auto simp add: runs_to.rep_eq)

lemma runs_to_assert_opt_iff[runs_to_iff]:
  "assert_opt x ∙ s ⦃Q⦄ ⟷ (∃v. x = Some v ∧ Q (Result v) s)"
  by (auto simp add: runs_to.rep_eq split: option.split)

lemma runs_to_partial_assert_opt[runs_to_vcg]:
  "(⋀v. x = Some v ⟹ Q (Result v) s) ⟹ assert_opt x ∙ s ?⦃Q⦄"
  by (auto simp add: runs_to_partial.rep_eq split: option.split)

subsection "\<^const>‹gets_the›"

lemma run_gets_the':
  "run (gets_the f) s = (case f s of Some v ⇒ pure_post_state (Result v, s) | None ⇒ ⊤)"
  by (simp add: gets_the_def run_bind top_post_state_def pure_post_state_def split: option.split)

lemma run_gets_the[run_spec_monad, simp]:
  "run (gets_the f) s = (case (f s) of Some v ⇒ pure_post_state (Result v, s) | None ⇒ ⊤)"
  by (simp add: gets_the_def run_bind)

lemma always_progress_gets_the[always_progress_intros]: "always_progress (gets_the f)"
  by (simp add: always_progress_intros gets_the_def split: option.splits)

lemma runs_to_gets_the[runs_to_vcg]: "(∃v. f s = Some v ∧ Q (Result v) s) ⟹ gets_the f ∙ s ⦃Q⦄"
  by (auto simp add: runs_to.rep_eq)

lemma runs_to_gets_the_iff[runs_to_iff]:
  "gets_the f ∙ s ⦃Q⦄ ⟷ (∃v. f s = Some v ∧ Q (Result v) s)"
  by (auto simp add: runs_to.rep_eq split: option.split)

lemma runs_to_partial_gets_the[runs_to_vcg]:
  "(⋀v. f s = Some v ⟹ Q (Result v) s) ⟹ gets_the f ∙ s ?⦃Q⦄"
  by (auto simp add: runs_to_partial.rep_eq split: option.split)

subsection ‹\<^const>‹modify››

lemma run_modify[run_spec_monad, simp]: "run (modify f) s = pure_post_state (Result (), f s)"
  by (simp add: modify_def run_bind)

lemma always_progress_modifiy[always_progress_intros]: "always_progress (modify f)"
  by (simp add: modify_def always_progress_intros)

lemma runs_to_modify[runs_to_vcg]: "modify f ∙ s ⦃ λr t. r = Result () ∧ t = f s ⦄"
  by (simp add: runs_to.rep_eq)

lemma runs_to_modify_res[runs_to_vcg]: "((modify f)::(unit, 's) res_monad) ∙ s ⦃ λr t. t = f s ⦄"
  by (simp add: runs_to.rep_eq)

lemma runs_to_modify_iff[runs_to_iff]: "modify f ∙ s ⦃Q⦄ ⟷ Q (Result ()) (f s)"
  by (simp add: runs_to.rep_eq)

lemma runs_to_partial_modify[runs_to_vcg]: "modify f ∙ s ?⦃ λr t. r = Result () ∧ t = f s ⦄"
  by (simp add: runs_to_partial.rep_eq)

lemma runs_to_partial_modify_res[runs_to_vcg]:
  "((modify f)::(unit, 's) res_monad) ∙ s ?⦃ λr t. t = f s ⦄"
  by (simp add: runs_to_partial.rep_eq)

lemma refines_modify:
  "R (Result (), f s) (Result (), g t) ⟹ refines (modify f) (modify g) s t R"
  by (auto simp add: refines.rep_eq)

lemma rel_spec_modify:
  "R (Result (), f s) (Result (), g t) ⟹ rel_spec (modify f) (modify g) s t R"
  by (auto simp add: rel_spec_iff_refines intro: refines_modify)

subsection ‹condition›

lemma run_condition[run_spec_monad]: "run (condition c f g) s = (if c s then run f s else run g s)"
  by (simp add: condition_def run_bind)

lemma run_condition_True[run_spec_monad, simp]: "c s ⟹ run (condition c f g) s = run f s"
  by (simp add: run_condition)

lemma run_condition_False[run_spec_monad, simp]: "¬c s ⟹ run (condition c f g) s = run g s"
  by (simp add: run_condition)

lemma always_progress_condition[always_progress_intros]:
  "always_progress f ⟹ always_progress g ⟹ always_progress (condition c f g)"
  by (auto simp add: always_progress_def run_condition)

lemma condition_swap: "(condition C A B) = (condition (λs. ¬ C s) B A)"
  by (rule spec_monad_ext) (clarsimp simp add: run_condition)

lemma condition_fail_rhs: "(condition C X fail) = (guard C >>= (λ_. X))"
  by (rule spec_monad_ext) (simp add: run_bind run_guard run_condition)

lemma condition_fail_lhs: "(condition C fail X) = (guard (λs. ¬ C s) >>= (λ_. X))"
  by (metis condition_fail_rhs condition_swap)

lemma condition_bind_fail[simp]:
  "(condition C A B >>= (λ_. fail)) = condition C (A >>= (λ_. fail)) (B >>= (λ_. fail))"
  apply (rule spec_monad_ext)
  apply (clarsimp simp add: run_condition run_bind)
  done

lemma condition_True[simp]: "condition (λ_. True) f g = f"
  apply (rule spec_monad_ext)
  apply (clarsimp simp add: run_condition run_bind)
  done

lemma condition_False[simp]: "condition (λ_. False) f g = g"
  apply (rule spec_monad_ext)
  apply (clarsimp simp add: run_condition run_bind)
  done

lemma le_condition_runI:
  "(⋀s. c s ⟹ run h s ≤ run f s) ⟹ (⋀s. ¬ c s ⟹ run h s ≤ run g s)
  ⟹ h ≤ condition c f g"
  by (simp add: le_spec_monad_runI run_condition)

lemma mono_condition_spec_monad:
  "mono T ⟹ mono F ⟹ mono (λx. condition C (F x) (T x))"
  by (auto simp: condition_def intro!: mono_bind_spec_monad mono_const)

lemma mono_condition: "f ≤ f' ⟹ g ≤ g' ⟹ condition c f g ≤ condition c f' g'"
  by (simp add: le_fun_def less_eq_spec_monad.rep_eq run_condition)

lemma monotone_condition_le[partial_function_mono]:
  "monotone R (≤) (λf'. f f') ⟹ (monotone R (≤) (λf'. g f'))
  ⟹ monotone R (≤) (λf'. condition c (f f') (g f'))"
  by (auto simp add: monotone_def intro!: mono_condition)

lemma monotone_condition_ge[partial_function_mono]:
  "monotone R (≥) (λf'. f f') ⟹ (monotone R (≥) (λf'. g f'))
  ⟹ monotone R (≥) (λf'. condition c (f f') (g f'))"
  by (auto simp add: monotone_def intro!: mono_condition)

lemma runs_to_condition[runs_to_vcg]:
  "(c s ⟹ f ∙ s ⦃ Q ⦄) ⟹ (¬ c s ⟹ g ∙ s ⦃ Q ⦄) ⟹ condition c f g ∙ s ⦃ Q ⦄"
  by (simp add: runs_to.rep_eq run_condition)

lemma runs_to_condition_iff[runs_to_iff]:
  "condition c f g ∙ s ⦃ Q ⦄ ⟷ (if c s then f ∙ s ⦃ Q ⦄ else g ∙ s ⦃ Q ⦄)"
  by (simp add: runs_to.rep_eq run_condition)

lemma runs_to_partial_condition[runs_to_vcg]:
  "(c s ⟹ f ∙ s ?⦃ Q ⦄) ⟹ (¬ c s ⟹ g ∙ s ?⦃ Q ⦄) ⟹ condition c f g ∙ s ?⦃ Q ⦄"
  by (simp add: runs_to_partial.rep_eq run_condition)

lemma refines_condition_iff:
  assumes "c' s' ⟷ c s"
  shows "refines (condition c f g) (condition c' f' g') s s' R ⟷
   (if c' s' then refines f f' s s' R else refines g g' s s' R)"
  using assms
  by (auto simp add: refines.rep_eq run_condition)

lemma refines_condition:
  "P s ⟷ P' s' ⟹
    (P s ⟹ P' s' ⟹ refines f f' s s' R) ⟹
    (¬ P s ⟹ ¬ P' s' ⟹ refines g g' s s' R) ⟹
    refines (condition P f g) (condition P' f' g') s s' R"
  using refines_condition_iff
  by metis

lemma refines_condition_TrueI:
  assumes "c' s' = c s" and "c' s'" "refines f f' s s' R"
  shows "refines (condition c f g) (condition c' f' g') s s' R"
  by (simp add:  refines_condition_iff[where c'=c' and c=c and s'=s' and s=s, OF assms(1)] assms(2, 3))

lemma refines_condition_FalseI:
  assumes "c' s' = c s" and "¬c' s'" "refines g g' s s' R"
  shows "refines (condition c f g) (condition c' f' g') s s' R"
  by (simp add:  refines_condition_iff[where c'=c' and c=c and s'=s' and s=s, OF assms(1)] assms(2, 3))

lemma refines_condition_bind_left:
  "refines (condition C T F ⤜ X) Y s t R ⟷
    (C s ⟶ refines (T ⤜ X) Y s t R) ∧ (¬ C s ⟶ refines (F ⤜ X) Y s t R)"
  by (simp add: refines.rep_eq run_bind run_condition)

lemma refines_condition_bind_right:
  "refines X (condition C T F ⤜ Y) s t R ⟷
    (C t ⟶ refines X (T ⤜ Y) s t R) ∧ (¬ C t ⟶ refines X (F ⤜ Y) s t R)"
  by (simp add: refines.rep_eq run_bind run_condition)

lemma rel_spec_condition_iff:
  assumes "c' s' ⟷ c s"
  shows "rel_spec (condition c f g) (condition c' f' g') s s' R ⟷
   (if c' s' then rel_spec f f' s s' R else rel_spec g g' s s' R)"
  using assms
  by (auto simp add: rel_spec_def run_condition)

lemma rel_spec_condition:
  "P s ⟷ P' s' ⟹
    (P s ⟹ P' s' ⟹ rel_spec f f' s s' R) ⟹
    (¬ P s ⟹ ¬ P' s' ⟹ rel_spec g g' s s' R) ⟹
    rel_spec (condition P f g) (condition P' f' g') s s' R"
  using rel_spec_condition_iff
  by metis

lemma rel_spec_condition_TrueI:
  assumes "c' s' = c s" and "c' s'" "rel_spec f f' s s' R"
  shows "rel_spec (condition c f g) (condition c' f' g') s s' R"
  by (simp add:  rel_spec_condition_iff[where c'=c' and c=c and s'=s' and s=s, OF assms(1)] assms(2, 3))

lemma rel_spec_condition_FalseI:
  assumes "c' s' = c s" and "¬c' s'" "rel_spec g g' s s' R"
  shows "rel_spec (condition c f g) (condition c' f' g') s s' R"
  by (simp add:  rel_spec_condition_iff[where c'=c' and c=c and s'=s' and s=s, OF assms(1)] assms(2, 3))

lemma refines_condition_left:
  "(P s ⟹ refines f h s t R) ⟹ (¬ P s ⟹ refines g h s t R) ⟹
    refines (condition P f g) h s t R"
  by (simp add: refines.rep_eq run_condition)

lemma rel_spec_condition_left:
  "(P s ⟹ rel_spec f h s t R) ⟹ (¬ P s ⟹ rel_spec g h s t R) ⟹
    rel_spec (condition P f g) h s t R"
  by (auto simp add: rel_spec_def run_condition)

lemma refines_condition_true:
  "P t ⟹ refines f g s t R ⟹ refines f (condition P g h) s t R"
  by (simp add: refines.rep_eq run_condition)

lemma rel_spec_condition_true:
  "P t ⟹ rel_spec f g s t R ⟹
    rel_spec f (condition P g h) s t R"
  by (auto simp add: rel_spec_def run_condition)

lemma refines_condition_false:
  "¬ P t ⟹ refines f h s t R ⟹
    refines f (condition P g h) s t R"
  by (simp add: refines.rep_eq run_condition)

lemma rel_spec_condition_false:
  "¬ P t ⟹ rel_spec f h s t R ⟹
    rel_spec f (condition P g h) s t R"
  by (auto simp add: rel_spec_def run_condition)

lemma condition_bind:
  "(condition P f g >>= h) = condition P (f >>= h) (g >>= h)"
  by (simp add: spec_monad_ext_iff run_condition run_bind)

lemma rel_spec_monad_condition:
  assumes "rel_fun R (=) P P'"
    and "rel_spec_monad R Q f f'"
    and "rel_spec_monad R Q g g'"
  shows "rel_spec_monad R Q (condition P f g) (condition P' f' g')"
  using assms
  by (auto simp add: rel_spec_monad_def run_condition rel_fun_def)

lemma rel_spec_monad_condition_const:
  "P ⟷ P' ⟹ (P ⟹ rel_spec_monad R Q f f') ⟹
    (¬ P ⟹ rel_spec_monad R Q g g') ⟹
  rel_spec_monad R Q (condition (λ_. P) f g) (condition (λ_. P') f' g')"
  by (cases P) (auto simp add: condition_def)

subsection "\<^const>‹when›"

lemma run_when[run_spec_monad]:
  "run (when c f) s = (if c then run f s else pure_post_state (Result (), s))"
  unfolding when_def by simp

lemma always_progress_when[always_progress_intros]:
  "always_progress f ⟹ always_progress (when c f)"
  unfolding when_def by (simp add: always_progress_intros)

lemma runs_to_when[runs_to_vcg]:
  "(c ⟹ f ∙ s ⦃ Q ⦄) ⟹ (¬ c ⟹ Q (Result ()) s) ⟹ when c f ∙ s ⦃ Q ⦄"
  by (auto simp add: runs_to.rep_eq run_when)

lemma runs_to_when_iff[runs_to_iff]: "
  (when c f) ∙ s ⦃ Q ⦄ ⟷ (if c then f ∙ s ⦃ Q ⦄ else Q (Result ()) s)"
  by (auto simp add: runs_to.rep_eq run_when)

lemma runs_to_partial_when[runs_to_vcg]:
  "(c ⟹ f ∙ s ?⦃ Q ⦄) ⟹ (¬ c ⟹ Q (Result ()) s) ⟹ when c f ∙ s ?⦃ Q ⦄"
  by (auto simp add: runs_to_partial.rep_eq run_when)

