Up to index of Isabelle/HOL
theory Dense_Linear_Order(* ID: $Id: Dense_Linear_Order.thy,v 1.16 2007/10/19 21:21:06 wenzelm Exp $ Author: Amine Chaieb, TU Muenchen *) header {* Dense linear order without endpoints and a quantifier elimination procedure in Ferrante and Rackoff style *} theory Dense_Linear_Order imports Finite_Set uses "Tools/Qelim/qelim.ML" "Tools/Qelim/langford_data.ML" "Tools/Qelim/ferrante_rackoff_data.ML" ("Tools/Qelim/langford.ML") ("Tools/Qelim/ferrante_rackoff.ML") begin setup Langford_Data.setup setup Ferrante_Rackoff_Data.setup context linorder begin lemma less_not_permute: "¬ (x < y ∧ y < x)" by (simp add: not_less linear) lemma gather_simps: shows "(∃x. (∀y ∈ L. y < x) ∧ (∀y ∈ U. x < y) ∧ x < u ∧ P x) <-> (∃x. (∀y ∈ L. y < x) ∧ (∀y ∈ (insert u U). x < y) ∧ P x)" and "(∃x. (∀y ∈ L. y < x) ∧ (∀y ∈ U. x < y) ∧ l < x ∧ P x) <-> (∃x. (∀y ∈ (insert l L). y < x) ∧ (∀y ∈ U. x < y) ∧ P x)" "(∃x. (∀y ∈ L. y < x) ∧ (∀y ∈ U. x < y) ∧ x < u) <-> (∃x. (∀y ∈ L. y < x) ∧ (∀y ∈ (insert u U). x < y))" and "(∃x. (∀y ∈ L. y < x) ∧ (∀y ∈ U. x < y) ∧ l < x) <-> (∃x. (∀y ∈ (insert l L). y < x) ∧ (∀y ∈ U. x < y))" by auto lemma gather_start: "(∃x. P x) ≡ (∃x. (∀y ∈ {}. y < x) ∧ (∀y∈ {}. x < y) ∧ P x)" by simp text{* Theorems for @{text "∃z. ∀x. x < z --> (P x <-> P-∞)"}*} lemma minf_lt: "∃z . ∀x. x < z --> (x < t <-> True)" by auto lemma minf_gt: "∃z . ∀x. x < z --> (t < x <-> False)" by (simp add: not_less) (rule exI[where x="t"], auto simp add: less_le) lemma minf_le: "∃z. ∀x. x < z --> (x ≤ t <-> True)" by (auto simp add: less_le) lemma minf_ge: "∃z. ∀x. x < z --> (t ≤ x <-> False)" by (auto simp add: less_le not_less not_le) lemma minf_eq: "∃z. ∀x. x < z --> (x = t <-> False)" by auto lemma minf_neq: "∃z. ∀x. x < z --> (x ≠ t <-> True)" by auto lemma minf_P: "∃z. ∀x. x < z --> (P <-> P)" by blast text{* Theorems for @{text "∃z. ∀x. x < z --> (P x <-> P+∞)"}*} lemma pinf_gt: "∃z . ∀x. z < x --> (t < x <-> True)" by auto lemma pinf_lt: "∃z . ∀x. z < x --> (x < t <-> False)" by (simp add: not_less) (rule exI[where x="t"], auto simp add: less_le) lemma pinf_ge: "∃z. ∀x. z < x --> (t ≤ x <-> True)" by (auto simp add: less_le) lemma pinf_le: "∃z. ∀x. z < x --> (x ≤ t <-> False)" by (auto simp add: less_le not_less not_le) lemma pinf_eq: "∃z. ∀x. z < x --> (x = t <-> False)" by auto lemma pinf_neq: "∃z. ∀x. z < x --> (x ≠ t <-> True)" by auto lemma pinf_P: "∃z. ∀x. z < x --> (P <-> P)" by blast lemma nmi_lt: "t ∈ U ==> ∀x. ¬True ∧ x < t --> (∃ u∈ U. u ≤ x)" by auto lemma nmi_gt: "t ∈ U ==> ∀x. ¬False ∧ t < x --> (∃ u∈ U. u ≤ x)" by (auto simp add: le_less) lemma nmi_le: "t ∈ U ==> ∀x. ¬True ∧ x≤ t --> (∃ u∈ U. u ≤ x)" by auto lemma nmi_ge: "t ∈ U ==> ∀x. ¬False ∧ t≤ x --> (∃ u∈ U. u ≤ x)" by auto lemma nmi_eq: "t ∈ U ==> ∀x. ¬False ∧ x = t --> (∃ u∈ U. u ≤ x)" by auto lemma nmi_neq: "t ∈ U ==>∀x. ¬True ∧ x ≠ t --> (∃ u∈ U. u ≤ x)" by auto lemma nmi_P: "∀ x. ~P ∧ P --> (∃ u∈ U. u ≤ x)" by auto lemma nmi_conj: "[|∀x. ¬P1' ∧ P1 x --> (∃ u∈ U. u ≤ x) ; ∀x. ¬P2' ∧ P2 x --> (∃ u∈ U. u ≤ x)|] ==> ∀x. ¬(P1' ∧ P2') ∧ (P1 x ∧ P2 x) --> (∃ u∈ U. u ≤ x)" by auto lemma nmi_disj: "[|∀x. ¬P1' ∧ P1 x --> (∃ u∈ U. u ≤ x) ; ∀x. ¬P2' ∧ P2 x --> (∃ u∈ U. u ≤ x)|] ==> ∀x. ¬(P1' ∨ P2') ∧ (P1 x ∨ P2 x) --> (∃ u∈ U. u ≤ x)" by auto lemma npi_lt: "t ∈ U ==> ∀x. ¬False ∧ x < t --> (∃ u∈ U. x ≤ u)" by (auto simp add: le_less) lemma npi_gt: "t ∈ U ==> ∀x. ¬True ∧ t < x --> (∃ u∈ U. x ≤ u)" by auto lemma npi_le: "t ∈ U ==> ∀x. ¬False ∧ x ≤ t --> (∃ u∈ U. x ≤ u)" by auto lemma npi_ge: "t ∈ U ==> ∀x. ¬True ∧ t ≤ x --> (∃ u∈ U. x ≤ u)" by auto lemma npi_eq: "t ∈ U ==> ∀x. ¬False ∧ x = t --> (∃ u∈ U. x ≤ u)" by auto lemma npi_neq: "t ∈ U ==> ∀x. ¬True ∧ x ≠ t --> (∃ u∈ U. x ≤ u )" by auto lemma npi_P: "∀ x. ~P ∧ P --> (∃ u∈ U. x ≤ u)" by auto lemma npi_conj: "[|∀x. ¬P1' ∧ P1 x --> (∃ u∈ U. x ≤ u) ; ∀x. ¬P2' ∧ P2 x --> (∃ u∈ U. x ≤ u)|] ==> ∀x. ¬(P1' ∧ P2') ∧ (P1 x ∧ P2 x) --> (∃ u∈ U. x ≤ u)" by auto lemma npi_disj: "[|∀x. ¬P1' ∧ P1 x --> (∃ u∈ U. x ≤ u) ; ∀x. ¬P2' ∧ P2 x --> (∃ u∈ U. x ≤ u)|] ==> ∀x. ¬(P1' ∨ P2') ∧ (P1 x ∨ P2 x) --> (∃ u∈ U. x ≤ u)" by auto lemma lin_dense_lt: "t ∈ U ==> ∀x l u. (∀ t. l < t ∧ t < u --> t ∉ U) ∧ l< x ∧ x < u ∧ x < t --> (∀ y. l < y ∧ y < u --> y < t)" proof(clarsimp) fix x l u y assume tU: "t ∈ U" and noU: "∀t. l < t ∧ t < u --> t ∉ U" and lx: "l < x" and xu: "x<u" and px: "x < t" and ly: "l<y" and yu:"y < u" from tU noU ly yu have tny: "t≠y" by auto {assume H: "t < y" from less_trans[OF lx px] less_trans[OF H yu] have "l < t ∧ t < u" by simp with tU noU have "False" by auto} hence "¬ t < y" by auto hence "y ≤ t" by (simp add: not_less) thus "y < t" using tny by (simp add: less_le) qed lemma lin_dense_gt: "t ∈ U ==> ∀x l u. (∀ t. l < t ∧ t< u --> t ∉ U) ∧ l < x ∧ x < u ∧ t < x --> (∀ y. l < y ∧ y < u --> t < y)" proof(clarsimp) fix x l u y assume tU: "t ∈ U" and noU: "∀t. l < t ∧ t < u --> t ∉ U" and lx: "l < x" and xu: "x<u" and px: "t < x" and ly: "l<y" and yu:"y < u" from tU noU ly yu have tny: "t≠y" by auto {assume H: "y< t" from less_trans[OF ly H] less_trans[OF px xu] have "l < t ∧ t < u" by simp with tU noU have "False" by auto} hence "¬ y<t" by auto hence "t ≤ y" by (auto simp add: not_less) thus "t < y" using tny by (simp add:less_le) qed lemma lin_dense_le: "t ∈ U ==> ∀x l u. (∀ t. l < t ∧ t< u --> t ∉ U) ∧ l< x ∧ x < u ∧ x ≤ t --> (∀ y. l < y ∧ y < u --> y≤ t)" proof(clarsimp) fix x l u y assume tU: "t ∈ U" and noU: "∀t. l < t ∧ t < u --> t ∉ U" and lx: "l < x" and xu: "x<u" and px: "x ≤ t" and ly: "l<y" and yu:"y < u" from tU noU ly yu have tny: "t≠y" by auto {assume H: "t < y" from less_le_trans[OF lx px] less_trans[OF H yu] have "l < t ∧ t < u" by simp with tU noU have "False" by auto} hence "¬ t < y" by auto thus "y ≤ t" by (simp add: not_less) qed lemma lin_dense_ge: "t ∈ U ==> ∀x l u. (∀ t. l < t ∧ t< u --> t ∉ U) ∧ l< x ∧ x < u ∧ t ≤ x --> (∀ y. l < y ∧ y < u --> t ≤ y)" proof(clarsimp) fix x l u y assume tU: "t ∈ U" and noU: "∀t. l < t ∧ t < u --> t ∉ U" and lx: "l < x" and xu: "x<u" and px: "t ≤ x" and ly: "l<y" and yu:"y < u" from tU noU ly yu have tny: "t≠y" by auto {assume H: "y< t" from less_trans[OF ly H] le_less_trans[OF px xu] have "l < t ∧ t < u" by simp with tU noU have "False" by auto} hence "¬ y<t" by auto thus "t ≤ y" by (simp add: not_less) qed lemma lin_dense_eq: "t ∈ U ==> ∀x l u. (∀ t. l < t ∧ t< u --> t ∉ U) ∧ l< x ∧ x < u ∧ x = t --> (∀ y. l < y ∧ y < u --> y= t)" by auto lemma lin_dense_neq: "t ∈ U ==> ∀x l u. (∀ t. l < t ∧ t< u --> t ∉ U) ∧ l< x ∧ x < u ∧ x ≠ t --> (∀ y. l < y ∧ y < u --> y≠ t)" by auto lemma lin_dense_P: "∀x l u. (∀ t. l < t ∧ t< u --> t ∉ U) ∧ l< x ∧ x < u ∧ P --> (∀ y. l < y ∧ y < u --> P)" by auto lemma lin_dense_conj: "[|∀x l u. (∀ t. l < t ∧ t< u --> t ∉ U) ∧ l< x ∧ x < u ∧ P1 x --> (∀ y. l < y ∧ y < u --> P1 y) ; ∀x l u. (∀ t. l < t ∧ t< u --> t ∉ U) ∧ l< x ∧ x < u ∧ P2 x --> (∀ y. l < y ∧ y < u --> P2 y)|] ==> ∀x l u. (∀ t. l < t ∧ t< u --> t ∉ U) ∧ l< x ∧ x < u ∧ (P1 x ∧ P2 x) --> (∀ y. l < y ∧ y < u --> (P1 y ∧ P2 y))" by blast lemma lin_dense_disj: "[|∀x l u. (∀ t. l < t ∧ t< u --> t ∉ U) ∧ l< x ∧ x < u ∧ P1 x --> (∀ y. l < y ∧ y < u --> P1 y) ; ∀x l u. (∀ t. l < t ∧ t< u --> t ∉ U) ∧ l< x ∧ x < u ∧ P2 x --> (∀ y. l < y ∧ y < u --> P2 y)|] ==> ∀x l u. (∀ t. l < t ∧ t< u --> t ∉ U) ∧ l< x ∧ x < u ∧ (P1 x ∨ P2 x) --> (∀ y. l < y ∧ y < u --> (P1 y ∨ P2 y))" by blast lemma npmibnd: "[|∀x. ¬ MP ∧ P x --> (∃ u∈ U. u ≤ x); ∀x. ¬PP ∧ P x --> (∃ u∈ U. x ≤ u)|] ==> ∀x. ¬ MP ∧ ¬PP ∧ P x --> (∃ u∈ U. ∃ u' ∈ U. u ≤ x ∧ x ≤ u')" by auto lemma finite_set_intervals: assumes px: "P x" and lx: "l ≤ x" and xu: "x ≤ u" and linS: "l∈ S" and uinS: "u ∈ S" and fS:"finite S" and lS: "∀ x∈ S. l ≤ x" and Su: "∀ x∈ S. x ≤ u" shows "∃ a ∈ S. ∃ b ∈ S. (∀ y. a < y ∧ y < b --> y ∉ S) ∧ a ≤ x ∧ x ≤ b ∧ P x" proof- let ?Mx = "{y. y∈ S ∧ y ≤ x}" let ?xM = "{y. y∈ S ∧ x ≤ y}" let ?a = "Max ?Mx" let ?b = "Min ?xM" have MxS: "?Mx ⊆ S" by blast hence fMx: "finite ?Mx" using fS finite_subset by auto from lx linS have linMx: "l ∈ ?Mx" by blast hence Mxne: "?Mx ≠ {}" by blast have xMS: "?xM ⊆ S" by blast hence fxM: "finite ?xM" using fS finite_subset by auto from xu uinS have linxM: "u ∈ ?xM" by blast hence xMne: "?xM ≠ {}" by blast have ax:"?a ≤ x" using Mxne fMx by auto have xb:"x ≤ ?b" using xMne fxM by auto have "?a ∈ ?Mx" using Max_in[OF fMx Mxne] by simp hence ainS: "?a ∈ S" using MxS by blast have "?b ∈ ?xM" using Min_in[OF fxM xMne] by simp hence binS: "?b ∈ S" using xMS by blast have noy:"∀ y. ?a < y ∧ y < ?b --> y ∉ S" proof(clarsimp) fix y assume ay: "?a < y" and yb: "y < ?b" and yS: "y ∈ S" from yS have "y∈ ?Mx ∨ y∈ ?xM" by (auto simp add: linear) moreover {assume "y ∈ ?Mx" hence "y ≤ ?a" using Mxne fMx by auto with ay have "False" by (simp add: not_le[symmetric])} moreover {assume "y ∈ ?xM" hence "?b ≤ y" using xMne fxM by auto with yb have "False" by (simp add: not_le[symmetric])} ultimately show "False" by blast qed from ainS binS noy ax xb px show ?thesis by blast qed lemma finite_set_intervals2: assumes px: "P x" and lx: "l ≤ x" and xu: "x ≤ u" and linS: "l∈ S" and uinS: "u ∈ S" and fS:"finite S" and lS: "∀ x∈ S. l ≤ x" and Su: "∀ x∈ S. x ≤ u" shows "(∃ s∈ S. P s) ∨ (∃ a ∈ S. ∃ b ∈ S. (∀ y. a < y ∧ y < b --> y ∉ S) ∧ a < x ∧ x < b ∧ P x)" proof- from finite_set_intervals[where P="P", OF px lx xu linS uinS fS lS Su] obtain a and b where as: "a∈ S" and bs: "b∈ S" and noS:"∀y. a < y ∧ y < b --> y ∉ S" and axb: "a ≤ x ∧ x ≤ b ∧ P x" by auto from axb have "x= a ∨ x= b ∨ (a < x ∧ x < b)" by (auto simp add: le_less) thus ?thesis using px as bs noS by blast qed end section {* The classical QE after Langford for dense linear orders *} context dense_linear_order begin lemma dlo_qe_bnds: assumes ne: "L ≠ {}" and neU: "U ≠ {}" and fL: "finite L" and fU: "finite U" shows "(∃x. (∀y ∈ L. y < x) ∧ (∀y ∈ U. x < y)) ≡ (∀ l ∈ L. ∀u ∈ U. l < u)" proof (simp only: atomize_eq, rule iffI) assume H: "∃x. (∀y∈L. y < x) ∧ (∀y∈U. x < y)" then obtain x where xL: "∀y∈L. y < x" and xU: "∀y∈U. x < y" by blast {fix l u assume l: "l ∈ L" and u: "u ∈ U" have "l < x" using xL l by blast also have "x < u" using xU u by blast finally (less_trans) have "l < u" .} thus "∀l∈L. ∀u∈U. l < u" by blast next assume H: "∀l∈L. ∀u∈U. l < u" let ?ML = "Max L" let ?MU = "Min U" from fL ne have th1: "?ML ∈ L" and th1': "∀l∈L. l ≤ ?ML" by auto from fU neU have th2: "?MU ∈ U" and th2': "∀u∈U. ?MU ≤ u" by auto from th1 th2 H have "?ML < ?MU" by auto with dense obtain w where th3: "?ML < w" and th4: "w < ?MU" by blast from th3 th1' have "∀l ∈ L. l < w" by auto moreover from th4 th2' have "∀u ∈ U. w < u" by auto ultimately show "∃x. (∀y∈L. y < x) ∧ (∀y∈U. x < y)" by auto qed lemma dlo_qe_noub: assumes ne: "L ≠ {}" and fL: "finite L" shows "(∃x. (∀y ∈ L. y < x) ∧ (∀y ∈ {}. x < y)) ≡ True" proof(simp add: atomize_eq) from gt_ex[of "Max L"] obtain M where M: "Max L < M" by blast from ne fL have "∀x ∈ L. x ≤ Max L" by simp with M have "∀x∈L. x < M" by (auto intro: le_less_trans) thus "∃x. ∀y∈L. y < x" by blast qed lemma dlo_qe_nolb: assumes ne: "U ≠ {}" and fU: "finite U" shows "(∃x. (∀y ∈ {}. y < x) ∧ (∀y ∈ U. x < y)) ≡ True" proof(simp add: atomize_eq) from lt_ex[of "Min U"] obtain M where M: "M < Min U" by blast from ne fU have "∀x ∈ U. Min U ≤ x" by simp with M have "∀x∈U. M < x" by (auto intro: less_le_trans) thus "∃x. ∀y∈U. x < y" by blast qed lemma exists_neq: "∃(x::'a). x ≠ t" "∃(x::'a). t ≠ x" using gt_ex[of t] by auto lemmas dlo_simps = order_refl less_irrefl not_less not_le exists_neq le_less neq_iff linear less_not_permute lemma axiom: "dense_linear_order (op ≤) (op <)" by fact lemma atoms: includes meta_term_syntax shows "TERM (less :: 'a => _)" and "TERM (less_eq :: 'a => _)" and "TERM (op = :: 'a => _)" . declare axiom[langford qe: dlo_qe_bnds dlo_qe_nolb dlo_qe_noub gather: gather_start gather_simps atoms: atoms] declare dlo_simps[langfordsimp] end (* FIXME: Move to HOL -- together with the conj_aci_rule in langford.ML *) lemma dnf: "(P & (Q | R)) = ((P&Q) | (P&R))" "((Q | R) & P) = ((Q&P) | (R&P))" by blast+ lemmas weak_dnf_simps = simp_thms dnf lemma nnf_simps: "(¬(P ∧ Q)) = (¬P ∨ ¬Q)" "(¬(P ∨ Q)) = (¬P ∧ ¬Q)" "(P --> Q) = (¬P ∨ Q)" "(P = Q) = ((P ∧ Q) ∨ (¬P ∧ ¬ Q))" "(¬ ¬(P)) = P" by blast+ lemma ex_distrib: "(∃x. P x ∨ Q x) <-> ((∃x. P x) ∨ (∃x. Q x))" by blast lemmas dnf_simps = weak_dnf_simps nnf_simps ex_distrib use "Tools/Qelim/langford.ML" method_setup dlo = {* Method.ctxt_args (Method.SIMPLE_METHOD' o LangfordQE.dlo_tac) *} "Langford's algorithm for quantifier elimination in dense linear orders" section {* Contructive dense linear orders yield QE for linear arithmetic over ordered Fields -- see @{text "Arith_Tools.thy"} *} text {* Linear order without upper bounds *} class linorder_no_ub = linorder + assumes gt_ex: "∃y. x < y" begin lemma ge_ex: "∃y. x ≤ y" using gt_ex by auto text {* Theorems for @{text "∃z. ∀x. z < x --> (P x <-> P+∞)"} *} lemma pinf_conj: assumes ex1: "∃z1. ∀x. z1 < x --> (P1 x <-> P1')" and ex2: "∃z2. ∀x. z2 < x --> (P2 x <-> P2')" shows "∃z. ∀x. z < x --> ((P1 x ∧ P2 x) <-> (P1' ∧ P2'))" proof- from ex1 ex2 obtain z1 and z2 where z1: "∀x. z1 < x --> (P1 x <-> P1')" and z2: "∀x. z2 < x --> (P2 x <-> P2')" by blast from gt_ex obtain z where z:"max z1 z2 < z" by blast from z have zz1: "z1 < z" and zz2: "z2 < z" by simp_all {fix x assume H: "z < x" from less_trans[OF zz1 H] less_trans[OF zz2 H] have "(P1 x ∧ P2 x) <-> (P1' ∧ P2')" using z1 zz1 z2 zz2 by auto } thus ?thesis by blast qed lemma pinf_disj: assumes ex1: "∃z1. ∀x. z1 < x --> (P1 x <-> P1')" and ex2: "∃z2. ∀x. z2 < x --> (P2 x <-> P2')" shows "∃z. ∀x. z < x --> ((P1 x ∨ P2 x) <-> (P1' ∨ P2'))" proof- from ex1 ex2 obtain z1 and z2 where z1: "∀x. z1 < x --> (P1 x <-> P1')" and z2: "∀x. z2 < x --> (P2 x <-> P2')" by blast from gt_ex obtain z where z:"max z1 z2 < z" by blast from z have zz1: "z1 < z" and zz2: "z2 < z" by simp_all {fix x assume H: "z < x" from less_trans[OF zz1 H] less_trans[OF zz2 H] have "(P1 x ∨ P2 x) <-> (P1' ∨ P2')" using z1 zz1 z2 zz2 by auto } thus ?thesis by blast qed lemma pinf_ex: assumes ex:"∃z. ∀x. z < x --> (P x <-> P1)" and p1: P1 shows "∃ x. P x" proof- from ex obtain z where z: "∀x. z < x --> (P x <-> P1)" by blast from gt_ex obtain x where x: "z < x" by blast from z x p1 show ?thesis by blast qed end text {* Linear order without upper bounds *} class linorder_no_lb = linorder + assumes lt_ex: "∃y. y < x" begin lemma le_ex: "∃y. y ≤ x" using lt_ex by auto text {* Theorems for @{text "∃z. ∀x. x < z --> (P x <-> P-∞)"} *} lemma minf_conj: assumes ex1: "∃z1. ∀x. x < z1 --> (P1 x <-> P1')" and ex2: "∃z2. ∀x. x < z2 --> (P2 x <-> P2')" shows "∃z. ∀x. x < z --> ((P1 x ∧ P2 x) <-> (P1' ∧ P2'))" proof- from ex1 ex2 obtain z1 and z2 where z1: "∀x. x < z1 --> (P1 x <-> P1')"and z2: "∀x. x < z2 --> (P2 x <-> P2')" by blast from lt_ex obtain z where z:"z < min z1 z2" by blast from z have zz1: "z < z1" and zz2: "z < z2" by simp_all {fix x assume H: "x < z" from less_trans[OF H zz1] less_trans[OF H zz2] have "(P1 x ∧ P2 x) <-> (P1' ∧ P2')" using z1 zz1 z2 zz2 by auto } thus ?thesis by blast qed lemma minf_disj: assumes ex1: "∃z1. ∀x. x < z1 --> (P1 x <-> P1')" and ex2: "∃z2. ∀x. x < z2 --> (P2 x <-> P2')" shows "∃z. ∀x. x < z --> ((P1 x ∨ P2 x) <-> (P1' ∨ P2'))" proof- from ex1 ex2 obtain z1 and z2 where z1: "∀x. x < z1 --> (P1 x <-> P1')"and z2: "∀x. x < z2 --> (P2 x <-> P2')" by blast from lt_ex obtain z where z:"z < min z1 z2" by blast from z have zz1: "z < z1" and zz2: "z < z2" by simp_all {fix x assume H: "x < z" from less_trans[OF H zz1] less_trans[OF H zz2] have "(P1 x ∨ P2 x) <-> (P1' ∨ P2')" using z1 zz1 z2 zz2 by auto } thus ?thesis by blast qed lemma minf_ex: assumes ex:"∃z. ∀x. x < z --> (P x <-> P1)" and p1: P1 shows "∃ x. P x" proof- from ex obtain z where z: "∀x. x < z --> (P x <-> P1)" by blast from lt_ex obtain x where x: "x < z" by blast from z x p1 show ?thesis by blast qed end class constr_dense_linear_order = linorder_no_lb + linorder_no_ub + fixes between assumes between_less: "x < y ==> x < between x y ∧ between x y < y" and between_same: "between x x = x" begin subclass dense_linear_order apply unfold_locales using gt_ex lt_ex between_less by (auto, rule_tac x="between x y" in exI, simp) lemma rinf_U: assumes fU: "finite U" and lin_dense: "∀x l u. (∀ t. l < t ∧ t< u --> t ∉ U) ∧ l< x ∧ x < u ∧ P x --> (∀ y. l < y ∧ y < u --> P y )" and nmpiU: "∀x. ¬ MP ∧ ¬PP ∧ P x --> (∃ u∈ U. ∃ u' ∈ U. u ≤ x ∧ x ≤ u')" and nmi: "¬ MP" and npi: "¬ PP" and ex: "∃ x. P x" shows "∃ u∈ U. ∃ u' ∈ U. P (between u u')" proof- from ex obtain x where px: "P x" by blast from px nmi npi nmpiU have "∃ u∈ U. ∃ u' ∈ U. u ≤ x ∧ x ≤ u'" by auto then obtain u and u' where uU:"u∈ U" and uU': "u' ∈ U" and ux:"u ≤ x" and xu':"x ≤ u'" by auto from uU have Une: "U ≠ {}" by auto let ?l = "Min U" let ?u = "Max U" have linM: "?l ∈ U" using fU Une by simp have uinM: "?u ∈ U" using fU Une by simp have lM: "∀ t∈ U. ?l ≤ t" using Une fU by auto have Mu: "∀ t∈ U. t ≤ ?u" using Une fU by auto have th:"?l ≤ u" using uU Une lM by auto from order_trans[OF th ux] have lx: "?l ≤ x" . have th: "u' ≤ ?u" using uU' Une Mu by simp from order_trans[OF xu' th] have xu: "x ≤ ?u" . from finite_set_intervals2[where P="P",OF px lx xu linM uinM fU lM Mu] have "(∃ s∈ U. P s) ∨ (∃ t1∈ U. ∃ t2 ∈ U. (∀ y. t1 < y ∧ y < t2 --> y ∉ U) ∧ t1 < x ∧ x < t2 ∧ P x)" . moreover { fix u assume um: "u∈U" and pu: "P u" have "between u u = u" by (simp add: between_same) with um pu have "P (between u u)" by simp with um have ?thesis by blast} moreover{ assume "∃ t1∈ U. ∃ t2 ∈ U. (∀ y. t1 < y ∧ y < t2 --> y ∉ U) ∧ t1 < x ∧ x < t2 ∧ P x" then obtain t1 and t2 where t1M: "t1 ∈ U" and t2M: "t2∈ U" and noM: "∀ y. t1 < y ∧ y < t2 --> y ∉ U" and t1x: "t1 < x" and xt2: "x < t2" and px: "P x" by blast from less_trans[OF t1x xt2] have t1t2: "t1 < t2" . let ?u = "between t1 t2" from between_less t1t2 have t1lu: "t1 < ?u" and ut2: "?u < t2" by auto from lin_dense noM t1x xt2 px t1lu ut2 have "P ?u" by blast with t1M t2M have ?thesis by blast} ultimately show ?thesis by blast qed theorem fr_eq: assumes fU: "finite U" and lin_dense: "∀x l u. (∀ t. l < t ∧ t< u --> t ∉ U) ∧ l< x ∧ x < u ∧ P x --> (∀ y. l < y ∧ y < u --> P y )" and nmibnd: "∀x. ¬ MP ∧ P x --> (∃ u∈ U. u ≤ x)" and npibnd: "∀x. ¬PP ∧ P x --> (∃ u∈ U. x ≤ u)" and mi: "∃z. ∀x. x < z --> (P x = MP)" and pi: "∃z. ∀x. z < x --> (P x = PP)" shows "(∃ x. P x) ≡ (MP ∨ PP ∨ (∃ u ∈ U. ∃ u'∈ U. P (between u u')))" (is "_ ≡ (_ ∨ _ ∨ ?F)" is "?E ≡ ?D") proof- { assume px: "∃ x. P x" have "MP ∨ PP ∨ (¬ MP ∧ ¬ PP)" by blast moreover {assume "MP ∨ PP" hence "?D" by blast} moreover {assume nmi: "¬ MP" and npi: "¬ PP" from npmibnd[OF nmibnd npibnd] have nmpiU: "∀x. ¬ MP ∧ ¬PP ∧ P x --> (∃ u∈ U. ∃ u' ∈ U. u ≤ x ∧ x ≤ u')" . from rinf_U[OF fU lin_dense nmpiU nmi npi px] have "?D" by blast} ultimately have "?D" by blast} moreover { assume "?D" moreover {assume m:"MP" from minf_ex[OF mi m] have "?E" .