author | nipkow |
Tue, 13 Aug 2002 21:57:15 +0200 | |
changeset 13499 | f95f5818f24f |
parent 13485 | acf39e924091 |
child 14113 | 7b3513ba0f86 |
permissions | -rw-r--r-- |
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(* Title: HOL/Integ/int_arith1.ML |
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ID: $Id$ |
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Authors: Larry Paulson and Tobias Nipkow |
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Simprocs and decision procedure for linear arithmetic. |
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*) |
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Addsimps [int_numeral_0_eq_0, int_numeral_1_eq_1]; |
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(*** Simprocs for numeric literals ***) |
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(** Combining of literal coefficients in sums of products **) |
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Goal "(x < y) = (x-y < (0::int))"; |
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by (simp_tac (simpset() addsimps zcompare_rls) 1); |
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qed "zless_iff_zdiff_zless_0"; |
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Goal "(x = y) = (x-y = (0::int))"; |
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by (simp_tac (simpset() addsimps zcompare_rls) 1); |
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qed "eq_iff_zdiff_eq_0"; |
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Goal "(x <= y) = (x-y <= (0::int))"; |
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by (simp_tac (simpset() addsimps zcompare_rls) 1); |
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qed "zle_iff_zdiff_zle_0"; |
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(** For combine_numerals **) |
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Goal "i*u + (j*u + k) = (i+j)*u + (k::int)"; |
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by (asm_simp_tac (simpset() addsimps [zadd_zmult_distrib]) 1); |
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qed "left_zadd_zmult_distrib"; |
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(** For cancel_numerals **) |
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val rel_iff_rel_0_rls = map (inst "y" "?u+?v") |
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[zless_iff_zdiff_zless_0, eq_iff_zdiff_eq_0, |
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zle_iff_zdiff_zle_0] @ |
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map (inst "y" "n") |
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[zless_iff_zdiff_zless_0, eq_iff_zdiff_eq_0, |
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zle_iff_zdiff_zle_0]; |
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Goal "!!i::int. (i*u + m = j*u + n) = ((i-j)*u + m = n)"; |
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by (asm_simp_tac (simpset() addsimps [zdiff_def, zadd_zmult_distrib]@ |
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zadd_ac@rel_iff_rel_0_rls) 1); |
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qed "eq_add_iff1"; |
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Goal "!!i::int. (i*u + m = j*u + n) = (m = (j-i)*u + n)"; |
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by (asm_simp_tac (simpset() addsimps [zdiff_def, zadd_zmult_distrib]@ |
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zadd_ac@rel_iff_rel_0_rls) 1); |
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qed "eq_add_iff2"; |
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Goal "!!i::int. (i*u + m < j*u + n) = ((i-j)*u + m < n)"; |
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by (asm_simp_tac (simpset() addsimps [zdiff_def, zadd_zmult_distrib]@ |
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zadd_ac@rel_iff_rel_0_rls) 1); |
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qed "less_add_iff1"; |
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Goal "!!i::int. (i*u + m < j*u + n) = (m < (j-i)*u + n)"; |
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by (asm_simp_tac (simpset() addsimps [zdiff_def, zadd_zmult_distrib]@ |
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zadd_ac@rel_iff_rel_0_rls) 1); |
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qed "less_add_iff2"; |
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Goal "!!i::int. (i*u + m <= j*u + n) = ((i-j)*u + m <= n)"; |
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by (asm_simp_tac (simpset() addsimps [zdiff_def, zadd_zmult_distrib]@ |
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zadd_ac@rel_iff_rel_0_rls) 1); |
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qed "le_add_iff1"; |
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Goal "!!i::int. (i*u + m <= j*u + n) = (m <= (j-i)*u + n)"; |
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by (asm_simp_tac (simpset() addsimps [zdiff_def, zadd_zmult_distrib] |
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@zadd_ac@rel_iff_rel_0_rls) 1); |
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qed "le_add_iff2"; |
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structure Int_Numeral_Simprocs = |
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struct |
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(*Maps 0 to Numeral0 and 1 to Numeral1 so that arithmetic in simprocs |
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isn't complicated by the abstract 0 and 1.*) |
