author | clasohm |
Sun, 24 Apr 1994 11:27:38 +0200 | |
changeset 70 | 9459592608e2 |
parent 66 | 14b9286ed036 |
child 111 | 361521bc7c47 |
permissions | -rw-r--r-- |
0 | 1 |
(* Title: HOL/univ |
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ID: $Id$ |
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Author: Lawrence C Paulson, Cambridge University Computer Laboratory |
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Copyright 1991 University of Cambridge |
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For univ.thy |
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*) |
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open Univ; |
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(** LEAST -- the least number operator **) |
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val [prem1,prem2] = goalw Univ.thy [Least_def] |
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"[| P(k); !!x. x<k ==> ~P(x) |] ==> (LEAST x.P(x)) = k"; |
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by (rtac select_equality 1); |
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by (fast_tac (HOL_cs addSIs [prem1,prem2]) 1); |
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by (cut_facts_tac [less_linear] 1); |
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by (fast_tac (HOL_cs addSIs [prem1] addSDs [prem2]) 1); |
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val Least_equality = result(); |
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val [prem] = goal Univ.thy "P(k) ==> P(LEAST x.P(x))"; |
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by (rtac (prem RS rev_mp) 1); |
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by (res_inst_tac [("n","k")] less_induct 1); |
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by (rtac impI 1); |
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by (rtac classical 1); |
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by (res_inst_tac [("s","n")] (Least_equality RS ssubst) 1); |
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by (assume_tac 1); |
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by (assume_tac 2); |
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by (fast_tac HOL_cs 1); |
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val LeastI = result(); |
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(*Proof is almost identical to the one above!*) |
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val [prem] = goal Univ.thy "P(k) ==> (LEAST x.P(x)) <= k"; |
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by (rtac (prem RS rev_mp) 1); |
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by (res_inst_tac [("n","k")] less_induct 1); |
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by (rtac impI 1); |
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by (rtac classical 1); |
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by (res_inst_tac [("s","n")] (Least_equality RS ssubst) 1); |
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by (assume_tac 1); |
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by (rtac le_refl 2); |
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by (fast_tac (HOL_cs addIs [less_imp_le,le_trans]) 1); |
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val Least_le = result(); |
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val [prem] = goal Univ.thy "k < (LEAST x.P(x)) ==> ~P(k)"; |
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by (rtac notI 1); |
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by (etac (rewrite_rule [le_def] Least_le RS notE) 1); |
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by (rtac prem 1); |
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val not_less_Least = result(); |
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(** apfst -- can be used in similar type definitions **) |
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goalw Univ.thy [apfst_def] "apfst(f,<a,b>) = <f(a),b>"; |
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by (rtac split 1); |
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val apfst = result(); |
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val [major,minor] = goal Univ.thy |
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"[| q = apfst(f,p); !!x y. [| p = <x,y>; q = <f(x),y> |] ==> R \ |
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\ |] ==> R"; |
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by (rtac PairE 1); |
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by (rtac minor 1); |
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by (assume_tac 1); |
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by (rtac (major RS trans) 1); |
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by (etac ssubst 1); |
