src/HOL/Univ.ML
author paulson
Thu, 12 Sep 1996 10:40:05 +0200
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Tidied many proofs, using AddIffs to let equivalences take the place of separate Intr and Elim rules. Also deleted most named clasets.
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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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(** 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_conv";
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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_conv 1);
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qed "apfst_convE";
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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 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 (!claset addSIs [apfst_conv, 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 apfst_convE, 
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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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(*** Injectiveness ***)
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(** Atomic nodes **)
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goalw Univ.thy [Atom_def, inj_def] "inj(Atom)";
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by (fast_tac (!claset addSIs [Node_K0_I] addSDs [Abs_Node_inject]) 1);
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qed "inj_Atom";
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val Atom_inject = inj_Atom RS injD;
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goal Univ.thy "(Atom(a)=Atom(b)) = (a=b)";
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by (fast_tac (!claset addSEs [Atom_inject]) 1);
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qed "Atom_Atom_eq";
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AddIffs [Atom_Atom_eq];
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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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AddSDs [Leaf_inject];
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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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AddSDs [Numb_inject];
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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 apfst_convE) 1);
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by (REPEAT (resolve_tac [Rep_Node RS Node_Push_I] 1));
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by (etac (sym RS apfst_convE) 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 (!claset addSEs [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'$N' ==> M<=M'";
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by (cut_facts_tac [major] 1);
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by (fast_tac (!claset addSEs [Push_Node_inject]) 1);
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qed "Scons_inject_lemma1";
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val [major] = goalw Univ.thy [Scons_def] "M$N <= M'$N' ==> N<=N'";
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by (cut_facts_tac [major] 1);
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by (fast_tac (!claset addSEs [Push_Node_inject]) 1);
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qed "Scons_inject_lemma2";
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val [major] = goal Univ.thy "M$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'$N';  [| M=M';  N=N' |] ==> P \
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\    |] ==> P";
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by (rtac ((major RS Scons_inject2) RS ((major RS Scons_inject1) RS minor)) 1);
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qed "Scons_inject";
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AddSDs [Scons_inject];
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goal Univ.thy "(M$N = M'$N') = (M=M' & N=N')";
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by (fast_tac (!claset addSEs [Scons_inject]) 1);
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qed "Scons_Scons_eq";
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(*** Distinctness involving Leaf and Numb ***)
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(** Scons vs Leaf **)
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goalw Univ.thy [Leaf_def,o_def] "(M$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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AddIffs [Scons_not_Leaf, Leaf_not_Scons];
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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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AddIffs [Scons_not_Numb, Numb_not_Scons];
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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 (!simpset addsimps [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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AddIffs [Leaf_not_Numb, Numb_not_Leaf];
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(*** ndepth -- the depth of a node ***)
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Addsimps [apfst_conv];
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AddIffs  [Scons_not_Atom, Atom_not_Scons, Scons_Scons_eq];
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goalw Univ.thy [ndepth_def] "ndepth (Abs_Node((%k.0, x))) = 0";
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by (EVERY1[stac (Node_K0_I RS Abs_Node_inverse), stac split]);
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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);
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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 (!claset));
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by (etac ssubst 1);  (*instantiates type variables!*)
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by (Simp_tac 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 (Fast_tac 1);
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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 (fast_tac (!claset addss (!simpset addsimps [ndepth_K0])) 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 ((claset_of "Fun") 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 (!simpset addsimps [ntrunc_Scons,ntrunc_0]) 1);
