src/HOL/Univ.ML
author paulson
Fri Apr 04 11:18:52 1997 +0200 (1997-04-04)
changeset 2891 d8f254ad1ab9
parent 1985 84cf16192e03
child 2935 998cb95fdd43
permissions -rw-r--r--
Calls Blast_tac
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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 (Blast_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 (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 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 (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 Univ.thy "(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 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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goalw Univ.thy [Scons_def] "!!M. M$N <= M'$N' ==> M<=M'";
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by (blast_tac (!claset addSDs [Push_Node_inject]) 1);
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qed "Scons_inject_lemma1";
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goalw Univ.thy [Scons_def] "!!M. M$N <= M'$N' ==> N<=N'";
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by (fast_tac (!claset addSDs [Push_Node_inject]) 1);
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qed "Scons_inject_lemma2";
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val [major] = goal Univ.thy "M$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'$N';  [| M=M';  N=N' |] ==> P \
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\    |] ==> P";
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by (rtac ((major RS Scons_inject2) RS ((major RS Scons_inject1) RS minor)) 1);
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qed "Scons_inject";
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AddSDs [Scons_inject];
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goal Univ.thy "(M$N = M'$N') = (M=M' & N=N')";
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by (blast_tac (!claset addSEs [Scons_inject]) 1);
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qed "Scons_Scons_eq";
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(*** Distinctness involving Leaf and Numb ***)
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(** Scons vs Leaf **)
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goalw Univ.thy [Leaf_def,o_def] "(M$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 (Blast_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 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 (Blast_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 (Blast_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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by (cut_facts_tac [major] 1);
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by (REPEAT (eresolve_tac [asm_rl,singletonE,UN_E] 1
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     ORELSE resolve_tac prems 1));
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qed "uprodE";
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   310
(*Elimination of a pair -- introduces no eigenvariables*)
clasohm@923
   311
val prems = goal Univ.thy
clasohm@923
   312
    "[| (M$N) : A<*>B;      [| M:A;  N:B |] ==> P   \
clasohm@923
   313
\    |] ==> P";
clasohm@923
   314
by (rtac uprodE 1);
clasohm@923
   315
by (REPEAT (ares_tac prems 1 ORELSE eresolve_tac [Scons_inject,ssubst] 1));
clasohm@923
   316
qed "uprodE2";
clasohm@923
   317
clasohm@923
   318
clasohm@923
   319
(*** Disjoint Sum ***)
clasohm@923
   320
clasohm@923
   321
goalw Univ.thy [usum_def] "!!M. M:A ==> In0(M) : A<+>B";
paulson@2891
   322
by (Blast_tac 1);
clasohm@923
   323
qed "usum_In0I";
clasohm@923
   324