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by (rtac apfst 1); |
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val apfstE = result(); |
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(** Push -- an injection, analogous to Cons on lists **) |
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val [major] = goalw Univ.thy [Push_def] "Push(i,f)=Push(j,g) ==> i=j"; |
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by (rtac (major RS fun_cong RS box_equals RS Suc_inject) 1); |
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by (rtac nat_case_0 1); |
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by (rtac nat_case_0 1); |
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val Push_inject1 = result(); |
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val [major] = goalw Univ.thy [Push_def] "Push(i,f)=Push(j,g) ==> f=g"; |
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by (rtac (major RS fun_cong RS ext RS box_equals) 1); |
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by (rtac (nat_case_Suc RS ext) 1); |
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by (rtac (nat_case_Suc RS ext) 1); |
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val Push_inject2 = result(); |
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val [major,minor] = goal Univ.thy |
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"[| Push(i,f)=Push(j,g); [| i=j; f=g |] ==> P \ |
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\ |] ==> P"; |
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by (rtac ((major RS Push_inject2) RS ((major RS Push_inject1) RS minor)) 1); |
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val Push_inject = result(); |
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val [major] = goalw Univ.thy [Push_def] "Push(k,f)=(%z.0) ==> P"; |
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by (rtac (major RS fun_cong RS box_equals RS Suc_neq_Zero) 1); |
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by (rtac nat_case_0 1); |
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by (rtac refl 1); |
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val Push_neq_K0 = result(); |
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(*** Isomorphisms ***) |
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goal Univ.thy "inj(Rep_Node)"; |
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by (rtac inj_inverseI 1); (*cannot combine by RS: multiple unifiers*) |
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by (rtac Rep_Node_inverse 1); |
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val inj_Rep_Node = result(); |
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goal Univ.thy "inj_onto(Abs_Node,Node)"; |
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by (rtac inj_onto_inverseI 1); |
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by (etac Abs_Node_inverse 1); |
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val inj_onto_Abs_Node = result(); |
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val Abs_Node_inject = inj_onto_Abs_Node RS inj_ontoD; |
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(*** Introduction rules for Node ***) |
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goalw Univ.thy [Node_def] "<%k. 0,a> : Node"; |
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by (fast_tac set_cs 1); |
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val Node_K0_I = result(); |
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goalw Univ.thy [Node_def,Push_def] |
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"!!p. p: Node ==> apfst(Push(i), p) : Node"; |
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by (fast_tac (set_cs addSIs [apfst, nat_case_Suc RS trans]) 1); |
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val Node_Push_I = result(); |
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(*** Distinctness of constructors ***) |
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(** Scons vs Atom **) |
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48
21291189b51e
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clasohm
parents:
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changeset
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goalw Univ.thy [Atom_def,Scons_def,Push_Node_def] "(M$N) ~= Atom(a)"; |
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by (rtac notI 1); |
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by (etac (equalityD2 RS subsetD RS UnE) 1); |
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by (rtac singletonI 1); |
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by (REPEAT (eresolve_tac [imageE, Abs_Node_inject RS apfstE, |
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Pair_inject, sym RS Push_neq_K0] 1 |