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val numeral_syms = [int_numeral_0_eq_0 RS sym, int_numeral_1_eq_1 RS sym]; |
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val numeral_sym_ss = HOL_ss addsimps numeral_syms; |
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fun rename_numerals th = |
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simplify numeral_sym_ss (Thm.transfer (the_context ()) th); |
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(*Utilities*) |
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fun mk_numeral n = HOLogic.number_of_const HOLogic.intT $ HOLogic.mk_bin n; |
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(*Decodes a binary INTEGER*) |
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fun dest_numeral (Const("0", _)) = 0 |
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| dest_numeral (Const("1", _)) = 1 |
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| dest_numeral (Const("Numeral.number_of", _) $ w) = |
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(HOLogic.dest_binum w |
94 |
handle TERM _ => raise TERM("Int_Numeral_Simprocs.dest_numeral:1", [w])) |
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| dest_numeral t = raise TERM("Int_Numeral_Simprocs.dest_numeral:2", [t]); |
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fun find_first_numeral past (t::terms) = |
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((dest_numeral t, rev past @ terms) |
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handle TERM _ => find_first_numeral (t::past) terms) |
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| find_first_numeral past [] = raise TERM("find_first_numeral", []); |
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val zero = mk_numeral 0; |
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val mk_plus = HOLogic.mk_binop "op +"; |
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val uminus_const = Const ("uminus", HOLogic.intT --> HOLogic.intT); |
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(*Thus mk_sum[t] yields t+0; longer sums don't have a trailing zero*) |
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fun mk_sum [] = zero |
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| mk_sum [t,u] = mk_plus (t, u) |
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| mk_sum (t :: ts) = mk_plus (t, mk_sum ts); |
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|
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(*this version ALWAYS includes a trailing zero*) |
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fun long_mk_sum [] = zero |
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| long_mk_sum (t :: ts) = mk_plus (t, mk_sum ts); |
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|
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val dest_plus = HOLogic.dest_bin "op +" HOLogic.intT; |
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|
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(*decompose additions AND subtractions as a sum*) |
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fun dest_summing (pos, Const ("op +", _) $ t $ u, ts) = |
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dest_summing (pos, t, dest_summing (pos, u, ts)) |
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| dest_summing (pos, Const ("op -", _) $ t $ u, ts) = |
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dest_summing (pos, t, dest_summing (not pos, u, ts)) |
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| dest_summing (pos, t, ts) = |
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if pos then t::ts else uminus_const$t :: ts; |
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|
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fun dest_sum t = dest_summing (true, t, []); |
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val mk_diff = HOLogic.mk_binop "op -"; |
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129 |
val dest_diff = HOLogic.dest_bin "op -" HOLogic.intT; |
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val one = mk_numeral 1; |
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132 |
val mk_times = HOLogic.mk_binop "op *"; |
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|
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fun mk_prod [] = one |
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| mk_prod [t] = t |
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| mk_prod (t :: ts) = if t = one then mk_prod ts |
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else mk_times (t, mk_prod ts); |
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138 |
|
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val dest_times = HOLogic.dest_bin "op *" HOLogic.intT; |
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140 |
|
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fun dest_prod t = |
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let val (t,u) = dest_times t |
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in dest_prod t @ dest_prod u end |
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handle TERM _ => [t]; |
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145 |
|
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(*DON'T do the obvious simplifications; that would create special cases*) |
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fun mk_coeff (k, ts) = mk_times (mk_numeral k, ts); |
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|
