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theory Dense_Linear_Order(* Title : HOL/Decision_Procs/Dense_Linear_Order.thy 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 Main uses "~~/src/HOL/Tools/Qelim/langford_data.ML" "~~/src/HOL/Tools/Qelim/ferrante_rackoff_data.ML" ("~~/src/HOL/Tools/Qelim/langford.ML") ("~~/src/HOL/Tools/Qelim/ferrante_rackoff.ML") begin setup {* Langford_Data.setup #> Ferrante_Rackoff_Data.setup *} context linorder begin lemma less_not_permute[noatp]: "¬ (x < y ∧ y < x)" by (simp add: not_less linear) lemma gather_simps[noatp]: 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[noatp]: "(∃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[noatp]: "∃z . ∀x. x < z --> (x < t <-> True)" by auto lemma minf_gt[noatp]: "∃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[noatp]: "∃z. ∀x. x < z --> (x ≤ t <-> True)" by (auto simp add: less_le) lemma minf_ge[noatp]: "∃z. ∀x. x < z --> (t ≤ x <-> False)" by (auto simp add: less_le not_less not_le) lemma minf_eq[noatp]: "∃z. ∀x. x < z --> (x = t <-> False)" by auto lemma minf_neq[noatp]: "∃z. ∀x. x < z --> (x ≠ t <-> True)" by auto lemma minf_P[noatp]: "∃z. ∀x. x < z --> (P <-> P)" by blast text{* Theorems for @{text "∃z. ∀x. x < z --> (P x <-> P+∞)"}*} lemma pinf_gt[noatp]: "∃z . ∀x. z < x --> (t < x <-> True)" by auto lemma pinf_lt[noatp]: "∃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[noatp]: "∃z. ∀x. z < x --> (t ≤ x <-> True)" by (auto simp add: less_le) lemma pinf_le[noatp]: "∃z. ∀x. z < x --> (x ≤ t <-> False)" by (auto simp add: less_le not_less not_le) lemma pinf_eq[noatp]: "∃z. ∀x. z < x --> (x = t <-> False)" by auto lemma pinf_neq[noatp]: "∃z. ∀x. z < x --> (x ≠ t <-> True)" by auto lemma pinf_P[noatp]: "∃z. ∀x. z < x --> (P <-> P)" by blast lemma nmi_lt[noatp]: "t ∈ U ==> ∀x. ¬True ∧ x < t --> (∃ u∈ U. u ≤ x)" by auto lemma nmi_gt[noatp]: "t ∈ U ==> ∀x. ¬False ∧ t < x --> (∃ u∈ U. u ≤ x)" by (auto simp add: le_less) lemma nmi_le[noatp]: "t ∈ U ==> ∀x. ¬True ∧ x≤ t --> (∃ u∈ U. u ≤ x)" by auto lemma nmi_ge[noatp]: "t ∈ U ==> ∀x. ¬False ∧ t≤ x --> (∃ u∈ U. u ≤ x)" by auto lemma nmi_eq[noatp]: "t ∈ U ==> ∀x. ¬False ∧ x = t --> (∃ u∈ U. u ≤ x)" by auto lemma nmi_neq[noatp]: "t ∈ U ==>∀x. ¬True ∧ x ≠ t --> (∃ u∈ U. u ≤ x)" by auto lemma nmi_P[noatp]: "∀ x. ~P ∧ P --> (∃ u∈ U. u ≤ x)" by auto lemma nmi_conj[noatp]: "[|∀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[noatp]: "[|∀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[noatp]: "t ∈ U ==> ∀x. ¬False ∧ x < t --> (∃ u∈ U. x ≤ u)" by (auto simp add: le_less) lemma npi_gt[noatp]: "t ∈ U ==> ∀x. ¬True ∧ t < x --> (∃ u∈ U. x ≤ u)" by auto lemma npi_le[noatp]: "t ∈ U ==> ∀x. ¬False ∧ x ≤ t --> (∃ u∈ U. x ≤ u)" by auto lemma npi_ge[noatp]: "t ∈ U ==> ∀x. ¬True ∧ t ≤ x --> (∃ u∈ U. x ≤ u)" by auto lemma npi_eq[noatp]: "t ∈ U ==> ∀x. ¬False ∧ x = t --> (∃ u∈ U. x ≤ u)" by auto lemma npi_neq[noatp]: "t ∈ U ==> ∀x. ¬True ∧ x ≠ t --> (∃ u∈ U. x ≤ u )" by auto lemma npi_P[noatp]: "∀ x. ~P ∧ P --> (∃ u∈ U. x ≤ u)" by auto lemma npi_conj[noatp]: "[|∀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[noatp]: "[|∀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[noatp]: "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[noatp]: "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[noatp]: "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[noatp]: "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[noatp]: "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[noatp]: "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[noatp]: "∀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[noatp]: "[|∀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[noatp]: "[|∀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[noatp]: "[|∀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[noatp]: 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[noatp]: 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 interval_empty_iff: "{y. x < y ∧ y < z} = {} <-> ¬ x < z" by (auto dest: dense) lemma dlo_qe_bnds[noatp]: 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[noatp]: 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[noatp]: 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[noatp]: "∃(x::'a). x ≠ t" "∃(x::'a). t ≠ x" using gt_ex[of t] by auto lemmas dlo_simps[noatp] = order_refl less_irrefl not_less not_le exists_neq le_less neq_iff linear less_not_permute lemma axiom[noatp]: "dense_linear_order (op ≤) (op <)" by (rule dense_linear_order_axioms) lemma atoms[noatp]: 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[noatp]: "(P & (Q | R)) = ((P&Q) | (P&R))" "((Q | R) & P) = ((Q&P) | (R&P))" by blast+ lemmas weak_dnf_simps[noatp] = simp_thms dnf lemma nnf_simps[noatp]: "(¬(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[noatp]: "(∃x. P x ∨ Q x) <-> ((∃x. P x) ∨ (∃x. Q x))" by blast lemmas dnf_simps[noatp] = weak_dnf_simps nnf_simps ex_distrib use "~~/src/HOL/Tools/Qelim/langford.ML" method_setup dlo = {* Scan.succeed (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 *} text {* Linear order without upper bounds *} locale linorder_stupid_syntax = linorder begin notation less_eq ("op \<sqsubseteq>") and less_eq ("(_/ \<sqsubseteq> _)" [51, 51] 50) and less ("op \<sqsubset>") and less ("(_/ \<sqsubset> _)" [51, 51] 50) end locale linorder_no_ub = linorder_stupid_syntax + assumes gt_ex: "∃y. less x y" begin lemma ge_ex[noatp]: "∃y. x \<sqsubseteq> y" using gt_ex by auto text {* Theorems for @{text "∃z. ∀x. z \<sqsubset> x --> (P x <-> P+∞)"} *} lemma pinf_conj[noatp]: assumes ex1: "∃z1. ∀x. z1 \<sqsubset> x --> (P1 x <-> P1')" and ex2: "∃z2. ∀x. z2 \<sqsubset> x --> (P2 x <-> P2')" shows "∃z. ∀x. z \<sqsubset> x --> ((P1 x ∧ P2 x) <-> (P1' ∧ P2'))" proof- from ex1 ex2 obtain z1 and z2 where z1: "∀x. z1 \<sqsubset> x --> (P1 x <-> P1')" and z2: "∀x. z2 \<sqsubset> x --> (P2 x <-> P2')" by blast from gt_ex obtain z where z:"ord.max less_eq z1 z2 \<sqsubset> z" by blast from z have zz1: "z1 \<sqsubset> z" and zz2: "z2 \<sqsubset> z" by simp_all {fix x assume H: "z \<sqsubset> 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[noatp]: assumes ex1: "∃z1. ∀x. z1 \<sqsubset> x --> (P1 x <-> P1')" and ex2: "∃z2. ∀x. z2 \<sqsubset> x --> (P2 x <-> P2')" shows "∃z. ∀x. z \<sqsubset> x --> ((P1 x ∨ P2 x) <-> (P1' ∨ P2'))" proof- from ex1 ex2 obtain z1 and z2 where z1: "∀x. z1 \<sqsubset> x --> (P1 x <-> P1')" and z2: "∀x. z2 \<sqsubset> x --> (P2 x <-> P2')" by blast from gt_ex obtain z where z:"ord.max less_eq z1 z2 \<sqsubset> z" by blast from z have zz1: "z1 \<sqsubset> z" and zz2: "z2 \<sqsubset> z" by simp_all {fix x assume H: "z \<sqsubset> 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[noatp]: assumes ex:"∃z. ∀x. z \<sqsubset> x --> (P x <-> P1)" and p1: P1 shows "∃ x. P x" proof- from ex obtain z where z: "∀x. z \<sqsubset> x --> (P x <-> P1)" by blast from gt_ex obtain x where x: "z \<sqsubset> x" by blast from z x p1 show ?thesis by blast qed end text {* Linear order without upper bounds *} locale linorder_no_lb = linorder_stupid_syntax + assumes lt_ex: "∃y. less y x" begin lemma le_ex[noatp]: "∃y. y \<sqsubseteq> x" using lt_ex by auto text {* Theorems for @{text "∃z. ∀x. x \<sqsubset> z --> (P x <-> P-∞)"} *} lemma minf_conj[noatp]: assumes ex1: "∃z1. ∀x. x \<sqsubset> z1 --> (P1 x <-> P1')" and ex2: "∃z2. ∀x. x \<sqsubset> z2 --> (P2 x <-> P2')" shows "∃z. ∀x. x \<sqsubset> z --> ((P1 x ∧ P2 x) <-> (P1' ∧ P2'))" proof- from ex1 ex2 obtain z1 and z2 where z1: "∀x. x \<sqsubset> z1 --> (P1 x <-> P1')"and z2: "∀x. x \<sqsubset> z2 --> (P2 x <-> P2')" by blast from lt_ex obtain z where z:"z \<sqsubset> ord.min less_eq z1 z2" by blast from z have zz1: "z \<sqsubset> z1" and zz2: "z \<sqsubset> z2" by simp_all {fix x assume H: "x \<sqsubset> 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[noatp]: assumes ex1: "∃z1. ∀x. x \<sqsubset> z1 --> (P1 x <-> P1')" and ex2: "∃z2. ∀x. x \<sqsubset> z2 --> (P2 x <-> P2')" shows "∃z. ∀x. x \<sqsubset> z --> ((P1 x ∨ P2 x) <-> (P1' ∨ P2'))" proof- from ex1 ex2 obtain z1 and z2 where z1: "∀x. x \<sqsubset> z1 --> (P1 x <-> P1')"and z2: "∀x. x \<sqsubset> z2 --> (P2 x <-> P2')" by blast from lt_ex obtain z where z:"z \<sqsubset> ord.min less_eq z1 z2" by blast from z have zz1: "z \<sqsubset> z1" and zz2: "z \<sqsubset> z2" by simp_all {fix x assume H: "x \<sqsubset> 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[noatp]: assumes ex:"∃z. ∀x. x \<sqsubset> z --> (P x <-> P1)" and p1: P1 shows "∃ x. P x" proof- from ex obtain z where z: "∀x. x \<sqsubset> z --> (P x <-> P1)" by blast from lt_ex obtain x where x: "x \<sqsubset> z" by blast from z x p1 show ?thesis by blast qed end locale constr_dense_linear_order = linorder_no_lb + linorder_no_ub + fixes between assumes between_less: "less x y ==> less x (between x y) ∧ less (between x y) y" and between_same: "between x x = x" sublocale constr_dense_linear_order < dense_linear_order apply unfold_locales using gt_ex lt_ex between_less by (auto, rule_tac x="between