wenzelm@23465: (* Title: HOL/Presburger.thy wenzelm@23465: ID: $Id$ wenzelm@23465: Author: Amine Chaieb, TU Muenchen wenzelm@23465: *) wenzelm@23465: huffman@23472: header {* Decision Procedure for Presburger Arithmetic *} huffman@23472: wenzelm@23465: theory Presburger wenzelm@23465: imports Arith_Tools SetInterval wenzelm@23465: uses wenzelm@23465: "Tools/Qelim/cooper_data.ML" wenzelm@23465: "Tools/Qelim/generated_cooper.ML" wenzelm@23465: ("Tools/Qelim/cooper.ML") wenzelm@23465: ("Tools/Qelim/presburger.ML") wenzelm@23465: begin wenzelm@23465: wenzelm@23465: setup CooperData.setup wenzelm@23465: wenzelm@23465: subsection{* The @{text "-\"} and @{text "+\"} Properties *} wenzelm@23465: chaieb@24404: wenzelm@23465: lemma minf: wenzelm@23465: "\\(z ::'a::linorder).\xz.\x wenzelm@23465: \ \z.\x Q x) = (P' x \ Q' x)" wenzelm@23465: "\\(z ::'a::linorder).\xz.\x wenzelm@23465: \ \z.\x Q x) = (P' x \ Q' x)" wenzelm@23465: "\(z ::'a::{linorder}).\x(z ::'a::{linorder}).\x t) = True" wenzelm@23465: "\(z ::'a::{linorder}).\x(z ::'a::{linorder}).\x t) = True" wenzelm@23465: "\(z ::'a::{linorder}).\x t) = False" wenzelm@23465: "\(z ::'a::{linorder}).\x t) = False" haftmann@24993: "\z.\(x::'a::{linorder,plus,Divides.div})z.\(x::'a::{linorder,plus,Divides.div}) d dvd x + s) = (\ d dvd x + s)" wenzelm@23465: "\z.\x\(z ::'a::linorder).\x>z. P x = P' x; \z.\x>z. Q x = Q' x\ wenzelm@23465: \ \z.\x>z. (P x \ Q x) = (P' x \ Q' x)" wenzelm@23465: "\\(z ::'a::linorder).\x>z. P x = P' x; \z.\x>z. Q x = Q' x\ wenzelm@23465: \ \z.\x>z. (P x \ Q x) = (P' x \ Q' x)" wenzelm@23465: "\(z ::'a::{linorder}).\x>z.(x = t) = False" wenzelm@23465: "\(z ::'a::{linorder}).\x>z.(x \ t) = True" wenzelm@23465: "\(z ::'a::{linorder}).\x>z.(x < t) = False" wenzelm@23465: "\(z ::'a::{linorder}).\x>z.(x \ t) = False" wenzelm@23465: "\(z ::'a::{linorder}).\x>z.(x > t) = True" wenzelm@23465: "\(z ::'a::{linorder}).\x>z.(x \ t) = True" haftmann@24993: "\z.\(x::'a::{linorder,plus,Divides.div})>z. (d dvd x + s) = (d dvd x + s)" haftmann@24993: "\z.\(x::'a::{linorder,plus,Divides.div})>z. (\ d dvd x + s) = (\ d dvd x + s)" wenzelm@23465: "\z.\x>z. F = F" wenzelm@23465: by ((erule exE, erule exE,rule_tac x="max z za" in exI,simp)+,(rule_tac x="t" in exI,fastsimp)+) simp_all wenzelm@23465: wenzelm@23465: lemma inf_period: wenzelm@23465: "\\x k. P x = P (x - k*D); \x k. Q x = Q (x - k*D)\ wenzelm@23465: \ \x k. (P x \ Q x) = (P (x - k*D) \ Q (x - k*D))" wenzelm@23465: "\\x k. P x = P (x - k*D); \x k. Q x = Q (x - k*D)\ wenzelm@23465: \ \x k. (P x \ Q x) = (P (x - k*D) \ Q (x - k*D))" haftmann@24993: "(d::'a::{comm_ring,Divides.div}) dvd D \ \x k. (d dvd x + t) = (d dvd (x - k*D) + t)" haftmann@24993: "(d::'a::{comm_ring,Divides.div}) dvd D \ \x k. (\d dvd x + t) = (\d dvd (x - k*D) + t)" wenzelm@23465: "\x k. F = F" wenzelm@23465: by simp_all wenzelm@23465: (clarsimp simp add: dvd_def, rule iffI, clarsimp,rule_tac x = "kb - ka*k" in exI, nipkow@23477: simp add: ring_simps, clarsimp,rule_tac x = "kb + ka*k" in exI,simp add: ring_simps)+ wenzelm@23465: huffman@23472: subsection{* The A and B sets *} wenzelm@23465: lemma bset: wenzelm@23465: "\\x.(\j \ {1 .. D}. \b\B. x \ b + j)\ P x \ P(x - D) ; wenzelm@23465: \x.(\j\{1 .. D}. \b\B. x \ b + j)\ Q x \ Q(x - D)\ \ wenzelm@23465: \x.(\j\{1 .. D}. \b\B. x \ b + j) \ (P x \ Q x) \ (P(x - D) \ Q (x - D))" wenzelm@23465: "\\x.(\j\{1 .. D}. \b\B. x \ b + j)\ P x \ P(x - D) ; wenzelm@23465: \x.(\j\{1 .. D}. \b\B. x \ b + j)\ Q x \ Q(x - D)\ \ wenzelm@23465: \x.(\j\{1 .. D}. \b\B. x \ b + j)\ (P x \ Q x) \ (P(x - D) \ Q (x - D))" wenzelm@23465: "\D>0; t - 1\ B\ \ (\x.(\j\{1 .. D}. \b\B. x \ b + j)\ (x = t) \ (x - D = t))" wenzelm@23465: "\D>0 ; t \ B\ \(\(x::int).(\j\{1 .. D}. \b\B. x \ b + j)\ (x \ t) \ (x - D \ t))" wenzelm@23465: "D>0 \ (\(x::int).(\j\{1 .. D}. \b\B. x \ b + j)\ (x < t) \ (x - D < t))" wenzelm@23465: "D>0 \ (\(x::int).(\j\{1 .. D}. \b\B. x \ b + j)\ (x \ t) \ (x - D \ t))" wenzelm@23465: "\D>0 ; t \ B\ \(\(x::int).(\j\{1 .. D}. \b\B. x \ b + j)\ (x > t) \ (x - D > t))" wenzelm@23465: "\D>0 ; t - 1 \ B\ \(\(x::int).(\j\{1 .. D}. \b\B. x \ b + j)\ (x \ t) \ (x - D \ t))" wenzelm@23465: "d dvd D \(\(x::int).(\j\{1 .. D}. \b\B. x \ b + j)\ (d dvd x+t) \ (d dvd (x - D) + t))" wenzelm@23465: "d dvd D \(\(x::int).(\j\{1 .. D}. \b\B. x \ b + j)\ (\d dvd x+t) \ (\ d dvd (x - D) + t))" wenzelm@23465: "\x.(\j\{1 .. D}. \b\B. x \ b + j) \ F \ F" wenzelm@23465: proof (blast, blast) wenzelm@23465: assume dp: "D > 0" and tB: "t - 1\ B" wenzelm@23465: show "(\x.