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Tarski_Archimedes.thy
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Tarski_Archimedes.thy
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(* IsageoCoq2_R1
Tarski_Archimedes.thy
Version 2.2.0 IsaGeoCoq2_R1, Port part of GeoCoq 3.4.0
[X] equivalence Grad (function) \<longleftrightarrow> GradI (induction)
[X] local: smt \<longrightarrow> metis/meson
[x] angle_archimedes.v
Version 2.1.0 IsaGeoCoq2_R1, Port part of GeoCoq 3.4.0
Copyright (C) 2021-2023 Roland Coghetto roland.coghetto ( a t ) cafr-msa2p.be
License: LGPGL
History
Version 1.0.0: IsaGeoCoq
Port part of GeoCoq 3.4.0 (https://geocoq.github.io/GeoCoq/) in Isabelle/Hol (Isabelle2021)
Copyright (C) 2021 Roland Coghetto roland_coghetto (at) hotmail.com
License: LGPL
This library is free software; you can redistribute it and/or
modify it under the terms of the GNU Lesser General Public
License as published by the Free Software Foundation; either
version 2.1 of the License, or (at your option) any later version.
This library is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
Lesser General Public License for more details.
You should have received a copy of the GNU Lesser General Public
License along with this library; if not, write to the Free Software
Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA
*)
(*>*)
theory Tarski_Archimedes
imports
Tarski_Neutral
begin
(*>*)
context Tarski_neutral_dimensionless
begin
subsection "Graduation"
subsubsection "Définitions"
definition PreGrad :: "TPoint \<Rightarrow> TPoint \<Rightarrow> TPoint \<Rightarrow> TPoint \<Rightarrow> bool" where
"PreGrad A B C D \<equiv> (A \<noteq> B \<and> Bet A B C \<and> Bet A C D \<and> Cong A B C D)"
fun Sym :: "TPoint \<Rightarrow> TPoint \<Rightarrow> TPoint \<Rightarrow> TPoint" where
"Sym A B C = (if (A \<noteq> B \<and> Bet A B C) then
(SOME x::TPoint. PreGrad A B C x)
else
A)"
fun Gradn :: "[TPoint,TPoint] \<Rightarrow> nat \<Rightarrow>TPoint"where
"Gradn A B n = (if (A = B) then
A
else
(if (n = 0) then
A
else
(if (n = 1) then
B
else
(Sym A B (Gradn A B (n-1))))))"
definition Grad :: "[TPoint,TPoint,TPoint] \<Rightarrow> bool" where
"Grad A B C \<equiv> \<exists> n. (n \<noteq> 0) \<and> (C = Gradn A B n)"
definition Reach :: "[TPoint,TPoint,TPoint,TPoint] \<Rightarrow> bool" where
"Reach A B C D \<equiv> \<exists> B'. Grad A B B' \<and> C D Le A B'"
definition Grad2 :: "[TPoint,TPoint,TPoint,TPoint,TPoint,TPoint] \<Rightarrow> bool" where
"Grad2 A B C D E F \<equiv> \<exists> n. (n \<noteq> 0) \<and> (C = Gradn A B n) \<and> (F = Gradn D E n)"
fun SymR :: "TPoint \<Rightarrow> TPoint \<Rightarrow> TPoint" where
"SymR A B = (SOME x::TPoint. B Midpoint A x)"
fun GradExpn :: "TPoint \<Rightarrow> TPoint \<Rightarrow> nat \<Rightarrow> TPoint" where
"(GradExpn A B n) = (if (A = B) then
A
else
(if (n = 0) then
A
else
(if (n = 1) then
B
else
(SymR A (GradExpn A B (n-1))))))"
definition GradExp :: "TPoint \<Rightarrow> TPoint \<Rightarrow> TPoint \<Rightarrow> bool" where
"GradExp A B C \<equiv> \<exists> n. (n \<noteq> 0) \<and> C = GradExpn A B n"
definition GradExp2 :: "[TPoint,TPoint,TPoint,TPoint,TPoint,TPoint] \<Rightarrow> bool" where
"GradExp2 A B C D E F \<equiv> \<exists> n. (n \<noteq> 0) \<and> (C = GradExpn A B n) \<and> (F = GradExpn D E n)"
fun MidR :: "TPoint \<Rightarrow> TPoint \<Rightarrow> TPoint" where
"MidR A B = (SOME x. x Midpoint A B)"
(* Je peux encore reduire
(if (n = 1) then (MidR A B)
else (MidR A (GradExpInvn A B (n-1))))))
par
(if (n = 1) then (MidR A (GradExpInvn A B (n-1))))))"
car si n - 1 =0 alors = MidR A B
*)
fun GradExpInvn :: "TPoint \<Rightarrow> TPoint \<Rightarrow> nat \<Rightarrow> TPoint" where
"(GradExpInvn A B n) = (if (A = B) then
A
else
(if (n = 0) then
B
else
(if (n = 1) then
(MidR A B)
else
(MidR A (GradExpInvn A B (n-1))))))"
definition GradExpInv :: "TPoint \<Rightarrow> TPoint \<Rightarrow> TPoint \<Rightarrow> bool" where
"GradExpInv A B C \<equiv> \<exists> n. B = GradExpInvn A C n"
subsubsection "Continuity Axioms"
definition archimedes_axiom ::
"bool"
("ArchimedesAxiom") where
"archimedes_axiom \<equiv> \<forall> A B C D::TPoint.
