diff -r 4a4f6ad607a1 -r fdf4638bf726 src/HOL/arith_data.ML --- 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-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));