--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/ZF/IMP/Equiv.ML Thu Jul 21 14:27:00 1994 +0200
@@ -0,0 +1,226 @@
+(* Title: ZF/IMP/Equiv.ML
+ ID: $Id$
+ Author: Heiko Loetzbeyer & Robert Sandner, TUM
+ Copyright 1994 TUM
+*)
+
+val type_cs = ZF_cs addSEs [make_elim(Evala.dom_subset RS subsetD),
+ make_elim(Evalb.dom_subset RS subsetD),
+ make_elim(Evalc.dom_subset RS subsetD)];
+
+(********** type_intrs fuer Evala **********)
+
+val Evala_type_intrs =
+ map (fn s => prove_goal Evala.thy s (fn _ => [(fast_tac type_cs 1)]))
+ ["!!a.<a,sigma> -a-> n ==> a:aexp","!!a.<a,sigma> -a-> n ==> sigma:loc->nat",
+ "!!a.<a,sigma> -a-> n ==> n:nat"];
+
+
+(********** type_intrs fuer Evalb **********)
+
+val Evalb_type_intrs =
+ map (fn s => prove_goal Evalb.thy s (fn _ => [(fast_tac type_cs 1)]))
+ ["!!b.<b,sigma> -b-> w ==> b:bexp","!!b.<b,sigma> -b-> w ==> sigma:loc->nat",
+ "!!b.<b,sigma> -b-> w ==> w:bool"];
+
+
+(********** type_intrs fuer Evalb **********)
+
+val Evalc_type_intrs =
+ map (fn s => prove_goal Evalc.thy s (fn _ => [(fast_tac type_cs 1)]))
+ ["!!c.<c,sigma> -c-> sigma' ==> c:com",
+ "!!c.<c,sigma> -c-> sigma' ==> sigma:loc->nat",
+ "!!c.<c,sigma> -c-> sigma' ==> sigma':loc->nat"];
+
+
+val op_type_intrs = Evala_type_intrs@Evalb_type_intrs@Evalc_type_intrs;
+
+val Aexp_elim_cases =
+ [
+ Evala.mk_cases Aexp.con_defs "<N(n),sigma,i> : evala",
+ Evala.mk_cases Aexp.con_defs "<X(x),sigma,i> : evala",
+ Evala.mk_cases Aexp.con_defs "<Op1(f,e),sigma,i> : evala",
+ Evala.mk_cases Aexp.con_defs "<Op2(f,a1,a2),sigma,i> : evala"
+ ];
+
+
+val prems = goal Equiv.thy "[| a: aexp; sigma: loc -> nat |] ==> \
+\ <a,sigma> -a-> n <-> n = A(a,sigma) ";
+
+by (res_inst_tac [("x","n")] spec 1); (* quantify n *)
+by (res_inst_tac [("x","a")] Aexp.induct 1); (* struct. ind. *)
+by (resolve_tac prems 1); (* type prem. *)
+by (safe_tac ZF_cs); (* allI,-->,<-- *)
+by (rewrite_goals_tac A_rewrite_rules); (* rewr. Den. *)
+by (TRYALL (fast_tac (ZF_cs addSIs (Evala.intrs@prems)) )); (* <== *)
+by (TRYALL (fast_tac (ZF_cs addSEs Aexp_elim_cases))); (* ==> *)
+
+val aexp_iff = result();
+
+
+val Aexp_rew_rules_cs = ZF_cs addIs op_type_intrs@[aexp_iff RS iffD1 RS sym];
+
+val aexp1 = prove_goal Equiv.thy (* elim the prems *)
+ "<a,sigma> -a-> n ==> A(a,sigma) = n" (* destruction rule *)
+ (fn prems => [(fast_tac (Aexp_rew_rules_cs addSIs prems) 1)]);
+
+val aexp2 = aexp_iff RS iffD2;
+
+
+val Bexp_elim_cases =
+ [
+ Evalb.mk_cases Bexp.con_defs "<true,sigma,x> : evalb",
+ Evalb.mk_cases Bexp.con_defs "<false,sigma,x> : evalb",
+ Evalb.mk_cases Bexp.con_defs "<ROp(f,a0,a1),sigma,x> : evalb",
+ Evalb.mk_cases Bexp.con_defs "<noti(b),sigma,x> : evalb",
+ Evalb.mk_cases Bexp.con_defs "<b0 andi b1,sigma,x> : evalb",
+ Evalb.mk_cases Bexp.con_defs "<b0 ori b1,sigma,x> : evalb"
+ ];
+
+
+val prems = goal Equiv.thy "[| b: bexp; sigma: loc -> nat |] ==> \
+\ <b,sigma> -b-> w <-> w = B(b,sigma) ";
+
+by (res_inst_tac [("x","w")] spec 1); (* quantify w *)
+by (res_inst_tac [("x","b")] Bexp.induct 1); (* struct. ind. *)
+by (resolve_tac prems 1); (* type prem. *)
+by (safe_tac ZF_cs); (* allI,-->,<-- *)
