src/HOL/Univ.ML
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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 [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 "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 "inj_on Abs_Node Node";
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by (rtac inj_on_inverseI 1);
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by (etac Abs_Node_inverse 1);
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qed "inj_on_Abs_Node";
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val Abs_Node_inject = inj_on_Abs_Node RS inj_onD;
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(*** Introduction rules for Node ***)
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Goalw [Node_def] "(%k. 0,a) : Node";
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by (Blast_tac 1);
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qed "Node_K0_I";
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Goalw [Node_def,Push_def]
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    "!!p. p: Node ==> apfst (Push i) p : Node";
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by (blast_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 [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 [Atom_def, inj_def] "inj(Atom)";
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by (blast_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 "(Atom(a)=Atom(b)) = (a=b)";
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by (blast_tac (claset() addSDs [Atom_inject]) 1);
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qed "Atom_Atom_eq";
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AddIffs [Atom_Atom_eq];
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Goalw [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 [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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Goalw [Scons_def] "!!M. M$N <= M'$N' ==> M<=M'";
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by (blast_tac (claset() addSDs [Push_Node_inject]) 1);
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qed "Scons_inject_lemma1";
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Goalw [Scons_def] "!!M. M$N <= M'$N' ==> N<=N'";
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by (blast_tac (claset() addSDs [Push_Node_inject]) 1);
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qed "Scons_inject_lemma2";
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val [major] = goal Univ.thy "M$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'$N';  [| M=M';  N=N' |] ==> P \
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\    |] ==> P";
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by (rtac ((major RS Scons_inject2) RS ((major RS Scons_inject1) RS minor)) 1);
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qed "Scons_inject";
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Goal "(M$N = M'$N') = (M=M' & N=N')";
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by (blast_tac (claset() addSEs [Scons_inject]) 1);
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qed "Scons_Scons_eq";
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(*** Distinctness involving Leaf and Numb ***)
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(** Scons vs Leaf **)
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Goalw [Leaf_def,o_def] "(M$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 [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 [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 [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 "k < Suc(LEAST x. f(x)=0) --> 0 < nat_case (Suc i) f k";
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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 (dtac not_less_Least 1);
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by (Asm_full_simp_tac 1);
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val lemma = result();
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Goalw [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;
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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 (asm_simp_tac (simpset() addsimps [lemma]) 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 [ntrunc_def] "ntrunc 0 M = {}";
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by (Blast_tac 1);
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qed "ntrunc_0";
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Goalw [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 [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 [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 [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() addSIs [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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Addsimps [ntrunc_0, ntrunc_Atom, ntrunc_Leaf, ntrunc_Numb, ntrunc_Scons];
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(** Injection nodes **)
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Goalw [In0_def] "ntrunc (Suc 0) (In0 M) = {}";
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by (Simp_tac 1);
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by (rewtac Scons_def);
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by (Blast_tac 1);
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qed "ntrunc_one_In0";
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Goalw [In0_def]
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    "ntrunc (Suc (Suc k)) (In0 M) = In0 (ntrunc (Suc k) M)";
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by (Simp_tac 1);
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qed "ntrunc_In0";
