Theory SimCorrectness

(*  Title:      HOL/HOLCF/IOA/SimCorrectness.thy
    Author:     Olaf Müller
*)

section ‹Correctness of Simulations in HOLCF/IOA›

theory SimCorrectness
imports Simulations
begin

(*Note: s2 instead of s1 in last argument type!*)
definition corresp_ex_simC ::
    "('a, 's2) ioa ⇒ ('s1 × 's2) set ⇒ ('a, 's1) pairs → ('s2 ⇒ ('a, 's2) pairs)"
  where "corresp_ex_simC A R =
    (fix ⋅ (LAM h ex. (λs. case ex of
      nil ⇒ nil
    | x ## xs ⇒
        (flift1
          (λpr.
            let
              a = fst pr;
              t = snd pr;
              T' = SOME t'. ∃ex1. (t, t') ∈ R ∧ move A ex1 s a t'
            in (SOME cex. move A cex s a T') @@ ((h ⋅ xs) T')) ⋅ x))))"

definition corresp_ex_sim ::
    "('a, 's2) ioa ⇒ ('s1 × 's2) set ⇒ ('a, 's1) execution ⇒ ('a, 's2) execution"
  where "corresp_ex_sim A R ex ≡
    let S' = SOME s'. (fst ex, s') ∈ R ∧ s' ∈ starts_of A
    in (S', (corresp_ex_simC A R ⋅ (snd ex)) S')"


subsection ‹‹corresp_ex_sim››

lemma corresp_ex_simC_unfold:
  "corresp_ex_simC A R =
    (LAM ex. (λs. case ex of
      nil ⇒ nil
    | x ## xs ⇒
        (flift1
          (λpr.
            let
              a = fst pr;
              t = snd pr;
              T' = SOME t'. ∃ex1. (t, t') ∈ R ∧ move A ex1 s a t'
            in (SOME cex. move A cex s a T') @@ ((corresp_ex_simC A R ⋅ xs) T')) ⋅ x)))"
  apply (rule trans)
  apply (rule fix_eq2)
  apply (simp only: corresp_ex_simC_def)
  apply (rule beta_cfun)
  apply (simp add: flift1_def)
  done

lemma corresp_ex_simC_UU [simp]: "(corresp_ex_simC A R ⋅ UU) s = UU"
  apply (subst corresp_ex_simC_unfold)
  apply simp
  done

lemma corresp_ex_simC_nil [simp]: "(corresp_ex_simC A R ⋅ nil) s = nil"
  apply (subst corresp_ex_simC_unfold)
  apply simp
  done

lemma corresp_ex_simC_cons [simp]:
  "(corresp_ex_simC A R ⋅ ((a, t) ↝ xs)) s =
    (let T' = SOME t'. ∃ex1. (t, t') ∈ R ∧ move A ex1 s a t'
     in (SOME cex. move A cex s a T') @@ ((corresp_ex_simC A R ⋅ xs) T'))"
  apply (rule trans)
  apply (subst corresp_ex_simC_unfold)
  apply (simp add: Consq_def flift1_def)
  apply simp
  done


subsection ‹Properties of move›

lemma move_is_move_sim:
   "is_simulation R C A ⟹ reachable C s ⟹ s ─a─C→ t ⟹ (s, s') ∈ R ⟹
     let T' = SOME t'. ∃ex1. (t, t') ∈ R ∧ move A ex1 s' a t'
     in (t, T') ∈ R ∧ move A (SOME ex2. move A ex2 s' a T') s' a T'"
  supply Let_def [simp del]
  apply (unfold is_simulation_def)
  (* Does not perform conditional rewriting on assumptions automatically as
     usual. Instantiate all variables per hand. Ask Tobias?? *)
  apply (subgoal_tac "∃t' ex. (t, t') ∈ R ∧ move A ex s' a t'")
  prefer 2
  apply simp
  apply (erule conjE)
  apply (erule_tac x = "s" in allE)
  apply (erule_tac x = "s'" in allE)
  apply (erule_tac x = "t" in allE)
  apply (erule_tac x = "a" in allE)
  apply simp
  (* Go on as usual *)
  apply (erule exE)
  apply (drule_tac x = "t'" and P = "λt'. ∃ex. (t, t') ∈ R ∧ move A ex s' a t'" in someI)
  apply (erule exE)
  apply (erule conjE)
  apply (simp add: Let_def)
  apply (rule_tac x = "ex" in someI)
  apply assumption
  done

