author | oheimb |
Tue, 23 Apr 1996 16:58:21 +0200 | |
changeset 1672 | 2c109cd2fdd0 |
parent 1666 | 5183de4c496d |
child 1754 | 852093aeb0ab |
permissions | -rw-r--r-- |
1465 | 1 |
(* 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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(*** Range of a function ***) |
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(*Frequently b does not have the syntactic form of f(x).*) |
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val [prem] = goalw Fun.thy [range_def] "b=f(x) ==> b : range(f)"; |
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by (EVERY1 [rtac CollectI, rtac exI, rtac prem]); |
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qed "range_eqI"; |
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val rangeI = refl RS range_eqI; |
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val [major,minor] = goalw Fun.thy [range_def] |
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"[| b : range(%x.f(x)); !!x. b=f(x) ==> P |] ==> P"; |
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by (rtac (major RS CollectD RS exE) 1); |
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by (etac minor 1); |
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qed "rangeE"; |
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(*** Image of a set under a function ***) |
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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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val 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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goalw Fun.thy [o_def] "(f o g)``r = f``(g``r)"; |
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by (rtac set_ext 1); |
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by (fast_tac (HOL_cs addIs [imageI] addSEs [imageE]) 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 (rtac set_ext 1); |
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by (fast_tac (HOL_cs addIs [imageI,UnCI] addSEs [imageE,UnE]) 1); |
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qed "image_Un"; |
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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 (HOL_cs 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 (HOL_cs addIs prems addSIs [ballI]) 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 (HOL_cs 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 (HOL_cs addIs [injI,rangeI] |
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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 (HOL_cs 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 (HOL_cs addIs [inj_ontoI] |
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addEs [Inv_injective,injD,subsetD]) 1); |
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qed "inj_onto_Inv"; |
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(*** Set reasoning tools ***) |
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val set_cs = HOL_cs |
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addSIs [ballI, PowI, subsetI, InterI, INT_I, INT1_I, CollectI, |
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ComplI, IntI, DiffI, UnCI, insertCI] |
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addIs [bexI, UnionI, UN_I, UN1_I, imageI, rangeI] |
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addSEs [bexE, make_elim PowD, UnionE, UN_E, UN1_E, DiffE, |
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make_elim singleton_inject, |
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CollectE, ComplE, IntE, UnE, insertE, imageE, rangeE, emptyE] |
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addEs [ballE, InterD, InterE, INT_D, INT_E, make_elim INT1_D, |
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subsetD, subsetCE]; |
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fun cfast_tac prems = cut_facts_tac prems THEN' fast_tac set_cs; |
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fun prover s = prove_goal Fun.thy s (fn _=>[fast_tac set_cs 1]); |
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val mem_simps = map prover |
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1672
2c109cd2fdd0
repaired critical proofs depending on the order inside non-confluent SimpSets,
oheimb
parents:
1666
diff
changeset
|
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[ "(a : A Un B) = (a:A | a:B)", (* Un_iff *) |
2c109cd2fdd0
repaired critical proofs depending on the order inside non-confluent SimpSets,
oheimb
parents:
1666
diff
changeset
|
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"(a : A Int B) = (a:A & a:B)", (* Int_iff *) |
2c109cd2fdd0
repaired critical proofs depending on the order inside non-confluent SimpSets,
oheimb
parents:
1666
diff
changeset
|
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"(a : Compl(B)) = (~a:B)", (* Compl_iff *) |
2c109cd2fdd0
repaired critical proofs depending on the order inside non-confluent SimpSets,
oheimb
parents:
1666
diff
changeset
|
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"(a : A-B) = (a:A & ~a:B)", (* Diff_iff *) |
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"(a : {b}) = (a=b)", |
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"(a : {x.P(x)}) = P(a)" ]; |
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val mksimps_pairs = ("Ball",[bspec]) :: mksimps_pairs; |
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simpset := !simpset addsimps mem_simps |
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addcongs [ball_cong,bex_cong] |
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setmksimps (mksimps mksimps_pairs); |