src/HOL/Fun.ML
author nipkow
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(*  Title:      HOL/Fun
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    ID:         $Id$
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    Author:     Tobias Nipkow, Cambridge University Computer Laboratory
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    Copyright   1993  University of Cambridge
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Lemmas about functions.
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*)
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Goal "(f = g) = (!x. f(x)=g(x))";
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by (rtac iffI 1);
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by (Asm_simp_tac 1);
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by (rtac ext 1 THEN Asm_simp_tac 1);
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qed "expand_fun_eq";
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val prems = Goal
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    "[| f(x)=u;  !!x. P(x) ==> g(f(x)) = x;  P(x) |] ==> x=g(u)";
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by (rtac (arg_cong RS box_equals) 1);
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by (REPEAT (resolve_tac (prems@[refl]) 1));
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qed "apply_inverse";
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(** "Axiom" of Choice, proved using the description operator **)
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Goal "!!Q. ALL x. EX y. Q x y ==> EX f. ALL x. Q x (f x)";
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by (fast_tac (claset() addEs [selectI]) 1);
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qed "choice";
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Goal "!!S. ALL x:S. EX y. Q x y ==> EX f. ALL x:S. Q x (f x)";
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by (fast_tac (claset() addEs [selectI]) 1);
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qed "bchoice";
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section "id";
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qed_goalw "id_apply" thy [id_def] "id x = x" (K [rtac refl 1]);
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Addsimps [id_apply];
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section "o";
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qed_goalw "o_apply" thy [o_def] "(f o g) x = f (g x)"
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 (K [rtac refl 1]);
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Addsimps [o_apply];
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qed_goalw "o_assoc" thy [o_def] "f o (g o h) = f o g o h"
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  (K [rtac ext 1, rtac refl 1]);
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qed_goalw "id_o" thy [id_def] "id o g = g"
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 (K [rtac ext 1, Simp_tac 1]);
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Addsimps [id_o];
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qed_goalw "o_id" thy [id_def] "f o id = f"
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 (K [rtac ext 1, Simp_tac 1]);
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Addsimps [o_id];
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Goalw [o_def] "(f o g)``r = f``(g``r)";
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by (Blast_tac 1);
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qed "image_compose";
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Goalw [o_def] "UNION A (g o f) = UNION (f``A) g";
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by (Blast_tac 1);
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qed "UN_o";
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(** lemma for proving injectivity of representation functions for **)
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(** datatypes involving function types                            **)
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Goalw [o_def]
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  "[| !x y. g (f x) = g y --> f x = y; g o f = g o fa |] ==> f = fa";
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br ext 1;
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be allE 1;
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be allE 1;
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be mp 1;
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be fun_cong 1;
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qed "inj_fun_lemma";
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section "inj";
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(**NB: inj now just translates to inj_on**)
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(*** inj(f): f is a one-to-one function ***)
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(*for Tools/datatype_rep_proofs*)
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val [prem] = Goalw [inj_on_def]
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    "(!! x. ALL y. f(x) = f(y) --> x=y) ==> inj(f)";
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by (blast_tac (claset() addIs [prem RS spec RS mp]) 1);
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qed "datatype_injI";
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Goalw [inj_on_def] "[| inj(f); f(x) = f(y) |] ==> x=y";
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by (Blast_tac 1);
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qed "injD";
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(*Useful with the simplifier*)
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Goal "inj(f) ==> (f(x) = f(y)) = (x=y)";
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by (rtac iffI 1);
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by (etac arg_cong 2);
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by (etac injD 1);
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by (assume_tac 1);
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qed "inj_eq";
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Goal "inj(f) ==> (@x. f(x)=f(y)) = y";
