author | wenzelm |
Sat, 13 Mar 2010 14:44:47 +0100 | |
changeset 35743 | c506c029a082 |
parent 35584 | 768f8d92b767 |
child 36176 | 3fe7e97ccca8 |
permissions | -rw-r--r-- |
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(* Title: HOL/Fun.thy |
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Author: Tobias Nipkow, Cambridge University Computer Laboratory |
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Copyright 1994 University of Cambridge |
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*) |
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|
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header {* Notions about functions *} |
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|
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theory Fun |
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imports Complete_Lattice |
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begin |
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|
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text{*As a simplification rule, it replaces all function equalities by |
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first-order equalities.*} |
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lemma expand_fun_eq: "f = g \<longleftrightarrow> (\<forall>x. f x = g x)" |
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apply (rule iffI) |
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apply (simp (no_asm_simp)) |
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apply (rule ext) |
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apply (simp (no_asm_simp)) |
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done |
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lemma apply_inverse: |
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"f x = u \<Longrightarrow> (\<And>x. P x \<Longrightarrow> g (f x) = x) \<Longrightarrow> P x \<Longrightarrow> x = g u" |
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by auto |
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subsection {* The Identity Function @{text id} *} |
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definition |
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id :: "'a \<Rightarrow> 'a" |
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where |
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"id = (\<lambda>x. x)" |
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lemma id_apply [simp]: "id x = x" |
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by (simp add: id_def) |
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lemma image_ident [simp]: "(%x. x) ` Y = Y" |
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by blast |
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lemma image_id [simp]: "id ` Y = Y" |
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by (simp add: id_def) |
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lemma vimage_ident [simp]: "(%x. x) -` Y = Y" |
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by blast |
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lemma vimage_id [simp]: "id -` A = A" |
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by (simp add: id_def) |
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subsection {* The Composition Operator @{text "f \<circ> g"} *} |
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||
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definition |
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comp :: "('b \<Rightarrow> 'c) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'c" (infixl "o" 55) |
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where |
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"f o g = (\<lambda>x. f (g x))" |
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notation (xsymbols) |
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comp (infixl "\<circ>" 55) |
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notation (HTML output) |
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comp (infixl "\<circ>" 55) |
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text{*compatibility*} |
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lemmas o_def = comp_def |
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lemma o_apply [simp]: "(f o g) x = f (g x)" |
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by (simp add: comp_def) |
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lemma o_assoc: "f o (g o h) = f o g o h" |
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by (simp add: comp_def) |
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lemma id_o [simp]: "id o g = g" |
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by (simp add: comp_def) |
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lemma o_id [simp]: "f o id = f" |
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by (simp add: comp_def) |
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||