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149 |
(*Express t as a product of (possibly) a numeral with other sorted terms*) |
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fun dest_coeff sign (Const ("uminus", _) $ t) = dest_coeff (~sign) t |
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| dest_coeff sign t = |
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let val ts = sort Term.term_ord (dest_prod t) |
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val (n, ts') = find_first_numeral [] ts |
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handle TERM _ => (1, ts) |
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155 |
in (sign*n, mk_prod ts') end; |
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|
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(*Find first coefficient-term THAT MATCHES u*) |
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fun find_first_coeff past u [] = raise TERM("find_first_coeff", []) |
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| find_first_coeff past u (t::terms) = |
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let val (n,u') = dest_coeff 1 t |
161 |
in if u aconv u' then (n, rev past @ terms) |
|
162 |
else find_first_coeff (t::past) u terms |
|
163 |
end |
|
164 |
handle TERM _ => find_first_coeff (t::past) u terms; |
|
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|
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166 |
|
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167 |
(*Simplify Numeral0+n, n+Numeral0, Numeral1*n, n*Numeral1*) |
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val add_0s = map rename_numerals [zadd_0, zadd_0_right]; |
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val mult_1s = map rename_numerals [zmult_1, zmult_1_right] @ |
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[zmult_minus1, zmult_minus1_right]; |
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|
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(*To perform binary arithmetic. The "left" rewriting handles patterns |
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created by the simprocs, such as 3 * (5 * x). *) |
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val bin_simps = [int_numeral_0_eq_0 RS sym, int_numeral_1_eq_1 RS sym, |
13462 | 175 |
add_number_of_left, mult_number_of_left] @ |
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bin_arith_simps @ bin_rel_simps; |
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|
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(*To evaluate binary negations of coefficients*) |
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val zminus_simps = NCons_simps @ |
13462 | 180 |
[zminus_1_eq_m1, number_of_minus RS sym, |
181 |
bin_minus_1, bin_minus_0, bin_minus_Pls, bin_minus_Min, |
|
182 |
bin_pred_1, bin_pred_0, bin_pred_Pls, bin_pred_Min]; |
|
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183 |
|
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184 |
(*To let us treat subtraction as addition*) |
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val diff_simps = [zdiff_def, zminus_zadd_distrib, zminus_zminus]; |
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186 |
|
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187 |
(*push the unary minus down: - x * y = x * - y *) |
13462 | 188 |
val int_minus_mult_eq_1_to_2 = |
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189 |
[zmult_zminus, zmult_zminus_right RS sym] MRS trans |> standard; |
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190 |
|
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191 |
(*to extract again any uncancelled minuses*) |
13462 | 192 |
val int_minus_from_mult_simps = |
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193 |
[zminus_zminus, zmult_zminus, zmult_zminus_right]; |
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|
194 |
|
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195 |
(*combine unary minus with numeric literals, however nested within a product*) |
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196 |
val int_mult_minus_simps = |
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197 |
[zmult_assoc, zmult_zminus RS sym, int_minus_mult_eq_1_to_2]; |
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|
198 |
|
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199 |
(*Apply the given rewrite (if present) just once*) |
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200 |
fun trans_tac None = all_tac |
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201 |
| trans_tac (Some th) = ALLGOALS (rtac (th RS trans)); |
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|
202 |
|
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|
203 |
fun simplify_meta_eq rules = |
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|
204 |
simplify (HOL_basic_ss addeqcongs[eq_cong2] addsimps rules) |
12975 | 205 |
o mk_meta_eq; |
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|
206 |
|
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|
207 |
structure CancelNumeralsCommon = |
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|
208 |
struct |
13462 | 209 |
val mk_sum = mk_sum |
210 |
val dest_sum = dest_sum |
|
211 |
val mk_coeff = mk_coeff |
|
212 |
val dest_coeff = dest_coeff 1 |
|
213 |
val find_first_coeff = find_first_coeff [] |
|
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|
214 |
val trans_tac = trans_tac |
13462 | 215 |
val norm_tac = |
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216 |