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by (rewtac Scons_def);
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by (Fast_tac 1);
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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 (!simpset 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 (!simpset addsimps [ntrunc_Scons,ntrunc_0]) 1);
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by (rewtac Scons_def);
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by (Fast_tac 1);
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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 (!simpset 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$N) : A<*>B";
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by (REPEAT (ares_tac [singletonI,UN_I] 1));
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qed "uprodI";
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(*The general elimination rule*)
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val major::prems = goalw Univ.thy [uprod_def]
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    "[| c : A<*>B;  \
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\       !!x y. [| x:A;  y:B;  c=x$y |] ==> P \
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\    |] ==> P";
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   307
by (cut_facts_tac [major] 1);
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clasohm
parents:
diff changeset
   308
by (REPEAT (eresolve_tac [asm_rl,singletonE,UN_E] 1
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parents:
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   309
     ORELSE resolve_tac prems 1));
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   310
qed "uprodE";
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parents:
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   311
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parents:
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   312
(*Elimination of a pair -- introduces no eigenvariables*)
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parents:
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   313
val prems = goal Univ.thy
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parents:
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   314
    "[| (M$N) : A<*>B;      [| M:A;  N:B |] ==> P   \
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parents:
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   315
\    |] ==> P";
ff1574a81019 new version of HOL with curried function application
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parents:
diff changeset
   316
by (rtac uprodE 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   317
by (REPEAT (ares_tac prems 1 ORELSE eresolve_tac [Scons_inject,ssubst] 1));
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parents:
diff changeset
   318
qed "uprodE2";
ff1574a81019 new version of HOL with curried function application
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parents:
diff changeset
   319
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   320
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   321
(*** Disjoint Sum ***)
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parents:
diff changeset
   322
ff1574a81019 new version of HOL with curried function application
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parents:
diff changeset
   323
goalw Univ.thy [usum_def] "!!M. M:A ==> In0(M) : A<+>B";
1760
6f41a494f3b1 Replaced fast_tac by Fast_tac (which uses default claset)
berghofe
parents: 1642
diff changeset
   324
by (Fast_tac 1);
923
ff1574a81019 new version of HOL with curried function application
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parents:
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   325
qed "usum_In0I";
ff1574a81019 new version of HOL with curried function application
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parents:
diff changeset
   326
ff1574a81019 new version of HOL with curried function application
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parents:
diff changeset
   327
goalw Univ.thy [usum_def] "!!N. N:B ==> In1(N) : A<+>B";
1760
6f41a494f3b1 Replaced fast_tac by Fast_tac (which uses default claset)
berghofe
parents: 1642
diff changeset
   328
by (Fast_tac 1);
923
ff1574a81019 new version of HOL with curried function application
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parents:
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   329
qed "usum_In1I";
ff1574a81019 new version of HOL with curried function application
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parents:
diff changeset
   330
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   331
val major::prems = goalw Univ.thy [usum_def]
ff1574a81019 new version of HOL with curried function application
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parents:
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   332
    "[| u : A<+>B;  \
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   333
\       !!x. [| x:A;  u=In0(x) |] ==> P; \
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   334
\       !!y. [| y:B;  u=In1(y) |] ==> P \
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   335
\    |] ==> P";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   336
by (rtac (major RS UnE) 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   337
by (REPEAT (rtac refl 1 
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   338
     ORELSE eresolve_tac (prems@[imageE,ssubst]) 1));
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   339
qed "usumE";
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parents:
diff changeset
   340
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   341
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
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   342
(** Injection **)
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parents:
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   343
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parents:
