author | clasohm |
Fri, 03 Mar 1995 12:04:45 +0100 | |
changeset 925 | 15539deb6863 |
child 972 | e61b058d58d2 |
permissions | -rw-r--r-- |
925
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(* Title: Equiv.ML |
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ID: $Id$ |
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Authors: Riccardo Mattolini, Dip. Sistemi e Informatica |
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Lawrence C Paulson, Cambridge University Computer Laboratory |
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Copyright 1994 Universita' di Firenze |
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Copyright 1993 University of Cambridge |
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Equivalence relations in HOL Set Theory |
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*) |
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open Equiv; |
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(*** Suppes, Theorem 70: r is an equiv relation iff converse(r) O r = r ***) |
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(** first half: equiv A r ==> converse(r) O r = r **) |
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goalw Equiv.thy [trans_def,sym_def,converse_def] |
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"!!r. [| sym(r); trans(r) |] ==> converse(r) O r <= r"; |
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by (fast_tac (comp_cs addSEs [converseD]) 1); |
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qed "sym_trans_comp_subset"; |
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val [major,minor]=goal Equiv.thy "[|<x,y>:r; z=<x,y>|] ==> z:r"; |
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by (simp_tac (prod_ss addsimps [minor]) 1); |
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by (rtac major 1); |
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qed "BreakPair"; |
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val [major]=goal Equiv.thy "[|? x y. <x,y>:r & z=<x,y>|] ==> z:r"; |
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by (resolve_tac [major RS exE] 1); |
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by (etac exE 1); |
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by (etac conjE 1); |
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by (asm_simp_tac (prod_ss addsimps [minor]) 1); |
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qed "BreakPair1"; |
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val [major,minor]=goal Equiv.thy "[|z:r; z=<x,y>|] ==> <x,y>:r"; |
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by (simp_tac (prod_ss addsimps [minor RS sym]) 1); |
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by (rtac major 1); |
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qed "BuildPair"; |
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val [major]=goal Equiv.thy "[|? z:r. <x,y>=z|] ==> <x,y>:r"; |
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by (resolve_tac [major RS bexE] 1); |
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by (asm_simp_tac (prod_ss addsimps []) 1); |
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qed "BuildPair1"; |
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val rel_pair_cs = rel_cs addIs [BuildPair1] |
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addEs [BreakPair1]; |
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goalw Equiv.thy [refl_def,converse_def] |
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"!!A r. refl A r ==> r <= converse(r) O r"; |
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by (step_tac comp_cs 1); |
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by (dtac subsetD 1); |
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by (assume_tac 1); |
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by (etac SigmaE 1); |
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by (rtac BreakPair1 1); |
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by (fast_tac comp_cs 1); |
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qed "refl_comp_subset"; |
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goalw Equiv.thy [equiv_def] |
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"!!A r. equiv A r ==> converse(r) O r = r"; |
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by (rtac equalityI 1); |
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by (REPEAT (ares_tac [sym_trans_comp_subset, refl_comp_subset] 1 |
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ORELSE etac conjE 1)); |
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qed "equiv_comp_eq"; |
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(*second half*) |
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goalw Equiv.thy [equiv_def,refl_def,sym_def,trans_def] |
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"!!A r. [| converse(r) O r = r; Domain(r) = A |] ==> equiv A r"; |
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by (etac equalityE 1); |
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by (subgoal_tac "ALL x y. <x,y> : r --> <y,x> : r" 1); |
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by (safe_tac set_cs); |
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by (fast_tac (set_cs addSIs [converseI] addIs [compI]) 3); |
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by (fast_tac (set_cs addSIs [converseI] addIs [compI] addSEs [DomainE]) 2); |
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by (fast_tac (rel_pair_cs addSEs [SigmaE] addSIs [SigmaI]) 1); |
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by (dtac subsetD 1); |
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by (dtac subsetD 1); |
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by (fast_tac rel_cs 1); |
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by (fast_tac rel_cs 1); |
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by flexflex_tac; |
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by (dtac subsetD 1); |
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by (fast_tac converse_cs 2); |
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by (fast_tac converse_cs 1); |
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qed "comp_equivI"; |
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(** Equivalence classes **) |
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(*Lemma for the next result*) |
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goalw Equiv.thy [equiv_def,trans_def,sym_def] |
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"!!A r. [| equiv A r; <a,b>: r |] ==> r^^{a} <= r^^{b}"; |
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by (safe_tac rel_cs); |
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by (rtac ImageI 1); |