x y" in exI, simp) context constr_dense_linear_order begin lemma rinf_U[noatp]: assumes fU: "finite U" and lin_dense: "∀x l u. (∀ t. l \<sqsubset> t ∧ t\<sqsubset> u --> t ∉ U) ∧ l\<sqsubset> x ∧ x \<sqsubset> u ∧ P x --> (∀ y. l \<sqsubset> y ∧ y \<sqsubset> u --> P y )" and nmpiU: "∀x. ¬ MP ∧ ¬PP ∧ P x --> (∃ u∈ U. ∃ u' ∈ U. u \<sqsubseteq> x ∧ x \<sqsubseteq> 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 \<sqsubseteq> x ∧ x \<sqsubseteq> u'" by auto then obtain u and u' where uU:"u∈ U" and uU': "u' ∈ U" and ux:"u \<sqsubseteq> x" and xu':"x \<sqsubseteq> u'" by auto from uU have Une: "U ≠ {}" by auto term "linorder.Min less_eq" let ?l = "linorder.Min less_eq U" let ?u = "linorder.Max less_eq U" have linM: "?l ∈ U" using fU Une by simp have uinM: "?u ∈ U" using fU Une by simp have lM: "∀ t∈ U. ?l \<sqsubseteq> t" using Une fU by auto have Mu: "∀ t∈ U. t \<sqsubseteq> ?u" using Une fU by auto have th:"?l \<sqsubseteq> u" using uU Une lM by auto from order_trans[OF th ux] have lx: "?l \<sqsubseteq> x" . have th: "u' \<sqsubseteq> ?u" using uU' Une Mu by simp from order_trans[OF xu' th] have xu: "x \<sqsubseteq> ?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 \<sqsubset> y ∧ y \<sqsubset> t2 --> y ∉ U) ∧ t1 \<sqsubset> x ∧ x \<sqsubset> 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 \<sqsubset> y ∧ y \<sqsubset> t2 --> y ∉ U) ∧ t1 \<sqsubset> x ∧ x \<sqsubset> t2 ∧ P x" then obtain t1 and t2 where t1M: "t1 ∈ U" and t2M: "t2∈ U" and noM: "∀ y. t1 \<sqsubset> y ∧ y \<sqsubset> t2 --> y ∉ U" and t1x: "t1 \<sqsubset> x" and xt2: "x \<sqsubset> t2" and px: "P x" by blast from less_trans[OF t1x xt2] have t1t2: "t1 \<sqsubset> t2" . let ?u = "between t1 t2" from between_less t1t2 have t1lu: "t1 \<sqsubset> ?u" and ut2: "?u \<sqsubset> 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[noatp]: assumes fU: "finite U" and lin_dense: "∀x l u. (∀ t. l \<sqsubset> t ∧ t\<sqsubset> u --> t ∉ U) ∧ l\<sqsubset> x ∧ x \<sqsubset> u ∧ P x --> (∀ y. l \<sqsubset> y ∧ y \<sqsubset> u --> P y )" and nmibnd: "∀x. ¬ MP ∧ P x --> (∃ u∈ U. u \<sqsubseteq> x)" and npibnd: "∀x. ¬PP ∧ P x --> (∃ u∈ U. x \<sqsubseteq> u)" and mi: "∃z. ∀x. x \<sqsubset> z --> (P x = MP)" and pi: "∃z. ∀x. z \<sqsubset> 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 \<sqsubseteq> x ∧ x \<sqsubseteq> 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[noatp] = minf_conj minf_disj minf_eq minf_neq minf_lt minf_le minf_gt minf_ge minf_P lemmas pinf_thms[noatp] = pinf_conj pinf_disj pinf_eq pinf_neq pinf_lt pinf_le pinf_gt pinf_ge pinf_P lemmas nmi_thms[noatp] = nmi_conj nmi_disj nmi_eq nmi_neq nmi_lt nmi_le nmi_gt nmi_ge nmi_P lemmas npi_thms[noatp] = npi_conj npi_disj npi_eq npi_neq npi_lt npi_le npi_gt npi_ge npi_P lemmas lin_dense_thms[noatp] = 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[noatp]: "constr_dense_linear_order less_eq less between" by (rule constr_dense_linear_order_axioms) lemma atoms[noatp]: 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 \<sqsubset>"}, @{term "op \<sqsubseteq>"}] 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 "~~/src/HOL/Tools/Qelim/ferrante_rackoff.ML" method_setup ferrack = {* Scan.succeed (SIMPLE_METHOD' o FerranteRackoff.dlo_tac) *} "Ferrante and Rackoff's algorithm for quantifier elimination in dense linear orders" 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] algebra_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] algebra_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] algebra_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] algebra_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] algebra_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] algebra_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] algebra_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] algebra_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] algebra_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)) | Const(@{const_name inverse}, _)$a => Rat.rat_of_quotient(1, HOLogic.dest_number a |> snd) | t => Rat.rat_of_int (snd (HOLogic.dest_number t)) fun mk_frac phi cT x = let val (a, b) = Rat.quotient_of_rat x in if b = 1 then Numeral.mk_cnumber cT a else Thm.capply (Thm.capply (Drule.cterm_rule (instantiate' [SOME cT] []) @{cpat "op /"}) (Numeral.mk_cnumber cT a)) (Numeral.mk_cnumber cT b) end fun whatis x ct = case term_of ct of Const(@{const_name "HOL.plus"}, _)$(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 *} lemma upper_bound_finite_set: assumes fS: "finite S" shows "∃(a::'a::linorder). ∀x ∈ S. f x ≤ a" proof(induct rule: finite_induct[OF fS]) case 1 thus ?case by simp next case (2 x F) from "2.hyps" obtain a where a:"∀x ∈F. f x ≤ a" by blast let ?a = "max a (f x)" have m: "a ≤ ?a" "f x ≤ ?a" by simp_all {fix y assume y: "y ∈ insert x F" {assume "y = x" hence "f y ≤ ?a" using m by simp} moreover {assume yF: "y∈ F" from a[rule_format, OF yF] m have "f y ≤ ?a" by (simp add: max_def)} ultimately have "f y ≤ ?a" using y by blast} then show ?case by blast qed lemma lower_bound_finite_set: assumes fS: "finite S" shows "∃(a::'a::linorder). ∀x ∈ S. f x ≥ a" proof(induct rule: finite_induct[OF fS]) case 1 thus ?case by simp next case (2 x F) from "2.hyps" obtain a where a:"∀x ∈F. f x ≥ a" by blast let ?a = "min a (f x)" have m: "a ≥ ?a" "f x ≥ ?a" by simp_all {fix y assume y: "y ∈ insert x F" {assume "y = x" hence "f y ≥ ?a" using m by simp} moreover {assume yF: "y∈ F" from a[rule_format, OF yF] m have "f y ≥ ?a" by (simp add: min_def)} ultimately have "f y ≥ ?a" using y by blast} then show ?case by blast qed lemma bound_finite_set: assumes f: "finite S" shows "∃a. ∀x ∈S. (f x:: 'a::linorder) ≤ a" proof- let ?F = "f ` S" from f have fF: "finite ?F" by simp let ?a = "Max ?F" {assume "S = {}" hence ?thesis by blast} moreover {assume Se: "S ≠ {}" hence Fe: "?F ≠ {}" by simp {fix x assume x: "x ∈ S" hence th0: "f x ∈ ?F" by simp hence "f x ≤ ?a" using Max_ge[OF fF th0] ..} hence ?thesis by blast} ultimately show ?thesis by blast qed end