--- a/src/HOL/arith_data.ML Mon Dec 28 17:03:47 1998 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,257 +0,0 @@
-(* Title: HOL/arith_data.ML
- ID: $Id$
- Author: Markus Wenzel and Stefan Berghofer, TU Muenchen
-
-Setup various arithmetic proof procedures.
-*)
-
-signature ARITH_DATA =
-sig
- val nat_cancel_sums: simproc list
- val nat_cancel_factor: simproc list
- val nat_cancel: simproc list
-end;
-
-structure ArithData: ARITH_DATA =
-struct
-
-
-(** abstract syntax of structure nat: 0, Suc, + **)
-
-(* mk_sum, mk_norm_sum *)
-
-val one = HOLogic.mk_nat 1;
-val mk_plus = HOLogic.mk_binop "op +";
-
-fun mk_sum [] = HOLogic.zero
- | mk_sum [t] = t
- | mk_sum (t :: ts) = mk_plus (t, mk_sum ts);
-
-(*normal form of sums: Suc (... (Suc (a + (b + ...))))*)
-fun mk_norm_sum ts =
- let val (ones, sums) = partition (equal one) ts in
- funpow (length ones) HOLogic.mk_Suc (mk_sum sums)
- end;
-
-
-(* dest_sum *)
-
-val dest_plus = HOLogic.dest_bin "op +" HOLogic.natT;
-
-fun dest_sum tm =
- if HOLogic.is_zero tm then []
- else
- (case try HOLogic.dest_Suc tm of
- Some t => one :: dest_sum t
- | None =>
- (case try dest_plus tm of
- Some (t, u) => dest_sum t @ dest_sum u
- | None => [tm]));
-
-
-(** generic proof tools **)
-
-(* prove conversions *)
-
-val mk_eqv = HOLogic.mk_Trueprop o HOLogic.mk_eq;
-
-fun prove_conv expand_tac norm_tac sg (t, u) =
- mk_meta_eq (prove_goalw_cterm_nocheck [] (cterm_of sg (mk_eqv (t, u)))
- (K [expand_tac, norm_tac]))
- handle ERROR => error ("The error(s) above occurred while trying to prove " ^
- (string_of_cterm (cterm_of sg (mk_eqv (t, u)))));
-
-val subst_equals = prove_goal HOL.thy "[| t = s; u = t |] ==> u = s"
- (fn prems => [cut_facts_tac prems 1, SIMPSET' asm_simp_tac 1]);
-
-
-(* rewriting *)
-
-fun simp_all rules = ALLGOALS (simp_tac (HOL_ss addsimps rules));
-
-val add_rules = [add_Suc, add_Suc_right, add_0, add_0_right];
-val mult_rules = [mult_Suc, mult_Suc_right, mult_0, mult_0_right];
-
-
-
-(** cancel common summands **)
-
-structure Sum =
-struct
- val mk_sum = mk_norm_sum;
- val dest_sum = dest_sum;
- val prove_conv = prove_conv;
- val norm_tac = simp_all add_rules THEN simp_all add_ac;
-end;
-
-fun gen_uncancel_tac rule ct =
- rtac (instantiate' [] [None, Some ct] (rule RS subst_equals)) 1;
-
-
-(* nat eq *)
-
-structure EqCancelSums = CancelSumsFun
-(struct
- open Sum;
- val mk_bal = HOLogic.mk_eq;
- val dest_bal = HOLogic.dest_bin "op =" HOLogic.natT;
- val uncancel_tac = gen_uncancel_tac add_left_cancel;
-end);
-
-
-(* nat less *)
-
-structure LessCancelSums = CancelSumsFun
-(struct
- open Sum;
- val mk_bal = HOLogic.mk_binrel "op <";
- val dest_bal = HOLogic.dest_bin "op <" HOLogic.natT;
- val uncancel_tac = gen_uncancel_tac add_left_cancel_less;
-end);
-
-
-(* nat le *)
-
-structure LeCancelSums = CancelSumsFun
-(struct
- open Sum;
- val mk_bal = HOLogic.mk_binrel "op <=";
- val dest_bal = HOLogic.dest_bin "op <=" HOLogic.natT;
- val uncancel_tac = gen_uncancel_tac add_left_cancel_le;
-end);
-
-
-(* nat diff *)
-
-structure DiffCancelSums = CancelSumsFun
-(struct
- open Sum;
- val mk_bal = HOLogic.mk_binop "op -";
- val dest_bal = HOLogic.dest_bin "op -" HOLogic.natT;
- val uncancel_tac = gen_uncancel_tac diff_cancel;
-end);
-
-
-
-(** cancel common factor **)
-
-structure Factor =
-struct
- val mk_sum = mk_norm_sum;
- val dest_sum = dest_sum;
- val prove_conv = prove_conv;