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   344
goalw Univ.thy [In0_def,In1_def] "In0(M) ~= In1(N)";
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parents:
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   345
by (rtac notI 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   346
by (etac (Scons_inject1 RS Numb_inject RS Zero_neq_Suc) 1);
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parents:
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   347
qed "In0_not_In1";
ff1574a81019 new version of HOL with curried function application
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parents:
diff changeset
   348
1985
84cf16192e03 Tidied many proofs, using AddIffs to let equivalences take
paulson
parents: 1786
diff changeset
   349
bind_thm ("In1_not_In0", In0_not_In1 RS not_sym);
84cf16192e03 Tidied many proofs, using AddIffs to let equivalences take
paulson
parents: 1786
diff changeset
   350
84cf16192e03 Tidied many proofs, using AddIffs to let equivalences take
paulson
parents: 1786
diff changeset
   351
AddIffs [In0_not_In1, In1_not_In0];
923
ff1574a81019 new version of HOL with curried function application
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parents:
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   352
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   353
val [major] = goalw Univ.thy [In0_def] "In0(M) = In0(N) ==>  M=N";
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clasohm
parents:
diff changeset
   354
by (rtac (major RS Scons_inject2) 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   355
qed "In0_inject";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   356
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   357
val [major] = goalw Univ.thy [In1_def] "In1(M) = In1(N) ==>  M=N";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   358
by (rtac (major RS Scons_inject2) 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   359
qed "In1_inject";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   360
1985
84cf16192e03 Tidied many proofs, using AddIffs to let equivalences take
paulson
parents: 1786
diff changeset
   361
AddSDs [In0_inject, In1_inject];
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   362
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   363
(*** proving equality of sets and functions using ntrunc ***)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   364
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   365
goalw Univ.thy [ntrunc_def] "ntrunc k M <= M";
1760
6f41a494f3b1 Replaced fast_tac by Fast_tac (which uses default claset)
berghofe
parents: 1642
diff changeset
   366
by (Fast_tac 1);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   367
qed "ntrunc_subsetI";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   368
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   369
val [major] = goalw Univ.thy [ntrunc_def]
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   370
    "(!!k. ntrunc k M <= N) ==> M<=N";
1760
6f41a494f3b1 Replaced fast_tac by Fast_tac (which uses default claset)
berghofe
parents: 1642
diff changeset
   371
by (fast_tac (!claset addIs [less_add_Suc1, less_add_Suc2, 
1465
5d7a7e439cec expanded tabs
clasohm
parents: 1264
diff changeset
   372
                            major RS subsetD]) 1);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   373
qed "ntrunc_subsetD";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   374
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   375
(*A generalized form of the take-lemma*)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   376
val [major] = goal Univ.thy "(!!k. ntrunc k M = ntrunc k N) ==> M=N";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   377
by (rtac equalityI 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   378
by (ALLGOALS (rtac ntrunc_subsetD));
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   379
by (ALLGOALS (rtac (ntrunc_subsetI RSN (2, subset_trans))));
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   380
by (rtac (major RS equalityD1) 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   381
by (rtac (major RS equalityD2) 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   382
qed "ntrunc_equality";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   383
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   384
val [major] = goalw Univ.thy [o_def]
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   385
    "[| !!k. (ntrunc(k) o h1) = (ntrunc(k) o h2) |] ==> h1=h2";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   386
by (rtac (ntrunc_equality RS ext) 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   387
by (rtac (major RS fun_cong) 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   388
qed "ntrunc_o_equality";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   389
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   390
(*** Monotonicity ***)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   391
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   392
goalw Univ.thy [uprod_def] "!!A B. [| A<=A';  B<=B' |] ==> A<*>B <= A'<*>B'";
1760
6f41a494f3b1 Replaced fast_tac by Fast_tac (which uses default claset)
berghofe
parents: 1642
diff changeset
   393
by (Fast_tac 1);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   394
qed "uprod_mono";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   395
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   396
goalw Univ.thy [usum_def] "!!A B. [| A<=A';  B<=B' |] ==> A<+>B <= A'<+>B'";
1760
6f41a494f3b1 Replaced fast_tac by Fast_tac (which uses default claset)
berghofe
parents: 1642
diff changeset
   397
by (Fast_tac 1);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   398
qed "usum_mono";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   399
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   400
goalw Univ.thy [Scons_def] "!!M N. [| M<=M';  N<=N' |] ==> M$N <= M'$N'";
1760
6f41a494f3b1 Replaced fast_tac by Fast_tac (which uses default claset)
berghofe
parents: 1642
diff changeset
   401
by (Fast_tac 1);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   402
qed "Scons_mono";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   403
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   404