lemma move_subprop1_sim:
  "is_simulation R C A ⟹ reachable C s ⟹ s ─a─C→ t ⟹ (s, s') ∈ R ⟹
    let T' = SOME t'. ∃ex1. (t, t') ∈ R ∧ move A ex1 s' a t'
    in is_exec_frag A (s', SOME x. move A x s' a T')"
  apply (cut_tac move_is_move_sim)
  defer
  apply assumption+
  apply (simp add: move_def)
  done

lemma move_subprop2_sim:
  "is_simulation R C A ⟹ reachable C s ⟹ s ─a─C→ t ⟹ (s, s') ∈ R ⟹
    let T' = SOME t'. ∃ex1. (t, t') ∈ R ∧ move A ex1 s' a t'
    in Finite (SOME x. move A x s' a T')"
  apply (cut_tac move_is_move_sim)
  defer
  apply assumption+
  apply (simp add: move_def)
  done

lemma move_subprop3_sim:
  "is_simulation R C A ⟹ reachable C s ⟹ s ─a─C→ t ⟹ (s, s') ∈ R ⟹
    let T' = SOME t'. ∃ex1. (t, t') ∈ R ∧ move A ex1 s' a t'
    in laststate (s', SOME x. move A x s' a T') = T'"
  apply (cut_tac move_is_move_sim)
  defer
  apply assumption+
  apply (simp add: move_def)
  done

lemma move_subprop4_sim:
  "is_simulation R C A ⟹ reachable C s ⟹ s ─a─C→ t ⟹ (s, s') ∈ R ⟹
    let T' = SOME t'. ∃ex1. (t, t') ∈ R ∧ move A ex1 s' a t'
    in mk_trace A ⋅ (SOME x. move A x s' a T') = (if a ∈ ext A then a ↝ nil else nil)"
  apply (cut_tac move_is_move_sim)
  defer
  apply assumption+
  apply (simp add: move_def)
  done

lemma move_subprop5_sim:
  "is_simulation R C A ⟹ reachable C s ⟹ s ─a─C→ t ⟹ (s, s') ∈ R ⟹
    let T' = SOME t'. ∃ex1. (t, t') ∈ R ∧ move A ex1 s' a t'
    in (t, T') ∈ R"
  apply (cut_tac move_is_move_sim)
  defer
  apply assumption+
  apply (simp add: move_def)
  done


subsection ‹TRACE INCLUSION Part 1: Traces coincide›

subsubsection "Lemmata for <=="

text ‹Lemma 1: Traces coincide›

lemma traces_coincide_sim [rule_format (no_asm)]:
  "is_simulation R C A ⟹ ext C = ext A ⟹
    ∀s s'. reachable C s ∧ is_exec_frag C (s, ex) ∧ (s, s') ∈ R ⟶
      mk_trace C ⋅ ex = mk_trace A ⋅ ((corresp_ex_simC A R ⋅ ex) s')"
  supply if_split [split del]
  apply (pair_induct ex simp: is_exec_frag_def)
  text ‹cons case›
  apply auto
  apply (rename_tac ex a t s s')
  apply (simp add: mk_traceConc)
  apply (frule reachable.reachable_n)
  apply assumption
  apply (erule_tac x = "t" in allE)
  apply (erule_tac x = "SOME t'. ∃ex1. (t, t') ∈ R ∧ move A ex1 s' a t'" in allE)
  apply (simp add: move_subprop5_sim [unfolded Let_def]
    move_subprop4_sim [unfolded Let_def] split: if_split)
  done

