src/HOL/Fun.thy

     Copyright   1994  University of Cambridge
 
 Notions about functions.
 *)
 
-Fun = Typedef +
+theory Fun = Typedef:
 
 instance set :: (type) order
-                       (subset_refl,subset_trans,subset_antisym,psubset_eq)
-consts
-  fun_upd  :: "('a => 'b) => 'a => 'b => ('a => 'b)"
+  by (intro_classes,
+      (assumption | rule subset_refl subset_trans subset_antisym psubset_eq)+)
+
+constdefs
+  fun_upd :: "('a => 'b) => 'a => 'b => ('a => 'b)"
+   "fun_upd f a b == % x. if x=a then b else f x"
 
 nonterminals
   updbinds updbind
 syntax
-  "_updbind"       :: ['a, 'a] => updbind             ("(2_ :=/ _)")
-  ""               :: updbind => updbinds             ("_")
-  "_updbinds"      :: [updbind, updbinds] => updbinds ("_,/ _")
-  "_Update"        :: ['a, updbinds] => 'a            ("_/'((_)')" [1000,0] 900)
+  "_updbind" :: "['a, 'a] => updbind"             ("(2_ :=/ _)")
+  ""         :: "updbind => updbinds"             ("_")
+  "_updbinds":: "[updbind, updbinds] => updbinds" ("_,/ _")
+  "_Update"  :: "['a, updbinds] => 'a"            ("_/'((_)')" [1000,0] 900)
 
 translations
   "_Update f (_updbinds b bs)"  == "_Update (_Update f b) bs"
   "f(x:=y)"                     == "fun_upd f x y"
-
-defs
-  fun_upd_def "f(a:=b) == % x. if x=a then b else f x"
 
 (* Hint: to define the sum of two functions (or maps), use sum_case.
          A nice infix syntax could be defined (in Datatype.thy or below) by
 consts
   fun_sum :: "('a => 'c) => ('b => 'c) => (('a+'b) => 'c)" (infixr "'(+')"80)
 translations
- "fun_sum" == "sum_case"
+ "fun_sum" == sum_case
 *)
 
 constdefs
-  id ::  'a => 'a
+  id :: "'a => 'a"
     "id == %x. x"
 
-  o  :: ['b => 'c, 'a => 'b, 'a] => 'c   (infixl 55)
+  comp :: "['b => 'c, 'a => 'b, 'a] => 'c"   (infixl "o" 55)
     "f o g == %x. f(g(x))"
 
-  inj_on :: ['a => 'b, 'a set] => bool
+text{*compatibility*}
+lemmas o_def = comp_def
+
+syntax (xsymbols)
+  comp :: "['b => 'c, 'a => 'b, 'a] => 'c"        (infixl "\<circ>" 55)
+
+
+constdefs
+  inj_on :: "['a => 'b, 'a set] => bool"         (*injective*)
     "inj_on f A == ! x:A. ! y:A. f(x)=f(y) --> x=y"
 
-syntax (xsymbols)
-  "op o"        :: "['b => 'c, 'a => 'b, 'a] => 'c"      (infixl "\\<circ>" 55)
-
-syntax
-  inj   :: ('a => 'b) => bool                   (*injective*)
-
+text{*A common special case: functions injective over the entire domain type.*}
+syntax inj   :: "('a => 'b) => bool"
 translations
   "inj f" == "inj_on f UNIV"
 
 constdefs
-  surj :: ('a => 'b) => bool                   (*surjective*)
+  surj :: "('a => 'b) => bool"                   (*surjective*)
     "surj f == ! y. ? x. y=f(x)"
 
-  bij :: ('a => 'b) => bool                    (*bijective*)
+  bij :: "('a => 'b) => bool"                    (*bijective*)
     "bij f == inj f & surj f"
 
