Theory Hoare

(*  Title:      HOL/HOLCF/ex/Hoare.thy
    Author:     Franz Regensburger

Theory for an example by C.A.R. Hoare

p x = if b1 x
         then p (g x)
         else x fi

q x = if b1 x orelse b2 x
         then q (g x)
         else x fi

Prove: for all b1 b2 g .
            q o p  = q

In order to get a nice notation we fix the functions b1,b2 and g in the
signature of this example

*)

theory Hoare
imports HOLCF
begin

axiomatization
  b1 :: "'a → tr" and
  b2 :: "'a → tr" and
  g :: "'a → 'a"

definition
  p :: "'a → 'a" where
  "p = fix⋅(LAM f. LAM x. If b1⋅x then f⋅(g⋅x) else x)"

definition
  q :: "'a → 'a" where
  "q = fix⋅(LAM f. LAM x. If b1⋅x orelse b2⋅x then f⋅(g⋅x) else x)"


(* --------- pure HOLCF logic, some little lemmas ------ *)

lemma hoare_lemma2: "b ≠ TT ⟹ b = FF ∨ b = UU"
apply (rule Exh_tr [THEN disjE])
apply blast+
done

lemma hoare_lemma3: "(∀k. b1⋅(iterate k⋅g⋅x) = TT) ∨ (∃k. b1⋅(iterate k⋅g⋅x) ≠ TT)"
apply blast
done

lemma hoare_lemma4: "(∃k. b1⋅(iterate k⋅g⋅x) ≠ TT) ⟹
  ∃k. b1⋅(iterate k⋅g⋅x) = FF ∨ b1⋅(iterate k⋅g⋅x) = UU"
apply (erule exE)
apply (rule exI)
apply (rule hoare_lemma2)
apply assumption
done

lemma hoare_lemma5: "⟦(∃k. b1⋅(iterate k⋅g⋅x) ≠ TT);
    k = Least (λn. b1⋅(iterate n⋅g⋅x) ≠ TT)⟧ ⟹
  b1⋅(iterate k⋅g⋅x) = FF ∨ b1⋅(iterate k⋅g⋅x) = UU"
apply hypsubst
apply (rule hoare_lemma2)
apply (erule exE)
apply (erule LeastI)
done

lemma hoare_lemma6: "b = UU ⟹ b ≠ TT"
apply hypsubst
apply (rule dist_eq_tr)
done

lemma hoare_lemma7: "b = FF ⟹ b ≠ TT"
apply hypsubst
apply (rule dist_eq_tr)
done

lemma hoare_lemma8: "⟦(∃k. b1⋅(iterate k⋅g⋅x) ≠ TT);
    k = Least (λn. b1⋅(iterate n⋅g⋅x) ≠ TT)⟧ ⟹
  ∀m. m < k ⟶ b1⋅(iterate m⋅g⋅x) = TT"
apply hypsubst
apply (erule exE)
apply (intro strip)
apply (rule_tac p = "b1⋅(iterate m⋅g⋅x)" in trE)
prefer 2 apply (assumption)
apply (rule le_less_trans [THEN less_irrefl [THEN notE]])
prefer 2 apply (assumption)
apply (rule Least_le)
apply (erule hoare_lemma6)
apply (rule le_less_trans [THEN less_irrefl [THEN notE]])
prefer 2 apply (assumption)
apply (rule Least_le)
apply (erule hoare_lemma7)
done


lemma hoare_lemma28: "f⋅(y::'a) = (UU::tr) ⟹ f⋅UU = UU"
by (rule strictI)


(* ----- access to definitions ----- *)

lemma p_def3: "p⋅x = If b1⋅x then p⋅(g⋅x) else x"
apply (rule trans)
apply (rule p_def [THEN eq_reflection, THEN fix_eq3])
apply simp
done

lemma q_def3: "q⋅x = If b1⋅x orelse b2⋅x then q⋅(g⋅x) else x"
apply (rule trans)
apply (rule q_def [THEN eq_reflection, THEN fix_eq3])
apply simp
done

