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(* Title: HOL/Datatype_Universe.ML
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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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*)
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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
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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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Goalw [Push_def] "Push i f = Push j g ==> i=j";
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by (etac (fun_cong RS box_equals) 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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Goalw [Push_def] "Push i f = Push j g ==> f=g";
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by (rtac (ext RS box_equals) 1);
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by (etac fun_cong 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
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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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Goalw [Push_def] "Push (Inr (Suc k)) f = (%z. Inr 0) ==> P";
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by (rtac Suc_neq_Zero 1);
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by (etac (fun_cong RS box_equals RS Inr_inject) 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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bind_thm ("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. Inr 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: 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 [Atom_def,Scons_def,Push_Node_def] "Scons 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(Atom)";
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by (blast_tac (claset() addSIs [injI, Node_K0_I] addSDs [Abs_Node_inject]) 1);
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qed "inj_Atom";
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bind_thm ("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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bind_thm ("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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bind_thm ("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 [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] "Scons M N <= Scons 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] "Scons M N <= Scons 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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Goal "Scons M N = Scons M' N' ==> M=M'";
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by (etac 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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Goal "Scons M N = Scons M' N' ==> N=N'";
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by (etac 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
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"[| Scons M N = Scons 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 "(Scons M N = Scons 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] "Scons 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] "Scons 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. Inr 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 = Inr 0) --> nat_case (Inr (Suc i)) f k ~= Inr 0";
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by (induct_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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val lemma = result();
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Goalw [ndepth_def,Push_Node_def]
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"ndepth (Push_Node (Inr (Suc 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) (Scons M N) = Scons (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 1 (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 1 (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:A; N:B |] ==> Scons M N : uprod 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 [uprod_def]