A \<noteq> B \<longrightarrow> Reach A B C D"
definition greenberg_s_axiom ::
"bool"
("GreenBergsAxiom")
where
"greenberg_s_axiom \<equiv> \<forall> P Q R A B C.
\<not> Col A B C \<and> Acute A B C \<and> Q \<noteq> R \<and> Per P Q R \<longrightarrow>
(\<exists> S. P S Q LtA A B C \<and> Q Out S R)"
definition aristotle_s_axiom ::
"bool"
("AristotleAxiom") where
"aristotle_s_axiom \<equiv> \<forall> P Q A B C.
\<not> Col A B C \<and> Acute A B C \<longrightarrow>
(\<exists> X Y. B Out A X \<and> B Out C Y \<and> Per B X Y \<and> P Q Lt X Y)"
definition Axiom1:: "bool" where "Axiom1 \<equiv> \<forall> A B C D.
(\<exists> I. Col I A B \<and> Col I C D) \<or> \<not> (\<exists> I. Col I A B \<and> Col I C D)"
subsubsection "Propositions"
lemma PreGrad_lem1:
assumes "A \<noteq> B" and
"Bet A B C"
shows "\<exists> x. PreGrad A B C x"
by (meson PreGrad_def assms(1) assms(2) not_cong_3412 segment_construction)
lemma PreGrad_uniq:
assumes "PreGrad A B C x" and
"PreGrad A B C y"
shows "x = y"
by (metis (no_types, lifting) PreGrad_def assms(1) assms(2)
bet_neq12__neq between_cong_3 cong_inner_transitivity)
lemma Diff_Mid__PreGrad:
assumes "A \<noteq> B" and
"B Midpoint A C"
shows "PreGrad A B B C"
by (simp add: PreGrad_def assms(1) assms(2) between_trivial midpoint_bet midpoint_cong)
lemma Diff_Mid_Mid_PreGrad:
assumes "A \<noteq> B" and
"B Midpoint A C" and
"C Midpoint B D"
shows "PreGrad A B C D"
proof -
have "Bet A B C"
using Midpoint_def assms(2) by presburger
moreover have "Bet A C D"
using Midpoint_def assms(3) calculation is_midpoint_id outer_transitivity_between2 by blast
moreover have "Cong A B C D"
using assms(2) assms(3) cong_transitivity midpoint_cong by blast
ultimately show ?thesis
by (simp add: assms(1) PreGrad_def)
qed
lemma Sym_Diff__Diff:
assumes "Sym A B C = D" and
"A \<noteq> D"
shows "A \<noteq> B"
using assms(1) assms(2) by force
lemma Sym_Refl:
"Sym A A A = A"
by simp
lemma Diff_Mid__Sym:
assumes "A \<noteq> B" and
"B Midpoint A C"
shows "Sym A B B = C"
using someI_ex by (metis Diff_Mid__PreGrad Sym.elims PreGrad_uniq
assms(1) assms(2) between_trivial)
lemma Mid_Mid__Sym:
assumes "A \<noteq> B" and
"B Midpoint A C" and
"C Midpoint B D"
shows "Sym A B C = D"
proof -
have "PreGrad A B C D"
by (simp add: Diff_Mid_Mid_PreGrad assms(1) assms(2) assms(3))
thus ?thesis
using someI_ex assms(1)
by (metis PreGrad_uniq Sym.elims assms(2) midpoint_bet)
qed
lemma Sym_Bet__Bet_Bet:
assumes "Sym A B C = D" and
"A \<noteq> B" and
"Bet A B C"
shows "Bet A B D \<and> Bet A C D"
proof -
have "(SOME x::TPoint. PreGrad A B C x) = D"
using assms(1) assms(2) assms(3) by auto
hence "PreGrad A B C D"
by (metis PreGrad_lem1 assms(2) assms(3) someI2)
thus ?thesis
by (meson PreGrad_def between_exchange4)
qed
lemma Sym_Bet__Cong:
assumes "Sym A B C = D" and
"A \<noteq> B" and
"Bet A B C"
shows "Cong A B C D"
proof -
have "(SOME x::TPoint. PreGrad A B C x) = D"
using assms(1) assms(2) assms(3) by auto
hence "PreGrad A B C D"
by (metis PreGrad_lem1 assms(2) assms(3) someI2)
thus ?thesis
by (meson PreGrad_def between_exchange4)
qed
lemma LemSym_aux:
assumes "A \<noteq> B" and
"Bet A B C" and
"Bet A C D" and
"Cong A B C D"
shows "Sym A B C = D"
proof -
have "PreGrad A B C D"