lemma mono_when: "f ≤ f' ⟹ when c f ≤ when c f'"
  unfolding when_def by (simp add: mono_condition)

lemma monotone_when_le[partial_function_mono]:
  "monotone R (≤) (λf'. f f')
  ⟹ monotone R (≤) (λf'. when c (f f'))"
  unfolding when_def by (simp add: monotone_condition_le)

lemma monotone_when_ge[partial_function_mono]:
  "monotone R (≥) (λf'. f f')
  ⟹ monotone R (≥) (λf'. when c (f f'))"
  unfolding when_def by (simp add: monotone_condition_ge)

lemma when_True[simp]: "when True f = f"
  apply (rule spec_monad_ext)
  apply (simp add: run_when)
  done

lemma when_False[simp]: "when False f = return ()"
  apply (rule spec_monad_ext)
  apply (simp add: run_when)
  done

subsection ‹While›

context fixes C :: "'a ⇒ 's ⇒ bool" and B :: "'a ⇒ ('e::default, 'a, 's) spec_monad"
begin

definition whileLoop :: "'a ⇒ ('e, 'a, 's) spec_monad" where
  "whileLoop =
    gfp (λW a. condition (C a) (bind (B a) W) (return a))"
  ― ‹Collapses to \<^const>‹Failure› in case of any non terminating computation.›

definition whileLoop_finite :: "'a ⇒ ('e, 'a, 's) spec_monad" where
  "whileLoop_finite =
    lfp (λW a. condition (C a) (bind (B a) W) (return a))"
  ― ‹Does not collapse to \<^const>‹Failure› in presence of a non terminating computation.
    \<^const>‹Failure› can still occur when the body fails in some iteration. It
    captures the outcomes of all terminating and thus finite computations.›

inductive whileLoop_terminates :: "'a ⇒ 's ⇒ bool" where
  step: "⋀a s. (C a s ⟹ B a ∙ s ?⦃ λv s. ∀a. v = Result a ⟶ whileLoop_terminates a s ⦄) ⟹
    whileLoop_terminates a s"
  ― ‹ This is weaker than ‹run (whileLoop a) s ≠ ⊤›: as it uses partial correctness ›

lemma mono_whileLoop_functional:
  "mono (λW a. condition (C a) (bind (B a) W) (return a))"
  by (intro mono_app mono_lam mono_const mono_bind_spec_monad mono_condition_spec_monad)

lemma whileLoop_unroll:
  "whileLoop a =
    condition (C a) (bind (B a) whileLoop) (return a)"
  unfolding whileLoop_def
  by (subst gfp_unfold[OF mono_whileLoop_functional]) simp

lemma whileLoop_finite_unfold:
  "whileLoop_finite a =
    condition (C a) (bind (B a) whileLoop_finite) (return a)"
  unfolding whileLoop_finite_def
  by (subst lfp_unfold[OF mono_whileLoop_functional]) simp

lemma whileLoop_ne_Failure:
  "(C a s ⟹ B a ∙ s ⦃ λx s. ∀a. x = Result a ⟶ run (whileLoop a) s ≠ Failure ⦄) ⟹
    run (whileLoop a) s ≠ Failure"
  by (subst whileLoop_unroll)
     (simp add: run_condition run_bind_eq_top_iff flip: top_post_state_def)

lemma whileLoop_ne_top_induct[consumes 1, case_names step]:
  assumes a_s: "run (whileLoop a) s ≠ ⊤"
    and step: "⋀a s. (C a s ⟹ B a ∙ s ⦃ λx s. ∀a. x = Result a ⟶ P a s ⦄) ⟹ P a s"
  shows "P a s"
proof -
  have "(λa. Spec (λs. if P a s then ⊥ else ⊤)) ≤ whileLoop"
    unfolding whileLoop_def
    by (intro gfp_upperbound)
       (auto intro: step runs_to_weaken
         simp: le_fun_def less_eq_spec_monad.rep_eq run_condition top_unique run_bind_eq_top_iff)
  with a_s show ?thesis
    by (auto simp add: le_fun_def less_eq_spec_monad.rep_eq top_unique top_post_state_def
             split: if_splits)
qed

lemma runs_to_whileLoop:
  assumes R: "wf R"
  assumes *: "I (Result a) s"
  assumes P_Result: "⋀a s. ¬ C a s ⟹ I (Result a) s ⟹ P (Result a) s"
  assumes P_Exception: "⋀a s. a ≠ default ⟹ I (Exception a) s ⟹ P (Exception a) s"
  assumes B: "⋀a s. C a s ⟹ I (Result a) s ⟹
    B a  ∙ s ⦃λr t. I r t ∧ (∀b. r = Result b ⟶ ((b, t), (a, s)) ∈ R)⦄"
  shows "whileLoop a ∙ s ⦃P⦄"
proof (use R * in ‹induction x≡"(a, s)" arbitrary: a s›)
  case (less a s)
  show ?case
    by (auto simp: P_Result P_Exception whileLoop_unroll[of a]
                   runs_to_condition_iff runs_to_bind_iff less
             intro!: runs_to_weaken[OF B])
qed

lemma runs_to_whileLoop_finite:
  assumes *: "I (Result a) s"
  assumes P_Result: "⋀a s. ¬ C a s ⟹ I (Result a) s ⟹ P (Result a) s"
  assumes P_Exception: "⋀a s. a ≠ default ⟹ I (Exception a) s ⟹ P (Exception a) s"
  assumes B: "⋀a s. C a s ⟹ I (Result a) s ⟹ B a ∙ s ⦃ I ⦄"
  shows "whileLoop_finite a ∙ s ⦃P⦄"
proof -
  have "whileLoop_finite a ∙ s
    ⦃ λx s. I x s ∧ (∀a. x = Result a ⟶ C a s ⟶ P (Result a) s) ⦄"
    unfolding whileLoop_finite_def
    apply (rule runs_to_lfp[OF mono_whileLoop_functional, of "λa. I (Result a)", OF *])
    subgoal premises prems for W x s
      using prems(2)
      by (intro runs_to_condition runs_to_bind runs_to_yield_iff
                runs_to_weaken[OF B] runs_to_weaken[OF prems(1)] conjI allI impI)
         simp_all
    done
  then show ?thesis
    apply (rule runs_to_weaken)
    subgoal for r s
      by (cases r; cases "∃a. r = Result a ∧ C a s"; simp add: P_Result P_Exception)
    done
qed

lemma runs_to_partial_whileLoop_finite:
  assumes *: "I (Result a) s"
  assumes B: "⋀a s. C a s ⟹ I (Result a) s ⟹ (B a) ∙ s ?⦃ I ⦄"
  assumes P_Result: "⋀a s. ¬ C a s ⟹ I (Result a) s ⟹ P (Result a) s"
  assumes P_Exception: "⋀a s. a ≠ default ⟹ I (Exception a) s ⟹ P (Exception a) s"
  shows "whileLoop_finite a ∙ s ?⦃ P ⦄"
proof -
  have "whileLoop_finite a ∙ s ⦃P⦄" if **: "run (whileLoop_finite a) s ≠ ⊤"
    apply (rule runs_to_whileLoop_finite[where
          I="λx s. I x s ∧ (∀a. x = Result a ⟶ run (whileLoop_finite a) s ≠ ⊤)"])
       apply (auto simp: * ** P_Result P_Exception)[3]
    subgoal for a s using B[of a s]
      by (subst (asm) whileLoop_finite_unfold)
        (auto simp: runs_to_conj run_bind_eq_top_iff  dest: runs_to_of_runs_to_partial_runs_to')
    done
  then show ?thesis
    by (auto simp: runs_to_partial_alt top_post_state_def)
qed

lemma whileLoop_finite_eq_whileLoop_of_whileLoop_terminates:
  assumes "whileLoop_terminates a s"
  shows "run (whileLoop_finite a) s = run (whileLoop a) s"
proof (use assms in ‹induction›)
  case (step a s)
  show ?case
    apply (subst whileLoop_unroll)
    apply (subst whileLoop_finite_unfold)
    apply (auto simp: run_condition intro!: run_bind_cong[OF refl runs_to_partial_weaken, OF step])
    done
qed

lemma whileLoop_terminates_of_succeeds:
  "run (whileLoop a) s ≠ ⊤ ⟹ whileLoop_terminates a s"
  by (induction rule: whileLoop_ne_top_induct)
     (auto intro: whileLoop_terminates.step runs_to_partial_of_runs_to)

lemma whileLoop_finite_eq_whileLoop:
  "run (whileLoop a) s ≠ ⊤ ⟹ run (whileLoop_finite a) s = run (whileLoop a) s"
  by (rule whileLoop_finite_eq_whileLoop_of_whileLoop_terminates[OF
    whileLoop_terminates_of_succeeds])

lemma runs_to_whileLoop_of_runs_to_whileLoop_finite_if_terminates:
  "whileLoop_terminates i s ⟹ whileLoop_finite i ∙ s ⦃Q⦄ ⟹ whileLoop i ∙ s ⦃Q⦄"
  unfolding runs_to.rep_eq  whileLoop_finite_eq_whileLoop_of_whileLoop_terminates .

lemma whileLoop_finite_le_whileLoop: "whileLoop_finite a ≤ whileLoop a"
  using le_fun_def lfp_le_gfp mono_whileLoop_functional
  unfolding whileLoop_finite_def whileLoop_def
  by fastforce

lemma runs_to_partial_whileLoop_finite_whileLoop:
  "whileLoop_finite i ∙ s ?⦃Q⦄ ⟹ whileLoop i ∙ s ?⦃Q⦄"
  by (cases "run (whileLoop i) s = ⊤")
     (auto simp:
       runs_to.rep_eq runs_to_partial_alt whileLoop_finite_eq_whileLoop top_post_state_def)

lemma runs_to_partial_whileLoop:
  assumes "I (Result a) s"
  assumes "⋀a s. ¬ C a s ⟹ I (Result a) s ⟹ P (Result a) s"
  assumes "⋀a s. a ≠ default ⟹ I (Exception a) s ⟹ P (Exception a) s"
  assumes "⋀a s. C a s ⟹ I (Result a) s ⟹ (B a) ∙ s ?⦃ I ⦄"
  shows "whileLoop a ∙ s ?⦃ P ⦄"
  using assms(1,4,2,3)
  by (rule runs_to_partial_whileLoop_finite_whileLoop[OF runs_to_partial_whileLoop_finite])

lemma always_progress_whileLoop[always_progress_intros]:
  assumes B: "(⋀v. always_progress (B v))"
  shows "always_progress (whileLoop a)"
  unfolding always_progress.rep_eq
proof
  fix s have "run (whileLoop a) s ≠ ⊥" if *: "run (whileLoop a) s ≠ ⊤"
  proof (use * in ‹induction rule: whileLoop_ne_top_induct›)
    case (step a s) then show ?case
      apply (subst whileLoop_unroll)
      apply (simp add: run_condition run_bind bind_post_state_eq_bot prod_eq_iff runs_to.rep_eq
                  split: prod.splits exception_or_result_splits)
      apply clarsimp
      subgoal premises prems
        using holds_post_state_combine[OF prems(1,3), where R="λx. False"] B
        by (auto simp: holds_post_state_False exception_or_result_nchotomy always_progress.rep_eq)
      done
  qed
  then show "run (whileLoop a) s ≠ ⊥"
    by (cases "run (whileLoop a) s = Failure") (auto simp: bot_post_state_def)
qed

lemma runs_to_partial_whileLoop_cond_false:
  "(whileLoop I) ∙ s ?⦃ λr t. ∀a. r = Result a ⟶ ¬ C a t ⦄"
  by (rule runs_to_partial_whileLoop[of "λ_ _. True"]) simp_all

end

context
  fixes R
  fixes C :: "'a ⇒ 's ⇒ bool" and B :: "'a ⇒ ('e::default, 'a, 's) spec_monad"
    and C' :: "'b ⇒ 't ⇒ bool" and B' :: "'b ⇒ ('f::default, 'b, 't) spec_monad"
  assumes C: "⋀x s x' s'. R (Result x, s) (Result x', s') ⟹ C x s ⟷ C' x' s'"
  assumes B: "⋀x s x' s'. R (Result x, s) (Result x', s') ⟹ C x s ⟹ C' x' s' ⟹
    refines (B x) (B' x') s s' R"
  assumes R: "⋀v s v' s'. R (v, s) (v', s') ⟹ (∃x. v = Result x) ⟷ (∃x'. v' = Result x')"
begin

lemma refines_whileLoop_finite_strong:
  assumes x_x': "R (Result x, s) (Result x', s')"
  shows "refines (whileLoop_finite C B x) (whileLoop_finite C' B' x') s s'
    (λ(r, s) (r', s'). (∀v. r = Result v ⟶ (∃v'. r' = Result v' ∧ ¬ C v s ∧ ¬ C' v' s')) ∧
     R (r, s) (r', s'))"
    (is "refines _ _ _ _ ?R")
proof -
  let ?P = "λp. ∀x s x' s'. R (Result x, s) (Result x', s') ⟶
    refines (p x) (whileLoop_finite C' B' x') s s' ?R"

  have "?P (whileLoop_finite C B)"
    unfolding whileLoop_finite_def[of C B]
  proof (rule lfp_ordinal_induct[OF mono_whileLoop_functional], intro allI impI)
    fix S x s x' s' assume S: "?P S" and x_x': "R (Result x, s) (Result x', s')"
    from x_x' show
      "refines (condition (C x) (B x ⤜ S) (return x)) (whileLoop_finite C' B' x') s s' ?R"
      by (subst whileLoop_finite_unfold)
         (auto dest: R simp: C[OF x_x'] refines_condition_iff
               intro!: refines_bind'[OF refines_mono[OF _ B[OF x_x']]] S[rule_format])
  qed (simp add: refines_Sup1)
  with x_x' show ?thesis by simp
qed

lemma refines_whileLoop_finite:
  assumes x_x': "R (Result x, s) (Result x', s')"
  shows "refines (whileLoop_finite C B x) (whileLoop_finite C' B' x') s s' R"
  by (rule refines_mono[OF _ refines_whileLoop_finite_strong, OF _ x_x']) simp

lemma whileLoop_succeeds_terminates_of_refines:
  assumes "run (whileLoop C' B' x') s' ≠ ⊤"
  shows "R (Result x, s) (Result x', s') ⟹ run (whileLoop C B x) s ≠ Failure"
proof (use assms in ‹induction arbitrary: x s rule: whileLoop_ne_top_induct›)
  case (step a' s' a s)