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by (fast_tac rel_cs 2); |
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by (fast_tac rel_cs 1); |
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qed "equiv_class_subset"; |
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goal Equiv.thy "!!A r. [| equiv A r; <a,b>: r |] ==> r^^{a} = r^^{b}"; |
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by (REPEAT (ares_tac [equalityI, equiv_class_subset] 1)); |
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by (rewrite_goals_tac [equiv_def,sym_def]); |
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by (fast_tac rel_cs 1); |
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qed "equiv_class_eq"; |
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val prems = goalw Equiv.thy [equiv_def,refl_def] |
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"[| equiv A r; a: A |] ==> a: r^^{a}"; |
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by (cut_facts_tac prems 1); |
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by (fast_tac rel_cs 1); |
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qed "equiv_class_self"; |
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|
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(*Lemma for the next result*) |
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goalw Equiv.thy [equiv_def,refl_def] |
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"!!A r. [| equiv A r; r^^{b} <= r^^{a}; b: A |] ==> <a,b>: r"; |
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by (fast_tac rel_cs 1); |
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qed "subset_equiv_class"; |
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|
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val prems = goal Equiv.thy |
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"[| r^^{a} = r^^{b}; equiv A r; b: A |] ==> <a,b>: r"; |
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by (REPEAT (resolve_tac (prems @ [equalityD2, subset_equiv_class]) 1)); |
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qed "eq_equiv_class"; |
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|
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(*thus r^^{a} = r^^{b} as well*) |
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118 |
goalw Equiv.thy [equiv_def,trans_def,sym_def] |
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"!!A r. [| equiv A r; x: (r^^{a} Int r^^{b}) |] ==> <a,b>: r"; |
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by (fast_tac rel_cs 1); |
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qed "equiv_class_nondisjoint"; |
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|
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val [major] = goalw Equiv.thy [equiv_def,refl_def] |
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"equiv A r ==> r <= Sigma A (%x.A)"; |
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by (rtac (major RS conjunct1 RS conjunct1) 1); |
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qed "equiv_type"; |
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|
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goal Equiv.thy |
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"!!A r. equiv A r ==> (<x,y>: r) = (r^^{x} = r^^{y} & x:A & y:A)"; |
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by (safe_tac rel_cs); |
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by ((rtac equiv_class_eq 1) THEN (assume_tac 1) THEN (assume_tac 1)); |
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132 |
by ((rtac eq_equiv_class 3) THEN |
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133 |
(assume_tac 4) THEN (assume_tac 4) THEN (assume_tac 3)); |
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134 |
by ((dtac equiv_type 1) THEN (dtac rev_subsetD 1) THEN |
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135 |
(assume_tac 1) THEN (dtac SigmaD1 1) THEN (assume_tac 1)); |
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136 |
by ((dtac equiv_type 1) THEN (dtac rev_subsetD 1) THEN |
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137 |
(assume_tac 1) THEN (dtac SigmaD2 1) THEN (assume_tac 1)); |
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138 |
qed "equiv_class_eq_iff"; |
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139 |
|
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140 |
goal Equiv.thy |
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|
141 |
"!!A r. [| equiv A r; x: A; y: A |] ==> (r^^{x} = r^^{y}) = (<x,y>: r)"; |
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142 |
by (safe_tac rel_cs); |
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143 |
by ((rtac eq_equiv_class 1) THEN |
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144 |
(assume_tac 1) THEN (assume_tac 1) THEN (assume_tac 1)); |
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145 |
by ((rtac equiv_class_eq 1) THEN |
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|
146 |
(assume_tac 1) THEN (assume_tac 1)); |
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147 |
qed "eq_equiv_class_iff"; |
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|
148 |
|
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149 |
(*** Quotients ***) |
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150 |
|
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151 |
(** Introduction/elimination rules -- needed? **) |
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152 |
|
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153 |
val prems = goalw Equiv.thy [quotient_def] "x:A ==> r^^{x}: A/r"; |
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154 |
by (rtac UN_I 1); |
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|
155 |
by (resolve_tac prems 1); |
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156 |
by (rtac singletonI 1); |
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|
157 |
qed "quotientI"; |
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|
158 |
|
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159 |
val [major,minor] = goalw Equiv.thy [quotient_def] |
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160 |
"[| X:(A/r); !!x. [| X = r^^{x}; x:A |] ==> P |] \ |
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161 |
\ ==> P"; |
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162 |
by (resolve_tac [major RS UN_E] 1); |
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163 |
by (rtac minor 1); |
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|
164 |
by (assume_tac 2); |
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|
165 |
by (fast_tac rel_cs 1); |
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166 |