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ORELSE resolve_tac [Node_K0_I, Rep_Node RS Node_Push_I] 1)); |
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val Scons_not_Atom = result(); |
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val Atom_not_Scons = standard (Scons_not_Atom RS not_sym); |
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val Scons_neq_Atom = standard (Scons_not_Atom RS notE); |
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val Atom_neq_Scons = sym RS Scons_neq_Atom; |
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(*** Injectiveness ***) |
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(** Atomic nodes **) |
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goalw Univ.thy [Atom_def] "inj(Atom)"; |
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by (rtac injI 1); |
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by (etac (singleton_inject RS Abs_Node_inject RS Pair_inject) 1); |
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by (REPEAT (ares_tac [Node_K0_I] 1)); |
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val inj_Atom = result(); |
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val Atom_inject = inj_Atom RS injD; |
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goalw Univ.thy [Leaf_def,o_def] "inj(Leaf)"; |
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by (rtac injI 1); |
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by (etac (Atom_inject RS Inl_inject) 1); |
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val inj_Leaf = result(); |
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val Leaf_inject = inj_Leaf RS injD; |
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goalw Univ.thy [Numb_def,o_def] "inj(Numb)"; |
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by (rtac injI 1); |
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by (etac (Atom_inject RS Inr_inject) 1); |
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val inj_Numb = result(); |
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val Numb_inject = inj_Numb RS injD; |
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(** Injectiveness of Push_Node **) |
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val [major,minor] = goalw Univ.thy [Push_Node_def] |
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"[| Push_Node(i,m)=Push_Node(j,n); [| i=j; m=n |] ==> P \ |
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\ |] ==> P"; |
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by (rtac (major RS Abs_Node_inject RS apfstE) 1); |
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by (REPEAT (resolve_tac [Rep_Node RS Node_Push_I] 1)); |
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by (etac (sym RS apfstE) 1); |
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by (rtac minor 1); |
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by (etac Pair_inject 1); |
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by (etac (Push_inject1 RS sym) 1); |
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by (rtac (inj_Rep_Node RS injD) 1); |
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by (etac trans 1); |
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by (safe_tac (HOL_cs addSEs [Pair_inject,Push_inject,sym])); |
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val Push_Node_inject = result(); |
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(** Injectiveness of Scons **) |
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21291189b51e
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parents:
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diff
changeset
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val [major] = goalw Univ.thy [Scons_def] "M$N <= M'$N' ==> M<=M'"; |
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by (cut_facts_tac [major] 1); |
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by (fast_tac (set_cs addSDs [Suc_inject] |
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addSEs [Push_Node_inject, Zero_neq_Suc]) 1); |
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val Scons_inject_lemma1 = result(); |
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48
21291189b51e
changed "." to "$" and Cons to infix "#" to eliminate ambiguity
clasohm
parents:
5
diff
changeset
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val [major] = goalw Univ.thy [Scons_def] "M$N <= M'$N' ==> N<=N'"; |
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by (cut_facts_tac [major] 1); |
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by (fast_tac (set_cs addSDs [Suc_inject] |
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addSEs [Push_Node_inject, Suc_neq_Zero]) 1); |
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val Scons_inject_lemma2 = result(); |