  show ?case
  proof (rule whileLoop_ne_Failure)
    assume "C a s"
    with C[of a s a' s'] step.prems have "C' a' s'" by simp

    show "B a ∙ s ⦃ λx s. ∀a. x = Result a ⟶ run (whileLoop C B a) s ≠ Failure ⦄"
      using step.IH[OF ‹C' a' s'›] B[OF step.prems ‹C a s› ‹C' a' s'›]
      by (rule runs_to_refines) (metis R)
  qed
qed

lemma refines_whileLoop_strong:
  assumes x_x': "R (Result x, s) (Result x', s')"
  shows "refines (whileLoop C B x) (whileLoop C' B' x') s s'
      (λ(r, s) (r', s'). (∀v. r = Result v ⟶ (∃v'. r' = Result v' ∧ ¬ C v s ∧ ¬ C' v' s')) ∧
     R (r, s) (r', s'))"
  apply (cases "run (whileLoop C' B' x') s' = Failure")
  subgoal by (simp add: refines.rep_eq )
  subgoal
    using
      whileLoop_succeeds_terminates_of_refines[OF _ x_x']
      refines_whileLoop_finite_strong[OF x_x']
    by (simp add: refines.rep_eq whileLoop_finite_eq_whileLoop top_post_state_def)
  done

lemma refines_whileLoop:
  assumes x_x': "R (Result x, s) (Result x', s')"
  shows "refines (whileLoop C B x) (whileLoop C' B' x') s s' R"
  by (rule refines_mono[OF _ refines_whileLoop_strong, OF _ x_x']) simp

end

lemma runs_to_whileLoop_res':
  assumes R: "wf R"
  assumes *: "I a s"
  assumes P_Result: "⋀a s. ¬ C a s ⟹ I a s ⟹ P (Result a) s"
  assumes B: "⋀a s. C a s ⟹ I a s ⟹
    B a  ∙ s ⦃λr t. (∀b. r = Result b ⟶ I b t ∧ ((b, t), (a, s)) ∈ R)⦄"
  shows "(whileLoop C B a::('a, 's) res_monad) ∙ s ⦃P⦄"
  apply (rule runs_to_whileLoop[OF R, where I = "λException _ ⇒ (λ_. False) | Result v ⇒ I v"])
  subgoal using * by auto
  subgoal using P_Result by auto
  subgoal by auto
  subgoal for a b by (rule B[of a b, THEN runs_to_weaken]) auto
  done

lemma runs_to_whileLoop_res:
  assumes B: "⋀a s. C a s ⟹ I a s ⟹
    B a  ∙ s ⦃λRes r t. I r t ∧ ((r, t), (a, s)) ∈ R⦄"
  assumes P_Result: "⋀a s. I a s ⟹ ¬ C a s ⟹ P a s"
  assumes R: "wf R"
  assumes *: "I a s"
  shows "(whileLoop C B a::('a, 's) res_monad) ∙ s ⦃λRes r. P r⦄"
  apply (rule runs_to_whileLoop_res' [where R = R and I = I and B = B,  OF R *])
  using B P_Result by auto

lemma runs_to_whileLoop_variant_res:
  assumes I: "⋀r s c. I r s c ⟹
    C r s ⟹ (B r) ∙ s ⦃ λRes q t. ∃c'. I q t c' ∧ (c', c) ∈ R ⦄"
  assumes Q: "⋀r s c. I r s c ⟹ ¬ C r s ⟹ Q r s"
  assumes R: "wf R"
  shows "I r s c ⟹ (whileLoop C B r::('a, 's) res_monad) ∙ s ⦃λRes r. Q r ⦄"
proof (induction arbitrary: r s rule: wf_induct[OF R])
  case (1 c) show ?case
    apply (subst whileLoop_unroll)
    supply I[where c=c, runs_to_vcg]
    apply (runs_to_vcg)
    subgoal by (simp add: 1)
    subgoal using 1 by simp
    subgoal using Q 1 by blast
    done
qed

lemma runs_to_whileLoop_inc_res:
  assumes *: "⋀i. i < M ⟹ (B (F i)) ∙ (S i) ⦃ λRes r' t'. r' = (F (Suc i)) ∧ t' = (S (Suc i)) ⦄"
      and [simp]: "⋀i. i ≤ M ⟹ C (F i) (S i) ⟷ i < M"
      and [simp]: "i = F 0" "si = S 0" "t = F M" "st = S M"
    shows "(whileLoop C B i::('a, 's) res_monad) ∙ si ⦃ λRes r' t'. r' = t ∧ t' = st ⦄"
  apply (rule runs_to_whileLoop_variant_res[where I="λr t i. r = F i ∧ t = S i ∧ i ≤ M" and c=0,
    OF _ _ wf_nat_bound[of M]])
  apply simp_all
  apply (rule runs_to_weaken[OF *])
   apply auto
  done

lemma runs_to_whileLoop_dec_res:
  assumes *: "⋀i::nat. i > 0 ⟹ i ≤ M ⟹
        (B (F i)) ∙ (S i) ⦃ λRes r' t'. r' = (F (i - 1)) ∧ t' = (S (i - 1)) ⦄"
      and [simp]: "⋀i. C (F i) (S i) ⟷ i > 0"
      and [simp]: "i = F M" "si = S M" "t = F 0" "st = S 0"
    shows "(whileLoop C B i::('a, 's) res_monad) ∙ si ⦃ λRes r' t'. r' = t ∧ t' = st ⦄"
  apply (rule runs_to_whileLoop_variant_res[where I="λr t i. r = F i ∧ t = S i ∧ i ≤ M" and c=M,
        OF _ _ wf_measure[of "λx. x"]])
    apply (fastforce intro!: runs_to_weaken[OF *])+
  done


lemma runs_to_whileLoop_exn:
  assumes R: "wf R"
  assumes *: "I (Result a) s"
  assumes P_Result: "⋀a s. ¬ C a s ⟹ I (Result a) s ⟹ P (Result a) s"
  assumes P_Exn: "⋀a s. I (Exn a) s ⟹ P (Exn a) s"
  assumes B: "⋀a s. C a s ⟹ I (Result a) s ⟹
    B a  ∙ s ⦃λr t. I r t ∧ (∀b. r = Result b ⟶ ((b, t), (a, s)) ∈ R)⦄"
  shows "whileLoop C B a ∙ s ⦃P⦄"
  apply (rule runs_to_whileLoop[where I = I, OF R *])
  subgoal using P_Result by auto
  subgoal using P_Exn by (auto simp add: default_option_def Exn_def)
  subgoal using B by blast
  done

lemma runs_to_whileLoop_exn':
  assumes B: "⋀a s. I (Result a) s ⟹ C a s ⟹
    B a  ∙ s ⦃λr t. I r t ∧ (∀b. r = Result b ⟶ ((b, t), (a, s)) ∈ R)⦄"
  assumes P_Result: "⋀a s.  I (Result a) s  ⟹ ¬ C a s ⟹ P (Result a) s"
  assumes P_Exn: "⋀a s. I (Exn a) s ⟹ P (Exn a) s"
  assumes R: "wf R"
  assumes *: "I (Result a) s"
  shows "whileLoop C B a ∙ s ⦃P⦄"
  by (rule runs_to_whileLoop_exn[where R=R and I=I and B=B and C=C and P=P,
        OF R * P_Result P_Exn B])

lemma runs_to_partial_whileLoop_res:
  assumes *: "I a s"
  assumes P_Result: "⋀a s. ¬ C a s ⟹ I a s ⟹ P (Result a) s"
  assumes B: "⋀a s. C a s ⟹ I a s ⟹
    (B a) ∙ s ?⦃λr t. (∀b. r = Result b ⟶ I b t)⦄"
  shows "(whileLoop C B a::('a, 's) res_monad) ∙ s ?⦃P⦄"
  apply (rule runs_to_partial_whileLoop[where I = "λException _ ⇒ (λ_. False) | Result v ⇒ I v"])
  subgoal using * by auto
  subgoal using P_Result by auto
  subgoal by auto
  subgoal by (rule B[THEN runs_to_partial_weaken]) auto
  done

lemma runs_to_partial_whileLoop_exn:
  assumes *: "I (Result a) s"
  assumes P_Result: "⋀a s. ¬ C a s ⟹ I (Result a) s ⟹ P (Result a) s"
  assumes P_Exn: "⋀a s. I (Exn a) s ⟹ P (Exn a) s"
  assumes B: "⋀a s. C a s ⟹ I (Result a) s ⟹
    (B a) ∙ s ?⦃λr t. I r t⦄"
  shows "whileLoop C B a ∙ s ?⦃P⦄"
  apply (rule runs_to_partial_whileLoop[where I = I, OF *])
  subgoal using P_Result by auto
  subgoal using P_Exn by (auto simp add: default_option_def Exn_def)
  subgoal using B by blast
  done

notation (output)
  whileLoop  ("(whileLoop (_)//  (_))" [1000, 1000] 1000)

lemma whileLoop_mono: "b ≤ b' ⟹ whileLoop c b i ≤ whileLoop c b' i"
  unfolding whileLoop_def
  by (simp add: gfp_mono le_funD le_funI mono_bind mono_condition)

lemma whileLoop_finite_mono: "b ≤ b' ⟹ whileLoop_finite c b i ≤ whileLoop_finite c b' i"
  unfolding whileLoop_finite_def
  by (simp add: le_funD le_funI lfp_mono mono_bind mono_condition)

lemma monotone_whileLoop_le[partial_function_mono]:
  "(⋀x. monotone R (≤) (λf. b f x)) ⟹ monotone R (≤) (λf. whileLoop c (b f) i)"
  using whileLoop_mono unfolding monotone_def
  by (metis le_fun_def)

lemma monotone_whileLoop_ge[partial_function_mono]:
  "(⋀x. monotone R (≥) (λf. b f x)) ⟹ monotone R (≥) (λf. whileLoop c (b f) i)"
  using whileLoop_mono unfolding monotone_def
  by (metis le_fun_def)

lemma monotone_whileLoop_finite_le[partial_function_mono]:
  "(⋀x. monotone R (≤) (λf. b f x)) ⟹ monotone R (≤) (λf. whileLoop_finite c (b f) i)"
  using whileLoop_finite_mono unfolding monotone_def
  by (metis le_fun_def)

lemma monotone_whileLoop_finite_ge[partial_function_mono]:
  "(⋀x. monotone R (≥) (λf. b f x)) ⟹ monotone R (≥) (λf. whileLoop_finite c (b f) i)"
  using whileLoop_finite_mono unfolding monotone_def
  by (metis le_fun_def)

lemma run_whileLoop_le_invariant_cong:
  assumes I: "I (Result i) s"
  assumes invariant: "⋀r s. C r s ⟹ I (Result r) s ⟹ B r ∙ s ?⦃I⦄"
  assumes C: "⋀r s. I (Result r) s ⟹ C r s = C' r s"
  assumes B: "⋀r s. C r s ⟹ I (Result r) s ⟹ run (B r) s = run (B' r) s"
  shows "run (whileLoop C B i) s ≤ run (whileLoop C' B' i) s"
  unfolding le_run_refines_iff
  apply (rule refines_mono[of "λa b. a = b ∧ I (fst a) (snd a)"])
  apply simp
  apply (rule refines_whileLoop)
  subgoal using C by auto
  subgoal for x s y t using invariant[of x s] B[of x t]
    by (cases "run (B' y) t")
       (auto simp add: refines.rep_eq runs_to_partial_alt runs_to.rep_eq split_beta')
  subgoal by simp
  subgoal using I by simp
  done

lemma run_whileLoop_invariant_cong:
  assumes I: "I (Result i) s"
  assumes invariant: "⋀r s. C r s ⟹ I (Result r) s ⟹ B r ∙ s ?⦃I⦄"
  assumes C: "⋀r s. I (Result r) s ⟹ C r s = C' r s"
  assumes B: "⋀r s. C r s ⟹ I (Result r) s ⟹ run (B r) s = run (B' r) s"
  shows "run (whileLoop C B i) s = run (whileLoop C' B' i) s"
proof (rule antisym)
  from I invariant C B
  show "run (whileLoop C B i) s ≤ run (whileLoop C' B' i) s"
    by (rule run_whileLoop_le_invariant_cong)
next
  from B invariant have invariant_sym: "⋀r s. C r s ⟹ I (Result r) s ⟹ B' r ∙ s ?⦃I⦄"
    by (simp add: runs_to_partial.rep_eq)
  show "run (whileLoop C' B' i) s ≤ run (whileLoop C B i) s"
    apply (rule run_whileLoop_le_invariant_cong
      [where I=I, OF I invariant_sym C[symmetric] B[symmetric]])
    apply (auto simp add: C)
    done
qed

lemma whileLoop_cong:
  assumes C: "⋀r s. C r s = C' r s"
  assumes B: "⋀r s. C r s ⟹ run (B r) s = run (B' r) s"
  shows "whileLoop C B = whileLoop C' B'"
proof (rule ext)
  fix i
  show "whileLoop C B i = whileLoop C' B' i"
  proof (rule spec_monad_ext)
    fix s

    show "run (whileLoop C B i) s = run (whileLoop C' B' i) s"
      by (rule run_whileLoop_invariant_cong[where I = "λ_ _. True"])
         (simp_all add: C B)
  qed
qed

lemma refines_whileLoop':
  assumes C: "⋀a s b t. R (Result a, s) (Result b, t) ⟹ C a s ⟷ C' b t"
    and B:
      "⋀a s b t. R (Result a, s) (Result b, t) ⟹ C a s ⟹ C' b t ⟹ refines (B a) (B' b) s t R"
    and I: "R (Result I, s) (Result I', s')"
    and R:
      "⋀r s r' s'. R (r, s) (r', s') ⟹ rel_exception_or_result (λ_ _ . True) (λ_ _ . True) r r'"
  shows "refines (whileLoop C B I) (whileLoop C' B' I') s s'
    (λ(r, s) (r', s'). (∀v. r = Result v ⟶ (∃v'. r' = Result v' ∧ ¬ C v s ∧ ¬ C' v' s')) ∧
     R (r, s) (r', s'))"
  apply (rule refines_whileLoop_strong)
  subgoal using C by simp
  subgoal using B by simp
  subgoal for v s v' s' using R[of v s v' s'] by (auto elim!: rel_exception_or_result.cases)
  subgoal using I by simp
  done

