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(* 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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qed "Least_equality";
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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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qed "LeastI";
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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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qed "Least_le";
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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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qed "not_less_Least";
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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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qed "apfst";
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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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qed "apfstE";
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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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qed "Push_inject1";
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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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qed "Push_inject2";
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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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qed "Push_inject";
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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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qed "Push_neq_K0";
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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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qed "inj_Rep_Node";
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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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qed "inj_onto_Abs_Node";
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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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qed "Node_K0_I";
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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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qed "Node_Push_I";
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(*** Distinctness of constructors ***)
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(** Scons vs Atom **)
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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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qed "Scons_not_Atom";
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bind_thm ("Atom_not_Scons", (Scons_not_Atom RS not_sym));
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bind_thm ("Scons_neq_Atom", (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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qed "inj_Atom";
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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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qed "inj_Leaf";
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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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qed "inj_Numb";
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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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qed "Push_Node_inject";
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(** Injectiveness of Scons **)
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val [major] = goalw Univ.thy [Scons_def] "M$N <= M'$N' ==> M<=M'";
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by (cut_facts_tac [major] 1);
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by (fast_tac (set_cs addSDs [Suc_inject]
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addSEs [Push_Node_inject, Zero_neq_Suc]) 1);
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qed "Scons_inject_lemma1";
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val [major] = goalw Univ.thy [Scons_def] "M$N <= M'$N' ==> N<=N'";
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by (cut_facts_tac [major] 1);
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by (fast_tac (set_cs addSDs [Suc_inject]
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addSEs [Push_Node_inject, Suc_neq_Zero]) 1);
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qed "Scons_inject_lemma2";
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val [major] = goal Univ.thy "M$N = M'$N' ==> M=M'";
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by (rtac (major RS equalityE) 1);
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by (REPEAT (ares_tac [equalityI, Scons_inject_lemma1] 1));
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qed "Scons_inject1";
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val [major] = goal Univ.thy "M$N = M'$N' ==> N=N'";
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by (rtac (major RS equalityE) 1);
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by (REPEAT (ares_tac [equalityI, Scons_inject_lemma2] 1));
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qed "Scons_inject2";
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val [major,minor] = goal Univ.thy
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"[| M$N = M'$N'; [| M=M'; N=N' |] ==> P \
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\ |] ==> P";
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by (rtac ((major RS Scons_inject2) RS ((major RS Scons_inject1) RS minor)) 1);
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qed "Scons_inject";
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(*rewrite rules*)
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goal Univ.thy "(Atom(a)=Atom(b)) = (a=b)";
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by (fast_tac (HOL_cs addSEs [Atom_inject]) 1);
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qed "Atom_Atom_eq";
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goal Univ.thy "(M$N = M'$N') = (M=M' & N=N')";
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by (fast_tac (HOL_cs addSEs [Scons_inject]) 1);
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qed "Scons_Scons_eq";
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(*** Distinctness involving Leaf and Numb ***)
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(** Scons vs Leaf **)
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goalw Univ.thy [Leaf_def,o_def] "(M$N) ~= Leaf(a)";
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by (rtac Scons_not_Atom 1);
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qed "Scons_not_Leaf";
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bind_thm ("Leaf_not_Scons", (Scons_not_Leaf RS not_sym));
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bind_thm ("Scons_neq_Leaf", (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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qed "Scons_not_Numb";
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bind_thm ("Numb_not_Scons", (Scons_not_Numb RS not_sym));
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bind_thm ("Scons_neq_Numb", (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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qed "Leaf_not_Numb";
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bind_thm ("Numb_not_Leaf", (Leaf_not_Numb RS not_sym));
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bind_thm ("Leaf_neq_Numb", (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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qed "ndepth_K0";
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goal Univ.thy "k < Suc(LEAST x. f(x)=0) --> nat_case (Suc i) f k ~= 0";
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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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qed "ndepth_Push_lemma";