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lemma o_eq_dest: |
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"a o b = c o d \<Longrightarrow> a (b v) = c (d v)" |
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by (simp only: o_def) (fact fun_cong) |
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lemma o_eq_elim: |
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"a o b = c o d \<Longrightarrow> ((\<And>v. a (b v) = c (d v)) \<Longrightarrow> R) \<Longrightarrow> R" |
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by (erule meta_mp) (fact o_eq_dest) |
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lemma image_compose: "(f o g) ` r = f`(g`r)" |
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by (simp add: comp_def, blast) |
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lemma vimage_compose: "(g \<circ> f) -` x = f -` (g -` x)" |
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by auto |
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||
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lemma UN_o: "UNION A (g o f) = UNION (f`A) g" |
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by (unfold comp_def, blast) |
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subsection {* The Forward Composition Operator @{text fcomp} *} |
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definition |
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fcomp :: "('a \<Rightarrow> 'b) \<Rightarrow> ('b \<Rightarrow> 'c) \<Rightarrow> 'a \<Rightarrow> 'c" (infixl "o>" 60) |
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where |
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"f o> g = (\<lambda>x. g (f x))" |
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lemma fcomp_apply: "(f o> g) x = g (f x)" |
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by (simp add: fcomp_def) |
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lemma fcomp_assoc: "(f o> g) o> h = f o> (g o> h)" |
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by (simp add: fcomp_def) |
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lemma id_fcomp [simp]: "id o> g = g" |
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by (simp add: fcomp_def) |
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lemma fcomp_id [simp]: "f o> id = f" |
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by (simp add: fcomp_def) |
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code_const fcomp |
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(Eval infixl 1 "#>") |
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no_notation fcomp (infixl "o>" 60) |
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subsection {* Injectivity and Surjectivity *} |
121 |
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definition |
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inj_on :: "['a => 'b, 'a set] => bool" where |
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-- "injective" |
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"inj_on f A == ! x:A. ! y:A. f(x)=f(y) --> x=y" |
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text{*A common special case: functions injective over the entire domain type.*} |
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abbreviation |
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"inj f == inj_on f UNIV" |
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definition |
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bij_betw :: "('a => 'b) => 'a set => 'b set => bool" where -- "bijective" |
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[code del]: "bij_betw f A B \<longleftrightarrow> inj_on f A & f ` A = B" |
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definition |
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surj :: "('a => 'b) => bool" where |
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-- "surjective" |
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"surj f == ! y. ? x. y=f(x)" |
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definition |
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bij :: "('a => 'b) => bool" where |
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-- "bijective" |
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"bij f == inj f & surj f" |
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lemma injI: |
|
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assumes "\<And>x y. f x = f y \<Longrightarrow> x = y" |
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shows "inj f" |
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using assms unfolding inj_on_def by auto |
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text{*For Proofs in @{text "Tools/Datatype/datatype_rep_proofs"}*} |
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lemma datatype_injI: |
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"(!! x. ALL y. f(x) = f(y) --> x=y) ==> inj(f)" |
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by (simp add: inj_on_def) |