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Goalw [In1_def] "ntrunc (Suc 0) (In1 M) = {}";
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by (Simp_tac 1);
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by (rewtac Scons_def);
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by (Blast_tac 1);
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qed "ntrunc_one_In1";
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Goalw [In1_def]
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    "ntrunc (Suc (Suc k)) (In1 M) = In1 (ntrunc (Suc k) M)";
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by (Simp_tac 1);
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qed "ntrunc_In1";
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Addsimps [ntrunc_one_In0, ntrunc_In0, ntrunc_one_In1, ntrunc_In1];
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(*** Cartesian Product ***)
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Goalw [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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by (cut_facts_tac [major] 1);
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   310
by (REPEAT (eresolve_tac [asm_rl,singletonE,UN_E] 1
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   311
     ORELSE resolve_tac prems 1));
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
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   312
qed "uprodE";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   313
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   314
(*Elimination of a pair -- introduces no eigenvariables*)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   315
val prems = goal Univ.thy
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   316
    "[| (M$N) : A<*>B;      [| M:A;  N:B |] ==> P   \
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   317
\    |] ==> P";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   318
by (rtac uprodE 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   319
by (REPEAT (ares_tac prems 1 ORELSE eresolve_tac [Scons_inject,ssubst] 1));
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   320
qed "uprodE2";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   321
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   322
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   323
(*** Disjoint Sum ***)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   324
5069
3ea049f7979d isatool fixgoal;
wenzelm
parents: 4830
diff changeset
   325
Goalw [usum_def] "!!M. M:A ==> In0(M) : A<+>B";
2891
d8f254ad1ab9 Calls Blast_tac
paulson
parents: 1985
diff changeset
   326
by (Blast_tac 1);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   327
qed "usum_In0I";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   328
5069
3ea049f7979d isatool fixgoal;
wenzelm
parents: 4830
diff changeset
   329
Goalw [usum_def] "!!N. N:B ==> In1(N) : A<+>B";
2891
d8f254ad1ab9 Calls Blast_tac
paulson
parents: 1985
diff changeset
   330
by (Blast_tac 1);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   331
qed "usum_In1I";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   332
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   333
val major::prems = goalw Univ.thy [usum_def]
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   334
    "[| u : A<+>B;  \
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   335
\       !!x. [| x:A;  u=In0(x) |] ==> P; \
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   336
\       !!y. [| y:B;  u=In1(y) |] ==> P \
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   337
\    |] ==> P";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   338
by (rtac (major RS UnE) 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   339
by (REPEAT (rtac refl 1 
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   340
     ORELSE eresolve_tac (prems@[imageE,ssubst]) 1));
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   341
qed "usumE";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   342
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   343
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   344
(** Injection **)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   345
5069
3ea049f7979d isatool fixgoal;
wenzelm
parents: 4830
diff changeset
   346
Goalw [In0_def,In1_def] "In0(M) ~= In1(N)";
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   347
by (rtac notI 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   348
by (etac (Scons_inject1 RS Numb_inject RS Zero_neq_Suc) 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   349
qed "In0_not_In1";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   350
1985
84cf16192e03 Tidied many proofs, using AddIffs to let equivalences take
paulson
parents: 1786
diff changeset
   351
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
   352
84cf16192e03 Tidied many proofs, using AddIffs to let equivalences take
paulson
parents: 1786
diff changeset
   353
AddIffs [In0_not_In1, In1_not_In0];
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   354
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   355
val [major] = goalw Univ.thy [In0_def] "In0(M) = In0(N) ==>  M=N";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   356
by (rtac (major RS Scons_inject2) 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   357
qed "In0_inject";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   358
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   359
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
   360
by (rtac (major RS Scons_inject2) 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   361
