author | wenzelm |
Sun, 26 Nov 2006 23:43:53 +0100 | |
changeset 21539 | c5cf9243ad62 |
parent 20342 | 4392003fcbfa |
child 23071 | bf058e6405f8 |
permissions | -rw-r--r-- |
9570 | 1 |
(* Title: ZF/Integ/int_arith.ML |
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ID: $Id$ |
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Author: Larry Paulson |
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Copyright 2000 University of Cambridge |
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Simprocs for linear arithmetic. |
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*) |
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(** To simplify inequalities involving integer negation and literals, |
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such as -x = #3 |
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**) |
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Addsimps [inst "y" "integ_of(?w)" zminus_equation, |
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inst "x" "integ_of(?w)" equation_zminus]; |
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AddIffs [inst "y" "integ_of(?w)" zminus_zless, |
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inst "x" "integ_of(?w)" zless_zminus]; |
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AddIffs [inst "y" "integ_of(?w)" zminus_zle, |
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inst "x" "integ_of(?w)" zle_zminus]; |
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Addsimps [inst "s" "integ_of(?w)" (thm "Let_def")]; |
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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 $< #0)"; |
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by (simp_tac (simpset() addsimps zcompare_rls) 1); |
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qed "zless_iff_zdiff_zless_0"; |
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Goal "[| x: int; y: int |] ==> (x = y) <-> (x$-y = #0)"; |
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by (asm_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 $<= #0)"; |
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by (asm_simp_tac (simpset() addsimps zcompare_rls) 1); |
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qed "zle_iff_zdiff_zle_0"; |
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(** For combine_numerals **) |
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Goal "i$*u $+ (j$*u $+ k) = (i$+j)$*u $+ k"; |
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by (simp_tac (simpset() addsimps [zadd_zmult_distrib]@zadd_ac) 1); |
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qed "left_zadd_zmult_distrib"; |
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(** For cancel_numerals **) |
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val rel_iff_rel_0_rls = map (inst "y" "?u$+?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$*u $+ m = j$*u $+ n) <-> ((i$-j)$*u $+ m = intify(n))"; |
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by (simp_tac (simpset() addsimps [zdiff_def, zadd_zmult_distrib]) 1); |
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by (simp_tac (simpset() addsimps zcompare_rls) 1); |
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by (simp_tac (simpset() addsimps zadd_ac) 1); |
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qed "eq_add_iff1"; |
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Goal "(i$*u $+ m = j$*u $+ n) <-> (intify(m) = (j$-i)$*u $+ n)"; |
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by (simp_tac (simpset() addsimps [zdiff_def, zadd_zmult_distrib]) 1); |
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by (simp_tac (simpset() addsimps zcompare_rls) 1); |
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by (simp_tac (simpset() addsimps zadd_ac) 1); |
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qed "eq_add_iff2"; |
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Goal "(i$*u $+ m $< j$*u $+ n) <-> ((i$-j)$*u $+ m $< n)"; |
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by (asm_simp_tac (simpset() addsimps [zdiff_def, zadd_zmult_distrib]@ |
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zadd_ac@rel_iff_rel_0_rls) 1); |
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qed "less_add_iff1"; |
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Goal "(i$*u $+ m $< j$*u $+ n) <-> (m $< (j$-i)$*u $+ n)"; |
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by (asm_simp_tac (simpset() addsimps [zdiff_def, zadd_zmult_distrib]@ |
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zadd_ac@rel_iff_rel_0_rls) 1); |
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qed "less_add_iff2"; |
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Goal "(i$*u $+ m $<= j$*u $+ n) <-> ((i$-j)$*u $+ m $<= n)"; |
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by (simp_tac (simpset() addsimps [zdiff_def, zadd_zmult_distrib]) 1); |
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by (simp_tac (simpset() addsimps zcompare_rls) 1); |
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by (simp_tac (simpset() addsimps zadd_ac) 1); |
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qed "le_add_iff1"; |
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||
12089
34e7693271a9
Sidi Ehmety's port of the fold_set operator and multisets to ZF.
