(* This file is a part of IsarMathLib - a library of formalized mathematics written for Isabelle/Isar. Copyright (C) 2022 Daniel de la Concepcion This program is free software; Redistribution and use in source and binary forms, with or without modification, are permitted provided that the following conditions are met: 1. Redistributions of source code must retain the above copyright notice, this list of conditions and the following disclaimer. 2. Redistributions in binary form must reproduce the above copyright notice, this list of conditions and the following disclaimer in the documentation and/or other materials provided with the distribution. 3. The name of the author may not be used to endorse or promote products derived from this software without specific prior written permission. THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES LOSS OF USE, DATA, OR PROFITS OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. *) section ‹Rings - Zariski Topology - Properties› theory Ring_Zariski_ZF_2 imports Ring_Zariski_ZF Topology_ZF_1 begin theorem (in ring0) zariski_t0: shows "{D(I). I∈ℐ}{is T⇩_{0}}" unfolding isT0_def proof- { fix x y assume ass:"x:Spec" "y:Spec" "x≠y" from ass(3) have "¬(x⊆y) ∨ ¬(y⊆x)" by auto then have "y∈D(x) ∨ x∈D(y)" using ass(1,2) unfolding Spec_def using ass(1,2) openBasic_def by auto then have "(x ∈ D(y) ∧ y ∉ D(y)) ∨ (y ∈ D(x) ∧ x ∉ D(x))" using ass(1,2) unfolding Spec_def using openBasic_def by auto then have "∃U∈{D(I). I∈ℐ}. (x ∈ U ∧ y ∉ U) ∨ (y ∈ U ∧ x ∉ U)" using ass(1,2) unfolding Spec_def by auto } then show "∀x y. x ∈ ⋃RepFun(ℐ, D) ∧ y ∈ ⋃RepFun(ℐ, D) ∧ x ≠ y ⟶ (∃U∈RepFun(ℐ, D). x ∈ U ∧ y ∉ U ∨ y ∈ U ∧ x ∉ U)" using total_spec by auto qed text‹Noetherian rings have compact Zariski topology› theorem (in ring0) zariski_compact: assumes "∀I∈ℐ. (I{is finitely generated})" shows "Spec {is compact in} {D(I). I∈ℐ}" unfolding IsCompact_def proof(safe) show "⋀x. x ∈ Spec ⟹ x ∈ ⋃RepFun(ℐ, D)" using total_spec by auto fix M assume M:"M ⊆ RepFun(ℐ, D)" "Spec ⊆ ⋃M" let ?J ="{J∈ℐ. D(J)∈M}" have m:"M = RepFun(?J,D)" using M(1) by auto then have mm:"⋃M = D(⊕⇩_{I}?J)" using union_open_basic[of ?J] by auto obtain T where T:"T∈FinPow(?J)" "(⊕⇩_{I}?J) = ⊕⇩_{I}T" using sum_ideals_noetherian[OF assms(1), of ?J] by blast from T(2) have "D(⊕⇩_{I}?J) = D(⊕⇩_{I}T)" by auto moreover have "T⊆ℐ" using T(1) unfolding FinPow_def by auto ultimately have "D(⊕⇩_{I}?J) = ⋃RepFun(T,D)" using union_open_basic[of T] by auto with mm have "⋃M = ⋃RepFun(T,D)" by auto then have "Spec ⊆ ⋃RepFun(T,D)" using M(2) by auto moreover from T(1) have "RepFun(T,D) ⊆ RepFun(?J,D)" unfolding FinPow_def by auto with m have "RepFun(T,D) ⊆ M" by auto moreover from T(1) have "Finite(RepFun(T,D))" unfolding FinPow_def using Finite_RepFun by auto ultimately show "∃N∈FinPow(M). Spec ⊆ ⋃N" unfolding FinPow_def by auto qed end