qed "quotientE"; |
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|
167 |
|
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|
168 |
(** Not needed by Theory Integ --> bypassed **) |
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169 |
(**goalw Equiv.thy [equiv_def,refl_def,quotient_def] |
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170 |
"!!A r. equiv A r ==> Union(A/r) = A"; |
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171 |
by (fast_tac eq_cs 1); |
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172 |
qed "Union_quotient"; |
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|
173 |
**) |
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|
174 |
|
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|
175 |
(** Not needed by Theory Integ --> bypassed **) |
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|
176 |
(*goalw Equiv.thy [quotient_def] |
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177 |
"!!A r. [| equiv A r; X: A/r; Y: A/r |] ==> X=Y | (X Int Y <= 0)"; |
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178 |
by (safe_tac (ZF_cs addSIs [equiv_class_eq])); |
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179 |
by (assume_tac 1); |
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|
180 |
by (rewrite_goals_tac [equiv_def,trans_def,sym_def]); |
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181 |
by (fast_tac ZF_cs 1); |
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|
182 |
qed "quotient_disj"; |
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|
183 |
**) |
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|
184 |
|
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185 |
(**** Defining unary operations upon equivalence classes ****) |
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|
186 |
|
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|
187 |
(* theorem needed to prove UN_equiv_class *) |
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|
188 |
goal Set.thy "!!A. [| a:A; ! y:A. b(y)=b(a) |] ==> (UN y:A. b(y))=b(a)"; |
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189 |
by (fast_tac (eq_cs addSEs [equalityE]) 1); |
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|
190 |
qed "UN_singleton_lemma"; |
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|
191 |
val UN_singleton = ballI RSN (2,UN_singleton_lemma); |
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|
192 |
|
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|
193 |
|
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|
194 |
(** These proofs really require as local premises |
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195 |
equiv A r; congruent r b |
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|
196 |
**) |
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|
197 |
|
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|
198 |
(*Conversion rule*) |
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199 |
val prems as [equivA,bcong,_] = goal Equiv.thy |
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200 |
"[| equiv A r; congruent r b; a: A |] ==> (UN x:r^^{a}. b(x)) = b(a)"; |
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|
201 |
by (cut_facts_tac prems 1); |
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|
202 |
by (rtac UN_singleton 1); |
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|
203 |
by (rtac equiv_class_self 1); |
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|
204 |
by (assume_tac 1); |
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|
205 |
by (assume_tac 1); |
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|
206 |
by (rewrite_goals_tac [equiv_def,congruent_def,sym_def]); |
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|
207 |
by (fast_tac rel_cs 1); |
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|
208 |
qed "UN_equiv_class"; |
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|
209 |
|
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|
210 |
(*Resolve th against the "local" premises*) |
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|
211 |
val localize = RSLIST [equivA,bcong]; |
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|
212 |
|
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|
213 |
(*type checking of UN x:r``{a}. b(x) *) |
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|
214 |
val _::_::prems = goalw Equiv.thy [quotient_def] |
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|
215 |
"[| equiv A r; congruent r b; X: A/r; \ |
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|
216 |
\ !!x. x : A ==> b(x) : B |] \ |
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217 |
\ ==> (UN x:X. b(x)) : B"; |
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|
218 |
by (cut_facts_tac prems 1); |
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|
219 |
by (safe_tac rel_cs); |
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220 |
by (rtac (localize UN_equiv_class RS ssubst) 1); |
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|
221 |
by (REPEAT (ares_tac prems 1)); |
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|
222 |
qed "UN_equiv_class_type"; |
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|
223 |
|
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|
224 |
(*Sufficient conditions for injectiveness. Could weaken premises! |
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|
225 |
major premise could be an inclusion; bcong could be !!y. y:A ==> b(y):B |
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|
226 |
*) |
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|
227 |
val _::_::prems = goalw Equiv.thy [quotient_def] |
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|
228 |
"[| equiv A r; congruent r b; \ |
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|
229 |
\ (UN x:X. b(x))=(UN y:Y. b(y)); X: A/r; Y: A/r; \ |
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|
230 |
\ !!x y. [| x:A; y:A; b(x)=b(y) |] ==> <x,y>:r |] \ |
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|
231 |
\ ==> X=Y"; |
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|
232 |
by (cut_facts_tac prems 1); |
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|
233 |
by (safe_tac rel_cs); |
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parents:
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|
234 |
by (rtac (equivA RS equiv_class_eq) 1); |
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|
235 |
by (REPEAT (ares_tac prems 1)); |
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parents:
diff
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|
236 |
by (etac box_equals 1); |
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|
237 |