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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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qed "ndepth_Push_Node";
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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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qed "ntrunc_0";
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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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qed "ntrunc_Atom";
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goalw Univ.thy [Leaf_def,o_def] "ntrunc (Suc k) (Leaf a) = Leaf(a)";
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by (rtac ntrunc_Atom 1);
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qed "ntrunc_Leaf";
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goalw Univ.thy [Numb_def,o_def] "ntrunc (Suc k) (Numb i) = Numb(i)";
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by (rtac ntrunc_Atom 1);
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qed "ntrunc_Numb";
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goalw Univ.thy [Scons_def,ntrunc_def]
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"ntrunc (Suc k) (M$N) = ntrunc k M $ ntrunc k N";
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by (safe_tac (set_cs addSIs [equalityI,imageI]));
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by (REPEAT (stac ndepth_Push_Node 3 THEN etac Suc_mono 3));
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by (REPEAT (rtac Suc_less_SucD 1 THEN
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rtac (ndepth_Push_Node RS subst) 1 THEN
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assume_tac 1));
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qed "ntrunc_Scons";
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(** Injection nodes **)
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goalw Univ.thy [In0_def] "ntrunc (Suc 0) (In0 M) = {}";
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by (simp_tac (univ_ss addsimps [ntrunc_Scons,ntrunc_0]) 1);
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by (rewtac Scons_def);
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by (safe_tac (set_cs addSIs [equalityI]));
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qed "ntrunc_one_In0";
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goalw Univ.thy [In0_def]
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"ntrunc (Suc (Suc k)) (In0 M) = In0 (ntrunc (Suc k) M)";
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by (simp_tac (univ_ss addsimps [ntrunc_Scons,ntrunc_Numb]) 1);
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qed "ntrunc_In0";
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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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by (rewtac Scons_def);
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by (safe_tac (set_cs addSIs [equalityI]));
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qed "ntrunc_one_In1";
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goalw Univ.thy [In1_def]
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"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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qed "ntrunc_In1";
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(*** Cartesian Product ***)
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goalw Univ.thy [uprod_def] "!!M N. [| M:A; N:B |] ==> (M$N) : A<*>B";
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345 |
by (REPEAT (ares_tac [singletonI,UN_I] 1));
|
|
346 |
qed "uprodI";
|
|
347 |
|
|
348 |
(*The general elimination rule*)
|
|
349 |
val major::prems = goalw Univ.thy [uprod_def]
|
|
350 |
"[| c : A<*>B; \
|
|
351 |
\ !!x y. [| x:A; y:B; c=x$y |] ==> P \
|
|
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 |
qed "uprodE";
|
|
357 |
|
|
358 |
(*Elimination of a pair -- introduces no eigenvariables*)
|
|
359 |