clasohm@923
   325
goalw Univ.thy [usum_def] "!!N. N:B ==> In1(N) : A<+>B";
paulson@2891
   326
by (Blast_tac 1);
clasohm@923
   327
qed "usum_In1I";
clasohm@923
   328
clasohm@923
   329
val major::prems = goalw Univ.thy [usum_def]
clasohm@923
   330
    "[| u : A<+>B;  \
clasohm@923
   331
\       !!x. [| x:A;  u=In0(x) |] ==> P; \
clasohm@923
   332
\       !!y. [| y:B;  u=In1(y) |] ==> P \
clasohm@923
   333
\    |] ==> P";
clasohm@923
   334
by (rtac (major RS UnE) 1);
clasohm@923
   335
by (REPEAT (rtac refl 1 
clasohm@923
   336
     ORELSE eresolve_tac (prems@[imageE,ssubst]) 1));
clasohm@923
   337
qed "usumE";
clasohm@923
   338
clasohm@923
   339
clasohm@923
   340
(** Injection **)
clasohm@923
   341
clasohm@923
   342
goalw Univ.thy [In0_def,In1_def] "In0(M) ~= In1(N)";
clasohm@923
   343
by (rtac notI 1);
clasohm@923
   344
by (etac (Scons_inject1 RS Numb_inject RS Zero_neq_Suc) 1);
clasohm@923
   345
qed "In0_not_In1";
clasohm@923
   346
paulson@1985
   347
bind_thm ("In1_not_In0", In0_not_In1 RS not_sym);
paulson@1985
   348
paulson@1985
   349
AddIffs [In0_not_In1, In1_not_In0];
clasohm@923
   350
clasohm@923
   351
val [major] = goalw Univ.thy [In0_def] "In0(M) = In0(N) ==>  M=N";
clasohm@923
   352
by (rtac (major RS Scons_inject2) 1);
clasohm@923
   353
qed "In0_inject";
clasohm@923
   354
clasohm@923
   355
val [major] = goalw Univ.thy [In1_def] "In1(M) = In1(N) ==>  M=N";
clasohm@923
   356
by (rtac (major RS Scons_inject2) 1);
clasohm@923
   357
qed "In1_inject";
clasohm@923
   358
paulson@1985
   359
AddSDs [In0_inject, In1_inject];
clasohm@923
   360
clasohm@923
   361
(*** proving equality of sets and functions using ntrunc ***)
clasohm@923
   362
clasohm@923
   363
goalw Univ.thy [ntrunc_def] "ntrunc k M <= M";
paulson@2891
   364
by (Blast_tac 1);
clasohm@923
   365
qed "ntrunc_subsetI";
clasohm@923
   366
clasohm@923
   367
val [major] = goalw Univ.thy [ntrunc_def]
clasohm@923
   368
    "(!!k. ntrunc k M <= N) ==> M<=N";
paulson@2891
   369
by (blast_tac (!claset addIs [less_add_Suc1, less_add_Suc2, 
clasohm@1465
   370
                            major RS subsetD]) 1);
clasohm@923
   371
qed "ntrunc_subsetD";
clasohm@923
   372
clasohm@923
   373
(*A generalized form of the take-lemma*)
clasohm@923
   374
val [major] = goal Univ.thy "(!!k. ntrunc k M = ntrunc k N) ==> M=N";
clasohm@923
   375
by (rtac equalityI 1);
clasohm@923
   376
by (ALLGOALS (rtac ntrunc_subsetD));
clasohm@923
   377
by (ALLGOALS (rtac (ntrunc_subsetI RSN (2, subset_trans))));
clasohm@923
   378
by (rtac (major RS equalityD1) 1);
clasohm@923
   379
by (rtac (major RS equalityD2) 1);
clasohm@923
   380
qed "ntrunc_equality";
clasohm@923
   381
clasohm@923
   382
val [major] = goalw Univ.thy [o_def]
clasohm@923
   383
    "[| !!k. (ntrunc(k) o h1) = (ntrunc(k) o h2) |] ==> h1=h2";
clasohm@923
   384
by (rtac (ntrunc_equality RS ext) 1);
clasohm@923
   385
by (rtac (major RS fun_cong) 1);
clasohm@923
   386
qed "ntrunc_o_equality";
clasohm@923
   387
clasohm@923
   388
(*** Monotonicity ***)
clasohm@923
   389
clasohm@923
   390
goalw Univ.thy [uprod_def] "!!A B. [| A<=A';  B<=B' |] ==> A<*>B <= A'<*>B'";
paulson@2891
   391
by (Blast_tac 1);
clasohm@923
   392
qed "uprod_mono";
clasohm@923
   393
clasohm@923
   394
goalw Univ.thy [usum_def] "!!A B. [| A<=A';  B<=B' |] ==> A<+>B <= A'<+>B'";
paulson@2891
   395
by (Blast_tac 1);
clasohm@923
   396
qed "usum_mono";
clasohm@923
   397
clasohm@923
   398
goalw Univ.thy [Scons_def] "!!M N. [| M<=M';  N<=N' |] ==> M$N <= M'$N'";
paulson@2891
   399
by (Blast_tac 1);
clasohm@923
   400
qed "Scons_mono";
clasohm@923
   401
clasohm@923