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theorem range_ex1_eq: "inj f \<Longrightarrow> b : range f = (EX! x. b = f x)" |
157 |
by (unfold inj_on_def, blast) |
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||
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lemma injD: "[| inj(f); f(x) = f(y) |] ==> x=y" |
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by (simp add: inj_on_def) |
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||
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lemma inj_on_eq_iff: "inj_on f A ==> x:A ==> y:A ==> (f(x) = f(y)) = (x=y)" |
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by (force simp add: inj_on_def) |
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||
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lemma inj_eq: "inj f ==> (f(x) = f(y)) = (x=y)" |
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by (simp add: inj_on_eq_iff) |
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lemma inj_on_id[simp]: "inj_on id A" |
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by (simp add: inj_on_def) |
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lemma inj_on_id2[simp]: "inj_on (%x. x) A" |
172 |
by (simp add: inj_on_def) |
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lemma surj_id[simp]: "surj id" |
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by (simp add: surj_def) |
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lemma bij_id[simp]: "bij id" |
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by (simp add: bij_def) |
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lemma inj_onI: |
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"(!! x y. [| x:A; y:A; f(x) = f(y) |] ==> x=y) ==> inj_on f A" |
|
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by (simp add: inj_on_def) |
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lemma inj_on_inverseI: "(!!x. x:A ==> g(f(x)) = x) ==> inj_on f A" |
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by (auto dest: arg_cong [of concl: g] simp add: inj_on_def) |
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lemma inj_onD: "[| inj_on f A; f(x)=f(y); x:A; y:A |] ==> x=y" |
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by (unfold inj_on_def, blast) |
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lemma inj_on_iff: "[| inj_on f A; x:A; y:A |] ==> (f(x)=f(y)) = (x=y)" |
|
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by (blast dest!: inj_onD) |
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192 |
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lemma comp_inj_on: |
|
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"[| inj_on f A; inj_on g (f`A) |] ==> inj_on (g o f) A" |
|
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by (simp add: comp_def inj_on_def) |
|
196 |
||
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lemma inj_on_imageI: "inj_on (g o f) A \<Longrightarrow> inj_on g (f ` A)" |
198 |
apply(simp add:inj_on_def image_def) |
|
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apply blast |
|
200 |
done |
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||
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lemma inj_on_image_iff: "\<lbrakk> ALL x:A. ALL y:A. (g(f x) = g(f y)) = (g x = g y); |
203 |
inj_on f A \<rbrakk> \<Longrightarrow> inj_on g (f ` A) = inj_on g A" |
|
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apply(unfold inj_on_def) |
|
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apply blast |
|
206 |
done |
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207 |
||
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lemma inj_on_contraD: "[| inj_on f A; ~x=y; x:A; y:A |] ==> ~ f(x)=f(y)" |
209 |
by (unfold inj_on_def, blast) |
|
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lemma inj_singleton: "inj (%s. {s})" |
212 |
by (simp add: inj_on_def) |
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||
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lemma inj_on_empty[iff]: "inj_on f {}" |
215 |
by(simp add: inj_on_def) |
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216 |
||
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lemma subset_inj_on: "[| inj_on f B; A <= B |] ==> inj_on f A" |
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by (unfold inj_on_def, blast) |
219 |
||
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lemma inj_on_Un: |
221 |
"inj_on f (A Un B) = |
|
222 |
(inj_on f A & inj_on f B & f`(A-B) Int f`(B-A) = {})" |
|
223 |
apply(unfold inj_on_def) |
|
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apply (blast intro:sym) |
|
225 |
done |
|
226 |
||
227 |
lemma inj_on_insert[iff]: |
|
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"inj_on f (insert a A) = (inj_on f A & f a ~: f`(A-{a}))" |
|
229 |
apply(unfold inj_on_def) |
|
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apply (blast intro:sym) |
|
231 |
done |
|
232 |
||
233 |
lemma inj_on_diff: "inj_on f A ==> inj_on f (A-B)" |
|