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48
21291189b51e
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clasohm
parents:
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diff
changeset
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val [major] = goal Univ.thy "M$N = M'$N' ==> M=M'"; |
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by (rtac (major RS equalityE) 1); |
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by (REPEAT (ares_tac [equalityI, Scons_inject_lemma1] 1)); |
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val Scons_inject1 = result(); |
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48
21291189b51e
changed "." to "$" and Cons to infix "#" to eliminate ambiguity
clasohm
parents:
5
diff
changeset
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val [major] = goal Univ.thy "M$N = M'$N' ==> N=N'"; |
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by (rtac (major RS equalityE) 1); |
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by (REPEAT (ares_tac [equalityI, Scons_inject_lemma2] 1)); |
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val Scons_inject2 = result(); |
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val [major,minor] = goal Univ.thy |
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21291189b51e
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clasohm
parents:
5
diff
changeset
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"[| M$N = M'$N'; [| M=M'; N=N' |] ==> P \ |
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\ |] ==> P"; |
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by (rtac ((major RS Scons_inject2) RS ((major RS Scons_inject1) RS minor)) 1); |
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val Scons_inject = result(); |
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(*rewrite rules*) |
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goal Univ.thy "(Atom(a)=Atom(b)) = (a=b)"; |
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by (fast_tac (HOL_cs addSEs [Atom_inject]) 1); |
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val Atom_Atom_eq = result(); |
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48
21291189b51e
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clasohm
parents:
5
diff
changeset
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goal Univ.thy "(M$N = M'$N') = (M=M' & N=N')"; |
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by (fast_tac (HOL_cs addSEs [Scons_inject]) 1); |
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val Scons_Scons_eq = result(); |
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(*** Distinctness involving Leaf and Numb ***) |
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(** Scons vs Leaf **) |
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goalw Univ.thy [Leaf_def,o_def] "(M$N) ~= Leaf(a)"; |
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by (rtac Scons_not_Atom 1); |
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val Scons_not_Leaf = result(); |
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val Leaf_not_Scons = standard (Scons_not_Leaf RS not_sym); |
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val Scons_neq_Leaf = standard (Scons_not_Leaf RS notE); |
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val Leaf_neq_Scons = sym RS Scons_neq_Leaf; |
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(** Scons vs Numb **) |
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goalw Univ.thy [Numb_def,o_def] "(M$N) ~= Numb(k)"; |
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by (rtac Scons_not_Atom 1); |
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val Scons_not_Numb = result(); |
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val Numb_not_Scons = standard (Scons_not_Numb RS not_sym); |
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val Scons_neq_Numb = standard (Scons_not_Numb RS notE); |
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val Numb_neq_Scons = sym RS Scons_neq_Numb; |
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(** Leaf vs Numb **) |
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goalw Univ.thy [Leaf_def,Numb_def] "Leaf(a) ~= Numb(k)"; |
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by (simp_tac (HOL_ss addsimps [Atom_Atom_eq,Inl_not_Inr]) 1); |
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val Leaf_not_Numb = result(); |
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val Numb_not_Leaf = standard (Leaf_not_Numb RS not_sym); |
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val Leaf_neq_Numb = standard (Leaf_not_Numb RS notE); |
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val Numb_neq_Leaf = sym RS Leaf_neq_Numb; |
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(*** ndepth -- the depth of a node ***) |