using PreGrad_def assms(1) assms(2) assms(3) assms(4) by blast
thus ?thesis
by (metis PreGrad_def PreGrad_uniq Sym_Bet__Bet_Bet Sym_Bet__Cong)
qed
lemma Lem_Gradn_id_n:
"Gradn A A n = A"
by simp
lemma Lem_Gradn_0:
"Gradn A B 0 = A"
by simp
lemma Lem_Gradn_1:
"Gradn A B 1 = B"
by simp
lemma Diff__Gradn_Sym:
assumes "A \<noteq> B" and
"n > 1"
shows "Gradn A B n = Sym A B (Gradn A B (n-1))"
proof -
have "\<not> (n = 0 \<and> n = 1)"
by auto
thus ?thesis
using assms(1) assms(2) by simp
qed
lemma Diff__Bet_Gradn_Suc:
assumes "A \<noteq> B"
shows "Bet A B (Gradn A B (Suc n))"
proof (induction n)
case 0
hence "Gradn A B (Suc 0) = B"
using assms(1) by simp
thus ?case
by (simp add: between_trivial)
next
case (Suc n)
{
assume 1: "Bet A B (Gradn A B (Suc n))"
have "Gradn A B (Suc (Suc n)) = Sym A B (Gradn A B (Suc n))"
by simp
hence "Bet A B (Gradn A B (Suc (Suc n))) \<and>
Bet A (Gradn A B (Suc n)) (Gradn A B (Suc (Suc n)))"
using 1 Sym_Bet__Bet_Bet assms by presburger
hence "Bet A B (Gradn A B (Suc (Suc n)))"
by simp
}
thus ?case
using Suc.IH by blast
qed
lemma Diff_Le_Gradn_Suc:
assumes "A \<noteq> B"
shows "A B Le A (Gradn A B (Suc n))"
by (meson Diff__Bet_Gradn_Suc assms bet__le1213)
lemma Diff__Bet_Gradn:
assumes "A \<noteq> B" and
"n \<noteq> 0"
shows "Bet A B (Gradn A B n)"
using assms(1) assms(2) Diff__Bet_Gradn_Suc not0_implies_Suc by blast
lemma Diff_Le_Gradn_n:
assumes "A \<noteq> B" and
"n \<noteq> 0"
shows "A B Le A (Gradn A B n)"
by (meson Diff__Bet_Gradn assms(1) assms(2) l5_12_a)
lemma Diff_Bet_Gradn_Suc_Gradn_Suc2:
assumes "A \<noteq> B"
shows "Bet A (Gradn A B (Suc n)) (Gradn A B (Suc (Suc n)))"
proof (induction n)
case 0
hence 1: "Gradn A B (Suc 0) = B"
using assms(1) by simp
from assms(1)
have "(Gradn A B (Suc (Suc 0))) = (Sym A B (Gradn A B (Suc 0)))"
by simp
thus ?case
by (metis "1" Diff__Bet_Gradn_Suc assms)
next
case (Suc n)
{
assume 1: "Bet A (Gradn A B (Suc n)) (Gradn A B (Suc (Suc n)))"
have "Gradn A B (Suc (Suc n)) = Sym A B (Gradn A B (Suc n))"
by simp
hence "Bet A B (Gradn A B (Suc (Suc n)))"
using Diff__Bet_Gradn_Suc assms by blast
have "Gradn A B (Suc(Suc (Suc n))) = Sym A B (Gradn A B (Suc(Suc n)))"
by simp
hence "PreGrad A B (Gradn A B (Suc(Suc n))) (Gradn A B (Suc(Suc (Suc n))))"
using PreGrad_def Sym_Bet__Bet_Bet Sym_Bet__Cong
\<open>Bet A B (Gradn A B (Suc (Suc n)))\<close> assms by presburger
hence "Bet A B (Gradn A B (Suc(Suc (Suc n)))) \<and>
Bet A (Gradn A B (Suc(Suc n))) (Gradn A B (Suc(Suc (Suc n))))"
by (metis Diff__Bet_Gradn_Suc Sym_Bet__Bet_Bet
\<open>Gradn A B (Suc (Suc (Suc n))) = Sym A B (Gradn A B (Suc (Suc n)))\<close> assms)
hence "Bet A (Gradn A B (Suc(Suc n))) (Gradn A B (Suc(Suc (Suc n))))"
by blast
}
thus ?case
using Suc.IH by blast
qed
lemma Diff__Bet_Gradn_Gradn_SucA:
assumes "A \<noteq> B"
shows "A (Gradn A B (Suc n)) Le A (Gradn A B (Suc (Suc n)))"
by (meson Diff_Bet_Gradn_Suc_Gradn_Suc2 assms bet__le1213)
lemma Diff__Bet_Gradn_Gradn_Suc:
assumes "A \<noteq> B"
shows "Bet A (Gradn A B n) (Gradn A B (Suc n))"
proof (induction n)
case 0
hence "Gradn A B 0 = A" by simp
thus ?case
using between_trivial2 by presburger
next
case (Suc n)
{
assume 1: "Bet A (Gradn A B n) (Gradn A B (Suc n))"
have "Gradn A B (Suc (Suc n)) = Sym A B (Gradn A B (Suc n))"