} moreover {assume p: "PP" from pinf_ex[OF pi p] have "?E" . } moreover {assume f:"?F" hence "?E" by blast} ultimately have "?E" by blast} ultimately have "?E = ?D" by blast thus "?E ≡ ?D" by simp qed lemmas minf_thms = minf_conj minf_disj minf_eq minf_neq minf_lt minf_le minf_gt minf_ge minf_P lemmas pinf_thms = pinf_conj pinf_disj pinf_eq pinf_neq pinf_lt pinf_le pinf_gt pinf_ge pinf_P lemmas nmi_thms = nmi_conj nmi_disj nmi_eq nmi_neq nmi_lt nmi_le nmi_gt nmi_ge nmi_P lemmas npi_thms = npi_conj npi_disj npi_eq npi_neq npi_lt npi_le npi_gt npi_ge npi_P lemmas lin_dense_thms = lin_dense_conj lin_dense_disj lin_dense_eq lin_dense_neq lin_dense_lt lin_dense_le lin_dense_gt lin_dense_ge lin_dense_P lemma ferrack_axiom: "constr_dense_linear_order less_eq less between" by fact lemma atoms: includes meta_term_syntax shows "TERM (less :: 'a => _)" and "TERM (less_eq :: 'a => _)" and "TERM (op = :: 'a => _)" . declare ferrack_axiom [ferrack minf: minf_thms pinf: pinf_thms nmi: nmi_thms npi: npi_thms lindense: lin_dense_thms qe: fr_eq atoms: atoms] declaration {* let fun simps phi = map (Morphism.thm phi) [@{thm "not_less"}, @{thm "not_le"}] fun generic_whatis phi = let val [lt, le] = map (Morphism.term phi) [@{term "op <"}, @{term "op ≤"}] 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$z) => if term_of x aconv y then Ferrante_Rackoff_Data.NEq else Ferrante_Rackoff_Data.Nox | b$y$z => if Term.could_unify (b, lt) then 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 else if Term.could_unify (b, le) then 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 else Ferrante_Rackoff_Data.Nox | _ => Ferrante_Rackoff_Data.Nox in h end fun ss phi = HOL_ss addsimps (simps phi) in Ferrante_Rackoff_Data.funs @{thm "ferrack_axiom"} {isolate_conv = K (K (K Thm.reflexive)), whatis = generic_whatis, simpset = ss} end *} end use "Tools/Qelim/ferrante_rackoff.ML" method_setup ferrack = {* Method.ctxt_args (Method.SIMPLE_METHOD' o FerranteRackoff.dlo_tac) *} "Ferrante and Rackoff's algorithm for quantifier elimination in dense linear orders" end
lemma less_not_permute:
¬ (x < y ∧ y < x)
lemma gather_simps(1):
(∃x. (∀y∈L. y < x) ∧ (∀y∈U. x < y) ∧ x < u ∧ P x) =
(∃x. (∀y∈L. y < x) ∧ (∀y∈insert u U. x < y) ∧ P x)
and gather_simps(2-3):
(∃x. (∀y∈L. y < x) ∧ (∀y∈U. x < y) ∧ l < x ∧ P x) =
(∃x. (∀y∈insert l L. y < x) ∧ (∀y∈U. x < y) ∧ P x)
(∃x. (∀y∈L. y < x) ∧ (∀y∈U. x < y) ∧ x < u) =
(∃x. (∀y∈L. y < x) ∧ (∀y∈insert u U. x < y))
and gather_simps(4):
(∃x. (∀y∈L. y < x) ∧ (∀y∈U. x < y) ∧ l < x) =
(∃x. (∀y∈insert l L. y < x) ∧ (∀y∈U. x < y))
lemma gather_start:
∃x. P x == ∃x. (∀y∈{}. y < x) ∧ (∀y∈{}. x < y) ∧ P x
lemma minf_lt:
∃z. ∀x<z. (x < t) = True
lemma minf_gt:
∃z. ∀x<z. (t < x) = False
lemma minf_le:
∃z. ∀x<z. (x ≤ t) = True
lemma minf_ge:
∃z. ∀x<z. (t ≤ x) = False
lemma minf_eq:
∃z. ∀x<z. (x = t) = False
lemma minf_neq:
∃z. ∀x<z. (x ≠ t) = True
lemma minf_P:
∃z. ∀x<z. P = P
lemma pinf_gt:
∃z. ∀x>z. (t < x) = True
lemma pinf_lt:
∃z. ∀x>z. (x < t) = False
lemma pinf_ge:
∃z. ∀x>z. (t ≤ x) = True
lemma pinf_le:
∃z. ∀x>z. (x ≤ t) = False
lemma pinf_eq:
∃z. ∀x>z. (x = t) = False
lemma pinf_neq:
∃z. ∀x>z. (x ≠ t) = True
lemma pinf_P:
∃z. ∀x>z. P = P
lemma nmi_lt:
t ∈ U ==> ∀x. ¬ True ∧ x < t --> (∃u∈U. u ≤ x)
lemma nmi_gt:
t ∈ U ==> ∀x. ¬ False ∧ t < x --> (∃u∈U. u ≤ x)
lemma nmi_le:
t ∈ U ==> ∀x. ¬ True ∧ x ≤ t --> (∃u∈U. u ≤ x)
lemma nmi_ge:
t ∈ U ==> ∀x. ¬ False ∧ t ≤ x --> (∃u∈U. u ≤ x)
lemma nmi_eq:
t ∈ U ==> ∀x. ¬ False ∧ x = t --> (∃u∈U. u ≤ x)
lemma nmi_neq:
t ∈ U ==> ∀x. ¬ True ∧ x ≠ t --> (∃u∈U. u ≤ x)
lemma nmi_P:
∀x. ¬ P ∧ P --> (∃u∈U. u ≤ x)
lemma nmi_conj:
[| ∀x. ¬ P1' ∧ P1.0 x --> (∃u∈U. u ≤ x); ∀x. ¬ P2' ∧ P2.0 x --> (∃u∈U. u ≤ x) |]
==> ∀x. ¬ (P1' ∧ P2') ∧ P1.0 x ∧ P2.0 x --> (∃u∈U. u ≤ x)
lemma nmi_disj:
[| ∀x. ¬ P1' ∧ P1.0 x --> (∃u∈U. u ≤ x); ∀x. ¬ P2' ∧ P2.0 x --> (∃u∈U. u ≤ x) |]
==> ∀x. ¬ (P1' ∨ P2') ∧ (P1.0 x ∨ P2.0 x) --> (∃u∈U. u ≤ x)
lemma npi_lt:
t ∈ U ==> ∀x. ¬ False ∧ x < t --> (∃u∈U. x ≤ u)
lemma npi_gt:
t ∈ U ==> ∀x. ¬ True ∧ t < x --> (∃u∈U. x ≤ u)
lemma npi_le:
t ∈ U ==> ∀x. ¬ False ∧ x ≤ t --> (∃u∈U. x ≤ u)
lemma npi_ge:
t ∈ U ==> ∀x. ¬ True ∧ t ≤ x --> (∃u∈U. x ≤ u)
lemma npi_eq:
t ∈ U ==> ∀x. ¬ False ∧ x = t --> (∃u∈U. x ≤ u)
lemma npi_neq:
t ∈ U ==> ∀x. ¬ True ∧ x ≠ t --> (∃u∈U. x ≤ u)
lemma npi_P:
∀x. ¬ P ∧ P --> (∃u∈U. x ≤ u)
lemma npi_conj:
[| ∀x. ¬ P1' ∧ P1.0 x --> (∃u∈U. x ≤ u); ∀x. ¬ P2' ∧ P2.0 x --> (∃u∈U. x ≤ u) |]
==> ∀x. ¬ (P1' ∧ P2') ∧ P1.0 x ∧ P2.0 x --> (∃u∈U. x ≤ u)
lemma npi_disj:
[| ∀x. ¬ P1' ∧ P1.0 x --> (∃u∈U. x ≤ u); ∀x. ¬ P2' ∧ P2.0 x --> (∃u∈U. x ≤ u) |]
==> ∀x. ¬ (P1' ∨ P2') ∧ (P1.0 x ∨ P2.0 x) --> (∃u∈U. x ≤ u)
lemma lin_dense_lt:
t ∈ U
==> ∀x l u.