(\j\{1 .. D}. \b\B. x \ b + j)\ (x = t) \ (x - D = t))" wenzelm@23465: apply (rule allI, rule impI,erule ballE[where x="1"],erule ballE[where x="t - 1"]) wenzelm@23465: using dp tB by simp_all wenzelm@23465: next wenzelm@23465: assume dp: "D > 0" and tB: "t \ B" wenzelm@23465: show "(\x.(\j\{1 .. D}. \b\B. x \ b + j)\ (x \ t) \ (x - D \ t))" wenzelm@23465: apply (rule allI, rule impI,erule ballE[where x="D"],erule ballE[where x="t"]) wenzelm@23465: using dp tB by simp_all wenzelm@23465: next wenzelm@23465: assume dp: "D > 0" thus "(\x.(\j\{1 .. D}. \b\B. x \ b + j)\ (x < t) \ (x - D < t))" by arith wenzelm@23465: next wenzelm@23465: assume dp: "D > 0" thus "\x.(\j\{1 .. D}. \b\B. x \ b + j)\ (x \ t) \ (x - D \ t)" by arith wenzelm@23465: next wenzelm@23465: assume dp: "D > 0" and tB:"t \ B" wenzelm@23465: {fix x assume nob: "\j\{1 .. D}. \b\B. x \ b + j" and g: "x > t" and ng: "\ (x - D) > t" wenzelm@23465: hence "x -t \ D" and "1 \ x - t" by simp+ wenzelm@23465: hence "\j \ {1 .. D}. x - t = j" by auto nipkow@23477: hence "\j \ {1 .. D}. x = t + j" by (simp add: ring_simps) wenzelm@23465: with nob tB have "False" by simp} wenzelm@23465: thus "\x.(\j\{1 .. D}. \b\B. x \ b + j)\ (x > t) \ (x - D > t)" by blast wenzelm@23465: next wenzelm@23465: assume dp: "D > 0" and tB:"t - 1\ B" wenzelm@23465: {fix x assume nob: "\j\{1 .. D}. \b\B. x \ b + j" and g: "x \ t" and ng: "\ (x - D) \ t" wenzelm@23465: hence "x - (t - 1) \ D" and "1 \ x - (t - 1)" by simp+ wenzelm@23465: hence "\j \ {1 .. D}. x - (t - 1) = j" by auto nipkow@23477: hence "\j \ {1 .. D}. x = (t - 1) + j" by (simp add: ring_simps) wenzelm@23465: with nob tB have "False" by simp} wenzelm@23465: thus "\x.(\j\{1 .. D}. \b\B. x \ b + j)\ (x \ t) \ (x - D \ t)" by blast wenzelm@23465: next wenzelm@23465: assume d: "d dvd D" wenzelm@23465: {fix x assume H: "d dvd x + t" with d have "d dvd (x - D) + t" nipkow@23477: by (clarsimp simp add: dvd_def,rule_tac x= "ka - k" in exI,simp add: ring_simps)} wenzelm@23465: thus "\(x::int).(\j\{1 .. D}. \b\B. x \ b + j)\ (d dvd x+t) \ (d dvd (x - D) + t)" by simp wenzelm@23465: next wenzelm@23465: assume d: "d dvd D" wenzelm@23465: {fix x assume H: "\(d dvd x + t)" with d have "\d dvd (x - D) + t" nipkow@23477: by (clarsimp simp add: dvd_def,erule_tac x= "ka + k" in allE,simp add: ring_simps)} wenzelm@23465: thus "\(x::int).(\j\{1 .. D}. \b\B. x \ b + j)\ (\d dvd x+t) \ (\d dvd (x - D) + t)" by auto wenzelm@23465: qed blast wenzelm@23465: wenzelm@23465: lemma aset: wenzelm@23465: "\\x.(\j\{1 .. D}. \b\A. x \ b - j)\ P x \ P(x + D) ; wenzelm@23465: \x.(\j\{1 .. D}. \b\A. x \ b - j)\ Q x \ Q(x + D)\ \ wenzelm@23465: \x.(\j\{1 .. D}. \b\A. x \ b - j) \ (P x \ Q x) \ (P(x + D) \ Q (x + D))" wenzelm@23465: "\\x.(\j\{1 .. D}. \b\A. x \ b - j)\ P x \ P(x + D) ; wenzelm@23465: \x.(\j\{1 .. D}. \b\A. x \ b - j)\ Q x \ Q(x + D)\ \ wenzelm@23465: \x.(\j\{1 .. D}. \b\A. x \ b - j)\ (P x \ Q x) \ (P(x + D) \ Q (x + D))" wenzelm@23465: "\D>0; t + 1\ A\ \ (\x.(\j\{1 .. D}. \b\A. x \ b - j)\ (x = t) \ (x + D = t))" wenzelm@23465: "\D>0 ; t \ A\ \(\(x::int).(\j\{1 .. D}. \b\A. x \ b - j)\ (x \ t) \ (x + D \ t))" wenzelm@23465: "\D>0; t\ A\ \(\(x::int). (\j\{1 .. D}. \b\A. x \ b - j)\ (x < t) \ (x + D < t))" wenzelm@23465: "\D>0; t + 1 \ A\ \ (\(x::int).(\j\{1 .. D}. \b\A. x \ b - j)\ (x \ t) \ (x + D \ t))" wenzelm@23465: "D>0 \(\(x::int).(\j\{1 .. D}. \b\A. x \ b - j)\ (x > t) \ (x + D > t))" wenzelm@23465: "D>0 \(\(x::int).(\j\{1 .. D}. \b\A. x \ b - j)\ (x \ t) \ (x + D \ t))" wenzelm@23465: "d dvd D \(\(x::int).(\j\{1 .. D}. \b\A. x \ b - j)\ (d dvd x+t) \ (d dvd (x + D) + t))" wenzelm@23465: "d dvd D \(\(x::int).(\j\{1 .. D}. \b\A. x \ b - j)\ (\d dvd x+t) \ (\ d dvd (x + D) + t))" wenzelm@23465: "\x.(\j\{1 .. D}. \b\A. x \ b - j) \ F \ F" wenzelm@23465: proof (blast, blast) wenzelm@23465: assume dp: "D > 0" and tA: "t + 1 \ A" wenzelm@23465: show "(\x.(\j\{1 .. D}. \b\A. x \ b - j)\ (x = t) \ (x + D = t))" wenzelm@23465: apply (rule allI, rule impI,erule ballE[where x="1"],erule ballE[where x="t + 1"]) wenzelm@23465: using dp tA by simp_all wenzelm@23465: next wenzelm@23465: assume dp: "D > 0" and tA: "t \ A" wenzelm@23465: show "(\x.(\j\{1 .. D}. \b\A. x \ b - j)\ (x \ t) \ (x + D \ t))" wenzelm@23465: apply (rule allI, rule impI,erule ballE[where x="D"],erule ballE[where x="t"]) wenzelm@23465: using dp tA by simp_all wenzelm@23465: next wenzelm@23465: assume dp: "D > 0" thus "(\x.(\j\{1 .. D}. \b\A. x \ b - j)\ (x > t) \ (x + D > t))" by arith wenzelm@23465: next wenzelm@23465: assume dp: "D > 0" thus "\x.