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apply(unfold inj_on_def) |
|
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apply (blast) |
|
236 |
done |
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237 |
||
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lemma surjI: "(!! x. g(f x) = x) ==> surj g" |
239 |
apply (simp add: surj_def) |
|
240 |
apply (blast intro: sym) |
|
241 |
done |
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242 |
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243 |
lemma surj_range: "surj f ==> range f = UNIV" |
|
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by (auto simp add: surj_def) |
|
245 |
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246 |
lemma surjD: "surj f ==> EX x. y = f x" |
|
247 |
by (simp add: surj_def) |
|
248 |
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249 |
lemma surjE: "surj f ==> (!!x. y = f x ==> C) ==> C" |
|
250 |
by (simp add: surj_def, blast) |
|
251 |
||
252 |
lemma comp_surj: "[| surj f; surj g |] ==> surj (g o f)" |
|
253 |
apply (simp add: comp_def surj_def, clarify) |
|
254 |
apply (drule_tac x = y in spec, clarify) |
|
255 |
apply (drule_tac x = x in spec, blast) |
|
256 |
done |
|
257 |
||
258 |
lemma bijI: "[| inj f; surj f |] ==> bij f" |
|
259 |
by (simp add: bij_def) |
|
260 |
||
261 |
lemma bij_is_inj: "bij f ==> inj f" |
|
262 |
by (simp add: bij_def) |
|
263 |
||
264 |
lemma bij_is_surj: "bij f ==> surj f" |
|
265 |
by (simp add: bij_def) |
|
266 |
||
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lemma bij_betw_imp_inj_on: "bij_betw f A B \<Longrightarrow> inj_on f A" |
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by (simp add: bij_betw_def) |
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269 |
|
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lemma bij_comp: "bij f \<Longrightarrow> bij g \<Longrightarrow> bij (g o f)" |
271 |
by(fastsimp intro: comp_inj_on comp_surj simp: bij_def surj_range) |
|
272 |
||
31438 | 273 |
lemma bij_betw_trans: |
274 |
"bij_betw f A B \<Longrightarrow> bij_betw g B C \<Longrightarrow> bij_betw (g o f) A C" |
|
275 |
by(auto simp add:bij_betw_def comp_inj_on) |
|
276 |
||
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277 |
lemma bij_betw_inv: assumes "bij_betw f A B" shows "EX g. bij_betw g B A" |
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|
278 |
proof - |
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|
279 |
have i: "inj_on f A" and s: "f ` A = B" |
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|
280 |
using assms by(auto simp:bij_betw_def) |
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|
281 |
let ?P = "%b a. a:A \<and> f a = b" let ?g = "%b. The (?P b)" |
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282 |
{ fix a b assume P: "?P b a" |
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283 |
hence ex1: "\<exists>a. ?P b a" using s unfolding image_def by blast |
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284 |
hence uex1: "\<exists>!a. ?P b a" by(blast dest:inj_onD[OF i]) |
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|
285 |
hence " ?g b = a" using the1_equality[OF uex1, OF P] P by simp |
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|
286 |
} note g = this |
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|
287 |
have "inj_on ?g B" |
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288 |
proof(rule inj_onI) |
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289 |
fix x y assume "x:B" "y:B" "?g x = ?g y" |
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290 |
from s `x:B` obtain a1 where a1: "?P x a1" unfolding image_def by blast |
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291 |
from s `y:B` obtain a2 where a2: "?P y a2" unfolding image_def by blast |
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changeset
|
292 |
from g[OF a1] a1 g[OF a2] a2 `?g x = ?g y` show "x=y" by simp |
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changeset
|
293 |
qed |
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diff
changeset
|
294 |
moreover have "?g ` B = A" |
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changeset
|
295 |
proof(auto simp:image_def) |
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diff
changeset
|
296 |
fix b assume "b:B" |
ae06618225ec
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25886
diff
changeset
|
297 |
with s obtain a where P: "?P b a" unfolding image_def by blast |
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changeset
|
298 |
thus "?g b \<in> A" using g[OF P] by auto |
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changeset
|
299 |
next |
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moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
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diff
changeset
|
300 |
fix a assume "a:A" |
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changeset
|
301 |
then obtain b where P: "?P b a" using s unfolding image_def by blast |