by simp
hence "Bet A B (Gradn A B (Suc (Suc n))) \<and>
Bet A (Gradn A B (Suc n)) (Gradn A B (Suc (Suc n)))"
using 1 Sym_Bet__Bet_Bet assms Diff__Bet_Gradn_Suc by presburger
hence "Bet A (Gradn A B (Suc n)) (Gradn A B (Suc (Suc n)))"
by blast
}
thus ?case
using Suc.IH by simp
qed
lemma Bet_Gradn_Gradn_Suc:
shows "Bet A (Gradn A B n) (Gradn A B (Suc n))"
by (metis Lem_Gradn_id_n Diff__Bet_Gradn_Gradn_Suc not_bet_distincts)
lemma Gradn_Le_Gradn_Suc:
shows "A (Gradn A B n) Le A (Gradn A B (Suc n))"
using Bet_Gradn_Gradn_Suc bet__le1213 by blast
lemma Bet_Gradn_Suc_Gradn_Suc2:
shows "Bet B (Gradn A B (Suc n)) (Gradn A B (Suc(Suc n)))"
by (metis Bet_Gradn_Gradn_Suc Diff__Bet_Gradn_Suc between_exchange3)
lemma Gradn_Suc_Le_Gradn_Suc2:
shows "B (Gradn A B (Suc n)) Le B (Gradn A B (Suc(Suc n)))"
using Bet_Gradn_Suc_Gradn_Suc2 bet__le1213 by blast
lemma Diff_Le__Bet_Gradn_Plus:
assumes "A \<noteq> B" and
"n \<le> m"
shows "Bet A (Gradn A B n) (Gradn A B (k + n))"
proof (induction k)
case 0
thus ?case
using between_trivial by auto
next
case (Suc k)
{
assume "Bet A (Gradn A B n) (Gradn A B (k + n))"
have "Bet A (Gradn A B (k + n)) (Gradn A B (Suc (k + n)))"
using Diff__Bet_Gradn_Gradn_Suc assms(1) by presburger
hence "Bet A (Gradn A B n) (Gradn A B ((Suc k) + n))"
by (metis \<open>Bet A (Gradn A B n) (Gradn A B (k + n))\<close>
add_Suc between_exchange4)
}
thus ?case
using Suc.IH by blast
qed
lemma Diff_Le_Gradn_Plus:
assumes "A \<noteq> B" and
"n \<le> m"
shows "A (Gradn A B n) Le A (Gradn A B (k + n))"
by (meson Diff_Le__Bet_Gradn_Plus assms(1) assms(2) l5_12_a)
lemma Diff_Le_Bet__Gradn_Gradn:
assumes "A \<noteq> B" and
"n \<le> m"
shows "Bet A (Gradn A B n) (Gradn A B m)"
proof (cases "n = 0")
case True
thus ?thesis
using Lem_Gradn_0 between_trivial2 by presburger
next
case False
hence 1: "n \<noteq> 0"
by auto
show "Bet A (Gradn A B n) (Gradn A B m)"
proof (cases "n = m")
case True
thus ?thesis
using between_trivial by presburger
next
case False
hence "n < m"
using assms(2) le_neq_implies_less by blast
then obtain k where "m = k + n"
using add.commute assms(2) le_Suc_ex by blast
have "Bet A (Gradn A B n) (Gradn A B (k + n))"
using Diff_Le__Bet_Gradn_Plus assms(1) by blast
thus ?thesis
using \<open>m = k + n\<close> by blast
qed
qed
lemma Diff_Le_Gradn:
assumes "A \<noteq> B" and
"n \<le> m"
shows "A (Gradn A B n) Le A (Gradn A B m)"
by (metis Diff_Le_Bet__Gradn_Gradn bet__le1213 assms(1) assms(2))
lemma Diff__Cong_Gradn_Suc_Gradn_Suc2:
assumes "A \<noteq> B"
shows "Cong A B (Gradn A B (Suc n)) (Gradn A B (Suc (Suc n)))"
proof (induction n)
case 0
hence 1: "Gradn A B (Suc 0) = B"
using assms(1) by simp
from assms(1)
have "(Gradn A B (Suc (Suc 0))) = (Sym A B (Gradn A B (Suc 0)))" by simp
hence "(Gradn A B (Suc (Suc 0))) = (Sym A B B)"
using "1" by presburger
obtain C where "B Midpoint A C"
using symmetric_point_construction by blast
hence "C = Sym A B B"
using Diff_Mid__Sym assms by blast
hence "(Gradn A B (Suc (Suc 0))) = C"
using "1" \<open>Gradn A B (Suc (Suc 0)) = Sym A B (Gradn A B (Suc 0))\<close> by presburger
have "Cong A B (Gradn A B (Suc 0)) (Gradn A B (Suc (Suc 0)))"
using "1" \<open>B Midpoint A C\<close> \<open>Gradn A B (Suc (Suc 0)) = C\<close> midpoint_cong
