added Linear.Order (#300)

* upd congr_ex_6&7

* Update Congruence_exercise_ygr_7.lean

* finished prove of congr_exercise_ygr_6

* modify several thm in trash

* upd proof of 7

* upd on Congr_ex_ygr_7

* merge from upstream and lean upd  20240112

* upd problem_5 line_trash

add theorems about linear order into line_trash

* Update Problem_5.lean

* upd on problem 5 and open Order.lean

* Update Order.lean

* Update Problem_5.lean

* upd 20240115

* upd on problem_11 and Order.lean

* Update Order.lean

* upd order.lean

* Update Order.lean

* Update Order.lean

deleted theorems starting with MEOW

* Update Order.lean

updated theorems on linear order and LiesInt/LiesOn ray

* Update Order.lean

updated corollaries OVO and QWQ

* update Order.lean

added corollaries OVO_1 OVO_2 QWQ_1 QWQ_2

* Update Order.lean

* Update Order.lean

updated proofs of corollaries

* Update Order.lean

renamed theorems with HAHAHA, HOHOHO, QAQ and QWQ

* Update Order.lean

added several theorems to prove according to the plan

* Update Order.lean

added theorems to prove according to plan

* Update Order.lean

updated a part of the proof for lies_int_seg_nd_ext_iff_lies_int

* Update Order.lean

proved theorem lies_int_seg_nd_ext_iff_lies_int and lies_on_seg_nd_ext_iff_lies_on

