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theory Arith_Tools(* Title: HOL/Arith_Tools.thy ID: $Id: Arith_Tools.thy,v 1.10 2007/10/31 11:19:33 chaieb Exp $ Author: Lawrence C Paulson, Cambridge University Computer Laboratory Author: Amine Chaieb, TU Muenchen *) header {* Setup of arithmetic tools *} theory Arith_Tools imports Groebner_Basis Dense_Linear_Order uses "~~/src/Provers/Arith/cancel_numeral_factor.ML" "~~/src/Provers/Arith/extract_common_term.ML" "int_factor_simprocs.ML" "nat_simprocs.ML" begin subsection {* Simprocs for the Naturals *} declaration {* K nat_simprocs_setup *} subsubsection{*For simplifying @{term "Suc m - K"} and @{term "K - Suc m"}*} text{*Where K above is a literal*} lemma Suc_diff_eq_diff_pred: "Numeral0 < n ==> Suc m - n = m - (n - Numeral1)" by (simp add: numeral_0_eq_0 numeral_1_eq_1 split add: nat_diff_split) text {*Now just instantiating @{text n} to @{text "number_of v"} does the right simplification, but with some redundant inequality tests.*} lemma neg_number_of_pred_iff_0: "neg (number_of (Numeral.pred v)::int) = (number_of v = (0::nat))" apply (subgoal_tac "neg (number_of (Numeral.pred v)) = (number_of v < Suc 0) ") apply (simp only: less_Suc_eq_le le_0_eq) apply (subst less_number_of_Suc, simp) done text{*No longer required as a simprule because of the @{text inverse_fold} simproc*} lemma Suc_diff_number_of: "neg (number_of (uminus v)::int) ==> Suc m - (number_of v) = m - (number_of (Numeral.pred v))" apply (subst Suc_diff_eq_diff_pred) apply simp apply (simp del: nat_numeral_1_eq_1) apply (auto simp only: diff_nat_number_of less_0_number_of [symmetric] neg_number_of_pred_iff_0) done lemma diff_Suc_eq_diff_pred: "m - Suc n = (m - 1) - n" by (simp add: numerals split add: nat_diff_split) subsubsection{*For @{term nat_case} and @{term nat_rec}*} lemma nat_case_number_of [simp]: "nat_case a f (number_of v) = (let pv = number_of (Numeral.pred v) in if neg pv then a else f (nat pv))" by (simp split add: nat.split add: Let_def neg_number_of_pred_iff_0) lemma nat_case_add_eq_if [simp]: "nat_case a f ((number_of v) + n) = (let pv = number_of (Numeral.pred v) in if neg pv then nat_case a f n else f (nat pv + n))" apply (subst add_eq_if) apply (simp split add: nat.split del: nat_numeral_1_eq_1 add: numeral_1_eq_Suc_0 [symmetric] Let_def neg_imp_number_of_eq_0 neg_number_of_pred_iff_0) done lemma nat_rec_number_of [simp]: "nat_rec a f (number_of v) = (let pv = number_of (Numeral.pred v) in if neg pv then a else f (nat pv) (nat_rec a f (nat pv)))" apply (case_tac " (number_of v) ::nat") apply (simp_all (no_asm_simp) add: Let_def neg_number_of_pred_iff_0) apply (simp split add: split_if_asm) done lemma nat_rec_add_eq_if [simp]: "nat_rec a f (number_of v + n) = (let pv = number_of (Numeral.pred v) in if neg pv then nat_rec a f n else f (nat pv + n) (nat_rec a f (nat pv + n)))" apply (subst add_eq_if) apply (simp split add: nat.split del: nat_numeral_1_eq_1 add: numeral_1_eq_Suc_0 [symmetric] Let_def neg_imp_number_of_eq_0 neg_number_of_pred_iff_0) done subsubsection{*Various Other Lemmas*} text {*Evens and Odds, for Mutilated Chess Board*} text{*Lemmas for specialist use, NOT as default simprules*} lemma nat_mult_2: "2 * z = (z+z::nat)" proof - have "2*z = (1 + 1)*z" by simp also have "... = z+z" by (simp add: left_distrib) finally show ?thesis . qed lemma nat_mult_2_right: "z * 2 = (z+z::nat)" by (subst mult_commute, rule nat_mult_2) text{*Case analysis on @{term "n<2"}*} lemma less_2_cases: "(n::nat) < 2 ==> n = 0 | n = Suc 0" by arith lemma div2_Suc_Suc [simp]: "Suc(Suc m) div 2 = Suc (m div 2)" by arith lemma add_self_div_2 [simp]: "(m + m) div 2 = (m::nat)" by (simp add: nat_mult_2 [symmetric]) lemma mod2_Suc_Suc [simp]: "Suc(Suc(m)) mod 2 = m mod 2" apply (subgoal_tac "m mod 2 < 2") apply (erule less_2_cases [THEN disjE]) apply (simp_all (no_asm_simp) add: Let_def mod_Suc nat_1) done lemma mod2_gr_0 [simp]: "!!m::nat. (0 < m mod 2) = (m mod 2 = 1)" apply (subgoal_tac "m mod 2 < 2") apply (force simp del: mod_less_divisor, simp) done text{*Removal of Small Numerals: 0, 1 and (in additive positions) 2*} lemma add_2_eq_Suc [simp]: "2 + n = Suc (Suc n)" by simp lemma add_2_eq_Suc' [simp]: "n + 2 = Suc (Suc n)" by simp text{*Can be used to eliminate long strings of Sucs, but not by default*} lemma Suc3_eq_add_3: "Suc (Suc (Suc n)) = 3 + n" by simp text{*These lemmas collapse some needless occurrences of Suc: at least three Sucs, since two and fewer are rewritten back to Suc again! We already have some rules to simplify operands smaller than 3.*} lemma div_Suc_eq_div_add3 [simp]: "m div (Suc (Suc (Suc n))) = m div (3+n)" by (simp add: Suc3_eq_add_3) lemma mod_Suc_eq_mod_add3 [simp]: "m mod (Suc (Suc (Suc n))) = m mod (3+n)" by (simp add: Suc3_eq_add_3) lemma Suc_div_eq_add3_div: "(Suc (Suc (Suc m))) div n = (3+m) div n" by (simp add: Suc3_eq_add_3) lemma Suc_mod_eq_add3_mod: "(Suc (Suc (Suc m))) mod n = (3+m) mod n" by (simp add: Suc3_eq_add_3) lemmas Suc_div_eq_add3_div_number_of = Suc_div_eq_add3_div [of _ "number_of v", standard] declare Suc_div_eq_add3_div_number_of [simp] lemmas Suc_mod_eq_add3_mod_number_of = Suc_mod_eq_add3_mod [of _ "number_of v", standard] declare Suc_mod_eq_add3_mod_number_of [simp] subsubsection{*Special Simplification for Constants*} text{*These belong here, late in the development of HOL, to prevent their interfering with proofs of abstract properties of instances of the function @{term number_of}*} text{*These distributive laws move literals inside sums and differences.