ALLGOALS (simp_tac (HOL_ss addsimps numeral_syms@add_0s@mult_1s@ |
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|
217 |
diff_simps@zminus_simps@zadd_ac)) |
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|
218 |
THEN ALLGOALS (simp_tac (HOL_ss addsimps bin_simps@int_mult_minus_simps)) |
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219 |
THEN ALLGOALS (simp_tac (HOL_ss addsimps int_minus_from_mult_simps@ |
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|
220 |
zadd_ac@zmult_ac)) |
13462 | 221 |
val numeral_simp_tac = ALLGOALS (simp_tac (HOL_ss addsimps add_0s@bin_simps)) |
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|
222 |
val simplify_meta_eq = simplify_meta_eq (add_0s@mult_1s) |
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|
223 |
end; |
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|
224 |
|
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|
225 |
|
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|
226 |
structure EqCancelNumerals = CancelNumeralsFun |
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|
227 |
(open CancelNumeralsCommon |
13485
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|
228 |
val prove_conv = Bin_Simprocs.prove_conv |
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|
229 |
val mk_bal = HOLogic.mk_eq |
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|
230 |
val dest_bal = HOLogic.dest_bin "op =" HOLogic.intT |
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|
231 |
val bal_add1 = eq_add_iff1 RS trans |
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|
232 |
val bal_add2 = eq_add_iff2 RS trans |
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|
233 |
); |
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|
234 |
|
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|
235 |
structure LessCancelNumerals = CancelNumeralsFun |
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|
236 |
(open CancelNumeralsCommon |
13485
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|
237 |
val prove_conv = Bin_Simprocs.prove_conv |
9436
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|
238 |
val mk_bal = HOLogic.mk_binrel "op <" |
62bb04ab4b01
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parents:
diff
changeset
|
239 |
val dest_bal = HOLogic.dest_bin "op <" HOLogic.intT |
62bb04ab4b01
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parents:
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|
240 |
val bal_add1 = less_add_iff1 RS trans |
62bb04ab4b01
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parents:
diff
changeset
|
241 |
val bal_add2 = less_add_iff2 RS trans |
62bb04ab4b01
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|
242 |
); |
62bb04ab4b01
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changeset
|
243 |
|
62bb04ab4b01
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parents:
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|
244 |
structure LeCancelNumerals = CancelNumeralsFun |
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|
245 |
(open CancelNumeralsCommon |
13485
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|
246 |
val prove_conv = Bin_Simprocs.prove_conv |
9436
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parents:
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|
247 |
val mk_bal = HOLogic.mk_binrel "op <=" |
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parents:
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|
248 |
val dest_bal = HOLogic.dest_bin "op <=" HOLogic.intT |
62bb04ab4b01
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|
249 |
val bal_add1 = le_add_iff1 RS trans |
62bb04ab4b01
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parents:
diff
changeset
|
250 |
val bal_add2 = le_add_iff2 RS trans |
62bb04ab4b01
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parents:
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|
251 |
); |
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|
252 |
|
13462 | 253 |
val cancel_numerals = |
11868
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paulson
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diff
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|
254 |
map Bin_Simprocs.prep_simproc |
9436
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diff
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|
255 |
[("inteq_cancel_numerals", |
13462 | 256 |
["(l::int) + m = n", "(l::int) = m + n", |
257 |
"(l::int) - m = n", "(l::int) = m - n", |
|
258 |
"(l::int) * m = n", "(l::int) = m * n"], |
|
9436
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|
259 |
EqCancelNumerals.proc), |
13462 | 260 |
("intless_cancel_numerals", |
261 |
["(l::int) + m < n", "(l::int) < m + n", |
|
262 |
"(l::int) - m < n", "(l::int) < m - n", |
|
263 |
"(l::int) * m < n", "(l::int) < m * n"], |
|
9436
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parents:
diff
changeset
|
264 |
LessCancelNumerals.proc), |
13462 | 265 |
("intle_cancel_numerals", |
266 |
["(l::int) + m <= n", "(l::int) <= m + n", |
|
267 |
"(l::int) - m <= n", "(l::int) <= m - n", |
|
268 |
"(l::int) * m <= n", "(l::int) <= m * n"], |
|
9436
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|
269 |
LeCancelNumerals.proc)]; |
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|
270 |
|
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|
271 |
|
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|
272 |
structure CombineNumeralsData = |
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diff