* Update Order.lean

finished Order.lean

* upd before PR

updated Angle_ex_trash and moved two theorems to IsocelesTriangle_trash

moved lemmas about collinearity in Linear.Order to Linear.Line

updated Index

* Update Line.lean

EuclideanGeometry/Example/Congruence_Exercise_ygr/Problem_11.lean

@@ -0,0 +1,103 @@
+import EuclideanGeometry.Foundation.Index
+import EuclideanGeometry.Foundation.Axiom.Linear.Order
+
+noncomputable section
+
+namespace EuclidGeom
+
+namespace Problem_11
+
+/-
+Let $l$ be a directed line on the plane.
+Let $B, E, C, F$ be four points on $l$ in this order.
+Let $A, D$ be two points on the plane such that they lies on the same side of $l$.
+If $AB = DF$, $AC = DE$ and $BE = CF$, prove that $\angle BAC = - \angle FDE$.
+-/
+
+structure Setting (Plane : Type _) [EuclideanPlane Plane] where
+-- Let $l$ be a directed line on the plane
+  l : DirLine Plane
+-- Let $B, E, C, F$ be four points on $l$ in this order.
+  B : Plane
+  E : Plane
+  C : Plane
+  F : Plane
+  B_on_l : B LiesOn l
+  E_on_l : E LiesOn l
+  C_on_l : C LiesOn l
+  F_on_l : F LiesOn l
+  B_lt_E_on_l : (⟨B, B_on_l⟩ : l.carrier.Elem) < ⟨E, E_on_l⟩
+  E_lt_C_on_l : (⟨E, E_on_l⟩ : l.carrier.Elem) < ⟨C, C_on_l⟩
+  C_lt_F_on_l : (⟨C, C_on_l⟩ : l.carrier.Elem) < ⟨F, F_on_l⟩
+-- Let $A, D$ be two points on the plane,
+  A : Plane
+  D : Plane
+-- such that they lies on different sides of $l$,
+  A_D_IsOnSameSide_l : IsOnSameSide A D l
+-- $AB = DF$,
+  AB_eq_DF : (SEG A B).length = (SEG D F).length
+-- $AC = DE$,
+  AC_eq_DE : (SEG A C).length = (SEG D E).length
+-- $BE = CF$.
+  BE_eq_CF : (SEG B E).length = (SEG C F).length
+lemma B_ne_C' {Plane : Type _} [EuclideanPlane Plane] {e : Setting Plane} : e.B ≠ e.C := by
+  apply (DirLine.ne_iff_ne_as_line_elem e.B_on_l e.C_on_l).mpr; apply ne_of_lt;
+  calc
+  (⟨e.B, e.B_on_l⟩ : e.l.carrier.Elem)
+  _< ⟨e.E, e.E_on_l⟩ := e.B_lt_E_on_l
+  _< ⟨e.C, e.C_on_l⟩ := e.E_lt_C_on_l
+instance B_ne_C {Plane : Type _} [EuclideanPlane Plane] {e : Setting Plane} : PtNe e.B e.C := ⟨B_ne_C'⟩
+lemma E_ne_F' {Plane : Type _} [EuclideanPlane Plane] {e : Setting Plane} : e.E ≠ e.F := by
+  apply (DirLine.ne_iff_ne_as_line_elem e.E_on_l e.F_on_l).mpr; apply ne_of_lt;
+  calc
+  (⟨e.E, e.E_on_l⟩ : e.l.carrier.Elem)
+  _< ⟨e.C, e.C_on_l⟩ := e.E_lt_C_on_l
+  _< ⟨e.F, e.F_on_l⟩ := e.C_lt_F_on_l
+instance E_ne_F {Plane : Type _} [EuclideanPlane Plane] {e : Setting Plane} : PtNe e.E e.F := ⟨E_ne_F'⟩
+lemma not_colinear_ACB {Plane : Type _} [EuclideanPlane Plane] {e : Setting Plane} : ¬ colinear e.A e.C e.B := by sorry