lemma rel_spec_whileLoop':
  assumes C: "⋀a s b t. R (Result a, s) (Result b, t) ⟹ C a s ⟷ C' b t"
    and B: "⋀a s b t. R (Result a, s) (Result b, t) ⟹ C a s ⟹ C' b t ⟹ rel_spec (B a) (B' b) s t R"
    and I: "R (Result I, s) (Result I', s')"
    and R: "⋀r s r' s'. R (r, s) (r', s') ⟹ rel_exception_or_result (λ_ _ . True) (λ_ _ . True) r r'"
  shows "rel_spec (whileLoop C B I) (whileLoop C' B' I') s s'
    (λ(r, s) (r', s'). (∀v. r = Result v ⟶ (∃v'. r' = Result v' ∧ ¬ C v s ∧ ¬ C' v' s')) ∧
     R (r, s) (r', s'))"
  using C B I R
  unfolding rel_spec_iff_refines
  apply (intro conjI)
  subgoal
    by (intro refines_whileLoop') auto
  subgoal
    apply (intro refines_mono[OF _ refines_whileLoop'[where R="R¯¯"]])
    subgoal by simp (smt (verit) Exception_eq_Result rel_exception_or_result.cases)
    subgoal by auto
    subgoal by auto
    subgoal by auto
    subgoal by simp (smt (verit, best) rel_exception_or_result.simps)
    done
  done

lemma rel_spec_whileLoop:
  assumes C: "⋀a s b t. R (Result a, s) (Result b, t) ⟹ C a s ⟷ C' b t"
    and B: "⋀a s b t. R (Result a, s) (Result b, t) ⟹ C a s ⟹ C' b t ⟹ rel_spec (B a) (B' b) s t R"
    and I: "R (Result I, s) (Result I', s')"
    and R: "⋀r s r' s'. R (r, s) (r', s') ⟹ rel_exception_or_result (λ_ _ . True) (λ_ _ . True) r r'"
  shows "rel_spec (whileLoop C B I) (whileLoop C' B' I') s s' R"
  by (rule rel_spec_mono[OF _ rel_spec_whileLoop'[OF assms]]) auto

lemma do_whileLoop_combine:
  "do { x1 ← body x0; whileLoop P body x1 } =
    do {
      (b, y) ← whileLoop (λ(b, x) s. b ⟶ P x s) (λ(b, x). do { y ← body x; return (True, y) })
        (False, x0);
      return y
    }"
  apply (subst (2) whileLoop_unroll)
  apply (simp add: bind_assoc)
  apply (rule arg_cong[where f="bind _", OF ext])
  apply (subst bind_return[symmetric])
  apply (rule rel_spec_eqD)
  apply (rule rel_spec_bind_bind[where
    Q="rel_prod (rel_exception_or_result (=) (λx (b, y). b = True ∧ x = y)) (=)"])
  subgoal for x s
    apply (rule rel_spec_whileLoop)
    subgoal by auto
    subgoal 
      apply (simp split: prod.splits) [1]
      apply (subst bind_return[symmetric])
      apply clarify
      apply (rule rel_spec_bind_bind[where Q="(=)"])
          apply (simp_all add: rel_spec_refl)
      done
    by (auto simp: rel_exception_or_result.simps split: prod.splits)
     apply (simp_all split: prod.splits)
  done

lemma rel_spec_whileLoop_res:
  assumes C: "⋀a s b t. R (Result a, s) (Result b, t) ⟹ C a s ⟷ C' b t"
    and B: "⋀a s b t. R (Result a, s) (Result b, t) ⟹ C a s ⟹ C' b t ⟹ rel_spec (B a) (B' b) s t R"
    and I: "R (Result I, s) (Result I', s')"
  shows "rel_spec ((whileLoop C B I)::('a, 's) res_monad) ((whileLoop C' B' I')::('b, 't) res_monad) s s' R"
  by (rule rel_spec_whileLoop[OF C B I]) auto

lemma rel_spec_monad_whileLoop:
  assumes init: "R I I'"
  assumes cond: "⋀x y. R x y ⟹ (rel_fun S (=)) (C x) (C' y)"
  assumes body: "⋀x y. R x y ⟹ rel_spec_monad S (rel_exception_or_result E R) (B x) (B' y)"
  shows "rel_spec_monad S (rel_exception_or_result E R) (whileLoop C B I) (whileLoop C' B' I')"
  apply (clarsimp simp add: rel_spec_monad_iff_rel_spec)
  apply (rule rel_spec_whileLoop)
  subgoal for s t x s' y t' using cond by (simp add: rel_fun_def)
  subgoal for s t x s' y t' using body[of x y] by (simp add: rel_spec_monad_iff_rel_spec)
  subgoal using init by simp
  subgoal by (auto elim!: rel_exception_or_result.cases)
  done

lemma rel_spec_monad_whileLoop_res':
  assumes init: "R I I'"
  assumes cond: "⋀x y. R x y ⟹ (rel_fun S (=)) (C x) (C' y)"
  assumes body: "⋀x y. R x y ⟹ rel_spec_monad S (λRes x Res y. R x y) (B x) (B' y)"
  shows "rel_spec_monad S (λRes x Res y. R x y)
            ((whileLoop C B I)::('a, 's) res_monad)
            ((whileLoop C' B' I')::('b, 't) res_monad)"
  using assms
  by (auto intro: rel_spec_monad_whileLoop simp add: rel_exception_or_result_Res_val)

lemma rel_spec_monad_whileLoop_res:
  assumes init: "R I I'"
  assumes cond: "rel_fun R (rel_fun S (=)) C C'"
  assumes body: "rel_fun R (rel_spec_monad S (λRes x Res y. R x y)) B B'"
  shows "rel_spec_monad S (λRes x Res y. R x y)
            ((whileLoop C B I)::('a, 's) res_monad)
            ((whileLoop C' B' I')::('b, 't) res_monad)"
  using assms
  by (auto intro: rel_spec_monad_whileLoop_res' simp add: rel_fun_def)

lemma runs_to_whileLoop_finite_exn:
  assumes B: "⋀r s. I (Result r) s ⟹ C r s ⟹ (B r) ∙ s ⦃ I ⦄"
  assumes Qr: "⋀r s. I (Result r) s ⟹ ¬ C r s ⟹ Q (Result r) s"
  assumes Ql: "⋀e s. I (Exn e) s ⟹ Q (Exn e) s"
  assumes I: "I (Result r) s"
  shows "(whileLoop_finite C B r) ∙ s ⦃ Q ⦄"
  using assms by (intro runs_to_whileLoop_finite[of I]) (auto simp: Exn_def default_option_def)

lemma runs_to_whileLoop_finite_res:
  assumes B: "⋀r s. I r s ⟹ C r s ⟹ (B r) ∙ s ⦃λRes r. I r ⦄"
  assumes Q: "⋀r s. I r s ⟹ ¬ C r s ⟹ Q r s"
  assumes I: "I r s"
  shows "((whileLoop_finite C B r)::('a, 's) res_monad) ∙ s ⦃λRes r. Q r ⦄"
  using assms by (intro runs_to_whileLoop_finite[of "λRes r. I r"]) simp_all

lemma runs_to_whileLoop_eq_whileLoop_finite:
  "run (whileLoop C B r) s ≠ ⊤ ⟹
    (whileLoop C B r) ∙ s ⦃ Q ⦄ ⟷ (whileLoop_finite C B r) ∙ s ⦃ Q ⦄"
  by (simp add: whileLoop_finite_eq_whileLoop runs_to.rep_eq)

lemma whileLoop_finite_cond_fail:
    "¬ C r s ⟹ (run (whileLoop_finite C B r) s) = (run (return r) s)"
  apply (subst whileLoop_finite_unfold)
  apply simp
  done

lemma runs_to_whileLoop_finite_cond_fail:
  "¬ C r s ⟹ (whileLoop_finite C B r) ∙ s ⦃ Q ⦄ ⟷ (return r) ∙ s ⦃ Q ⦄"
  apply (subst whileLoop_finite_unfold)
  apply (simp add: runs_to.rep_eq)
  done

lemma runs_to_whileLoop_cond_fail:
  "¬ C r s ⟹ (whileLoop C B r) ∙ s ⦃ Q ⦄ ⟷ (return r) ∙ s ⦃ Q ⦄"
  apply (subst whileLoop_unroll)
  apply (simp add: runs_to.rep_eq)
  done

lemma whileLoop_finite_unfold':
  "(whileLoop_finite C B r) =
     ((condition (C r) (B r) (return r)) >>= (whileLoop_finite C B))"
  apply (rule spec_monad_ext)
  apply (subst (1) whileLoop_finite_unfold)
  subgoal for s
    apply (cases "C r s")
    subgoal by (simp add: run_bind)
    subgoal by (simp add: whileLoop_finite_cond_fail run_bind)
    done
  done

lemma runs_to_whileLoop_unroll:
  assumes "¬ C r s ⟹ P r s"
  assumes [runs_to_vcg]: "C r s ⟹ (B r) ∙ s ⦃ λRes r t. ((whileLoop C B r) ∙ t ⦃λRes r. P r ⦄) ⦄"
  shows "(whileLoop C B r) ∙ s ⦃λRes r. P r ⦄"
  using assms(1)
  by (subst whileLoop_unroll) runs_to_vcg

lemma runs_to_partial_whileLoop_unroll:
  assumes "¬ C r s ⟹ P r s"
  assumes "C r s ⟹ (B r) ∙ s ?⦃ λRes r t. ((whileLoop C B r) ∙ t ?⦃λRes r. P r ⦄) ⦄"
  shows "(whileLoop C B r) ∙ s ?⦃λRes r. P r ⦄"
  using assms
  by (subst whileLoop_unroll) runs_to_vcg


lemma runs_to_whileLoop_unroll_exn:
  assumes "¬ C r s ⟹ P (Result r) s"
  assumes [runs_to_vcg]: "C r s ⟹ (B r) ∙ s ⦃ λr t.
    (∀a. r = Result a ⟶ ((whileLoop C B a) ∙ t ⦃ P ⦄)) ∧
    (∀e. r = Exn e ⟶ P (Exn e) t) ⦄"
  shows "(whileLoop C B r) ∙ s ⦃λr s'. P r s' ⦄"
  using assms(1)
  by (subst whileLoop_unroll) (runs_to_vcg, auto simp: Exn_def default_option_def)


lemma runs_to_partial_whileLoop_unroll_exn:
  assumes "¬ C r s ⟹ P (Result r) s"
  assumes "C r s ⟹ (B r) ∙ s ?⦃ λr t.
    (∀a. r = Result a ⟶ ((whileLoop C B a) ∙ t ?⦃ P ⦄)) ∧
    (∀e. r = Exn e ⟶ P (Exn e) t) ⦄"
  shows "(whileLoop C B r) ∙ s ?⦃λr s'. P r s' ⦄"
  using assms
  by (subst whileLoop_unroll) runs_to_vcg

lemma refines_whileLoop_exn:
  assumes C: "⋀a s b t. R (Result a, s) (Result b, t) ⟹ C a s ⟷ C' b t"
    and B: "⋀a s b t. R (Result a, s) (Result b, t) ⟹ C a s ⟹ C' b t ⟹ refines (B a) (B' b) s t R"
    and I: "R (Result I, s) (Result I', s')"
    and R1: "⋀a s b t. R (Exn a, s) (Result b, t) ⟹ False"
    and R2: "⋀a s b t. R (Result a, s) (Exn b, t) ⟹ False"
  shows "refines (whileLoop C B I) (whileLoop C' B' I') s s' R"
proof -
  have ref: "refines (whileLoop C B I) (whileLoop C' B' I') s s'
         (λ(r, s) (r', s'). (∀v. r = Result v ⟶ (∃v'. r' = Result v' ∧ ¬ C v s ∧ ¬ C' v' s')) ∧ 
          R (r, s) (r', s'))"

    apply (rule refines_whileLoop' [OF C])
    subgoal by assumption
    subgoal by (rule B)
    subgoal by (rule I)
    subgoal using R1 R2
      by (auto simp add: rel_exception_or_result.simps Exn_def default_option_def)
        (metis default_option_def exception_or_result_cases not_None_eq)+
    done
  show ?thesis
    apply (rule refines_mono [OF _ ref])
    apply (auto)
    done
qed

lemma refines_whileLoop'':
  assumes C: "⋀a s b t. R (Result a, s) (Result b, t) ⟹ C a s ⟷ C' b t"
    and B: "⋀a s b t. R (Result a, s) (Result b, t) ⟹ C a s ⟹ C' b t ⟹ refines (B a) (B' b) s t R"
    and I: "R (Result I, s) (Result I', s')"
    and R: "⋀r s r' s'. R (r, s) (r', s') ⟹ rel_exception_or_result (λ_ _ . True) (λ_ _ . True) r r'"
  shows "refines (whileLoop C B I) (whileLoop C' B' I') s s' R"
proof -
  have "refines (whileLoop C B I) (whileLoop C' B' I') s s'
    (λ(r, s) (r', s'). (∀v. r = Result v ⟶ (∃v'. r' = Result v' ∧ ¬ C v s ∧ ¬ C' v' s')) ∧ 
     R (r, s) (r', s'))"
    by (rule refines_whileLoop' [OF C B I R])
  then show ?thesis
    apply (rule refines_weaken)
    apply auto
    done
qed

lemma rel_spec_monad_whileLoop_exn:
  assumes init: "R I I'"
  assumes cond: "⋀x y. R x y ⟹ (rel_fun S (=)) (C x) (C' y)"
  assumes body: "⋀x y. R x y ⟹ rel_spec_monad S (rel_xval E R) (B x) (B' y)"
  shows "rel_spec_monad S (rel_xval E R) (whileLoop C B I) (whileLoop C' B' I')"
  unfolding rel_xval_rel_exception_or_result_conv
  by (rule rel_spec_monad_whileLoop [OF init cond body [simplified rel_xval_rel_exception_or_result_conv]])