(\j\{1 .. D}. \b\A. x \ b - j)\ (x \ t) \ (x + D \ t)" by arith wenzelm@23465: next wenzelm@23465: assume dp: "D > 0" and tA:"t \ A" wenzelm@23465: {fix x assume nob: "\j\{1 .. D}. \b\A. x \ b - j" and g: "x < t" and ng: "\ (x + D) < t" wenzelm@23465: hence "t - x \ D" and "1 \ t - x" by simp+ wenzelm@23465: hence "\j \ {1 .. D}. t - x = j" by auto nipkow@23477: hence "\j \ {1 .. D}. x = t - j" by (auto simp add: ring_simps) wenzelm@23465: with nob tA have "False" by simp} wenzelm@23465: thus "\x.(\j\{1 .. D}. \b\A. x \ b - j)\ (x < t) \ (x + D < t)" by blast wenzelm@23465: next wenzelm@23465: assume dp: "D > 0" and tA:"t + 1\ A" wenzelm@23465: {fix x assume nob: "\j\{1 .. D}. \b\A. x \ b - j" and g: "x \ t" and ng: "\ (x + D) \ t" nipkow@23477: hence "(t + 1) - x \ D" and "1 \ (t + 1) - x" by (simp_all add: ring_simps) wenzelm@23465: hence "\j \ {1 .. D}. (t + 1) - x = j" by auto nipkow@23477: hence "\j \ {1 .. D}. x = (t + 1) - j" by (auto simp add: ring_simps) wenzelm@23465: with nob tA have "False" by simp} wenzelm@23465: thus "\x.(\j\{1 .. D}. \b\A. x \ b - j)\ (x \ t) \ (x + D \ t)" by blast wenzelm@23465: next wenzelm@23465: assume d: "d dvd D" wenzelm@23465: {fix x assume H: "d dvd x + t" with d have "d dvd (x + D) + t" nipkow@23477: by (clarsimp simp add: dvd_def,rule_tac x= "ka + k" in exI,simp add: ring_simps)} wenzelm@23465: thus "\(x::int).(\j\{1 .. D}. \b\A. x \ b - j)\ (d dvd x+t) \ (d dvd (x + D) + t)" by simp wenzelm@23465: next wenzelm@23465: assume d: "d dvd D" wenzelm@23465: {fix x assume H: "\(d dvd x + t)" with d have "\d dvd (x + D) + t" nipkow@23477: by (clarsimp simp add: dvd_def,erule_tac x= "ka - k" in allE,simp add: ring_simps)} wenzelm@23465: thus "\(x::int).(\j\{1 .. D}. \b\A. x \ b - j)\ (\d dvd x+t) \ (\d dvd (x + D) + t)" by auto wenzelm@23465: qed blast wenzelm@23465: wenzelm@23465: subsection{* Cooper's Theorem @{text "-\"} and @{text "+\"} Version *} wenzelm@23465: wenzelm@23465: subsubsection{* First some trivial facts about periodic sets or predicates *} wenzelm@23465: lemma periodic_finite_ex: wenzelm@23465: assumes dpos: "(0::int) < d" and modd: "ALL x k. P x = P(x - k*d)" wenzelm@23465: shows "(EX x. P x) = (EX j : {1..d}. P j)" wenzelm@23465: (is "?LHS = ?RHS") wenzelm@23465: proof wenzelm@23465: assume ?LHS wenzelm@23465: then obtain x where P: "P x" .. wenzelm@23465: have "x mod d = x - (x div d)*d" by(simp add:zmod_zdiv_equality mult_ac eq_diff_eq) wenzelm@23465: hence Pmod: "P x = P(x mod d)" using modd by simp wenzelm@23465: show ?RHS wenzelm@23465: proof (cases) wenzelm@23465: assume "x mod d = 0" wenzelm@23465: hence "P 0" using P Pmod by simp wenzelm@23465: moreover have "P 0 = P(0 - (-1)*d)" using modd by blast wenzelm@23465: ultimately have "P d" by simp wenzelm@23465: moreover have "d : {1..d}" using dpos by(simp add:atLeastAtMost_iff) wenzelm@23465: ultimately show ?RHS .. wenzelm@23465: next wenzelm@23465: assume not0: "x mod d \ 0" wenzelm@23465: have "P(x mod d)" using dpos P Pmod by(simp add:pos_mod_sign pos_mod_bound) wenzelm@23465: moreover have "x mod d : {1..d}" wenzelm@23465: proof - wenzelm@23465: from dpos have "0 \ x mod d" by(rule pos_mod_sign) wenzelm@23465: moreover from dpos have "x mod d < d" by(rule pos_mod_bound) wenzelm@23465: ultimately show ?thesis using not0 by(simp add:atLeastAtMost_iff) wenzelm@23465: qed wenzelm@23465: ultimately show ?RHS .. wenzelm@23465: qed wenzelm@23465: qed auto wenzelm@23465: wenzelm@23465: subsubsection{* The @{text "-\"} Version*} wenzelm@23465: wenzelm@23465: lemma decr_lemma: "0 < (d::int) \ x - (abs(x-z)+1) * d < z" wenzelm@23465: by(induct rule: int_gr_induct,simp_all add:int_distrib) wenzelm@23465: wenzelm@23465: lemma incr_lemma: "0 < (d::int) \ z < x + (abs(x-z)+1) * d" wenzelm@23465: by(induct rule: int_gr_induct, simp_all add:int_distrib) wenzelm@23465: wenzelm@23465: theorem int_induct[case_names base step1 step2]: wenzelm@23465: assumes wenzelm@23465: base: "P(k::int)" and step1: "\i. \k \ i; P i\ \ P(i+1)" and wenzelm@23465: step2: "\i. \k \ i; P i\ \ P(i - 1)" wenzelm@23465: shows "P i" wenzelm@23465: proof - wenzelm@23465: have "i \ k \ i\ k" by arith wenzelm@23465: thus ?thesis using prems int_ge_induct[where P="P" and k="k" and i="i"] int_le_induct[where P="P" and k="k" and i="i"] by blast wenzelm@23465: qed wenzelm@23465: wenzelm@23465: lemma decr_mult_lemma: wenzelm@23465: assumes dpos: "(0::int) < d" and minus: "\x. P x \ P(x - d)" and knneg: "0 <= k" wenzelm@23465: shows "ALL x. P x \ P(x - k*d)" wenzelm@23465: using knneg wenzelm@23465: proof (induct rule:int_ge_induct) wenzelm@23465: case base thus ?case by simp wenzelm@23465: next wenzelm@23465: case (step i) wenzelm@23465: {fix x wenzelm@23465: have "P x \ P (x - i * d)" using step.hyps by blast wenzelm@23465: also have "\ \ P(x - (i + 1) * d)" using minus[THEN spec, of "x - i * d"] wenzelm@23465: by (simp add:int_distrib OrderedGroup.diff_diff_eq[symmetric]) wenzelm@23465: ultimately have "P x \ P(x - (i + 1) * d)" by blast} wenzelm@23465: thus ?case .. wenzelm@23465: qed wenzelm@23465: wenzelm@23465: lemma minusinfinity: wenzelm@23465: assumes dpos: "0 < d" and wenzelm@23465: P1eqP1: "ALL x k. P1 x = P1(x - k*d)" and ePeqP1: "EX z::int. ALL x. x < z \ (P x = P1 x)" wenzelm@23465: shows "(EX x. P1 x) \ (EX x. P x)" wenzelm@23465: proof wenzelm@23465: assume eP1: "EX x. P1 x" wenzelm@23465: then obtain x where P1: "P1 x" .. wenzelm@23465: from ePeqP1 obtain z where P1eqP: "ALL x. x < z \ (P x = P1 x)" .. wenzelm@23465: let ?w = "x - (abs(x-z)+1) * d" wenzelm@23465: from dpos have w: "?w < z" by(rule decr_lemma) wenzelm@23465: have "P1 x = P1 ?w" using P1eqP1 by blast wenzelm@23465: also have "\ = P(?w)" using w P1eqP by blast wenzelm@23465: finally have "P ?w" using P1 by blast wenzelm@23465: thus "EX x. P x" .. wenzelm@23465: qed wenzelm@23465: wenzelm@23465: lemma cpmi: wenzelm@23465: assumes dp: "0 < D" and p1:"\z. \ x< z. P x = P' x" wenzelm@23465: and nb:"\x.(\ j\ {1..D}. \(b::int) \ B. x \ b+j) --> P (x) --> P (x - D)" wenzelm@23465: and pd: "\ x k. P' x = P' (x-k*D)" wenzelm@23465: shows "(\x. P x) = ((\ j\ {1..D} . P' j) | (\ j \ {1..D}.\ b\ B. P (b+j)))" wenzelm@23465: (is "?L = (?R1 \ ?R2)") wenzelm@23465: proof- wenzelm@23465: {assume "?R2" hence "?L" by blast} wenzelm@23465: moreover wenzelm@23465: {assume H:"?R1" hence "?L" using minusinfinity[OF dp pd p1] periodic_finite_ex[OF dp pd] by simp} wenzelm@23465: moreover wenzelm@23465: { fix x wenzelm@23465: assume P: "P x" and H: "\ ?R2" wenzelm@23465: {fix y assume "\ (\j\{1..D}. \b\B. P (b + j))" and P: "P y" wenzelm@23465: hence "~(EX (j::int) : {1..D}. EX (b::int) : B. y = b+j)" by auto wenzelm@23465: with nb P have "P (y - D)" by auto } wenzelm@23465: hence "ALL x.~(EX (j::int) : {1..D}. EX (b::int) : B. P(b+j)) --> P (x) --> P (x - D)" by blast wenzelm@23465: with H P have th: " \x. P x \ P (x - D)" by auto wenzelm@23465: from p1 obtain z where z: "ALL x. x < z --> (P x = P' x)" by blast wenzelm@23465: let ?y = "x - (\x - z\ + 1)*D" wenzelm@23465: have zp: "0 <= (\x - z\ + 1)" by arith wenzelm@23465: from dp have yz: "?y < z" using decr_lemma[OF dp] by simp wenzelm@23465: from z[rule_format, OF yz] decr_mult_lemma[OF dp th zp, rule_format, OF P] have th2: " P' ?y" by auto wenzelm@23465: with periodic_finite_ex[OF dp pd] wenzelm@23465: have "?R1" by blast} wenzelm@23465: ultimately show ?thesis by blast wenzelm@23465: qed wenzelm@23465: wenzelm@23465: subsubsection {* The @{text "+\"} Version*} wenzelm@23465: wenzelm@23465: lemma plusinfinity: wenzelm@23465: assumes dpos: "(0::int) < d" and wenzelm@23465: P1eqP1: "\x k. P' x = P'(x - k*d)" and ePeqP1: "\ z. \ x>z. P x = P' x" wenzelm@23465: shows "(\ x. P' x) \ (\ x. P x)" wenzelm@23465: proof wenzelm@23465: assume eP1: "EX x. P' x" wenzelm@23465: then obtain x where P1: "P' x" .. wenzelm@23465: from ePeqP1 obtain z where P1eqP: "\x>z. P x = P' x" .. wenzelm@23465: let ?w' = "x + (abs(x-z)+1) * d" wenzelm@23465: let ?w = "x - (-(abs(x-z) + 1))*d" nipkow@23477: have ww'[simp]: "?w = ?w'" by (simp add: ring_simps) wenzelm@23465: from dpos have w: "?w > z" by(simp only: ww' incr_lemma) wenzelm@23465: hence "P' x = P' ?w" using P1eqP1 by blast wenzelm@23465: also have "\ = P(?w)" using w P1eqP by blast wenzelm@23465: finally have "P ?w" using P1 by blast wenzelm@23465: thus "EX x. P x" .. wenzelm@23465: qed wenzelm@23465: wenzelm@23465: lemma incr_mult_lemma: wenzelm@23465: assumes dpos: "(0::int) < d" and plus: "ALL x::int. P x \ P(x + d)" and knneg: "0 <= k" wenzelm@23465: shows "ALL x. P x \ P(x + k*d)" wenzelm@23465: using knneg wenzelm@23465: proof (induct rule:int_ge_induct) wenzelm@23465: case base thus ?case by simp wenzelm@23465: next wenzelm@23465: case (step i) wenzelm@23465: {fix x wenzelm@23465: have "P x \ P (x + i * d)" using step.hyps by blast wenzelm@23465: also have "\ \ P(x + (i + 1) * d)" using plus[THEN spec, of "x + i * d"] wenzelm@23465: by (simp add:int_distrib zadd_ac) wenzelm@23465: ultimately have "P x \ P(x + (i + 1) * d)" by blast} wenzelm@23465: thus ?case .. wenzelm@23465: qed wenzelm@23465: wenzelm@23465: lemma cppi: wenzelm@23465: assumes dp: "0 < D" and p1:"\z. \ x> z. P x = P' x" wenzelm@23465: and nb:"\x.