+lemma not_colinear_DEF {Plane : Type _} [EuclideanPlane Plane] {e : Setting Plane} : ¬ colinear e.D e.E e.F := by sorry
+instance B_ne_A {Plane : Type _} [EuclideanPlane Plane] {e : Setting Plane} : PtNe e.B e.A := ⟨(ne_of_not_colinear not_colinear_ACB).2.1.symm⟩
+instance C_ne_A {Plane : Type _} [EuclideanPlane Plane] {e : Setting Plane} : PtNe e.C e.A := ⟨(ne_of_not_colinear not_colinear_ACB).2.2⟩
+instance F_ne_D {Plane : Type _} [EuclideanPlane Plane] {e : Setting Plane} : PtNe e.F e.D := ⟨(ne_of_not_colinear not_colinear_DEF).2.1.symm⟩
+instance E_ne_D {Plane : Type _} [EuclideanPlane Plane] {e : Setting Plane} : PtNe e.E e.D := ⟨(ne_of_not_colinear not_colinear_DEF).2.2⟩
+
+structure Setting2 (Plane : Type _) [EuclideanPlane Plane] extends Setting Plane where
+  not_colinear_ACB : ¬ Collinear A C B := not_colinear_ACB
+  not_colinear_DEF : ¬ Collinear D E F := not_colinear_DEF
+
+-- Prove that $\angle BAC = - \angle FDE$.
+theorem result {Plane : Type _} [EuclideanPlane Plane] (e : Setting2 Plane) : (∠ e.B e.A e.C) = - (∠ e.F e.D e.E) := by
+  have E_int_BC : e.E LiesInt (SEG e.B e.C) := by
+    apply DirLine.lies_int_seg_of_lt_and_lt e.B_on_l e.E_on_l e.C_on_l
+    exact e.B_lt_E_on_l; exact e.E_lt_C_on_l
+  have C_int_EF : e.C LiesInt (SEG e.E e.F) := by
+    apply DirLine.lies_int_seg_of_lt_and_lt e.E_on_l e.C_on_l e.F_on_l
+    exact e.E_lt_C_on_l; exact e.C_lt_F_on_l
+  have CB_eq_EF : (SEG e.C e.B).length = (SEG e.E e.F).length := by
+    calc
+    _= (SEG e.B e.C).length := by apply length_of_rev_eq_length'
+    _= (SEG e.B e.E).length + (SEG e.E e.C).length := by
+      apply length_eq_length_add_length
+      apply Seg.lies_on_of_lies_int
+      exact E_int_BC
+    _= (SEG e.E e.C).length + (SEG e.C e.F).length := by simp only [e.BE_eq_CF]; ring
+    _= _ := by
+      symm;
+      apply length_eq_length_add_length
+      apply Seg.lies_on_of_lies_int
+      exact C_int_EF
+  have BA_eq_FD : (SEG e.B e.A).length = (SEG e.F e.D).length := by
+    calc
+    _= (SEG e.A e.B).length := by apply length_of_rev_eq_length'
+    _= (SEG e.D e.F).length := e.AB_eq_DF
+    _=_ := by apply length_of_rev_eq_length'
+  have clockwise_ACB_eq_neg_clockwise_DEF : (TRI_nd e.A e.C e.B e.not_colinear_ACB).is_cclock = ¬(TRI_nd e.D e.E e.F e.not_colinear_DEF).is_cclock := by sorry
+  have triangle_ACB_acongr_triangle_DEF : (TRI_nd e.A e.C e.B e.not_colinear_ACB) ≅ₐ (TRI_nd e.D e.E e.F e.not_colinear_DEF) := by
+    apply TriangleND.acongr_of_SSS_of_ne_orientation
+    · exact CB_eq_EF
+    · exact BA_eq_FD
+    · exact e.AC_eq_DE
+    · exact clockwise_ACB_eq_neg_clockwise_DEF
+end Problem_11
+
+end EuclidGeom