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by (etac injD 1);
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by (rtac selectI 1);
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by (rtac refl 1);
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qed "inj_select";
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(*A one-to-one function has an inverse (given using select).*)
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Goalw [inv_def] "inj(f) ==> inv f (f x) = x";
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by (etac inj_select 1);
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qed "inv_f_f";
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Addsimps [inv_f_f];
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(* Useful??? *)
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val [oneone,minor] = Goal
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    "[| inj(f); !!y. y: range(f) ==> P(inv f y) |] ==> P(x)";
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by (res_inst_tac [("t", "x")] (oneone RS (inv_f_f RS subst)) 1);
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by (rtac (rangeI RS minor) 1);
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qed "inj_transfer";
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Goalw [o_def] "[| inj f; f o g = f o h |] ==> g = h";
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by (rtac ext 1);
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by (etac injD 1);
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by (etac fun_cong 1);
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qed "inj_o";
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(*** inj_on f A: f is one-to-one over A ***)
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val prems = Goalw [inj_on_def]
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    "(!! x y. [| f(x) = f(y);  x:A;  y:A |] ==> x=y) ==> inj_on f A";
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by (blast_tac (claset() addIs prems) 1);
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qed "inj_onI";
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val injI = inj_onI;                  (*for compatibility*)
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val [major] = Goal 
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    "(!!x. x:A ==> g(f(x)) = x) ==> inj_on f A";
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by (rtac inj_onI 1);
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by (etac (apply_inverse RS trans) 1);
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by (REPEAT (eresolve_tac [asm_rl,major] 1));
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qed "inj_on_inverseI";
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val inj_inverseI = inj_on_inverseI;   (*for compatibility*)
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Goalw [inj_on_def] "[| inj_on f A;  f(x)=f(y);  x:A;  y:A |] ==> x=y";
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by (Blast_tac 1);
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qed "inj_onD";
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Goal "[| inj_on f A;  x:A;  y:A |] ==> (f(x)=f(y)) = (x=y)";
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by (blast_tac (claset() addSDs [inj_onD]) 1);
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qed "inj_on_iff";
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Goalw [inj_on_def] "[| inj_on f A;  ~x=y;  x:A;  y:A |] ==> ~ f(x)=f(y)";
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by (Blast_tac 1);
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qed "inj_on_contraD";
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Goalw [inj_on_def] "[| A<=B; inj_on f B |] ==> inj_on f A";
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by (Blast_tac 1);
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qed "subset_inj_on";
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(** surj **)
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val [prem] = Goalw [surj_def] "(!! x. g(f x) = x) ==> surj g";
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by (blast_tac (claset() addIs [prem RS sym]) 1);
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qed "surjI";
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Goalw [surj_def] "surj f ==> range f = UNIV";
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by Auto_tac;
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qed "surj_range";
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Goalw [surj_def] "surj f ==> EX x. y = f x";
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by (Blast_tac 1);
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qed "surjD";
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(*** Lemmas about injective functions and inv ***)
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Goalw [o_def] "[| inj_on f A;  inj_on g (f``A) |] ==> inj_on (g o f) A";
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by (fast_tac (claset() addIs [inj_onI] addEs [inj_onD]) 1);
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qed "comp_inj_on";
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Goalw [inv_def] "y : range(f) ==> f(inv f y) = y";
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by (fast_tac (claset() addIs [selectI]) 1);
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qed "f_inv_f";
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Goal "surj f ==> f(inv f y) = y";
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by (asm_simp_tac (simpset() addsimps [f_inv_f, surj_range]) 1);
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qed "surj_f_inv_f";
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Goal "[| inv f x = inv f y;  x: range(f);  y: range(f) |] ==> x=y";
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by (rtac (arg_cong RS box_equals) 1);
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by (REPEAT (ares_tac [f_inv_f] 1));
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qed "inv_injective";
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Goal "A <= range(f) ==> inj_on (inv f) A";