lemma refines_whileLoop_guard_right: 
  assumes "⋀x s x' s'. R (Result x, s) (Result x', s') ⟹ G' x' s' ⟹ C x s = C' x' s'"
  assumes "⋀x s x' s'. R (Result x, s) (Result x', s') ⟹ C x s ⟹ C' x' s' ⟹ G' x' s' ⟹ refines (B x) (B' x') s s' R" 
  assumes "⋀v s v' s'. R (v, s) (v', s') ⟹ (∃x. v = Result x) = (∃x'. v' = Result x')"
  assumes "R (Result x, s) (Result x', s')"
  assumes "G s' = G' x' s'"
  shows "refines (whileLoop C B x) (guard G ⤜ (λ_. whileLoop C' (λr. do {res <- B' r; guard (G' res); return res}) x')) s s' R"
  apply (rule refines_bind_guard_right)
  apply (rule refines_mono [where R="λ(r, s) (q, t). R (r, s) (q, t) ∧ 
                (case q of Exception e ⇒ True | Result v ⇒ G' v t)"])
  subgoal by simp
  subgoal
    apply (rule refines_whileLoop)
    subgoal using assms(1) by auto
    subgoal apply (rule  refines_bind_guard_right_end')
      using assms(2)
      by auto
    subgoal using assms(3) by (auto split: exception_or_result_splits)
    subgoal using assms(4,5) by auto
    done
  done

subsection ‹\<^const>‹map_value››

lemma map_value_lift_state: "map_value f (lift_state R g) = lift_state R (map_value f g)"
  apply transfer
  apply (simp add: map_post_state_eq_lift_post_state lift_post_state_comp
      lift_post_state_Sup image_image OO_def prod_eq_iff rel_prod.simps)
  apply (simp add: ac_simps)
  done

lemma run_map_value[run_spec_monad]: "run (map_value f g) s =
  map_post_state (λ(v, s). (f v, s)) (run g s)"
  by transfer (auto simp add: apfst_def map_prod_def map_post_state_def
      split: post_state.splits prod.splits)

lemma always_progress_map_value[always_progress_intros]:
  "always_progress g ⟹ always_progress (map_value f g)"
  by (simp add: always_progress_def run_map_value)

lemma runs_to_map_value_iff[runs_to_iff]: "map_value f g ∙ s ⦃ Q ⦄ ⟷ g ∙ s ⦃λr t. Q (f r) t⦄"
  by (auto simp add: runs_to.rep_eq run_map_value split_beta')

lemma runs_to_partial_map_value_iff[runs_to_iff]:
  "map_value f g ∙ s ?⦃ Q ⦄ ⟷ g ∙ s ?⦃λr t. Q (f r) t⦄"
  by (auto simp add: runs_to_partial.rep_eq run_map_value split_beta')

lemma runs_to_map_value[runs_to_vcg]: "g ∙ s ⦃λr t. Q (f r) t⦄ ⟹ map_value f g ∙ s ⦃ Q ⦄"
  by (simp add: runs_to_map_value_iff)

lemma runs_to_partial_map_value[runs_to_vcg]:
  "g ∙ s ?⦃λr t. Q (f r) t⦄ ⟹ map_value f g ∙ s ?⦃ Q ⦄"
  by (auto simp add: runs_to_partial.rep_eq run_map_value split_beta')

lemma mono_map_value: "g ≤ g' ⟹ map_value f g ≤ map_value f g'"
  by transfer (simp add: le_fun_def mono_map_post_state)

lemma monotone_map_value_le[partial_function_mono]:
  "monotone R (≤) (λf'. g f') ⟹ monotone R (≤) (λf'. map_value f (g f'))"
  by (auto simp: monotone_def mono_map_value)

lemma monotone_map_value_ge[partial_function_mono]:
  "monotone R (≥) (λf'. g f') ⟹ monotone R (≥) (λf'. map_value f (g f'))"
  by (auto simp: monotone_def mono_map_value)

lemma map_value_fail[simp]: "map_value f fail = fail"
  by transfer simp

lemma map_value_map_exn_gets[simp]: "map_value (map_exn emb) (gets x) = gets x"
  by (simp add: spec_monad_ext_iff run_map_value)

lemma refines_map_value_right_iff:
  "refines f (map_value m g) s t R ⟷ refines f g s t (λ(x, s) (y, t). R (x, s) (m y, t))"
  by transfer (simp add: sim_post_state_map_post_state2 split_beta' apfst_def map_prod_def)

lemma refines_map_value_left_iff:
  "refines (map_value m f) g s t R ⟷ refines f g s t (λ(x, s) (y, t). R (m x, s) (y, t))"
  by transfer (simp add: sim_post_state_map_post_state1 split_beta' apfst_def map_prod_def)

lemma rel_spec_map_value_right_iff:
  "rel_spec f (map_value m g) s t R ⟷ rel_spec f g s t (λ(x, s) (y, t). R (x, s) (m y, t))"
  by (simp add: rel_spec_iff_refines refines_map_value_right_iff refines_map_value_left_iff
                conversep.simps[abs_def] split_beta')

lemma rel_spec_map_value_left_iff:
  "rel_spec (map_value m f) g s t R ⟷ rel_spec f g s t (λ(x, s) (y, t). R (m x, s) (y, t))"
  by (simp add: rel_spec_iff_refines refines_map_value_right_iff refines_map_value_left_iff
                conversep.simps[abs_def] split_beta')

lemma refines_map_value_right:
  "refines f (map_value m f) s s (λ(x, s) (y, t). y = m x ∧ s = t)"
  by (simp add: refines_map_value_right_iff refines_refl)

lemma refines_map_value: 
  assumes "refines f f' s t Q"
  assumes "⋀r s' w t'. Q (r, s') (w, t') ⟹ R (g r, s') (g' w, t')"
  shows "refines (map_value g f) (map_value g' f') s t R"
  apply (simp add: refines_map_value_left_iff refines_map_value_right_iff)
  apply (rule refines_weaken [OF assms(1)])
  using assms(2) by auto

lemma map_value_id[simp]: "map_value (λx. x) = (λx. x)"
  apply (simp add: fun_eq_iff, intro allI iffI)
  apply (rule spec_monad_eqI)
  apply (rule runs_to_iff)
  done

subsection ‹\<^const>‹liftE››

lemma run_liftE[run_spec_monad]:
  "run (liftE f) s =
    map_post_state (λ(v, s). (map_exception_or_result (λx. undefined) id v, s)) (run f s)"
  by (simp add: run_map_value liftE_def)

lemma always_progress_liftE[always_progress_intros]:
  "always_progress f ⟹ always_progress (liftE f)"
  by (simp add: liftE_def always_progress_intros)

lemma runs_to_liftE_iff:
  "liftE f ∙ s ⦃ Q ⦄ ⟷ f ∙ s ⦃λr t. Q (map_exception_or_result (λx. undefined) id r) t⦄"
  by (simp add: liftE_def runs_to_map_value_iff)

lemma runs_to_liftE_iff_Res[runs_to_iff]:
  "liftE f ∙ s ⦃ Q ⦄ ⟷ f ∙ s ⦃ λRes r. Q (Result r)⦄"
  by (auto intro!: runs_to_cong_pred_only simp: runs_to_liftE_iff)

lemma runs_to_liftE':
  "f ∙ s ⦃λr t. Q (map_exception_or_result (λx. undefined) id r) t⦄ ⟹ liftE f ∙ s ⦃ Q ⦄"
  by (simp add: runs_to_liftE_iff)

lemma runs_to_liftE[runs_to_vcg]: "f ∙ s ⦃ λRes r. Q (Result r)⦄  ⟹ liftE f ∙ s ⦃ Q ⦄"
  by (simp add: runs_to_liftE_iff_Res)

lemma refines_liftE_left_iff:
  "refines (liftE f) g s t R ⟷
    refines f g s t (λ(x, s') (y, t'). ∀v. x = Result v ⟶ R (Result v, s') (y, t'))"
  by (auto simp add: liftE_def refines_map_value_left_iff intro!: refines_cong_cases del: iffI)

lemma refines_liftE_right_iff:
  "refines f (liftE g) s t R ⟷
    refines f g s t (λ(x, s') (y, t'). ∀v. y = Result v ⟶ R (x, s') (Result v, t'))"
  by (auto simp add: liftE_def refines_map_value_right_iff intro!: refines_cong_cases del: iffI)

lemma rel_spec_liftE:
  "rel_spec (liftE f) g s t R ⟷
    rel_spec f g s t (λ(x, s') (y, t'). ∀v. x = Result v ⟶ R (Result v, s') (y, t'))"
  by (simp add: rel_spec_iff_refines refines_liftE_right_iff refines_liftE_left_iff
                conversep.simps[abs_def] split_beta')

lemma rel_spec_monad_bind_liftE:
  "rel_spec_monad R (λRes v1 Res v2. P v1 v2) m n ⟹ rel_fun P (rel_spec_monad R Q) f g ⟹
    rel_spec_monad R Q ((liftE m) >>= f) ((liftE n) >>= g)"
  apply (auto intro!: rel_bind_post_state' rel_post_state_refl
    simp: rel_spec_monad_def run_bind run_liftE bind_post_state_map_post_state rel_fun_def)[1]
  subgoal premises prems for s t
    by (rule rel_post_state_weaken[OF prems(1)[rule_format, OF prems(3)]])
       (auto simp add: rel_prod.simps prems(2))
  done

lemma rel_spec_monad_bind_liftE':
  "(⋀x. rel_spec_monad (=) Q (f x) (g x)) ⟹ rel_spec_monad (=) Q (liftE m >>= f) (liftE m >>= g)"
  by (auto simp add: rel_spec_monad_def run_bind run_liftE bind_post_state_map_post_state
           intro!: rel_bind_post_state' rel_post_state_refl)

lemma runs_to_partial_liftE':
  "f ∙ s ?⦃λr t. Q (map_exception_or_result (λx. undefined) id r) t⦄ ⟹ liftE f ∙ s ?⦃ Q ⦄"
  by (simp add: runs_to_partial.rep_eq run_liftE split_beta')

lemma runs_to_partial_liftE[runs_to_vcg]:
  assumes [runs_to_vcg]: "f ∙ s ?⦃ λRes r. Q (Result r)⦄" shows "liftE f ∙ s ?⦃ Q ⦄"
  supply runs_to_partial_liftE'[runs_to_vcg] by runs_to_vcg

lemma mono_lifE: "f ≤ f' ⟹ liftE f  ≤ liftE f'"
  unfolding liftE_def by (rule mono_map_value)

lemma monotone_liftE_le[partial_function_mono]:
  "monotone R (≤) (λf'. f f') ⟹ monotone R (≤) (λf'. liftE (f f'))"
  unfolding liftE_def by (rule monotone_map_value_le)

lemma monotone_liftE_ge[partial_function_mono]:
  "monotone R (≥) (λf'. f f') ⟹ monotone R (≥) (λf'. liftE (f f'))"
  unfolding liftE_def by (rule monotone_map_value_ge)

lemma bind_handle_liftE: "bind_handle (liftE f) g h = bind f g"
  by (simp add: spec_monad_eq_iff runs_to_iff)

lemma liftE_top[simp]: "liftE ⊤ = ⊤"
  by (rule spec_monad_ext) (simp add: run_liftE)

lemma liftE_bot[simp]: "liftE bot = bot"
  by (rule spec_monad_ext) (simp add: run_liftE)

lemma liftE_fail[simp]: "liftE fail = fail"
  by (rule spec_monad_ext) (simp add: run_liftE)

lemma liftE_return[simp]: "liftE (return x) = return x"
  by (rule spec_monad_ext) (simp add: run_liftE)

lemma liftE_yield_Exception[simp]:
  "liftE (yield (Exception x)) = skip"
  by (rule spec_monad_ext) (simp add: run_liftE)

lemma liftE_throw_exception_or_result[simp]:
  "liftE (throw_exception_or_result x) = return undefined"
  by (rule spec_monad_ext) (simp add: run_liftE map_exception_or_result_def)

lemma liftE_get_state[simp]: "liftE (get_state) = get_state"
  by (rule spec_monad_ext) (simp add: run_liftE)

lemma liftE_set_state[simp]: "liftE (set_state s) = set_state s"
  by (rule spec_monad_ext) (simp add: run_liftE)

lemma liftE_select[simp]: "liftE (select S) = select S"
  by (rule spec_monad_ext) (simp add: run_liftE)

lemma liftE_unknown[simp]: "liftE (unknown) = unknown"
  by (rule spec_monad_ext) (simp add: run_liftE)

lemma liftE_lift_state: "liftE (lift_state R f) = lift_state R (liftE f)"
  unfolding liftE_def by (rule map_value_lift_state)

lemma liftE_exec_concrete: "liftE (exec_concrete st f) = exec_concrete st (liftE f)"
  unfolding exec_concrete_def by (rule liftE_lift_state)

lemma liftE_exec_abstract: "liftE (exec_abstract st f) = exec_abstract st (liftE f)"
  unfolding exec_abstract_def by (rule liftE_lift_state)

lemma liftE_assert[simp]: "liftE (assert P) = assert P"
  by (rule spec_monad_ext) (simp add: run_liftE run_assert)

lemma liftE_assume[simp]: "liftE (assume P) = assume P"
  by (rule spec_monad_ext) (simp add: run_liftE run_assume)

lemma liftE_gets[simp]: "liftE (gets f) = gets f"
  by (rule spec_monad_ext) (simp add: run_liftE)

lemma liftE_guard[simp]: "liftE (guard P) = guard P"
  by (rule spec_monad_ext) (simp add: run_liftE run_guard)

lemma liftE_assert_opt[simp]: "liftE (assert_opt v) = assert_opt v"
  by (rule spec_monad_ext) (auto simp add: run_liftE split: option.splits)

lemma liftE_gets_the[simp]: "liftE (gets_the f) = gets_the f"
  by (rule spec_monad_ext) (auto simp add: run_liftE split: option.splits)

lemma liftE_modify[simp]: "liftE (modify f) = modify f"
  by (rule spec_monad_ext) (simp add: run_liftE)

lemma liftE_bind: "liftE x >>= (λa. liftE (y a)) = liftE (x >>= y)"
  by (simp add: spec_monad_eq_iff runs_to_iff)

lemma bindE_liftE_skip: "liftE (f ⤜ (λy. skip)) ⤜ g = liftE f ⤜ (λ_. g ())"
  by (auto simp add: spec_monad_eq_iff runs_to_iff intro!: arg_cong[where f="runs_to f _"])

lemma liftE_state_select[simp]: "liftE (state_select f) = state_select f"
  apply (rule spec_monad_eqI)
  apply (auto simp add: runs_to_iff)
  done

lemma liftE_assume_result_and_state[simp]:
  "liftE (assume_result_and_state f) = assume_result_and_state f"
  apply (rule spec_monad_eqI)
  apply (auto simp add: runs_to_iff)
  done

lemma map_value_map_exn_liftE [simp]:
  "map_value (map_exn emb) (liftE f) = liftE f"
  by (simp add: spec_monad_eq_iff runs_to_iff)