qed "In1_inject";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   362
5069
3ea049f7979d isatool fixgoal;
wenzelm
parents: 4830
diff changeset
   363
Goal "(In0 M = In0 N) = (M=N)";
4089
96fba19bcbe2 isatool fixclasimp;
wenzelm
parents: 3842
diff changeset
   364
by (blast_tac (claset() addSDs [In0_inject]) 1);
3421
be777156c7e9 New facts about In0/1 by Burkhart Wolff
paulson
parents: 2949
diff changeset
   365
qed "In0_eq";
be777156c7e9 New facts about In0/1 by Burkhart Wolff
paulson
parents: 2949
diff changeset
   366
5069
3ea049f7979d isatool fixgoal;
wenzelm
parents: 4830
diff changeset
   367
Goal "(In1 M = In1 N) = (M=N)";
4089
96fba19bcbe2 isatool fixclasimp;
wenzelm
parents: 3842
diff changeset
   368
by (blast_tac (claset() addSDs [In1_inject]) 1);
3421
be777156c7e9 New facts about In0/1 by Burkhart Wolff
paulson
parents: 2949
diff changeset
   369
qed "In1_eq";
be777156c7e9 New facts about In0/1 by Burkhart Wolff
paulson
parents: 2949
diff changeset
   370
be777156c7e9 New facts about In0/1 by Burkhart Wolff
paulson
parents: 2949
diff changeset
   371
AddIffs [In0_eq, In1_eq];
be777156c7e9 New facts about In0/1 by Burkhart Wolff
paulson
parents: 2949
diff changeset
   372
5069
3ea049f7979d isatool fixgoal;
wenzelm
parents: 4830
diff changeset
   373
Goalw [inj_def] "inj In0";
3421
be777156c7e9 New facts about In0/1 by Burkhart Wolff
paulson
parents: 2949
diff changeset
   374
by (Blast_tac 1);
be777156c7e9 New facts about In0/1 by Burkhart Wolff
paulson
parents: 2949
diff changeset
   375
qed "inj_In0";
be777156c7e9 New facts about In0/1 by Burkhart Wolff
paulson
parents: 2949
diff changeset
   376
5069
3ea049f7979d isatool fixgoal;
wenzelm
parents: 4830
diff changeset
   377
Goalw [inj_def] "inj In1";
3421
be777156c7e9 New facts about In0/1 by Burkhart Wolff
paulson
parents: 2949
diff changeset
   378
by (Blast_tac 1);
be777156c7e9 New facts about In0/1 by Burkhart Wolff
paulson
parents: 2949
diff changeset
   379
qed "inj_In1";
be777156c7e9 New facts about In0/1 by Burkhart Wolff
paulson
parents: 2949
diff changeset
   380
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   381
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   382
(*** proving equality of sets and functions using ntrunc ***)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   383
5069
3ea049f7979d isatool fixgoal;
wenzelm
parents: 4830
diff changeset
   384
Goalw [ntrunc_def] "ntrunc k M <= M";
2891
d8f254ad1ab9 Calls Blast_tac
paulson
parents: 1985
diff changeset
   385
by (Blast_tac 1);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   386
qed "ntrunc_subsetI";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   387
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   388
val [major] = goalw Univ.thy [ntrunc_def]
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   389
    "(!!k. ntrunc k M <= N) ==> M<=N";
4089
96fba19bcbe2 isatool fixclasimp;
wenzelm
parents: 3842
diff changeset
   390
by (blast_tac (claset() addIs [less_add_Suc1, less_add_Suc2, 
4521
c7f56322a84b Tidied by adding more default simprules
paulson
parents: 4356
diff changeset
   391
			       major RS subsetD]) 1);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   392
qed "ntrunc_subsetD";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   393
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   394
(*A generalized form of the take-lemma*)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   395
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
   396
by (rtac equalityI 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   397
by (ALLGOALS (rtac ntrunc_subsetD));
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   398
by (ALLGOALS (rtac (ntrunc_subsetI RSN (2, subset_trans))));
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   399
by (rtac (major RS equalityD1) 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   400
by (rtac (major RS equalityD2) 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   401
qed "ntrunc_equality";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   402
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   403
val [major] = goalw Univ.thy [o_def]
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   404
    "[| !!k. (ntrunc(k) o h1) = (ntrunc(k) o h2) |] ==> h1=h2";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   405
by (rtac (ntrunc_equality RS ext) 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   406
by (rtac (major RS fun_cong) 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   407
qed "ntrunc_o_equality";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   408
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   409
(*** Monotonicity ***)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   410
5069
3ea049f7979d isatool fixgoal;
wenzelm
parents: 4830
diff changeset
   411
Goalw [uprod_def] "!!A B. [| A<=A';  B<=B' |] ==> A<*>B <= A'<*>B'";
2891
d8f254ad1ab9 Calls Blast_tac
paulson
parents: 1985
diff changeset
   412
by (Blast_tac 1);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   413
qed "uprod_mono";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   414
5069
3ea049f7979d isatool fixgoal;
wenzelm
parents: 4830
diff changeset
   415
Goalw [usum_def] "!!A B. [| A<=A';  B<=B' |] ==> A<+>B <= A'<+>B'";
2891
d8f254ad1ab9 Calls Blast_tac
paulson
parents: 1985
diff changeset
   416
by (Blast_tac 1);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   417
qed "usum_mono";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   418
5069
3ea049f7979d isatool fixgoal;
wenzelm
parents: 4830
diff changeset
   419
Goalw [Scons_def] "!!M N. [| M<=M';  N<=N' |] ==> M$N <= M'$N'";
2891
d8f254ad1ab9 Calls Blast_tac
paulson
parents: 1985
diff changeset