(** --------- proofs about iterations of p and q ---------- **)

lemma hoare_lemma9: "(∀m. m < Suc k ⟶ b1⋅(iterate m⋅g⋅x) = TT) ⟶
   p⋅(iterate k⋅g⋅x) = p⋅x"
apply (induct_tac k)
apply (simp (no_asm))
apply (simp (no_asm))
apply (intro strip)
apply (rule_tac s = "p⋅(iterate n⋅g⋅x)" in trans)
apply (rule trans)
apply (rule_tac [2] p_def3 [symmetric])
apply (rule_tac s = "TT" and t = "b1⋅(iterate n⋅g⋅x)" in ssubst)
apply (rule mp)
apply (erule spec)
apply (simp (no_asm) add: less_Suc_eq)
apply simp
apply (erule mp)
apply (intro strip)
apply (rule mp)
apply (erule spec)
apply (erule less_trans)
apply simp
done

lemma hoare_lemma24: "(∀m. m < Suc k ⟶ b1⋅(iterate m⋅g⋅x) = TT) ⟶
  q⋅(iterate k⋅g⋅x) = q⋅x"
apply (induct_tac k)
apply (simp (no_asm))
apply (simp (no_asm) add: less_Suc_eq)
apply (intro strip)
apply (rule_tac s = "q⋅(iterate n⋅g⋅x)" in trans)
apply (rule trans)
apply (rule_tac [2] q_def3 [symmetric])
apply (rule_tac s = "TT" and t = "b1⋅(iterate n⋅g⋅x)" in ssubst)
apply blast
apply simp
apply (erule mp)
apply (intro strip)
apply (fast dest!: less_Suc_eq [THEN iffD1])
done

(* -------- results about p for case (∃k. b1⋅(iterate k⋅g⋅x) ≠ TT) ------- *)

lemma hoare_lemma10:
  "∃k. b1⋅(iterate k⋅g⋅x) ≠ TT
    ⟹ Suc k = (LEAST n. b1⋅(iterate n⋅g⋅x) ≠ TT) ⟹ p⋅(iterate k⋅g⋅x) = p⋅x"
  by (rule hoare_lemma8 [THEN hoare_lemma9 [THEN mp]])

lemma hoare_lemma11: "(∃n. b1⋅(iterate n⋅g⋅x) ≠ TT) ⟹
  k = (LEAST n. b1⋅(iterate n⋅g⋅x) ≠ TT) ∧ b1⋅(iterate k⋅g⋅x) = FF
  ⟶ p⋅x = iterate k⋅g⋅x"
apply (case_tac "k")
apply hypsubst
apply (simp (no_asm))
apply (intro strip)
apply (erule conjE)
apply (rule trans)
apply (rule p_def3)
apply simp
apply hypsubst
apply (intro strip)
apply (erule conjE)
apply (rule trans)
apply (erule hoare_lemma10 [symmetric])
apply assumption
apply (rule trans)
apply (rule p_def3)
apply (rule_tac s = "TT" and t = "b1⋅(iterate nat⋅g⋅x)" in ssubst)
apply (rule hoare_lemma8 [THEN spec, THEN mp])
apply assumption
apply assumption
apply (simp (no_asm))
apply (simp (no_asm))
apply (rule trans)
apply (rule p_def3)
apply (simp (no_asm) del: iterate_Suc add: iterate_Suc [symmetric])
apply (erule_tac s = "FF" in ssubst)
apply simp
done

lemma hoare_lemma12: "(∃n. b1⋅(iterate n⋅g⋅x) ≠ TT) ⟹
  k = Least (λn. b1⋅(iterate n⋅g⋅x) ≠ TT) ∧ b1⋅(iterate k⋅g⋅x) = UU
  ⟶ p⋅x = UU"
apply (case_tac "k")
apply hypsubst
apply (simp (no_asm))
apply (intro strip)
apply (erule conjE)
apply (rule trans)
apply (rule p_def3)
apply simp
apply hypsubst
apply (simp (no_asm))
apply (intro strip)
apply (erule conjE)
apply (rule trans)
apply (rule hoare_lemma10 [symmetric])
apply assumption
apply assumption
apply (rule trans)
apply (rule p_def3)
apply (rule_tac s = "TT" and t = "b1⋅(iterate nat⋅g⋅x)" in ssubst)
apply (rule hoare_lemma8 [THEN spec, THEN mp])
apply assumption
apply assumption
apply (simp (no_asm))
apply (simp)
apply (rule trans)
apply (rule p_def3)
apply simp
done