paulson
parents:
11321
diff
changeset
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Goal "(i$*u $+ m $<= j$*u $+ n) <-> (m $<= (j$-i)$*u $+ n)"; |
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by (simp_tac (simpset() addsimps [zdiff_def, zadd_zmult_distrib]) 1); |
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by (simp_tac (simpset() addsimps zcompare_rls) 1); |
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by (simp_tac (simpset() addsimps zadd_ac) 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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(*Utilities*) |
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val integ_of_const = Const ("Bin.integ_of", iT --> iT); |
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fun mk_numeral n = integ_of_const $ NumeralSyntax.mk_bin n; |
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(*Decodes a binary INTEGER*) |
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fun dest_numeral (Const("Bin.integ_of", _) $ w) = |
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(NumeralSyntax.dest_bin w |
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handle Match => 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 = FOLogic.mk_binop "Int.zadd"; |
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val iT = Ind_Syntax.iT; |
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val zminus_const = Const ("Int.zminus", iT --> iT); |
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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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(*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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val dest_plus = FOLogic.dest_bin "Int.zadd" iT; |
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(*decompose additions AND subtractions as a sum*) |
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fun dest_summing (pos, Const ("Int.zadd", _) $ t $ u, ts) = |
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dest_summing (pos, t, dest_summing (pos, u, ts)) |
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| dest_summing (pos, Const ("Int.zdiff", _) $ 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 zminus_const$t :: ts; |
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fun dest_sum t = dest_summing (true, t, []); |
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val mk_diff = FOLogic.mk_binop "Int.zdiff"; |
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val dest_diff = FOLogic.dest_bin "Int.zdiff" iT; |
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val one = mk_numeral 1; |
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val mk_times = FOLogic.mk_binop "Int.zmult"; |
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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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val dest_times = FOLogic.dest_bin "Int.zmult" iT; |
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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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(*DON'T do the obvious simplifications; that would create special cases*) |
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fun mk_coeff (k, t) = mk_times (mk_numeral k, t); |
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(*Express t as a product of (possibly) a numeral with other sorted terms*) |
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fun dest_coeff sign (Const ("Int.zminus", _) $ 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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in (sign*n, mk_prod ts') end; |
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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 |
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in if u aconv u' then (n, rev past @ terms) |
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else find_first_coeff (t::past) u terms |
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end |
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handle TERM _ => find_first_coeff (t::past) u terms; |
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(*Simplify #1*n and n*#1 to n*) |
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val add_0s = [zadd_0_intify, zadd_0_right_intify]; |
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val mult_1s = [zmult_1_intify, zmult_1_right_intify, |
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zmult_minus1, zmult_minus1_right]; |
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val tc_rules = [integ_of_type, intify_in_int, |
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int_of_type, zadd_type, zdiff_type, zmult_type] @ |
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thms "bin.intros"; |
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val intifys = [intify_ident, zadd_intify1, zadd_intify2, |
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zdiff_intify1, zdiff_intify2, zmult_intify1, zmult_intify2, |
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zless_intify1, zless_intify2, zle_intify1, zle_intify2]; |
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(*To perform binary arithmetic*) |
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val bin_simps = [add_integ_of_left] @ bin_arith_simps @ bin_rel_simps; |
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(*To evaluate binary negations of coefficients*) |
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val zminus_simps = NCons_simps @ |
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[integ_of_minus RS sym, |
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bin_minus_1, bin_minus_0, bin_minus_Pls, bin_minus_Min, |
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bin_pred_1, bin_pred_0, bin_pred_Pls, bin_pred_Min]; |
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(*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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(*push the unary minus down: - x * y = x * - y *) |
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val int_minus_mult_eq_1_to_2 = |
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[zmult_zminus, zmult_zminus_right RS sym] MRS trans |> standard; |