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"[| c : uprod A B; \
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\ !!x y. [| x:A; y:B; c = Scons 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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(*Elimination of a pair -- introduces no eigenvariables*)
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val prems = Goal
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"[| Scons M N : uprod A B; [| M:A; N:B |] ==> P \
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\ |] ==> P";
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by (rtac uprodE 1);
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by (REPEAT (ares_tac prems 1 ORELSE eresolve_tac [Scons_inject,ssubst] 1));
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qed "uprodE2";
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(*** Disjoint Sum ***)
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Goalw [usum_def] "M:A ==> In0(M) : usum A B";
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by (Blast_tac 1);
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qed "usum_In0I";
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Goalw [usum_def] "N:B ==> In1(N) : usum A B";
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by (Blast_tac 1);
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qed "usum_In1I";
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val major::prems = Goalw [usum_def]
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"[| u : usum A B; \
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\ !!x. [| x:A; u=In0(x) |] ==> P; \
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\ !!y. [| y:B; u=In1(y) |] ==> P \
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\ |] ==> P";
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by (rtac (major RS UnE) 1);
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by (REPEAT (rtac refl 1
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ORELSE eresolve_tac (prems@[imageE,ssubst]) 1));
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qed "usumE";
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(** Injection **)
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Goalw [In0_def,In1_def] "In0(M) ~= In1(N)";
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by (rtac notI 1);
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by (etac (Scons_inject1 RS Numb_inject RS Zero_neq_Suc) 1);
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qed "In0_not_In1";
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bind_thm ("In1_not_In0", In0_not_In1 RS not_sym);
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AddIffs [In0_not_In1, In1_not_In0];
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Goalw [In0_def] "In0(M) = In0(N) ==> M=N";
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by (etac (Scons_inject2) 1);
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qed "In0_inject";
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Goalw [In1_def] "In1(M) = In1(N) ==> M=N";
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by (etac (Scons_inject2) 1);
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qed "In1_inject";
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360 |
Goal "(In0 M = In0 N) = (M=N)";
|
|
361 |
by (blast_tac (claset() addSDs [In0_inject]) 1);
|
|
362 |
qed "In0_eq";
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|
363 |
|
|
364 |
Goal "(In1 M = In1 N) = (M=N)";
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|
365 |
by (blast_tac (claset() addSDs [In1_inject]) 1);
|
|
366 |
qed "In1_eq";
|
|
367 |
|
|
368 |
AddIffs [In0_eq, In1_eq];
|
|
369 |
|
|
370 |
Goal "inj In0";
|
|