by (REPEAT (ares_tac [localize UN_equiv_class] 1)); |
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|
238 |
qed "UN_equiv_class_inject"; |
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|
239 |
|
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|
240 |
|
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|
241 |
(**** Defining binary operations upon equivalence classes ****) |
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|
242 |
|
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|
243 |
|
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|
244 |
goalw Equiv.thy [congruent_def,congruent2_def,equiv_def,refl_def] |
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|
245 |
"!!A r. [| equiv A r; congruent2 r b; a: A |] ==> congruent r (b a)"; |
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|
246 |
by (fast_tac rel_cs 1); |
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|
247 |
qed "congruent2_implies_congruent"; |
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|
248 |
|
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|
249 |
val equivA::prems = goalw Equiv.thy [congruent_def] |
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|
250 |
"[| equiv A r; congruent2 r b; a: A |] ==> \ |
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|
251 |
\ congruent r (%x1. UN x2:r^^{a}. b x1 x2)"; |
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|
252 |
by (cut_facts_tac (equivA::prems) 1); |
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|
253 |
by (safe_tac rel_cs); |
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|
254 |
by (rtac (equivA RS equiv_type RS subsetD RS SigmaE2) 1); |
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|
255 |
by (assume_tac 1); |
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|
256 |
by (asm_simp_tac (prod_ss addsimps [equivA RS UN_equiv_class, |
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|
257 |
congruent2_implies_congruent]) 1); |
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|
258 |
by (rewrite_goals_tac [congruent2_def,equiv_def,refl_def]); |
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|
259 |
by (fast_tac rel_cs 1); |
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|
260 |
qed "congruent2_implies_congruent_UN"; |
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|
261 |
|
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|
262 |
val prems as equivA::_ = goal Equiv.thy |
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|
263 |
"[| equiv A r; congruent2 r b; a1: A; a2: A |] \ |
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\ ==> (UN x1:r^^{a1}. UN x2:r^^{a2}. b x1 x2) = b a1 a2"; |
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by (cut_facts_tac prems 1); |
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by (asm_simp_tac (prod_ss addsimps [equivA RS UN_equiv_class, |
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congruent2_implies_congruent, |
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congruent2_implies_congruent_UN]) 1); |
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qed "UN_equiv_class2"; |
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|
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(*type checking*) |
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val prems = goalw Equiv.thy [quotient_def] |
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"[| equiv A r; congruent2 r b; \ |
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\ X1: A/r; X2: A/r; \ |
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\ !!x1 x2. [| x1: A; x2: A |] ==> b x1 x2 : B |] \ |
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\ ==> (UN x1:X1. UN x2:X2. b x1 x2) : B"; |
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by (cut_facts_tac prems 1); |
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by (safe_tac rel_cs); |
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by (REPEAT (ares_tac (prems@[UN_equiv_class_type, |
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congruent2_implies_congruent_UN, |
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congruent2_implies_congruent, quotientI]) 1)); |
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qed "UN_equiv_class_type2"; |
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|
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|
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(*Suggested by John Harrison -- the two subproofs may be MUCH simpler |
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than the direct proof*) |
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val prems = goalw Equiv.thy [congruent2_def,equiv_def,refl_def] |
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"[| equiv A r; \ |
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\ !! y z w. [| w: A; <y,z> : r |] ==> b y w = b z w; \ |
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\ !! y z w. [| w: A; <y,z> : r |] ==> b w y = b w z \ |
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\ |] ==> congruent2 r b"; |
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by (cut_facts_tac prems 1); |
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by (safe_tac rel_cs); |
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by (rtac trans 1); |
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by (REPEAT (ares_tac prems 1 |
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ORELSE etac (subsetD RS SigmaE2) 1 THEN assume_tac 2 THEN assume_tac 1)); |
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qed "congruent2I"; |
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|
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val [equivA,commute,congt] = goal Equiv.thy |
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"[| equiv A r; \ |
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\ !! y z. [| y: A; z: A |] ==> b y z = b z y; \ |
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\ !! y z w. [| w: A; <y,z>: r |] ==> b w y = b w z \ |
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\ |] ==> congruent2 r b"; |
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by (resolve_tac [equivA RS congruent2I] 1); |
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by (rtac (commute RS trans) 1); |
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by (rtac (commute RS trans RS sym) 3); |
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by (rtac sym 5); |
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by (REPEAT (ares_tac [congt] 1 |
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ORELSE etac (equivA RS equiv_type RS subsetD RS SigmaE2) 1)); |
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qed "congruent2_commuteI"; |
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