(* This file is a part of IsarMathLib - a library of formalized mathematics for Isabelle/Isar. Copyright (C) 2005, 2006, 2007 Slawomir Kolodynski 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 ‹More on order relations› theory Order_ZF_1 imports Order ZF1 begin text{*In @{text "Order_ZF"} we define some notions related to order relations based on the nonstrict orders ($\leq $ type). Some people however prefer to talk about these notions in terms of the strict order relation ($<$ type). This is the case for the standard Isabelle @{text "Order.thy"} and also for Metamath. In this theory file we repeat some developments from @{text "Order_ZF"} using the strict order relation as a basis.This is mostly useful for Metamath translation, but is also of some general interest. The names of theorems are copied from Metamath.*} subsection{*Definitions and basic properties*} text{*In this section we introduce some definitions taken from Metamath and relate them to the ones used by the standard Isabelle @{text "Order.thy"}. *} text{* The next definition is the strict version of the linear order. What we write as @{text "R Orders A"} is written $R Ord A$ in Metamath. *} definition StrictOrder (infix "Orders" 65) where "R Orders A ≡ ∀x y z. (x∈A ∧ y∈A ∧ z∈A) ⟶ (⟨x,y⟩ ∈ R ⟷ ¬(x=y ∨ ⟨y,x⟩ ∈ R)) ∧ (⟨x,y⟩ ∈ R ∧ ⟨y,z⟩ ∈ R ⟶ ⟨x,z⟩ ∈ R)" text{*The definition of supremum for a (strict) linear order.*} definition "Sup(B,A,R) ≡ ⋃ {x ∈ A. (∀y∈B. ⟨x,y⟩ ∉ R) ∧ (∀y∈A. ⟨y,x⟩ ∈ R ⟶ (∃z∈B. ⟨y,z⟩ ∈ R))}" text{*Definition of infimum for a linear order. It is defined in terms of supremum.*} definition "Infim(B,A,R) ≡ Sup(B,A,converse(R))" text{*If relation $R$ orders a set $A$, (in Metamath sense) then $R$ is irreflexive, transitive and linear therefore is a total order on $A$ (in Isabelle sense).*} lemma orders_imp_tot_ord: assumes A1: "R Orders A" shows "irrefl(A,R)" "trans[A](R)" "part_ord(A,R)" "linear(A,R)" "tot_ord(A,R)" proof - from A1 have I: "∀x y z. (x∈A ∧ y∈A ∧ z∈A) ⟶ (⟨x,y⟩ ∈ R ⟷ ¬(x=y ∨ ⟨y,x⟩ ∈ R)) ∧ (⟨x,y⟩ ∈ R ∧ ⟨y,z⟩ ∈ R ⟶ ⟨x,z⟩ ∈ R)" unfolding StrictOrder_def by simp then have "∀x∈A. ⟨x,x⟩ ∉ R" by blast then show "irrefl(A,R)" using irrefl_def by simp moreover from I have "∀x∈A. ∀y∈A. ∀z∈A. ⟨x,y⟩ ∈ R ⟶ ⟨y,z⟩ ∈ R ⟶ ⟨x,z⟩ ∈ R" by blast then show "trans[A](R)" unfolding trans_on_def by blast ultimately show "part_ord(A,R)" using part_ord_def by simp moreover from I have "∀x∈A. ∀y∈A. ⟨x,y⟩ ∈ R ∨ x=y ∨ ⟨y,x⟩ ∈ R" by blast then show "linear(A,R)" unfolding linear_def by blast ultimately show "tot_ord(A,R)" using tot_ord_def by simp qed text{*A converse of @{text "orders_imp_tot_ord"}. Together with that theorem this shows that Metamath's notion of an order relation is equivalent to Isabelles @{text "tot_ord"} predicate. *} lemma tot_ord_imp_orders: assumes A1: "tot_ord(A,R)" shows "R Orders A" proof - from A1 have I: "linear(A,R)" and II: "irrefl(A,R)" and III: "trans[A](R)" and IV: "part_ord(A,R)" using tot_ord_def part_ord_def by auto from IV have "asym(R ∩ A×A)" using part_ord_Imp_asym by simp then have V: "∀x y. ⟨x,y⟩ ∈ (R ∩ A×A) ⟶ ¬(⟨y,x⟩ ∈ (R ∩ A×A))" unfolding asym_def by blast from I have VI: "∀x∈A. ∀y∈A. ⟨x,y⟩ ∈ R ∨ x=y ∨ ⟨y,x⟩ ∈ R" unfolding linear_def by blast from III have VII: "∀x∈A. ∀y∈A. ∀z∈A. ⟨x,y⟩ ∈ R ⟶ ⟨y,z⟩ ∈ R ⟶ ⟨x,z⟩ ∈ R" unfolding trans_on_def by blast { fix x y z assume T: "x∈A" "y∈A" "z∈A" have "⟨x,y⟩ ∈ R ⟷ ¬(x=y ∨ ⟨y,x⟩ ∈ R)" proof assume A2: "⟨x,y⟩ ∈ R" with V T have "¬(⟨y,x⟩ ∈ R)" by blast moreover from II T A2 have "x≠y" using irrefl_def by auto ultimately show "¬(x=y ∨ ⟨y,x⟩ ∈ R)" by simp next assume "¬(x=y ∨ ⟨y,x⟩ ∈ R)" with VI T show "⟨x,y⟩ ∈ R" by auto qed moreover from VII T have "⟨x,y⟩ ∈ R ∧ ⟨y,z⟩ ∈ R ⟶ ⟨x,z⟩ ∈ R" by blast ultimately have "(⟨x,y⟩ ∈ R ⟷ ¬(x=y ∨ ⟨y,x⟩ ∈ R)) ∧ (⟨x,y⟩ ∈ R ∧ ⟨y,z⟩ ∈ R ⟶ ⟨x,z⟩ ∈ R)" by simp } then have "∀x y z. (x∈A ∧ y∈A ∧ z∈A) ⟶ (⟨x,y⟩ ∈ R ⟷ ¬(x=y ∨ ⟨y,x⟩ ∈ R)) ∧ (⟨x,y⟩ ∈ R ∧ ⟨y,z⟩ ∈ R ⟶ ⟨x,z⟩ ∈ R)" by auto then show "R Orders A" using StrictOrder_def by simp qed subsection{*Properties of (strict) total orders *} text{*In this section we discuss the properties of strict order relations. This continues the development contained in the standard Isabelle's @{text "Order.thy"} with a view towards using the theorems translated from Metamath.*} text{*A relation orders a set iff the converse relation orders a set. Going one way we can use the the lemma @{text "tot_od_converse"} from the standard Isabelle's @{text "Order.thy"}.The other way is a bit more complicated (note that in Isabelle for @{text "converse(converse(r)) = r"} one needs $r$ to consist of ordered pairs, which does not follow from the @{text "StrictOrder"} definition above).*} lemma cnvso: shows "R Orders A ⟷ converse(R) Orders A" proof let ?r = "converse(R)" assume "R Orders A" then have "tot_ord(A,?r)" using orders_imp_tot_ord tot_ord_converse by simp then show "?r Orders A" using tot_ord_imp_orders by simp next let ?r = "converse(R)" assume "?r Orders A" then have A2: "∀x y z. (x∈A ∧ y∈A ∧ z∈A) ⟶ (⟨x,y⟩ ∈ ?r ⟷ ¬(x=y ∨ ⟨y,x⟩ ∈ ?r)) ∧ (⟨x,y⟩ ∈ ?r ∧ ⟨y,z⟩ ∈ ?r ⟶ ⟨x,z⟩ ∈ ?r)" using StrictOrder_def by simp { fix x y z assume "x∈A ∧ y∈A ∧ z∈A" with A2 have I: "⟨y,x⟩ ∈ ?r ⟷ ¬(x=y ∨ ⟨x,y⟩ ∈ ?r)" and II: "⟨y,x⟩ ∈ ?r ∧ ⟨z,y⟩ ∈ ?r ⟶ ⟨z,x⟩ ∈ ?r" by auto from I have "⟨x,y⟩ ∈ R ⟷ ¬(x=y ∨ ⟨y,x⟩ ∈ R)" by auto moreover from II have "⟨x,y⟩ ∈ R ∧ ⟨y,z⟩ ∈ R ⟶ ⟨x,z⟩ ∈ R" by auto ultimately have "(⟨x,y⟩ ∈ R ⟷ ¬(x=y ∨ ⟨y,x⟩ ∈ R)) ∧ (⟨x,y⟩ ∈ R ∧ ⟨y,z⟩ ∈ R ⟶ ⟨x,z⟩ ∈ R)" by simp } then have "∀x y z. (x∈A ∧ y∈A ∧ z∈A) ⟶ (⟨x,y⟩ ∈ R ⟷ ¬(x=y ∨ ⟨y,x⟩ ∈ R)) ∧ (⟨x,y⟩ ∈ R ∧ ⟨y,z⟩ ∈ R ⟶ ⟨x,z⟩ ∈ R)" by auto then show "R Orders A" using StrictOrder_def by simp qed text{*Supremum is unique, if it exists.