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|
273 |
struct |
13462 | 274 |
val add = op + : int*int -> int |
275 |
val mk_sum = long_mk_sum (*to work for e.g. 2*x + 3*x *) |
|
276 |
val dest_sum = dest_sum |
|
277 |
val mk_coeff = mk_coeff |
|
278 |
val dest_coeff = dest_coeff 1 |
|
279 |
val left_distrib = left_zadd_zmult_distrib RS trans |
|
13485
acf39e924091
tuned prove_conv (error reporting done within meta_simplifier.ML);
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parents:
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diff
changeset
|
280 |
val prove_conv = Bin_Simprocs.prove_conv_nohyps |
9436
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diff
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|
281 |
val trans_tac = trans_tac |
13462 | 282 |
val norm_tac = |
11868
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paulson
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diff
changeset
|
283 |
ALLGOALS (simp_tac (HOL_ss addsimps numeral_syms@add_0s@mult_1s@ |
56db9f3a6b3e
Numerals now work for the integers: the binary numerals for 0 and 1 rewrite
paulson
parents:
11713
diff
changeset
|
284 |
diff_simps@zminus_simps@zadd_ac)) |
10713
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|
285 |
THEN ALLGOALS (simp_tac (HOL_ss addsimps bin_simps@int_mult_minus_simps)) |
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|
286 |
THEN ALLGOALS (simp_tac (HOL_ss addsimps int_minus_from_mult_simps@ |
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|
287 |
zadd_ac@zmult_ac)) |
13462 | 288 |
val numeral_simp_tac = ALLGOALS |
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|
289 |
(simp_tac (HOL_ss addsimps add_0s@bin_simps)) |
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|
290 |
val simplify_meta_eq = simplify_meta_eq (add_0s@mult_1s) |
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|
291 |
end; |
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|
292 |
|
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|
293 |
structure CombineNumerals = CombineNumeralsFun(CombineNumeralsData); |
13462 | 294 |
|
295 |
val combine_numerals = |
|
296 |
Bin_Simprocs.prep_simproc |
|
297 |
("int_combine_numerals", ["(i::int) + j", "(i::int) - j"], CombineNumerals.proc); |
|
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|
298 |
|
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|
299 |
end; |
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|
300 |
|
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|
301 |
Addsimprocs Int_Numeral_Simprocs.cancel_numerals; |
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changeset
|
302 |
Addsimprocs [Int_Numeral_Simprocs.combine_numerals]; |
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changeset
|
303 |
|
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parents:
diff
changeset
|
304 |
(*The Abel_Cancel simprocs are now obsolete*) |
62bb04ab4b01
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parents:
diff
changeset
|
305 |
Delsimprocs [Int_Cancel.sum_conv, Int_Cancel.rel_conv]; |
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parents:
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changeset
|
306 |
|
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parents:
diff
changeset
|
307 |
(*examples: |
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parents:
diff
changeset
|
308 |
print_depth 22; |
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parents:
diff
changeset
|
309 |
set timing; |
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parents:
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|
310 |
set trace_simp; |
13462 | 311 |
fun test s = (Goal s, by (Simp_tac 1)); |
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|
312 |
|
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* sane numerals (stage 2): plain "num" syntax (removed "#");
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changeset
|
313 |
test "l + 2 + 2 + 2 + (l + 2) + (oo + 2) = (uu::int)"; |
9436
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|
314 |
|
11704
3c50a2cd6f00
* sane numerals (stage 2): plain "num" syntax (removed "#");
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parents:
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changeset
|
315 |
test "2*u = (u::int)"; |
3c50a2cd6f00
* sane numerals (stage 2): plain "num" syntax (removed "#");
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parents:
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diff
changeset
|
316 |
test "(i + j + 12 + (k::int)) - 15 = y"; |
3c50a2cd6f00
* sane numerals (stage 2): plain "num" syntax (removed "#");
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parents:
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diff
changeset
|
317 |
test "(i + j + 12 + (k::int)) - 5 = y"; |
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changeset
|
318 |
|
62bb04ab4b01
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parents:
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|
319 |
test "y - b < (b::int)"; |
11704
3c50a2cd6f00
* sane numerals (stage 2): plain "num" syntax (removed "#");
wenzelm
parents:
11701
diff
changeset
|
320 |
test "y - (3*b + c) < (b::int) - 2*c"; |
9436
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changeset
|
321 |
|
11704
3c50a2cd6f00
* sane numerals (stage 2): plain "num" syntax (removed "#");
wenzelm
parents:
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diff
changeset
|
322 |
test "(2*x - (u*v) + y) - v*3*u = (w::int)"; |