(∀t. l < t ∧ t < u --> t ∉ U) ∧ l < x ∧ x < u ∧ x < t -->
(∀y. l < y ∧ y < u --> y < t)
lemma lin_dense_gt:
t ∈ U
==> ∀x l u.
(∀t. l < t ∧ t < u --> t ∉ U) ∧ l < x ∧ x < u ∧ t < x -->
(∀y. l < y ∧ y < u --> t < y)
lemma lin_dense_le:
t ∈ U
==> ∀x l u.
(∀t. l < t ∧ t < u --> t ∉ U) ∧ l < x ∧ x < u ∧ x ≤ t -->
(∀y. l < y ∧ y < u --> y ≤ t)
lemma lin_dense_ge:
t ∈ U
==> ∀x l u.
(∀t. l < t ∧ t < u --> t ∉ U) ∧ l < x ∧ x < u ∧ t ≤ x -->
(∀y. l < y ∧ y < u --> t ≤ y)
lemma lin_dense_eq:
t ∈ U
==> ∀x l u.
(∀t. l < t ∧ t < u --> t ∉ U) ∧ l < x ∧ x < u ∧ x = t -->
(∀y. l < y ∧ y < u --> y = t)
lemma lin_dense_neq:
t ∈ U
==> ∀x l u.
(∀t. l < t ∧ t < u --> t ∉ U) ∧ l < x ∧ x < u ∧ x ≠ t -->
(∀y. l < y ∧ y < u --> y ≠ t)
lemma lin_dense_P:
∀x l u.
(∀t. l < t ∧ t < u --> t ∉ U) ∧ l < x ∧ x < u ∧ P -->
(∀y. l < y ∧ y < u --> P)
lemma lin_dense_conj:
[| ∀x l u.
(∀t. l < t ∧ t < u --> t ∉ U) ∧ l < x ∧ x < u ∧ P1.0 x -->
(∀y. l < y ∧ y < u --> P1.0 y);
∀x l u.
(∀t. l < t ∧ t < u --> t ∉ U) ∧ l < x ∧ x < u ∧ P2.0 x -->
(∀y. l < y ∧ y < u --> P2.0 y) |]
==> ∀x l u.
(∀t. l < t ∧ t < u --> t ∉ U) ∧ l < x ∧ x < u ∧ P1.0 x ∧ P2.0 x -->
(∀y. l < y ∧ y < u --> P1.0 y ∧ P2.0 y)
lemma lin_dense_disj:
[| ∀x l u.
(∀t. l < t ∧ t < u --> t ∉ U) ∧ l < x ∧ x < u ∧ P1.0 x -->
(∀y. l < y ∧ y < u --> P1.0 y);
∀x l u.
(∀t. l < t ∧ t < u --> t ∉ U) ∧ l < x ∧ x < u ∧ P2.0 x -->
(∀y. l < y ∧ y < u --> P2.0 y) |]
==> ∀x l u.
(∀t. l < t ∧ t < u --> t ∉ U) ∧ l < x ∧ x < u ∧ (P1.0 x ∨ P2.0 x) -->
(∀y. l < y ∧ y < u --> P1.0 y ∨ P2.0 y)
lemma npmibnd:
[| ∀x. ¬ MP ∧ P x --> (∃u∈U. u ≤ x); ∀x. ¬ PP ∧ P x --> (∃u∈U. x ≤ u) |]
==> ∀x. ¬ MP ∧ ¬ PP ∧ P x --> (∃u∈U. ∃u'∈U. u ≤ x ∧ x ≤ u')
lemma finite_set_intervals:
[| P x; l ≤ x; x ≤ u; l ∈ S; u ∈ S; finite S; ∀x∈S. l ≤ x; ∀x∈S. x ≤ u |]
==> ∃a∈S. ∃b∈S. (∀y. a < y ∧ y < b --> y ∉ S) ∧ a ≤ x ∧ x ≤ b ∧ P x
lemma finite_set_intervals2:
[| P x; l ≤ x; x ≤ u; l ∈ S; u ∈ S; finite S; ∀x∈S. l ≤ x; ∀x∈S. x ≤ u |]
==> (∃s∈S. P s) ∨
(∃a∈S. ∃b∈S. (∀y. a < y ∧ y < b --> y ∉ S) ∧ a < x ∧ x < b ∧ P x)
lemma dlo_qe_bnds:
[| L ≠ {}; U ≠ {}; finite L; finite U |]
==> ∃x. (∀y∈L. y < x) ∧ (∀y∈U. x < y) == ∀l∈L. ∀u∈U. l < u
lemma dlo_qe_noub:
[| L ≠ {}; finite L |] ==> ∃x. (∀y∈L. y < x) ∧ (∀y∈{}. x < y) == True
lemma dlo_qe_nolb:
[| U ≠ {}; finite U |] ==> ∃x. (∀y∈{}. y < x) ∧ (∀y∈U. x < y) == True
lemma exists_neq:
∃x. x ≠ t
∃x. t ≠ x
lemma dlo_simps:
x ≤ x
¬ x < x
(¬ x < y) = (y ≤ x)
(¬ x ≤ y) = (y < x)
∃x. x ≠ t
∃x. t ≠ x
(x ≤ y) = (x < y ∨ x = y)
(x ≠ y) = (x < y ∨ y < x)
x ≤ y ∨ y ≤ x
¬ (x < y ∧ y < x)
lemma axiom:
dense_linear_order op ≤ op <
lemma atoms(1):
TERM op <
and atoms(2):
TERM op ≤
and atoms(3):
TERM op =
lemma dnf:
(P ∧ (Q ∨ R)) = (P ∧ Q ∨ P ∧ R)
((Q ∨ R) ∧ P) = (Q ∧ P ∨ R ∧ P)
lemma weak_dnf_simps:
(¬ ¬ P) = P
((¬ P) = (¬ Q)) = (P = Q)
(P ≠ Q) = (P = (¬ Q))
(P ∨ ¬ P) = True