+by (rewrite_goals_tac B_rewrite_rules); (* rewr. Den. *)
+by (TRYALL (fast_tac (* <== *)
+ (ZF_cs addSIs (Evalb.intrs@prems@[aexp2])) ));
+by (TRYALL (fast_tac ((ZF_cs addSDs [aexp1]) addSEs Bexp_elim_cases)));
+ (* ==> *)
+
+val bexp_iff = result();
+
+
+val Bexp_rew_rules_cs = ZF_cs addIs op_type_intrs@[bexp_iff RS iffD1 RS sym];
+
+val bexp1 = prove_goal Equiv.thy
+ "<b,sigma> -b-> w ==> B(b,sigma) = w"
+ (fn prems => [(fast_tac (Bexp_rew_rules_cs addSIs prems) 1)]);
+
+val bexp2 = prove_goal Equiv.thy
+ "[| B(b,sigma) = w; b : bexp; sigma : loc -> nat |] ==> <b,sigma> -b-> w"
+ (fn prems =>
+ [(cut_facts_tac prems 1),
+ (fast_tac (ZF_cs addIs ([bexp_iff RS iffD2])) 1)]);
+
+
+
+
+val prems = goal Equiv.thy
+ "<c,sigma> -c-> sigma' ==> <sigma,sigma'> : C(c)";
+by (cut_facts_tac prems 1);
+
+by
+(EVERY [(rtac (Evalc.mutual_induct RS spec RS spec RS spec RSN (2,rev_mp)) 1),
+(atac 1)]);
+
+by (rewrite_tac C_rewrite_rules);
+(* skip *)
+by (fast_tac (ZF_cs addSIs [idI]) 1);
+(* assign *)
+by (asm_full_simp_tac (ZF_ss addsimps [aexp1,assign_type] @
+ op_type_intrs) 1);
+(* comp *)
+by (fast_tac comp_cs 1);
+
+(* if *)
+by (fast_tac (ZF_cs addSIs [bexp1]
+ addIs [(fst_conv RS ssubst),UnI1]) 1);
+by (fast_tac (ZF_cs addSIs [bexp1]
+ addIs [(fst_conv RS ssubst),UnI2]) 1);
+
+(* while *)
+by (rtac (lfp_Tarski RS ssubst) 1);
+by (fast_tac (ZF_cs addSIs [Gamma_bnd_mono]) 1);
+by (rewrite_tac [Gamma_def]);
+by (fast_tac (ZF_cs addSIs [bexp1,idI]@Evalb_type_intrs
+ addIs [(fst_conv RS ssubst),UnI2]) 1);
+
+by (rtac (lfp_Tarski RS ssubst) 1);
+by (fold_tac [Gamma_def]);
+by (fast_tac (ZF_cs addSIs [Gamma_bnd_mono]@Evalc_type_intrs) 1);
+by (fast_tac (comp_cs addSIs [bexp1,compI]@Evalb_type_intrs
+ addIs [(fst_conv RS ssubst)]) 1);
+
+val Lemma_5_6 = result();
+
+
+
+
+
+val com_cs = ZF_cs addSIs [aexp2,bexp2,B_type,A_type]
+ addIs Evalc.intrs
+ addSEs [idE,compE]
+ addEs [C_type,C_type_fst];
+
+val [prem] = goal Equiv.thy "c : com ==> ALL x. x:C(c) \
+\ --> <c,fst(x)> -c-> snd(x)";
+
+br (prem RS Com.induct) 1;
+by (rewrite_tac C_rewrite_rules);
+by (safe_tac com_cs);
+
+by (ALLGOALS (asm_full_simp_tac ZF_ss));
+(* skip *)
+by (fast_tac com_cs 1);
+(* assign *)
+by (fast_tac com_cs 1);
+(* comp *)
+by (REPEAT (EVERY [(etac allE 1),(etac impE 1),(atac 1)]));
+by (asm_full_simp_tac ZF_ss 1);
+by (fast_tac com_cs 1);
+(* while *)
+by (EVERY [(forward_tac [Gamma_bnd_mono] 1),(etac induct 1),(atac 1)]);
+
+by (rewrite_goals_tac [Gamma_def]);
+by (safe_tac com_cs);
+
+by (EVERY [(etac allE 1),(etac impE 1),(atac 1)]);
+by (ALLGOALS (asm_full_simp_tac ZF_ss));
+
+(* while und if *)
+by (ALLGOALS (fast_tac com_cs));
+val com2 = result();
+
+
+
+(**** Beweis der Aequivalenz ****)
+
+
+val com_iff_cs = ZF_cs addIs [C_subset RS subsetD]
+ addEs [com2 RS spec RS impE]
+ addDs [Lemma_5_6];
+
+goal Equiv.thy "ALL c:com.\
+\ C(c) = {io:(loc->nat)*(loc->nat). <c,fst(io)> -c-> snd(io)}";
+by (rtac ballI 1);
+by (rtac equalityI 1);
+by (fast_tac com_iff_cs 1);
+(* Gegenrichtung ! *)
+by (safe_tac com_iff_cs);
+bd Lemma_5_6 1;
+by (asm_full_simp_tac ZF_ss 1);
+
+val Com_equivalence = result();
+
+
+
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+