3c50a2cd6f00
* sane numerals (stage 2): plain "num" syntax (removed "#");
wenzelm
parents:
11701
diff
changeset
|
323 |
test "(2*x*u*v + (u*v)*4 + y) - v*u*4 = (w::int)"; |
3c50a2cd6f00
* sane numerals (stage 2): plain "num" syntax (removed "#");
wenzelm
parents:
11701
diff
changeset
|
324 |
test "(2*x*u*v + (u*v)*4 + y) - v*u = (w::int)"; |
3c50a2cd6f00
* sane numerals (stage 2): plain "num" syntax (removed "#");
wenzelm
parents:
11701
diff
changeset
|
325 |
test "u*v - (x*u*v + (u*v)*4 + y) = (w::int)"; |
9436
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parents:
diff
changeset
|
326 |
|
11704
3c50a2cd6f00
* sane numerals (stage 2): plain "num" syntax (removed "#");
wenzelm
parents:
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diff
changeset
|
327 |
test "(i + j + 12 + (k::int)) = u + 15 + y"; |
3c50a2cd6f00
* sane numerals (stage 2): plain "num" syntax (removed "#");
wenzelm
parents:
11701
diff
changeset
|
328 |
test "(i + j*2 + 12 + (k::int)) = j + 5 + y"; |
9436
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parents:
diff
changeset
|
329 |
|
11704
3c50a2cd6f00
* sane numerals (stage 2): plain "num" syntax (removed "#");
wenzelm
parents:
11701
diff
changeset
|
330 |
test "2*y + 3*z + 6*w + 2*y + 3*z + 2*u = 2*y' + 3*z' + 6*w' + 2*y' + 3*z' + u + (vv::int)"; |
9436
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
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parents:
diff
changeset
|
331 |
|
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset
|
332 |
test "a + -(b+c) + b = (d::int)"; |
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset
|
333 |
test "a + -(b+c) - b = (d::int)"; |
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset
|
334 |
|
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset
|
335 |
(*negative numerals*) |
11704
3c50a2cd6f00
* sane numerals (stage 2): plain "num" syntax (removed "#");
wenzelm
parents:
11701
diff
changeset
|
336 |
test "(i + j + -2 + (k::int)) - (u + 5 + y) = zz"; |
3c50a2cd6f00
* sane numerals (stage 2): plain "num" syntax (removed "#");
wenzelm
parents:
11701
diff
changeset
|
337 |
test "(i + j + -3 + (k::int)) < u + 5 + y"; |
3c50a2cd6f00
* sane numerals (stage 2): plain "num" syntax (removed "#");
wenzelm
parents:
11701
diff
changeset
|
338 |
test "(i + j + 3 + (k::int)) < u + -6 + y"; |
3c50a2cd6f00
* sane numerals (stage 2): plain "num" syntax (removed "#");
wenzelm
parents:
11701
diff
changeset
|
339 |
test "(i + j + -12 + (k::int)) - 15 = y"; |
3c50a2cd6f00
* sane numerals (stage 2): plain "num" syntax (removed "#");
wenzelm
parents:
11701
diff
changeset
|
340 |
test "(i + j + 12 + (k::int)) - -15 = y"; |
3c50a2cd6f00
* sane numerals (stage 2): plain "num" syntax (removed "#");
wenzelm
parents:
11701
diff
changeset
|
341 |
test "(i + j + -12 + (k::int)) - -15 = y"; |
9436
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
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parents:
diff
changeset
|
342 |
*) |
62bb04ab4b01
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wenzelm
parents:
diff
changeset
|
343 |
|
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset
|
344 |
|
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset
|
345 |
(** Constant folding for integer plus and times **) |
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset
|
346 |
|
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset
|
347 |
(*We do not need |
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset
|
348 |
structure Nat_Plus_Assoc = Assoc_Fold (Nat_Plus_Assoc_Data); |
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset
|
349 |
structure Int_Plus_Assoc = Assoc_Fold (Int_Plus_Assoc_Data); |
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset
|
350 |
because combine_numerals does the same thing*) |
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset
|
351 |
|
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset
|
352 |
structure Int_Times_Assoc_Data : ASSOC_FOLD_DATA = |
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset
|
353 |
struct |
13462 | 354 |
val ss = HOL_ss |
355 |
val eq_reflection = eq_reflection |
|
9436
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset
|
356 |
val sg_ref = Sign.self_ref (Theory.sign_of (the_context ())) |
13462 | 357 |
val T = HOLogic.intT |
9436
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset
|
358 |
val plus = Const ("op *", [HOLogic.intT,HOLogic.intT] ---> HOLogic.intT); |
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset
|
359 |
val add_ac = zmult_ac |
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset
|
360 |
end; |
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset
|
361 |
|
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset
|
362 |
structure Int_Times_Assoc = Assoc_Fold (Int_Times_Assoc_Data); |
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset
|
363 |
|
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset
|
364 |
Addsimprocs [Int_Times_Assoc.conv]; |
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset
|
365 |
|
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset
|
366 |
|
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset
|
367 |
(** The same for the naturals **) |
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset
|
368 |
|
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset
|
369 |
structure Nat_Times_Assoc_Data : ASSOC_FOLD_DATA = |
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset
|
370 |
struct |
13462 | 371 |
val ss = HOL_ss |
372 |
val eq_reflection = eq_reflection |
|
9436