by presburger
thus ?case
by blast
next
case (Suc n)
{
assume "Cong A B (Gradn A B (Suc n)) (Gradn A B (Suc (Suc n)))"
have 1: "Gradn A B (Suc(Suc (Suc n))) = Sym A B (Gradn A B (Suc(Suc n)))"
by simp
have "Bet A B (Gradn A B (Suc (Suc n)))"
using Diff__Bet_Gradn_Suc assms by blast
hence "PreGrad A B (Gradn A B (Suc(Suc n))) (Gradn A B (Suc(Suc (Suc n))))"
using 1 assms
by (metis PreGrad_def Sym_Bet__Cong Diff_Bet_Gradn_Suc_Gradn_Suc2)
hence "Cong A B (Gradn A B (Suc (Suc n))) (Gradn A B (Suc(Suc (Suc n))))"
using "1" Sym_Bet__Cong \<open>Bet A B (Gradn A B (Suc (Suc n)))\<close> assms
by presburger
}
thus ?case
using Suc.IH by blast
qed
lemma Cong_Gradn_Suc_Gradn_Suc2:
shows "Cong A B (Gradn A B (Suc n)) (Gradn A B (Suc (Suc n)))"
using Diff__Cong_Gradn_Suc_Gradn_Suc2 cong_reflexivity by auto
lemma Cong_Gradn_Gradn_Suc:
shows "Cong a b (Gradn a b n) (Gradn a b (Suc n))"
proof (cases "a = b")
case True
thus ?thesis
by (simp add: cong_trivial_identity)
next
case False
have 1: "Gradn a b 0 = a"
by auto
have 2: "(Gradn a b (Suc 0)) = b"
by auto
{
assume 3: "n = 0"
hence "Cong a b a b \<and> (Suc n = 1)"
by (simp add: cong_reflexivity)
hence "Cong a b (Gradn a b n) (Gradn a b (Suc n))"
using 1 2 3 by presburger
}
moreover
{
assume "n \<noteq> 0"
hence "Cong a b (Gradn a b n) (Gradn a b (Suc n))"
by (metis False Diff__Cong_Gradn_Suc_Gradn_Suc2 old.nat.exhaust)
}
ultimately
show ?thesis
by blast
qed
lemma Diff_Bet_Bet_Cong_Gradn_Suc:
assumes "A \<noteq> B" and
"Bet A B C" and
"Bet A (Gradn A B n) C" and
"Cong A B (Gradn A B n) C"
shows "C = (Gradn A B (Suc n))"
proof (cases "n = 0")
case True
thus ?thesis
by (metis Lem_Gradn_0 Diff__Bet_Gradn_Suc Cong_Gradn_Gradn_Suc
assms(1) assms(2) assms(4) between_cong)
next
case False
hence "Bet A B (Gradn A B n)"
using Diff__Bet_Gradn assms(1) by blast
thus ?thesis
by (metis LemSym_aux Diff__Bet_Gradn_Gradn_Suc Cong_Gradn_Gradn_Suc
assms(1) assms(3) assms(4))
qed
lemma grad_rec_0_1:
shows "Cong a b (Gradn a b 0) (Gradn a b 1)"
by (simp add: cong_reflexivity)
lemma grad_rec_1_2:
shows "Cong a b (Gradn a b 1) (Gradn a b 2)"
by (metis Cong_Gradn_Gradn_Suc Suc_1)
lemma grad_rec_2_3:
shows "Cong a b (Gradn a b 2) (Gradn a b 3)"
proof (cases "a = b")
case True
thus ?thesis
using Lem_Gradn_id_n cong_reflexivity by presburger
next
case False
thus ?thesis
using Diff__Cong_Gradn_Suc_Gradn_Suc2 numeral_2_eq_2 numeral_3_eq_3
by presburger
qed
lemma grad_rec_a_a:
shows "(Gradn a a n) = a"
by simp
lemma Gradn_uniq_aux_1:
assumes "A \<noteq> B"
shows "Gradn A B n \<noteq> Gradn A B (Suc n)"
proof -
have "Gradn A B 0 \<noteq> Gradn A B (Suc 0)"
by (simp add: assms)
moreover
have "n > 0 \<longrightarrow> (Gradn A B n \<noteq> Gradn A B (Suc n))"
by (metis Diff__Cong_Gradn_Suc_Gradn_Suc2 assms cong_diff_2 gr0_implies_Suc)
ultimately
show ?thesis
by blast
qed
lemma Gradn_uniq_aux_1_aa:
assumes "A \<noteq> B"
shows "Gradn A B (k + n) \<noteq> Gradn A B (k + (Suc n))"
proof (induction k)
case 0
show "Gradn A B (0 + n) \<noteq> Gradn A B (0 + (Suc n))"
using Gradn_uniq_aux_1 assms plus_nat.add_0 by presburger
next
case (Suc k)
show "Gradn A B (Suc k + n) \<noteq> Gradn A B (Suc k + (Suc n))"
using Gradn_uniq_aux_1 add_Suc_right assms by presburger