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val univ_simps = [apfst,Scons_not_Atom,Atom_not_Scons,Scons_Scons_eq]; |
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val univ_ss = nat_ss addsimps univ_simps; |
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goalw Univ.thy [ndepth_def] "ndepth (Abs_Node(<%k.0, x>)) = 0"; |
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by (sstac [Node_K0_I RS Abs_Node_inverse, split] 1); |
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by (rtac Least_equality 1); |
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by (rtac refl 1); |
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by (etac less_zeroE 1); |
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val ndepth_K0 = result(); |
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5 | 266 |
goal Univ.thy "k < Suc(LEAST x. f(x)=0) --> nat_case(k, Suc(i), f) ~= 0"; |
0 | 267 |
by (nat_ind_tac "k" 1); |
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by (ALLGOALS (simp_tac nat_ss)); |
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by (rtac impI 1); |
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by (etac not_less_Least 1); |
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val ndepth_Push_lemma = result(); |
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goalw Univ.thy [ndepth_def,Push_Node_def] |
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"ndepth (Push_Node(i,n)) = Suc(ndepth(n))"; |
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by (stac (Rep_Node RS Node_Push_I RS Abs_Node_inverse) 1); |
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by (cut_facts_tac [rewrite_rule [Node_def] Rep_Node] 1); |
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by (safe_tac set_cs); |
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be ssubst 1; (*instantiates type variables!*) |
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by (simp_tac univ_ss 1); |
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by (rtac Least_equality 1); |
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by (rewtac Push_def); |
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by (rtac (nat_case_Suc RS trans) 1); |
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by (etac LeastI 1); |
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by (etac (ndepth_Push_lemma RS mp) 1); |
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val ndepth_Push_Node = result(); |
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287 |
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(*** ntrunc applied to the various node sets ***) |
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goalw Univ.thy [ntrunc_def] "ntrunc(0, M) = {}"; |
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by (safe_tac (set_cs addSIs [equalityI] addSEs [less_zeroE])); |
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val ntrunc_0 = result(); |
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goalw Univ.thy [Atom_def,ntrunc_def] "ntrunc(Suc(k), Atom(a)) = Atom(a)"; |
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by (safe_tac (set_cs addSIs [equalityI])); |
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by (stac ndepth_K0 1); |
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by (rtac zero_less_Suc 1); |
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val ntrunc_Atom = result(); |
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299 |
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66 | 300 |
goalw Univ.thy [Leaf_def,o_def] "ntrunc(Suc(k), Leaf(a)) = Leaf(a)"; |
0 | 301 |
by (rtac ntrunc_Atom 1); |
302 |
val ntrunc_Leaf = result(); |
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303 |
||
66 | 304 |
goalw Univ.thy [Numb_def,o_def] "ntrunc(Suc(k), Numb(i)) = Numb(i)"; |
0 | 305 |
by (rtac ntrunc_Atom 1); |
306 |
val ntrunc_Numb = result(); |
|
307 |
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308 |
goalw Univ.thy [Scons_def,ntrunc_def] |
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48
21291189b51e
changed "." to "$" and Cons to infix "#" to eliminate ambiguity
clasohm
parents:
5
diff
changeset
|
309 |
"ntrunc(Suc(k), M$N) = ntrunc(k,M) $ ntrunc(k,N)"; |
0 | 310 |
by (safe_tac (set_cs addSIs [equalityI,imageI])); |
311 |
by (REPEAT (stac ndepth_Push_Node 3 THEN etac Suc_mono 3)); |
|
312 |
by (REPEAT (rtac Suc_less_SucD 1 THEN |
|
313 |
rtac (ndepth_Push_Node RS subst) 1 THEN |
|
314 |
assume_tac 1)); |
|
315 |
val ntrunc_Scons = result(); |
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316 |
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317 |
(** Injection nodes **) |
|
318 |
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319 |
goalw Univ.thy [In0_def] "ntrunc(Suc(0), In0(M)) = {}"; |
|
