diff -r f04b33ce250f -r a4dc62a46ee4 Fun.ML --- a/Fun.ML Tue Oct 24 14:59:17 1995 +0100 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,200 +0,0 @@ -(* Title: HOL/Fun - ID: $Id$ - Author: Tobias Nipkow, Cambridge University Computer Laboratory - Copyright 1993 University of Cambridge - -Lemmas about functions. -*) - -goal Fun.thy "(f = g) = (!x. f(x)=g(x))"; -by (rtac iffI 1); -by(asm_simp_tac HOL_ss 1); -by(rtac ext 1 THEN asm_simp_tac HOL_ss 1); -qed "expand_fun_eq"; - -val prems = goal Fun.thy - "[| f(x)=u; !!x. P(x) ==> g(f(x)) = x; P(x) |] ==> x=g(u)"; -by (rtac (arg_cong RS box_equals) 1); -by (REPEAT (resolve_tac (prems@[refl]) 1)); -qed "apply_inverse"; - - -(*** Range of a function ***) - -(*Frequently b does not have the syntactic form of f(x).*) -val [prem] = goalw Fun.thy [range_def] "b=f(x) ==> b : range(f)"; -by (EVERY1 [rtac CollectI, rtac exI, rtac prem]); -qed "range_eqI"; - -val rangeI = refl RS range_eqI; - -val [major,minor] = goalw Fun.thy [range_def] - "[| b : range(%x.f(x)); !!x. b=f(x) ==> P |] ==> P"; -by (rtac (major RS CollectD RS exE) 1); -by (etac minor 1); -qed "rangeE"; - -(*** Image of a set under a function ***) - -val prems = goalw Fun.thy [image_def] "[| b=f(x); x:A |] ==> b : f``A"; -by (REPEAT (resolve_tac (prems @ [CollectI,bexI,prem]) 1)); -qed "image_eqI"; - -val imageI = refl RS image_eqI; - -(*The eta-expansion gives variable-name preservation.*) -val major::prems = goalw Fun.thy [image_def] - "[| b : (%x.f(x))``A; !!x.[| b=f(x); x:A |] ==> P |] ==> P"; -by (rtac (major RS CollectD RS bexE) 1); -by (REPEAT (ares_tac prems 1)); -qed "imageE"; - -goalw Fun.thy [o_def] "(f o g)``r = f``(g``r)"; -by (rtac set_ext 1); -by (fast_tac (HOL_cs addIs [imageI] addSEs [imageE]) 1); -qed "image_compose"; - -goal Fun.thy "f``(A Un B) = f``A Un f``B"; -by (rtac set_ext 1); -by (fast_tac (HOL_cs addIs [imageI,UnCI] addSEs [imageE,UnE]) 1); -qed "image_Un"; - -(*** inj(f): f is a one-to-one function ***) - -val prems = goalw Fun.thy [inj_def] - "[| !! x y. f(x) = f(y) ==> x=y |] ==> inj(f)"; -by (fast_tac (HOL_cs addIs prems) 1); -qed "injI"; - -val [major] = goal Fun.thy "(!!x. g(f(x)) = x) ==> inj(f)"; -by (rtac injI 1); -by (etac (arg_cong RS box_equals) 1); -by (rtac major 1); -by (rtac major 1); -qed "inj_inverseI"; - -val [major,minor] = goalw Fun.thy [inj_def] - "[| inj(f); f(x) = f(y) |] ==> x=y"; -by (rtac (major RS spec RS spec RS mp) 1); -by (rtac minor 1); -qed "injD"; - -(*Useful with the simplifier*) -val [major] = goal Fun.thy "inj(f) ==> (f(x) = f(y)) = (x=y)"; -by (rtac iffI 1); -by (etac (major RS injD) 1); -by (etac arg_cong 1); -qed "inj_eq"; - -val [major] = goal Fun.thy "inj(f) ==> (@x.f(x)=f(y)) = y"; -by (rtac (major RS injD) 1); -by (rtac selectI 1); -by (rtac refl 1); -qed "inj_select"; - -(*A one-to-one function has an inverse (given using select).