 
-(*The Pi-operator, by Florian Kammueller*)
-
-constdefs
-  Pi      :: "['a set, 'a => 'b set] => ('a => 'b) set"
-    "Pi A B == {f. ! x. if x:A then f(x) : B(x) else f(x) = arbitrary}"
-
-  restrict :: "['a => 'b, 'a set] => ('a => 'b)"
-    "restrict f A == (%x. if x : A then f x else arbitrary)"
-
-syntax
-  "@Pi"  :: "[pttrn, 'a set, 'b set] => ('a => 'b) set"  ("(3PI _:_./ _)" 10)
-  funcset :: "['a set, 'b set] => ('a => 'b) set"      (infixr 60)
-  "@lam" :: "[pttrn, 'a set, 'a => 'b] => ('a => 'b)"  ("(3%_:_./ _)" [0, 0, 3] 3)
-syntax (xsymbols)
-  "@lam" :: "[pttrn, 'a set, 'a => 'b] => ('a => 'b)"  ("(3\\<lambda>_\\<in>_./ _)" [0, 0, 3] 3)
-
-  (*Giving funcset an arrow syntax (-> or =>) clashes with many existing theories*)
-
-syntax (xsymbols)
-  "@Pi" :: "[pttrn, 'a set, 'b set] => ('a => 'b) set"  ("(3\\<Pi> _\\<in>_./ _)"   10)
-
-translations
-  "PI x:A. B" => "Pi A (%x. B)"
-  "A funcset B"    => "Pi A (_K B)"
-  "%x:A. f"  == "restrict (%x. f) A"
-
-constdefs
-  compose :: "['a set, 'b => 'c, 'a => 'b] => ('a => 'c)"
-  "compose A g f == %x:A. g (f x)"
+
+text{*As a simplification rule, it replaces all function equalities by
+  first-order equalities.*}
+lemma expand_fun_eq: "(f = g) = (! x. f(x)=g(x))"
+apply (rule iffI)
+apply (simp (no_asm_simp))
+apply (rule ext, simp (no_asm_simp))
+done
+
+lemma apply_inverse:
+    "[| f(x)=u;  !!x. P(x) ==> g(f(x)) = x;  P(x) |] ==> x=g(u)"
+by auto
+
+
+text{*The Identity Function: @{term id}*}
+lemma id_apply [simp]: "id x = x"
+by (simp add: id_def)
+
+
+subsection{*The Composition Operator: @{term "f \<circ> g"}*}
+
+lemma o_apply [simp]: "(f o g) x = f (g x)"
+by (simp add: comp_def)
+
+lemma o_assoc: "f o (g o h) = f o g o h"
+by (simp add: comp_def)
+
+lemma id_o [simp]: "id o g = g"
+by (simp add: comp_def)
+
+lemma o_id [simp]: "f o id = f"
+by (simp add: comp_def)
+
+lemma image_compose: "(f o g) ` r = f`(g`r)"
+by (simp add: comp_def, blast)
+
+lemma image_eq_UN: "f`A = (UN x:A. {f x})"
+by blast
+
+lemma UN_o: "UNION A (g o f) = UNION (f`A) g"
+by (unfold comp_def, blast)
+
+text{*Lemma for proving injectivity of representation functions for
+datatypes involving function types*}
+lemma inj_fun_lemma:
+  "[| ! x y. g (f x) = g y --> f x = y; g o f = g o fa |] ==> f = fa"
+by (simp add: comp_def expand_fun_eq)
+
+
+subsection{*The Injectivity Predicate, @{term inj}*}
+
+text{*NB: @{term inj} now just translates to @{term inj_on}*}
+
+
+text{*For Proofs in @{text "Tools/datatype_rep_proofs"}*}
+lemma datatype_injI:
+    "(!! x. ALL y. f(x) = f(y) --> x=y) ==> inj(f)"
+by (simp add: inj_on_def)
+
+lemma injD: "[| inj(f); f(x) = f(y) |] ==> x=y"
+by (simp add: inj_on_def)
+
+(*Useful with the simplifier*)
+lemma inj_eq: "inj(f) ==> (f(x) = f(y)) = (x=y)"
+by (force simp add: inj_on_def)
+
+lemma inj_o: "[| inj f; f o g = f o h |] ==> g = h"
+by (simp add: comp_def inj_on_def expand_fun_eq)
+
+
+subsection{*The Predicate @{term inj_on}: Injectivity On A Restricted Domain*}
+
+lemma inj_onI:
+    "(!! x y. [|  x:A;  y:A;  f(x) = f(y) |] ==> x=y) ==> inj_on f A"
+by (simp add: inj_on_def)
+
+lemma inj_on_inverseI: "(!!x. x:A ==> g(f(x)) = x) ==> inj_on f A"
+by (auto dest:  arg_cong [of concl: g] simp add: inj_on_def)