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25886
diff
changeset
|
302 |
then have "b:B" using s unfolding image_def by blast |
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25886
diff
changeset
|
303 |
with g[OF P] show "\<exists>b\<in>B. a = ?g b" by blast |
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diff
changeset
|
304 |
qed |
ae06618225ec
moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
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diff
changeset
|
305 |
ultimately show ?thesis by(auto simp:bij_betw_def) |
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parents:
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diff
changeset
|
306 |
qed |
ae06618225ec
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parents:
25886
diff
changeset
|
307 |
|
13585 | 308 |
lemma surj_image_vimage_eq: "surj f ==> f ` (f -` A) = A" |
309 |
by (simp add: surj_range) |
|
310 |
||
311 |
lemma inj_vimage_image_eq: "inj f ==> f -` (f ` A) = A" |
|
312 |
by (simp add: inj_on_def, blast) |
|
313 |
||
314 |
lemma vimage_subsetD: "surj f ==> f -` B <= A ==> B <= f ` A" |
|
315 |
apply (unfold surj_def) |
|
316 |
apply (blast intro: sym) |
|
317 |
done |
|
318 |
||
319 |
lemma vimage_subsetI: "inj f ==> B <= f ` A ==> f -` B <= A" |
|
320 |
by (unfold inj_on_def, blast) |
|
321 |
||
322 |
lemma vimage_subset_eq: "bij f ==> (f -` B <= A) = (B <= f ` A)" |
|
323 |
apply (unfold bij_def) |
|
324 |
apply (blast del: subsetI intro: vimage_subsetI vimage_subsetD) |
|
325 |
done |
|
326 |
||
31438 | 327 |
lemma inj_on_Un_image_eq_iff: "inj_on f (A \<union> B) \<Longrightarrow> f ` A = f ` B \<longleftrightarrow> A = B" |
328 |
by(blast dest: inj_onD) |
|
329 |
||
13585 | 330 |
lemma inj_on_image_Int: |
331 |
"[| inj_on f C; A<=C; B<=C |] ==> f`(A Int B) = f`A Int f`B" |
|
332 |
apply (simp add: inj_on_def, blast) |
|
333 |
done |
|
334 |
||
335 |
lemma inj_on_image_set_diff: |
|
336 |
"[| inj_on f C; A<=C; B<=C |] ==> f`(A-B) = f`A - f`B" |
|
337 |
apply (simp add: inj_on_def, blast) |
|
338 |
done |
|
339 |
||
340 |
lemma image_Int: "inj f ==> f`(A Int B) = f`A Int f`B" |
|
341 |
by (simp add: inj_on_def, blast) |
|
342 |
||
343 |
lemma image_set_diff: "inj f ==> f`(A-B) = f`A - f`B" |
|
344 |
by (simp add: inj_on_def, blast) |
|
345 |
||
346 |
lemma inj_image_mem_iff: "inj f ==> (f a : f`A) = (a : A)" |
|
347 |
by (blast dest: injD) |
|
348 |
||
349 |
lemma inj_image_subset_iff: "inj f ==> (f`A <= f`B) = (A<=B)" |
|
350 |
by (simp add: inj_on_def, blast) |
|
351 |
||
352 |
lemma inj_image_eq_iff: "inj f ==> (f`A = f`B) = (A = B)" |
|
353 |
by (blast dest: injD) |
|
354 |
||
355 |
(*injectivity's required. Left-to-right inclusion holds even if A is empty*) |
|
356 |
lemma image_INT: |
|
357 |
"[| inj_on f C; ALL x:A. B x <= C; j:A |] |
|
358 |
==> f ` (INTER A B) = (INT x:A. f ` B x)" |
|
359 |
apply (simp add: inj_on_def, blast) |
|
360 |
done |
|
361 |
||
362 |
(*Compare with image_INT: no use of inj_on, and if f is surjective then |
|
363 |
it doesn't matter whether A is empty*) |
|
364 |
lemma bij_image_INT: "bij f ==> f ` (INTER A B) = (INT x:A. f ` B x)" |
|
365 |
apply (simp add: bij_def) |
|
366 |
apply (simp add: inj_on_def surj_def, blast) |
|
367 |
done |
|
368 |
||
369 |
lemma surj_Compl_image_subset: "surj f ==> -(f`A) <= f`(-A)" |
|
370 |
by (auto simp add: surj_def) |
|
371 |
||
372 |
lemma inj_image_Compl_subset: "inj f ==> f`(-A) <= -(f`A)" |
|
373 |
by (auto simp add: inj_on_def) |
|
5852 | 374 |
|
13585 | 375 |
lemma bij_image_Compl_eq: "bij f ==> f`(-A) = -(f`A)" |
376 |
apply (simp add: bij_def) |
|
377 |
apply (rule equalityI) |
|
378 |
apply (simp_all (no_asm_simp) add: inj_image_Compl_subset surj_Compl_image_subset) |
|
379 |
done |
|
380 |
||
35584
768f8d92b767
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parents:
35580
diff
changeset
|
381 |
lemma (in ordered_ab_group_add) inj_uminus[simp, intro]: "inj_on uminus A" |
35580 | 382 |
by (auto intro!: inj_onI) |
13585 | 383 |
|
35584
768f8d92b767
generalized inj_uminus; added strict_mono_imp_inj_on
hoelzl
parents:
35580
diff
changeset
|
384 |
lemma (in linorder) strict_mono_imp_inj_on: "strict_mono f \<Longrightarrow> inj_on f A" |
768f8d92b767
generalized inj_uminus; added strict_mono_imp_inj_on
hoelzl
parents:
35580
diff
changeset
|
385 |
by (auto intro!: inj_onI dest: strict_mono_eq) |
768f8d92b767
generalized inj_uminus; added strict_mono_imp_inj_on
hoelzl
parents:
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diff
changeset
|
386 |
|
13585 | 387 |
subsection{*Function Updating*} |
388 |
||
35416
d8d7d1b785af
replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
haftmann
parents:
35115
diff
changeset
|
389 |
definition |
d8d7d1b785af
replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
haftmann
parents:
35115
diff
changeset
|
390 |
fun_upd :: "('a => 'b) => 'a => 'b => ('a => 'b)" where |
26147 | 391 |
"fun_upd f a b == % x. if x=a then b else f x" |
392 |
||
393 |
nonterminals |
|
394 |
updbinds updbind |