goalw Univ.thy [In0_def] "!!M N. M<=N ==> In0(M) <= In0(N)";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   405
by (REPEAT (ares_tac [subset_refl,Scons_mono] 1));
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   406
qed "In0_mono";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   407
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   408
goalw Univ.thy [In1_def] "!!M N. M<=N ==> In1(M) <= In1(N)";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   409
by (REPEAT (ares_tac [subset_refl,Scons_mono] 1));
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   410
qed "In1_mono";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   411
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   412
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   413
(*** Split and Case ***)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   414
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   415
goalw Univ.thy [Split_def] "Split c (M$N) = c M N";
1985
84cf16192e03 Tidied many proofs, using AddIffs to let equivalences take
paulson
parents: 1786
diff changeset
   416
by (fast_tac (!claset addIs [select_equality]) 1);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   417
qed "Split";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   418
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   419
goalw Univ.thy [Case_def] "Case c d (In0 M) = c(M)";
1985
84cf16192e03 Tidied many proofs, using AddIffs to let equivalences take
paulson
parents: 1786
diff changeset
   420
by (fast_tac (!claset addIs [select_equality]) 1);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   421
qed "Case_In0";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   422
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   423
goalw Univ.thy [Case_def] "Case c d (In1 N) = d(N)";
1985
84cf16192e03 Tidied many proofs, using AddIffs to let equivalences take
paulson
parents: 1786
diff changeset
   424
by (fast_tac (!claset addIs [select_equality]) 1);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   425
qed "Case_In1";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   426
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   427
(**** UN x. B(x) rules ****)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   428
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   429
goalw Univ.thy [ntrunc_def] "ntrunc k (UN x.f(x)) = (UN x. ntrunc k (f x))";
1761
29e08d527ba1 Removed equalityI from some proofs (because it is now included
berghofe
parents: 1760
diff changeset
   430
by (Fast_tac 1);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   431
qed "ntrunc_UN1";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   432
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   433
goalw Univ.thy [Scons_def] "(UN x.f(x)) $ M = (UN x. f(x) $ M)";
1761
29e08d527ba1 Removed equalityI from some proofs (because it is now included
berghofe
parents: 1760
diff changeset
   434
by (Fast_tac 1);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   435
qed "Scons_UN1_x";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   436
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   437
goalw Univ.thy [Scons_def] "M $ (UN x.f(x)) = (UN x. M $ f(x))";
1761
29e08d527ba1 Removed equalityI from some proofs (because it is now included
berghofe
parents: 1760
diff changeset
   438
by (Fast_tac 1);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   439
qed "Scons_UN1_y";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   440
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   441
goalw Univ.thy [In0_def] "In0(UN x.f(x)) = (UN x. In0(f(x)))";
1465
5d7a7e439cec expanded tabs
clasohm
parents: 1264
diff changeset
   442
by (rtac Scons_UN1_y 1);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   443
qed "In0_UN1";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   444
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   445
goalw Univ.thy [In1_def] "In1(UN x.f(x)) = (UN x. In1(f(x)))";
1465
5d7a7e439cec expanded tabs
clasohm
parents: 1264
diff changeset
   446
by (rtac Scons_UN1_y 1);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   447
qed "In1_UN1";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   448
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   449
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   450
(*** Equality : the diagonal relation ***)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   451
972
e61b058d58d2 changed syntax of tuples from <..., ...> to (..., ...)
clasohm
parents: 923
diff changeset
   452
goalw Univ.thy [diag_def] "!!a A. [| a=b;  a:A |] ==> (a,b) : diag(A)";
1760
6f41a494f3b1 Replaced fast_tac by Fast_tac (which uses default claset)
berghofe
parents: 1642
diff changeset
   453
by (Fast_tac 1);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   454
qed "diag_eqI";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   455
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   456
val diagI = refl RS diag_eqI |> standard;
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   457
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   458
(*The general elimination rule*)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   459
val major::prems = goalw Univ.thy [diag_def]
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   460
    "[| c : diag(A);  \
972
e61b058d58d2 changed syntax of tuples from <..., ...> to (..., ...)
clasohm
parents: 923
diff changeset
   461
\       !!x y. [| x:A;  c = (x,x) |] ==> P \
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   462
\    |] ==> P";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   463
by (rtac (major RS UN_E) 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   464
by (REPEAT (eresolve_tac [asm_rl,singletonE] 1 ORELSE resolve_tac prems 1));
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   465
qed "diagE";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   466
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   467
(*** Equality for Cartesian Product ***)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   468
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   469
goalw Univ.thy [dprod_def]
972
e61b058d58d2 changed syntax of tuples from <..., ...> to (..., ...)
