Numerals now work for the integers: the binary numerals for 0 and 1 rewrite
to their abstract counterparts, while other binary numerals work correctly.
(* Title: HOL/Integ/Ring.ML
ID: $Id$
Author: Tobias Nipkow
Copyright 1996 TU Muenchen
Derives a few equational consequences about rings
and defines cring_simpl, a simplification tactic for commutative rings.
*)
Goal "!!x::'a::cring. x*(y*z)=y*(x*z)";
by (rtac trans 1);
by (rtac times_commute 1);
by (rtac trans 1);
by (rtac times_assoc 1);
by (simp_tac (HOL_basic_ss addsimps [times_commute]) 1);
qed "times_commuteL";
val times_cong = read_instantiate [("f1","op *")] (arg_cong RS cong);
Goal "!!x::'a::ring. 0*x = 0";
by (rtac trans 1);
by (rtac right_inv 2);
by (rtac trans 1);
by (rtac plus_cong 2);
by (rtac refl 3);
by (rtac trans 2);
by (rtac times_cong 3);
by (rtac zeroL 3);
by (rtac refl 3);
by (rtac (distribR RS sym) 2);
by (rtac trans 1);
by (rtac (plus_assoc RS sym) 2);
by (rtac trans 1);
by (rtac plus_cong 2);
by (rtac refl 2);
by (rtac (right_inv RS sym) 2);
by (rtac (zeroR RS sym) 1);
qed "mult_zeroL";
Goal "!!x::'a::ring. x*0 = 0";
by (rtac trans 1);
by (rtac right_inv 2);
by (rtac trans 1);
by (rtac plus_cong 2);
by (rtac refl 3);
by (rtac trans 2);
by (rtac times_cong 3);
by (rtac zeroL 4);
by (rtac refl 3);
by (rtac (distribL RS sym) 2);
by (rtac trans 1);
by (rtac (plus_assoc RS sym) 2);
by (rtac trans 1);
by (rtac plus_cong 2);
by (rtac refl 2);
by (rtac (right_inv RS sym) 2);
by (rtac (zeroR RS sym) 1);
qed "mult_zeroR";
Goal "!!x::'a::ring. (0-x)*y = 0-(x*y)";
by (rtac trans 1);
by (rtac zeroL 2);
by (rtac trans 1);
by (rtac plus_cong 2);
by (rtac refl 3);
by (rtac mult_zeroL 2);
by (rtac trans 1);
by (rtac plus_cong 2);
by (rtac refl 3);
by (rtac times_cong 2);
by (rtac left_inv 2);
by (rtac refl 2);
by (rtac trans 1);
by (rtac plus_cong 2);
by (rtac refl 3);
by (rtac (distribR RS sym) 2);
by (rtac trans 1);
by (rtac (plus_assoc RS sym) 2);
by (rtac trans 1);
by (rtac plus_cong 2);
by (rtac refl 2);
by (rtac (right_inv RS sym) 2);
by (rtac (zeroR RS sym) 1);
qed "mult_invL";
Goal "!!x::'a::ring. x*(0-y) = 0-(x*y)";
by (rtac trans 1);
by (rtac zeroL 2);
by (rtac trans 1);
by (rtac plus_cong 2);
by (rtac refl 3);
by (rtac mult_zeroR 2);
by (rtac trans 1);
by (rtac plus_cong 2);
by (rtac refl 3);
by (rtac times_cong 2);
by (rtac refl 2);
by (rtac left_inv 2);
by (rtac trans 1);
by (rtac plus_cong 2);
by (rtac refl 3);
by (rtac (distribL RS sym) 2);
by (rtac trans 1);
by (rtac (plus_assoc RS sym) 2);
by (rtac trans 1);
by (rtac plus_cong 2);
by (rtac refl 2);
by (rtac (right_inv RS sym) 2);
by (rtac (zeroR RS sym) 1);
qed "mult_invR";
Goal "x*(y-z) = (x*y - x*z::'a::ring)";
by (mk_group1_tac 1);
by (simp_tac (HOL_basic_ss addsimps [distribL,mult_invR]) 1);
qed "minus_distribL";
Goal "(x-y)*z = (x*z - y*z::'a::ring)";
by (mk_group1_tac 1);
by (simp_tac (HOL_basic_ss addsimps [distribR,mult_invL]) 1);
qed "minus_distribR";
val cring_simps = [times_assoc,times_commute,times_commuteL,
distribL,distribR,minus_distribL,minus_distribR]
@ agroup2_simps;
val cring_tac =
let val ss = HOL_basic_ss addsimps cring_simps
in simp_tac ss end;
(*** The order [minus_plusL3,minus_plusL2] is important because minus_plusL3
MUST be tried first
val cring_simp =
let val phase1 = simpset() addsimps
[plus_minusL,minus_plusR,minus_minusR,plus_minusR]
val phase2 = HOL_ss addsimps [minus_plusL3,minus_plusL2,
zeroL,zeroR,mult_zeroL,mult_zeroR]
in simp_tac phase1 THEN' simp_tac phase2 end;
***)