lemma liftE_condition: "liftE (condition c f g) = condition c (liftE f) (liftE g)"
  by (simp add: spec_monad_eq_iff runs_to_iff)

lemma liftE_whileLoop: "liftE (whileLoop C B I) = whileLoop C (λr. liftE (B r)) I"
  apply (rule rel_spec_eqD)
  apply (subst rel_spec_liftE)
  apply (rule rel_spec_whileLoop)
  subgoal by simp
  apply (subst rel_spec_symm)
  apply (subst rel_spec_liftE)
  apply (simp add: rel_spec_refl)
  apply auto
  done

subsection ‹\<^const>‹try››

lemma run_try[run_spec_monad]:
  "run (try f) s = map_post_state (λ(v, s). (unnest_exn v, s)) (run f s)"
  by (simp add: try_def run_map_value)

lemma always_progress_try[always_progress_intros]: "always_progress f ⟹ always_progress (try f)"
  by (simp add: try_def always_progress_intros)

lemma runs_to_try[runs_to_vcg]: "f ∙ s ⦃λr t. Q (unnest_exn r) t⦄ ⟹ try f ∙ s ⦃ Q ⦄"
  by (simp add: try_def runs_to_map_value)

lemma runs_to_try_iff[runs_to_iff]: "try f ∙ s ⦃ Q ⦄ ⟷ f ∙ s ⦃λr t. Q (unnest_exn r) t⦄"
  by (auto simp add: try_def runs_to_iff)

lemma runs_to_partial_try[runs_to_vcg]: "f ∙ s ?⦃λr t. Q (unnest_exn r) t⦄ ⟹ try f ∙ s ?⦃ Q ⦄"
  by (simp add: try_def runs_to_partial_map_value)

lemma mono_try: "f ≤ f' ⟹ try f ≤ try f'"
  unfolding try_def
  by (rule mono_map_value)

lemma monotone_try_le[partial_function_mono]:
  "monotone R (≤) (λf'. f f') ⟹ monotone R (≤) (λf'. try (f f'))"
  unfolding try_def by (rule monotone_map_value_le)

lemma monotone_try_ge[partial_function_mono]:
  "monotone R (≥) (λf'. f f') ⟹ monotone R (≥) (λf'. try (f f'))"
  unfolding try_def by (rule monotone_map_value_ge)

lemma refines_try_right:
  "refines f (try f) s s (λ(x, s) (y, t). y = unnest_exn x ∧ s = t)"
  by (auto simp add: try_def refines_map_value_right)

subsection ‹\<^const>‹finally››

lemma run_finally[run_spec_monad]:
  "run (finally f) s = map_post_state (λ(v, s). (unite v, s)) (run f s)"
  by (simp add: finally_def run_map_value)

lemma always_progress_finally[always_progress_intros]:
  "always_progress f ⟹ always_progress (finally f)"
  by (simp add: finally_def always_progress_intros)

lemma runs_to_finally_iff[runs_to_iff]:
  "finally f ∙ s ⦃ Q ⦄ ⟷
    f ∙ s ⦃λr t. (∀v. r = Result v ⟶ Q (Result v) t) ∧ (∀v. r = Exn v ⟶ Q (Result v) t)⦄"
  by (auto simp: finally_def runs_to_map_value_iff unite_def
           intro!: arg_cong[where f="runs_to _ _"] split: xval_splits)

lemma runs_to_finally': "f ∙ s ⦃λr t. Q (unite r) t⦄ ⟹ finally f ∙ s ⦃ Q ⦄"
  by (simp add: finally_def runs_to_map_value)

lemma runs_to_finally[runs_to_vcg]:
  "f ∙ s ⦃λr t. (∀v. r = Result v ⟶ Q (Result v) t) ∧ (∀v. r = Exn v ⟶ Q (Result v) t)⦄ ⟹
  finally f ∙ s ⦃ Q ⦄"
  by (simp add: runs_to_finally_iff)

lemma runs_to_partial_finally': "f ∙ s ?⦃λr t. Q (unite r) t⦄ ⟹ finally f ∙ s ?⦃ Q ⦄"
  by (simp add: finally_def runs_to_partial_map_value)

lemma runs_to_partial_finally_iff:
  "finally f ∙ s ?⦃ Q ⦄ ⟷
    f ∙ s ?⦃λr t. (∀v. r = Result v ⟶ Q (Result v) t) ∧ (∀v. r = Exn v ⟶ Q (Result v) t)⦄"
  by (auto simp: finally_def runs_to_partial_map_value_iff unite_def
           intro!: arg_cong[where f="runs_to_partial _ _"] split: xval_splits)

lemma runs_to_partial_finally[runs_to_vcg]:
  "f ∙ s ?⦃λr t. (∀v. r = Result v ⟶ Q (Result v) t) ∧ (∀v. r = Exn v ⟶ Q (Result v) t)⦄ ⟹
  finally f ∙ s ?⦃ Q ⦄"
  by (simp add: runs_to_partial_finally_iff)

lemma mono_finally: "f ≤ f' ⟹ finally f ≤ finally f'"
  unfolding finally_def
  by (rule mono_map_value)

lemma monotone_finally_le[partial_function_mono]:
  "monotone R (≤) (λf'. f f') ⟹ monotone R (≤) (λf'. finally (f f'))"
  unfolding finally_def by (rule monotone_map_value_le)

lemma monotone_finally_ge[partial_function_mono]:
  "monotone R (≥) (λf'. f f') ⟹ monotone R (≥) (λf'. finally (f f'))"
  unfolding finally_def by (rule monotone_map_value_ge)

subsection ‹\<^const>‹catch››

lemma run_catch[run_spec_monad]:
  "run (f <catch> h) s =
    bind_post_state (run f s)
      (λ(r, t). case r of Exn e ⇒ run (h e) t | Result v ⇒ run (return v) t)"
  apply (simp add: catch_def run_bind_handle)
  apply (rule arg_cong[where f="bind_post_state _", OF ext])
  apply (auto simp add: Exn_def default_option_def split: exception_or_result_splits xval_splits)
  done

lemma always_progress_catch[always_progress_intros]:
  "always_progress f ⟹ (⋀e. always_progress (h e)) ⟹ always_progress (f <catch> h)"
  unfolding catch_def
  by (auto intro: always_progress_intros)

lemma runs_to_catch_iff[runs_to_iff]:
  "(f <catch> h) ∙ s ⦃ Q ⦄ ⟷
    f ∙ s ⦃ λr t. (∀v. r = Result v ⟶ Q (Result v) t) ∧ (∀e. r = Exn e ⟶ h e ∙ t ⦃ Q ⦄)⦄"
  by (simp add: runs_to.rep_eq run_catch split_beta' ac_simps split: xval_splits prod.splits)

lemma runs_to_catch[runs_to_vcg]:
  "f ∙ s ⦃ λr t. (∀v. r = Result v ⟶  Q (Result v) t) ∧
                 (∀e. r = Exn e ⟶ h e ∙ t ⦃ Q ⦄)⦄ ⟹
      (f <catch> h) ∙ s ⦃ Q ⦄"
  by (simp add: runs_to_catch_iff)

lemma runs_to_partial_catch[runs_to_vcg]:
  "f ∙ s ?⦃ λr t. (∀v. r = Result v ⟶  Q (Result v) t) ∧
                 (∀e. r = Exn e ⟶ h e ∙ t ?⦃ Q ⦄)⦄ ⟹
      (f <catch> h) ∙ s ?⦃ Q ⦄"
  unfolding runs_to_partial.rep_eq run_catch
  by (rule holds_partial_bind_post_state)
     (simp add: split_beta' ac_simps split: xval_splits prod.splits)

lemma mono_catch: "f ≤ f' ⟹ h ≤ h' ⟹ catch f h ≤ catch f' h'"
  unfolding catch_def
  apply (rule mono_bind_handle)
  by (simp_all add: le_fun_def)

lemma monotone_catch_le[partial_function_mono]:
  "monotone R (≤) (λf'. f f') ⟹ (⋀e. monotone R (≤) (λf'. h f' e))
  ⟹ monotone R (≤) (λf'. catch (f f') (h f'))"
  unfolding catch_def
  apply (rule monotone_bind_handle_le)
  by auto

lemma monotone_catch_ge[partial_function_mono]:
  "monotone R (≥) (λf'. f f') ⟹ (⋀e. monotone R (≥) (λf'. h f' e))
  ⟹ monotone R (≥) (λf'. catch (f f') (h f'))"
  unfolding catch_def
  apply (rule monotone_bind_handle_ge)
  by auto

lemma catch_liftE: "catch (liftE g) h = g"
  by (simp add: catch_def bind_handle_liftE)

lemma refines_catch:
  assumes f: "refines f f' s s' Q"
  assumes ll: "⋀e e' t t'. Q (Exn e, t) (Exn e', t') ⟹
    refines (h e) (h' e') t t' R"
  assumes lr: "⋀e v' t t'. Q (Exn e, t) (Result v', t') ⟹
    refines (h e) (return v') t t' R"
  assumes rl: "⋀v e' t t'. Q (Result v, t) (Exn e', t') ⟹
    refines (return v) (h' e') t t' R"
  assumes rr: "⋀v v' t t'. Q (Result v, t) (Result v', t') ⟹
    R (Result v, t) (Result v', t')"
  shows "refines (catch f h) (catch f' h') s s' R"
  unfolding catch_def
  apply (rule refines_bind_handle_bind_handle_exn[OF f])
  subgoal using ll by auto
  subgoal using lr by auto
  subgoal using rl by auto
  subgoal using rr by (auto simp add: refines_yield)
  done

lemma rel_spec_catch:
  assumes f: "rel_spec f f' s s' Q"
  assumes ll: "⋀e e' t t'. Q (Exn e, t) (Exn e', t') ⟹
    rel_spec (h e) (h' e') t t' R"
  assumes lr: "⋀e v' t t'. Q (Exn e, t) (Result v', t') ⟹
    rel_spec (h e) (return v') t t' R"
  assumes rl: "⋀v e' t t'. Q (Result v, t) (Exn e', t') ⟹
    rel_spec (return v) (h' e') t t' R"
  assumes rr: "⋀v v' t t'. Q (Result v, t) (Result v', t') ⟹
    R (Result v, t) (Result v', t')"
  shows "rel_spec (catch f h) (catch f' h') s s' R"
  using assms by (auto simp: rel_spec_iff_refines intro!: refines_catch)

subsection ‹\<^const>‹check››

lemma run_check[run_spec_monad]: "run (check e p) s =
  (if (∃x. p s x) then Success ((λx. (x, s)) ` (Result ` {x. p s x})) else
    pure_post_state (Exn e, s))"
  by (auto simp add: check_def run_bind)

lemma always_progress_check[always_progress_intros, simp]: "always_progress (check e p)"
  by (auto simp add: always_progress_def run_check)

lemma runs_to_check_iff[runs_to_iff]:
  "(check e p) ∙ s ⦃Q⦄ ⟷ (if (∃x. p s x) then (∀x. p s x ⟶ Q (Result x) s) else Q (Exn e) s)"
  by (auto simp add: runs_to.rep_eq run_check)

lemma runs_to_check[runs_to_vcg]:
  "(∃x. p s x ⟹ (∀x. p s x ⟶ Q (Result x) s)) ⟹ (∀x. ¬ p s x) ⟹ Q (Exn e) s ⟹
    check e p ∙ s ⦃Q⦄"
  by (simp add: runs_to_check_iff)

lemma runs_to_partial_check[runs_to_vcg]:
  "(∃x. p s x ⟹ (∀x. p s x ⟶ Q (Result x) s)) ⟹ (∀x. ¬ p s x) ⟹ Q (Exn e) s ⟹
    check e p ∙ s ?⦃Q⦄"
  by (simp add: runs_to_partial.rep_eq run_check)

lemma refines_check_right_ok:
  "Q t a ⟹ refines f (g a) s t R ⟹ refines f (check q Q >>= g) s t R"
  by (rule refines_bind_right_single[of a t])
     (auto simp add: run_check)

lemma refines_check_right_fail:
  "(∀a. ¬ Q t a) ⟹ f ∙ s ⦃ λr s'. R (r, s') (Exn q, t) ⦄ ⟹ refines f (check q Q >>= g) s t R"
  apply (rule refines_bind_right_runs_to_partialI[OF runs_to_partial_of_runs_to])
  apply (auto simp: runs_to_check_iff Exn_def refines_yield_right_iff)
  done

lemma refines_throwError_check:
  "(∀a. ¬ P t a) ⟹ R (Exn r, s) (Exn q, t) ⟹
    refines (throw r) (check q P >>= f) s t R"
  by (intro refines_check_right_fail) auto

lemma refines_condition_neg_check:
  "P s ⟷ (∀a. ¬ Q t a) ⟹
    (P s ⟹ ∀a. ¬ Q t a ⟹ R (Result r, s) (Exn q, t)) ⟹
    (⋀a. ¬ P s ⟹ Q t a ⟹ refines f (g a) s t R) ⟹
    refines (condition P (return r) f) (check q Q >>= g) s t R"
 by (intro refines_condition_left)
    (auto intro: refines_check_right_ok refines_check_right_fail)

subsection ‹\<^const>‹ignoreE››

named_theorems ignoreE_simps ‹Rewrite rules to push \<^const>‹ignoreE› inside.›

lemma ignoreE_eq: "ignoreE f = vmap_value (map_exception_or_result (λx. undefined) id) f"
  unfolding ignoreE_def catch_def
  by transfer
     (simp add: fun_eq_iff post_state_eq_iff eq_commute all_comm[where 'a='a]
           split: prod.splits exception_or_result_splits)

lemma run_ignoreE: "run (ignoreE f) s =
  bind_post_state (run f s)
    (λ(r, t). case r of Exn e ⇒ ⊥ | Result v ⇒ pure_post_state (Result v, t))"
  unfolding ignoreE_def by (simp add: run_catch)

lemma runs_to_ignoreE_iff[runs_to_iff]:
  "(ignoreE f) ∙ s ⦃ Q ⦄ ⟷ f ∙ s ⦃ λr t. (∀v. r = Result v ⟶  Q (Result v) t)⦄"
  unfolding ignoreE_eq runs_to.rep_eq
  by transfer
     (simp add: split_beta' eq_commute prod_eq_iff split: prod.splits exception_or_result_splits)

lemma runs_to_ignoreE[runs_to_vcg]:
  "f ∙ s ⦃ λr t. (∀v. r = Result v ⟶  Q (Result v) t)⦄ ⟹ (ignoreE f) ∙ s ⦃ Q ⦄"
  by (simp add: runs_to_ignoreE_iff)

