--- a/src/HOLCF/ex/Hoare.thy Sat Nov 27 14:34:54 2010 -0800
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,425 +0,0 @@
-(* Title: 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$x) else x)"
-
-
-(* --------- pure HOLCF logic, some little lemmas ------ *)
-
-lemma hoare_lemma2: "b~=TT ==> b=FF | b=UU"
-apply (rule Exh_tr [THEN disjE])
-apply blast+
-done
-
-lemma hoare_lemma3: " (ALL k. b1$(iterate k$g$x) = TT) | (EX k. b1$(iterate k$g$x)~=TT)"
-apply blast
-done
-
-lemma hoare_lemma4: "(EX k. b1$(iterate k$g$x) ~= TT) ==>
- EX 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: "[|(EX k. b1$(iterate k$g$x) ~= TT);
- k=Least(%n. b1$(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: "[|(EX k. b1$(iterate k$g$x) ~= TT);
- k=Least(%n. b1$(iterate n$g$x) ~= TT)|] ==>
- ALL 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$x) else x"
-apply (rule trans)
-apply (rule q_def [THEN eq_reflection, THEN fix_eq3])
-apply simp
-done
-
-(** --------- proofs about iterations of p and q ---------- **)
-
-lemma hoare_lemma9: "(ALL m. m< Suc k --> b1$(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: "(ALL 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 (EX k. b1$(iterate k$g$x)~=TT) ------- *)
-
-thm hoare_lemma8 [THEN hoare_lemma9 [THEN mp], standard]
-
-lemma hoare_lemma10:
- "EX k. b1$(iterate k$g$x) ~= TT
- ==> Suc k = (LEAST n. b1$(iterate n$g$x) ~= TT) ==> p$(iterate k$g$x) = p$x"
- by (rule hoare_lemma8 [THEN hoare_lemma9 [THEN mp]])
-
-lemma hoare_lemma11: "(EX n. b1$(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: "(EX n. b1$(iterate n$g$x) ~= TT) ==>
- k=Least(%n. b1$(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 (ALL k. b1$(iterate k$g$x)=TT) ------- *)
-
-lemma fernpass_lemma: "(ALL k. b1$(iterate k$g$x)=TT) ==> ALL 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: "(ALL 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 (ALL k. b1$(iterate k$g$x)=TT) ------- *)
-
-lemma hoare_lemma17: "(ALL k. b1$(iterate k$g$x)=TT) ==> ALL 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: "(ALL 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:
- "(ALL k. (b1::'a->tr)$(iterate k$g$x)=TT) ==> b1$(UU::'a) = UU | (ALL y. b1$(y::'a)=TT)"
-apply (rule flat_codom)
-apply (rule_tac t = "x1" in iterate_0 [THEN subst])
-apply (erule spec)
-done
-
-lemma hoare_lemma20: "(ALL y. b1$(y::'a)=TT) ==> ALL 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: "(ALL 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 (EX k. b1$(iterate k$g$x) ~= TT) ------- *)
-
-lemma hoare_lemma25: "EX k. b1$(iterate k$g$x) ~= TT
- ==> Suc k = (LEAST n. b1$(iterate n$g$x) ~= TT) ==> q$(iterate k$g$x) = q$x"
- by (rule hoare_lemma8 [THEN hoare_lemma24 [THEN mp]])
-
-lemma hoare_lemma26: "(EX n. b1$(iterate n$g$x)~=TT) ==>
- k=Least(%n. 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: "(EX n. b1$(iterate n$g$x) ~= TT) ==>
- k=Least(%n. b1$(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
-
-(* ------- (ALL k. b1$(iterate k$g$x)=TT) ==> q o p = q ----- *)
-
-lemma hoare_lemma23: "(ALL k. b1$(iterate k$g$x)=TT) ==> q$(p$x) = q$x"
-apply (subst hoare_lemma16)
-apply assumption
-apply (rule hoare_lemma19 [THEN disjE])
-apply assumption
-apply (simplesubst hoare_lemma18)
-apply assumption
-apply (simplesubst hoare_lemma22)
-apply assumption
-apply (rule refl)
-apply (simplesubst hoare_lemma21)
-apply assumption
-apply (simplesubst hoare_lemma21)
-apply assumption
-apply (rule refl)
-done
-
-(* ------------ EX k. b1~(iterate k$g$x) ~= TT ==> q o p = q ----- *)
-
-lemma hoare_lemma29: "EX k. b1$(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 o 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