(¬ P ∨ P) = True
(x = x) = True
(¬ True) = False
(¬ False) = True
(¬ P) ≠ P
P ≠ (¬ P)
(True = P) = P
(P = True) = P
(False = P) = (¬ P)
(P = False) = (¬ P)
(True --> P) = P
(False --> P) = True
(P --> True) = True
(P --> P) = True
(P --> False) = (¬ P)
(P --> ¬ P) = (¬ P)
(P ∧ True) = P
(True ∧ P) = P
(P ∧ False) = False
(False ∧ P) = False
(P ∧ P) = P
(P ∧ P ∧ Q) = (P ∧ Q)
(P ∧ ¬ P) = False
(¬ P ∧ P) = False
(P ∨ True) = True
(True ∨ P) = True
(P ∨ False) = P
(False ∨ P) = P
(P ∨ P) = P
(P ∨ P ∨ Q) = (P ∨ Q)
(∀x. P) = P
(∃x. P) = P
∃x. x = t
∃x. t = x
(∃x. x = t ∧ P x) = P t
(∃x. t = x ∧ P x) = P t
(∀x. x = t --> P x) = P t
(∀x. t = x --> P x) = P t
(P ∧ (Q ∨ R)) = (P ∧ Q ∨ P ∧ R)
((Q ∨ R) ∧ P) = (Q ∧ P ∨ R ∧ P)
lemma nnf_simps:
(¬ (P ∧ Q)) = (¬ P ∨ ¬ Q)
(¬ (P ∨ Q)) = (¬ P ∧ ¬ Q)
(P --> Q) = (¬ P ∨ Q)
(P = Q) = (P ∧ Q ∨ ¬ P ∧ ¬ Q)
(¬ ¬ P) = P
lemma ex_distrib:
(∃x. P x ∨ Q x) = ((∃x. P x) ∨ (∃x. Q x))
lemma dnf_simps:
(¬ ¬ P) = P
((¬ P) = (¬ Q)) = (P = Q)
(P ≠ Q) = (P = (¬ Q))
(P ∨ ¬ P) = True
(¬ P ∨ P) = True
(x = x) = True
(¬ True) = False
(¬ False) = True
(¬ P) ≠ P
P ≠ (¬ P)
(True = P) = P
(P = True) = P
(False = P) = (¬ P)
(P = False) = (¬ P)
(True --> P) = P
(False --> P) = True
(P --> True) = True
(P --> P) = True
(P --> False) = (¬ P)
(P --> ¬ P) = (¬ P)
(P ∧ True) = P
(True ∧ P) = P
(P ∧ False) = False
(False ∧ P) = False
(P ∧ P) = P
(P ∧ P ∧ Q) = (P ∧ Q)
(P ∧ ¬ P) = False
(¬ P ∧ P) = False
(P ∨ True) = True
(True ∨ P) = True
(P ∨ False) = P
(False ∨ P) = P
(P ∨ P) = P
(P ∨ P ∨ Q) = (P ∨ Q)
(∀x. P) = P
(∃x. P) = P
∃x. x = t
∃x. t = x
(∃x. x = t ∧ P x) = P t
(∃x. t = x ∧ P x) = P t
(∀x. x = t --> P x) = P t
(∀x. t = x --> P x) = P t
(P ∧ (Q ∨ R)) = (P ∧ Q ∨ P ∧ R)
((Q ∨ R) ∧ P) = (Q ∧ P ∨ R ∧ P)
(¬ (P ∧ Q)) = (¬ P ∨ ¬ Q)
(¬ (P ∨ Q)) = (¬ P ∧ ¬ Q)
(P --> Q) = (¬ P ∨ Q)
(P = Q) = (P ∧ Q ∨ ¬ P ∧ ¬ Q)
(¬ ¬ P) = P
(∃x. P x ∨ Q x) = ((∃x. P x) ∨ (∃x. Q x))
lemma ge_ex:
∃y. x ≤ y
lemma pinf_conj:
[| ∃z1. ∀x>z1. P1.0 x = P1'; ∃z2. ∀x>z2. P2.0 x = P2' |]
==> ∃z. ∀x>z. (P1.0 x ∧ P2.0 x) = (P1' ∧ P2')
lemma pinf_disj:
[| ∃z1. ∀x>z1. P1.0 x = P1'; ∃z2. ∀x>z2. P2.0 x = P2' |]
==> ∃z. ∀x>z. (P1.0 x ∨ P2.0 x) = (P1' ∨ P2')
lemma pinf_ex:
[| ∃z. ∀x>z. P x = P1.0; P1.0 |] ==> ∃x. P x
lemma le_ex:
∃y. y ≤ x
lemma minf_conj:
[| ∃z1. ∀x<z1. P1.0 x = P1'; ∃z2. ∀x<z2. P2.0 x = P2' |]
==> ∃z. ∀x<z. (P1.0 x ∧ P2.0 x) = (P1' ∧ P2')
lemma minf_disj:
[| ∃z1. ∀x<z1. P1.0 x = P1'; ∃z2. ∀x<z2. P2.0 x = P2' |]
==> ∃z. ∀x<z. (P1.0 x ∨ P2.0 x) = (P1' ∨ P2')
lemma minf_ex:
[| ∃z. ∀x<z. P x = P1.0; P1.0 |] ==> ∃x. P x
lemma rinf_U:
[| finite U;
∀x l u.
(∀t. l < t ∧ t < u --> t ∉ U) ∧ l < x ∧ x < u ∧ P x -->
(∀y. l < y ∧ y < u --> P y);
∀x. ¬ MP ∧ ¬ PP ∧ P x --> (∃u∈U. ∃u'∈U. u ≤ x ∧ x ≤ u'); ¬ MP; ¬ PP;
∃x. P x |]
==> ∃u∈U. ∃u'∈U. P (between u u')
theorem fr_eq:
[| finite U;
∀x l u.
