(* Ring homomorphism $Id: RingHomo.thy,v 1.5 2006/11/19 22:48:56 wenzelm Exp $ Author: Clemens Ballarin, started 15 April 1997 *) header {* Ring homomorphism *} theory RingHomo imports Ring2 begin definition homo :: "('a::ring => 'b::ring) => bool" where "homo f <-> (ALL a b. f (a + b) = f a + f b & f (a * b) = f a * f b) & f 1 = 1" lemma homoI: "!! f. [| !! a b. f (a + b) = f a + f b; !! a b. f (a * b) = f a * f b; f 1 = 1 |] ==> homo f" unfolding homo_def by blast lemma homo_add [simp]: "!! f. homo f ==> f (a + b) = f a + f b" unfolding homo_def by blast lemma homo_mult [simp]: "!! f. homo f ==> f (a * b) = f a * f b" unfolding homo_def by blast lemma homo_one [simp]: "!! f. homo f ==> f 1 = 1" unfolding homo_def by blast lemma homo_zero [simp]: "!! f::('a::ring=>'b::ring). homo f ==> f 0 = 0" apply (rule_tac a = "f 0" in a_lcancel) apply (simp (no_asm_simp) add: homo_add [symmetric]) done lemma homo_uminus [simp]: "!! f::('a::ring=>'b::ring). homo f ==> f (-a) = - f a" apply (rule_tac a = "f a" in a_lcancel) apply (frule homo_zero) apply (simp (no_asm_simp) add: homo_add [symmetric]) done lemma homo_power [simp]: "!! f::('a::ring=>'b::ring). homo f ==> f (a ^ n) = f a ^ n" apply (induct_tac n) apply (drule homo_one) apply simp apply (drule_tac a = "a^n" and b = "a" in homo_mult) apply simp done lemma homo_SUM [simp]: "!! f::('a::ring=>'b::ring). homo f ==> f (setsum g {..n::nat}) = setsum (f o g) {..n}" apply (induct_tac n) apply simp apply simp done lemma id_homo [simp]: "homo (%x. x)" by (blast intro!: homoI) end
lemma homoI:
[| !!a b. f (a + b) = f a + f b; !!a b. f (a * b) = f a * f b;
f (1::'a) = (1::'b) |]
==> homo f
lemma homo_add:
homo f ==> f (a + b) = f a + f b
lemma homo_mult:
homo f ==> f (a * b) = f a * f b
lemma homo_one:
homo f ==> f (1::'a) = (1::'b)
lemma homo_zero:
homo f ==> f (0::'a) = (0::'b)
lemma homo_uminus:
homo f ==> f (- a) = - f a
lemma homo_power:
homo f ==> f (a ^ n) = f a ^ n
lemma homo_SUM:
homo f ==> f (setsum g {..n}) = setsum (f o g) {..n}
lemma id_homo:
homo (λx. x)