   420
by (Blast_tac 1);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   421
qed "Scons_mono";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   422
5069
3ea049f7979d isatool fixgoal;
wenzelm
parents: 4830
diff changeset
   423
Goalw [In0_def] "!!M N. M<=N ==> In0(M) <= In0(N)";
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   424
by (REPEAT (ares_tac [subset_refl,Scons_mono] 1));
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   425
qed "In0_mono";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   426
5069
3ea049f7979d isatool fixgoal;
wenzelm
parents: 4830
diff changeset
   427
Goalw [In1_def] "!!M N. M<=N ==> In1(M) <= In1(N)";
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   428
by (REPEAT (ares_tac [subset_refl,Scons_mono] 1));
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   429
qed "In1_mono";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   430
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   431
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   432
(*** Split and Case ***)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   433
5069
3ea049f7979d isatool fixgoal;
wenzelm
parents: 4830
diff changeset
   434
Goalw [Split_def] "Split c (M$N) = c M N";
4535
f24cebc299e4 added select_equality to the implicit claset
oheimb
parents: 4521
diff changeset
   435
by (Blast_tac  1);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   436
qed "Split";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   437
5069
3ea049f7979d isatool fixgoal;
wenzelm
parents: 4830
diff changeset
   438
Goalw [Case_def] "Case c d (In0 M) = c(M)";
4535
f24cebc299e4 added select_equality to the implicit claset
oheimb
parents: 4521
diff changeset
   439
by (Blast_tac 1);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   440
qed "Case_In0";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   441
5069
3ea049f7979d isatool fixgoal;
wenzelm
parents: 4830
diff changeset
   442
Goalw [Case_def] "Case c d (In1 N) = d(N)";
4535
f24cebc299e4 added select_equality to the implicit claset
oheimb
parents: 4521
diff changeset
   443
by (Blast_tac 1);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   444
qed "Case_In1";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   445
4521
c7f56322a84b Tidied by adding more default simprules
paulson
parents: 4356
diff changeset
   446
Addsimps [Split, Case_In0, Case_In1];
c7f56322a84b Tidied by adding more default simprules
paulson
parents: 4356
diff changeset
   447
c7f56322a84b Tidied by adding more default simprules
paulson
parents: 4356
diff changeset
   448
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   449
(**** UN x. B(x) rules ****)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   450
5069
3ea049f7979d isatool fixgoal;
wenzelm
parents: 4830
diff changeset
   451
Goalw [ntrunc_def] "ntrunc k (UN x. f(x)) = (UN x. ntrunc k (f x))";
2891
d8f254ad1ab9 Calls Blast_tac
paulson
parents: 1985
diff changeset
   452
by (Blast_tac 1);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   453
qed "ntrunc_UN1";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   454
5069
3ea049f7979d isatool fixgoal;
wenzelm
parents: 4830
diff changeset
   455
Goalw [Scons_def] "(UN x. f(x)) $ M = (UN x. f(x) $ M)";
2891
d8f254ad1ab9 Calls Blast_tac
paulson
parents: 1985
diff changeset
   456
by (Blast_tac 1);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   457
qed "Scons_UN1_x";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   458
5069
3ea049f7979d isatool fixgoal;
wenzelm
parents: 4830
diff changeset
   459
Goalw [Scons_def] "M $ (UN x. f(x)) = (UN x. M $ f(x))";
2891
d8f254ad1ab9 Calls Blast_tac
paulson
parents: 1985
diff changeset
   460
by (Blast_tac 1);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   461
qed "Scons_UN1_y";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   462
5069
3ea049f7979d isatool fixgoal;
wenzelm
parents: 4830
diff changeset
   463
Goalw [In0_def] "In0(UN x. f(x)) = (UN x. In0(f(x)))";
1465
5d7a7e439cec expanded tabs
clasohm
parents: 1264
diff changeset
   464
by (rtac Scons_UN1_y 1);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   465
qed "In0_UN1";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   466
5069
3ea049f7979d isatool fixgoal;
wenzelm
parents: 4830
diff changeset
   467
Goalw [In1_def] "In1(UN x. f(x)) = (UN x. In1(f(x)))";
1465
5d7a7e439cec expanded tabs
clasohm
parents: 1264
diff changeset
   468
by (rtac Scons_UN1_y 1);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   469
qed "In1_UN1";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   470
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   471
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   472
(*** Equality : the diagonal relation ***)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   473
5069
3ea049f7979d isatool fixgoal;
wenzelm
parents: 4830
diff changeset
   474
Goalw [diag_def] "!!a A. [| a=b;  a:A |] ==> (a,b) : diag(A)";
2891
d8f254ad1ab9 Calls Blast_tac
paulson
parents: 1985
diff changeset
   475
by (Blast_tac 1);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   476
qed "diag_eqI";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   477
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   478
val diagI = refl RS diag_eqI |> standard;
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   479
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   480
(*The general elimination rule*)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   481
val major::prems = goalw Univ.thy [diag_def]
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   482
    "[| c : diag(A);  \
972
e61b058d58d2 changed syntax of tuples from <..., ...> to (..., ...)