- val norm_tac = simp_all (add_rules @ mult_rules) THEN simp_all add_ac;
-end;
-
-fun mk_cnat n = cterm_of (sign_of Nat.thy) (HOLogic.mk_nat n);
-
-fun gen_multiply_tac rule k =
- if k > 0 then
- rtac (instantiate' [] [None, Some (mk_cnat (k - 1))] (rule RS subst_equals)) 1
- else no_tac;
-
-
-(* nat eq *)
-
-structure EqCancelFactor = CancelFactorFun
-(struct
- open Factor;
- val mk_bal = HOLogic.mk_eq;
- val dest_bal = HOLogic.dest_bin "op =" HOLogic.natT;
- val multiply_tac = gen_multiply_tac Suc_mult_cancel1;
-end);
-
-
-(* nat less *)
-
-structure LessCancelFactor = CancelFactorFun
-(struct
- open Factor;
- val mk_bal = HOLogic.mk_binrel "op <";
- val dest_bal = HOLogic.dest_bin "op <" HOLogic.natT;
- val multiply_tac = gen_multiply_tac Suc_mult_less_cancel1;
-end);
-
-
-(* nat le *)
-
-structure LeCancelFactor = CancelFactorFun
-(struct
- open Factor;
- val mk_bal = HOLogic.mk_binrel "op <=";
- val dest_bal = HOLogic.dest_bin "op <=" HOLogic.natT;
- val multiply_tac = gen_multiply_tac Suc_mult_le_cancel1;
-end);
-
-
-
-(** prepare nat_cancel simprocs **)
-
-fun prep_pat s = Thm.read_cterm (sign_of Arith.thy) (s, HOLogic.termTVar);
-val prep_pats = map prep_pat;
-
-fun prep_simproc (name, pats, proc) = Simplifier.mk_simproc name pats proc;
-
-val eq_pats = prep_pats ["(l::nat) + m = n", "(l::nat) = m + n", "Suc m = n", "m = Suc n"];
-val less_pats = prep_pats ["(l::nat) + m < n", "(l::nat) < m + n", "Suc m < n", "m < Suc n"];
-val le_pats = prep_pats ["(l::nat) + m <= n", "(l::nat) <= m + n", "Suc m <= n", "m <= Suc n"];
-val diff_pats = prep_pats ["((l::nat) + m) - n", "(l::nat) - (m + n)", "Suc m - n", "m - Suc n"];
-
-val nat_cancel_sums = map prep_simproc
- [("nateq_cancel_sums", eq_pats, EqCancelSums.proc),
- ("natless_cancel_sums", less_pats, LessCancelSums.proc),
- ("natle_cancel_sums", le_pats, LeCancelSums.proc),
- ("natdiff_cancel_sums", diff_pats, DiffCancelSums.proc)];
-
-val nat_cancel_factor = map prep_simproc
- [("nateq_cancel_factor", eq_pats, EqCancelFactor.proc),
- ("natless_cancel_factor", less_pats, LessCancelFactor.proc),
- ("natle_cancel_factor", le_pats, LeCancelFactor.proc)];
-
-val nat_cancel = nat_cancel_factor @ nat_cancel_sums;
-
-
-end;
-
-
-open ArithData;
-
-
-context Arith.thy;
-Addsimprocs nat_cancel;
-
-
-(*This proof requires natdiff_cancel_sums*)
-Goal "m < (n::nat) --> m<l --> (l-n) < (l-m)";
-by (induct_tac "l" 1);
-by (Simp_tac 1);
-by (Clarify_tac 1);
-by (etac less_SucE 1);
-by (Clarify_tac 2);
-by (asm_simp_tac (simpset() addsimps [Suc_le_eq]) 2);
-by (asm_simp_tac (simpset() addsimps [diff_Suc_le_Suc_diff RS le_less_trans,
- Suc_diff_le, less_imp_le]) 1);
-qed_spec_mp "diff_less_mono2";
-
-(** Elimination of - on nat due to John Harrison **)
-(*This proof requires natle_cancel_sums*)
-
-Goal "P(a - b::nat) = (!d. (b = a + d --> P 0) & (a = b + d --> P d))";
-by(case_tac "a <= b" 1);
-by(Auto_tac);
-by(eres_inst_tac [("x","b-a")] allE 1);
-by(Auto_tac);
-qed "nat_diff_split";
-
-(*
-This is an example of the power of nat_diff_split. Many of the `-' thms in
-Arith.ML could take advantage of this, but would need to be moved.
-*)
-Goal "m-n = 0 --> n-m = 0 --> m=n";
-by(simp_tac (simpset() addsplits [nat_diff_split]) 1);
-qed_spec_mp "diffs0_imp_equal";
-
-use"fast_nat_decider";
-
-simpset_ref () := (simpset() addSolver
- (fn thms => cut_facts_tac thms THEN' Fast_Nat_Decider.nat_arith_tac));