320 |
by (simp_tac (univ_ss addsimps [ntrunc_Scons,ntrunc_0]) 1); |
|
321 |
by (rewtac Scons_def); |
|
322 |
by (safe_tac (set_cs addSIs [equalityI])); |
|
323 |
val ntrunc_one_In0 = result(); |
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324 |
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325 |
goalw Univ.thy [In0_def] |
|
326 |
"ntrunc(Suc(Suc(k)), In0(M)) = In0 (ntrunc(Suc(k),M))"; |
|
327 |
by (simp_tac (univ_ss addsimps [ntrunc_Scons,ntrunc_Numb]) 1); |
|
328 |
val ntrunc_In0 = result(); |
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329 |
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330 |
goalw Univ.thy [In1_def] "ntrunc(Suc(0), In1(M)) = {}"; |
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by (simp_tac (univ_ss addsimps [ntrunc_Scons,ntrunc_0]) 1); |
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332 |
by (rewtac Scons_def); |
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by (safe_tac (set_cs addSIs [equalityI])); |
|
334 |
val ntrunc_one_In1 = result(); |
|
335 |
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336 |
goalw Univ.thy [In1_def] |
|
337 |
"ntrunc(Suc(Suc(k)), In1(M)) = In1 (ntrunc(Suc(k),M))"; |
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by (simp_tac (univ_ss addsimps [ntrunc_Scons,ntrunc_Numb]) 1); |
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339 |
val ntrunc_In1 = result(); |
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340 |
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341 |
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342 |
(*** Cartesian Product ***) |
|
343 |
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48
21291189b51e
changed "." to "$" and Cons to infix "#" to eliminate ambiguity
clasohm
parents:
5
diff
changeset
|
344 |
goalw Univ.thy [uprod_def] "!!M N. [| M:A; N:B |] ==> (M$N) : A<*>B"; |
0 | 345 |
by (REPEAT (ares_tac [singletonI,UN_I] 1)); |
346 |
val uprodI = result(); |
|
347 |
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348 |
(*The general elimination rule*) |
|
349 |
val major::prems = goalw Univ.thy [uprod_def] |
|
350 |
"[| c : A<*>B; \ |
|
48
21291189b51e
changed "." to "$" and Cons to infix "#" to eliminate ambiguity
clasohm
parents:
5
diff
changeset
|
351 |
\ !!x y. [| x:A; y:B; c=x$y |] ==> P \ |
0 | 352 |
\ |] ==> P"; |
353 |
by (cut_facts_tac [major] 1); |
|
354 |
by (REPEAT (eresolve_tac [asm_rl,singletonE,UN_E] 1 |
|
355 |
ORELSE resolve_tac prems 1)); |
|
356 |
val uprodE = result(); |
|
357 |
||
358 |
(*Elimination of a pair -- introduces no eigenvariables*) |
|
359 |
val prems = goal Univ.thy |
|
48
21291189b51e
changed "." to "$" and Cons to infix "#" to eliminate ambiguity
clasohm
parents:
5
diff
changeset
|
360 |
"[| (M$N) : A<*>B; [| M:A; N:B |] ==> P \ |
0 | 361 |
\ |] ==> P"; |
362 |
by (rtac uprodE 1); |
|
363 |
by (REPEAT (ares_tac prems 1 ORELSE eresolve_tac [Scons_inject,ssubst] 1)); |
|
364 |
val uprodE2 = result(); |
|
365 |
||
366 |
||
367 |
(*** Disjoint Sum ***) |
|
368 |
||
369 |
goalw Univ.thy [usum_def] "!!M. M:A ==> In0(M) : A<+>B"; |
|
370 |
by (fast_tac set_cs 1); |
|
371 |
val usum_In0I = result(); |
|
372 |
||
373 |
goalw Univ.thy [usum_def] "!!N. N:B ==> In1(N) : A<+>B"; |
|
374 |
by (fast_tac set_cs 1); |
|
375 |
val usum_In1I = result(); |
|
376 |
||
377 |
val major::prems = goalw Univ.thy [usum_def] |
|
378 |
"[| u : A<+>B; \ |
|
379 |
\ !!x. [| x:A; u=In0(x) |] ==> P; \ |
|
380 |
\ !!y. [| y:B; u=In1(y) |] ==> P \ |
|
381 |
\ |] ==> P"; |
|
382 |
by (rtac (major RS UnE) 1); |
|
383 |
by (REPEAT (rtac refl 1 |
|
384 |
ORELSE eresolve_tac (prems@[imageE,ssubst]) 1)); |
|
385 |
val usumE = result(); |
|
386 |
||
387 |
||
388 |
(** Injection **) |
|
389 |
||
5 | 390 |
goalw Univ.thy [In0_def,In1_def] "In0(M) ~= In1(N)"; |
0 | 391 |
by (rtac notI 1); |
392 |
by (etac (Scons_inject1 RS Numb_inject RS Zero_neq_Suc) 1); |
|
393 |
val In0_not_In1 = result(); |
|
394 |
||
395 |
val In1_not_In0 = standard (In0_not_In1 RS not_sym); |
|
396 |
val In0_neq_In1 = standard (In0_not_In1 RS notE); |
|
397 |
val In1_neq_In0 = sym RS In0_neq_In1; |
|
398 |
||
399 |
val [major] = goalw Univ.thy [In0_def] "In0(M) = In0(N) ==> M=N"; |
|
400 |
by (rtac (major RS Scons_inject2) 1); |
|
401 |
val In0_inject = result(); |
|
402 |
||
403 |
val [major] = goalw Univ.thy [In1_def] "In1(M) = In1(N) ==> M=N"; |
|
404 |
by (rtac (major RS Scons_inject2) 1); |
|
405 |
val In1_inject = result(); |
|
406 |
||
407 |
||
408 |
(*** proving equality of sets and functions using ntrunc ***) |
|
409 |
||
410 |
goalw Univ.thy [ntrunc_def] "ntrunc(k,M) <= M"; |
|
411 |
by (fast_tac set_cs 1); |
|
412 |
val ntrunc_subsetI = result(); |
|
413 |
||
414 |
val [major] = goalw Univ.thy [ntrunc_def] |
|
415 |
"(!!k. ntrunc(k,M) <= N) ==> M<=N"; |
|
416 |
by (fast_tac (set_cs addIs [less_add_Suc1, less_add_Suc2, |