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(*to extract again any uncancelled minuses*) |
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val int_minus_from_mult_simps = |
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[zminus_zminus, zmult_zminus, zmult_zminus_right]; |
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(*combine unary minus with numeric literals, however nested within a product*) |
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val int_mult_minus_simps = |
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[zmult_assoc, zmult_zminus RS sym, int_minus_mult_eq_1_to_2]; |
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fun prep_simproc (name, pats, proc) = |
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Simplifier.simproc (the_context ()) name pats proc; |
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structure CancelNumeralsCommon = |
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struct |
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val mk_sum = (fn T:typ => mk_sum) |
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val dest_sum = dest_sum |
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val mk_coeff = mk_coeff |
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val dest_coeff = dest_coeff 1 |
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val find_first_coeff = find_first_coeff [] |
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fun trans_tac _ = ArithData.gen_trans_tac iff_trans |
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val norm_ss1 = ZF_ss addsimps add_0s @ mult_1s @ diff_simps @ zminus_simps @ zadd_ac |
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val norm_ss2 = ZF_ss addsimps bin_simps @ int_mult_minus_simps @ intifys |
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val norm_ss3 = ZF_ss addsimps int_minus_from_mult_simps @ zadd_ac @ zmult_ac @ tc_rules @ intifys |
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fun norm_tac ss = |
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ALLGOALS (asm_simp_tac (Simplifier.inherit_context ss norm_ss1)) |
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THEN ALLGOALS (asm_simp_tac (Simplifier.inherit_context ss norm_ss2)) |
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THEN ALLGOALS (asm_simp_tac (Simplifier.inherit_context ss norm_ss3)) |
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val numeral_simp_ss = ZF_ss addsimps add_0s @ bin_simps @ tc_rules @ intifys |
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fun numeral_simp_tac ss = |
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ALLGOALS (simp_tac (Simplifier.inherit_context ss numeral_simp_ss)) |
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Simplifier.inherit_context instead of Simplifier.inherit_bounds;
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THEN ALLGOALS (SIMPSET' (fn simpset => asm_simp_tac (Simplifier.inherit_context ss simpset))) |
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val simplify_meta_eq = ArithData.simplify_meta_eq (add_0s @ mult_1s) |
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end; |
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structure EqCancelNumerals = CancelNumeralsFun |
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(open CancelNumeralsCommon |
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val prove_conv = ArithData.prove_conv "inteq_cancel_numerals" |
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val mk_bal = FOLogic.mk_eq |
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val dest_bal = FOLogic.dest_eq |
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val bal_add1 = eq_add_iff1 RS iff_trans |
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val bal_add2 = eq_add_iff2 RS iff_trans |
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); |
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structure LessCancelNumerals = CancelNumeralsFun |
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(open CancelNumeralsCommon |
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val prove_conv = ArithData.prove_conv "intless_cancel_numerals" |
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val mk_bal = FOLogic.mk_binrel "Int.zless" |
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val dest_bal = FOLogic.dest_bin "Int.zless" iT |
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val bal_add1 = less_add_iff1 RS iff_trans |
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val bal_add2 = less_add_iff2 RS iff_trans |
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); |
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structure LeCancelNumerals = CancelNumeralsFun |
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(open CancelNumeralsCommon |
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val prove_conv = ArithData.prove_conv "intle_cancel_numerals" |
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val mk_bal = FOLogic.mk_binrel "Int.zle" |
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val dest_bal = FOLogic.dest_bin "Int.zle" iT |
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val bal_add1 = le_add_iff1 RS iff_trans |
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val bal_add2 = le_add_iff2 RS iff_trans |
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); |
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val cancel_numerals = |
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map prep_simproc |
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[("inteq_cancel_numerals", |
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["l $+ m = n", "l = m $+ n", |
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"l $- m = n", "l = m $- n", |
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"l $* m = n", "l = m $* n"], |
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wenzelm
parents:
18328
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K EqCancelNumerals.proc), |
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("intless_cancel_numerals", |
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["l $+ m $< n", "l $< m $+ n", |