   402
goalw Univ.thy [In0_def] "!!M N. M<=N ==> In0(M) <= In0(N)";
clasohm@923
   403
by (REPEAT (ares_tac [subset_refl,Scons_mono] 1));
clasohm@923
   404
qed "In0_mono";
clasohm@923
   405
clasohm@923
   406
goalw Univ.thy [In1_def] "!!M N. M<=N ==> In1(M) <= In1(N)";
clasohm@923
   407
by (REPEAT (ares_tac [subset_refl,Scons_mono] 1));
clasohm@923
   408
qed "In1_mono";
clasohm@923
   409
clasohm@923
   410
clasohm@923
   411
(*** Split and Case ***)
clasohm@923
   412
clasohm@923
   413
goalw Univ.thy [Split_def] "Split c (M$N) = c M N";
paulson@2891
   414
by (blast_tac (!claset addIs [select_equality]) 1);
clasohm@923
   415
qed "Split";
clasohm@923
   416
clasohm@923
   417
goalw Univ.thy [Case_def] "Case c d (In0 M) = c(M)";
paulson@2891
   418
by (blast_tac (!claset addIs [select_equality]) 1);
clasohm@923
   419
qed "Case_In0";
clasohm@923
   420
clasohm@923
   421
goalw Univ.thy [Case_def] "Case c d (In1 N) = d(N)";
paulson@2891
   422
by (blast_tac (!claset addIs [select_equality]) 1);
clasohm@923
   423
qed "Case_In1";
clasohm@923
   424
clasohm@923
   425
(**** UN x. B(x) rules ****)
clasohm@923
   426
clasohm@923
   427
goalw Univ.thy [ntrunc_def] "ntrunc k (UN x.f(x)) = (UN x. ntrunc k (f x))";
paulson@2891
   428
by (Blast_tac 1);
clasohm@923
   429
qed "ntrunc_UN1";
clasohm@923
   430
clasohm@923
   431
goalw Univ.thy [Scons_def] "(UN x.f(x)) $ M = (UN x. f(x) $ M)";
paulson@2891
   432
by (Blast_tac 1);
clasohm@923
   433
qed "Scons_UN1_x";
clasohm@923
   434
clasohm@923
   435
goalw Univ.thy [Scons_def] "M $ (UN x.f(x)) = (UN x. M $ f(x))";
paulson@2891
   436
by (Blast_tac 1);
clasohm@923
   437
qed "Scons_UN1_y";
clasohm@923
   438
clasohm@923
   439
goalw Univ.thy [In0_def] "In0(UN x.f(x)) = (UN x. In0(f(x)))";
clasohm@1465
   440
by (rtac Scons_UN1_y 1);
clasohm@923
   441
qed "In0_UN1";
clasohm@923
   442
clasohm@923
   443
goalw Univ.thy [In1_def] "In1(UN x.f(x)) = (UN x. In1(f(x)))";
clasohm@1465
   444
by (rtac Scons_UN1_y 1);
clasohm@923
   445
qed "In1_UN1";
clasohm@923
   446
clasohm@923
   447
clasohm@923
   448
(*** Equality : the diagonal relation ***)
clasohm@923
   449
clasohm@972
   450
goalw Univ.thy [diag_def] "!!a A. [| a=b;  a:A |] ==> (a,b) : diag(A)";
paulson@2891
   451
by (Blast_tac 1);
clasohm@923
   452
qed "diag_eqI";
clasohm@923
   453
clasohm@923
   454
val diagI = refl RS diag_eqI |> standard;
clasohm@923
   455
clasohm@923
   456
(*The general elimination rule*)
clasohm@923
   457
val major::prems = goalw Univ.thy [diag_def]
clasohm@923
   458
    "[| c : diag(A);  \
clasohm@972
   459
\       !!x y. [| x:A;  c = (x,x) |] ==> P \
clasohm@923
   460
\    |] ==> P";
clasohm@923
   461
by (rtac (major RS UN_E) 1);
clasohm@923
   462
by (REPEAT (eresolve_tac [asm_rl,singletonE] 1 ORELSE resolve_tac prems 1));
clasohm@923
   463
qed "diagE";
clasohm@923
   464
clasohm@923
   465
(*** Equality for Cartesian Product ***)
clasohm@923
   466
clasohm@923
   467
goalw Univ.thy [dprod_def]
clasohm@972
   468
    "!!r s. [| (M,M'):r;  (N,N'):s |] ==> (M$N, M'$N') : r<**>s";
paulson@2891
   469
by (Blast_tac 1);
clasohm@923
   470
qed "dprodI";
clasohm@923
   471
clasohm@923
   472
(*The general elimination rule*)
clasohm@923
   473
val major::prems = goalw Univ.thy [dprod_def]
clasohm@923
   474
    "[| c : r<**>s;  \
clasohm@972
   475
\       !!x y x' y'. [| (x,x') : r;  (y,y') : s;  c = (x$y,x'$y') |] ==> P \
clasohm@923
   476
\    |] ==> P";
clasohm@923
   477
by (cut_facts_tac [major] 1);
clasohm@923
   478
by (REPEAT_FIRST (eresolve_tac [asm_rl, UN_E, mem_splitE, singletonE]));
clasohm@923
   479
by (REPEAT (ares_tac prems 1 ORELSE hyp_subst_tac 1));
