--- a/Integ/Equiv.ML Tue Oct 24 14:59:17 1995 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,272 +0,0 @@
-(* Title: Equiv.ML
- ID: $Id$
- Authors: Riccardo Mattolini, Dip. Sistemi e Informatica
- Lawrence C Paulson, Cambridge University Computer Laboratory
- Copyright 1994 Universita' di Firenze
- Copyright 1993 University of Cambridge
-
-Equivalence relations in HOL Set Theory
-*)
-
-open Equiv;
-
-(*** Suppes, Theorem 70: r is an equiv relation iff converse(r) O r = r ***)
-
-(** first half: equiv(A,r) ==> converse(r) O r = r **)
-
-goalw Equiv.thy [trans_def,sym_def,converse_def]
- "!!r. [| sym(r); trans(r) |] ==> converse(r) O r <= r";
-by (fast_tac (comp_cs addSEs [converseD]) 1);
-qed "sym_trans_comp_subset";
-
-goalw Equiv.thy [refl_def]
- "!!A r. refl(A,r) ==> r <= converse(r) O r";
-by (fast_tac (rel_cs addIs [compI]) 1);
-qed "refl_comp_subset";
-
-goalw Equiv.thy [equiv_def]
- "!!A r. equiv(A,r) ==> converse(r) O r = r";
-by (rtac equalityI 1);
-by (REPEAT (ares_tac [sym_trans_comp_subset, refl_comp_subset] 1
- ORELSE etac conjE 1));
-qed "equiv_comp_eq";
-
-(*second half*)
-goalw Equiv.thy [equiv_def,refl_def,sym_def,trans_def]
- "!!A r. [| converse(r) O r = r; Domain(r) = A |] ==> equiv(A,r)";
-by (etac equalityE 1);
-by (subgoal_tac "ALL x y. <x,y> : r --> <y,x> : r" 1);
-by (safe_tac set_cs);
-by (fast_tac (set_cs addSIs [converseI] addIs [compI]) 3);
-by (ALLGOALS (fast_tac (rel_cs addIs [compI] addSEs [compE])));
-qed "comp_equivI";
-
-(** Equivalence classes **)
-
-(*Lemma for the next result*)
-goalw Equiv.thy [equiv_def,trans_def,sym_def]
- "!!A r. [| equiv(A,r); <a,b>: r |] ==> r^^{a} <= r^^{b}";
-by (safe_tac rel_cs);
-by (rtac ImageI 1);
-by (fast_tac rel_cs 2);
-by (fast_tac rel_cs 1);
-qed "equiv_class_subset";
-
-goal Equiv.thy "!!A r. [| equiv(A,r); <a,b>: r |] ==> r^^{a} = r^^{b}";
-by (REPEAT (ares_tac [equalityI, equiv_class_subset] 1));
-by (rewrite_goals_tac [equiv_def,sym_def]);
-by (fast_tac rel_cs 1);
-qed "equiv_class_eq";
-
-val prems = goalw Equiv.thy [equiv_def,refl_def]
- "[| equiv(A,r); a: A |] ==> a: r^^{a}";
-by (cut_facts_tac prems 1);
-by (fast_tac rel_cs 1);
-qed "equiv_class_self";
-
-(*Lemma for the next result*)
-goalw Equiv.thy [equiv_def,refl_def]
- "!!A r. [| equiv(A,r); r^^{b} <= r^^{a}; b: A |] ==> <a,b>: r";
-by (fast_tac rel_cs 1);
-qed "subset_equiv_class";
-
-val prems = goal Equiv.thy
- "[| r^^{a} = r^^{b}; equiv(A,r); b: A |] ==> <a,b>: r";
-by (REPEAT (resolve_tac (prems @ [equalityD2, subset_equiv_class]) 1));
-qed "eq_equiv_class";
-
-(*thus r^^{a} = r^^{b} as well*)
-goalw Equiv.thy [equiv_def,trans_def,sym_def]
- "!!A r. [| equiv(A,r); x: (r^^{a} Int r^^{b}) |] ==> <a,b>: r";
-by (fast_tac rel_cs 1);
-qed "equiv_class_nondisjoint";
-
-val [major] = goalw Equiv.thy [equiv_def,refl_def]
- "equiv(A,r) ==> r <= Sigma(A,%x.A)";
-by (rtac (major RS conjunct1 RS conjunct1) 1);
-qed "equiv_type";
-
-goal Equiv.thy
- "!!A r. equiv(A,r) ==> (<x,y>: r) = (r^^{x} = r^^{y} & x:A & y:A)";
-by (safe_tac rel_cs);