+
+lemma inj_onD: "[| inj_on f A;  f(x)=f(y);  x:A;  y:A |] ==> x=y"
+by (unfold inj_on_def, blast)
+
+lemma inj_on_iff: "[| inj_on f A;  x:A;  y:A |] ==> (f(x)=f(y)) = (x=y)"
+by (blast dest!: inj_onD)
+
+lemma comp_inj_on:
+     "[| inj_on f A;  inj_on g (f`A) |] ==> inj_on (g o f) A"
+by (simp add: comp_def inj_on_def)
+
+lemma inj_on_contraD: "[| inj_on f A;  ~x=y;  x:A;  y:A |] ==> ~ f(x)=f(y)"
+by (unfold inj_on_def, blast)
+
+lemma inj_singleton: "inj (%s. {s})"
+by (simp add: inj_on_def)
+
+lemma subset_inj_on: "[| A<=B; inj_on f B |] ==> inj_on f A"
+by (unfold inj_on_def, blast)
+
+
+subsection{*The Predicate @{term surj}: Surjectivity*}
+
+lemma surjI: "(!! x. g(f x) = x) ==> surj g"
+apply (simp add: surj_def)
+apply (blast intro: sym)
+done
+
+lemma surj_range: "surj f ==> range f = UNIV"
+by (auto simp add: surj_def)
+
+lemma surjD: "surj f ==> EX x. y = f x"
+by (simp add: surj_def)
+
+lemma surjE: "surj f ==> (!!x. y = f x ==> C) ==> C"
+by (simp add: surj_def, blast)
+
+lemma comp_surj: "[| surj f;  surj g |] ==> surj (g o f)"
+apply (simp add: comp_def surj_def, clarify)
+apply (drule_tac x = y in spec, clarify)
+apply (drule_tac x = x in spec, blast)
+done
+
+
+
+subsection{*The Predicate @{term bij}: Bijectivity*}
+
+lemma bijI: "[| inj f; surj f |] ==> bij f"
+by (simp add: bij_def)
+
+lemma bij_is_inj: "bij f ==> inj f"
+by (simp add: bij_def)
+
+lemma bij_is_surj: "bij f ==> surj f"
+by (simp add: bij_def)
+
+
+subsection{*Facts About the Identity Function*}
+
+text{*We seem to need both the @{term id} forms and the @{term "\<lambda>x. x"}
+forms. The latter can arise by rewriting, while @{term id} may be used
+explicitly.*}
+
+lemma image_ident [simp]: "(%x. x) ` Y = Y"
+by blast
+
+lemma image_id [simp]: "id ` Y = Y"
+by (simp add: id_def)
+
+lemma vimage_ident [simp]: "(%x. x) -` Y = Y"
+by blast
+
+lemma vimage_id [simp]: "id -` A = A"
+by (simp add: id_def)
+
+lemma vimage_image_eq: "f -` (f ` A) = {y. EX x:A. f x = f y}"
+by (blast intro: sym)
+
+lemma image_vimage_subset: "f ` (f -` A) <= A"
+by blast
+
+lemma image_vimage_eq [simp]: "f ` (f -` A) = A Int range f"
+by blast
+
+lemma surj_image_vimage_eq: "surj f ==> f ` (f -` A) = A"
+by (simp add: surj_range)
+
+lemma inj_vimage_image_eq: "inj f ==> f -` (f ` A) = A"
+by (simp add: inj_on_def, blast)
+
+lemma vimage_subsetD: "surj f ==> f -` B <= A ==> B <= f ` A"
+apply (unfold surj_def)
+apply (blast intro: sym)
+done
+
+lemma vimage_subsetI: "inj f ==> B <= f ` A ==> f -` B <= A"
+by (unfold inj_on_def, blast)
+
+lemma vimage_subset_eq: "bij f ==> (f -` B <= A) = (B <= f ` A)"
+apply (unfold bij_def)
+apply (blast del: subsetI intro: vimage_subsetI vimage_subsetD)
+done
+
+lemma image_Int_subset: "f`(A Int B) <= f`A Int f`B"
+by blast
+
+lemma image_diff_subset: "f`A - f`B <= f`(A - B)"
+by blast
+
+lemma inj_on_image_Int:
+   "[| inj_on f C;  A<=C;  B<=C |] ==> f`(A Int B) = f`A Int f`B"
+apply (simp add: inj_on_def, blast)
+done
+
+lemma inj_on_image_set_diff:
+   "[| inj_on f C;  A<=C;  B<=C |] ==> f`(A-B) = f`A - f`B"
+apply (simp add: inj_on_def, blast)
+done
+
+lemma image_Int: "inj f ==> f`(A Int B) = f`A Int f`B"
+by (simp add: inj_on_def, blast)