clasohm
parents: 923
diff changeset
   483
\       !!x y. [| x:A;  c = (x,x) |] ==> P \
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   484
\    |] ==> P";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   485
by (rtac (major RS UN_E) 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   486
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
   487
qed "diagE";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   488
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   489
(*** Equality for Cartesian Product ***)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   490
5069
3ea049f7979d isatool fixgoal;
wenzelm
parents: 4830
diff changeset
   491
Goalw [dprod_def]
972
e61b058d58d2 changed syntax of tuples from <..., ...> to (..., ...)
clasohm
parents: 923
diff changeset
   492
    "!!r s. [| (M,M'):r;  (N,N'):s |] ==> (M$N, M'$N') : r<**>s";
2891
d8f254ad1ab9 Calls Blast_tac
paulson
parents: 1985
diff changeset
   493
by (Blast_tac 1);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   494
qed "dprodI";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   495
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   496
(*The general elimination rule*)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   497
val major::prems = goalw Univ.thy [dprod_def]
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   498
    "[| c : r<**>s;  \
972
e61b058d58d2 changed syntax of tuples from <..., ...> to (..., ...)
clasohm
parents: 923
diff changeset
   499
\       !!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
   500
\    |] ==> P";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   501
by (cut_facts_tac [major] 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   502
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
   503
by (REPEAT (ares_tac prems 1 ORELSE hyp_subst_tac 1));
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   504
qed "dprodE";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   505
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   506
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   507
(*** Equality for Disjoint Sum ***)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   508
5069
3ea049f7979d isatool fixgoal;
wenzelm
parents: 4830
diff changeset
   509
Goalw [dsum_def]  "!!r. (M,M'):r ==> (In0(M), In0(M')) : r<++>s";
2891
d8f254ad1ab9 Calls Blast_tac
paulson
parents: 1985
diff changeset
   510
by (Blast_tac 1);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   511
qed "dsum_In0I";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   512
5069
3ea049f7979d isatool fixgoal;
wenzelm
parents: 4830
diff changeset
   513
Goalw [dsum_def]  "!!r. (N,N'):s ==> (In1(N), In1(N')) : r<++>s";
2891
d8f254ad1ab9 Calls Blast_tac
paulson
parents: 1985
diff changeset
   514
by (Blast_tac 1);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   515
qed "dsum_In1I";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   516
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   517
val major::prems = goalw Univ.thy [dsum_def]
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   518
    "[| w : r<++>s;  \
972
e61b058d58d2 changed syntax of tuples from <..., ...> to (..., ...)
clasohm
parents: 923
diff changeset
   519
\       !!x x'. [| (x,x') : r;  w = (In0(x), In0(x')) |] ==> P; \
e61b058d58d2 changed syntax of tuples from <..., ...> to (..., ...)