|
395 |
syntax |
|
396 |
"_updbind" :: "['a, 'a] => updbind" ("(2_ :=/ _)") |
|
397 |
"" :: "updbind => updbinds" ("_") |
|
398 |
"_updbinds":: "[updbind, updbinds] => updbinds" ("_,/ _") |
|
35115 | 399 |
"_Update" :: "['a, updbinds] => 'a" ("_/'((_)')" [1000, 0] 900) |
26147 | 400 |
|
401 |
translations |
|
35115 | 402 |
"_Update f (_updbinds b bs)" == "_Update (_Update f b) bs" |
403 |
"f(x:=y)" == "CONST fun_upd f x y" |
|
26147 | 404 |
|
405 |
(* Hint: to define the sum of two functions (or maps), use sum_case. |
|
406 |
A nice infix syntax could be defined (in Datatype.thy or below) by |
|
35115 | 407 |
notation |
408 |
sum_case (infixr "'(+')"80) |
|
26147 | 409 |
*) |
410 |
||
13585 | 411 |
lemma fun_upd_idem_iff: "(f(x:=y) = f) = (f x = y)" |
412 |
apply (simp add: fun_upd_def, safe) |
|
413 |
apply (erule subst) |
|
414 |
apply (rule_tac [2] ext, auto) |
|
415 |
done |
|
416 |
||
417 |
(* f x = y ==> f(x:=y) = f *) |
|
418 |
lemmas fun_upd_idem = fun_upd_idem_iff [THEN iffD2, standard] |
|
419 |
||
420 |
(* f(x := f x) = f *) |
|
17084
fb0a80aef0be
classical rules must have names for ATP integration
paulson
parents:
16973
diff
changeset
|
421 |
lemmas fun_upd_triv = refl [THEN fun_upd_idem] |
fb0a80aef0be
classical rules must have names for ATP integration
paulson
parents:
16973
diff
changeset
|
422 |
declare fun_upd_triv [iff] |
13585 | 423 |
|
424 |
lemma fun_upd_apply [simp]: "(f(x:=y))z = (if z=x then y else f z)" |
|
17084
fb0a80aef0be
classical rules must have names for ATP integration
paulson
parents:
16973
diff
changeset
|
425 |
by (simp add: fun_upd_def) |
13585 | 426 |
|
427 |
(* fun_upd_apply supersedes these two, but they are useful |
|
428 |
if fun_upd_apply is intentionally removed from the simpset *) |
|
429 |
lemma fun_upd_same: "(f(x:=y)) x = y" |
|
430 |
by simp |
|
431 |
||
432 |
lemma fun_upd_other: "z~=x ==> (f(x:=y)) z = f z" |
|
433 |
by simp |
|
434 |
||
435 |
lemma fun_upd_upd [simp]: "f(x:=y,x:=z) = f(x:=z)" |
|
436 |
by (simp add: expand_fun_eq) |
|
437 |
||
438 |
lemma fun_upd_twist: "a ~= c ==> (m(a:=b))(c:=d) = (m(c:=d))(a:=b)" |
|
439 |
by (rule ext, auto) |
|
440 |
||
15303 | 441 |
lemma inj_on_fun_updI: "\<lbrakk> inj_on f A; y \<notin> f`A \<rbrakk> \<Longrightarrow> inj_on (f(x:=y)) A" |
34209 | 442 |
by (fastsimp simp:inj_on_def image_def) |
15303 | 443 |
|
15510 | 444 |
lemma fun_upd_image: |
445 |
"f(x:=y) ` A = (if x \<in> A then insert y (f ` (A-{x})) else f ` A)" |
|
446 |
by auto |
|
447 |
||
31080 | 448 |
lemma fun_upd_comp: "f \<circ> (g(x := y)) = (f \<circ> g)(x := f y)" |
34209 | 449 |
by (auto intro: ext) |
31080 | 450 |
|
26147 | 451 |
|
452 |
subsection {* @{text override_on} *} |
|
453 |
||
454 |
definition |
|
455 |
override_on :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> 'a \<Rightarrow> 'b" |
|
456 |
where |
|
457 |
"override_on f g A = (\<lambda>a. if a \<in> A then g a else f a)" |
|
13910 | 458 |
|
15691 | 459 |
lemma override_on_emptyset[simp]: "override_on f g {} = f" |
460 |
by(simp add:override_on_def) |
|
13910 | 461 |
|
15691 | 462 |
lemma override_on_apply_notin[simp]: "a ~: A ==> (override_on f g A) a = f a" |
463 |
by(simp add:override_on_def) |
|
13910 | 464 |
|
15691 | 465 |
lemma override_on_apply_in[simp]: "a : A ==> (override_on f g A) a = g a" |
466 |
by(simp add:override_on_def) |
|
13910 | 467 |
|
26147 | 468 |
|
469 |
subsection {* @{text swap} *} |
|
15510 | 470 |
|
22744
5cbe966d67a2
Isar definitions are now added explicitly to code theorem table
haftmann
parents:
22577
diff
changeset
|
471 |
definition |
5cbe966d67a2
Isar definitions are now added explicitly to code theorem table
haftmann
parents:
22577
diff
changeset
|
472 |
swap :: "'a \<Rightarrow> 'a \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b)" |
5cbe966d67a2
Isar definitions are now added explicitly to code theorem table
haftmann
parents:
22577
diff
changeset
|
473 |
where |
5cbe966d67a2
Isar definitions are now added explicitly to code theorem table
haftmann
parents:
22577
diff
changeset
|
474 |
"swap a b f = f (a := f b, b:= f a)" |
15510 | 475 |
|
34101 | 476 |
lemma swap_self [simp]: "swap a a f = f" |
15691 | 477 |
by (simp add: swap_def) |
15510 | 478 |
|
479 |
lemma swap_commute: "swap a b f = swap b a f" |
|
480 |
by (rule ext, simp add: fun_upd_def swap_def) |
|
481 |
||
482 |
lemma swap_nilpotent [simp]: "swap a b (swap a b f) = f" |
|
483 |
by (rule ext, simp add: fun_upd_def swap_def) |
|
484 |
||
34145 | 485 |
lemma swap_triple: |
486 |
assumes "a \<noteq> c" and "b \<noteq> c" |
|
487 |
shows "swap a b (swap b c (swap a b f)) = swap a c f" |
|
488 |
using assms by (simp add: expand_fun_eq swap_def) |
|
489 |
||
34101 | 490 |
lemma comp_swap: "f \<circ> swap a b g = swap a b (f \<circ> g)" |
491 |
by (rule ext, simp add: fun_upd_def swap_def) |
|
492 |
||
15510 | 493 |
lemma inj_on_imp_inj_on_swap: |
22744
5cbe966d67a2
Isar definitions are now added explicitly to code theorem table
haftmann
parents:
22577
diff
changeset
|
494 |