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset
|
373 |
val sg_ref = Sign.self_ref (Theory.sign_of (the_context ())) |
13462 | 374 |
val T = HOLogic.natT |
9436
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset
|
375 |
val plus = Const ("op *", [HOLogic.natT,HOLogic.natT] ---> HOLogic.natT); |
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset
|
376 |
val add_ac = mult_ac |
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset
|
377 |
end; |
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset
|
378 |
|
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset
|
379 |
structure Nat_Times_Assoc = Assoc_Fold (Nat_Times_Assoc_Data); |
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset
|
380 |
|
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset
|
381 |
Addsimprocs [Nat_Times_Assoc.conv]; |
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset
|
382 |
|
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset
|
383 |
|
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset
|
384 |
(*** decision procedure for linear arithmetic ***) |
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset
|
385 |
|
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset
|
386 |
(*---------------------------------------------------------------------------*) |
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset
|
387 |
(* Linear arithmetic *) |
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset
|
388 |
(*---------------------------------------------------------------------------*) |
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset
|
389 |
|
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset
|
390 |
(* |
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset
|
391 |
Instantiation of the generic linear arithmetic package for int. |
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset
|
392 |
*) |
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset
|
393 |
|
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset
|
394 |
(* Update parameters of arithmetic prover *) |
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset
|
395 |
local |
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset
|
396 |
|
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset
|
397 |
(* reduce contradictory <= to False *) |
13462 | 398 |
val add_rules = |
399 |
simp_thms @ bin_arith_simps @ bin_rel_simps @ |
|
11868
56db9f3a6b3e
Numerals now work for the integers: the binary numerals for 0 and 1 rewrite
paulson
parents:
11713
diff
changeset
|
400 |
[int_numeral_0_eq_0, int_numeral_1_eq_1, |
12018
ec054019c910
Numerals and simprocs for types real and hypreal. The abstract
paulson
parents:
11868
diff
changeset
|
401 |
zminus_0, zadd_0, zadd_0_right, zdiff_def, |
13462 | 402 |
zadd_zminus_inverse, zadd_zminus_inverse2, |
403 |
zmult_0, zmult_0_right, |
|
12482
d2848ccc9757
new rewrite rules for use by arith_tac to take care of uminus.
nipkow
parents:
12018
diff
changeset
|
404 |
zmult_1, zmult_1_right, |
d2848ccc9757
new rewrite rules for use by arith_tac to take care of uminus.
nipkow
parents:
12018
diff
changeset
|
405 |
zmult_zminus, zmult_zminus_right, |
11868
56db9f3a6b3e
Numerals now work for the integers: the binary numerals for 0 and 1 rewrite
paulson
parents:
11713
diff
changeset
|
406 |
zminus_zadd_distrib, zminus_zminus, zmult_assoc, |
13499 | 407 |
int_0, int_1, int_Suc, zadd_int RS sym, zmult_int RS sym]; |
9436
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset
|
408 |
|
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset
|
409 |
val simprocs = [Int_Times_Assoc.conv, Int_Numeral_Simprocs.combine_numerals]@ |
13462 | 410 |
Int_Numeral_Simprocs.cancel_numerals @ |
11868
56db9f3a6b3e
Numerals now work for the integers: the binary numerals for 0 and 1 rewrite
paulson
parents:
11713
diff
changeset
|
411 |
Bin_Simprocs.eval_numerals; |
9436
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset
|
412 |
|
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset
|
413 |
val add_mono_thms_int = |
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset
|
414 |
map (fn s => prove_goal (the_context ()) s |
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset
|
415 |
(fn prems => [cut_facts_tac prems 1, |
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset
|
416 |
asm_simp_tac (simpset() addsimps [zadd_zle_mono]) 1])) |
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset
|
417 |
["(i <= j) & (k <= l) ==> i + k <= j + (l::int)", |
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset
|
418 |
"(i = j) & (k <= l) ==> i + k <= j + (l::int)", |
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset
|
419 |
"(i <= j) & (k = l) ==> i + k <= j + (l::int)", |
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset
|
420 |
"(i = j) & (k = l) ==> i + k = j + (l::int)" |
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset
|
421 |
]; |
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset
|
422 |
|
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset
|
423 |
in |
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset
|
424 |
|
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset
|
425 |
val int_arith_setup = |
10693 | 426 |
[Fast_Arith.map_data (fn {add_mono_thms, mult_mono_thms, inj_thms, lessD, simpset} => |
9436
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset
|
427 |
{add_mono_thms = add_mono_thms @ add_mono_thms_int, |
10693 | 428 |
mult_mono_thms = mult_mono_thms, |
10574
8f98f0301d67
Linear arithmetic now copes with mixed nat/int formulae.