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"l $- m $< n", "l $< m $- n", |
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"l $* m $< n", "l $< m $* n"], |
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simprocs: no theory argument -- use simpset context instead;
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parents:
18328
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K LessCancelNumerals.proc), |
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("intle_cancel_numerals", |
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["l $+ m $<= n", "l $<= m $+ n", |
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"l $- m $<= n", "l $<= m $- n", |
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"l $* m $<= n", "l $<= m $* n"], |
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parents:
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K LeCancelNumerals.proc)]; |
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291 |
(*version without the hyps argument*) |
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fun prove_conv_nohyps name tacs sg = ArithData.prove_conv name tacs sg []; |
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structure CombineNumeralsData = |
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struct |
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Use of IntInf.int instead of int in most numeric simprocs; avoids
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val add = IntInf.+ |
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val mk_sum = (fn T:typ => long_mk_sum) (*to work for #2*x $+ #3*x *) |
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val dest_sum = dest_sum |
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val mk_coeff = mk_coeff |
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val dest_coeff = dest_coeff 1 |
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val left_distrib = left_zadd_zmult_distrib RS trans |
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val prove_conv = prove_conv_nohyps "int_combine_numerals" |
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fun trans_tac _ = ArithData.gen_trans_tac trans |
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val norm_ss1 = ZF_ss addsimps add_0s @ mult_1s @ diff_simps @ zminus_simps @ zadd_ac @ intifys |
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val norm_ss2 = ZF_ss addsimps bin_simps @ int_mult_minus_simps @ intifys |
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val norm_ss3 = ZF_ss addsimps int_minus_from_mult_simps @ zadd_ac @ zmult_ac @ tc_rules @ intifys |
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fun norm_tac ss = |
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ALLGOALS (asm_simp_tac (Simplifier.inherit_context ss norm_ss1)) |
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THEN ALLGOALS (asm_simp_tac (Simplifier.inherit_context ss norm_ss2)) |
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THEN ALLGOALS (asm_simp_tac (Simplifier.inherit_context ss norm_ss3)) |
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313 |
val numeral_simp_ss = ZF_ss addsimps add_0s @ bin_simps @ tc_rules @ intifys |
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16973 | 314 |
fun numeral_simp_tac ss = |
18328 | 315 |
ALLGOALS (simp_tac (Simplifier.inherit_context ss numeral_simp_ss)) |
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val simplify_meta_eq = ArithData.simplify_meta_eq (add_0s @ mult_1s) |
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end; |
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structure CombineNumerals = CombineNumeralsFun(CombineNumeralsData); |
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13462 | 320 |
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321 |
val combine_numerals = |
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parents:
18328
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322 |
prep_simproc ("int_combine_numerals", ["i $+ j", "i $- j"], K CombineNumerals.proc); |
9570 | 323 |
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(** Constant folding for integer multiplication **) |
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(*The trick is to regard products as sums, e.g. #3 $* x $* #4 as |
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329 |
the "sum" of #3, x, #4; the literals are then multiplied*) |
|
9648
35d761c7d934
better rules for cancellation of common factors across comparisons
paulson
parents:
9576
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changeset
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330 |
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35d761c7d934
better rules for cancellation of common factors across comparisons
paulson
parents:
9576
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structure CombineNumeralsProdData = |
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struct |
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val add = IntInf.* |
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val mk_sum = (fn T:typ => mk_prod) |
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val dest_sum = dest_prod |
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fun mk_coeff(k,t) = if t=one then mk_numeral k |
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else raise TERM("mk_coeff", []) |
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fun dest_coeff t = (dest_numeral t, one) (*We ONLY want pure numerals.*) |
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val left_distrib = zmult_assoc RS sym RS trans |
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val prove_conv = prove_conv_nohyps "int_combine_numerals_prod" |
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fun trans_tac _ = ArithData.gen_trans_tac trans |
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val norm_ss1 = ZF_ss addsimps mult_1s @ diff_simps @ zminus_simps |
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val norm_ss2 = ZF_ss addsimps [zmult_zminus_right RS sym] @ |
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bin_simps @ zmult_ac @ tc_rules @ intifys |
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fun norm_tac ss = |