"[|inj_on f A; a \<in> A; b \<in> A|] ==> inj_on (swap a b f) A" |
15510 | 495 |
by (simp add: inj_on_def swap_def, blast) |
496 |
||
497 |
lemma inj_on_swap_iff [simp]: |
|
498 |
assumes A: "a \<in> A" "b \<in> A" shows "inj_on (swap a b f) A = inj_on f A" |
|
499 |
proof |
|
500 |
assume "inj_on (swap a b f) A" |
|
501 |
with A have "inj_on (swap a b (swap a b f)) A" |
|
17589 | 502 |
by (iprover intro: inj_on_imp_inj_on_swap) |
15510 | 503 |
thus "inj_on f A" by simp |
504 |
next |
|
505 |
assume "inj_on f A" |
|
34209 | 506 |
with A show "inj_on (swap a b f) A" by (iprover intro: inj_on_imp_inj_on_swap) |
15510 | 507 |
qed |
508 |
||
509 |
lemma surj_imp_surj_swap: "surj f ==> surj (swap a b f)" |
|
510 |
apply (simp add: surj_def swap_def, clarify) |
|
27125 | 511 |
apply (case_tac "y = f b", blast) |
512 |
apply (case_tac "y = f a", auto) |
|
15510 | 513 |
done |
514 |
||
515 |
lemma surj_swap_iff [simp]: "surj (swap a b f) = surj f" |
|
516 |
proof |
|
517 |
assume "surj (swap a b f)" |
|
518 |
hence "surj (swap a b (swap a b f))" by (rule surj_imp_surj_swap) |
|
519 |
thus "surj f" by simp |
|
520 |
next |
|
521 |
assume "surj f" |
|
522 |
thus "surj (swap a b f)" by (rule surj_imp_surj_swap) |
|
523 |
qed |
|
524 |
||
525 |
lemma bij_swap_iff: "bij (swap a b f) = bij f" |
|
526 |
by (simp add: bij_def) |
|
21547
9c9fdf4c2949
moved order arities for fun and bool to Fun/Orderings
haftmann
parents:
21327
diff
changeset
|
527 |
|
27188 | 528 |
hide (open) const swap |
21547
9c9fdf4c2949
moved order arities for fun and bool to Fun/Orderings
haftmann
parents:
21327
diff
changeset
|
529 |
|
31949 | 530 |
|
531 |
subsection {* Inversion of injective functions *} |
|
532 |
||
33057 | 533 |
definition the_inv_into :: "'a set => ('a => 'b) => ('b => 'a)" where |
534 |
"the_inv_into A f == %x. THE y. y : A & f y = x" |
|
32961 | 535 |
|
33057 | 536 |
lemma the_inv_into_f_f: |
537 |
"[| inj_on f A; x : A |] ==> the_inv_into A f (f x) = x" |
|
538 |
apply (simp add: the_inv_into_def inj_on_def) |
|
34209 | 539 |
apply blast |
32961 | 540 |
done |
541 |
||
33057 | 542 |
lemma f_the_inv_into_f: |
543 |
"inj_on f A ==> y : f`A ==> f (the_inv_into A f y) = y" |
|
544 |
apply (simp add: the_inv_into_def) |
|
32961 | 545 |
apply (rule the1I2) |
546 |
apply(blast dest: inj_onD) |
|
547 |
apply blast |
|
548 |
done |
|
549 |
||
33057 | 550 |
lemma the_inv_into_into: |
551 |
"[| inj_on f A; x : f ` A; A <= B |] ==> the_inv_into A f x : B" |
|
552 |
apply (simp add: the_inv_into_def) |
|
32961 | 553 |
apply (rule the1I2) |
554 |
apply(blast dest: inj_onD) |
|
555 |
apply blast |
|
556 |
done |
|
557 |
||
33057 | 558 |
lemma the_inv_into_onto[simp]: |
559 |
"inj_on f A ==> the_inv_into A f ` (f ` A) = A" |
|
560 |
by (fast intro:the_inv_into_into the_inv_into_f_f[symmetric]) |
|
32961 | 561 |
|
33057 | 562 |
lemma the_inv_into_f_eq: |
563 |
"[| inj_on f A; f x = y; x : A |] ==> the_inv_into A f y = x" |
|
32961 | 564 |
apply (erule subst) |
33057 | 565 |
apply (erule the_inv_into_f_f, assumption) |
32961 | 566 |
done |
567 |
||
33057 | 568 |
lemma the_inv_into_comp: |
32961 | 569 |
"[| inj_on f (g ` A); inj_on g A; x : f ` g ` A |] ==> |
33057 | 570 |
the_inv_into A (f o g) x = (the_inv_into A g o the_inv_into (g ` A) f) x" |
571 |
apply (rule the_inv_into_f_eq) |
|
32961 | 572 |
apply (fast intro: comp_inj_on) |
33057 | 573 |
apply (simp add: f_the_inv_into_f the_inv_into_into) |
574 |
apply (simp add: the_inv_into_into) |
|
32961 | 575 |
done |
576 |
||
33057 | 577 |
lemma inj_on_the_inv_into: |
578 |
"inj_on f A \<Longrightarrow> inj_on (the_inv_into A f) (f ` A)" |
|
579 |
by (auto intro: inj_onI simp: image_def the_inv_into_f_f) |
|
32961 | 580 |
|
33057 | 581 |
lemma bij_betw_the_inv_into: |
582 |
"bij_betw f A B \<Longrightarrow> bij_betw (the_inv_into A f) B A" |
|
583 |
by (auto simp add: bij_betw_def inj_on_the_inv_into the_inv_into_into) |
|
32961 | 584 |
|
32998
31b19fa0de0b
Renamed inv to the_inv and turned it into an abbreviation (based on the_inv_onto).
berghofe
parents:
32988
diff
changeset
|
585 |
abbreviation the_inv :: "('a \<Rightarrow> 'b) \<Rightarrow> ('b \<Rightarrow> 'a)" where |
33057 | 586 |
"the_inv f \<equiv> the_inv_into UNIV f" |
32998
31b19fa0de0b
Renamed inv to the_inv and turned it into an abbreviation (based on the_inv_onto).
berghofe
parents:
32988
diff
changeset
|
587 |
|
31b19fa0de0b
Renamed inv to the_inv and turned it into an abbreviation (based on the_inv_onto).
berghofe
parents:
32988
diff
changeset
|
588 |
lemma the_inv_f_f: |
31b19fa0de0b
Renamed inv to the_inv and turned it into an abbreviation (based on the_inv_onto).
berghofe
parents:
32988
diff
changeset
|
589 |
assumes "inj f" |
31b19fa0de0b
Renamed inv to the_inv and turned it into an abbreviation (based on the_inv_onto).
berghofe
parents:
32988
diff
changeset
|
590 |
shows "the_inv f (f x) = x" using assms UNIV_I |
33057 | 591 |
by (rule the_inv_into_f_f) |
32998
31b19fa0de0b
Renamed inv to the_inv and turned it into an abbreviation (based on the_inv_onto).
