src/HOL/Fun.ML
author paulson
Thu Jan 09 10:23:39 1997 +0100 (1997-01-09)
changeset 2499 0bc87b063447
parent 2031 03a843f0f447
child 2890 f27002fc531d
permissions -rw-r--r--
Tidying of proofs. New theorems are enterred immediately into the
relevant clasets or simpsets.
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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 Fun.thy "(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 Fun.thy
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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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(*** Image of a set under a function ***)
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(*Frequently b does not have the syntactic form of f(x).*)
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val prems = goalw Fun.thy [image_def] "[| b=f(x);  x:A |] ==> b : f``A";
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by (REPEAT (resolve_tac (prems @ [CollectI,bexI,prem]) 1));
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qed "image_eqI";
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bind_thm ("imageI", refl RS image_eqI);
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(*The eta-expansion gives variable-name preservation.*)
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val major::prems = goalw Fun.thy [image_def]
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    "[| b : (%x.f(x))``A;  !!x.[| b=f(x);  x:A |] ==> P |] ==> P"; 
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by (rtac (major RS CollectD RS bexE) 1);
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by (REPEAT (ares_tac prems 1));
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qed "imageE";
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AddIs  [imageI]; 
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AddSEs [imageE]; 
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goalw Fun.thy [o_def] "(f o g)``r = f``(g``r)";
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by (Fast_tac 1);
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qed "image_compose";
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goal Fun.thy "f``(A Un B) = f``A Un f``B";
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by (Fast_tac 1);
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qed "image_Un";
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(*** Range of a function -- just a translation for image! ***)
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goal Fun.thy "!!b. b=f(x) ==> b : range(f)";
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by (EVERY1 [etac image_eqI, rtac UNIV_I]);
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bind_thm ("range_eqI", UNIV_I RSN (2,image_eqI));
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bind_thm ("rangeI", UNIV_I RS imageI);
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val [major,minor] = goal Fun.thy 
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    "[| b : range(%x.f(x));  !!x. b=f(x) ==> P |] ==> P"; 
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by (rtac (major RS imageE) 1);
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by (etac minor 1);
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qed "rangeE";
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(*** inj(f): f is a one-to-one function ***)
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val prems = goalw Fun.thy [inj_def]
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    "[| !! x y. f(x) = f(y) ==> x=y |] ==> inj(f)";
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by (fast_tac (!claset addIs prems) 1);
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qed "injI";
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val [major] = goal Fun.thy "(!!x. g(f(x)) = x) ==> inj(f)";
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by (rtac injI 1);
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by (etac (arg_cong RS box_equals) 1);
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by (rtac major 1);
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by (rtac major 1);
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qed "inj_inverseI";
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val [major,minor] = goalw Fun.thy [inj_def]
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    "[| inj(f); f(x) = f(y) |] ==> x=y";
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by (rtac (major RS spec RS spec RS mp) 1);
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by (rtac minor 1);
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qed "injD";
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(*Useful with the simplifier*)
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val [major] = goal Fun.thy "inj(f) ==> (f(x) = f(y)) = (x=y)";
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by (rtac iffI 1);
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by (etac (major RS injD) 1);
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by (etac arg_cong 1);
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qed "inj_eq";
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val [major] = goal Fun.thy "inj(f) ==> (@x.f(x)=f(y)) = y";
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by (rtac (major RS 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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val [major] = goalw Fun.thy [Inv_def] "inj(f) ==> Inv f (f x) = x";
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by (EVERY1 [rtac (major RS inj_select)]);
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qed "Inv_f_f";
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(* Useful??? *)
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val [oneone,minor] = goal Fun.thy
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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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(*** inj_onto f A: f is one-to-one over A ***)
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val prems = goalw Fun.thy [inj_onto_def]
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    "(!! x y. [| f(x) = f(y);  x:A;  y:A |] ==> x=y) ==> inj_onto f A";
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by (fast_tac (!claset addIs prems) 1);
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qed "inj_ontoI";
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val [major] = goal Fun.thy 
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    "(!!x. x:A ==> g(f(x)) = x) ==> inj_onto f A";
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by (rtac inj_ontoI 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_onto_inverseI";
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val major::prems = goalw Fun.thy [inj_onto_def]
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    "[| inj_onto f A;  f(x)=f(y);  x:A;  y:A |] ==> x=y";
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by (rtac (major RS bspec RS bspec RS mp) 1);
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by (REPEAT (resolve_tac prems 1));
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qed "inj_ontoD";
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goal Fun.thy "!!x y.[| inj_onto f A;  x:A;  y:A |] ==> (f(x)=f(y)) = (x=y)";
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by (fast_tac (!claset addSEs [inj_ontoD]) 1);
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qed "inj_onto_iff";
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val major::prems = goal Fun.thy
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    "[| inj_onto f A;  ~x=y;  x:A;  y:A |] ==> ~ f(x)=f(y)";
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by (rtac contrapos 1);
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by (etac (major RS inj_ontoD) 2);
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by (REPEAT (resolve_tac prems 1));
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qed "inj_onto_contraD";
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(*** Lemmas about inj ***)
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val prems = goalw Fun.thy [o_def]
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    "[| inj(f);  inj_onto g (range f) |] ==> inj(g o f)";
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by (cut_facts_tac prems 1);
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by (fast_tac (!claset addIs [injI]
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                     addEs [injD,inj_ontoD]) 1);
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qed "comp_inj";
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val [prem] = goal Fun.thy "inj(f) ==> inj_onto f A";
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by (fast_tac (!claset addIs [prem RS injD, inj_ontoI]) 1);
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qed "inj_imp";
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val [prem] = goalw Fun.thy [Inv_def] "y : range(f) ==> f(Inv f y) = y";
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by (EVERY1 [rtac (prem RS rangeE), rtac selectI, etac sym]);
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qed "f_Inv_f";
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val prems = goal Fun.thy
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    "[| 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 (resolve_tac (prems @ [f_Inv_f]) 1));
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qed "Inv_injective";
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val prems = goal Fun.thy
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    "[| inj(f);  A<=range(f) |] ==> inj_onto (Inv f) A";
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by (cut_facts_tac prems 1);
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by (fast_tac (!claset addIs [inj_ontoI] 
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                      addEs [Inv_injective,injD]) 1);
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qed "inj_onto_Inv";
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AddIs  [rangeI]; 
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AddSEs [rangeE]; 
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val set_cs = !claset delrules [equalityI];
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