371 |
by (blast_tac (claset() addSIs [injI]) 1);
|
|
372 |
qed "inj_In0";
|
|
373 |
|
|
374 |
Goal "inj In1";
|
|
375 |
by (blast_tac (claset() addSIs [injI]) 1);
|
|
376 |
qed "inj_In1";
|
|
377 |
|
|
378 |
|
|
379 |
(*** Function spaces ***)
|
|
380 |
|
|
381 |
Goalw [Lim_def] "Lim f = Lim g ==> f = g";
|
|
382 |
by (rtac ext 1);
|
|
383 |
by (blast_tac (claset() addSEs [Push_Node_inject]) 1);
|
|
384 |
qed "Lim_inject";
|
|
385 |
|
|
386 |
Goalw [Funs_def] "S <= T ==> Funs S <= Funs T";
|
|
387 |
by (Blast_tac 1);
|
|
388 |
qed "Funs_mono";
|
|
389 |
|
|
390 |
val [prem] = Goalw [Funs_def] "(!!x. f x : S) ==> f : Funs S";
|
|
391 |
by (blast_tac (claset() addIs [prem]) 1);
|
|
392 |
qed "FunsI";
|
|
393 |
|
|
394 |
Goalw [Funs_def] "f : Funs S ==> f x : S";
|
|
395 |
by (etac CollectE 1);
|
|
396 |
by (etac subsetD 1);
|
|
397 |
by (rtac rangeI 1);
|
|
398 |
qed "FunsD";
|
|
399 |
|
|
400 |
val [p1, p2] = Goalw [o_def]
|
|
401 |
"[| f : Funs R; !!x. x : R ==> r (a x) = x |] ==> r o (a o f) = f";
|
|
402 |
by (rtac (p2 RS ext) 1);
|
|
403 |
by (rtac (p1 RS FunsD) 1);
|
|
404 |
qed "Funs_inv";
|
|
405 |
|
|
406 |
val [p1, p2] = Goalw [o_def]
|
|
407 |
"[| f : Funs (range g); !!h. f = g o h ==> P |] ==> P";
|
|
408 |
by (res_inst_tac [("h", "%x. @y. (f::'a=>'b) x = g y")] p2 1);
|
|
409 |
by (rtac ext 1);
|
|
410 |
by (rtac (p1 RS FunsD RS rangeE) 1);
|
|
411 |
by (etac (exI RS (some_eq_ex RS iffD2)) 1);
|
|
412 |
qed "Funs_rangeE";
|
|
413 |
|
|
414 |
Goal "a : S ==> (%x. a) : Funs S";
|
|
415 |
by (rtac FunsI 1);
|
|
416 |
by (assume_tac 1);
|
|
417 |
qed "Funs_nonempty";
|
|
418 |
|
|
419 |
|
|
420 |
(*** proving equality of sets and functions using ntrunc ***)
|
|
421 |
|
|
422 |
Goalw [ntrunc_def] "ntrunc k M <= M";
|
|
423 |
by (Blast_tac 1);
|
|
424 |
qed "ntrunc_subsetI";
|
|
425 |
|
|
426 |
val [major] = Goalw [ntrunc_def] "(!!k. ntrunc k M <= N) ==> M<=N";
|
|
427 |
by (blast_tac (claset() addIs [less_add_Suc1, less_add_Suc2,
|
|
428 |
major RS subsetD]) 1);
|
|
429 |
qed "ntrunc_subsetD";
|
|
430 |
|
|
431 |
(*A generalized form of the take-lemma*)
|
|
432 |
val [major] = Goal "(!!k. ntrunc k M = ntrunc k N) ==> M=N";
|
|
433 |
by (rtac equalityI 1);
|
|
434 |
by (ALLGOALS (rtac ntrunc_subsetD));
|
|
435 |
by (ALLGOALS (rtac (ntrunc_subsetI RSN (2, subset_trans))));
|
|
436 |
by (rtac (major RS equalityD1) 1);
|
|
437 |
by (rtac (major RS equalityD2) 1);
|
|
438 |
qed "ntrunc_equality";
|
|
439 |
|
|
440 |
val [major] = Goalw [o_def]
|
|
441 |
"[| !!k. (ntrunc(k) o h1) = (ntrunc(k) o h2) |] ==> h1=h2";
|
|
442 |
by (rtac (ntrunc_equality RS ext) 1);
|
|
443 |
by (rtac (major RS fun_cong) 1);
|
|
444 |
qed "ntrunc_o_equality";
|
|
445 |
|
|
446 |
(*** Monotonicity ***)
|
|
447 |
|
|
448 |
Goalw [uprod_def] "[| A<=A'; B<=B' |] ==> uprod A B <= uprod A' B'";
|
|
449 |
by (Blast_tac 1);
|
|
450 |
qed "uprod_mono";
|
|
451 |
|
|
452 |
Goalw [usum_def] "[| A<=A'; B<=B' |] ==> usum A B <= usum A' B'";
|
|
453 |
by (Blast_tac 1);
|
|
454 |
qed "usum_mono";
|
|
455 |
|
|
456 |
Goalw [Scons_def] "[| M<=M'; N<=N' |] ==> Scons M N <= Scons M' N'";
|
|
457 |
by (Blast_tac 1);
|
|
458 |
qed "Scons_mono";
|
|
459 |
|
|
460 |
Goalw [In0_def] "M<=N ==> In0(M) <= In0(N)";
|
|
461 |
by (REPEAT (ares_tac [subset_refl,Scons_mono] 1));
|
|
462 |
qed "In0_mono";
|
|
463 |
|
|
464 |
Goalw [In1_def] "M<=N ==> In1(M) <= In1(N)";
|
|
465 |
by (REPEAT (ares_tac [subset_refl,Scons_mono] 1));
|
|
466 |
qed "In1_mono";
|
|
467 |
|
|
468 |
|
|
469 |
(*** Split and Case ***)
|
|
470 |
|
|
471 |
Goalw [Split_def] "Split c (Scons M N) = c M N";
|
|
472 |
by (Blast_tac 1);
|
|
473 |
qed "Split";
|
|
474 |
|
|
475 |
Goalw [Case_def] "Case c d (In0 M) = c(M)";
|
|
476 |
by (Blast_tac 1);
|
|
477 |
qed "Case_In0";
|
|
478 |
|
|
479 |
Goalw [Case_def] "Case c d (In1 N) = d(N)";
|
|
480 |
by (Blast_tac 1);
|
|
481 |
qed "Case_In1";
|
|
482 |
|
|
483 |
Addsimps [Split, Case_In0, Case_In1];