|
417 |
major RS subsetD]) 1); |
|
418 |
val ntrunc_subsetD = result(); |
|
419 |
||
420 |
(*A generalized form of the take-lemma*) |
|
421 |
val [major] = goal Univ.thy "(!!k. ntrunc(k,M) = ntrunc(k,N)) ==> M=N"; |
|
422 |
by (rtac equalityI 1); |
|
423 |
by (ALLGOALS (rtac ntrunc_subsetD)); |
|
424 |
by (ALLGOALS (rtac (ntrunc_subsetI RSN (2, subset_trans)))); |
|
425 |
by (rtac (major RS equalityD1) 1); |
|
426 |
by (rtac (major RS equalityD2) 1); |
|
427 |
val ntrunc_equality = result(); |
|
428 |
||
66 | 429 |
val [major] = goalw Univ.thy [o_def] |
0 | 430 |
"[| !!k. (ntrunc(k) o h1) = (ntrunc(k) o h2) |] ==> h1=h2"; |
431 |
by (rtac (ntrunc_equality RS ext) 1); |
|
66 | 432 |
by (rtac (major RS fun_cong) 1); |
0 | 433 |
val ntrunc_o_equality = result(); |
434 |
||
435 |
(*** Monotonicity ***) |
|
436 |
||
437 |
goalw Univ.thy [uprod_def] "!!A B. [| A<=A'; B<=B' |] ==> A<*>B <= A'<*>B'"; |
|
438 |
by (fast_tac set_cs 1); |
|
439 |
val uprod_mono = result(); |
|
440 |
||
441 |
goalw Univ.thy [usum_def] "!!A B. [| A<=A'; B<=B' |] ==> A<+>B <= A'<+>B'"; |
|
442 |
by (fast_tac set_cs 1); |
|
443 |
val usum_mono = result(); |
|
444 |
||
48
21291189b51e
changed "." to "$" and Cons to infix "#" to eliminate ambiguity
clasohm
parents:
5
diff
changeset
|
445 |
goalw Univ.thy [Scons_def] "!!M N. [| M<=M'; N<=N' |] ==> M$N <= M'$N'"; |
0 | 446 |
by (fast_tac set_cs 1); |
447 |
val Scons_mono = result(); |
|
448 |
||
449 |
goalw Univ.thy [In0_def] "!!M N. M<=N ==> In0(M) <= In0(N)"; |
|
450 |
by (REPEAT (ares_tac [subset_refl,Scons_mono] 1)); |
|
451 |
val In0_mono = result(); |
|
452 |
||
453 |
goalw Univ.thy [In1_def] "!!M N. M<=N ==> In1(M) <= In1(N)"; |
|
454 |
by (REPEAT (ares_tac [subset_refl,Scons_mono] 1)); |
|
455 |
val In1_mono = result(); |
|
456 |
||
457 |
||
458 |
(*** Split and Case ***) |
|
459 |
||
48
21291189b51e
changed "." to "$" and Cons to infix "#" to eliminate ambiguity
clasohm
parents:
5
diff
changeset
|
460 |
goalw Univ.thy [Split_def] "Split(M$N, c) = c(M,N)"; |
0 | 461 |
by (fast_tac (set_cs addIs [select_equality] addEs [Scons_inject]) 1); |
462 |
val Split = result(); |
|
463 |
||
464 |
goalw Univ.thy [Case_def] "Case(In0(M), c, d) = c(M)"; |
|
465 |
by (fast_tac (set_cs addIs [select_equality] |
|
466 |
addEs [make_elim In0_inject, In0_neq_In1]) 1); |
|
467 |
val Case_In0 = result(); |
|
468 |
||
469 |
goalw Univ.thy [Case_def] "Case(In1(N), c, d) = d(N)"; |
|
470 |
by (fast_tac (set_cs addIs [select_equality] |
|
471 |
addEs [make_elim In1_inject, In1_neq_In0]) 1); |
|
472 |
val Case_In1 = result(); |
|
473 |
||
474 |
(**** UN x. B(x) rules ****) |
|
475 |
||
476 |
goalw Univ.thy [ntrunc_def] "ntrunc(k, UN x.f(x)) = (UN x. ntrunc(k, f(x)))"; |
|
477 |
by (fast_tac (set_cs addIs [equalityI]) 1); |
|
478 |
val ntrunc_UN1 = result(); |
|
479 |
||
48
21291189b51e
changed "." to "$" and Cons to infix "#" to eliminate ambiguity
clasohm
parents:
5
diff
changeset
|
480 |
goalw Univ.thy [Scons_def] "(UN x.f(x)) $ M = (UN x. f(x) $ M)"; |
0 | 481 |
by (fast_tac (set_cs addIs [equalityI]) 1); |
482 |
val Scons_UN1_x = result(); |
|
483 |
||
48
21291189b51e
changed "." to "$" and Cons to infix "#" to eliminate ambiguity
clasohm
parents:
5
diff
changeset
|
484 |
goalw Univ.thy [Scons_def] "M $ (UN x.f(x)) = (UN x. M $ f(x))"; |
0 | 485 |
by (fast_tac (set_cs addIs [equalityI]) 1); |
486 |
val Scons_UN1_y = result(); |
|
487 |
||
488 |
goalw Univ.thy [In0_def] "In0(UN x.f(x)) = (UN x. In0(f(x)))"; |
|
489 |
br Scons_UN1_y 1; |
|
490 |
val In0_UN1 = result(); |
|
491 |
||
492 |
goalw Univ.thy [In1_def] "In1(UN x.f(x)) = (UN x. In1(f(x)))"; |
|
493 |
br Scons_UN1_y 1; |
|
494 |
val In1_UN1 = result(); |
|
495 |
||
496 |
||
497 |
(*** Equality : the diagonal relation ***) |
|
498 |
||
499 |
goalw Univ.thy [diag_def] "!!a A. a:A ==> <a,a> : diag(A)"; |
|
500 |
by (REPEAT (ares_tac [singletonI,UN_I] 1)); |
|
501 |
val diagI = result(); |
|
502 |
||
503 |
(*The general elimination rule*) |
|
504 |
val major::prems = goalw Univ.thy [diag_def] |
|
505 |
"[| c : diag(A); \ |
|
506 |
\ !!x y. [| x:A; c = <x,x> |] ==> P \ |
|
507 |
\ |] ==> P"; |
|
508 |
by (rtac (major RS UN_E) 1); |
|
509 |
by (REPEAT (eresolve_tac [asm_rl,singletonE] 1 ORELSE resolve_tac prems 1)); |
|
510 |
val diagE = result(); |
|
511 |
||
512 |
(*** Equality for Cartesian Product ***) |
|
513 |
||
514 |
goal Univ.thy |
|
48
21291189b51e
changed "." to "$" and Cons to infix "#" to eliminate ambiguity
clasohm
parents:
5
diff
changeset
|
515 |
"split(<M,M'>, %x x'. split(<N,N'>, %y y'. {<x$y,x'$y'>})) = {<M$N, M'$N'>}"; |
0 | 516 |
by (simp_tac univ_ss 1); |
517 |
val dprod_lemma = result(); |
|
518 |
||
519 |
goalw Univ.thy [dprod_def] |
|
48
21291189b51e
changed "." to "$" and Cons to infix "#" to eliminate ambiguity
clasohm
parents:
5
diff