clasohm@923
   480
qed "dprodE";
clasohm@923
   481
clasohm@923
   482
clasohm@923
   483
(*** Equality for Disjoint Sum ***)
clasohm@923
   484
clasohm@972
   485
goalw Univ.thy [dsum_def]  "!!r. (M,M'):r ==> (In0(M), In0(M')) : r<++>s";
paulson@2891
   486
by (Blast_tac 1);
clasohm@923
   487
qed "dsum_In0I";
clasohm@923
   488
clasohm@972
   489
goalw Univ.thy [dsum_def]  "!!r. (N,N'):s ==> (In1(N), In1(N')) : r<++>s";
paulson@2891
   490
by (Blast_tac 1);
clasohm@923
   491
qed "dsum_In1I";
clasohm@923
   492
clasohm@923
   493
val major::prems = goalw Univ.thy [dsum_def]
clasohm@923
   494
    "[| w : r<++>s;  \
clasohm@972
   495
\       !!x x'. [| (x,x') : r;  w = (In0(x), In0(x')) |] ==> P; \
clasohm@972
   496
\       !!y y'. [| (y,y') : s;  w = (In1(y), In1(y')) |] ==> P \
clasohm@923
   497
\    |] ==> P";
clasohm@923
   498
by (cut_facts_tac [major] 1);
clasohm@923
   499
by (REPEAT_FIRST (eresolve_tac [asm_rl, UN_E, UnE, mem_splitE, singletonE]));
clasohm@923
   500
by (DEPTH_SOLVE (ares_tac prems 1 ORELSE hyp_subst_tac 1));
clasohm@923
   501
qed "dsumE";
clasohm@923
   502
clasohm@923
   503
berghofe@1760
   504
AddSIs [diagI, uprodI, dprodI];
berghofe@1760
   505
AddIs  [usum_In0I, usum_In1I, dsum_In0I, dsum_In1I];
berghofe@1760
   506
AddSEs [diagE, uprodE, dprodE, usumE, dsumE];
clasohm@923
   507
clasohm@923
   508
(*** Monotonicity ***)
clasohm@923
   509
clasohm@923
   510
goal Univ.thy "!!r s. [| r<=r';  s<=s' |] ==> r<**>s <= r'<**>s'";
paulson@2891
   511
by (Blast_tac 1);
clasohm@923
   512
qed "dprod_mono";
clasohm@923
   513
clasohm@923
   514
goal Univ.thy "!!r s. [| r<=r';  s<=s' |] ==> r<++>s <= r'<++>s'";
paulson@2891
   515
by (Blast_tac 1);
clasohm@923
   516
qed "dsum_mono";
clasohm@923
   517
clasohm@923
   518
clasohm@923
   519
(*** Bounding theorems ***)
clasohm@923
   520
paulson@1642
   521
goal Univ.thy "diag(A) <= A Times A";
paulson@2891
   522
by (Blast_tac 1);
clasohm@923
   523
qed "diag_subset_Sigma";
clasohm@923
   524
paulson@1642
   525
goal Univ.thy "((A Times B) <**> (C Times D)) <= (A<*>C) Times (B<*>D)";
paulson@2891
   526
by (Blast_tac 1);
clasohm@923
   527
qed "dprod_Sigma";
clasohm@923
   528
clasohm@923
   529
val dprod_subset_Sigma = [dprod_mono, dprod_Sigma] MRS subset_trans |>standard;
clasohm@923
   530
clasohm@923
   531
(*Dependent version*)
clasohm@923
   532
goal Univ.thy
clasohm@923
   533
    "(Sigma A B <**> Sigma C D) <= Sigma (A<*>C) (Split(%x y. B(x)<*>D(y)))";
berghofe@1786
   534
by (safe_tac (!claset));
clasohm@923
   535
by (stac Split 1);
paulson@2891
   536
by (Blast_tac 1);
clasohm@923
   537
qed "dprod_subset_Sigma2";
clasohm@923
   538
paulson@1642
   539
goal Univ.thy "(A Times B <++> C Times D) <= (A<+>C) Times (B<+>D)";
paulson@2891
   540
by (Blast_tac 1);
clasohm@923
   541
qed "dsum_Sigma";
clasohm@923
   542
clasohm@923
   543
val dsum_subset_Sigma = [dsum_mono, dsum_Sigma] MRS subset_trans |> standard;
clasohm@923
   544
clasohm@923
   545
clasohm@923
   546
(*** Domain ***)
clasohm@923
   547
clasohm@923
   548
goal Univ.thy "fst `` diag(A) = A";
paulson@2891
   549
by (Blast_tac 1);
clasohm@923
   550
qed "fst_image_diag";
clasohm@923
   551
clasohm@923
   552
goal Univ.thy "fst `` (r<**>s) = (fst``r) <*> (fst``s)";
paulson@2891
   553
by (Blast_tac 1);
clasohm@923
   554
qed "fst_image_dprod";
clasohm@923
   555
clasohm@923
   556
goal Univ.thy "fst `` (r<++>s) = (fst``r) <+> (fst``s)";
paulson@2891
   557
by (Blast_tac 1);
clasohm@923
   558
qed "fst_image_dsum";
clasohm@923
   559
clasohm@1264
   560
Addsimps [fst_image_diag, fst_image_dprod, fst_image_dsum];