nipkow
parents:
9571
diff
changeset
|
429 |
inj_thms = [zle_int RS iffD2,int_int_eq RS iffD2] @ inj_thms, |
9436
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset
|
430 |
lessD = lessD @ [add1_zle_eq RS iffD2], |
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset
|
431 |
simpset = simpset addsimps add_rules |
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset
|
432 |
addsimprocs simprocs |
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset
|
433 |
addcongs [if_weak_cong]}), |
10834 | 434 |
arith_inj_const ("IntDef.int", HOLogic.natT --> HOLogic.intT), |
9436
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset
|
435 |
arith_discrete ("IntDef.int", true)]; |
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset
|
436 |
|
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset
|
437 |
end; |
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset
|
438 |
|
13462 | 439 |
val fast_int_arith_simproc = |
440 |
Simplifier.simproc (Theory.sign_of (the_context())) |
|
441 |
"fast_int_arith" ["(m::int) < n","(m::int) <= n", "(m::int) = n"] Fast_Arith.lin_arith_prover; |
|
9436
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset
|
442 |
|
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset
|
443 |
Addsimprocs [fast_int_arith_simproc] |
13462 | 444 |
|
9436
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset
|
445 |
|
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset
|
446 |
(* Some test data |
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset
|
447 |
Goal "!!a::int. [| a <= b; c <= d; x+y<z |] ==> a+c <= b+d"; |
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset
|
448 |
by (fast_arith_tac 1); |
11704
3c50a2cd6f00
* sane numerals (stage 2): plain "num" syntax (removed "#");
wenzelm
parents:
11701
diff
changeset
|
449 |
Goal "!!a::int. [| a < b; c < d |] ==> a-d+ 2 <= b+(-c)"; |
9436
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset
|
450 |
by (fast_arith_tac 1); |
11868
56db9f3a6b3e
Numerals now work for the integers: the binary numerals for 0 and 1 rewrite
paulson
parents:
11713
diff
changeset
|
451 |
Goal "!!a::int. [| a < b; c < d |] ==> a+c+ 1 < b+d"; |
9436
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset
|
452 |
by (fast_arith_tac 1); |
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset
|
453 |
Goal "!!a::int. [| a <= b; b+b <= c |] ==> a+a <= c"; |
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset
|
454 |
by (fast_arith_tac 1); |
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset
|
455 |
Goal "!!a::int. [| a+b <= i+j; a<=b; i<=j |] \ |
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset
|
456 |
\ ==> a+a <= j+j"; |
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset
|
457 |
by (fast_arith_tac 1); |
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset
|
458 |
Goal "!!a::int. [| a+b < i+j; a<b; i<j |] \ |
11704
3c50a2cd6f00
* sane numerals (stage 2): plain "num" syntax (removed "#");
wenzelm
parents:
11701
diff
changeset
|
459 |
\ ==> a+a - - -1 < j+j - 3"; |
9436
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset
|
460 |
by (fast_arith_tac 1); |
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset
|
461 |
Goal "!!a::int. a+b+c <= i+j+k & a<=b & b<=c & i<=j & j<=k --> a+a+a <= k+k+k"; |
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset
|
462 |
by (arith_tac 1); |
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset
|
463 |
Goal "!!a::int. [| a+b+c+d <= i+j+k+l; a<=b; b<=c; c<=d; i<=j; j<=k; k<=l |] \ |
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset
|
464 |
\ ==> a <= l"; |
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset
|
465 |
by (fast_arith_tac 1); |
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset
|
466 |
Goal "!!a::int. [| a+b+c+d <= i+j+k+l; a<=b; b<=c; c<=d; i<=j; j<=k; k<=l |] \ |
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset
|
467 |
\ ==> a+a+a+a <= l+l+l+l"; |
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset
|
468 |
by (fast_arith_tac 1); |
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset
|
469 |
Goal "!!a::int. [| a+b+c+d <= i+j+k+l; a<=b; b<=c; c<=d; i<=j; j<=k; k<=l |] \ |
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset
|
470 |
\ ==> a+a+a+a+a <= l+l+l+l+i"; |
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset
|
471 |
by (fast_arith_tac 1); |
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset
|
472 |
Goal "!!a::int. [| a+b+c+d <= i+j+k+l; a<=b; b<=c; c<=d; i<=j; j<=k; k<=l |] \ |
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset
|
473 |
\ ==> a+a+a+a+a+a <= l+l+l+l+i+l"; |
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset
|
474 |
by (fast_arith_tac 1); |
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset
|
475 |
Goal "!!a::int. [| a+b+c+d <= i+j+k+l; a<=b; b<=c; c<=d; i<=j; j<=k; k<=l |] \ |
11704
3c50a2cd6f00
* sane numerals (stage 2): plain "num" syntax (removed "#");
wenzelm
parents:
11701
diff
changeset
|
476 |
\ ==> 6*a <= 5*l+i"; |
9436
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset
|
477 |
by (fast_arith_tac 1); |
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset
|
478 |
*) |