-by ((rtac equiv_class_eq 1) THEN (assume_tac 1) THEN (assume_tac 1));
-by ((rtac eq_equiv_class 3) THEN
- (assume_tac 4) THEN (assume_tac 4) THEN (assume_tac 3));
-by ((dtac equiv_type 1) THEN (dtac rev_subsetD 1) THEN
- (assume_tac 1) THEN (dtac SigmaD1 1) THEN (assume_tac 1));
-by ((dtac equiv_type 1) THEN (dtac rev_subsetD 1) THEN
- (assume_tac 1) THEN (dtac SigmaD2 1) THEN (assume_tac 1));
-qed "equiv_class_eq_iff";
-
-goal Equiv.thy
- "!!A r. [| equiv(A,r); x: A; y: A |] ==> (r^^{x} = r^^{y}) = (<x,y>: r)";
-by (safe_tac rel_cs);
-by ((rtac eq_equiv_class 1) THEN
- (assume_tac 1) THEN (assume_tac 1) THEN (assume_tac 1));
-by ((rtac equiv_class_eq 1) THEN
- (assume_tac 1) THEN (assume_tac 1));
-qed "eq_equiv_class_iff";
-
-(*** Quotients ***)
-
-(** Introduction/elimination rules -- needed? **)
-
-val prems = goalw Equiv.thy [quotient_def] "x:A ==> r^^{x}: A/r";
-by (rtac UN_I 1);
-by (resolve_tac prems 1);
-by (rtac singletonI 1);
-qed "quotientI";
-
-val [major,minor] = goalw Equiv.thy [quotient_def]
- "[| X:(A/r); !!x. [| X = r^^{x}; x:A |] ==> P |] \
-\ ==> P";
-by (resolve_tac [major RS UN_E] 1);
-by (rtac minor 1);
-by (assume_tac 2);
-by (fast_tac rel_cs 1);
-qed "quotientE";
-
-(** Not needed by Theory Integ --> bypassed **)
-(**goalw Equiv.thy [equiv_def,refl_def,quotient_def]
- "!!A r. equiv(A,r) ==> Union(A/r) = A";
-by (fast_tac eq_cs 1);
-qed "Union_quotient";
-**)
-
-(** Not needed by Theory Integ --> bypassed **)
-(*goalw Equiv.thy [quotient_def]
- "!!A r. [| equiv(A,r); X: A/r; Y: A/r |] ==> X=Y | (X Int Y <= 0)";
-by (safe_tac (ZF_cs addSIs [equiv_class_eq]));
-by (assume_tac 1);
-by (rewrite_goals_tac [equiv_def,trans_def,sym_def]);
-by (fast_tac ZF_cs 1);
-qed "quotient_disj";
-**)
-
-(**** Defining unary operations upon equivalence classes ****)
-
-(* theorem needed to prove UN_equiv_class *)
-goal Set.thy "!!A. [| a:A; ! y:A. b(y)=b(a) |] ==> (UN y:A. b(y))=b(a)";
-by (fast_tac (eq_cs addSEs [equalityE]) 1);
-qed "UN_singleton_lemma";
-val UN_singleton = ballI RSN (2,UN_singleton_lemma);
-
-
-(** These proofs really require as local premises
- equiv(A,r); congruent(r,b)
-**)
-
-(*Conversion rule*)
-val prems as [equivA,bcong,_] = goal Equiv.thy
- "[| equiv(A,r); congruent(r,b); a: A |] ==> (UN x:r^^{a}. b(x)) = b(a)";
-by (cut_facts_tac prems 1);
-by (rtac UN_singleton 1);
-by (rtac equiv_class_self 1);
-by (assume_tac 1);
-by (assume_tac 1);
-by (rewrite_goals_tac [equiv_def,congruent_def,sym_def]);
-by (fast_tac rel_cs 1);
-qed "UN_equiv_class";
-
-(*Resolve th against the "local" premises*)
-val localize = RSLIST [equivA,bcong];
-
-(*type checking of UN x:r``{a}. b(x) *)
-val _::_::prems = goalw Equiv.thy [quotient_def]
- "[| equiv(A,r); congruent(r,b); X: A/r; \
-\ !!x. x : A ==> b(x) : B |] \
-\ ==> (UN x:X. b(x)) : B";
-by (cut_facts_tac prems 1);
-by (safe_tac rel_cs);
-by (rtac (localize UN_equiv_class RS ssubst) 1);
-by (REPEAT (ares_tac prems 1));
-qed "UN_equiv_class_type";
-
-(*Sufficient conditions for injectiveness. Could weaken premises!