qed
lemma Gradn_uniq_aux_1_bb:
assumes "A \<noteq> B"
shows "Gradn A B (k + n) \<noteq> Gradn A B (k + (Suc (Suc n)))"
proof (induction k)
case 0
show "Gradn A B (0 + n) \<noteq> Gradn A B (0 + (Suc(Suc n)))"
by (metis Gradn_uniq_aux_1 Diff__Bet_Gradn_Gradn_Suc add.left_neutral
assms between_equality_2)
next
case (Suc k)
show "Gradn A B ((Suc k) + n) \<noteq> Gradn A B ((Suc k) + (Suc(Suc n)))"
by (metis Gradn_uniq_aux_1_aa Diff_Bet_Gradn_Suc_Gradn_Suc2 add_Suc
add_Suc_shift assms between_equality_2)
qed
lemma Gradn_aux_1_0:
assumes "A \<noteq> B"
shows "Gradn A B (Suc n) \<noteq> A"
by (metis Diff__Bet_Gradn_Suc assms bet_neq32__neq)
lemma Gradn_aux_1_1:
assumes "A \<noteq> B" and
"n \<noteq> 0"
shows "Gradn A B (Suc n) \<noteq> B"
proof -
obtain m where "n = Suc m"
using assms(2) not0_implies_Suc by blast
have "Gradn A B (Suc(Suc m)) \<noteq> B"
proof (induction m)
show "Gradn A B (Suc(Suc 0)) \<noteq> B"
by (metis Gradn_uniq_aux_1 Diff__Bet_Gradn_Suc
Diff_Bet_Gradn_Suc_Gradn_Suc2 assms(1) between_equality_2)
next
fix m
assume "Gradn A B (Suc(Suc m)) \<noteq> B"
thus "Gradn A B (Suc(Suc(Suc m))) \<noteq> B"
by (metis Diff__Bet_Gradn_Suc Diff_Bet_Gradn_Suc_Gradn_Suc2
assms(1) between_equality_2)
qed
thus ?thesis
by (simp add: \<open>n = Suc m\<close>)
qed
lemma Gradn_aux_1_1_bis:
assumes "A \<noteq> B" and
"n \<noteq> 1"
shows "Gradn A B n \<noteq> B"
proof (cases "n = 0")
case True
thus ?thesis
using Lem_Gradn_0 assms(1) by presburger
next
case False
then obtain m where "n = Suc m"
using not0_implies_Suc by presburger
hence "m \<noteq> 0"
using assms(2) by force
thus ?thesis
using Gradn_aux_1_1 assms(1) \<open>n = Suc m\<close> by blast
qed
lemma Gradn_aux_1_2:
assumes "A \<noteq> B" and
"Gradn A B n = A"
shows "n = 0"
proof -
{
assume "n \<noteq> 0"
then obtain m where "n = Suc m"
using not0_implies_Suc by presburger
hence "Gradn A B n \<noteq> A"
using Gradn_aux_1_0 assms(1) by blast
hence "False"
using Gradn_aux_1_0 assms(1) assms(2) by blast
}
thus ?thesis by blast
qed
lemma Gradn_aux_1_3:
assumes "A \<noteq> B" and
"Gradn A B n = B"
shows "n = 1"
using Gradn_aux_1_1_bis assms(1) assms(2) by blast
lemma Gradn_uniq_aux_2_a:
assumes "A \<noteq> B" and
"n \<noteq> 0"
shows "Gradn A B 0 \<noteq> Gradn A B n"
by (metis Gradn_aux_1_2 Lem_Gradn_0 assms(1) assms(2))
lemma Gradn_uniq_aux_2:
assumes "A \<noteq> B" and
"n < m"
shows "Gradn A B n \<noteq> Gradn A B m"
proof -
obtain k where "m = (Suc k) + n"
by (metis Suc_diff_Suc add.commute add_diff_cancel_left'
assms(2) less_imp_add_positive)
have "Gradn A B n \<noteq> Gradn A B ((Suc k) + n)"
proof (induction k)
case 0
have "0 + n = n"
by simp
thus "Gradn A B n \<noteq> Gradn A B ((Suc 0) + n)"
using Gradn_uniq_aux_1_aa assms(1) by (metis add.commute)
next
case (Suc k)
hence "Gradn A B n \<noteq> Gradn A B ((Suc k) + n)"
by blast
have "Suc ((Suc k) + n) = Suc(Suc(k)) + n"
by simp
{
assume "Gradn A B n = Gradn A B ((Suc (Suc k)) + n)"
have "Gradn A B n = Gradn A B ((Suc k) + n)"
proof (cases "n = 0")
case True
thus ?thesis
by (metis Gradn_aux_1_2 Lem_Gradn_0
\<open>Gradn A B n = Gradn A B (Suc (Suc k) + n)\<close> add_cancel_left_right
assms(1) nat_neq_iff zero_less_Suc)
next
case False
have "(Suc k) + n = Suc(k + n)"