+
+lemma image_set_diff: "inj f ==> f`(A-B) = f`A - f`B"
+by (simp add: inj_on_def, blast)
+
+lemma inj_image_mem_iff: "inj f ==> (f a : f`A) = (a : A)"
+by (blast dest: injD)
+
+lemma inj_image_subset_iff: "inj f ==> (f`A <= f`B) = (A<=B)"
+by (simp add: inj_on_def, blast)
+
+lemma inj_image_eq_iff: "inj f ==> (f`A = f`B) = (A = B)"
+by (blast dest: injD)
+
+lemma image_UN: "(f ` (UNION A B)) = (UN x:A.(f ` (B x)))"
+by blast
+
+(*injectivity's required.  Left-to-right inclusion holds even if A is empty*)
+lemma image_INT:
+   "[| inj_on f C;  ALL x:A. B x <= C;  j:A |]
+    ==> f ` (INTER A B) = (INT x:A. f ` B x)"
+apply (simp add: inj_on_def, blast)
+done
+
+(*Compare with image_INT: no use of inj_on, and if f is surjective then
+  it doesn't matter whether A is empty*)
+lemma bij_image_INT: "bij f ==> f ` (INTER A B) = (INT x:A. f ` B x)"
+apply (simp add: bij_def)
+apply (simp add: inj_on_def surj_def, blast)
+done
+
+lemma surj_Compl_image_subset: "surj f ==> -(f`A) <= f`(-A)"
+by (auto simp add: surj_def)
+
+lemma inj_image_Compl_subset: "inj f ==> f`(-A) <= -(f`A)"
+by (auto simp add: inj_on_def)
+
+lemma bij_image_Compl_eq: "bij f ==> f`(-A) = -(f`A)"
+apply (simp add: bij_def)
+apply (rule equalityI)
+apply (simp_all (no_asm_simp) add: inj_image_Compl_subset surj_Compl_image_subset)
+done
+
+
+subsection{*Function Updating*}
+
+lemma fun_upd_idem_iff: "(f(x:=y) = f) = (f x = y)"
+apply (simp add: fun_upd_def, safe)
+apply (erule subst)
+apply (rule_tac [2] ext, auto)
+done
+
+(* f x = y ==> f(x:=y) = f *)
+lemmas fun_upd_idem = fun_upd_idem_iff [THEN iffD2, standard]
+
+(* f(x := f x) = f *)
+declare refl [THEN fun_upd_idem, iff]
+
+lemma fun_upd_apply [simp]: "(f(x:=y))z = (if z=x then y else f z)"
+apply (simp (no_asm) add: fun_upd_def)
+done
+
+(* fun_upd_apply supersedes these two,   but they are useful
+   if fun_upd_apply is intentionally removed from the simpset *)
+lemma fun_upd_same: "(f(x:=y)) x = y"
+by simp
+
+lemma fun_upd_other: "z~=x ==> (f(x:=y)) z = f z"
+by simp
+
+lemma fun_upd_upd [simp]: "f(x:=y,x:=z) = f(x:=z)"
+by (simp add: expand_fun_eq)
+
+lemma fun_upd_twist: "a ~= c ==> (m(a:=b))(c:=d) = (m(c:=d))(a:=b)"
+by (rule ext, auto)
+
+text{*The ML section includes some compatibility bindings and a simproc
+for function updates, in addition to the usual ML-bindings of theorems.*}
+ML
+{*
+val id_def = thm "id_def";
+val inj_on_def = thm "inj_on_def";
+val surj_def = thm "surj_def";
+val bij_def = thm "bij_def";
+val fun_upd_def = thm "fun_upd_def";
+
+val o_def = thm "comp_def";
+val injI = thm "inj_onI";
+val inj_inverseI = thm "inj_on_inverseI";
+val set_cs = claset() delrules [equalityI];
+
+val print_translation = [("Pi", dependent_tr' ("@Pi", "op funcset"))];
+
+(* simplifies terms of the form f(...,x:=y,...,x:=z,...) to f(...,x:=z,...) *)
+local
+  fun gen_fun_upd None T _ _ = None
+    | gen_fun_upd (Some f) T x y = Some (Const ("Fun.fun_upd",T) $ f $ x $ y)
+  fun dest_fun_T1 (Type (_, T :: Ts)) = T
+  fun find_double (t as Const ("Fun.fun_upd",T) $ f $ x $ y) =
+    let
+      fun find (Const ("Fun.fun_upd",T) $ g $ v $ w) =
+            if v aconv x then Some g else gen_fun_upd (find g) T v w