val prems = goal Univ.thy
|
|
360 |
"[| (M$N) : A<*>B; [| M:A; N:B |] ==> P \
|
|
361 |
\ |] ==> P";
|
|
362 |
by (rtac uprodE 1);
|
|
363 |
by (REPEAT (ares_tac prems 1 ORELSE eresolve_tac [Scons_inject,ssubst] 1));
|
|
364 |
qed "uprodE2";
|
|
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 |
qed "usum_In0I";
|
|
372 |
|
|
373 |
goalw Univ.thy [usum_def] "!!N. N:B ==> In1(N) : A<+>B";
|
|
374 |
by (fast_tac set_cs 1);
|
|
375 |
qed "usum_In1I";
|
|
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 |
qed "usumE";
|
|
386 |
|
|
387 |
|
|
388 |
(** Injection **)
|
|
389 |
|
|
390 |
goalw Univ.thy [In0_def,In1_def] "In0(M) ~= In1(N)";
|
|
391 |
by (rtac notI 1);
|
|
392 |
by (etac (Scons_inject1 RS Numb_inject RS Zero_neq_Suc) 1);
|
|
393 |
qed "In0_not_In1";
|
|
394 |
|
|
395 |
bind_thm ("In1_not_In0", (In0_not_In1 RS not_sym));
|
|
396 |
bind_thm ("In0_neq_In1", (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 |
qed "In0_inject";
|
|
402 |
|
|
403 |
val [major] = goalw Univ.thy [In1_def] "In1(M) = In1(N) ==> M=N";
|
|
404 |
by (rtac (major RS Scons_inject2) 1);
|
|
405 |
qed "In1_inject";
|
|
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 |
qed "ntrunc_subsetI";
|
|
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 |
qed "ntrunc_subsetD";
|
|
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 |
qed "ntrunc_equality";
|
|
428 |
|
|
429 |
val [major] = goalw Univ.thy [o_def]
|
|
430 |
"[| !!k. (ntrunc(k) o h1) = (ntrunc(k) o h2) |] ==> h1=h2";
|
|
431 |
by (rtac (ntrunc_equality RS ext) 1);
|
|
432 |
by (rtac (major RS fun_cong) 1);
|
|
433 |
qed "ntrunc_o_equality";
|
|
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 |
qed "uprod_mono";
|
|
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 |
qed "usum_mono";
|
|
444 |
|
|
445 |
goalw Univ.thy [Scons_def] "!!M N. [| M<=M'; N<=N' |] ==> M$N <= M'$N'";
|
|
446 |
by (fast_tac set_cs 1);
|
|
447 |
qed "Scons_mono";
|
|
448 |
|
|
449 |
goalw Univ.thy [In0_def] "!!M N. M<=N ==> In0(M) <= In0(N)";
|
|
450 |
by (REPEAT (ares_tac [subset_refl,Scons_mono] 1));
|
|
451 |
qed "In0_mono";
|
|
452 |
|
|
453 |
goalw Univ.thy [In1_def] "!!M N. M<=N ==> In1(M) <= In1(N)";
|
|
454 |
by (REPEAT (ares_tac [subset_refl,Scons_mono] 1));
|
|
455 |
qed "In1_mono";
|
|
456 |
|
|
457 |
|
|
458 |
(*** Split and Case ***)
|
|
459 |
|
|
460 |
goalw Univ.thy [Split_def] "Split c (M$N) = c M N";
|
|
461 |
by (fast_tac (set_cs addIs [select_equality] addEs [Scons_inject]) 1);
|
|
462 |
qed "Split";
|
|
463 |
|
|
464 |
goalw Univ.thy [Case_def] "Case c d (In0 M) = c(M)";
|
|
465 |
by (fast_tac (set_cs addIs [select_equality]
|
|
466 |
addEs [make_elim In0_inject, In0_neq_In1]) 1);
|
|
467 |
qed "Case_In0";
|
|
468 |
|
|
469 |
goalw Univ.thy [Case_def] "Case c d (In1 N) = d(N)";
|
|
470 |
by (fast_tac (set_cs addIs [select_equality]
|
|
471 |
addEs [make_elim In1_inject, In1_neq_In0]) 1);
|
|
472 |
qed "Case_In1";
|
|
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 |
qed "ntrunc_UN1";
|
|
479 |
|
|
480 |
goalw Univ.thy [Scons_def] "(UN x.f(x)) $ M = (UN x. f(x) $ M)";
|
|
481 |
by (fast_tac (set_cs addIs [equalityI]) 1);
|
|
482 |
qed "Scons_UN1_x";
|
|
483 |
|
|
484 |
goalw Univ.thy [Scons_def] "M $ (UN x.f(x)) = (UN x. M $ f(x))";
|
|
485 |
by (fast_tac (set_cs addIs [equalityI]) 1);
|
|
486 |
qed "Scons_UN1_y";
|
|
487 |
|
|
488 |
goalw Univ.thy [In0_def] "In0(UN x.f(x)) = (UN x. In0(f(x)))";
|
|
489 |
br Scons_UN1_y 1;
|
|
490 |
qed "In0_UN1";
|
|
491 |
|
|
492 |
goalw Univ.thy [In1_def] "In1(UN x.f(x)) = (UN x. In1(f(x)))";
|
|
493 |
br Scons_UN1_y 1;
|
|
494 |
qed "In1_UN1";
|
|
495 |
|
|
496 |
|
|
497 |
(*** Equality : the diagonal relation ***)
|
|
498 |
|
|
499 |
goalw Univ.thy [diag_def] "!!a A. [| a=b; a:A |] ==> <a,b> : diag(A)";
|
|
500 |
by (fast_tac set_cs 1);
|
|
501 |
qed "diag_eqI";
|
|
502 |
|
|
503 |
val diagI = refl RS diag_eqI |> standard;
|
|
504 |
|
|
505 |
(*The general elimination rule*)
|
|
506 |
val major::prems = goalw Univ.thy [diag_def]
|
|
507 |
"[| c : diag(A); \
|
|
508 |
\ !!x y. [| x:A; c = <x,x> |] ==> P \
|
|
509 |
\ |] ==> P";