clasohm
parents: 923
diff changeset
   520
\       !!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
   521
\    |] ==> P";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   522
by (cut_facts_tac [major] 1);
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   523
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
   524
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
   525
qed "dsumE";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   526
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   527
1760
6f41a494f3b1 Replaced fast_tac by Fast_tac (which uses default claset)
berghofe
parents: 1642
diff changeset
   528
AddSIs [diagI, uprodI, dprodI];
6f41a494f3b1 Replaced fast_tac by Fast_tac (which uses default claset)
berghofe
parents: 1642
diff changeset
   529
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
   530
AddSEs [diagE, uprodE, dprodE, usumE, dsumE];
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   531
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   532
(*** Monotonicity ***)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   533
5069
3ea049f7979d isatool fixgoal;
wenzelm
parents: 4830
diff changeset
   534
Goal "!!r s. [| r<=r';  s<=s' |] ==> r<**>s <= r'<**>s'";
2891
d8f254ad1ab9 Calls Blast_tac
paulson
parents: 1985
diff changeset
   535
by (Blast_tac 1);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   536
qed "dprod_mono";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   537
5069
3ea049f7979d isatool fixgoal;
wenzelm
parents: 4830
diff changeset
   538
Goal "!!r s. [| r<=r';  s<=s' |] ==> r<++>s <= r'<++>s'";
2891
d8f254ad1ab9 Calls Blast_tac
paulson
parents: 1985
diff changeset
   539
by (Blast_tac 1);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   540
qed "dsum_mono";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   541
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   542
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   543
(*** Bounding theorems ***)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   544
5069
3ea049f7979d isatool fixgoal;
wenzelm
parents: 4830
diff changeset
   545
Goal "diag(A) <= A Times A";
2891
d8f254ad1ab9 Calls Blast_tac
paulson
parents: 1985
diff changeset
   546
by (Blast_tac 1);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   547
qed "diag_subset_Sigma";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   548
5069
3ea049f7979d isatool fixgoal;
wenzelm
parents: 4830
diff changeset
   549
Goal "((A Times B) <**> (C Times D)) <= (A<*>C) Times (B<*>D)";
2891
d8f254ad1ab9 Calls Blast_tac
paulson
parents: 1985
diff changeset
   550
by (Blast_tac 1);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   551
qed "dprod_Sigma";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   552
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   553
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
   554
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   555
(*Dependent version*)
5069
3ea049f7979d isatool fixgoal;
wenzelm
parents: 4830
diff changeset
   556
Goal
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   557
    "(Sigma A B <**> Sigma C D) <= Sigma (A<*>C) (Split(%x y. B(x)<*>D(y)))";
4153
e534c4c32d54 Ran expandshort, especially to introduce Safe_tac
paulson
parents: 4089
diff changeset
   558
by Safe_tac;
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   559
by (stac Split 1);
2891
d8f254ad1ab9 Calls Blast_tac
paulson
parents: 1985
diff changeset
   560
by (Blast_tac 1);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   561
qed "dprod_subset_Sigma2";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   562
5069
3ea049f7979d isatool fixgoal;
wenzelm
parents: 4830
diff changeset
   563
Goal "(A Times B <++> C Times D) <= (A<+>C) Times (B<+>D)";
2891
d8f254ad1ab9 Calls Blast_tac
paulson
parents: 1985
diff changeset
   564
by (Blast_tac 1);
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   565
qed "dsum_Sigma";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   566
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   567
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
   568
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   569
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   570
(*** Domain ***)
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   571
5069
3ea049f7979d isatool fixgoal;
wenzelm
parents: 4830
diff changeset
   572
Goal "fst `` diag(A) = A";
4521
c7f56322a84b Tidied by adding more default simprules
paulson
parents: 4356
diff changeset
   573
by Auto_tac;
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   574
qed "fst_image_diag";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   575
5069
3ea049f7979d isatool fixgoal;
wenzelm
parents: 4830
diff changeset
   576
Goal "fst `` (r<**>s) = (fst``r) <*> (fst``s)";
4521
c7f56322a84b Tidied by adding more default simprules
paulson
parents: 4356
diff changeset
   577
by Auto_tac;
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   578
qed "fst_image_dprod";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   579
5069
3ea049f7979d isatool fixgoal;
wenzelm
parents: 4830
diff changeset
   580
Goal "fst `` (r<++>s) = (fst``r) <+> (fst``s)";
4521
c7f56322a84b Tidied by adding more default simprules
paulson
parents: 4356
diff changeset
   581
by Auto_tac;
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   582
qed "fst_image_dsum";
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
   583
1264
3eb91524b938 added local simpsets; removed IOA from 'make test'
clasohm
parents: 976
diff changeset
   584
Addsimps [fst_image_diag, fst_image_dprod, fst_image_dsum];