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by (fast_tac (claset() addIs [inj_onI] 
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                       addEs [inv_injective, injD]) 1);
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qed "inj_on_inv";
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Goal "surj f ==> inj (inv f)";
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by (asm_simp_tac (simpset() addsimps [inj_on_inv, surj_range]) 1);
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qed "surj_imp_inj_inv";
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Goal "f``(A Int B) <= f``A Int f``B";
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by (Blast_tac 1);
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qed "image_Int_subset";
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Goal "f``A - f``B <= f``(A - B)";
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by (Blast_tac 1);
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qed "image_diff_subset";
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Goalw [inj_on_def]
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   "[| inj_on f C;  A<=C;  B<=C |] ==> f``(A Int B) = f``A Int f``B";
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by (Blast_tac 1);
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qed "inj_on_image_Int";
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Goalw [inj_on_def]
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   "[| inj_on f C;  A<=C;  B<=C |] ==> f``(A-B) = f``A - f``B";
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by (Blast_tac 1);
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qed "inj_on_image_set_diff";
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Goalw [inj_on_def] "inj f ==> f``(A Int B) = f``A Int f``B";
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by (Blast_tac 1);
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qed "image_Int";
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Goalw [inj_on_def] "inj f ==> f``(A-B) = f``A - f``B";
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by (Blast_tac 1);
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qed "image_set_diff";
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Goalw [image_def] "inj(f) ==> inv(f)``(f``X) = X";
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by Auto_tac;
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qed "inv_image_comp";
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Goal "inj f ==> (f a : f``A) = (a : A)";
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by (blast_tac (claset() addDs [injD]) 1);
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qed "inj_image_mem_iff";
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Goal "inj f ==> (f``A = f``B) = (A = B)";
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by (blast_tac (claset() addSEs [equalityE] addDs [injD]) 1);
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qed "inj_image_eq_iff";
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Goal  "(f `` (UNION A B)) = (UN x:A.(f `` (B x)))";
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by (Blast_tac 1);
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qed "image_UN";
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(*injectivity's required.  Left-to-right inclusion holds even if A is empty*)
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Goalw [inj_on_def]
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   "[| inj_on f C;  ALL x:A. B x <= C;  j:A |] \
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\   ==> f `` (INTER A B) = (INT x:A. f `` B x)";
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by (Blast_tac 1);
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qed "image_INT";
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val set_cs = claset() delrules [equalityI];
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section "fun_upd";
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Goalw [fun_upd_def] "(f(x:=y) = f) = (f x = y)";
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by Safe_tac;
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by (etac subst 1);
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by (rtac ext 2);
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by Auto_tac;
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qed "fun_upd_idem_iff";
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(* f x = y ==> f(x:=y) = f *)
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bind_thm("fun_upd_idem", fun_upd_idem_iff RS iffD2);
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(* f(x := f x) = f *)
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AddIffs [refl RS fun_upd_idem];
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Goal "(f(x:=y))z = (if z=x then y else f z)";
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by (simp_tac (simpset() addsimps [fun_upd_def]) 1);
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qed "fun_upd_apply";
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Addsimps [fun_upd_apply];
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qed_goal "fun_upd_same" thy "(f(x:=y)) x = y" 
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	(K [Simp_tac 1]);
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qed_goal "fun_upd_other" thy "!!X. z~=x ==> (f(x:=y)) z = f z"
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	(K [Asm_simp_tac 1]);
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(*Addsimps [fun_upd_same, fun_upd_other];*)
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Goal "a ~= c ==> m(a:=b)(c:=d) = m(c:=d)(a:=b)";
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by (rtac ext 1);
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by (Auto_tac);
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qed "fun_upd_twist";
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(*** -> and Pi, by Florian Kammueller and LCP ***)
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val prems = Goalw [Pi_def]
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"[| !!x. x: A ==> f x: B x; !!x. x ~: A  ==> f(x) = (@ y. True)|] \