*} lemmas left_distrib_number_of = left_distrib [of _ _ "number_of v", standard] declare left_distrib_number_of [simp] lemmas right_distrib_number_of = right_distrib [of "number_of v", standard] declare right_distrib_number_of [simp] lemmas left_diff_distrib_number_of = left_diff_distrib [of _ _ "number_of v", standard] declare left_diff_distrib_number_of [simp] lemmas right_diff_distrib_number_of = right_diff_distrib [of "number_of v", standard] declare right_diff_distrib_number_of [simp] text{*These are actually for fields, like real: but where else to put them?*} lemmas zero_less_divide_iff_number_of = zero_less_divide_iff [of "number_of w", standard] declare zero_less_divide_iff_number_of [simp,noatp] lemmas divide_less_0_iff_number_of = divide_less_0_iff [of "number_of w", standard] declare divide_less_0_iff_number_of [simp,noatp] lemmas zero_le_divide_iff_number_of = zero_le_divide_iff [of "number_of w", standard] declare zero_le_divide_iff_number_of [simp,noatp] lemmas divide_le_0_iff_number_of = divide_le_0_iff [of "number_of w", standard] declare divide_le_0_iff_number_of [simp,noatp] (**** IF times_divide_eq_right and times_divide_eq_left are removed as simprules, then these special-case declarations may be useful. text{*These simprules move numerals into numerators and denominators.*} lemma times_recip_eq_right [simp]: "a * (1/c) = a / (c::'a::field)" by (simp add: times_divide_eq) lemma times_recip_eq_left [simp]: "(1/c) * a = a / (c::'a::field)" by (simp add: times_divide_eq) lemmas times_divide_eq_right_number_of = times_divide_eq_right [of "number_of w", standard] declare times_divide_eq_right_number_of [simp] lemmas times_divide_eq_right_number_of = times_divide_eq_right [of _ _ "number_of w", standard] declare times_divide_eq_right_number_of [simp] lemmas times_divide_eq_left_number_of = times_divide_eq_left [of _ "number_of w", standard] declare times_divide_eq_left_number_of [simp] lemmas times_divide_eq_left_number_of = times_divide_eq_left [of _ _ "number_of w", standard] declare times_divide_eq_left_number_of [simp] ****) text {*Replaces @{text "inverse #nn"} by @{text "1/#nn"}. It looks strange, but then other simprocs simplify the quotient.*} lemmas inverse_eq_divide_number_of = inverse_eq_divide [of "number_of w", standard] declare inverse_eq_divide_number_of [simp] text {*These laws simplify inequalities, moving unary minus from a term into the literal.*} lemmas less_minus_iff_number_of = less_minus_iff [of "number_of v", standard] declare less_minus_iff_number_of [simp,noatp] lemmas le_minus_iff_number_of = le_minus_iff [of "number_of v", standard] declare le_minus_iff_number_of [simp,noatp] lemmas equation_minus_iff_number_of = equation_minus_iff [of "number_of v", standard] declare equation_minus_iff_number_of [simp,noatp] lemmas minus_less_iff_number_of = minus_less_iff [of _ "number_of v", standard] declare minus_less_iff_number_of [simp,noatp] lemmas minus_le_iff_number_of = minus_le_iff [of _ "number_of v", standard] declare minus_le_iff_number_of [simp,noatp] lemmas minus_equation_iff_number_of = minus_equation_iff [of _ "number_of v", standard] declare minus_equation_iff_number_of [simp,noatp] text{*To Simplify Inequalities Where One Side is the Constant 1*} lemma less_minus_iff_1 [simp,noatp]: fixes b::"'b::{ordered_idom,number_ring}" shows "(1 < - b) = (b < -1)" by auto lemma le_minus_iff_1 [simp,noatp]: fixes b::"'b::{ordered_idom,number_ring}" shows "(1 ≤ - b) = (b ≤ -1)" by auto lemma equation_minus_iff_1 [simp,noatp]: fixes b::"'b::number_ring" shows "(1 = - b) = (b = -1)" by (subst equation_minus_iff, auto) lemma minus_less_iff_1 [simp,noatp]: fixes a::"'b::{ordered_idom,number_ring}" shows "(- a < 1) = (-1 < a)" by auto lemma minus_le_iff_1 [simp,noatp]: fixes a::"'b::{ordered_idom,number_ring}" shows "(- a ≤ 1) = (-1 ≤ a)" by auto lemma minus_equation_iff_1 [simp,noatp]: fixes a::"'b::number_ring" shows "(- a = 1) = (a = -1)" by (subst minus_equation_iff, auto) text {*Cancellation of constant factors in comparisons (@{text "<"} and @{text "≤"}) *} lemmas mult_less_cancel_left_number_of = mult_less_cancel_left [of "number_of v", standard] declare mult_less_cancel_left_number_of [simp,noatp] lemmas mult_less_cancel_right_number_of = mult_less_cancel_right [of _ "number_of v", standard] declare mult_less_cancel_right_number_of [simp,noatp] lemmas mult_le_cancel_left_number_of = mult_le_cancel_left [of "number_of v", standard] declare mult_le_cancel_left_number_of [simp,noatp] lemmas mult_le_cancel_right_number_of = mult_le_cancel_right [of _ "number_of v", standard] declare mult_le_cancel_right_number_of [simp,noatp] text {*Multiplying out constant divisors in comparisons (@{text "<"}, @{text "≤"} and @{text "="}) *} lemmas le_divide_eq_number_of = le_divide_eq [of _ _ "number_of w", standard] declare le_divide_eq_number_of [simp] lemmas divide_le_eq_number_of = divide_le_eq [of _ "number_of w", standard] declare divide_le_eq_number_of [simp] lemmas less_divide_eq_number_of = less_divide_eq [of _ _ "number_of w", standard] declare less_divide_eq_number_of [simp] lemmas divide_less_eq_number_of = divide_less_eq [of _ "number_of w", standard] declare divide_less_eq_number_of [simp] lemmas eq_divide_eq_number_of = eq_divide_eq [of _ _ "number_of w", standard] declare eq_divide_eq_number_of [simp] lemmas divide_eq_eq_number_of = divide_eq_eq [of _ "number_of w", standard] declare divide_eq_eq_number_of [simp] subsubsection{*Optional Simplification Rules Involving Constants*} text{*Simplify quotients that are compared with a literal constant.*} lemmas le_divide_eq_number_of = le_divide_eq [of "number_of w", standard] lemmas divide_le_eq_number_of = divide_le_eq [of _ _ "number_of w", standard] lemmas less_divide_eq_number_of = less_divide_eq [of "number_of w", standard] lemmas divide_less_eq_number_of = divide_less_eq [of _ _ "number_of w", standard] lemmas eq_divide_eq_number_of = eq_divide_eq [of "number_of w", standard] lemmas divide_eq_eq_number_of = divide_eq_eq [of _ _ "number_of w", standard] text{*Not good as automatic simprules because they cause case splits.*} lemmas divide_const_simps = le_divide_eq_number_of divide_le_eq_number_of less_divide_eq_number_of divide_less_eq_number_of eq_divide_eq_number_of divide_eq_eq_number_of le_divide_eq_1 divide_le_eq_1 less_divide_eq_1 divide_less_eq_1 text{*Division By @{text "-1"}*} lemma divide_minus1 [simp]: "x/-1 = -(x::'a::{field,division_by_zero,number_ring})" by simp lemma minus1_divide [simp]: "-1 / (x::'a::{field,division_by_zero,number_ring}) = - (1/x)" by (simp add: divide_inverse inverse_minus_eq) lemma half_gt_zero_iff: "(0 < r/2) = (0 < (r::'a::{ordered_field,division_by_zero,number_ring}))" by auto lemmas half_gt_zero = half_gt_zero_iff [THEN iffD2, standard] declare half_gt_zero [simp] (* The following lemma should appear in Divides.thy, but there the proof doesn't work. *) lemma nat_dvd_not_less: "[| 0 < m; m < n |] ==> ¬ n dvd (m::nat)" by (unfold dvd_def) auto ML {* val divide_minus1 = @{thm divide_minus1}; val minus1_divide = @{thm minus1_divide}; *} subsection{* Groebner Bases for fields *} interpretation class_fieldgb: fieldgb["op +" "op *" "op ^" "0::'a::{field,recpower,number_ring}" "1" "op -" "uminus" "op /" "inverse"] apply (unfold_locales) by (simp_all add: divide_inverse) lemma divide_Numeral1: "(x::'a::{field,number_ring}) / Numeral1 = x" by simp lemma divide_Numeral0: "(x::'a::{field,number_ring, division_by_zero}) / Numeral0 = 0" by simp lemma mult_frac_frac: "((x::'a::{field,division_by_zero}) / y) * (z / w) = (x*z) / (y*w)" by simp lemma mult_frac_num: "((x::'a::{field, division_by_zero}) / y) * z = (x*z) / y" by simp lemma mult_num_frac: "((x::'a::{field, division_by_zero}) / y) * z = (x*z) / y" by simp lemma Numeral1_eq1_nat: "(1::nat) = Numeral1" by simp lemma add_frac_num: "y≠ 0 ==> (x::'a::{field, division_by_zero}) / y + z = (x + z*y) / y" by (simp add: add_divide_distrib) lemma add_num_frac: "y≠ 0 ==> z + (x::'a::{field, division_by_zero}) / y = (x + z*y) / y" by (simp add: add_divide_distrib) ML{* local val zr = @{cpat "0"} val zT = ctyp_of_term zr val geq = @{cpat "op ="} val eqT = Thm.dest_ctyp (ctyp_of_term geq) |> hd val add_frac_eq = mk_meta_eq @{thm "add_frac_eq"} val add_frac_num = mk_meta_eq @{thm "add_frac_num"} val add_num_frac = mk_meta_eq @{thm "add_num_frac"} fun prove_nz ss T t = let val z = instantiate_cterm ([(zT,T)],[]) zr val eq = instantiate_cterm ([(eqT,T)],[]) geq val th = Simplifier.rewrite (ss addsimps simp_thms) (Thm.capply @{cterm "Trueprop"} (Thm.capply @{cterm "Not"} (Thm.capply (Thm.capply eq t) z))) in equal_elim (symmetric th) TrueI end fun proc phi ss ct = let val ((x,y),(w,z)) = (Thm.dest_binop #> (fn (a,b) => (Thm.dest_binop a, Thm.dest_binop b))) ct val _ = map (HOLogic.dest_number o term_of) [x,y,z,w] val T = ctyp_of_term x val [y_nz, z_nz] = map (prove_nz ss T) [y, z] val th = instantiate' [SOME T] (map SOME [y,z,x,w]) add_frac_eq in SOME (implies_elim (implies_elim th y_nz) z_nz) end handle CTERM _ => NONE | TERM _ => NONE | THM _ => NONE fun proc2 phi ss ct = let val (l,r) = Thm.dest_binop ct val T = ctyp_of_term l in (case (term_of l, term_of r) of (Const(@{const_name "HOL.divide"},_)$_$_, _) => let val (x,y) = Thm.dest_binop l val z = r val _ = map (HOLogic.dest_number o term_of) [x,y,z] val ynz = prove_nz ss T y in SOME (implies_elim (instantiate' [SOME T] (map SOME [y,x,z]) add_frac_num) ynz) end | (_, Const (@{const_name "HOL.divide"},_)$_$_) => let val (x,y) = Thm.dest_binop r val z = l val _ = map (HOLogic.dest_number o term_of) [x,y,z] val ynz = prove_nz ss T y in SOME (implies_elim (instantiate' [SOME T] (map SOME [y,z,x]) add_num_frac) ynz) end | _ => NONE) end handle CTERM _ => NONE | TERM _ => NONE | THM _ => NONE fun is_number (Const(@{const_name "HOL.divide"},_)$a$b) = is_number a andalso is_number b | is_number t = can HOLogic.dest_number t val is_number = is_number o term_of fun proc3 phi ss ct = (case term_of ct of Const(@{const_name HOL.less},_)$(Const(@{const_name "HOL.divide"},_)$_$_)$_ => let val ((a,b),c) = Thm.dest_binop ct |>> Thm.dest_binop val _ = map is_number [a,b,c] val T = ctyp_of_term c val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "divide_less_eq"} in SOME (mk_meta_eq th) end | Const(@{const_name HOL.less_eq},_)$(Const(@{const_name "HOL.divide"},_)$_$_)$_ => let val ((a,b),c) = Thm.dest_binop ct |>> Thm.dest_binop val _ = map is_number [a,b,c] val T = ctyp_of_term c val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "divide_le_eq"} in SOME (mk_meta_eq th) end | Const("op =",_)$(Const(@{const_name "HOL.divide"},_)$_$_)$_ => let val ((a,b),c) = Thm.dest_binop ct |>> Thm.dest_binop val _ = map is_number [a,b,c] val T = ctyp_of_term c val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "divide_eq_eq"} in SOME (mk_meta_eq th) end | Const(@{const_name HOL.less},_)$_$(Const(@{const_name "HOL.divide"},_)$_$_) => let val (a,(b,c)) = Thm.dest_binop ct ||> Thm.dest_binop val _ = map is_number [a,b,c] val T = ctyp_of_term c val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "less_divide_eq"} in SOME (mk_meta_eq th) end | Const(@{const_name HOL.less_eq},_)$_$(Const(@{const_name "HOL.divide"},_)$_$_) => let val (a,(b,c)) = Thm.dest_binop ct ||> Thm.dest_binop val _ = map is_number [a,b,c] val T = ctyp_of_term c val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "le_divide_eq"} in SOME (mk_meta_eq th) end | Const("op =",_)$_$(Const(@{const_name "HOL.divide"},_)$_$_) => let val (a,(b,c)) = Thm.dest_binop ct ||> Thm.dest_binop val _ = map is_number [a,b,c] val T = ctyp_of_term c val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "eq_divide_eq"} in SOME (mk_meta_eq th) end | _ => NONE) handle TERM _ => NONE | CTERM _ => NONE | THM _ => NONE val add_frac_frac_simproc = make_simproc {lhss = [@{cpat "(?x::?'a::field)/?y + (?w::?'a::field)/?z"}], name = "add_frac_frac_simproc", proc = proc, identifier = []} val add_frac_num_simproc = make_simproc {lhss = [@{cpat "(?x::?'a::field)/?y + ?z"}, @{cpat "?z + (?x::?'a::field)/?y"}], name = "add_frac_num_simproc", proc = proc2, identifier = []} val ord_frac_simproc = make_simproc {lhss = [@{cpat "(?a::(?'a::{field, ord}))/?b < ?c"}, @{cpat "(?a::(?'a::{field, ord}))/?b ≤ ?c"}, @{cpat "?c < (?a::(?'a::{field, ord}))/?b"}, @{cpat "?c ≤ (?a::(?'a::{field, ord}))/?b"}, @{cpat "?c = ((?a::(?'a::{field, ord}))/?b)"}, @{cpat "((?a::(?'a::{field, ord}))/ ?b) = ?c"}], name = "ord_frac_simproc", proc = proc3, identifier = []} val nat_arith = map thm ["add_nat_number_of", "diff_nat_number_of", "mult_nat_number_of", "eq_nat_number_of", "less_nat_number_of"] val comp_arith = (map thm ["Let_def", "if_False", "if_True", "add_0", "add_Suc", "add_number_of_left", "mult_number_of_left", "Suc_eq_add_numeral_1"])@ (map (fn s => thm s RS sym) ["numeral_1_eq_1", "numeral_0_eq_0"]) @ arith_simps@ nat_arith @ rel_simps val ths = [@{thm "mult_numeral_1"}, @{thm "mult_numeral_1_right"}, @{thm "divide_Numeral1"}, @{thm "Ring_and_Field.divide_zero"}, @{thm "divide_Numeral0"}, @{thm "divide_divide_eq_left"}, @{thm "mult_frac_frac"}, @{thm "mult_num_frac"}, @{thm "mult_frac_num"}, @{thm "mult_frac_frac"}, @{thm "times_divide_eq_right"}, @{thm "times_divide_eq_left"}, @{thm "divide_divide_eq_right"}, @{thm "diff_def"}, @{thm "minus_divide_left"}, @{thm "Numeral1_eq1_nat"}, @{thm "add_divide_distrib"} RS sym] local open Conv in val comp_conv = (Simplifier.rewrite (HOL_basic_ss addsimps @{thms "Groebner_Basis.comp_arith"} addsimps ths addsimps comp_arith addsimps simp_thms addsimprocs field_cancel_numeral_factors addsimprocs [add_frac_frac_simproc, add_frac_num_simproc, ord_frac_simproc] addcongs [@{thm "if_weak_cong"}])) then_conv (Simplifier.rewrite (HOL_basic_ss addsimps [@{thm numeral_1_eq_1},@{thm numeral_0_eq_0}] @ @{thms numerals(1-2)})) end fun numeral_is_const ct = case term_of ct of Const (@{const_name "HOL.divide"},_) $ a $ b => numeral_is_const (Thm.dest_arg1 ct) andalso numeral_is_const (Thm.dest_arg ct) | Const (@{const_name "HOL.uminus"},_)$t => numeral_is_const (Thm.dest_arg ct) | t => can HOLogic.dest_number t fun dest_const ct = ((case term_of ct of Const (@{const_name "HOL.divide"},_) $ a $ b=> Rat.rat_of_quotient (snd (HOLogic.dest_number a), snd (HOLogic.dest_number b)) | t => Rat.rat_of_int (snd (HOLogic.dest_number t))) handle TERM _ => error "ring_dest_const") fun mk_const phi cT x = let val (a, b) = Rat.quotient_of_rat x in if b = 1 then Numeral.mk_cnumber cT a else Thm.capply (Thm.capply (Drule.cterm_rule (instantiate' [SOME cT] []) @{cpat "op /"}) (Numeral.mk_cnumber cT a)) (Numeral.mk_cnumber cT b) end in val field_comp_conv = comp_conv; val fieldgb_declaration = NormalizerData.funs @{thm class_fieldgb.axioms} {is_const = K numeral_is_const, dest_const = K dest_const, mk_const = mk_const, conv = K (K comp_conv)} end; *} declaration{* fieldgb_declaration *} subsection {* Ferrante and Rackoff algorithm over ordered fields *} lemma neg_prod_lt:"(c::'a::ordered_field) < 0 ==> ((c*x < 0) == (x > 0))" proof- assume H: "c < 0" have "c*x < 0 = (0/c < x)" by (simp only: neg_divide_less_eq[OF H] ring_simps) also have "… = (0 < x)" by simp finally show "(c*x < 0) == (x > 0)" by simp qed lemma pos_prod_lt:"(c::'a::ordered_field) > 0 ==> ((c*x < 0) == (x < 0))" proof- assume H: "c > 0" hence "c*x < 0 = (0/c > x)" by (simp only: pos_less_divide_eq[OF H] ring_simps) also have "… = (0 > x)" by simp finally show "(c*x < 0) == (x < 0)" by simp qed lemma neg_prod_sum_lt: "(c::'a::ordered_field) < 0 ==> ((c*x + t< 0) == (x > (- 1/c)*t))" proof- assume H: "c < 0" have "c*x + t< 0 = (c*x < -t)" by (subst less_iff_diff_less_0 [of "c*x" "-t"], simp) also have "… = (-t/c < x)" by (simp only: neg_divide_less_eq[OF H] ring_simps) also have "… = ((- 1/c)*t < x)" by simp finally show "(c*x + t < 0) == (x > (- 1/c)*t)" by simp qed lemma pos_prod_sum_lt:"(c::'a::ordered_field) > 0 ==> ((c*x + t < 0) == (x < (- 1/c)*t))" proof- assume H: "c > 0" have "c*x + t< 0 = (c*x < -t)" by (subst less_iff_diff_less_0 [of "c*x" "-t"], simp) also have "… = (-t/c > x)" by (simp only: pos_less_divide_eq[OF H] ring_simps) also have "… = ((- 1/c)*t > x)" by simp finally show "(c*x + t < 0) == (x < (- 1/c)*t)" by simp qed lemma sum_lt:"((x::'a::pordered_ab_group_add) + t < 0) == (x < - t)" using less_diff_eq[where a= x and b=t and c=0] by simp lemma neg_prod_le:"(c::'a::ordered_field) < 0 ==> ((c*x <= 0) == (x >= 0))" proof- assume H: "c < 0" have "c*x <= 0 = (0/c <= x)" by (simp only: neg_divide_le_eq[OF H] ring_simps) also have "… = (0 <= x)" by simp finally show "(c*x <= 0) == (x >= 0)" by simp qed lemma pos_prod_le:"(c::'a::ordered_field) > 0 ==> ((c*x <= 0) == (x <= 0))" proof- assume H: "c > 0" hence "c*x <= 0 = (0/c >= x)" by (simp only: pos_le_divide_eq[OF H] ring_simps) also have "… = (0 >= x)" by simp finally show "(c*x <= 0) == (x <= 0)" by simp qed lemma neg_prod_sum_le: "(c::'a::ordered_field) < 0 ==> ((c*x + t <= 0) == (x >= (- 1/c)*t))" proof- assume H: "c < 0" have "c*x + t <= 0 = (c*x <= -t)" by (subst le_iff_diff_le_0 [of "c*x" "-t"], simp) also have "… = (-t/c <= x)" by (simp only: neg_divide_le_eq[OF H] ring_simps) also have "… = ((- 1/c)*t <= x)" by simp finally show "(c*x + t <= 0) == (x >= (- 1/c)*t)" by simp qed lemma pos_prod_sum_le:"(c::'a::ordered_field) > 0 ==> ((c*x + t <= 0) == (x <= (- 1/c)*t))" proof- assume H: "c > 0" have "c*x + t <= 0 = (c*x <= -t)" by (subst le_iff_diff_le_0 [of "c*x" "-t"], simp) also have "… = (-t/c >= x)" by (simp only: pos_le_divide_eq[OF H] ring_simps) also have "… = ((- 1/c)*t >= x)" by simp finally show "(c*x + t <= 0) == (x <= (- 1/c)*t)" by simp qed lemma sum_le:"((x::'a::pordered_ab_group_add) + t <= 0) == (x <= - t)" using le_diff_eq[where a= x and b=t and c=0] by simp lemma nz_prod_eq:"(c::'a::ordered_field) ≠ 0 ==> ((c*x = 0) == (x = 0))" by simp lemma nz_prod_sum_eq: "(c::'a::ordered_field) ≠ 0 ==> ((c*x + t = 0) == (x = (- 1/c)*t))" proof- assume H: "c ≠ 0" have "c*x + t = 0 = (c*x = -t)" by (subst eq_iff_diff_eq_0 [of "c*x" "-t"], simp) also have "… = (x = -t/c)" by (simp only: nonzero_eq_divide_eq[OF H] ring_simps) finally show "(c*x + t = 0) == (x = (- 1/c)*t)" by simp qed lemma sum_eq:"((x::'a::pordered_ab_group_add) + t = 0) == (x = - t)" using eq_diff_eq[where a= x and b=t and c=0] by simp interpretation class_ordered_field_dense_linear_order: constr_dense_linear_order ["op <=" "op <" "λ x y. 1/2 * ((x::'a::{ordered_field,recpower,number_ring}) + y)"] proof (unfold_locales, dlo, dlo, auto) fix x y::'a assume lt: "x < y" from less_half_sum[OF lt] show "x < (x + y) /2" by simp next fix x y::'a assume lt: "x < y" from gt_half_sum[OF lt] show "(x + y) /2 < y" by simp qed declaration{* let fun earlier [] x y = false | earlier (h::t) x y = if h aconvc y then false else if h aconvc x then true else earlier t x y; fun dest_frac ct = case term_of ct of Const (@{const_name "HOL.divide"},_) $ a $ b=> Rat.rat_of_quotient (snd (HOLogic.dest_number a), snd (HOLogic.dest_number b)) | t => Rat.rat_of_int (snd (HOLogic.dest_number t)) fun mk_frac phi cT x = let val (a, b) = Rat.quotient_of_rat x in if b = 1 then Numeral.mk_cnumber cT a else Thm.capply (Thm.capply (Drule.cterm_rule (instantiate' [SOME cT] []) @{cpat "op /"}) (Numeral.mk_cnumber cT a)) (Numeral.mk_cnumber cT b) end fun whatis x ct = case term_of ct of Const(@{const_name "HOL.plus"}, _)$(Const(@{const_name "HOL.times"},_)$_$y)$_ => if y aconv term_of x then ("c*x+t",[(funpow 2 Thm.dest_arg1) ct, Thm.dest_arg ct]) else ("Nox",[]) | Const(@{const_name "HOL.plus"}, _)$y$_ => if y aconv term_of x then ("x+t",[Thm.dest_arg ct]) else ("Nox",[]) | Const(@{const_name "HOL.times"}, _)$_$y => if y aconv term_of x then ("c*x",[Thm.dest_arg1 ct]) else ("Nox",[]) | t => if t aconv term_of x then ("x",[]) else ("Nox",[]); fun xnormalize_conv ctxt [] ct = reflexive ct | xnormalize_conv ctxt (vs as (x::_)) ct = case term_of ct of Const(@{const_name HOL.less},_)$_$Const(@{const_name "HOL.zero"},_) => (case whatis x (Thm.dest_arg1 ct) of ("c*x+t",[c,t]) => let val cr = dest_frac c val clt = Thm.dest_fun2 ct val cz = Thm.dest_arg ct val neg = cr </ Rat.zero val cthp = Simplifier.rewrite (local_simpset_of ctxt) (Thm.capply @{cterm "Trueprop"} (if neg then Thm.capply (Thm.capply clt c) cz else Thm.capply (Thm.capply clt cz) c)) val cth = equal_elim (symmetric cthp) TrueI val th = implies_elim (instantiate' [SOME (ctyp_of_term x)] (map SOME [c,x,t]) (if neg then @{thm neg_prod_sum_lt} else @{thm pos_prod_sum_lt})) cth val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv (Normalizer.semiring_normalize_ord_conv ctxt (earlier vs)))) th in rth end | ("x+t",[t]) => let val T = ctyp_of_term x val th = instantiate' [SOME T] [SOME x, SOME t] @{thm "sum_lt"} val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv (Normalizer.semiring_normalize_ord_conv ctxt (earlier vs)))) th in rth end | ("c*x",[c]) => let val cr = dest_frac c val clt = Thm.dest_fun2 ct val cz = Thm.dest_arg ct val neg = cr </ Rat.zero val cthp = Simplifier.rewrite (local_simpset_of ctxt) (Thm.capply @{cterm "Trueprop"} (if neg then Thm.capply (Thm.capply clt c) cz else Thm.capply (Thm.capply clt cz) c)) val cth = equal_elim (symmetric cthp) TrueI val th = implies_elim (instantiate' [SOME (ctyp_of_term x)] (map SOME [c,x]) (if neg then @{thm neg_prod_lt} else @{thm pos_prod_lt})) cth val rth = th in rth end | _ => reflexive ct) | Const(@{const_name HOL.less_eq},_)$_$Const(@{const_name "HOL.zero"},_) => (case whatis x (Thm.dest_arg1 ct) of ("c*x+t",[c,t]) => let val T = ctyp_of_term x val cr = dest_frac c val clt = Drule.cterm_rule (instantiate' [SOME T] []) @{cpat "op <"} val cz = Thm.dest_arg ct val neg = cr </ Rat.zero val cthp = Simplifier.rewrite (local_simpset_of ctxt) (Thm.capply @{cterm "Trueprop"} (if neg then Thm.capply (Thm.capply clt c) cz else Thm.capply (Thm.capply clt cz) c)) val cth = equal_elim (symmetric cthp) TrueI val th = implies_elim (instantiate' [SOME T] (map SOME [c,x,t]) (if neg then @{thm neg_prod_sum_le} else @{thm pos_prod_sum_le})) cth val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv (Normalizer.semiring_normalize_ord_conv ctxt (earlier vs)))) th in rth end | ("x+t",[t]) => let val T = ctyp_of_term x val th = instantiate' [SOME T] [SOME x, SOME t] @{thm "sum_le"} val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv (Normalizer.semiring_normalize_ord_conv ctxt (earlier vs)))) th in rth end | ("c*x",[c]) => let val T = ctyp_of_term x val cr = dest_frac c val clt = Drule.cterm_rule (instantiate' [SOME T] []) @{cpat "op <"} val cz = Thm.dest_arg ct val neg = cr </ Rat.zero val cthp = Simplifier.rewrite (local_simpset_of ctxt) (Thm.capply @{cterm "Trueprop"} (if neg then Thm.capply (Thm.capply clt c) cz else Thm.capply (Thm.capply clt cz) c)) val cth = equal_elim (symmetric cthp) TrueI val th = implies_elim (instantiate' [SOME (ctyp_of_term x)] (map SOME [c,x]) (if neg then @{thm neg_prod_le} else @{thm pos_prod_le})) cth val rth = th in rth end | _ => reflexive ct) | Const("op =",_)$_$Const(@{const_name "HOL.zero"},_) => (case whatis x (Thm.dest_arg1 ct) of ("c*x+t",[c,t]) => let val T = ctyp_of_term x val cr = dest_frac c val ceq = Thm.dest_fun2 ct val cz = Thm.dest_arg ct val cthp = Simplifier.rewrite (local_simpset_of ctxt) (Thm.capply @{cterm "Trueprop"} (Thm.capply @{cterm "Not"} (Thm.capply (Thm.capply ceq c) cz))) val cth = equal_elim (symmetric cthp) TrueI val th = implies_elim (instantiate' [SOME T] (map SOME [c,x,t]) @{thm nz_prod_sum_eq}) cth val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv (Normalizer.semiring_normalize_ord_conv ctxt (earlier vs)))) th in rth end | ("x+t",[t]) => let val T = ctyp_of_term x val th = instantiate' [SOME T] [SOME x, SOME t] @{thm "sum_eq"} val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv (Normalizer.semiring_normalize_ord_conv ctxt (earlier vs)))) th in rth end | ("c*x",[c]) => let val T = ctyp_of_term x val cr = dest_frac c val ceq = Thm.dest_fun2 ct val cz = Thm.dest_arg ct val cthp = Simplifier.rewrite (local_simpset_of ctxt) (Thm.capply @{cterm "Trueprop"} (Thm.capply @{cterm "Not"} (Thm.capply (Thm.capply ceq c) cz))) val cth = equal_elim (symmetric cthp) TrueI val rth = implies_elim (instantiate' [SOME T] (map SOME [c,x]) @{thm nz_prod_eq}) cth in rth end | _ => reflexive ct); local val less_iff_diff_less_0 = mk_meta_eq @{thm "less_iff_diff_less_0"} val le_iff_diff_le_0 = mk_meta_eq @{thm "le_iff_diff_le_0"} val eq_iff_diff_eq_0 = mk_meta_eq @{thm "eq_iff_diff_eq_0"} in fun field_isolate_conv phi ctxt vs ct = case term_of ct of Const(@{const_name HOL.less},_)$a$b => let val (ca,cb) = Thm.dest_binop ct val T = ctyp_of_term ca val th = instantiate' [SOME T] [SOME ca, SOME cb] less_iff_diff_less_0 val nth = Conv.fconv_rule (Conv.arg_conv (Conv.arg1_conv (Normalizer.semiring_normalize_ord_conv @{context} (earlier vs)))) th val rth = transitive nth (xnormalize_conv ctxt vs (Thm.rhs_of nth)) in rth end | Const(@{const_name HOL.less_eq},_)$a$b => let val (ca,cb) = Thm.dest_binop ct val T = ctyp_of_term ca val th = instantiate' [SOME T] [SOME ca, SOME cb] le_iff_diff_le_0 val nth = Conv.fconv_rule (Conv.arg_conv (Conv.arg1_conv (Normalizer.semiring_normalize_ord_conv @{context} (earlier vs)))) th val rth = transitive nth (xnormalize_conv ctxt vs (Thm.rhs_of nth)) in rth end | Const("op =",_)$a$b => let val (ca,cb) = Thm.dest_binop ct val T = ctyp_of_term ca val th = instantiate' [SOME T] [SOME ca, SOME cb] eq_iff_diff_eq_0 val nth = Conv.fconv_rule (Conv.arg_conv (Conv.arg1_conv (Normalizer.semiring_normalize_ord_conv @{context} (earlier vs)))) th val rth = transitive nth (xnormalize_conv ctxt vs (Thm.rhs_of nth)) in rth end | @{term "Not"} $(Const("op =",_)$a$b) => Conv.arg_conv (field_isolate_conv phi ctxt vs) ct | _ => reflexive ct end; fun classfield_whatis phi = let fun h x t = case term_of t of Const("op =", _)$y$z => if term_of x aconv y then Ferrante_Rackoff_Data.Eq else Ferrante_Rackoff_Data.Nox | @{term "Not"}$(Const("op =", _)$y$z) => if term_of x aconv y then Ferrante_Rackoff_Data.NEq else Ferrante_Rackoff_Data.Nox | Const(@{const_name HOL.less},_)$y$z => if term_of x aconv y then Ferrante_Rackoff_Data.Lt else if term_of x aconv z then Ferrante_Rackoff_Data.Gt else Ferrante_Rackoff_Data.Nox | Const (@{const_name HOL.less_eq},_)$y$z => if term_of x aconv y then Ferrante_Rackoff_Data.Le else if term_of x aconv z then Ferrante_Rackoff_Data.Ge else Ferrante_Rackoff_Data.Nox | _ => Ferrante_Rackoff_Data.Nox in h end; fun class_field_ss phi = HOL_basic_ss addsimps ([@{thm "linorder_not_less"}, @{thm "linorder_not_le"}]) addsplits [@{thm "abs_split"},@{thm "split_max"}, @{thm "split_min"}] in Ferrante_Rackoff_Data.funs @{thm "class_ordered_field_dense_linear_order.ferrack_axiom"} {isolate_conv = field_isolate_conv, whatis = classfield_whatis, simpset = class_field_ss} end *} end
lemma Suc_diff_eq_diff_pred:
Numeral0 < n ==> Suc m - n = m - (n - Numeral1)
lemma neg_number_of_pred_iff_0:
neg (number_of (Numeral.pred v)) = (number_of v = 0)
lemma Suc_diff_number_of:
neg (number_of (- v)) ==> Suc m - number_of v = m - number_of (Numeral.pred v)
lemma diff_Suc_eq_diff_pred:
m - Suc n = m - 1 - n
lemma nat_case_number_of:
nat_case a f (number_of v) =
(let pv = number_of (Numeral.pred v) in if neg pv then a else f (nat pv))
lemma nat_case_add_eq_if:
nat_case a f (number_of v + n) =
(let pv = number_of (Numeral.pred v)
in if neg pv then nat_case a f n else f (nat pv + n))
lemma nat_rec_number_of:
nat_rec a f (number_of v) =
(let pv = number_of (Numeral.pred v)
in if neg pv then a else f (nat pv) (nat_rec a f (nat pv)))
lemma nat_rec_add_eq_if:
nat_rec a f (number_of v + n) =
(let pv = number_of (Numeral.pred v)
in if neg pv then nat_rec a f n else f (nat pv + n) (nat_rec a f (nat pv + n)))
lemma nat_mult_2:
2 * z = z + z
lemma nat_mult_2_right:
z * 2 = z + z
lemma less_2_cases:
n < 2 ==> n = 0 ∨ n = Suc 0
lemma div2_Suc_Suc:
Suc (Suc m) div 2 = Suc (m div 2)
lemma add_self_div_2:
(m + m) div 2 = m
lemma mod2_Suc_Suc:
Suc (Suc m) mod 2 = m mod 2
lemma mod2_gr_0:
(0 < m mod 2) = (m mod 2 = 1)
lemma add_2_eq_Suc:
2 + n = Suc (Suc n)
lemma add_2_eq_Suc':
n + 2 = Suc (Suc n)
lemma Suc3_eq_add_3:
Suc (Suc (Suc n)) = 3 + n
lemma div_Suc_eq_div_add3:
m div Suc (Suc (Suc n)) = m div (3 + n)
lemma mod_Suc_eq_mod_add3:
m mod Suc (Suc (Suc n)) = m mod (3 + n)
lemma Suc_div_eq_add3_div:
Suc (Suc (Suc m)) div n = (3 + m) div n
lemma Suc_mod_eq_add3_mod:
Suc (Suc (Suc m)) mod n = (3 + m) mod n
lemma Suc_div_eq_add3_div_number_of:
Suc (Suc (Suc m)) div number_of v = (3 + m) div number_of v
lemma Suc_mod_eq_add3_mod_number_of:
Suc (Suc (Suc m)) mod number_of v = (3 + m) mod number_of v
lemma left_distrib_number_of:
(a + b) * number_of v = a * number_of v + b * number_of v
lemma right_distrib_number_of:
number_of v * (b + c) = number_of v * b + number_of v * c
lemma left_diff_distrib_number_of:
(a - b) * number_of v = a * number_of v - b * number_of v
lemma right_diff_distrib_number_of:
number_of v * (b - c) = number_of v * b - number_of v * c
lemma zero_less_divide_iff_number_of:
((0::'b) < number_of w / b) =
((0::'b) < number_of w ∧ (0::'b) < b ∨ number_of w < (0::'b) ∧ b < (0::'b))
lemma divide_less_0_iff_number_of:
(number_of w / b < (0::'b)) =
((0::'b) < number_of w ∧ b < (0::'b) ∨ number_of w < (0::'b) ∧ (0::'b) < b)
lemma zero_le_divide_iff_number_of:
((0::'b) ≤ number_of w / b) =
((0::'b) ≤ number_of w ∧ (0::'b) ≤ b ∨ number_of w ≤ (0::'b) ∧ b ≤ (0::'b))
lemma divide_le_0_iff_number_of:
(number_of w / b ≤ (0::'b)) =
((0::'b) ≤ number_of w ∧ b ≤ (0::'b) ∨ number_of w ≤ (0::'b) ∧ (0::'b) ≤ b)
lemma inverse_eq_divide_number_of:
inverse (number_of w) = (1::'b) / number_of w
lemma less_minus_iff_number_of:
(number_of v < - b) = (b < - number_of v)
lemma le_minus_iff_number_of:
(number_of v ≤ - b) = (b ≤ - number_of v)
lemma equation_minus_iff_number_of:
(number_of v = - b) = (b = - number_of v)
lemma minus_less_iff_number_of:
(- a < number_of v) = (- number_of v < a)
lemma minus_le_iff_number_of:
(- a ≤ number_of v) = (- number_of v ≤ a)
lemma minus_equation_iff_number_of:
(- a = number_of v) = (- number_of v = a)
lemma less_minus_iff_1:
((1::'b) < - b) = (b < (-1::'b))
lemma le_minus_iff_1:
((1::'b) ≤ - b) = (b ≤ (-1::'b))
lemma equation_minus_iff_1:
((1::'b) = - b) = (b = (-1::'b))
lemma minus_less_iff_1:
(- a < (1::'b)) = ((-1::'b) < a)
lemma minus_le_iff_1:
(- a ≤ (1::'b)) = ((-1::'b) ≤ a)