berghofe
parents:
32988
diff
changeset
|
592 |
|
31949 | 593 |
|
22845 | 594 |
subsection {* Proof tool setup *} |
595 |
||
596 |
text {* simplifies terms of the form |
|
597 |
f(...,x:=y,...,x:=z,...) to f(...,x:=z,...) *} |
|
598 |
||
24017 | 599 |
simproc_setup fun_upd2 ("f(v := w, x := y)") = {* fn _ => |
22845 | 600 |
let |
601 |
fun gen_fun_upd NONE T _ _ = NONE |
|
24017 | 602 |
| gen_fun_upd (SOME f) T x y = SOME (Const (@{const_name fun_upd}, T) $ f $ x $ y) |
22845 | 603 |
fun dest_fun_T1 (Type (_, T :: Ts)) = T |
604 |
fun find_double (t as Const (@{const_name fun_upd},T) $ f $ x $ y) = |
|
605 |
let |
|
606 |
fun find (Const (@{const_name fun_upd},T) $ g $ v $ w) = |
|
607 |
if v aconv x then SOME g else gen_fun_upd (find g) T v w |
|
608 |
| find t = NONE |
|
609 |
in (dest_fun_T1 T, gen_fun_upd (find f) T x y) end |
|
24017 | 610 |
|
611 |
fun proc ss ct = |
|
612 |
let |
|
613 |
val ctxt = Simplifier.the_context ss |
|
614 |
val t = Thm.term_of ct |
|
615 |
in |
|
616 |
case find_double t of |
|
617 |
(T, NONE) => NONE |
|
618 |
| (T, SOME rhs) => |
|
27330 | 619 |
SOME (Goal.prove ctxt [] [] (Logic.mk_equals (t, rhs)) |
24017 | 620 |
(fn _ => |
621 |
rtac eq_reflection 1 THEN |
|
622 |
rtac ext 1 THEN |
|
623 |
simp_tac (Simplifier.inherit_context ss @{simpset}) 1)) |
|
624 |
end |
|
625 |
in proc end |
|
22845 | 626 |
*} |
627 |
||
628 |
||
21870 | 629 |
subsection {* Code generator setup *} |
630 |
||
25886
7753e0d81b7a
Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents:
24286
diff
changeset
|
631 |
types_code |
7753e0d81b7a
Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents:
24286
diff
changeset
|
632 |
"fun" ("(_ ->/ _)") |
7753e0d81b7a
Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents:
24286
diff
changeset
|
633 |
attach (term_of) {* |
7753e0d81b7a
Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents:
24286
diff
changeset
|
634 |
fun term_of_fun_type _ aT _ bT _ = Free ("<function>", aT --> bT); |
7753e0d81b7a
Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents:
24286
diff
changeset
|
635 |
*} |
7753e0d81b7a
Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents:
24286
diff
changeset
|
636 |
attach (test) {* |
7753e0d81b7a
Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents:
24286
diff
changeset
|
637 |
fun gen_fun_type aF aT bG bT i = |
7753e0d81b7a
Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents:
24286
diff
changeset
|
638 |
let |
32740 | 639 |
val tab = Unsynchronized.ref []; |
25886
7753e0d81b7a
Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents:
24286
diff
changeset
|
640 |
fun mk_upd (x, (_, y)) t = Const ("Fun.fun_upd", |
7753e0d81b7a
Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents:
24286
diff
changeset
|
641 |
(aT --> bT) --> aT --> bT --> aT --> bT) $ t $ aF x $ y () |
7753e0d81b7a
Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents:
24286
diff
changeset
|
642 |
in |
7753e0d81b7a
Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents:
24286
diff
changeset
|
643 |
(fn x => |
7753e0d81b7a
Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents:
24286
diff
changeset
|
644 |
case AList.lookup op = (!tab) x of |
7753e0d81b7a
Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents:
24286
diff
changeset
|
645 |
NONE => |
7753e0d81b7a
Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents:
24286
diff
changeset
|
646 |
let val p as (y, _) = bG i |
7753e0d81b7a
Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents:
24286
diff
changeset
|
647 |
in (tab := (x, p) :: !tab; y) end |
7753e0d81b7a
Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents:
24286
diff
changeset
|
648 |
| SOME (y, _) => y, |
28711 | 649 |
fn () => Basics.fold mk_upd (!tab) (Const ("HOL.undefined", aT --> bT))) |
25886
7753e0d81b7a
Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents:
24286
diff
changeset
|
650 |
end; |
7753e0d81b7a
Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents:
24286
diff
changeset
|
651 |
*} |
7753e0d81b7a
Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents:
24286
diff
changeset
|
652 |
|
21870 | 653 |
code_const "op \<circ>" |
654 |
(SML infixl 5 "o") |
|
655 |
(Haskell infixr 9 ".") |
|
656 |
||
21906 | 657 |
code_const "id" |
658 |
(Haskell "id") |
|
659 |
||
2912 | 660 |
end |