clasohm
parents: 923
diff changeset
   470
    "!!r s. [| (M,M'):r;  (N,N'):s |] ==> (M$N, M'$N') : r<**>s";
1760
6f41a494f3b1 Replaced fast_tac by Fast_tac (which uses default claset)
berghofe
parents: 1642
diff changeset
   471
by (Fast_tac 1);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   472
qed "dprodI";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   473
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   474
(*The general elimination rule*)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   475
val major::prems = goalw Univ.thy [dprod_def]
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   476
    "[| c : r<**>s;  \
972
e61b058d58d2 changed syntax of tuples from <..., ...> to (..., ...)
clasohm
parents: 923
diff changeset
   477
\       !!x y x' y'. [| (x,x') : r;  (y,y') : s;  c = (x$y,x'$y') |] ==> P \
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   478
\    |] ==> P";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   479
by (cut_facts_tac [major] 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   480
by (REPEAT_FIRST (eresolve_tac [asm_rl, UN_E, mem_splitE, singletonE]));
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   481
by (REPEAT (ares_tac prems 1 ORELSE hyp_subst_tac 1));
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   482
qed "dprodE";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   483
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   484
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   485
(*** Equality for Disjoint Sum ***)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   486
972
e61b058d58d2 changed syntax of tuples from <..., ...> to (..., ...)
clasohm
parents: 923
diff changeset
   487
goalw Univ.thy [dsum_def]  "!!r. (M,M'):r ==> (In0(M), In0(M')) : r<++>s";
1760
6f41a494f3b1 Replaced fast_tac by Fast_tac (which uses default claset)
berghofe
parents: 1642
diff changeset
   488
by (Fast_tac 1);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   489
qed "dsum_In0I";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   490
972
e61b058d58d2 changed syntax of tuples from <..., ...> to (..., ...)
clasohm
parents: 923
diff changeset
   491
goalw Univ.thy [dsum_def]  "!!r. (N,N'):s ==> (In1(N), In1(N')) : r<++>s";
1760
6f41a494f3b1 Replaced fast_tac by Fast_tac (which uses default claset)
berghofe
parents: 1642
diff changeset
   492
by (Fast_tac 1);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   493
qed "dsum_In1I";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   494
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   495
val major::prems = goalw Univ.thy [dsum_def]
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   496
    "[| w : r<++>s;  \
972
e61b058d58d2 changed syntax of tuples from <..., ...> to (..., ...)
clasohm
parents: 923
diff changeset
   497
\       !!x x'. [| (x,x') : r;  w = (In0(x), In0(x')) |] ==> P; \
e61b058d58d2 changed syntax of tuples from <..., ...> to (..., ...)
clasohm
parents: 923
diff changeset
   498
\       !!y y'. [| (y,y') : s;  w = (In1(y), In1(y')) |] ==> P \
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   499
\    |] ==> P";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   500
by (cut_facts_tac [major] 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   501
by (REPEAT_FIRST (eresolve_tac [asm_rl, UN_E, UnE, mem_splitE, singletonE]));
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   502
by (DEPTH_SOLVE (ares_tac prems 1 ORELSE hyp_subst_tac 1));
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   503
qed "dsumE";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   504
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   505
1760
6f41a494f3b1 Replaced fast_tac by Fast_tac (which uses default claset)
berghofe
parents: 1642
diff changeset
   506
AddSIs [diagI, uprodI, dprodI];
6f41a494f3b1 Replaced fast_tac by Fast_tac (which uses default claset)
berghofe
parents: 1642
diff changeset
   507
AddIs  [usum_In0I, usum_In1I, dsum_In0I, dsum_In1I];
6f41a494f3b1 Replaced fast_tac by Fast_tac (which uses default claset)
berghofe
parents: 1642
diff changeset
   508
AddSEs [diagE, uprodE, dprodE, usumE, dsumE];
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   509
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   510
(*** Monotonicity ***)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   511
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   512
goal Univ.thy "!!r s. [| r<=r';  s<=s' |] ==> r<**>s <= r'<**>s'";
1760
6f41a494f3b1 Replaced fast_tac by Fast_tac (which uses default claset)
berghofe
parents: 1642
diff changeset
   513
by (Fast_tac 1);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   514