by simp
hence "Bet A B (Gradn A B ((Suc k)+n))"
using assms(1) Diff__Bet_Gradn_Suc by presburger
hence "Bet A B (Gradn A B n)"
using assms(1) Diff__Bet_Gradn_Suc
\<open>Gradn A B n = Gradn A B (Suc (Suc k) + n)\<close> add_Suc by presburger
have "Bet A (Gradn A B ((Suc k)+n)) (Gradn A B (Suc((Suc k) +n)))"
using Diff__Bet_Gradn_Gradn_Suc assms(1) by blast
hence "Bet A (Gradn A B ((Suc k)+n)) (Gradn A B n)"
using \<open>Gradn A B n = Gradn A B ((Suc (Suc k)) + n)\<close>
\<open>Suc ((Suc k) + n) = Suc(Suc(k)) + n\<close> by simp
moreover
have "Bet A (Gradn A B n) (Gradn A B ((Suc k) + n))"
using Diff_Le__Bet_Gradn_Plus assms(1) by blast
ultimately
show ?thesis
using between_equality_2 by blast
qed
hence False
using Suc.IH by blast
}
thus "Gradn A B n \<noteq> Gradn A B ((Suc (Suc k)) + n)"
by blast
qed
thus ?thesis
using \<open>m = Suc k + n\<close> by blast
qed
lemma Gradn_uniq:
assumes "A \<noteq> B" and
"Gradn A B n = Gradn A B m"
shows "n = m"
proof -
{
assume "n \<noteq> m"
{
assume "n < m"
hence "False"
using Gradn_uniq_aux_2 assms(1) assms(2) by blast
}
moreover
{
assume "m < n"
hence "False"
by (metis Gradn_uniq_aux_2 assms(1) assms(2))
}
ultimately
have "False"
using \<open>n \<noteq> m\<close> nat_neq_iff by blast
}
thus ?thesis by blast
qed
lemma Gradn_le_suc_1:
shows "A (Gradn A B n) Le A (Gradn A B (Suc n))"
using Bet_Gradn_Gradn_Suc l5_12_a by presburger
lemma Gradn_le_1:
assumes "m \<le> n"
shows "A (Gradn A B m) Le A (Gradn A B (Suc n))"
by (metis Bet_Gradn_Gradn_Suc Lem_Gradn_id_n Diff_Le_Bet__Gradn_Gradn
assms bet__le1213 le_Suc_eq)
lemma Gradn_le_suc_2:
shows "B (Gradn A B (Suc n)) Le B (Gradn A B (Suc(Suc n)))"
by (metis Bet_Gradn_Gradn_Suc Diff__Bet_Gradn_Suc bet__le1213
between_exchange3)
lemma grad_equiv_coq_1:
shows "Grad A B B"
proof -
have "(Gradn A B (Suc 0)) = B"
by auto
thus ?thesis
by (metis Grad_def n_not_Suc_n)
qed
lemma grad_aab__ab:
assumes "Grad A A B"
shows "A = B"
proof -
obtain n where "B = Gradn A A n"
using Grad_def assms by blast
thus ?thesis
by simp
qed
lemma grad_stab:
assumes "Grad A B C" and
"Bet A C C'" and
"Cong A B C C'"
shows "Grad A B C'"
proof (cases "A = B")
case True
thus ?thesis
using assms(1) assms(3) cong_reverse_identity by blast
next
case False
obtain n where "n \<noteq> 0 \<and> C = Gradn A B n"
using Grad_def assms(1) by presburger
hence "Bet A B (Gradn A B n)"
using False Diff__Bet_Gradn by blast
hence "Bet A B C"
using \<open>n \<noteq> 0 \<and> C = Gradn A B n\<close> by blast
hence "C' = Gradn A B (Suc n)"
using False Diff_Bet_Bet_Cong_Gradn_Suc \<open>n \<noteq> 0 \<and> C = Gradn A B n\<close>
assms(2) assms(3) between_exchange4 by blast
thus ?thesis
using Grad_def by blast
qed
lemma grad__bet:
assumes "Grad A B C"
shows "Bet A B C"
proof (cases "A = B")
case True
thus ?thesis
by (simp add: between_trivial2)
next
case False
obtain n where "n \<noteq> 0 \<and> C = Gradn A B n"
using Grad_def assms(1) by presburger
hence "Bet A B (Gradn A B n)"
using False Diff__Bet_Gradn by blast
thus ?thesis
using \<open>n \<noteq> 0 \<and> C = Gradn A B n\<close> by blast
qed
lemma grad__col:
assumes "Grad A B C"
shows "Col A B C"
by (simp add: assms bet_col grad__bet)
lemma grad_neq__neq13:
assumes "Grad A B C" and