*} lemma supeu: assumes A1: "R Orders A" and A2: "x∈A" and A3: "∀y∈B. ⟨x,y⟩ ∉ R" and A4: "∀y∈A. ⟨y,x⟩ ∈ R ⟶ ( ∃z∈B. ⟨y,z⟩ ∈ R)" shows "∃!x. x∈A∧(∀y∈B. ⟨x,y⟩ ∉ R) ∧ (∀y∈A. ⟨y,x⟩ ∈ R ⟶ ( ∃z∈B. ⟨y,z⟩ ∈ R))" proof from A2 A3 A4 show "∃ x. x∈A∧(∀y∈B. ⟨x,y⟩ ∉ R) ∧ (∀y∈A. ⟨y,x⟩ ∈ R ⟶ ( ∃z∈B. ⟨y,z⟩ ∈ R))" by auto next fix x⇩_{1}x⇩_{2}assume A5: "x⇩_{1}∈ A ∧ (∀y∈B. ⟨x⇩_{1},y⟩ ∉ R) ∧ (∀y∈A. ⟨y,x⇩_{1}⟩ ∈ R ⟶ ( ∃z∈B. ⟨y,z⟩ ∈ R))" "x⇩_{2}∈ A ∧ (∀y∈B. ⟨x⇩_{2},y⟩ ∉ R) ∧ (∀y∈A. ⟨y,x⇩_{2}⟩ ∈ R ⟶ ( ∃z∈B. ⟨y,z⟩ ∈ R))" from A1 have "linear(A,R)" using orders_imp_tot_ord tot_ord_def by simp then have "∀x∈A. ∀y∈A. ⟨x,y⟩ ∈ R ∨ x=y ∨ ⟨y,x⟩ ∈ R" unfolding linear_def by blast with A5 have "⟨x⇩_{1},x⇩_{2}⟩ ∈ R ∨ x⇩_{1}=x⇩_{2}∨ ⟨x⇩_{2},x⇩_{1}⟩ ∈ R" by blast moreover { assume "⟨x⇩_{1},x⇩_{2}⟩ ∈ R" with A5 obtain z where "z∈B" and "⟨x⇩_{1},z⟩ ∈ R" by auto with A5 have False by auto } moreover { assume "⟨x⇩_{2},x⇩_{1}⟩ ∈ R" with A5 obtain z where "z∈B" and "⟨x⇩_{2},z⟩ ∈ R" by auto with A5 have False by auto } ultimately show "x⇩_{1}= x⇩_{2}" by auto qed text{*Supremum has expected properties if it exists.*} lemma sup_props: assumes A1: "R Orders A" and A2: "∃x∈A. (∀y∈B. ⟨x,y⟩ ∉ R) ∧ (∀y∈A. ⟨y,x⟩ ∈ R ⟶ ( ∃z∈B. ⟨y,z⟩ ∈ R))" shows "Sup(B,A,R) ∈ A" "∀y∈B. ⟨Sup(B,A,R),y⟩ ∉ R" "∀y∈A. ⟨y,Sup(B,A,R)⟩ ∈ R ⟶ ( ∃z∈B. ⟨y,z⟩ ∈ R )" proof - let ?S = "{x∈A. (∀y∈B. ⟨x,y⟩ ∉ R) ∧ (∀y∈A. ⟨y,x⟩ ∈ R ⟶ ( ∃z∈B. ⟨y,z⟩ ∈ R ) ) }" from A2 obtain x where "x∈A" and "(∀y∈B. ⟨x,y⟩ ∉ R)" and "∀y∈A. ⟨y,x⟩ ∈ R ⟶ ( ∃z∈B. ⟨y,z⟩ ∈ R)" by auto with A1 have I: "∃!x. x∈A∧(∀y∈B. ⟨x,y⟩ ∉ R) ∧ (∀y∈A. ⟨y,x⟩ ∈ R ⟶ ( ∃z∈B. ⟨y,z⟩ ∈ R))" using supeu by simp then have "( ⋃?S ) ∈ A" by (rule ZF1_1_L9) then show "Sup(B,A,R) ∈ A" using Sup_def by simp from I have II: "(∀y∈B. ⟨⋃?S ,y⟩ ∉ R) ∧ (∀y∈A. ⟨y,⋃?S⟩ ∈ R ⟶ ( ∃z∈B. ⟨y,z⟩ ∈ R))" by (rule ZF1_1_L9) hence "∀y∈B. ⟨⋃?S,y⟩ ∉ R" by blast moreover have III: "(⋃?S) = Sup(B,A,R)" using Sup_def by simp ultimately show "∀y∈B. ⟨Sup(B,A,R),y⟩ ∉ R" by simp from II have IV: "∀y∈A. ⟨y,⋃?S⟩ ∈ R ⟶ ( ∃z∈B. ⟨y,z⟩ ∈ R)" by blast { fix y assume A3: "y∈A" and "⟨y,Sup(B,A,R)⟩ ∈ R" with III have "⟨y,⋃?S⟩ ∈ R" by simp with IV A3 have "∃z∈B. ⟨y,z⟩ ∈ R" by blast } thus "∀y∈A. ⟨y,Sup(B,A,R)⟩ ∈ R ⟶ ( ∃z∈B. ⟨y,z⟩ ∈ R )" by simp qed text{*Elements greater or equal than any element of $B$ are greater or equal than supremum of $B$.*} lemma supnub: assumes A1: "R Orders A" and A2: "∃x∈A. (∀y∈B. ⟨x,y⟩ ∉ R) ∧ (∀y∈A. ⟨y,x⟩ ∈ R ⟶ ( ∃z∈B. ⟨y,z⟩ ∈ R))" and A3: "c ∈ A" and A4: "∀z∈B. ⟨c,z⟩ ∉ R" shows "⟨c, Sup(B,A,R)⟩ ∉ R" proof - from A1 A2 have "∀y∈A. ⟨y,Sup(B,A,R)⟩ ∈ R ⟶ ( ∃z∈B. ⟨y,z⟩ ∈ R )" by (rule sup_props) with A3 A4 show "⟨c, Sup(B,A,R)⟩ ∉ R" by auto qed end