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(* Title: SubUnion.ML
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ID: $Id$
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Author: Ole Rasmussen
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Copyright 1995 University of Cambridge
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Logic Image: ZF
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*)
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open SubUnion;
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fun rotate n i = EVERY(replicate n (etac revcut_rl i));
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(* ------------------------------------------------------------------------- *)
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(* Specialisation of comp-rules *)
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(* ------------------------------------------------------------------------- *)
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val compD1 =Scomp.dom_subset RS subsetD RS SigmaD1;
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val compD2 =Scomp.dom_subset RS subsetD RS SigmaD2;
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val regD =Sreg.dom_subset RS subsetD;
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(* ------------------------------------------------------------------------- *)
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(* Equality rules for union *)
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(* ------------------------------------------------------------------------- *)
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goalw SubUnion.thy [union_def]
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"!!u.n:nat==>Var(n) un Var(n)=Var(n)";
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by (Asm_simp_tac 1);
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by (simp_tac (rank_ss addsimps redexes.con_defs) 1);
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val union_Var = result();
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goalw SubUnion.thy [union_def]
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"!!u.[|u:redexes; v:redexes|]==>Fun(u) un Fun(v)=Fun(u un v)";
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by (Asm_simp_tac 1);
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by (simp_tac (rank_ss addsimps redexes.con_defs) 1);
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val union_Fun = result();
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goalw SubUnion.thy [union_def]
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"!!u.[|b1:bool; b2:bool; u1:redexes; v1:redexes; u2:redexes; v2:redexes|]==> \
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\ App(b1,u1,v1) un App(b2,u2,v2)=App(b1 or b2,u1 un u2,v1 un v2)";
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by (Asm_simp_tac 1);
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by (simp_tac (rank_ss addsimps redexes.con_defs) 1);
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val union_App = result();
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Addsimps (Ssub.intrs@bool_typechecks@
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Sreg.intrs@Scomp.intrs@
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[or_1 RSN (3,or_commute RS trans),
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or_0 RSN (3,or_commute RS trans),
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union_App,union_Fun,union_Var,compD2,compD1,regD]);
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AddIs Scomp.intrs;
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AddSEs [Sreg.mk_cases redexes.con_defs "regular(App(b,f,a))",
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Sreg.mk_cases redexes.con_defs "regular(Fun(b))",
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Sreg.mk_cases redexes.con_defs "regular(Var(b))",
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Scomp.mk_cases redexes.con_defs "Fun(u) ~ Fun(t)",
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Scomp.mk_cases redexes.con_defs "u ~ Fun(t)",
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Scomp.mk_cases redexes.con_defs "u ~ Var(n)",
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Scomp.mk_cases redexes.con_defs "u ~ App(b,t,a)",
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Scomp.mk_cases redexes.con_defs "Fun(t) ~ v",
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Scomp.mk_cases redexes.con_defs "App(b,f,a) ~ v",
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Scomp.mk_cases redexes.con_defs "Var(n) ~ u"
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];
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(* ------------------------------------------------------------------------- *)
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(* comp proofs *)
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(* ------------------------------------------------------------------------- *)
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goal SubUnion.thy "!!u.u:redexes ==> u ~ u";
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by (eresolve_tac [redexes.induct] 1);
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by (ALLGOALS Fast_tac);
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val comp_refl = result();
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goal SubUnion.thy
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"!!u.u ~ v ==> v ~ u";
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by (etac Scomp.induct 1);
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by (ALLGOALS Fast_tac);
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val comp_sym = result();
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goal SubUnion.thy
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"u ~ v <-> v ~ u";
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by (fast_tac (!claset addIs [comp_sym]) 1);
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val comp_sym_iff = result();
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goal SubUnion.thy
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"!!u.u ~ v ==> ALL w.v ~ w-->u ~ w";
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by (etac Scomp.induct 1);
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by (ALLGOALS Fast_tac);
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val comp_trans1 = result();
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val comp_trans = comp_trans1 RS spec RS mp;
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(* ------------------------------------------------------------------------- *)
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(* union proofs *)
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(* ------------------------------------------------------------------------- *)
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goal SubUnion.thy
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"!!u.u ~ v ==> u <== (u un v)";
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by (etac Scomp.induct 1);
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by (etac boolE 3);
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by (ALLGOALS Asm_full_simp_tac);
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val union_l = result();
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goal SubUnion.thy
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"!!u.u ~ v ==> v <== (u un v)";
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by (etac Scomp.induct 1);
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by (eres_inst_tac [("c","b2")] boolE 3);
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by (ALLGOALS Asm_full_simp_tac);
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val union_r = result();
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goal SubUnion.thy
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"!!u.u ~ v ==> u un v = v un u";
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by (etac Scomp.induct 1);
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by (ALLGOALS(asm_simp_tac (!simpset addsimps [or_commute])));
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val union_sym = result();
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(* ------------------------------------------------------------------------- *)
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(* regular proofs *)
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(* ------------------------------------------------------------------------- *)
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goal SubUnion.thy
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"!!u.u ~ v ==> regular(u)-->regular(v)-->regular(u un v)";
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by (etac Scomp.induct 1);
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by (ALLGOALS(asm_full_simp_tac
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(!simpset setloop(SELECT_GOAL (safe_tac (!claset))))));
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by (dres_inst_tac [("psi", "regular(Fun(?u) un ?v)")] asm_rl 1);
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by (Asm_full_simp_tac 1);
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val union_preserve_regular = result();
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