|
|
510 |
by (rtac (major RS UN_E) 1);
|
|
511 |
by (REPEAT (eresolve_tac [asm_rl,singletonE] 1 ORELSE resolve_tac prems 1));
|
|
512 |
qed "diagE";
|
|
513 |
|
|
514 |
(*** Equality for Cartesian Product ***)
|
|
515 |
|
|
516 |
goalw Univ.thy [dprod_def]
|
|
517 |
"!!r s. [| <M,M'>:r; <N,N'>:s |] ==> <M$N, M'$N'> : r<**>s";
|
|
518 |
by (fast_tac prod_cs 1);
|
|
519 |
qed "dprodI";
|
|
520 |
|
|
521 |
(*The general elimination rule*)
|
|
522 |
val major::prems = goalw Univ.thy [dprod_def]
|
|
523 |
"[| c : r<**>s; \
|
|
524 |
\ !!x y x' y'. [| <x,x'> : r; <y,y'> : s; c = <x$y,x'$y'> |] ==> P \
|
|
525 |
\ |] ==> P";
|
|
526 |
by (cut_facts_tac [major] 1);
|
|
527 |
by (REPEAT_FIRST (eresolve_tac [asm_rl, UN_E, mem_splitE, singletonE]));
|
|
528 |
by (REPEAT (ares_tac prems 1 ORELSE hyp_subst_tac 1));
|
|
529 |
qed "dprodE";
|
|
530 |
|
|
531 |
|
|
532 |
(*** Equality for Disjoint Sum ***)
|
|
533 |
|
|
534 |
goalw Univ.thy [dsum_def] "!!r. <M,M'>:r ==> <In0(M), In0(M')> : r<++>s";
|
|
535 |
by (fast_tac prod_cs 1);
|
|
536 |
qed "dsum_In0I";
|
|
537 |
|
|
538 |
goalw Univ.thy [dsum_def] "!!r. <N,N'>:s ==> <In1(N), In1(N')> : r<++>s";
|
|
539 |
by (fast_tac prod_cs 1);
|
|
540 |
qed "dsum_In1I";
|
|
541 |
|
|
542 |
val major::prems = goalw Univ.thy [dsum_def]
|
|
543 |
"[| w : r<++>s; \
|
|
544 |
\ !!x x'. [| <x,x'> : r; w = <In0(x), In0(x')> |] ==> P; \
|
|
545 |
\ !!y y'. [| <y,y'> : s; w = <In1(y), In1(y')> |] ==> P \
|
|
546 |
\ |] ==> P";
|
|
547 |
by (cut_facts_tac [major] 1);
|
|
548 |
by (REPEAT_FIRST (eresolve_tac [asm_rl, UN_E, UnE, mem_splitE, singletonE]));
|
|
549 |
by (DEPTH_SOLVE (ares_tac prems 1 ORELSE hyp_subst_tac 1));
|
|
550 |
qed "dsumE";
|
|
551 |
|
|
552 |
|
|
553 |
val univ_cs =
|
|
554 |
prod_cs addSIs [diagI, uprodI, dprodI]
|
|
555 |
addIs [usum_In0I, usum_In1I, dsum_In0I, dsum_In1I]
|
|
556 |
addSEs [diagE, uprodE, dprodE, usumE, dsumE];
|
|
557 |
|
|
558 |
|
|
559 |
(*** Monotonicity ***)
|
|
560 |
|
|
561 |
goal Univ.thy "!!r s. [| r<=r'; s<=s' |] ==> r<**>s <= r'<**>s'";
|
|
562 |
by (fast_tac univ_cs 1);
|
|
563 |
qed "dprod_mono";
|
|
564 |
|
|
565 |
goal Univ.thy "!!r s. [| r<=r'; s<=s' |] ==> r<++>s <= r'<++>s'";
|
|
566 |
by (fast_tac univ_cs 1);
|
|
567 |
qed "dsum_mono";
|
|
568 |
|
|
569 |
|
|
570 |
(*** Bounding theorems ***)
|
|
571 |
|
|
572 |
goal Univ.thy "diag(A) <= Sigma A (%x.A)";
|
|
573 |
by (fast_tac univ_cs 1);
|
|
574 |
qed "diag_subset_Sigma";
|
|
575 |
|
|
576 |
goal Univ.thy "(Sigma A (%x.B) <**> Sigma C (%x.D)) <= Sigma (A<*>C) (%z. B<*>D)";
|
|
577 |
by (fast_tac univ_cs 1);
|
|
578 |
qed "dprod_Sigma";
|
|
579 |
|
|
580 |
val dprod_subset_Sigma = [dprod_mono, dprod_Sigma] MRS subset_trans |>standard;
|
|
581 |
|
|
582 |
(*Dependent version*)
|
|
583 |
goal Univ.thy
|
|
584 |
"(Sigma A B <**> Sigma C D) <= Sigma (A<*>C) (Split(%x y. B(x)<*>D(y)))";
|
|
585 |
by (safe_tac univ_cs);
|
|
586 |
by (stac Split 1);
|
|
587 |
by (fast_tac univ_cs 1);
|
|
588 |
qed "dprod_subset_Sigma2";
|
|
589 |
|
|
590 |
goal Univ.thy "(Sigma A (%x.B) <++> Sigma C (%x.D)) <= Sigma (A<+>C) (%z. B<+>D)";
|
|
591 |
by (fast_tac univ_cs 1);
|
|
592 |
qed "dsum_Sigma";
|
|
593 |
|
|
594 |
val dsum_subset_Sigma = [dsum_mono, dsum_Sigma] MRS subset_trans |> standard;
|
|
595 |
|
|
596 |
|
|
597 |
(*** Domain ***)
|
|
598 |
|
|
599 |
goal Univ.thy "fst `` diag(A) = A";
|
|
600 |
by (fast_tac (prod_cs addIs [equalityI, diagI] addSEs [diagE]) 1);
|
|
601 |
qed "fst_image_diag";
|
|
602 |
|
|
603 |
goal Univ.thy "fst `` (r<**>s) = (fst``r) <*> (fst``s)";
|
|
604 |
by (fast_tac (prod_cs addIs [equalityI, uprodI, dprodI]
|
|
605 |
addSEs [uprodE, dprodE]) 1);
|
|
606 |
qed "fst_image_dprod";
|
|
607 |
|
|
608 |
goal Univ.thy "fst `` (r<++>s) = (fst``r) <+> (fst``s)";
|
|
609 |
by (fast_tac (prod_cs addIs [equalityI, usum_In0I, usum_In1I,
|
|
610 |
dsum_In0I, dsum_In1I]
|
|
611 |
addSEs [usumE, dsumE]) 1);
|
|
612 |
qed "fst_image_dsum";
|
|
613 |
|
|
614 |
val fst_image_simps = [fst_image_diag, fst_image_dprod, fst_image_dsum];
|
|
615 |
val fst_image_ss = univ_ss addsimps fst_image_simps;
|