lemma minus_equation_iff_1:
(- a = (1::'b)) = (a = (-1::'b))
lemma mult_less_cancel_left_number_of:
(number_of v * a < number_of v * b) =
(((0::'b) ≤ number_of v --> a < b) ∧ (number_of v ≤ (0::'b) --> b < a))
lemma mult_less_cancel_right_number_of:
(a * number_of v < b * number_of v) =
(((0::'b) ≤ number_of v --> a < b) ∧ (number_of v ≤ (0::'b) --> b < a))
lemma mult_le_cancel_left_number_of:
(number_of v * a ≤ number_of v * b) =
(((0::'b) < number_of v --> a ≤ b) ∧ (number_of v < (0::'b) --> b ≤ a))
lemma mult_le_cancel_right_number_of:
(a * number_of v ≤ b * number_of v) =
(((0::'b) < number_of v --> a ≤ b) ∧ (number_of v < (0::'b) --> b ≤ a))
lemma le_divide_eq_number_of:
(a ≤ b / number_of w) =
(if (0::'b) < number_of w then a * number_of w ≤ b
else if number_of w < (0::'b) then b ≤ a * number_of w else a ≤ (0::'b))
lemma divide_le_eq_number_of:
(b / number_of w ≤ a) =
(if (0::'b) < number_of w then b ≤ a * number_of w
else if number_of w < (0::'b) then a * number_of w ≤ b else (0::'b) ≤ a)
lemma less_divide_eq_number_of:
(a < b / number_of w) =
(if (0::'b) < number_of w then a * number_of w < b
else if number_of w < (0::'b) then b < a * number_of w else a < (0::'b))
lemma divide_less_eq_number_of:
(b / number_of w < a) =
(if (0::'b) < number_of w then b < a * number_of w
else if number_of w < (0::'b) then a * number_of w < b else (0::'b) < a)
lemma eq_divide_eq_number_of:
(a = b / number_of w) =
(if number_of w ≠ (0::'b) then a * number_of w = b else a = (0::'b))
lemma divide_eq_eq_number_of:
(b / number_of w = a) =
(if number_of w ≠ (0::'b) then b = a * number_of w else a = (0::'b))
lemma le_divide_eq_number_of:
(number_of w ≤ b / c) =
(if (0::'b) < c then number_of w * c ≤ b
else if c < (0::'b) then b ≤ number_of w * c else number_of w ≤ (0::'b))
lemma divide_le_eq_number_of:
(b / c ≤ number_of w) =
(if (0::'b) < c then b ≤ number_of w * c
else if c < (0::'b) then number_of w * c ≤ b else (0::'b) ≤ number_of w)
lemma less_divide_eq_number_of:
(number_of w < b / c) =
(if (0::'b) < c then number_of w * c < b
else if c < (0::'b) then b < number_of w * c else number_of w < (0::'b))
lemma divide_less_eq_number_of:
(b / c < number_of w) =
(if (0::'b) < c then b < number_of w * c
else if c < (0::'b) then number_of w * c < b else (0::'b) < number_of w)
lemma eq_divide_eq_number_of:
(number_of w = b / c) =
(if c ≠ (0::'b) then number_of w * c = b else number_of w = (0::'b))
lemma divide_eq_eq_number_of:
(b / c = number_of w) =
(if c ≠ (0::'b) then b = number_of w * c else number_of w = (0::'b))
lemma divide_const_simps:
(number_of w ≤ b / c) =
(if (0::'b) < c then number_of w * c ≤ b
else if c < (0::'b) then b ≤ number_of w * c else number_of w ≤ (0::'b))
(b / c ≤ number_of w) =
(if (0::'b) < c then b ≤ number_of w * c
else if c < (0::'b) then number_of w * c ≤ b else (0::'b) ≤ number_of w)
(number_of w < b / c) =
(if (0::'b) < c then number_of w * c < b
else if c < (0::'b) then b < number_of w * c else number_of w < (0::'b))
(b / c < number_of w) =
(if (0::'b) < c then b < number_of w * c
else if c < (0::'b) then number_of w * c < b else (0::'b) < number_of w)
(number_of w = b / c) =
(if c ≠ (0::'b) then number_of w * c = b else number_of w = (0::'b))
(b / c = number_of w) =
(if c ≠ (0::'b) then b = number_of w * c else number_of w = (0::'b))
((1::'a) ≤ b / a) = ((0::'a) < a ∧ a ≤ b ∨ a < (0::'a) ∧ b ≤ a)
(b / a ≤ (1::'a)) = ((0::'a) < a ∧ b ≤ a ∨ a < (0::'a) ∧ a ≤ b ∨ a = (0::'a))
((1::'a) < b / a) = ((0::'a) < a ∧ a < b ∨ a < (0::'a) ∧ b < a)
(b / a < (1::'a)) = ((0::'a) < a ∧ b < a ∨ a < (0::'a) ∧ a < b ∨ a = (0::'a))
lemma divide_minus1:
x / (-1::'a) = - x
lemma minus1_divide:
(-1::'a) / x = - ((1::'a) / x)
lemma half_gt_zero_iff:
((0::'a) < r / (2::'a)) = ((0::'a) < r)
lemma half_gt_zero:
(0::'a) < r ==> (0::'a) < r / (2::'a)
lemma nat_dvd_not_less:
[| 0 < m; m < n |] ==> ¬ n dvd m
lemma divide_Numeral1:
x / Numeral1 = x
lemma divide_Numeral0:
x / Numeral0 = (0::'a)
lemma mult_frac_frac:
x / y * (z / w) = x * z / (y * w)
lemma mult_frac_num:
x / y * z = x * z / y
lemma mult_num_frac:
x / y * z = x * z / y
lemma Numeral1_eq1_nat:
1 = Numeral1
lemma add_frac_num:
y ≠ (0::'a) ==> x / y + z = (x + z * y) / y
lemma add_num_frac:
y ≠ (0::'a) ==> z + x / y = (x + z * y) / y
lemma neg_prod_lt:
c < (0::'a) ==> c * x < (0::'a) == (0::'a) < x
lemma pos_prod_lt:
(0::'a) < c ==> c * x < (0::'a) == x < (0::'a)
lemma neg_prod_sum_lt:
c < (0::'a) ==> c * x + t < (0::'a) == - (1::'a) / c * t < x
lemma pos_prod_sum_lt:
(0::'a) < c ==> c * x + t < (0::'a) == x < - (1::'a) / c * t
lemma sum_lt:
x + t < (0::'a) == x < - t
lemma neg_prod_le:
c < (0::'a) ==> c * x ≤ (0::'a) == (0::'a) ≤ x
lemma pos_prod_le:
(0::'a) < c ==> c * x ≤ (0::'a) == x ≤ (0::'a)
lemma neg_prod_sum_le:
c < (0::'a) ==> c * x + t ≤ (0::'a) == - (1::'a) / c * t ≤ x
lemma pos_prod_sum_le:
(0::'a) < c ==> c * x + t ≤ (0::'a) == x ≤ - (1::'a) / c * t
lemma sum_le:
x + t ≤ (0::'a) == x ≤ - t
lemma nz_prod_eq:
c ≠ (0::'a) ==> c * x = (0::'a) == x = (0::'a)
lemma nz_prod_sum_eq:
c ≠ (0::'a) ==> c * x + t = (0::'a) == x = - (1::'a) / c * t
lemma sum_eq:
x + t = (0::'a) == x = - t