src/HOL/Sum.ML
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(*  Title:      HOL/Sum.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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The disjoint sum of two types
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*)
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(** Inl_Rep and Inr_Rep: Representations of the constructors **)
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(*This counts as a non-emptiness result for admitting 'a+'b as a type*)
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Goalw [Sum_def] "Inl_Rep(a) : Sum";
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by (EVERY1 [rtac CollectI, rtac disjI1, rtac exI, rtac refl]);
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qed "Inl_RepI";
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Goalw [Sum_def] "Inr_Rep(b) : Sum";
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by (EVERY1 [rtac CollectI, rtac disjI2, rtac exI, rtac refl]);
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qed "Inr_RepI";
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Goal "inj_on Abs_Sum Sum";
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by (rtac inj_on_inverseI 1);
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by (etac Abs_Sum_inverse 1);
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qed "inj_on_Abs_Sum";
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(** Distinctness of Inl and Inr **)
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Goalw [Inl_Rep_def, Inr_Rep_def] "Inl_Rep(a) ~= Inr_Rep(b)";
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by (EVERY1 [rtac notI,
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            etac (fun_cong RS fun_cong RS fun_cong RS iffE), 
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            rtac (notE RS ccontr),  etac (mp RS conjunct2), 
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            REPEAT o (ares_tac [refl,conjI]) ]);
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qed "Inl_Rep_not_Inr_Rep";
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Goalw [Inl_def,Inr_def] "Inl(a) ~= Inr(b)";
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by (rtac (inj_on_Abs_Sum RS inj_on_contraD) 1);
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by (rtac Inl_Rep_not_Inr_Rep 1);
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by (rtac Inl_RepI 1);
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by (rtac Inr_RepI 1);
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qed "Inl_not_Inr";
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bind_thm ("Inr_not_Inl", Inl_not_Inr RS not_sym);
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AddIffs [Inl_not_Inr, Inr_not_Inl];
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bind_thm ("Inl_neq_Inr", Inl_not_Inr RS notE);
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val Inr_neq_Inl = sym RS Inl_neq_Inr;
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(** Injectiveness of Inl and Inr **)
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Goalw [Inl_Rep_def] "Inl_Rep(a) = Inl_Rep(c) ==> a=c";
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by (etac (fun_cong RS fun_cong RS fun_cong RS iffE) 1);
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by (Blast_tac 1);
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qed "Inl_Rep_inject";
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Goalw [Inr_Rep_def] "Inr_Rep(b) = Inr_Rep(d) ==> b=d";
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by (etac (fun_cong RS fun_cong RS fun_cong RS iffE) 1);
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by (Blast_tac 1);
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qed "Inr_Rep_inject";
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Goalw [Inl_def] "inj(Inl)";
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by (rtac injI 1);
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by (etac (inj_on_Abs_Sum RS inj_onD RS Inl_Rep_inject) 1);
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by (rtac Inl_RepI 1);
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by (rtac Inl_RepI 1);
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qed "inj_Inl";
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val Inl_inject = inj_Inl RS injD;
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Goalw [Inr_def] "inj(Inr)";
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by (rtac injI 1);
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by (etac (inj_on_Abs_Sum RS inj_onD RS Inr_Rep_inject) 1);
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by (rtac Inr_RepI 1);
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by (rtac Inr_RepI 1);
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qed "inj_Inr";
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val Inr_inject = inj_Inr RS injD;
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Goal "(Inl(x)=Inl(y)) = (x=y)";
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by (blast_tac (claset() addSDs [Inl_inject]) 1);
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qed "Inl_eq";
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Goal "(Inr(x)=Inr(y)) = (x=y)";
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by (blast_tac (claset() addSDs [Inr_inject]) 1);
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qed "Inr_eq";
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AddIffs [Inl_eq, Inr_eq];
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(*** Rules for the disjoint sum of two SETS ***)
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(** Introduction rules for the injections **)
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Goalw [sum_def] "a : A ==> Inl(a) : A Plus B";
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by (Blast_tac 1);
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qed "InlI";
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Goalw [sum_def] "b : B ==> Inr(b) : A Plus B";
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by (Blast_tac 1);
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qed "InrI";
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(** Elimination rules **)
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val major::prems = Goalw [sum_def]
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    "[| u: A Plus B;  \
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\       !!x. [| x:A;  u=Inl(x) |] ==> P; \
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\       !!y. [| y:B;  u=Inr(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 "PlusE";
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AddSIs [InlI, InrI]; 
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AddSEs [PlusE];
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(** Exhaustion rule for sums -- a degenerate form of induction **)
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val prems = Goalw [Inl_def,Inr_def]
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    "[| !!x::'a. s = Inl(x) ==> P;  !!y::'b. s = Inr(y) ==> P \
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\    |] ==> P";
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by (rtac (rewrite_rule [Sum_def] Rep_Sum RS CollectE) 1);
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by (REPEAT (eresolve_tac [disjE,exE] 1
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     ORELSE EVERY1 [resolve_tac prems, 
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                    etac subst,
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                    rtac (Rep_Sum_inverse RS sym)]));
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qed "sumE";
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val prems = Goal "[| !!x. P (Inl x); !!x. P (Inr x) |] ==> P x";
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by (res_inst_tac [("s","x")] sumE 1);
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by (ALLGOALS (hyp_subst_tac THEN' (resolve_tac prems)));
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qed "sum_induct";
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(** Rules for the Part primitive **)
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Goalw [Part_def] "[| a : A;  a=h(b) |] ==> a : Part A h";
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by (Blast_tac 1);
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qed "Part_eqI";
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val PartI = refl RSN (2,Part_eqI);
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val major::prems = Goalw [Part_def]
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    "[| a : Part A h;  !!z. [| a : A;  a=h(z) |] ==> P  \
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\    |] ==> P";
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by (rtac (major RS IntE) 1);
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by (etac CollectE 1);
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by (etac exE 1);
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by (REPEAT (ares_tac prems 1));
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qed "PartE";
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AddIs  [Part_eqI];
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AddSEs [PartE];
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Goalw [Part_def] "Part A h <= A";
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by (rtac Int_lower1 1);
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qed "Part_subset";
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   158
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b94cd208f073 Removal of leading "\!\!..." from most Goal commands
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Goal "A<=B ==> Part A h <= Part B h";
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by (Blast_tac 1);
923
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qed "Part_mono";
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   162
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val basic_monos = basic_monos @ [Part_mono];
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   164
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Goalw [Part_def] "a : Part A h ==> a : A";
923
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by (etac IntD1 1);
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qed "PartD1";
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   168
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Goal "Part A (%x. x) = A";
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by (Blast_tac 1);
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qed "Part_id";
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   172
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Goal "Part (A Int B) h = (Part A h) Int (Part B h)";
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by (Blast_tac 1);
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qed "Part_Int";
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Goal "Part (A Int {x. P x}) h = (Part A h) Int {x. P x}";
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by (Blast_tac 1);
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qed "Part_Collect";