"A \<noteq> B"
shows "A \<noteq> C"
using assms(1) assms(2) between_identity grad__bet by blast
lemma grad_neq__neq12:
assumes "Grad A B C" and
"A \<noteq> C"
shows "A \<noteq> B"
using Grad_def assms(1) assms(2) grad_rec_a_a by force
lemma grad112__eq:
assumes "Grad A A B"
shows "A = B"
by (meson assms grad_neq__neq12)
lemma grad121__eq:
assumes "Grad A B A"
shows "A = B"
using assms grad_neq__neq13 by blast
lemma grad__le:
assumes "Grad A B C"
shows "A B Le A C"
using assms bet__le1213 grad__bet by blast
lemma grad2_init:
shows "Grad2 A B B C D D"
proof -
have "(B = Gradn A B (Suc 0)) \<and> (D = Gradn C D (Suc 0))"
using One_nat_def Lem_Gradn_1 by presburger
thus ?thesis
using Grad2_def by blast
qed
lemma Grad2_stab:
assumes "Grad2 A B C D E F" and
"Bet A C C'" and
"Cong A B C C'" and
"Bet D F F'" and
"Cong D E F F'"
shows "Grad2 A B C' D E F'"
proof -
obtain n where "(n \<noteq> 0) \<and> (C = Gradn A B n) \<and> (F = Gradn D E n)"
using Grad2_def assms(1) by presburger
have "C' = Gradn A B (Suc n)"
by (metis Diff_Bet_Bet_Cong_Gradn_Suc Lem_Gradn_id_n Diff__Bet_Gradn
\<open>n \<noteq> 0 \<and> C = Gradn A B n \<and> F = Gradn D E n\<close> assms(2) assms(3)
between_exchange4 cong_reverse_identity)
moreover
have "F' = Gradn D E (Suc n)"
by (metis Diff_Bet_Bet_Cong_Gradn_Suc Lem_Gradn_id_n Diff__Bet_Gradn
\<open>n \<noteq> 0 \<and> C = Gradn A B n \<and> F = Gradn D E n\<close> assms(4) assms(5)
between_exchange4 cong_reverse_identity)
ultimately
show ?thesis
using Grad2_def by blast
qed
lemma bet_cong2_grad__grad2_aux_1:
assumes "C = (Gradn A B 0)" and
"Bet D E F" and
"Cong A B D E" and
"Cong B C E F"
shows "F = Gradn D E 2"
proof -
have "(Gradn A B 0) = A"
using Lem_Gradn_0 by blast
hence "A = C"
using assms(1) by auto
hence "Cong D E E F"
using assms(3) assms(4) cong_transitivity not_cong_4312 by blast
thus ?thesis
by (metis Diff_Bet_Bet_Cong_Gradn_Suc Lem_Gradn_1 Suc_1
assms(2) cong_diff_3 grad_rec_1_2)
qed
lemma bet_cong2_grad__grad2_aux_2:
assumes "Bet D E F" and
"Cong A B D E" and
"Cong B (Gradn A B (Suc n)) E F"
shows "F = Gradn D E (Suc n)"
proof -
have "\<forall> A B D E F. (Bet D E F \<and> Cong A B D E \<and>
Cong B (Gradn A B (Suc n)) E F \<longrightarrow> F = Gradn D E (Suc n))"
proof (induction n)
case 0
{
fix A B C D E F
assume 1: "Bet D E F \<and> Cong A B D E \<and> Cong B (Gradn A B (Suc 0)) E F"
hence "Gradn A B (Suc 0) = B"
using One_nat_def Lem_Gradn_1 by presburger
hence "E = F"
by (metis "1" cong_diff_4)
hence "F = Gradn D E (Suc 0)"
by simp
}
thus ?case
by blast
next
case (Suc n)
{
assume 2: "\<forall> A B D E F. (Bet D E F \<and> Cong A B D E \<and>
Cong B (Gradn A B (Suc n)) E F) \<longrightarrow> F = Gradn D E (Suc n)"
{
fix A B D E F
assume "Bet D E F" and
"Cong A B D E" and
"Cong B (Gradn A B (Suc (Suc n))) E F"
have "Cong A B (Gradn A B (Suc n)) (Gradn A B (Suc(Suc n)))"
using Cong_Gradn_Suc_Gradn_Suc2 by auto
have "Bet A (Gradn A B (Suc n)) (Gradn A B (Suc(Suc n)))"
using Bet_Gradn_Gradn_Suc by auto
have "F = Gradn D E (Suc (Suc n))"
proof (cases "A = B")
case True
thus ?thesis
by (metis \<open>Cong A B D E\<close> \<open>Cong B (Gradn A B (Suc (Suc n))) E F\<close>
cong_reverse_identity grad_rec_a_a)
next
case False
have "D \<noteq> E"