text ‹Lemma 2: ‹corresp_ex_sim› is execution›

lemma correspsim_is_execution [rule_format]:
  "is_simulation R C A ⟹
    ∀s s'. reachable C s ∧ is_exec_frag C (s, ex) ∧ (s, s') ∈ R
      ⟶ is_exec_frag A (s', (corresp_ex_simC A R ⋅ ex) s')"
  apply (pair_induct ex simp: is_exec_frag_def)
  text ‹main case›
  apply auto
  apply (rename_tac ex a t s s')
  apply (rule_tac t = "SOME t'. ∃ex1. (t, t') ∈ R ∧ move A ex1 s' a t'" in lemma_2_1)

  text ‹Finite›
  apply (erule move_subprop2_sim [unfolded Let_def])
  apply assumption+
  apply (rule conjI)

  text ‹‹is_exec_frag››
  apply (erule move_subprop1_sim [unfolded Let_def])
  apply assumption+
  apply (rule conjI)

  text ‹Induction hypothesis›
  text ‹‹reachable_n› looping, therefore apply it manually›
  apply (erule_tac x = "t" in allE)
  apply (erule_tac x = "SOME t'. ∃ex1. (t, t') ∈ R ∧ move A ex1 s' a t'" in allE)
  apply simp
  apply (frule reachable.reachable_n)
  apply assumption
  apply (simp add: move_subprop5_sim [unfolded Let_def])
  text ‹laststate›
  apply (erule move_subprop3_sim [unfolded Let_def, symmetric])
  apply assumption+
  done


subsection ‹Main Theorem: TRACE - INCLUSION›

text ‹
  Generate condition ‹(s, S') ∈ R ∧ S' ∈ starts_of A›, the first being
  interesting for the induction cases concerning the two lemmas
  ‹correpsim_is_execution› and ‹traces_coincide_sim›, the second for the start
  state case.
  ‹S' := SOME s'. (s, s') ∈ R ∧ s' ∈ starts_of A›, where ‹s ∈ starts_of C›
›

lemma simulation_starts:
  "is_simulation R C A ⟹ s∈starts_of C ⟹
    let S' = SOME s'. (s, s') ∈ R ∧ s' ∈ starts_of A
    in (s, S') ∈ R ∧ S' ∈ starts_of A"
  apply (simp add: is_simulation_def corresp_ex_sim_def Int_non_empty Image_def)
  apply (erule conjE)+
  apply (erule ballE)
  prefer 2 apply blast
  apply (erule exE)
  apply (rule someI2)
  apply assumption
  apply blast
  done

lemmas sim_starts1 = simulation_starts [unfolded Let_def, THEN conjunct1]
lemmas sim_starts2 = simulation_starts [unfolded Let_def, THEN conjunct2]


lemma trace_inclusion_for_simulations:
  "ext C = ext A ⟹ is_simulation R C A ⟹ traces C ⊆ traces A"
  apply (unfold traces_def)
  apply (simp add: has_trace_def2)
  apply auto

  text ‹give execution of abstract automata›
  apply (rule_tac x = "corresp_ex_sim A R ex" in bexI)

  text ‹Traces coincide, Lemma 1›
  apply (pair ex)
  apply (rename_tac s ex)
  apply (simp add: corresp_ex_sim_def)
  apply (rule_tac s = "s" in traces_coincide_sim)
  apply assumption+
  apply (simp add: executions_def reachable.reachable_0 sim_starts1)

  text ‹‹corresp_ex_sim› is execution, Lemma 2›
  apply (pair ex)
  apply (simp add: executions_def)
  apply (rename_tac s ex)

  text ‹start state›
  apply (rule conjI)
  apply (simp add: sim_starts2 corresp_ex_sim_def)

  text ‹‹is_execution-fragment››
  apply (simp add: corresp_ex_sim_def)
  apply (rule_tac s = s in correspsim_is_execution)
  apply assumption
  apply (simp add: reachable.reachable_0 sim_starts1)
  done

end