(\ j\ {1..D}. \(b::int) \ A. x \ b - j) --> P (x) --> P (x + D)" wenzelm@23465: and pd: "\ x k. P' x= P' (x-k*D)" wenzelm@23465: shows "(\x. P x) = ((\ j\ {1..D} . P' j) | (\ j \ {1..D}.\ b\ A. P (b - j)))" (is "?L = (?R1 \ ?R2)") wenzelm@23465: proof- wenzelm@23465: {assume "?R2" hence "?L" by blast} wenzelm@23465: moreover wenzelm@23465: {assume H:"?R1" hence "?L" using plusinfinity[OF dp pd p1] periodic_finite_ex[OF dp pd] by simp} wenzelm@23465: moreover wenzelm@23465: { fix x wenzelm@23465: assume P: "P x" and H: "\ ?R2" wenzelm@23465: {fix y assume "\ (\j\{1..D}. \b\A. P (b - j))" and P: "P y" wenzelm@23465: hence "~(EX (j::int) : {1..D}. EX (b::int) : A. y = b - j)" by auto wenzelm@23465: with nb P have "P (y + D)" by auto } wenzelm@23465: hence "ALL x.~(EX (j::int) : {1..D}. EX (b::int) : A. P(b-j)) --> P (x) --> P (x + D)" by blast wenzelm@23465: with H P have th: " \x. P x \ P (x + D)" by auto wenzelm@23465: from p1 obtain z where z: "ALL x. x > z --> (P x = P' x)" by blast wenzelm@23465: let ?y = "x + (\x - z\ + 1)*D" wenzelm@23465: have zp: "0 <= (\x - z\ + 1)" by arith wenzelm@23465: from dp have yz: "?y > z" using incr_lemma[OF dp] by simp wenzelm@23465: from z[rule_format, OF yz] incr_mult_lemma[OF dp th zp, rule_format, OF P] have th2: " P' ?y" by auto wenzelm@23465: with periodic_finite_ex[OF dp pd] wenzelm@23465: have "?R1" by blast} wenzelm@23465: ultimately show ?thesis by blast wenzelm@23465: qed wenzelm@23465: wenzelm@23465: lemma simp_from_to: "{i..j::int} = (if j < i then {} else insert i {i+1..j})" wenzelm@23465: apply(simp add:atLeastAtMost_def atLeast_def atMost_def) wenzelm@23465: apply(fastsimp) wenzelm@23465: done wenzelm@23465: haftmann@24993: theorem unity_coeff_ex: "(\(x::'a::{semiring_0,Divides.div}). P (l * x)) \ (\x. l dvd (x + 0) \ P x)" wenzelm@23465: apply (rule eq_reflection[symmetric]) wenzelm@23465: apply (rule iffI) wenzelm@23465: defer wenzelm@23465: apply (erule exE) wenzelm@23465: apply (rule_tac x = "l * x" in exI) wenzelm@23465: apply (simp add: dvd_def) wenzelm@23465: apply (rule_tac x="x" in exI, simp) wenzelm@23465: apply (erule exE) wenzelm@23465: apply (erule conjE) wenzelm@23465: apply (erule dvdE) wenzelm@23465: apply (rule_tac x = k in exI) wenzelm@23465: apply simp wenzelm@23465: done wenzelm@23465: wenzelm@23465: lemma zdvd_mono: assumes not0: "(k::int) \ 0" wenzelm@23465: shows "((m::int) dvd t) \ (k*m dvd k*t)" wenzelm@23465: using not0 by (simp add: dvd_def) wenzelm@23465: wenzelm@23465: lemma uminus_dvd_conv: "(d dvd (t::int)) \ (-d dvd t)" "(d dvd (t::int)) \ (d dvd -t)" wenzelm@23465: by simp_all wenzelm@23465: text {* \bigskip Theorems for transforming predicates on nat to predicates on @{text int}*} wenzelm@23465: lemma all_nat: "(\x::nat. P x) = (\x::int. 0 <= x \ P (nat x))" wenzelm@23465: by (simp split add: split_nat) wenzelm@23465: wenzelm@23465: lemma ex_nat: "(\x::nat. P x) = (\x::int. 0 <= x \ P (nat x))" wenzelm@23465: apply (auto split add: split_nat) wenzelm@23465: apply (rule_tac x="int x" in exI, simp) wenzelm@23465: apply (rule_tac x = "nat x" in exI,erule_tac x = "nat x" in allE, simp) wenzelm@23465: done wenzelm@23465: wenzelm@23465: lemma zdiff_int_split: "P (int (x - y)) = wenzelm@23465: ((y \ x \ P (int x - int y)) \ (x < y \ P 0))" wenzelm@23465: by (case_tac "y \ x", simp_all add: zdiff_int) wenzelm@23465: wenzelm@23465: lemma number_of1: "(0::int) <= number_of n \ (0::int) <= number_of (n BIT b)" by simp wenzelm@23465: lemma number_of2: "(0::int) <= Numeral0" by simp wenzelm@23465: lemma Suc_plus1: "Suc n = n + 1" by simp wenzelm@23465: wenzelm@23465: text {* wenzelm@23465: \medskip Specific instances of congruence rules, to prevent wenzelm@23465: simplifier from looping. *} wenzelm@23465: wenzelm@23465: theorem imp_le_cong: "(0 <= x \ P = P') \ (0 <= (x::int) \ P) = (0 <= x \ P')" by simp wenzelm@23465: wenzelm@23465: theorem conj_le_cong: "(0 <= x \ P = P') \ (0 <= (x::int) \ P) = (0 <= x \ P')" wenzelm@23465: by (simp cong: conj_cong) wenzelm@23465: lemma int_eq_number_of_eq: wenzelm@23465: "(((number_of v)::int) = (number_of w)) = iszero ((number_of (v + (uminus w)))::int)" wenzelm@23465: by simp wenzelm@23465: wenzelm@23465: lemma mod_eq0_dvd_iff[presburger]: "(m::nat) mod n = 0 \ n dvd m" wenzelm@23465: unfolding dvd_eq_mod_eq_0[symmetric] .. wenzelm@23465: wenzelm@23465: lemma zmod_eq0_zdvd_iff[presburger]: "(m::int) mod n = 0 \ n dvd m" wenzelm@23465: unfolding zdvd_iff_zmod_eq_0[symmetric] .. wenzelm@23465: declare mod_1[presburger] wenzelm@23465: declare mod_0[presburger] wenzelm@23465: declare zmod_1[presburger] wenzelm@23465: declare zmod_zero[presburger] wenzelm@23465: declare zmod_self[presburger] wenzelm@23465: declare mod_self[presburger] wenzelm@23465: declare DIVISION_BY_ZERO_MOD[presburger] wenzelm@23465: declare nat_mod_div_trivial[presburger] wenzelm@23465: declare div_mod_equality2[presburger] wenzelm@23465: declare div_mod_equality[presburger] wenzelm@23465: declare mod_div_equality2[presburger] wenzelm@23465: declare mod_div_equality[presburger] wenzelm@23465: declare mod_mult_self1[presburger] wenzelm@23465: declare mod_mult_self2[presburger] wenzelm@23465: declare zdiv_zmod_equality2[presburger] wenzelm@23465: declare zdiv_zmod_equality[presburger] wenzelm@23465: declare mod2_Suc_Suc[presburger] wenzelm@23465: lemma [presburger]: "(a::int) div 0 = 0" and [presburger]: "a mod 0 = a" wenzelm@23465: using IntDiv.DIVISION_BY_ZERO by blast+ wenzelm@23465: wenzelm@23465: use "Tools/Qelim/cooper.ML" wenzelm@23465: oracle linzqe_oracle ("term") = Coopereif.cooper_oracle wenzelm@23465: wenzelm@23465: use "Tools/Qelim/presburger.ML" wenzelm@23465: wenzelm@24075: declaration {* fn _ => wenzelm@24075: arith_tactic_add wenzelm@24094: (mk_arith_tactic "presburger" (fn ctxt => fn i => fn st => wenzelm@23465: (warning "Trying Presburger arithmetic ..."; wenzelm@24094: Presburger.cooper_tac true [] [] ctxt i st))) wenzelm@23465: *} wenzelm@23465: wenzelm@23465: method_setup presburger = {* wenzelm@23465: let wenzelm@23465: fun keyword k = Scan.lift (Args.$$$ k -- Args.colon) >> K () wenzelm@23465: fun simple_keyword k = Scan.lift (Args.$$$ k) >> K () wenzelm@23465: val addN = "add" wenzelm@23465: val delN = "del" wenzelm@23465: val elimN = "elim" wenzelm@23465: val any_keyword = keyword addN || keyword delN || simple_keyword elimN wenzelm@23465: val thms = Scan.repeat (Scan.unless any_keyword Attrib.multi_thm) >> flat; wenzelm@23465: in wenzelm@23465: fn src => Method.syntax wenzelm@23465: ((Scan.optional (simple_keyword elimN >> K false) true) -- wenzelm@23465: (Scan.optional (keyword addN |-- thms) []) -- wenzelm@23465: (Scan.optional (keyword delN |-- thms) [])) src wenzelm@23465: #> (fn (((elim, add_ths), del_ths),ctxt) => wenzelm@23465: Method.SIMPLE_METHOD' (Presburger.cooper_tac elim add_ths del_ths ctxt)) wenzelm@23465: end wenzelm@23465: *} "Cooper's algorithm for Presburger arithmetic" wenzelm@23465: wenzelm@23465: lemma [presburger]: "m mod 2 = (1::nat) \ \ 2 dvd m " by presburger wenzelm@23465: lemma [presburger]: "m mod 2 = Suc 0 \ \ 2 dvd m " by presburger wenzelm@23465: lemma [presburger]: "m mod (Suc (Suc 0)) = (1::nat) \ \ 2 dvd m " by presburger wenzelm@23465: lemma [presburger]: "m mod (Suc (Suc 0)) = Suc 0 \ \ 2 dvd m " by presburger wenzelm@23465: lemma [presburger]: "m mod 2 = (1::int) \ \ 2 dvd m " by presburger wenzelm@23465: wenzelm@23465: haftmann@23685: lemma zdvd_period: haftmann@23685: fixes a d :: int haftmann@23685: assumes advdd: "a dvd d" haftmann@23685: shows "a dvd (x + t) \ a dvd ((x + c * d) + t)" haftmann@23685: proof- haftmann@23685: { haftmann@23685: fix x k haftmann@23685: from inf_period(3) [OF advdd, rule_format, where x=x and k="-k"] haftmann@23685: have "a dvd (x + t) \ a dvd (x + k * d + t)" by simp haftmann@23685: } haftmann@23685: hence "\x.\k. ((a::int) dvd (x + t)) = (a dvd (x+k*d + t))" by simp haftmann@23685: then show ?thesis by simp haftmann@23685: qed haftmann@23685: haftmann@23685: wenzelm@23465: subsection {* Code generator setup *} wenzelm@23465: wenzelm@23465: text {* wenzelm@23465: Presburger arithmetic is convenient to prove some wenzelm@23465: of the following code lemmas on integer numerals: wenzelm@23465: *} wenzelm@23465: wenzelm@23465: lemma eq_Pls_Pls: haftmann@25919: "Int.Pls = Int.Pls \ True" by presburger wenzelm@23465: wenzelm@23465: lemma eq_Pls_Min: haftmann@25919: "Int.Pls = Int.Min \ False" haftmann@25919: unfolding Pls_def Int.Min_def by presburger wenzelm@23465: wenzelm@23465: lemma eq_Pls_Bit0: haftmann@25919: "Int.Pls = Int.Bit k bit.B0 \ Int.Pls = k" wenzelm@23465: unfolding Pls_def Bit_def bit.cases by presburger wenzelm@23465: wenzelm@23465: lemma eq_Pls_Bit1: haftmann@25919: "Int.Pls = Int.Bit k bit.B1 \ False" wenzelm@23465: unfolding Pls_def Bit_def bit.cases by presburger wenzelm@23465: wenzelm@23465: lemma eq_Min_Pls: haftmann@25919: "Int.Min = Int.Pls \ False" haftmann@25919: unfolding Pls_def Int.Min_def by presburger wenzelm@23465: wenzelm@23465: lemma eq_Min_Min: haftmann@25919: "Int.Min = Int.Min \ True" by presburger wenzelm@23465: wenzelm@23465: lemma eq_Min_Bit0: haftmann@25919: "Int.Min = Int.Bit k bit.B0 \ False" haftmann@25919: unfolding Int.Min_def Bit_def bit.cases by presburger wenzelm@23465: wenzelm@23465: lemma eq_Min_Bit1: haftmann@25919: "Int.Min = Int.Bit k bit.B1 \ Int.Min = k" haftmann@25919: unfolding Int.Min_def Bit_def bit.cases by presburger wenzelm@23465: wenzelm@23465: lemma eq_Bit0_Pls: haftmann@25919: "Int.Bit k bit.B0 = Int.Pls \ Int.Pls = k" wenzelm@23465: unfolding Pls_def Bit_def bit.cases by presburger wenzelm@23465: wenzelm@23465: lemma eq_Bit1_Pls: haftmann@25919: "Int.Bit k bit.B1 = Int.Pls \ False" wenzelm@23465: unfolding Pls_def Bit_def bit.cases by presburger wenzelm@23465: wenzelm@23465: lemma eq_Bit0_Min: haftmann@25919: "Int.Bit k bit.B0 = Int.Min \ False" haftmann@25919: unfolding Int.Min_def Bit_def bit.cases by presburger wenzelm@23465: wenzelm@23465: lemma eq_Bit1_Min: haftmann@25919: "(Int.Bit k bit.B1) = Int.Min \ Int.Min = k" haftmann@25919: unfolding Int.Min_def Bit_def bit.cases by presburger wenzelm@23465: wenzelm@23465: lemma eq_Bit_Bit: haftmann@25919: "Int.Bit k1 v1 = Int.Bit k2 v2 \ wenzelm@23465: v1 = v2 \ k1 = k2" wenzelm@23465: unfolding Bit_def wenzelm@23465: apply (cases v1) wenzelm@23465: apply (cases v2) wenzelm@23465: apply auto wenzelm@23465: apply presburger wenzelm@23465: apply (cases v2) wenzelm@23465: apply auto wenzelm@23465: apply presburger wenzelm@23465: apply (cases v2) wenzelm@23465: apply auto wenzelm@23465: done wenzelm@23465: wenzelm@23465: lemma eq_number_of: wenzelm@23465: "(number_of k \ int) = number_of l \ k = l" wenzelm@23465: unfolding number_of_is_id .. wenzelm@23465: wenzelm@23465: wenzelm@23465: lemma less_eq_Pls_Pls: haftmann@25919: "Int.Pls \ Int.Pls \ True" by rule+ wenzelm@23465: wenzelm@23465: lemma less_eq_Pls_Min: haftmann@25919: "Int.Pls \ Int.Min \ False" haftmann@25919: unfolding Pls_def Int.Min_def by presburger wenzelm@23465: wenzelm@23465: lemma less_eq_Pls_Bit: haftmann@25919: "Int.Pls \ Int.Bit k v \ Int.Pls \ k" wenzelm@23465: unfolding Pls_def Bit_def by (cases v) auto wenzelm@23465: wenzelm@23465: lemma less_eq_Min_Pls: haftmann@25919: "Int.Min \ Int.Pls \ True" haftmann@25919: unfolding Pls_def Int.Min_def by presburger wenzelm@23465: wenzelm@23465: lemma less_eq_Min_Min: haftmann@25919: "Int.Min \ Int.Min \ True" by rule+ wenzelm@23465: wenzelm@23465: lemma less_eq_Min_Bit0: haftmann@25919: "Int.Min \ Int.Bit k bit.B0 \ Int.Min < k" haftmann@25919: unfolding Int.Min_def Bit_def by auto wenzelm@23465: wenzelm@23465: lemma less_eq_Min_Bit1: haftmann@25919: "Int.Min \ Int.Bit k bit.B1 \ Int.Min \ k" haftmann@25919: unfolding Int.Min_def Bit_def by auto wenzelm@23465: wenzelm@23465: lemma less_eq_Bit0_Pls: haftmann@25919: "Int.Bit k bit.B0 \ Int.Pls \ k \ Int.Pls" wenzelm@23465: unfolding Pls_def Bit_def by simp wenzelm@23465: wenzelm@23465: lemma less_eq_Bit1_Pls: haftmann@25919: "Int.Bit k bit.B1 \ Int.Pls \ k < Int.Pls" wenzelm@23465: unfolding Pls_def Bit_def by auto wenzelm@23465: wenzelm@23465: lemma less_eq_Bit_Min: haftmann@25919: "Int.Bit k v \ Int.Min \ k \ Int.Min" haftmann@25919: unfolding Int.Min_def Bit_def by (cases v) auto wenzelm@23465: wenzelm@23465: lemma less_eq_Bit0_Bit: haftmann@25919: "Int.Bit k1 bit.B0 \ Int.Bit k2 v \ k1 \ k2" wenzelm@23465: unfolding Bit_def bit.cases by (cases v) auto wenzelm@23465: wenzelm@23465: lemma less_eq_Bit_Bit1: haftmann@25919: "Int.Bit k1 v \ Int.Bit k2 bit.B1 \ k1 \ k2" wenzelm@23465: unfolding Bit_def bit.cases by (cases v) auto wenzelm@23465: wenzelm@23465: lemma less_eq_Bit1_Bit0: haftmann@25919: "Int.Bit k1 bit.B1 \ Int.Bit k2 bit.B0 \ k1 < k2" wenzelm@23465: unfolding Bit_def by (auto split: bit.split) wenzelm@23465: wenzelm@23465: lemma less_eq_number_of: wenzelm@23465: "(number_of k \ int) \ number_of l \ k \ l" wenzelm@23465: unfolding number_of_is_id .. wenzelm@23465: wenzelm@23465: wenzelm@23465: lemma less_Pls_Pls: haftmann@25919: "Int.Pls < Int.Pls \ False" by simp wenzelm@23465: wenzelm@23465: lemma less_Pls_Min: haftmann@25919: "Int.Pls < Int.Min \ False" haftmann@25919: unfolding Pls_def Int.Min_def by presburger wenzelm@23465: wenzelm@23465: lemma less_Pls_Bit0: haftmann@25919: "Int.Pls < Int.Bit k bit.B0 \ Int.Pls < k" wenzelm@23465: unfolding Pls_def Bit_def by auto wenzelm@23465: wenzelm@23465: lemma less_Pls_Bit1: haftmann@25919: "Int.Pls < Int.Bit k bit.B1 \ Int.Pls \ k" wenzelm@23465: unfolding Pls_def Bit_def by auto wenzelm@23465: wenzelm@23465: lemma less_Min_Pls: haftmann@25919: "Int.Min < Int.Pls \ True" haftmann@25919: unfolding