--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/ZF/ex/ParContract.ML Thu Sep 16 12:20:38 1993 +0200
@@ -0,0 +1,144 @@
+(* Title: ZF/ex/parcontract.ML
+ ID: $Id$
+ Author: Lawrence C Paulson
+ Copyright 1993 University of Cambridge
+
+Parallel contraction
+
+HOL system proofs may be found in
+/usr/groups/theory/hvg-aftp/contrib/rule-induction/cl.ml
+*)
+
+structure ParContract = Inductive_Fun
+ (val thy = Contract.thy;
+ val rec_doms = [("parcontract","comb*comb")];
+ val sintrs =
+ ["[| p:comb |] ==> p =1=> p",
+ "[| p:comb; q:comb |] ==> K#p#q =1=> p",
+ "[| p:comb; q:comb; r:comb |] ==> S#p#q#r =1=> (p#r)#(q#r)",
+ "[| p=1=>q; r=1=>s |] ==> p#r =1=> q#s"];
+ val monos = [];
+ val con_defs = [];
+ val type_intrs = Comb.intrs@[SigmaI];
+ val type_elims = [SigmaE2]);
+
+val [parcontract_refl,K_parcontract,S_parcontract,Ap_parcontract] = ParContract.intrs;
+
+val parcontract_induct = standard
+ (ParContract.mutual_induct RS spec RS spec RSN (2,rev_mp));
+
+(*For type checking: replaces a=1=>b by a,b:comb *)
+val parcontract_combE2 = ParContract.dom_subset RS subsetD RS SigmaE2;
+val parcontract_combD1 = ParContract.dom_subset RS subsetD RS SigmaD1;
+val parcontract_combD2 = ParContract.dom_subset RS subsetD RS SigmaD2;
+
+goal ParContract.thy "field(parcontract) = comb";
+by (fast_tac (ZF_cs addIs [equalityI,K_parcontract]
+ addSEs [parcontract_combE2]) 1);
+val field_parcontract_eq = result();
+
+val parcontract_caseE = standard
+ (ParContract.unfold RS equalityD1 RS subsetD RS CollectE);
+
+(*Derive a case for each combinator constructor*)
+val K_parcontract_case = ParContract.mk_cases Comb.con_defs "K =1=> r";
+val S_parcontract_case = ParContract.mk_cases Comb.con_defs "S =1=> r";
+val Ap_parcontract_case = ParContract.mk_cases Comb.con_defs "p#q =1=> r";
+
+val parcontract_cs =
+ ZF_cs addSIs Comb.intrs
+ addIs ParContract.intrs
+ addSEs [Ap_E, K_parcontract_case, S_parcontract_case,
+ Ap_parcontract_case]
+ addSEs [parcontract_combD1, parcontract_combD2] (*type checking*)
+ addSEs Comb.free_SEs;
+
+(*** Basic properties of parallel contraction ***)
+
+goal ParContract.thy "!!p r. K#p =1=> r ==> (EX p'. r = K#p' & p =1=> p')";
+by (fast_tac parcontract_cs 1);
+val K1_parcontractD = result();
+
+goal ParContract.thy "!!p r. S#p =1=> r ==> (EX p'. r = S#p' & p =1=> p')";
+by (fast_tac parcontract_cs 1);
+val S1_parcontractD = result();
+
+goal ParContract.thy
+ "!!p q r. S#p#q =1=> r ==> (EX p' q'. r = S#p'#q' & p =1=> p' & q =1=> q')";
+by (fast_tac (parcontract_cs addSDs [S1_parcontractD]) 1);
+val S2_parcontractD = result();
+
+(*Church-Rosser property for parallel contraction*)
+goalw ParContract.thy [diamond_def] "diamond(parcontract)";
+by (rtac (impI RS allI RS allI) 1);
+by (etac parcontract_induct 1);
+by (ALLGOALS
+ (fast_tac (parcontract_cs addSDs [K1_parcontractD,S2_parcontractD])));
+val diamond_parcontract = result();
+
+(*** Transitive closure preserves the Church-Rosser property ***)
+
+goalw ParContract.thy [diamond_def]
+ "!!x y r. [| diamond(r); <x,y>:r^+ |] ==> \
+\ ALL y'. <x,y'>:r --> (EX z. <y',z>: r^+ & <y,z>: r)";
+by (etac trancl_induct 1);
+by (fast_tac (ZF_cs addIs [r_into_trancl]) 1);
+by (slow_best_tac (ZF_cs addSDs [spec RS mp]
+ addIs [r_into_trancl, trans_trancl RS transD]) 1);
+val diamond_trancl_lemma = result();
+
+val diamond_lemmaE = diamond_trancl_lemma RS spec RS mp RS exE;
+
+val [major] = goal ParContract.thy "diamond(r) ==> diamond(r^+)";
+bw diamond_def; (*unfold only in goal, not in premise!*)
+by (rtac (impI RS allI RS allI) 1);
+by (etac trancl_induct 1);
+by (ALLGOALS
+ (slow_best_tac (ZF_cs addIs [r_into_trancl, trans_trancl RS transD]
+ addEs [major RS diamond_lemmaE])));
+val diamond_trancl = result();
+
+
+(*** Equivalence of p--->q and p===>q ***)
+
+goal ParContract.thy "!!p q. p-1->q ==> p=1=>q";
+by (etac contract_induct 1);
+by (ALLGOALS (fast_tac (parcontract_cs)));
+val contract_imp_parcontract = result();
+
+goal ParContract.thy "!!p q. p--->q ==> p===>q";
+by (forward_tac [rtrancl_type RS subsetD RS SigmaD1] 1);
+by (dtac (field_contract_eq RS equalityD1 RS subsetD) 1);
+by (etac rtrancl_induct 1);
+by (fast_tac (parcontract_cs addIs [r_into_trancl]) 1);
+by (fast_tac (ZF_cs addIs [contract_imp_parcontract,
+ r_into_trancl, trans_trancl RS transD]) 1);
+val reduce_imp_parreduce = result();
+
+
+goal ParContract.thy "!!p q. p=1=>q ==> p--->q";
+by (etac parcontract_induct 1);
+by (fast_tac (contract_cs addIs reduction_rls) 1);
+by (fast_tac (contract_cs addIs reduction_rls) 1);
+by (fast_tac (contract_cs addIs reduction_rls) 1);
+by (rtac (trans_rtrancl RS transD) 1);
+by (ALLGOALS
+ (fast_tac
+ (contract_cs addIs [Ap_reduce1, Ap_reduce2]
+ addSEs [parcontract_combD1,parcontract_combD2])));
+val parcontract_imp_reduce = result();
+
+goal ParContract.thy "!!p q. p===>q ==> p--->q";
+by (forward_tac [trancl_type RS subsetD RS SigmaD1] 1);
+by (dtac (field_parcontract_eq RS equalityD1 RS subsetD) 1);
+by (etac trancl_induct 1);
+by (etac parcontract_imp_reduce 1);
+by (etac (trans_rtrancl RS transD) 1);
+by (etac parcontract_imp_reduce 1);
+val parreduce_imp_reduce = result();
+
+goal ParContract.thy "p===>q <-> p--->q";
+by (REPEAT (ares_tac [iffI, parreduce_imp_reduce, reduce_imp_parreduce] 1));
+val parreduce_iff_reduce = result();
+
+writeln"Reached end of file.";