+        | find t = None
+    in (dest_fun_T1 T, gen_fun_upd (find f) T x y) end
+
+  val ss = simpset ()
+  val fun_upd_prover = K (rtac eq_reflection 1 THEN rtac ext 1 THEN simp_tac ss 1)
+in
+  val fun_upd2_simproc =
+    Simplifier.simproc (Theory.sign_of (the_context ()))
+      "fun_upd2" ["f(v := w, x := y)"]
+      (fn sg => fn _ => fn t =>
+        case find_double t of (T, None) => None
+        | (T, Some rhs) => Some (Tactic.prove sg [] [] (Term.equals T $ t $ rhs) fun_upd_prover))
+end;
+Addsimprocs[fun_upd2_simproc];
+
+val expand_fun_eq = thm "expand_fun_eq";
+val apply_inverse = thm "apply_inverse";
+val id_apply = thm "id_apply";
+val o_apply = thm "o_apply";
+val o_assoc = thm "o_assoc";
+val id_o = thm "id_o";
+val o_id = thm "o_id";
+val image_compose = thm "image_compose";
+val image_eq_UN = thm "image_eq_UN";
+val UN_o = thm "UN_o";
+val inj_fun_lemma = thm "inj_fun_lemma";
+val datatype_injI = thm "datatype_injI";
+val injD = thm "injD";
+val inj_eq = thm "inj_eq";
+val inj_o = thm "inj_o";
+val inj_onI = thm "inj_onI";
+val inj_on_inverseI = thm "inj_on_inverseI";
+val inj_onD = thm "inj_onD";
+val inj_on_iff = thm "inj_on_iff";
+val comp_inj_on = thm "comp_inj_on";
+val inj_on_contraD = thm "inj_on_contraD";
+val inj_singleton = thm "inj_singleton";
+val subset_inj_on = thm "subset_inj_on";
+val surjI = thm "surjI";
+val surj_range = thm "surj_range";
+val surjD = thm "surjD";
+val surjE = thm "surjE";
+val comp_surj = thm "comp_surj";
+val bijI = thm "bijI";
+val bij_is_inj = thm "bij_is_inj";
+val bij_is_surj = thm "bij_is_surj";
+val image_ident = thm "image_ident";
+val image_id = thm "image_id";
+val vimage_ident = thm "vimage_ident";
+val vimage_id = thm "vimage_id";
+val vimage_image_eq = thm "vimage_image_eq";
+val image_vimage_subset = thm "image_vimage_subset";
+val image_vimage_eq = thm "image_vimage_eq";
+val surj_image_vimage_eq = thm "surj_image_vimage_eq";
+val inj_vimage_image_eq = thm "inj_vimage_image_eq";
+val vimage_subsetD = thm "vimage_subsetD";
+val vimage_subsetI = thm "vimage_subsetI";
+val vimage_subset_eq = thm "vimage_subset_eq";
+val image_Int_subset = thm "image_Int_subset";
+val image_diff_subset = thm "image_diff_subset";
+val inj_on_image_Int = thm "inj_on_image_Int";
+val inj_on_image_set_diff = thm "inj_on_image_set_diff";
+val image_Int = thm "image_Int";
+val image_set_diff = thm "image_set_diff";
+val inj_image_mem_iff = thm "inj_image_mem_iff";
+val inj_image_subset_iff = thm "inj_image_subset_iff";
+val inj_image_eq_iff = thm "inj_image_eq_iff";
+val image_UN = thm "image_UN";
+val image_INT = thm "image_INT";
+val bij_image_INT = thm "bij_image_INT";
+val surj_Compl_image_subset = thm "surj_Compl_image_subset";
+val inj_image_Compl_subset = thm "inj_image_Compl_subset";
+val bij_image_Compl_eq = thm "bij_image_Compl_eq";
+val fun_upd_idem_iff = thm "fun_upd_idem_iff";
+val fun_upd_idem = thm "fun_upd_idem";
+val fun_upd_apply = thm "fun_upd_apply";
+val fun_upd_same = thm "fun_upd_same";
+val fun_upd_other = thm "fun_upd_other";
+val fun_upd_upd = thm "fun_upd_upd";
+val fun_upd_twist = thm "fun_upd_twist";
+*}
 
 end
-
-ML
-val print_translation = [("Pi", dependent_tr' ("@Pi", "op funcset"))];