diff -r b7f57e0ab47c -r 0ec63df3ae04 Fun.ML --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/Fun.ML Fri Aug 19 11:02:45 1994 +0200 @@ -0,0 +1,191 @@ +(* 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); +val expand_fun_eq = result(); + +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)); +val apply_inverse = result(); + + +(*** 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]); +val range_eqI = result(); + +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); +val rangeE = result(); + +(*** 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)); +val image_eqI = result(); + +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)); +val imageE = result(); + +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); +val image_compose = result(); + +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); +val image_Un = result(); + +(*** 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); +val injI = result(); + +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); +val inj_inverseI = result(); + +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); +val injD = result(); + +(*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); +val inj_eq = result(); + +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); +val inj_select = result(); + +(*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)]); +val Inv_f_f = result(); + +(* 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); +val inj_transfer = result(); + + +(*** 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); +val inj_ontoI = result(); + +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)); +val inj_onto_inverseI = result(); + +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)); +val inj_ontoD = result(); + +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)); +val inj_onto_contraD = result(); + + +(*** 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); +val comp_inj = result(); + +val [prem] = goal Fun.thy "inj(f) ==> inj_onto(f,A)"; +by (fast_tac (HOL_cs addIs [prem RS injD, inj_ontoI]) 1); +val inj_imp = result(); + +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]); +val f_Inv_f = result(); + +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)); +val Inv_injective = result(); + +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); +val inj_onto_Inv = result(); + + +(*** Set reasoning tools ***) + +val set_cs = HOL_cs + addSIs [ballI, subsetI, InterI, INT_I, INT1_I, CollectI, + ComplI, IntI, DiffI, UnCI, insertCI] + addIs [bexI, UnionI, UN_I, UN1_I, imageI, rangeI] + addSEs [bexE, 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 set_ss = HOL_ss addsimps mem_simps;