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ALLGOALS (asm_simp_tac (Simplifier.inherit_context ss norm_ss1)) |
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THEN ALLGOALS (asm_simp_tac (Simplifier.inherit_context ss norm_ss2)) |
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val numeral_simp_ss = ZF_ss addsimps bin_simps @ tc_rules @ intifys |
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fun numeral_simp_tac ss = |
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ALLGOALS (simp_tac (Simplifier.inherit_context ss numeral_simp_ss)) |
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val simplify_meta_eq = ArithData.simplify_meta_eq (mult_1s); |
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end; |
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structure CombineNumeralsProd = CombineNumeralsFun(CombineNumeralsProdData); |
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val combine_numerals_prod = |
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prep_simproc ("int_combine_numerals_prod", ["i $* j"], K CombineNumeralsProd.proc); |
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end; |
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Addsimprocs Int_Numeral_Simprocs.cancel_numerals; |
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Addsimprocs [Int_Numeral_Simprocs.combine_numerals, |
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Int_Numeral_Simprocs.combine_numerals_prod]; |
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(*examples:*) |
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(* |
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print_depth 22; |
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set timing; |
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set trace_simp; |
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fun test s = (Goal s; by (Asm_simp_tac 1)); |
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val sg = #sign (rep_thm (topthm())); |
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val t = FOLogic.dest_Trueprop (Logic.strip_assums_concl(getgoal 1)); |
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val (t,_) = FOLogic.dest_eq t; |
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(*combine_numerals_prod (products of separate literals) *) |
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test "#5 $* x $* #3 = y"; |
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test "y2 $+ ?x42 = y $+ y2"; |
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test "oo : int ==> l $+ (l $+ #2) $+ oo = oo"; |
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test "#9$*x $+ y = x$*#23 $+ z"; |
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test "y $+ x = x $+ z"; |
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test "x : int ==> x $+ y $+ z = x $+ z"; |
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test "x : int ==> y $+ (z $+ x) = z $+ x"; |
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test "z : int ==> x $+ y $+ z = (z $+ y) $+ (x $+ w)"; |
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test "z : int ==> x$*y $+ z = (z $+ y) $+ (y$*x $+ w)"; |
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test "#-3 $* x $+ y $<= x $* #2 $+ z"; |
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test "y $+ x $<= x $+ z"; |
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test "x $+ y $+ z $<= x $+ z"; |
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test "y $+ (z $+ x) $< z $+ x"; |
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test "x $+ y $+ z $< (z $+ y) $+ (x $+ w)"; |
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test "x$*y $+ z $< (z $+ y) $+ (y$*x $+ w)"; |
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test "l $+ #2 $+ #2 $+ #2 $+ (l $+ #2) $+ (oo $+ #2) = uu"; |
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test "u : int ==> #2 $* u = u"; |
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test "(i $+ j $+ #12 $+ k) $- #15 = y"; |
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test "(i $+ j $+ #12 $+ k) $- #5 = y"; |
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test "y $- b $< b"; |
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test "y $- (#3 $* b $+ c) $< b $- #2 $* c"; |
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test "(#2 $* x $- (u $* v) $+ y) $- v $* #3 $* u = w"; |
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test "(#2 $* x $* u $* v $+ (u $* v) $* #4 $+ y) $- v $* u $* #4 = w"; |
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test "(#2 $* x $* u $* v $+ (u $* v) $* #4 $+ y) $- v $* u = w"; |
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test "u $* v $- (x $* u $* v $+ (u $* v) $* #4 $+ y) = w"; |
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test "(i $+ j $+ #12 $+ k) = u $+ #15 $+ y"; |
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test "(i $+ j $* #2 $+ #12 $+ k) = j $+ #5 $+ y"; |
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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"; |
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test "a $+ $-(b$+c) $+ b = d"; |
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test "a $+ $-(b$+c) $- b = d"; |
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(*negative numerals*) |
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test "(i $+ j $+ #-2 $+ k) $- (u $+ #5 $+ y) = zz"; |
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test "(i $+ j $+ #-3 $+ k) $< u $+ #5 $+ y"; |
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test "(i $+ j $+ #3 $+ k) $< u $+ #-6 $+ y"; |
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test "(i $+ j $+ #-12 $+ k) $- #15 = y"; |
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test "(i $+ j $+ #12 $+ k) $- #-15 = y"; |
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test "(i $+ j $+ #-12 $+ k) $- #-15 = y"; |
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(*Multiplying separated numerals*) |
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Goal "#6 $* ($# x $* #2) = uu"; |
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Goal "#4 $* ($# x $* $# x) $* (#2 $* $# x) = uu"; |
9570 | 436 |
*) |
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