(* -------- results about p for case  (∀k. b1⋅(iterate k⋅g⋅x) = TT) ------- *)

lemma fernpass_lemma: "(∀k. b1⋅(iterate k⋅g⋅x) = TT) ⟹ ∀k. p⋅(iterate k⋅g⋅x) = UU"
apply (rule p_def [THEN eq_reflection, THEN def_fix_ind])
apply simp
apply simp
apply (simp (no_asm))
apply (rule allI)
apply (rule_tac s = "TT" and t = "b1⋅(iterate k⋅g⋅x)" in ssubst)
apply (erule spec)
apply (simp)
apply (rule iterate_Suc [THEN subst])
apply (erule spec)
done

lemma hoare_lemma16: "(∀k. b1⋅(iterate k⋅g⋅x) = TT) ⟹ p⋅x = UU"
apply (rule_tac F1 = "g" and t = "x" in iterate_0 [THEN subst])
apply (erule fernpass_lemma [THEN spec])
done

(* -------- results about q for case  (∀k. b1⋅(iterate k⋅g⋅x) = TT) ------- *)

lemma hoare_lemma17: "(∀k. b1⋅(iterate k⋅g⋅x) = TT) ⟹ ∀k. q⋅(iterate k⋅g⋅x) = UU"
apply (rule q_def [THEN eq_reflection, THEN def_fix_ind])
apply simp
apply simp
apply (rule allI)
apply (simp (no_asm))
apply (rule_tac s = "TT" and t = "b1⋅(iterate k⋅g⋅x)" in ssubst)
apply (erule spec)
apply (simp)
apply (rule iterate_Suc [THEN subst])
apply (erule spec)
done

lemma hoare_lemma18: "(∀k. b1⋅(iterate k⋅g⋅x) = TT) ⟹ q⋅x = UU"
apply (rule_tac F1 = "g" and t = "x" in iterate_0 [THEN subst])
apply (erule hoare_lemma17 [THEN spec])
done

lemma hoare_lemma19:
  "(∀k. (b1::'a→tr)⋅(iterate k⋅g⋅x) = TT) ⟹ b1⋅(UU::'a) = UU ∨ (∀y. b1⋅(y::'a) = TT)"
apply (rule flat_codom)
apply (rule_tac t = "x" in iterate_0 [THEN subst])
apply (erule spec)
done

lemma hoare_lemma20: "(∀y. b1⋅(y::'a) = TT) ⟹ ∀k. q⋅(iterate k⋅g⋅(x::'a)) = UU"
apply (rule q_def [THEN eq_reflection, THEN def_fix_ind])
apply simp
apply simp
apply (rule allI)
apply (simp (no_asm))
apply (rule_tac s = "TT" and t = "b1⋅(iterate k⋅g⋅(x::'a))" in ssubst)
apply (erule spec)
apply (simp)
apply (rule iterate_Suc [THEN subst])
apply (erule spec)
done

lemma hoare_lemma21: "(∀y. b1⋅(y::'a) = TT) ⟹ q⋅(x::'a) = UU"
apply (rule_tac F1 = "g" and t = "x" in iterate_0 [THEN subst])
apply (erule hoare_lemma20 [THEN spec])
done

lemma hoare_lemma22: "b1⋅(UU::'a) = UU ⟹ q⋅(UU::'a) = UU"
apply (subst q_def3)
apply simp
done