|
|
484 |
|
|
485 |
|
|
486 |
(**** UN x. B(x) rules ****)
|
|
487 |
|
|
488 |
Goalw [ntrunc_def] "ntrunc k (UN x. f(x)) = (UN x. ntrunc k (f x))";
|
|
489 |
by (Blast_tac 1);
|
|
490 |
qed "ntrunc_UN1";
|
|
491 |
|
|
492 |
Goalw [Scons_def] "Scons (UN x. f x) M = (UN x. Scons (f x) M)";
|
|
493 |
by (Blast_tac 1);
|
|
494 |
qed "Scons_UN1_x";
|
|
495 |
|
|
496 |
Goalw [Scons_def] "Scons M (UN x. f x) = (UN x. Scons M (f x))";
|
|
497 |
by (Blast_tac 1);
|
|
498 |
qed "Scons_UN1_y";
|
|
499 |
|
|
500 |
Goalw [In0_def] "In0(UN x. f(x)) = (UN x. In0(f(x)))";
|
|
501 |
by (rtac Scons_UN1_y 1);
|
|
502 |
qed "In0_UN1";
|
|
503 |
|
|
504 |
Goalw [In1_def] "In1(UN x. f(x)) = (UN x. In1(f(x)))";
|
|
505 |
by (rtac Scons_UN1_y 1);
|
|
506 |
qed "In1_UN1";
|
|
507 |
|
|
508 |
|
|
509 |
(*** Equality for Cartesian Product ***)
|
|
510 |
|
|
511 |
Goalw [dprod_def]
|
|
512 |
"[| (M,M'):r; (N,N'):s |] ==> (Scons M N, Scons M' N') : dprod r s";
|
|
513 |
by (Blast_tac 1);
|
|
514 |
qed "dprodI";
|
|
515 |
|
|
516 |
(*The general elimination rule*)
|
|
517 |
val major::prems = Goalw [dprod_def]
|
|
518 |
"[| c : dprod r s; \
|
|
519 |
\ !!x y x' y'. [| (x,x') : r; (y,y') : s; c = (Scons x y, Scons x' y') |] ==> P \
|
|
520 |
\ |] ==> P";
|
|
521 |
by (cut_facts_tac [major] 1);
|
|
522 |
by (REPEAT_FIRST (eresolve_tac [asm_rl, UN_E, mem_splitE, singletonE]));
|
|
523 |
by (REPEAT (ares_tac prems 1 ORELSE hyp_subst_tac 1));
|
|
524 |
qed "dprodE";
|
|
525 |
|
|
526 |
|
|
527 |
(*** Equality for Disjoint Sum ***)
|
|
528 |
|
|
529 |
Goalw [dsum_def] "(M,M'):r ==> (In0(M), In0(M')) : dsum r s";
|
|
530 |
by (Blast_tac 1);
|
|
531 |
qed "dsum_In0I";
|
|
532 |
|
|
533 |
Goalw [dsum_def] "(N,N'):s ==> (In1(N), In1(N')) : dsum r s";
|
|
534 |
by (Blast_tac 1);
|
|
535 |
qed "dsum_In1I";
|
|
536 |
|
|
537 |
val major::prems = Goalw [dsum_def]
|
|
538 |
"[| w : dsum r s; \
|
|
539 |
\ !!x x'. [| (x,x') : r; w = (In0(x), In0(x')) |] ==> P; \
|
|
540 |
\ !!y y'. [| (y,y') : s; w = (In1(y), In1(y')) |] ==> P \
|
|
541 |
\ |] ==> P";
|
|
542 |
by (cut_facts_tac [major] 1);
|
|
543 |
by (REPEAT_FIRST (eresolve_tac [asm_rl, UN_E, UnE, mem_splitE, singletonE]));
|
|
544 |
by (DEPTH_SOLVE (ares_tac prems 1 ORELSE hyp_subst_tac 1));
|
|
545 |
qed "dsumE";
|
|
546 |
|
|
547 |
AddSIs [uprodI, dprodI];
|
|
548 |
AddIs [usum_In0I, usum_In1I, dsum_In0I, dsum_In1I];
|
|
549 |
AddSEs [uprodE, dprodE, usumE, dsumE];
|
|
550 |
|
|
551 |
|
|
552 |
(*** Monotonicity ***)
|
|
553 |
|
|
554 |
Goal "[| r<=r'; s<=s' |] ==> dprod r s <= dprod r' s'";
|
|
555 |
by (Blast_tac 1);
|
|
556 |
qed "dprod_mono";
|
|
557 |
|
|
558 |
Goal "[| r<=r'; s<=s' |] ==> dsum r s <= dsum r' s'";
|
|
559 |
by (Blast_tac 1);
|
|
560 |
qed "dsum_mono";
|
|
561 |
|
|
562 |
|
|
563 |
(*** Bounding theorems ***)
|
|
564 |
|
|
565 |
Goal "(dprod (A <*> B) (C <*> D)) <= (uprod A C) <*> (uprod B D)";
|
|
566 |
by (Blast_tac 1);
|
|
567 |
qed "dprod_Sigma";
|
|
568 |
|
|
569 |
bind_thm ("dprod_subset_Sigma", [dprod_mono, dprod_Sigma] MRS subset_trans |> standard);
|
|
570 |
|
|
571 |
(*Dependent version*)
|
|
572 |
Goal "(dprod (Sigma A B) (Sigma C D)) <= Sigma (uprod A C) (Split (%x y. uprod (B x) (D y)))";
|
|
573 |
by Safe_tac;
|
|
574 |
by (stac Split 1);
|
|
575 |
by (Blast_tac 1);
|
|
576 |
qed "dprod_subset_Sigma2";
|
|
577 |
|
|
578 |
Goal "(dsum (A <*> B) (C <*> D)) <= (usum A C) <*> (usum B D)";
|
|
579 |
by (Blast_tac 1);
|
|
580 |
qed "dsum_Sigma";
|
|
581 |
|
|
582 |
bind_thm ("dsum_subset_Sigma", [dsum_mono, dsum_Sigma] MRS subset_trans |> standard);
|
|
583 |
|
|
584 |
|
|
585 |
(*** Domain ***)
|
|
586 |
|
|
587 |
Goal "Domain (dprod r s) = uprod (Domain r) (Domain s)";
|
|
588 |
by Auto_tac;
|
|
589 |
qed "Domain_dprod";
|
|
590 |
|
|
591 |
Goal "Domain (dsum r s) = usum (Domain r) (Domain s)";
|
|
592 |
by Auto_tac;
|
|
593 |
qed "Domain_dsum";
|
|
594 |
|
|
595 |
Addsimps [Domain_dprod, Domain_dsum];
|