- major premise could be an inclusion; bcong could be !!y. y:A ==> b(y):B
-*)
-val _::_::prems = goalw Equiv.thy [quotient_def]
- "[| equiv(A,r); congruent(r,b); \
-\ (UN x:X. b(x))=(UN y:Y. b(y)); X: A/r; Y: A/r; \
-\ !!x y. [| x:A; y:A; b(x)=b(y) |] ==> <x,y>:r |] \
-\ ==> X=Y";
-by (cut_facts_tac prems 1);
-by (safe_tac rel_cs);
-by (rtac (equivA RS equiv_class_eq) 1);
-by (REPEAT (ares_tac prems 1));
-by (etac box_equals 1);
-by (REPEAT (ares_tac [localize UN_equiv_class] 1));
-qed "UN_equiv_class_inject";
-
-
-(**** Defining binary operations upon equivalence classes ****)
-
-
-goalw Equiv.thy [congruent_def,congruent2_def,equiv_def,refl_def]
- "!!A r. [| equiv(A,r); congruent2(r,b); a: A |] ==> congruent(r,b(a))";
-by (fast_tac rel_cs 1);
-qed "congruent2_implies_congruent";
-
-val equivA::prems = goalw Equiv.thy [congruent_def]
- "[| equiv(A,r); congruent2(r,b); a: A |] ==> \
-\ congruent(r, %x1. UN x2:r^^{a}. b(x1,x2))";
-by (cut_facts_tac (equivA::prems) 1);
-by (safe_tac rel_cs);
-by (rtac (equivA RS equiv_type RS subsetD RS SigmaE2) 1);
-by (assume_tac 1);
-by (asm_simp_tac (prod_ss addsimps [equivA RS UN_equiv_class,
- congruent2_implies_congruent]) 1);
-by (rewrite_goals_tac [congruent2_def,equiv_def,refl_def]);
-by (fast_tac rel_cs 1);
-qed "congruent2_implies_congruent_UN";
-
-val prems as equivA::_ = goal Equiv.thy
- "[| equiv(A,r); congruent2(r,b); a1: A; a2: A |] \
-\ ==> (UN x1:r^^{a1}. UN x2:r^^{a2}. b(x1,x2)) = b(a1,a2)";
-by (cut_facts_tac prems 1);
-by (asm_simp_tac (prod_ss addsimps [equivA RS UN_equiv_class,
- congruent2_implies_congruent,
- congruent2_implies_congruent_UN]) 1);
-qed "UN_equiv_class2";
-
-(*type checking*)
-val prems = goalw Equiv.thy [quotient_def]
- "[| equiv(A,r); congruent2(r,b); \
-\ X1: A/r; X2: A/r; \
-\ !!x1 x2. [| x1: A; x2: A |] ==> b(x1,x2) : B |] \
-\ ==> (UN x1:X1. UN x2:X2. b(x1,x2)) : B";
-by (cut_facts_tac prems 1);
-by (safe_tac rel_cs);
-by (REPEAT (ares_tac (prems@[UN_equiv_class_type,
- congruent2_implies_congruent_UN,
- congruent2_implies_congruent, quotientI]) 1));
-qed "UN_equiv_class_type2";
-
-
-(*Suggested by John Harrison -- the two subproofs may be MUCH simpler
- than the direct proof*)
-val prems = goalw Equiv.thy [congruent2_def,equiv_def,refl_def]
- "[| equiv(A,r); \
-\ !! y z w. [| w: A; <y,z> : r |] ==> b(y,w) = b(z,w); \
-\ !! y z w. [| w: A; <y,z> : r |] ==> b(w,y) = b(w,z) \
-\ |] ==> congruent2(r,b)";
-by (cut_facts_tac prems 1);
-by (safe_tac rel_cs);
-by (rtac trans 1);
-by (REPEAT (ares_tac prems 1
- ORELSE etac (subsetD RS SigmaE2) 1 THEN assume_tac 2 THEN assume_tac 1));
-qed "congruent2I";
-
-val [equivA,commute,congt] = goal Equiv.thy
- "[| equiv(A,r); \
-\ !! y z. [| y: A; z: A |] ==> b(y,z) = b(z,y); \
-\ !! y z w. [| w: A; <y,z>: r |] ==> b(w,y) = b(w,z) \
-\ |] ==> congruent2(r,b)";
-by (resolve_tac [equivA RS congruent2I] 1);
-by (rtac (commute RS trans) 1);
-by (rtac (commute RS trans RS sym) 3);
-by (rtac sym 5);
-by (REPEAT (ares_tac [congt] 1
- ORELSE etac (equivA RS equiv_type RS subsetD RS SigmaE2) 1));
-qed "congruent2_commuteI";
-