lemma liftE_le_iff_le_ignoreE: "liftE f ≤ g ⟷ f ≤ ignoreE g"
  unfolding liftE_def ignoreE_eq[abs_def]
  by transfer (simp add: le_fun_def vmap_post_state_le_iff_le_map_post_state id_def)

lemma runs_to_partial_ignoreE[runs_to_vcg]:
  "f ∙ s ?⦃ λr t. (∀v. r = Result v ⟶  Q (Result v) t)⦄ ⟹
      (ignoreE f) ∙ s ?⦃ Q ⦄"
  unfolding ignoreE_def
  apply (runs_to_vcg)
  apply (fastforce simp add: runs_to_partial.rep_eq)
  done

lemma mono_ignoreE: "f ≤ f' ⟹ ignoreE f ≤ ignoreE f'"
  unfolding ignoreE_def
  apply (rule mono_catch)
  by simp_all

lemma monotone_ignoreE_le[partial_function_mono]:
  "monotone R (≤) (λf'. f f')
  ⟹ monotone R (≤) (λf'. ignoreE (f f'))"
  unfolding ignoreE_def
  apply (rule monotone_catch_le)
  by simp_all

lemma monotone_ignoreE_ge[partial_function_mono]:
  "monotone R (≥) (λf'. f f')
  ⟹ monotone R (≥) (λf'. ignoreE (f f'))"
  unfolding ignoreE_def
  apply (rule monotone_catch_ge)
  by simp_all

lemma ignoreE_liftE [simp]: "ignoreE (liftE f) = f"
  by (simp add: catch_liftE ignoreE_def)

lemma ignoreE_top[simp]: "ignoreE ⊤ = ⊤"
  by (rule spec_monad_eqI) (simp add: runs_to_iff)

lemma ignoreE_bot[simp]: "ignoreE bot = bot"
  by (rule spec_monad_eqI) (simp add: runs_to_iff)

lemma ignoreE_fail[simp]: "ignoreE fail = fail"
  by (rule spec_monad_eqI) (simp add: runs_to_iff)

lemma ignoreE_return[simp]: "ignoreE (return x) = return x"
  by (rule spec_monad_eqI) (simp add: runs_to_iff)

lemma ignoreE_throw[simp]: "ignoreE (throw x) = bot"
  by (rule spec_monad_eqI) (simp add: runs_to_iff)

lemma ignoreE_yield_Exception[simp]: "e ≠ default ⟹ ignoreE (yield (Exception e)) = bot"
  by (rule spec_monad_eqI) (simp add: runs_to_iff)

lemma ignoreE_guard[simp]: "ignoreE (guard P) = guard P"
  by (rule spec_monad_eqI) (simp add: runs_to_iff)

lemma ignoreE_get_state[simp]: "ignoreE (get_state) = get_state"
  by (rule spec_monad_eqI) (simp add: runs_to_iff)

lemma ignoreE_set_state[simp]: "ignoreE (set_state s) = set_state s"
  by (rule spec_monad_eqI) (simp add: runs_to_iff)

lemma ignoreE_select[simp]: "ignoreE (select S) = select S"
  by (rule spec_monad_eqI) (simp add: runs_to_iff)

lemma ignoreE_unknown[simp]: "ignoreE (unknown) = unknown"
  by (rule spec_monad_eqI) (simp add: runs_to_iff)

lemma ignoreE_exec_concrete[simp]: "ignoreE (exec_concrete st f) = exec_concrete st (ignoreE f)"
  by (rule spec_monad_eqI) (auto simp add: runs_to_iff)

lemma ignoreE_assert[simp]: "ignoreE (assert P) = assert P"
  by (rule spec_monad_eqI) (simp add: runs_to_iff)

lemma ignoreE_assume[simp]: "ignoreE (assume P) = assume P"
  by (rule spec_monad_eqI) (simp add: runs_to_iff)

lemma ignoreE_gets[simp]: "ignoreE (gets f) = gets f"
  by (rule spec_monad_eqI) (simp add: runs_to_iff)

lemma ignoreE_assert_opt[simp]: "ignoreE (assert_opt v) = assert_opt v"
  by (rule spec_monad_eqI) (simp add: runs_to_iff)

lemma ignoreE_gets_the[simp]: "ignoreE (gets_the f) = gets_the f"
  by (rule spec_monad_eqI) (auto simp add: runs_to_iff)

lemma ignoreE_modify[simp]: "ignoreE (modify f) = modify f"
  by (rule spec_monad_eqI) (simp add: runs_to_iff)

lemma ignoreE_condition[ignoreE_simps]:
  "ignoreE (condition c f g) = condition c (ignoreE f) (ignoreE g)"
  by (rule spec_monad_eqI) (simp add: runs_to_iff)

lemma ignoreE_when[simp]: "ignoreE (when P f) = when P (ignoreE f)"
  by (simp add: when_def ignoreE_condition)

lemma ignoreE_bind[ignoreE_simps]:
  "ignoreE (bind f g) = bind (ignoreE f) (λv. (ignoreE (g v)))"
  by (simp add: spec_monad_eq_iff runs_to_ignoreE_iff runs_to_bind_iff)

lemma ignoreE_map_exn[simp]: "ignoreE (map_value (map_exn f) g) = ignoreE g"
  by (simp add: spec_monad_eq_iff runs_to_iff)

lemma ignoreE_whileLoop[ignoreE_simps]:
  "ignoreE (whileLoop C B I) = whileLoop C (λx. ignoreE (B x)) I"
proof -
  have "(λI. ignoreE (whileLoop C B I)) = whileLoop C (λx. ignoreE (B x))"
    unfolding whileLoop_def
    apply (rule gfp_fusion[OF _ mono_whileLoop_functional mono_whileLoop_functional,
      where g="λx a. liftE (x a)"])
    subgoal by (simp add: le_fun_def liftE_le_iff_le_ignoreE)
    subgoal by (simp add: ignoreE_bind ignoreE_condition)
    done
  then show ?thesis by (simp add: fun_eq_iff)
qed

subsection ‹\<^const>‹on_exit'››

lemmas bind_finally_def = bind_exception_or_result_def

lemma bind_exception_or_result_liftE_assoc:
  "bind_exception_or_result (bind (liftE f) g) h = bind f (λv. bind_exception_or_result (g v) h)"
  by (rule spec_monad_eqI) (auto simp add: runs_to_iff)

lemma bind_exception_or_result_bind_guard_assoc:
  "bind_exception_or_result (bind (guard P) g) h =
    bind (guard P) (λv. bind_exception_or_result (g v) h)"
  by (rule spec_monad_eqI) (auto simp add: runs_to_iff)

lemma on_exit_bind_exception_or_result_conv: 
  "on_exit f cleanup = bind_exception_or_result f (λx. do {state_select cleanup; yield x})"
  by (simp add: on_exit_def on_exit'_def)

lemma guard_on_exit_bind_exception_or_result_conv: 
  "guard_on_exit f P cleanup =
    bind_exception_or_result f (λx. do {guard P; state_select cleanup; yield x})"
  by (simp add: on_exit'_def bind_assoc)

lemma assume_result_and_state_check_only_state:
  "assume_result_and_state (λs. {((), s'). s' = s ∧ P s}) = assuming P"
  by (simp add: spec_monad_eq_iff runs_to_iff)

lemma assume_on_exit_bind_exception_or_result_conv: 
  "assume_on_exit f P cleanup = bind_exception_or_result f
    (λx. do {assume_result_and_state (λs. {((), s'). s' = s ∧ P s}); state_select cleanup; yield x})"
  by (simp add: on_exit'_def bind_assoc assume_result_and_state_check_only_state)

lemma monotone_on_exit'_le[partial_function_mono]:
  "monotone R (≤) (λf'. f f') ⟹ monotone R (≤) (λf'. g f') ⟹
    monotone R (≤) (λf'. on_exit' (f f') (g f'))"
  unfolding on_exit'_def by (intro partial_function_mono)

lemma monotone_on_exit'_ge[partial_function_mono]:
  "monotone R (≥) (λf'. f f') ⟹ monotone R (≥) (λf'. g f') ⟹
    monotone R (≥) (λf'. on_exit' (f f') (g f'))"
  unfolding on_exit'_def by (intro partial_function_mono)

lemma runs_to_on_exit'_iff[runs_to_iff]:
  "on_exit' f c ∙ s ⦃ Q ⦄ ⟷
    f ∙ s ⦃ λr t. c ∙ t ⦃ λq t. Q (case q of Exception e ⇒ Exception e | Result _ ⇒ r) t ⦄ ⦄"
  by (auto simp add: on_exit'_def runs_to_iff fun_eq_iff split: exception_or_result_split
           intro!: arg_cong[where f="runs_to _ _"])

lemma runs_to_on_exit'[runs_to_vcg]:
  "f ∙ s ⦃ λr t. c ∙ t ⦃ λq t. Q (case q of Exception e ⇒ Exception e | Result _ ⇒ r) t ⦄ ⦄ ⟹ on_exit' f c ∙ s ⦃ Q ⦄"
  by (simp add: runs_to_on_exit'_iff)

lemma runs_to_partial_on_exit'[runs_to_vcg]:
  "f ∙ s ?⦃ λr t. c ∙ t ?⦃ λq t. Q (case q of Exception e ⇒ Exception e | Result _ ⇒ r) t ⦄ ⦄ ⟹ on_exit' f c ∙ s ?⦃ Q ⦄"
  unfolding on_exit'_def
  apply (rule runs_to_partial_bind_exception_or_result)
  apply (rule runs_to_partial_weaken, assumption)
  apply (rule runs_to_partial_bind)
  apply (rule runs_to_partial_weaken, assumption)
  apply auto
  done

lemma refines_on_exit':
  assumes f: "refines f f' s s'
    (λ(r, t) (r', t'). refines c c' t t' (λ(q, t) (q', t'). R
      (case q of Exception e ⇒ Exception e | Result _ ⇒ r, t) 
      (case q' of Exception e' ⇒ Exception e' | Result _ ⇒ r', t')))"
  shows "refines (on_exit' f c) (on_exit' f' c') s s' R"
  unfolding on_exit'_def
  apply (rule refines_bind_exception_or_result[OF f[THEN refines_weaken]])
  apply (clarsimp intro!: refines_bind')
  apply (rule refines_weaken, assumption)
  apply auto
  done

lemma on_exit'_skip: "on_exit' f skip = f"
  by (simp add: spec_monad_eq_iff runs_to_iff)

subsection ‹\<^const>‹run_bind››

lemma run_run_bind: "run (run_bind f t g) s = bind_post_state (run f t) (λ(r, t). run (g r t) s)"
  by (transfer) simp

lemma runs_to_run_bind_iff[runs_to_iff]:
  "(run_bind f t g) ∙ s ⦃Q⦄ ⟷ f ∙ t ⦃λr t. (g r t) ∙ s ⦃Q⦄ ⦄"
  by (simp add: runs_to.rep_eq run_run_bind split_beta')

lemma runs_to_run_bind[runs_to_vcg]:
  "f ∙ t ⦃λr t. (g r t) ∙ s ⦃Q⦄ ⦄ ⟹ (run_bind f t g) ∙ s ⦃Q⦄"
  by (simp add: runs_to_run_bind_iff)

lemma runs_to_partial_run_bind[runs_to_vcg]:
  "f ∙ t ?⦃λr t. (g r t) ∙ s ?⦃Q⦄ ⦄ ⟹ (run_bind f t g) ∙ s ?⦃Q⦄"
  by (auto simp: runs_to_partial.rep_eq run_run_bind split_beta'
           intro!: holds_partial_bind_post_state)

lemma mono_run_bind: "f ≤ f' ⟹ g ≤ g' ⟹ run_bind f t g ≤ run_bind f' t g' "
  apply (rule le_spec_monad_runI)
  apply (simp add: run_run_bind)
  apply (rule mono_bind_post_state)
   apply (auto simp add: le_fun_def less_eq_spec_monad.rep_eq split: exception_or_result_splits)
  done

lemma monotone_run_bind_le[partial_function_mono]:
  "monotone R (≤) (λf'. f f') ⟹ (⋀r t. monotone R (≤) (λf'. g f' r t))
  ⟹ monotone R (≤) (λf'. run_bind (f f') t (g f'))"
  apply (simp add: monotone_def)
  using mono_run_bind
  by (metis le_fun_def)

lemma monotone_run_bind_ge[partial_function_mono]:
  "monotone R (≥) (λf'. f f') ⟹ (⋀r t. monotone R (≥) (λf'. g f' r t))
  ⟹ monotone R (≥) (λf'. run_bind (f f') t (g f'))"
  apply (simp add: monotone_def)
  using mono_run_bind
  by (metis le_fun_def)

lemma liftE_run_bind: "liftE (run_bind f t g) = run_bind f t (λr t. liftE (g r t))"
  apply (rule spec_monad_eqI)
  apply (auto simp add: runs_to_iff)
  done

lemma exec_concrete_run_bind: "exec_concrete st f =
  do {
   s ← get_state;
   t ← select {t. st t = s};
   run_bind f t (λr' t'. do {set_state (st t'); yield r' })
  }"
  apply (rule spec_monad_eqI)
  apply (auto simp add: runs_to_iff)
  done

lemma exec_abstract_run_bind: "exec_abstract st f =
  do {
   s ← get_state;
   run_bind f (st s) (λr' t'. do {s' ← select {s'. t' = st s'}; set_state s'; yield r' })
  }"
  apply (rule spec_monad_eqI)
  apply (auto simp add: runs_to_iff)
  done

lemma refines_run_bind:
  "refines f f' x y (λ(r,x') (r', y'). refines (g r x') (g' r' y') s t R) ⟹
    refines (run_bind f x g) (run_bind f' y g') s t R"
  apply transfer
  apply (rule sim_bind_post_state, assumption)
  apply auto
  done


subsection ‹Iteration of monadic actions›

fun iter_spec_monad where
  "iter_spec_monad f 0       = skip" |
  "iter_spec_monad f (Suc n) = do {iter_spec_monad f n; f n }"

lemma iter_spec_monad_unfold:
  "0 < n ⟹ iter_spec_monad f n = do {iter_spec_monad f (n-1); f (n-1) }"
  by (metis Suc_pred' iter_spec_monad.simps(2))

lemma iter_spec_monad_cong:
  "(⋀j. j < i ⟹ f j = g j) ⟹ iter_spec_monad f i = iter_spec_monad g i"
  by (induction i) auto