(* -------- results about q for case (∃k. b1⋅(iterate k⋅g⋅x) ≠ TT) ------- *)

lemma hoare_lemma25: "∃k. b1⋅(iterate k⋅g⋅x) ≠ TT
  ⟹ Suc k = (LEAST n. b1⋅(iterate n⋅g⋅x) ≠ TT) ⟹ q⋅(iterate k⋅g⋅x) = q⋅x"
  by (rule hoare_lemma8 [THEN hoare_lemma24 [THEN mp]])

lemma hoare_lemma26: "(∃n. b1⋅(iterate n⋅g⋅x) ≠ TT) ⟹
  k = Least (λn. b1⋅(iterate n⋅g⋅x) ≠ TT) ∧ b1⋅(iterate k⋅g⋅x) = FF
  ⟶ q⋅x = q⋅(iterate k⋅g⋅x)"
apply (case_tac "k")
apply hypsubst
apply (intro strip)
apply (simp (no_asm))
apply hypsubst
apply (intro strip)
apply (erule conjE)
apply (rule trans)
apply (rule hoare_lemma25 [symmetric])
apply assumption
apply assumption
apply (rule trans)
apply (rule q_def3)
apply (rule_tac s = "TT" and t = "b1⋅(iterate nat⋅g⋅x)" in ssubst)
apply (rule hoare_lemma8 [THEN spec, THEN mp])
apply assumption
apply assumption
apply (simp (no_asm))
apply (simp (no_asm))
done


lemma hoare_lemma27: "(∃n. b1⋅(iterate n⋅g⋅x) ≠ TT) ⟹
  k = Least(λn. b1⋅(iterate n⋅g⋅x) ≠ TT) ∧ b1⋅(iterate k⋅g⋅x) = UU
  ⟶ q⋅x = UU"
apply (case_tac "k")
apply hypsubst
apply (simp (no_asm))
apply (intro strip)
apply (erule conjE)
apply (subst q_def3)
apply (simp)
apply hypsubst
apply (simp (no_asm))
apply (intro strip)
apply (erule conjE)
apply (rule trans)
apply (rule hoare_lemma25 [symmetric])
apply assumption
apply assumption
apply (rule trans)
apply (rule q_def3)
apply (rule_tac s = "TT" and t = "b1⋅(iterate nat⋅g⋅x)" in ssubst)
apply (rule hoare_lemma8 [THEN spec, THEN mp])
apply assumption
apply assumption
apply (simp (no_asm))
apply (simp)
apply (rule trans)
apply (rule q_def3)
apply (simp)
done

(* ------- (∀k. b1⋅(iterate k⋅g⋅x) = TT) ⟹ q ∘ p = q   ----- *)

lemma hoare_lemma23: "(∀k. b1⋅(iterate k⋅g⋅x) = TT) ⟹ q⋅(p⋅x) = q⋅x"
apply (subst hoare_lemma16)
apply assumption
apply (rule hoare_lemma19 [THEN disjE])
apply assumption
apply (simplesubst hoare_lemma18)
apply assumption
apply (simplesubst hoare_lemma22)
apply assumption
apply (rule refl)
apply (simplesubst hoare_lemma21)
apply assumption
apply (simplesubst hoare_lemma21)
apply assumption
apply (rule refl)
done

(* ------------  ∃k. b1⋅(iterate k⋅g⋅x) ≠ TT ⟹ q ∘ p = q   ----- *)

lemma hoare_lemma29: "∃k. b1⋅(iterate k⋅g⋅x) ≠ TT ⟹ q⋅(p⋅x) = q⋅x"
apply (rule hoare_lemma5 [THEN disjE])
apply assumption
apply (rule refl)
apply (subst hoare_lemma11 [THEN mp])
apply assumption
apply (rule conjI)
apply (rule refl)
apply assumption
apply (rule hoare_lemma26 [THEN mp, THEN subst])
apply assumption
apply (rule conjI)
apply (rule refl)
apply assumption
apply (rule refl)
apply (subst hoare_lemma12 [THEN mp])
apply assumption
apply (rule conjI)
apply (rule refl)
apply assumption
apply (subst hoare_lemma22)
apply (subst hoare_lemma28)
apply assumption
apply (rule refl)
apply (rule sym)
apply (subst hoare_lemma27 [THEN mp])
apply assumption
apply (rule conjI)
apply (rule refl)
apply assumption
apply (rule refl)
done

(* ------ the main proof q ∘ p = q ------ *)

theorem hoare_main: "q oo p = q"
apply (rule cfun_eqI)
apply (subst cfcomp2)
apply (rule hoare_lemma3 [THEN disjE])
apply (erule hoare_lemma23)
apply (erule hoare_lemma29)
done

end