qed "dprod_mono";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   515
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   516
goal Univ.thy "!!r s. [| r<=r';  s<=s' |] ==> r<++>s <= r'<++>s'";
1760
6f41a494f3b1 Replaced fast_tac by Fast_tac (which uses default claset)
berghofe
parents: 1642
diff changeset
   517
by (Fast_tac 1);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   518
qed "dsum_mono";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   519
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   520
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   521
(*** Bounding theorems ***)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   522
1642
21db0cf9a1a4 Using new "Times" infix
paulson
parents: 1563
diff changeset
   523
goal Univ.thy "diag(A) <= A Times A";
1760
6f41a494f3b1 Replaced fast_tac by Fast_tac (which uses default claset)
berghofe
parents: 1642
diff changeset
   524
by (Fast_tac 1);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   525
qed "diag_subset_Sigma";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   526
1642
21db0cf9a1a4 Using new "Times" infix
paulson
parents: 1563
diff changeset
   527
goal Univ.thy "((A Times B) <**> (C Times D)) <= (A<*>C) Times (B<*>D)";
1760
6f41a494f3b1 Replaced fast_tac by Fast_tac (which uses default claset)
berghofe
parents: 1642
diff changeset
   528
by (Fast_tac 1);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   529
qed "dprod_Sigma";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   530
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   531
val dprod_subset_Sigma = [dprod_mono, dprod_Sigma] MRS subset_trans |>standard;
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   532
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   533
(*Dependent version*)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   534
goal Univ.thy
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   535
    "(Sigma A B <**> Sigma C D) <= Sigma (A<*>C) (Split(%x y. B(x)<*>D(y)))";
1786
8a31d85d27b8 best_tac, deepen_tac and safe_tac now also use default claset.
berghofe
parents: 1761
diff changeset
   536
by (safe_tac (!claset));
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   537
by (stac Split 1);
1760
6f41a494f3b1 Replaced fast_tac by Fast_tac (which uses default claset)
berghofe
parents: 1642
diff changeset
   538
by (Fast_tac 1);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   539
qed "dprod_subset_Sigma2";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   540
1642
21db0cf9a1a4 Using new "Times" infix
paulson
parents: 1563
diff changeset
   541
goal Univ.thy "(A Times B <++> C Times D) <= (A<+>C) Times (B<+>D)";
1760
6f41a494f3b1 Replaced fast_tac by Fast_tac (which uses default claset)
berghofe
parents: 1642
diff changeset
   542
by (Fast_tac 1);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   543
qed "dsum_Sigma";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   544
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   545
val dsum_subset_Sigma = [dsum_mono, dsum_Sigma] MRS subset_trans |> standard;
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   546
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   547
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   548
(*** Domain ***)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   549
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   550
goal Univ.thy "fst `` diag(A) = A";
1985
84cf16192e03 Tidied many proofs, using AddIffs to let equivalences take
paulson
parents: 1786
diff changeset
   551
by (Fast_tac 1);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   552
qed "fst_image_diag";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   553
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   554
goal Univ.thy "fst `` (r<**>s) = (fst``r) <*> (fst``s)";
1985
84cf16192e03 Tidied many proofs, using AddIffs to let equivalences take
paulson
parents: 1786
diff changeset
   555
by (Fast_tac 1);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   556
qed "fst_image_dprod";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   557
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   558
goal Univ.thy "fst `` (r<++>s) = (fst``r) <+> (fst``s)";
1985
84cf16192e03 Tidied many proofs, using AddIffs to let equivalences take
paulson
parents: 1786
diff changeset
   559
by (Fast_tac 1);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   560
qed "fst_image_dsum";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   561
1264
3eb91524b938 added local simpsets; removed IOA from 'make test'
clasohm
parents: 976
diff changeset
   562
Addsimps [fst_image_diag, fst_image_dprod, fst_image_dsum];