Pls_def Int.Min_def by presburger wenzelm@23465: wenzelm@23465: lemma less_Min_Min: haftmann@25919: "Int.Min < Int.Min \ False" by simp wenzelm@23465: wenzelm@23465: lemma less_Min_Bit: haftmann@25919: "Int.Min < Int.Bit k v \ Int.Min < k" haftmann@25919: unfolding Int.Min_def Bit_def by (auto split: bit.split) wenzelm@23465: wenzelm@23465: lemma less_Bit_Pls: haftmann@25919: "Int.Bit k v < Int.Pls \ k < Int.Pls" wenzelm@23465: unfolding Pls_def Bit_def by (auto split: bit.split) wenzelm@23465: wenzelm@23465: lemma less_Bit0_Min: haftmann@25919: "Int.Bit k bit.B0 < Int.Min \ k \ Int.Min" haftmann@25919: unfolding Int.Min_def Bit_def by auto wenzelm@23465: wenzelm@23465: lemma less_Bit1_Min: haftmann@25919: "Int.Bit k bit.B1 < Int.Min \ k < Int.Min" haftmann@25919: unfolding Int.Min_def Bit_def by auto wenzelm@23465: wenzelm@23465: lemma less_Bit_Bit0: haftmann@25919: "Int.Bit k1 v < Int.Bit k2 bit.B0 \ k1 < k2" wenzelm@23465: unfolding Bit_def by (auto split: bit.split) wenzelm@23465: wenzelm@23465: lemma less_Bit1_Bit: haftmann@25919: "Int.Bit k1 bit.B1 < Int.Bit k2 v \ k1 < k2" wenzelm@23465: unfolding Bit_def by (auto split: bit.split) wenzelm@23465: wenzelm@23465: lemma less_Bit0_Bit1: haftmann@25919: "Int.Bit k1 bit.B0 < Int.Bit k2 bit.B1 \ k1 \ k2" wenzelm@23465: unfolding Bit_def bit.cases by arith wenzelm@23465: wenzelm@23465: lemma less_number_of: wenzelm@23465: "(number_of k \ int) < number_of l \ k < l" wenzelm@23465: unfolding number_of_is_id .. wenzelm@23465: wenzelm@23465: lemmas pred_succ_numeral_code [code func] = huffman@26075: pred_bin_simps succ_bin_simps wenzelm@23465: wenzelm@23465: lemmas plus_numeral_code [code func] = huffman@26075: add_bin_simps wenzelm@23465: arith_extra_simps(1) [where 'a = int] wenzelm@23465: wenzelm@23465: lemmas minus_numeral_code [code func] = huffman@26075: minus_bin_simps wenzelm@23465: arith_extra_simps(2) [where 'a = int] wenzelm@23465: arith_extra_simps(5) [where 'a = int] wenzelm@23465: wenzelm@23465: lemmas times_numeral_code [code func] = huffman@26075: mult_bin_simps wenzelm@23465: arith_extra_simps(4) [where 'a = int] wenzelm@23465: wenzelm@23465: lemmas eq_numeral_code [code func] = wenzelm@23465: eq_Pls_Pls eq_Pls_Min eq_Pls_Bit0 eq_Pls_Bit1 wenzelm@23465: eq_Min_Pls eq_Min_Min eq_Min_Bit0 eq_Min_Bit1 wenzelm@23465: eq_Bit0_Pls eq_Bit1_Pls eq_Bit0_Min eq_Bit1_Min eq_Bit_Bit wenzelm@23465: eq_number_of wenzelm@23465: wenzelm@23465: lemmas less_eq_numeral_code [code func] = less_eq_Pls_Pls less_eq_Pls_Min less_eq_Pls_Bit wenzelm@23465: less_eq_Min_Pls less_eq_Min_Min less_eq_Min_Bit0 less_eq_Min_Bit1 wenzelm@23465: less_eq_Bit0_Pls less_eq_Bit1_Pls less_eq_Bit_Min less_eq_Bit0_Bit less_eq_Bit_Bit1 less_eq_Bit1_Bit0 wenzelm@23465: less_eq_number_of wenzelm@23465: wenzelm@23465: lemmas less_numeral_code [code func] = less_Pls_Pls less_Pls_Min less_Pls_Bit0 wenzelm@23465: less_Pls_Bit1 less_Min_Pls less_Min_Min less_Min_Bit less_Bit_Pls wenzelm@23465: less_Bit0_Min less_Bit1_Min less_Bit_Bit0 less_Bit1_Bit less_Bit0_Bit1 wenzelm@23465: less_number_of wenzelm@23465: haftmann@25230: context ring_1 haftmann@25230: begin haftmann@23856: haftmann@23856: lemma of_int_num [code func]: haftmann@23856: "of_int k = (if k = 0 then 0 else if k < 0 then haftmann@23856: - of_int (- k) else let haftmann@23856: (l, m) = divAlg (k, 2); haftmann@23856: l' = of_int l haftmann@23856: in if m = 0 then l' + l' else l' + l' + 1)" haftmann@23856: proof - haftmann@23856: have aux1: "k mod (2\int) \ (0\int) \ haftmann@23856: of_int k = of_int (k div 2 * 2 + 1)" haftmann@23856: proof - haftmann@23856: assume "k mod 2 \ 0" haftmann@23856: then have "k mod 2 = 1" by arith haftmann@23856: moreover have "of_int k = of_int (k div 2 * 2 + k mod 2)" by simp haftmann@23856: ultimately show ?thesis by auto haftmann@23856: qed haftmann@23856: have aux2: "\x. of_int 2 * x = x + x" haftmann@23856: proof - haftmann@23856: fix x haftmann@23856: have int2: "(2::int) = 1 + 1" by arith haftmann@23856: show "of_int 2 * x = x + x" haftmann@23856: unfolding int2 of_int_add left_distrib by simp haftmann@23856: qed haftmann@23856: have aux3: "\x. x * of_int 2 = x + x" haftmann@23856: proof - haftmann@23856: fix x haftmann@23856: have int2: "(2::int) = 1 + 1" by arith haftmann@23856: show "x * of_int 2 = x + x" haftmann@23856: unfolding int2 of_int_add right_distrib by simp haftmann@23856: qed haftmann@23856: from aux1 show ?thesis by (auto simp add: divAlg_mod_div Let_def aux2 aux3) haftmann@23856: qed haftmann@23856: wenzelm@23465: end haftmann@25230: haftmann@25230: end