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\    ==> f: Pi A B";
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by (auto_tac (claset(), simpset() addsimps prems));
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qed "Pi_I";
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val prems = Goal 
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"[| !!x. x: A ==> f x: B; !!x. x ~: A  ==> f(x) = (@ y. True)|] ==> f: A funcset B";
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by (blast_tac (claset() addIs Pi_I::prems) 1);
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qed "funcsetI";
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Goalw [Pi_def] "[|f: Pi A B; x: A|] ==> f x: B x";
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by Auto_tac;
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qed "Pi_mem";
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Goalw [Pi_def] "[|f: A funcset B; x: A|] ==> f x: B";
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by Auto_tac;
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qed "funcset_mem";
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Goalw [Pi_def] "[|f: Pi A B; x~: A|] ==> f x = (@ y. True)";
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by Auto_tac;
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qed "apply_arb";
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Goalw [Pi_def] "[| f: Pi A B; g: Pi A B; ! x: A. f x = g x |] ==> f = g";
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by (rtac ext 1);
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by Auto_tac;
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val Pi_extensionality = ballI RSN (3, result());
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   316
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   317
(*** compose ***)
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   318
4d7320490be4 the function space operator
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   319
Goalw [Pi_def, compose_def, restrict_def]
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   320
     "[| f: A funcset B; g: B funcset C |]==> compose A g f: A funcset C";
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   321
by Auto_tac;
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   322
qed "funcset_compose";
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   323
4d7320490be4 the function space operator
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   324
Goal "[| f: A funcset B; g: B funcset C; h: C funcset D |]\
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   325
\     ==> compose A h (compose A g f) = compose A (compose B h g) f";
4d7320490be4 the function space operator
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diff changeset
   326
by (res_inst_tac [("A","A")] Pi_extensionality 1);
4d7320490be4 the function space operator
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diff changeset
   327
by (blast_tac (claset() addIs [funcset_compose]) 1);
4d7320490be4 the function space operator
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diff changeset
   328
by (blast_tac (claset() addIs [funcset_compose]) 1);
4d7320490be4 the function space operator
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diff changeset
   329
by (rewrite_goals_tac [Pi_def, compose_def, restrict_def]);  
4d7320490be4 the function space operator
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   330
by Auto_tac;
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   331
qed "compose_assoc";
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   332
4d7320490be4 the function space operator
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diff changeset
   333
Goal "[| f: A funcset B; g: B funcset C; x: A |]==> compose A g f x = g(f(x))";
4d7320490be4 the function space operator
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diff changeset
   334
by (asm_full_simp_tac (simpset() addsimps [compose_def, restrict_def]) 1);
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   335
qed "compose_eq";
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diff changeset
   336
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diff changeset
   337
Goal "[| f : A funcset B; f `` A = B; g: B funcset C; g `` B = C |]\
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   338
\     ==> compose A g f `` A = C";
4d7320490be4 the function space operator
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   339
by (auto_tac (claset(),
4d7320490be4 the function space operator
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   340
	      simpset() addsimps [image_def, compose_eq]));
4d7320490be4 the function space operator
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   341
qed "surj_compose";
4d7320490be4 the function space operator
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diff changeset
   342
4d7320490be4 the function space operator
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diff changeset
   343
4d7320490be4 the function space operator
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diff changeset
   344
Goal "[| f : A funcset B; g: B funcset C; f `` A = B; inj_on f A; inj_on g B |]\
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   345
\     ==> inj_on (compose A g f) A";
4d7320490be4 the function space operator
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diff changeset
   346
by (auto_tac (claset(),
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   347
	      simpset() addsimps [inj_on_def, compose_eq]));
4d7320490be4 the function space operator
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   348
qed "inj_on_compose";
4d7320490be4 the function space operator
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   349
4d7320490be4 the function space operator
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diff changeset
   350
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   351
(*** restrict / lam ***)
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   352
Goal "[| f `` A <= B |] ==> (lam x: A. f x) : A funcset B";