(∀t. l < t ∧ t < u --> t ∉ U) ∧ l < x ∧ x < u ∧ P x -->
(∀y. l < y ∧ y < u --> P y);
∀x. ¬ MP ∧ P x --> (∃u∈U. u ≤ x); ∀x. ¬ PP ∧ P x --> (∃u∈U. x ≤ u);
∃z. ∀x<z. P x = MP; ∃z. ∀x>z. P x = PP |]
==> ∃x. P x == MP ∨ PP ∨ (∃u∈U. ∃u'∈U. P (between u u'))
lemma minf_thms:
[| ∃z1. ∀x<z1. P1.0 x = P1'; ∃z2. ∀x<z2. P2.0 x = P2' |]
==> ∃z. ∀x<z. (P1.0 x ∧ P2.0 x) = (P1' ∧ P2')
[| ∃z1. ∀x<z1. P1.0 x = P1'; ∃z2. ∀x<z2. P2.0 x = P2' |]
==> ∃z. ∀x<z. (P1.0 x ∨ P2.0 x) = (P1' ∨ P2')
∃z. ∀x<z. (x = t) = False
∃z. ∀x<z. (x ≠ t) = True
∃z. ∀x<z. (x < t) = True
∃z. ∀x<z. (x ≤ t) = True
∃z. ∀x<z. (t < x) = False
∃z. ∀x<z. (t ≤ x) = False
∃z. ∀x<z. P = P
lemma pinf_thms:
[| ∃z1. ∀x>z1. P1.0 x = P1'; ∃z2. ∀x>z2. P2.0 x = P2' |]
==> ∃z. ∀x>z. (P1.0 x ∧ P2.0 x) = (P1' ∧ P2')
[| ∃z1. ∀x>z1. P1.0 x = P1'; ∃z2. ∀x>z2. P2.0 x = P2' |]
==> ∃z. ∀x>z. (P1.0 x ∨ P2.0 x) = (P1' ∨ P2')
∃z. ∀x>z. (x = t) = False
∃z. ∀x>z. (x ≠ t) = True
∃z. ∀x>z. (x < t) = False
∃z. ∀x>z. (x ≤ t) = False
∃z. ∀x>z. (t < x) = True
∃z. ∀x>z. (t ≤ x) = True
∃z. ∀x>z. P = P
lemma nmi_thms:
[| ∀x. ¬ P1' ∧ P1.0 x --> (∃u∈U. u ≤ x); ∀x. ¬ P2' ∧ P2.0 x --> (∃u∈U. u ≤ x) |]
==> ∀x. ¬ (P1' ∧ P2') ∧ P1.0 x ∧ P2.0 x --> (∃u∈U. u ≤ x)
[| ∀x. ¬ P1' ∧ P1.0 x --> (∃u∈U. u ≤ x); ∀x. ¬ P2' ∧ P2.0 x --> (∃u∈U. u ≤ x) |]
==> ∀x. ¬ (P1' ∨ P2') ∧ (P1.0 x ∨ P2.0 x) --> (∃u∈U. u ≤ x)
t ∈ U ==> ∀x. ¬ False ∧ x = t --> (∃u∈U. u ≤ x)
t ∈ U ==> ∀x. ¬ True ∧ x ≠ t --> (∃u∈U. u ≤ x)
t ∈ U ==> ∀x. ¬ True ∧ x < t --> (∃u∈U. u ≤ x)
t ∈ U ==> ∀x. ¬ True ∧ x ≤ t --> (∃u∈U. u ≤ x)
t ∈ U ==> ∀x. ¬ False ∧ t < x --> (∃u∈U. u ≤ x)
t ∈ U ==> ∀x. ¬ False ∧ t ≤ x --> (∃u∈U. u ≤ x)
∀x. ¬ P ∧ P --> (∃u∈U. u ≤ x)
lemma npi_thms:
[| ∀x. ¬ P1' ∧ P1.0 x --> (∃u∈U. x ≤ u); ∀x. ¬ P2' ∧ P2.0 x --> (∃u∈U. x ≤ u) |]
==> ∀x. ¬ (P1' ∧ P2') ∧ P1.0 x ∧ P2.0 x --> (∃u∈U. x ≤ u)
[| ∀x. ¬ P1' ∧ P1.0 x --> (∃u∈U. x ≤ u); ∀x. ¬ P2' ∧ P2.0 x --> (∃u∈U. x ≤ u) |]
==> ∀x. ¬ (P1' ∨ P2') ∧ (P1.0 x ∨ P2.0 x) --> (∃u∈U. x ≤ u)
t ∈ U ==> ∀x. ¬ False ∧ x = t --> (∃u∈U. x ≤ u)
t ∈ U ==> ∀x. ¬ True ∧ x ≠ t --> (∃u∈U. x ≤ u)
t ∈ U ==> ∀x. ¬ False ∧ x < t --> (∃u∈U. x ≤ u)
t ∈ U ==> ∀x. ¬ False ∧ x ≤ t --> (∃u∈U. x ≤ u)
t ∈ U ==> ∀x. ¬ True ∧ t < x --> (∃u∈U. x ≤ u)
t ∈ U ==> ∀x. ¬ True ∧ t ≤ x --> (∃u∈U. x ≤ u)
∀x. ¬ P ∧ P --> (∃u∈U. x ≤ u)
lemma lin_dense_thms:
[| ∀x l u.
(∀t. l < t ∧ t < u --> t ∉ U) ∧ l < x ∧ x < u ∧ P1.0 x -->
(∀y. l < y ∧ y < u --> P1.0 y);
∀x l u.
(∀t. l < t ∧ t < u --> t ∉ U) ∧ l < x ∧ x < u ∧ P2.0 x -->
(∀y. l < y ∧ y < u --> P2.0 y) |]
==> ∀x l u.
(∀t. l < t ∧ t < u --> t ∉ U) ∧ l < x ∧ x < u ∧ P1.0 x ∧ P2.0 x -->
(∀y. l < y ∧ y < u --> P1.0 y ∧ P2.0 y)
[| ∀x l u.
(∀t. l < t ∧ t < u --> t ∉ U) ∧ l < x ∧ x < u ∧ P1.0 x -->
(∀y. l < y ∧ y < u --> P1.0 y);
∀x l u.
(∀t. l < t ∧ t < u --> t ∉ U) ∧ l < x ∧ x < u ∧ P2.0 x -->
(∀y. l < y ∧ y < u --> P2.0 y) |]
==> ∀x l u.
(∀t. l < t ∧ t < u --> t ∉ U) ∧ l < x ∧ x < u ∧ (P1.0 x ∨ P2.0 x) -->
(∀y. l < y ∧ y < u --> P1.0 y ∨ P2.0 y)
t ∈ U
==> ∀x l u.
(∀t. l < t ∧ t < u --> t ∉ U) ∧ l < x ∧ x < u ∧ x = t -->
(∀y. l < y ∧ y < u --> y = t)
t ∈ U
==> ∀x l u.
(∀t. l < t ∧ t < u --> t ∉ U) ∧ l < x ∧ x < u ∧ x ≠ t -->
(∀y. l < y ∧ y < u --> y ≠ t)
t ∈ U
==> ∀x l u.
(∀t. l < t ∧ t < u --> t ∉ U) ∧ l < x ∧ x < u ∧ x < t -->
(∀y. l < y ∧ y < u --> y < t)
t ∈ U
==> ∀x l u.
(∀t. l < t ∧ t < u --> t ∉ U) ∧ l < x ∧ x < u ∧ x ≤ t -->
(∀y. l < y ∧ y < u --> y ≤ t)
t ∈ U
==> ∀x l u.
(∀t. l < t ∧ t < u --> t ∉ U) ∧ l < x ∧ x < u ∧ t < x -->
(∀y. l < y ∧ y < u --> t < y)
t ∈ U
==> ∀x l u.
(∀t. l < t ∧ t < u --> t ∉ U) ∧ l < x ∧ x < u ∧ t ≤ x -->
(∀y. l < y ∧ y < u --> t ≤ y)
∀x l u.
(∀t. l < t ∧ t < u --> t ∉ U) ∧ l < x ∧ x < u ∧ P -->
(∀y. l < y ∧ y < u --> P)
lemma ferrack_axiom:
constr_dense_linear_order op ≤ op < between
lemma atoms(1):
TERM op <
and atoms(2):
TERM op ≤
and atoms(3):
TERM op =