*) -val [major] = goalw Fun.thy [Inv_def] "inj(f) ==> Inv(f,f(x)) = x"; -by (EVERY1 [rtac (major RS inj_select)]); -qed "Inv_f_f"; - -(* Useful??? *) -val [oneone,minor] = goal Fun.thy - "[| inj(f); !!y. y: range(f) ==> P(Inv(f,y)) |] ==> P(x)"; -by (res_inst_tac [("t", "x")] (oneone RS (Inv_f_f RS subst)) 1); -by (rtac (rangeI RS minor) 1); -qed "inj_transfer"; - - -(*** inj_onto(f,A): f is one-to-one over A ***) - -val prems = goalw Fun.thy [inj_onto_def] - "(!! x y. [| f(x) = f(y); x:A; y:A |] ==> x=y) ==> inj_onto(f,A)"; -by (fast_tac (HOL_cs addIs prems addSIs [ballI]) 1); -qed "inj_ontoI"; - -val [major] = goal Fun.thy - "(!!x. x:A ==> g(f(x)) = x) ==> inj_onto(f,A)"; -by (rtac inj_ontoI 1); -by (etac (apply_inverse RS trans) 1); -by (REPEAT (eresolve_tac [asm_rl,major] 1)); -qed "inj_onto_inverseI"; - -val major::prems = goalw Fun.thy [inj_onto_def] - "[| inj_onto(f,A); f(x)=f(y); x:A; y:A |] ==> x=y"; -by (rtac (major RS bspec RS bspec RS mp) 1); -by (REPEAT (resolve_tac prems 1)); -qed "inj_ontoD"; - -goal Fun.thy "!!x y.[| inj_onto(f,A); x:A; y:A |] ==> (f(x)=f(y)) = (x=y)"; -by (fast_tac (HOL_cs addSEs [inj_ontoD]) 1); -qed "inj_onto_iff"; - -val major::prems = goal Fun.thy - "[| inj_onto(f,A); ~x=y; x:A; y:A |] ==> ~ f(x)=f(y)"; -by (rtac contrapos 1); -by (etac (major RS inj_ontoD) 2); -by (REPEAT (resolve_tac prems 1)); -qed "inj_onto_contraD"; - - -(*** Lemmas about inj ***) - -val prems = goalw Fun.thy [o_def] - "[| inj(f); inj_onto(g,range(f)) |] ==> inj(g o f)"; -by (cut_facts_tac prems 1); -by (fast_tac (HOL_cs addIs [injI,rangeI] - addEs [injD,inj_ontoD]) 1); -qed "comp_inj"; - -val [prem] = goal Fun.thy "inj(f) ==> inj_onto(f,A)"; -by (fast_tac (HOL_cs addIs [prem RS injD, inj_ontoI]) 1); -qed "inj_imp"; - -val [prem] = goalw Fun.thy [Inv_def] "y : range(f) ==> f(Inv(f,y)) = y"; -by (EVERY1 [rtac (prem RS rangeE), rtac selectI, etac sym]); -qed "f_Inv_f"; - -val prems = goal Fun.thy - "[| Inv(f,x)=Inv(f,y); x: range(f); y: range(f) |] ==> x=y"; -by (rtac (arg_cong RS box_equals) 1); -by (REPEAT (resolve_tac (prems @ [f_Inv_f]) 1)); -qed "Inv_injective"; - -val prems = goal Fun.thy - "[| inj(f); A<=range(f) |] ==> inj_onto(Inv(f), A)"; -by (cut_facts_tac prems 1); -by (fast_tac (HOL_cs addIs [inj_ontoI] - addEs [Inv_injective,injD,subsetD]) 1); -qed "inj_onto_Inv"; - - -(*** Set reasoning tools ***) - -val set_cs = HOL_cs - addSIs [ballI, PowI, subsetI, InterI, INT_I, INT1_I, CollectI, - ComplI, IntI, DiffI, UnCI, insertCI] - addIs [bexI, UnionI, UN_I, UN1_I, imageI, rangeI] - addSEs [bexE, make_elim PowD, UnionE, UN_E, UN1_E, DiffE, - CollectE, ComplE, IntE, UnE, insertE, imageE, rangeE, emptyE] - addEs [ballE, InterD, InterE, INT_D, INT_E, make_elim INT1_D, - subsetD, subsetCE]; - -fun cfast_tac prems = cut_facts_tac prems THEN' fast_tac set_cs; - - -fun prover s = prove_goal Fun.thy s (fn _=>[fast_tac set_cs 1]); - -val mem_simps = map prover - [ "(a : A Un B) = (a:A | a:B)", - "(a : A Int B) = (a:A & a:B)", - "(a : Compl(B)) = (~a:B)", - "(a : A-B) = (a:A & ~a:B)", - "(a : {b}) = (a=b)", - "(a : {x.P(x)}) = P(a)" ]; - -val mksimps_pairs = ("Ball",[bspec]) :: mksimps_pairs; - -val set_ss = - HOL_ss addsimps mem_simps - addcongs [ball_cong,bex_cong] - setmksimps (mksimps mksimps_pairs);