4d7320490be4 the function space operator
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diff changeset
   353
by (auto_tac (claset(),
4d7320490be4 the function space operator
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   354
	      simpset() addsimps [restrict_def, Pi_def]));
4d7320490be4 the function space operator
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   355
qed "restrict_in_funcset";
4d7320490be4 the function space operator
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diff changeset
   356
4d7320490be4 the function space operator
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   357
val prems = Goalw [restrict_def, Pi_def]
4d7320490be4 the function space operator
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diff changeset
   358
     "(!!x. x: A ==> f x: B x) ==> (lam x: A. f x) : Pi A B";
4d7320490be4 the function space operator
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diff changeset
   359
by (asm_simp_tac (simpset() addsimps prems) 1);
4d7320490be4 the function space operator
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   360
qed "restrictI";
4d7320490be4 the function space operator
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diff changeset
   361
4d7320490be4 the function space operator
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diff changeset
   362
4d7320490be4 the function space operator
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diff changeset
   363
Goal "x: A ==> (lam y: A. f y) x = f x";
4d7320490be4 the function space operator
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diff changeset
   364
by (asm_simp_tac (simpset() addsimps [restrict_def]) 1);
4d7320490be4 the function space operator
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diff changeset
   365
qed "restrict_apply1";
4d7320490be4 the function space operator
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diff changeset
   366
4d7320490be4 the function space operator
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diff changeset
   367
Goal "[| x: A; f : A funcset B |] ==> (lam y: A. f y) x : B";
4d7320490be4 the function space operator
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diff changeset
   368
by (asm_full_simp_tac (simpset() addsimps [restrict_apply1,Pi_def]) 1);
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   369
qed "restrict_apply1_mem";
4d7320490be4 the function space operator
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diff changeset
   370
4d7320490be4 the function space operator
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diff changeset
   371
Goal "x ~: A ==> (lam y: A. f y) x =  (@ y. True)";
4d7320490be4 the function space operator
paulson
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diff changeset
   372
by (asm_simp_tac (simpset() addsimps [restrict_def]) 1);
4d7320490be4 the function space operator
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diff changeset
   373
qed "restrict_apply2";
4d7320490be4 the function space operator
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diff changeset
   374
4d7320490be4 the function space operator
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diff changeset
   375
4d7320490be4 the function space operator
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   376
val prems = Goal
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   377
    "(!!x. x: A ==> f x = g x) ==> (lam x: A. f x) = (lam x: A. g x)";
4d7320490be4 the function space operator
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diff changeset
   378
by (rtac ext 1);
4d7320490be4 the function space operator
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diff changeset
   379
by (auto_tac (claset(),
4d7320490be4 the function space operator
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diff changeset
   380
	      simpset() addsimps prems@[restrict_def, Pi_def]));
4d7320490be4 the function space operator
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   381
qed "restrict_ext";
4d7320490be4 the function space operator
paulson
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diff changeset
   382
4d7320490be4 the function space operator
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diff changeset
   383
4d7320490be4 the function space operator
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diff changeset
   384
(*** Inverse ***)
4d7320490be4 the function space operator
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diff changeset
   385
4d7320490be4 the function space operator
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diff changeset
   386
Goal "[|f `` A = B;  x: B |] ==> ? y: A. f y = x";
4d7320490be4 the function space operator
paulson
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diff changeset
   387
by (Blast_tac 1);
4d7320490be4 the function space operator
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diff changeset
   388
qed "surj_image";
4d7320490be4 the function space operator
paulson
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diff changeset
   389
4d7320490be4 the function space operator
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diff changeset
   390
Goalw [Inv_def] "[| f `` A = B; f : A funcset B |] \
4d7320490be4 the function space operator
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diff changeset
   391
\                ==> (lam x: B. (Inv A f) x) : B funcset A";
4d7320490be4 the function space operator
paulson
parents: 5847
diff changeset
   392
by (fast_tac (claset() addIs [restrict_in_funcset, selectI2]) 1);
4d7320490be4 the function space operator
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diff changeset
   393
qed "Inv_funcset";
4d7320490be4 the function space operator
paulson
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diff changeset
   394
4d7320490be4 the function space operator
paulson
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diff changeset
   395
4d7320490be4 the function space operator
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diff changeset
   396
Goal "[| f: A funcset B;  inj_on f A;  f `` A = B;  x: A |] \
4d7320490be4 the function space operator