subsection ‹‹forLoop››

definition forLoop:: "int ⇒ int ⇒ (int ⇒ ('a, 's) res_monad) ⇒ (int, 's) res_monad" where
  "forLoop z (m::int) B = whileLoop (λi s. i < m) (λi. do {B i; return (i + 1) }) z"

lemma runs_to_forLoop:
  assumes "z ≤ m"
  assumes I: "⋀r s. I r s ⟹ z ≤ r ⟹ r < m ⟹ (B r) ∙ s ⦃ λ_ t. I (r + 1) t ⦄"
  assumes Q: "⋀s. I m s ⟹ Q m s"
  shows "I z s ⟹ (forLoop z m B) ∙ s ⦃λRes r. Q r⦄"
  unfolding forLoop_def
  using ‹z ≤ m› Q I
  by (intro runs_to_whileLoop_res[where I="λq t. I q t ∧ z ≤ q ∧ q ≤ m"
        and P=Q
        and R="measure (λ(i, _). nat (m - i))"])
    (auto intro!: runs_to_bind)

lemma whileLoop_cong_inv:
  "run ((whileLoop C f z)::('a, 's) res_monad) s = run (whileLoop D g z) s"
  if I: "⋀i s. I i s ⟹ C i s ⟹ f i ∙ s ?⦃ λv' s'. ∀i'. v' = Result i' ⟶ I i' s'⦄"
    and I_eq: "⋀i s. I i s ⟹ C i s ⟹ run (f i) s = run (g i) s"
    and C_eq: "⋀i s. I i s ⟹ C i s ⟷ D i s"
    and "I z s"
  for s::'s and z::'a and R::"('a × 's) rel"
proof -
  have "rel_spec (whileLoop C f z) (whileLoop D g z) s s
    (λ(Res i, s) (Res i', s'). i = i' ∧ s = s' ∧ I i s)"
    apply (intro rel_spec_whileLoop_res)
    subgoal for i s j t using C_eq[of i s] by auto
    subgoal premises prems for i s j t
      using prems I[of i s]
      apply (clarsimp simp: rel_spec.rep_eq runs_to_partial.rep_eq I_eq
                      intro!: rel_post_state_refl')
      apply (auto intro: holds_partial_post_state_weaken)
      done
    subgoal using ‹I z s› by simp
    done
  from rel_spec_mono[OF _ this, of "(=)"]
  show ?thesis
    by (auto simp: le_fun_def rel_spec_eq)
qed

lemma forLoop_skip: "forLoop z m f = return z" if "z ≥ m"
  using that unfolding forLoop_def
  by (subst whileLoop_unroll) simp

lemma forLoop_eq_whileLoop: "forLoop z m f = whileLoop C g z"
  if "⋀i. z ≤ i ⟹ i < m ⟹ g i = do { y ← f i; return (i + 1) }"
    "⋀i s. z ≤ i ⟹ i ≤ m ⟹ C i s ⟷ i < m"
    "⋀s. z > m ⟹ ¬ C z s"
proof cases
  assume "z ≤ m"
  then show ?thesis
    unfolding forLoop_def
    apply (intro spec_monad_ext ext whileLoop_cong_inv[where I="λi s. z ≤ i ∧ i ≤ m"])
    apply (auto simp: that intro!: runs_to_partial_bind)
    done
next
  assume "¬ z ≤ m"
  then show ?thesis
    apply (subst whileLoop_unroll)
    apply (simp add: forLoop_skip)
    apply (rule spec_monad_ext)
    apply (simp add: that)
    done
qed

lemma runs_to_partial_forLoopE: "forLoop z m f ∙ s ?⦃ λx s'. ∀v. x = Result v ⟶ v = m ⦄"
  if "z ≤ m"
proof -
  have *: "forLoop z m f = whileLoop (λi _. i ≠ m) (λi. do {f i; return (i + 1) }) z"
    using ‹z ≤ m›
    by (auto intro!: forLoop_eq_whileLoop)
  show ?thesis unfolding *
    by (rule runs_to_partial_whileLoop_cond_false[THEN runs_to_partial_weaken]) simp
qed

subsection ‹‹whileLoop_unroll_reachable››

context fixes C :: "'a ⇒ 's ⇒ bool" and B :: "'a ⇒ ('e::default, 'a, 's) spec_monad"
begin

inductive whileLoop_unroll_reachable :: "'a ⇒ 's ⇒ 'a ⇒ 's ⇒ bool" for a s where
  initial[intro, simp]: "whileLoop_unroll_reachable a s a s"
| step: "⋀b t X c u.
    whileLoop_unroll_reachable a s b t ⟹ C b t ⟹
    run (B b) t = Success X ⟹ (Result c, u) ∈ X ⟹
    whileLoop_unroll_reachable a s c u"

lemma whileLoop_unroll_reachable_trans:
  assumes a_b: "whileLoop_unroll_reachable a s b t" and b_c: "whileLoop_unroll_reachable b t c u"
  shows "whileLoop_unroll_reachable a s c u"
proof (use b_c in ‹induction arbitrary: ›)
  case (step c u X d v)
  show ?case
    by (rule whileLoop_unroll_reachable.step[OF step(5,2,3,4)])
qed (simp add: a_b)

end

lemma run_whileLoop_unroll_reachable_cong:
  assumes eq:
    "⋀b t. whileLoop_unroll_reachable C B a s b t ⟹ C b t ⟷ C' b t"
    "⋀b t. whileLoop_unroll_reachable C B a s b t ⟹ C b t ⟹ run (B b) t = run (B' b) t"
  shows "run (whileLoop C B a) s  = run (whileLoop C' B' a) s"
proof -
  have "rel_spec (whileLoop C B a) (whileLoop C' B' a) s s
    (λ(x', s') (y, t). y = x' ∧ t = s'∧
      (∀a'. x' = Result a' ⟶ whileLoop_unroll_reachable C B a s a' s'))"
    apply (rule rel_spec_whileLoop)
    subgoal by (simp add: eq)
    subgoal for a' s'
      using whileLoop_unroll_reachable.step[of C B a s a' s']
      by (cases "run (B' a') s'") (auto simp add: rel_spec_def eq intro!: rel_set_refl)
    subgoal by simp
    subgoal for a' s' b' t' by (cases a') (auto simp add: rel_exception_or_result.simps)
    done
  then have "rel_spec (whileLoop C B a) (whileLoop C' B' a) s s (=)"
    by (rule rel_spec_mono[rotated]) auto
  then show ?thesis
    by (simp add: rel_spec_eq)
qed

subsection ‹‹on_exit››

lemma refines_rel_prod_on_exit:
  assumes f: "refines f⇩c f⇩a s⇩c s⇩a (rel_prod R S')"
  assumes cleanup: "⋀s⇩c s⇩a t⇩c. S' s⇩c s⇩a ⟹ (s⇩c, t⇩c) ∈ cleanup⇩c ⟹ ∃t⇩a. (s⇩a, t⇩a) ∈ cleanup⇩a ∧ S t⇩c t⇩a"
  assumes emp: "⋀s⇩c s⇩a. S' s⇩c s⇩a ⟹ ∄t⇩c. (s⇩c, t⇩c) ∈ cleanup⇩c ⟹ ∄t⇩a. (s⇩a, t⇩a) ∈ cleanup⇩a"
  shows "refines (on_exit f⇩c cleanup⇩c) (on_exit f⇩a cleanup⇩a) s⇩c s⇩a (rel_prod R S)"
  unfolding on_exit_bind_exception_or_result_conv
  apply (rule refines_bind_exception_or_result) 
  apply (rule refines_mono [OF _ f])
  apply (clarsimp) 
  apply (rule refines_bind')
  apply (rule refines_state_select)
  using cleanup emp
   apply auto
  done

lemma refines_runs_to_partial_rel_prod_on_exit:
  assumes f: "refines f⇩c f⇩a s⇩c s⇩a (rel_prod R S')"
  assumes runs_to: "f⇩c ∙ s⇩c ?⦃λr t. P t⦄"
  assumes cleanup: "⋀s⇩c s⇩a t⇩c. S' s⇩c s⇩a ⟹ (s⇩c, t⇩c) ∈ cleanup⇩c ⟹ P s⇩c ⟹ ∃t⇩a. (s⇩a, t⇩a) ∈ cleanup⇩a ∧ S t⇩c t⇩a"
  assumes emp: "⋀s⇩c s⇩a. S' s⇩c s⇩a ⟹ P s⇩c ⟹ ∄t⇩c. (s⇩c, t⇩c) ∈ cleanup⇩c ⟹ ∄t⇩a. (s⇩a, t⇩a) ∈ cleanup⇩a"
  shows "refines (on_exit f⇩c cleanup⇩c) (on_exit f⇩a cleanup⇩a) s⇩c s⇩a (rel_prod R S)"
proof -
  from refines_runs_to_partial_fuse [OF f runs_to]
  have "refines f⇩c f⇩a s⇩c s⇩a (λ(r, t) (r', t'). rel_prod R S' (r, t) (r', t') ∧ P t)" .
  moreover have "(λ(r, t) (r', t'). rel_prod R S' (r, t) (r', t') ∧ P t) = rel_prod R (λt t'. S' t t' ∧ P t)"
    by (auto simp add: rel_prod_conv)
  ultimately have "refines f⇩c f⇩a s⇩c s⇩a (rel_prod R (λt t'. S' t t' ∧ P t))" by simp
  then show ?thesis
    apply (rule refines_rel_prod_on_exit) 
    subgoal using cleanup by blast
    subgoal using emp by blast
    done
qed


lemma rel_spec_monad_mono:
  assumes Q: "rel_spec_monad R Q f g" and QQ': "⋀x y. Q x y ⟹ Q' x y"
  shows "rel_spec_monad R Q' f g"
proof -
  have "rel_spec_monad R Q ≤ rel_spec_monad R Q'"
    unfolding rel_spec_monad_def using QQ' 
    by (auto simp add: rel_post_state.simps rel_set_def)
      (metis rel_prod_sel)+
  with Q show ?thesis by auto
qed

lemma gets_return: "gets (λ_. x) = return x"
  by (rule spec_monad_eqI) (auto simp add: runs_to_iff)

lemma bind_handle_bind_exception_or_result_conv: 
  "bind_handle f g h = 
         bind_exception_or_result f 
          (λException e ⇒ h e | Result v ⇒ g v)"
  apply (rule spec_monad_eqI)
  by (auto simp add: runs_to_iff)
    (auto elim!: runs_to_weaken split: exception_or_result_splits)

lemma bind_handle_bind_exception_or_result_conv_exn: "bind_handle f g (λSome e ⇒ h e) = 
         bind_exception_or_result f 
          (λExn e ⇒ h e | Result v ⇒ g v)"
  by (simp add: bind_handle_bind_exception_or_result_conv case_xval_def)

lemma try_nested_bind_exception_or_result_conv: 
  shows "try (f >>= g) = 
    (bind_exception_or_result f 
      (λExn e ⇒ (case e of Inl l ⇒ throw l | Inr r ⇒ return r )
      | Result v ⇒ try (g v)))"
  apply (rule spec_monad_eqI)
  by (auto simp add: runs_to_iff )
    (auto elim!: runs_to_weaken simp add: runs_to_iff  unnest_exn_def 
      split: xval_splits sum.splits)

lemma try_nested_bind_handle_conv:
  shows "try (f >>= g) = 
    (bind_handle f (λv. try (g v))
      (λSome e ⇒ (case e of Inl l ⇒ throw l | Inr r ⇒ return r )))"
  by (simp add: bind_handle_bind_exception_or_result_conv_exn try_nested_bind_exception_or_result_conv)

definition no_fail:: "('s ⇒ bool) ⇒ ('e::default, 'a, 's) spec_monad ⇒ bool" where
 "no_fail P f ≡ ∀s. P s ⟶ run f s ≠ ⊤"

definition no_throw:: "('s ⇒ bool) ⇒ ('e::default, 'a, 's) spec_monad ⇒ bool" where
 "no_throw P f ≡ ∀s. P s ⟶ f ∙ s ?⦃ λr t. ∃v. r = Result v⦄"

definition no_return:: "('s ⇒ bool) ⇒ ('e::default, 'a, 's) spec_monad ⇒ bool" where
 "no_return P f ≡ ∀s. P s ⟶ f ∙ s ?⦃ λr t. ∃e. r = Exception e ∧ e ≠ default⦄"

lemma no_return_exn_def: "no_return P f ⟷ (∀s. P s ⟶ f ∙ s ?⦃ λr t. ∃e. r = Exn e⦄)"
  by (auto simp add: no_return_def Exn_def default_option_def elim!: runs_to_partial_weaken ) 

lemma no_throw_gets[simp]: "no_throw P (gets f)"
  by (auto simp add: no_throw_def runs_to_partial_def)

lemma no_throw_modify[simp]: "no_throw P (modify f)"
  by (auto simp add: no_throw_def runs_to_partial_def)

lemma no_throw_select[simp]: "no_throw P (select f)"
  by (auto simp add: no_throw_def runs_to_partial_def)

lemma always_progress_select_UNIV[simp]: "always_progress (select UNIV)"
  by (auto simp add: always_progress_def bot_post_state_def)

lemma rel_spec_monad_rel_xval_catch:
  assumes fh: "rel_spec_monad R (rel_xval E Q) f h"
  assumes gi: "rel_fun E (rel_spec_monad R (rel_xval E2 Q)) g i"
  shows "rel_spec_monad R (rel_xval E2 Q) (f <catch> g) (h <catch> i)"
  using assms unfolding rel_spec_monad_iff_rel_spec
  by (auto intro!: rel_spec_catch simp: rel_fun_def)

section ‹Setup for Tagging›

lemma runs_to_tag:
  "(¶ tag ⟹ f ∙ s ⦃ P ⦄) ⟹ (tag ¦ f) ∙ s ⦃ P ⦄"
  unfolding TAG_def ASM_TAG_def by auto

lemma runs_to_tag_guard:
  fixes g :: "'a ⇒ bool"
    and s :: "'a"
  assumes "tag ¦ g s"
  assumes "P (Result ()) s"
  shows "(tag ¦ guard g) ∙ s ⦃ P ⦄"
  using assms unfolding TAG_def by - (rule runs_to_weaken, rule runs_to_guard; simp)

bundle runs_to_vcg_tagging_setup
begin

unbundle basic_vcg_tagging_setup

lemmas [runs_to_vcg] = runs_to_tag runs_to_tag_guard

end

end