changeset
|
520 |
"!!r s. [| <M,M'>:r; <N,N'>:s |] ==> <M$N, M'$N'> : r<**>s"; |
0 | 521 |
by (REPEAT (ares_tac [UN_I] 1)); |
522 |
by (rtac (singletonI RS (dprod_lemma RS equalityD2 RS subsetD)) 1); |
|
523 |
val dprodI = result(); |
|
524 |
||
525 |
(*The general elimination rule*) |
|
526 |
val major::prems = goalw Univ.thy [dprod_def] |
|
527 |
"[| c : r<**>s; \ |
|
48
21291189b51e
changed "." to "$" and Cons to infix "#" to eliminate ambiguity
clasohm
parents:
5
diff
changeset
|
528 |
\ !!x y x' y'. [| <x,x'> : r; <y,y'> : s; c = <x$y,x'$y'> |] ==> P \ |
0 | 529 |
\ |] ==> P"; |
530 |
by (cut_facts_tac [major] 1); |
|
531 |
by (REPEAT (eresolve_tac [asm_rl,singletonE,UN_E] 1)); |
|
532 |
by (res_inst_tac [("p","u")] PairE 1); |
|
533 |
by (res_inst_tac [("p","v")] PairE 1); |
|
534 |
by (safe_tac HOL_cs); |
|
535 |
by (REPEAT (ares_tac prems 1)); |
|
536 |
by (safe_tac (set_cs addSDs [dprod_lemma RS equalityD1 RS subsetD])); |
|
537 |
val dprodE = result(); |
|
538 |
||
539 |
||
540 |
(*** Equality for Disjoint Sum ***) |
|
541 |
||
542 |
goalw Univ.thy [dsum_def] "!!r. <M,M'>:r ==> <In0(M), In0(M')> : r<++>s"; |
|
543 |
by (fast_tac (set_cs addSIs [split RS equalityD2 RS subsetD]) 1); |
|
544 |
val dsum_In0I = result(); |
|
545 |
||
546 |
goalw Univ.thy [dsum_def] "!!r. <N,N'>:s ==> <In1(N), In1(N')> : r<++>s"; |
|
547 |
by (fast_tac (set_cs addSIs [split RS equalityD2 RS subsetD]) 1); |
|
548 |
val dsum_In1I = result(); |
|
549 |
||
550 |
val major::prems = goalw Univ.thy [dsum_def] |
|
551 |
"[| w : r<++>s; \ |
|
552 |
\ !!x x'. [| <x,x'> : r; w = <In0(x), In0(x')> |] ==> P; \ |
|
553 |
\ !!y y'. [| <y,y'> : s; w = <In1(y), In1(y')> |] ==> P \ |
|
554 |
\ |] ==> P"; |
|
555 |
by (rtac (major RS UnE) 1); |
|
556 |
by (safe_tac set_cs); |
|
557 |
by (res_inst_tac [("p","u")] PairE 1); |
|
558 |
by (res_inst_tac [("p","v")] PairE 2); |
|
559 |
by (safe_tac (set_cs addSEs prems |
|
560 |
addSDs [split RS equalityD1 RS subsetD])); |
|
561 |
val dsumE = result(); |
|
562 |
||
563 |
||
564 |
(*** Monotonicity ***) |
|
565 |
||
566 |
goalw Univ.thy [dprod_def] "!!r s. [| r<=r'; s<=s' |] ==> r<**>s <= r'<**>s'"; |
|
567 |
by (fast_tac set_cs 1); |
|
568 |
val dprod_mono = result(); |
|
569 |
||
570 |
goalw Univ.thy [dsum_def] "!!r s. [| r<=r'; s<=s' |] ==> r<++>s <= r'<++>s'"; |
|
571 |
by (fast_tac set_cs 1); |
|
572 |
val dsum_mono = result(); |
|
573 |
||
574 |
||
575 |
(*** Bounding theorems ***) |
|
576 |
||
577 |
goal Univ.thy "diag(A) <= Sigma(A,%x.A)"; |
|
578 |
by (fast_tac (set_cs addIs [SigmaI] addSEs [diagE]) 1); |
|
579 |
val diag_subset_Sigma = result(); |
|
580 |
||
581 |
val prems = goal Univ.thy |
|
582 |
"[| r <= Sigma(A,%x.B); s <= Sigma(C,%x.D) |] ==> \ |
|
583 |
\ (r<**>s) <= Sigma(A<*>C, %z. B<*>D)"; |
|
584 |
by (cut_facts_tac prems 1); |
|
585 |
by (fast_tac (set_cs addSIs [SigmaI,uprodI] |
|
586 |
addSEs [dprodE,SigmaE2]) 1); |
|
587 |
val dprod_subset_Sigma = result(); |
|
588 |
||
589 |
goal Univ.thy |
|
590 |
"!!r s. [| r <= Sigma(A,B); s <= Sigma(C,D) |] ==> \ |
|
591 |
\ (r<**>s) <= Sigma(A<*>C, %z. Split(z, %x y. B(x)<*>D(y)))"; |
|
592 |
by (safe_tac (set_cs addSIs [SigmaI,uprodI] addSEs [dprodE])); |
|
593 |
by (stac Split 3); |
|
594 |
by (ALLGOALS (fast_tac (set_cs addSIs [uprodI] addSEs [SigmaE2]))); |
|
595 |
val dprod_subset_Sigma2 = result(); |
|
596 |
||
597 |
goal Univ.thy |
|
598 |
"!!r s. [| r <= Sigma(A,%x.B); s <= Sigma(C,%x.D) |] ==> \ |
|
599 |
\ (r<++>s) <= Sigma(A<+>C, %z. B<+>D)"; |
|
600 |
by (fast_tac (set_cs addSIs [SigmaI,usum_In0I,usum_In1I] |
|
601 |
addSEs [dsumE,SigmaE2]) 1); |
|
602 |
val dsum_subset_Sigma = result(); |
|
603 |
||
604 |
||
605 |
(*** Domain ***) |
|
606 |
||
607 |
goal Univ.thy "fst `` diag(A) = A"; |
|
608 |
by (fast_tac (set_cs addIs [equalityI, fst_imageI, diagI] |
|
609 |
addSEs [fst_imageE, Pair_inject, diagE]) 1); |
|
610 |
val fst_image_diag = result(); |
|
611 |
||
612 |
goal Univ.thy "fst `` (r<**>s) = (fst``r) <*> (fst``s)"; |
|
613 |
by (fast_tac (set_cs addIs [equalityI, fst_imageI, uprodI, dprodI] |
|
614 |
addSEs [fst_imageE, Pair_inject, uprodE, dprodE]) 1); |
|
615 |
val fst_image_dprod = result(); |
|
616 |
||
617 |
goal Univ.thy "fst `` (r<++>s) = (fst``r) <+> (fst``s)"; |
|
618 |
by (fast_tac (set_cs addIs [equalityI, fst_imageI, usum_In0I, usum_In1I, |
|
619 |
dsum_In0I, dsum_In1I] |
|
620 |
addSEs [fst_imageE, Pair_inject, usumE, dsumE]) 1); |
|
621 |
val fst_image_dsum = result(); |
|
622 |
||
623 |
val fst_image_simps = [fst_image_diag, fst_image_dprod, fst_image_dsum]; |
|
624 |
val fst_image_ss = univ_ss addsimps fst_image_simps; |
|
625 |
||
626 |
val univ_cs = |
|
627 |
set_cs addSIs [SigmaI,uprodI,dprodI] |
|
628 |
addIs [usum_In0I,usum_In1I,dsum_In0I,dsum_In1I] |
|
629 |
addSEs [diagE,uprodE,dprodE,usumE,dsumE,SigmaE2,Pair_inject]; |