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diff changeset
   397
\     ==> (lam y: B. (Inv A f) y) (f x) = x";
4d7320490be4 the function space operator
paulson
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diff changeset
   398
by (asm_simp_tac (simpset() addsimps [restrict_apply1, funcset_mem]) 1);
4d7320490be4 the function space operator
paulson
parents: 5847
diff changeset
   399
by (asm_full_simp_tac (simpset() addsimps [Inv_def, inj_on_def]) 1);
4d7320490be4 the function space operator
paulson
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diff changeset
   400
by (rtac selectI2 1);
4d7320490be4 the function space operator
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diff changeset
   401
by Auto_tac;
4d7320490be4 the function space operator
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diff changeset
   402
qed "Inv_f_f";
4d7320490be4 the function space operator
paulson
parents: 5847
diff changeset
   403
4d7320490be4 the function space operator
paulson
parents: 5847
diff changeset
   404
Goal "[| f: A funcset B;  f `` A = B;  x: B |] \
4d7320490be4 the function space operator
paulson
parents: 5847
diff changeset
   405
\     ==> f ((lam y: B. (Inv A f y)) x) = x";
4d7320490be4 the function space operator
paulson
parents: 5847
diff changeset
   406
by (asm_simp_tac (simpset() addsimps [Inv_def, restrict_apply1]) 1);
4d7320490be4 the function space operator
paulson
parents: 5847
diff changeset
   407
by (fast_tac (claset() addIs [selectI2]) 1);
4d7320490be4 the function space operator
paulson
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diff changeset
   408
qed "f_Inv_f";
4d7320490be4 the function space operator
paulson
parents: 5847
diff changeset
   409
4d7320490be4 the function space operator
paulson
parents: 5847
diff changeset
   410
Goal "[| f: A funcset B;  inj_on f A;  f `` A = B |]\
4d7320490be4 the function space operator
paulson
parents: 5847
diff changeset
   411
\     ==> compose A (lam y:B. (Inv A f) y) f = (lam x: A. x)";
4d7320490be4 the function space operator
paulson
parents: 5847
diff changeset
   412
by (rtac Pi_extensionality 1);
4d7320490be4 the function space operator
paulson
parents: 5847
diff changeset
   413
by (blast_tac (claset() addIs [funcset_compose, Inv_funcset]) 1);
4d7320490be4 the function space operator
paulson
parents: 5847
diff changeset
   414
by (blast_tac (claset() addIs [restrict_in_funcset]) 1);
4d7320490be4 the function space operator
paulson
parents: 5847
diff changeset
   415
by (asm_simp_tac
4d7320490be4 the function space operator
paulson
parents: 5847
diff changeset
   416
    (simpset() addsimps [restrict_apply1, compose_def, Inv_f_f]) 1);
4d7320490be4 the function space operator
paulson
parents: 5847
diff changeset
   417
qed "compose_Inv_id";
4d7320490be4 the function space operator
paulson
parents: 5847
diff changeset
   418
4d7320490be4 the function space operator
paulson
parents: 5847
diff changeset
   419
4d7320490be4 the function space operator
paulson
parents: 5847
diff changeset
   420
(*** Pi and Applyall ***)
4d7320490be4 the function space operator
paulson
parents: 5847
diff changeset
   421
4d7320490be4 the function space operator
paulson
parents: 5847
diff changeset
   422
Goalw [Pi_def] "[| B(x) = {};  x: A |] ==> (PI x: A. B x) = {}";
4d7320490be4 the function space operator
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parents: 5847
diff changeset
   423
by Auto_tac;
4d7320490be4 the function space operator
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diff changeset
   424
qed "Pi_eq_empty";
4d7320490be4 the function space operator
paulson
parents: 5847
diff changeset
   425
4d7320490be4 the function space operator
paulson
parents: 5847
diff changeset
   426
Goal "[| (PI x: A. B x) ~= {};  x: A |] ==> B(x) ~= {}";
4d7320490be4 the function space operator
paulson
parents: 5847
diff changeset
   427
by (blast_tac (HOL_cs addIs [Pi_eq_empty]) 1);
4d7320490be4 the function space operator
paulson
parents: 5847
diff changeset
   428
qed "Pi_total1";
4d7320490be4 the function space operator
paulson
parents: 5847
diff changeset
   429
4d7320490be4 the function space operator
paulson
parents: 5847
diff changeset
   430
Goal "[| a : A; Pi A B ~= {} |] ==> Applyall (Pi A B) a = B a";
4d7320490be4 the function space operator
paulson
parents: 5847
diff changeset
   431
by (auto_tac (claset(), simpset() addsimps [Applyall_def, Pi_def]));
4d7320490be4 the function space operator
paulson
parents: 5847
diff changeset
   432
by (rename_tac "g z" 1);
4d7320490be4 the function space operator
paulson
parents: 5847
diff changeset
   433
by (res_inst_tac [("x","%y. if  (y = a) then z else g y")] exI 1);
4d7320490be4 the function space operator
paulson
parents: 5847
diff changeset
   434
by (auto_tac (claset(), simpset() addsimps [split_if_mem1, split_if_eq1]));
4d7320490be4 the function space operator
paulson
parents: 5847
diff changeset
   435
qed "Applyall_beta";
4d7320490be4 the function space operator
paulson
parents: 5847
diff changeset
   436
5865
2303f5a3036d moved some facts about Pi from ex/PiSets to Fun.ML
paulson
parents: 5852
diff changeset
   437
Goal "Pi {} B = { (%x. @ y. True) }";
2303f5a3036d moved some facts about Pi from ex/PiSets to Fun.ML
paulson
parents: 5852
diff changeset
   438
by (auto_tac (claset() addIs [ext], simpset() addsimps [Pi_def]));
2303f5a3036d moved some facts about Pi from ex/PiSets to Fun.ML
paulson
parents: 5852
diff changeset
   439
qed "Pi_empty";
5852
4d7320490be4 the function space operator
paulson
parents: 5847
diff changeset
   440
5865
2303f5a3036d moved some facts about Pi from ex/PiSets to Fun.ML
paulson
parents: 5852
diff changeset
   441
val [major] = Goalw [Pi_def] "(!!x. x: A ==> B x <= C x) ==> Pi A B <= Pi A C";
2303f5a3036d moved some facts about Pi from ex/PiSets to Fun.ML
paulson
parents: 5852
diff changeset
   442
by (auto_tac (claset(),
2303f5a3036d moved some facts about Pi from ex/PiSets to Fun.ML
paulson
parents: 5852
diff changeset
   443
	      simpset() addsimps [impOfSubs major]));
2303f5a3036d moved some facts about Pi from ex/PiSets to Fun.ML
paulson
parents: 5852
diff changeset
   444
qed "Pi_mono";