EuclideanGeometry/Example/Congruence_Exercise_ygr/Problem_5.lean

@@ -0,0 +1,190 @@
+import EuclideanGeometry.Foundation.Index
+import EuclideanGeometry.Foundation.Axiom.Linear.Order
+
+noncomputable section
+
+namespace EuclidGeom
+
+namespace Problem_5
+
+/-
+Let $l$ be a directed line on the plane.
+Let $A, F, C, D$ be four points on $l$ in this order.
+Let $B, E$ be two points on the plane such that they lies on different sides of $l$, and that $CB \parallel FE$.
+If $\angle BAC = \angle EDF$ and $AF = DC$, prove that $AB = DE$.
+-/
+
+structure Setting1 (Plane : Type _) [EuclideanPlane Plane] where
+-- Let $l$ be a directed line on the plane
+  l : DirLine Plane
+-- Let $A, F, C, D$ be four points on $l$ in this order.
+  A : Plane
+  F : Plane
+  C : Plane
+  D : Plane
+  A_on_l : A LiesOn l
+  F_on_l : F LiesOn l
+  C_on_l : C LiesOn l
+  D_on_l : D LiesOn l
+  A_lt_F_on_l : (⟨A, A_on_l⟩ : l.carrier.Elem) < ⟨F, F_on_l⟩
+  F_lt_C_on_l : (⟨F, F_on_l⟩ : l.carrier.Elem) < ⟨C, C_on_l⟩
+  C_lt_D_on_l : (⟨C, C_on_l⟩ : l.carrier.Elem) < ⟨D, D_on_l⟩
+-- Let $B, E$ be two points on the plane,
+  B : Plane
+  E : Plane
+-- such that they lies on different sides of $l$,
+  B_E_IsOnOppositeSide_l : IsOnOppositeSide B E l
+
+-- Claim : $A \ne C$.
+lemma A_ne_C' {Plane : Type _} [EuclideanPlane Plane] {e : Setting1 Plane} : e.A ≠ e.C := by
+  apply (DirLine.ne_iff_ne_as_line_elem e.A_on_l e.C_on_l).mpr; apply ne_of_lt;
+  calc
+  (⟨e.A, e.A_on_l⟩ : e.l.carrier.Elem)
+  _< ⟨e.F, e.F_on_l⟩ := e.A_lt_F_on_l
+  _< ⟨e.C, e.C_on_l⟩ := e.F_lt_C_on_l
+instance A_ne_C {Plane : Type _} [EuclideanPlane Plane] {e : Setting1 Plane} : PtNe e.A e.C := ⟨A_ne_C'⟩
+-- Claim : $D \ne F$.
+lemma D_ne_F' {Plane : Type _} [EuclideanPlane Plane] {e : Setting1 Plane} : e.D ≠ e.F := by
+  symm; apply (DirLine.ne_iff_ne_as_line_elem e.F_on_l e.D_on_l).mpr; apply ne_of_lt;
+  calc
+  (⟨e.F, e.F_on_l⟩ : e.l.carrier.Elem)
+  _< ⟨e.C, e.C_on_l⟩ := e.F_lt_C_on_l
+  _< ⟨e.D, e.D_on_l⟩ := e.C_lt_D_on_l
+instance D_ne_F {Plane : Type _} [EuclideanPlane Plane] {e : Setting1 Plane} : PtNe e.D e.F := ⟨D_ne_F'⟩
+lemma not_colinear_BCA {Plane : Type _} [EuclideanPlane Plane] {e : Setting1 Plane} : ¬ colinear e.B e.C e.A := by
+  have h : IsOnOppositeSide e.B e.E (RAY e.C e.A) := by
+    have h3 : (RAY e.C e.A).toLine = e.l.toLine := by
+      calc
+        _= (LIN e.C e.A) := by
+          apply Line.eq_line_of_pt_pt_of_ne
+          apply Line.fst_pt_lies_on_mk_pt_pt
+          apply Line.snd_pt_lies_on_mk_pt_pt
+        _=_ := by
+          apply Line.eq_line_of_pt_pt_of_ne
+          apply DirLine.lies_on_iff_lies_on_toLine.mpr; exact e.C_on_l
+          apply DirLine.lies_on_iff_lies_on_toLine.mpr; exact e.A_on_l
+    have : IsOnOppositeSide e.B e.E (RAY e.C e.A) = IsOnOppositeSide e.B e.E e.l := by
+      calc
+        _= IsOnOppositeSide e.B e.E (RAY e.C e.A).toLine := by rfl
+        _=_ := by congr
+    simp only [this]
+    exact e.B_E_IsOnOppositeSide_l
+  have h1 : ¬ colinear e.C e.A e.B := (not_colinear_of_IsOnOppositeSide e.C e.A e.B e.E h).1
+  by_contra h2; absurd h1
+  exact perm_colinear_snd_trd_fst h2
+lemma not_colinear_EFD {Plane : Type _} [EuclideanPlane Plane] {e : Setting1 Plane} : ¬ colinear e.E e.F e.D := by
+  have h : IsOnOppositeSide e.B e.E (RAY e.F e.D) := by
+    have h3 : (RAY e.F e.D).toLine = e.l.toLine := by
+      calc
+        _= (LIN e.F e.D) := by
+          apply Line.eq_line_of_pt_pt_of_ne
+          apply Line.fst_pt_lies_on_mk_pt_pt
+          apply Line.snd_pt_lies_on_mk_pt_pt
+        _=_ := by
+          apply Line.eq_line_of_pt_pt_of_ne
+          apply DirLine.lies_on_iff_lies_on_toLine.mpr; exact e.F_on_l
+          apply DirLine.lies_on_iff_lies_on_toLine.mpr; exact e.D_on_l
+    have : IsOnOppositeSide e.B e.E (RAY e.F e.D) = IsOnOppositeSide e.B e.E e.l := by
+      calc
+        _= IsOnOppositeSide e.B e.E (RAY e.F e.D).toLine := by rfl
+        _=_ := by congr
+    simp only [this]
+    exact e.B_E_IsOnOppositeSide_l
+  have h1 : ¬ colinear e.F e.D e.E := (not_colinear_of_IsOnOppositeSide e.F e.D e.B e.E h).2
+  by_contra h2; absurd h1
+  exact perm_colinear_snd_trd_fst h2
+-- Claim : $C \ne B$.
+instance C_ne_B {Plane : Type _} [EuclideanPlane Plane] {e : Setting1 Plane} : PtNe e.C e.B := ⟨(ne_of_not_colinear not_colinear_BCA).2.2⟩
+-- Claim : $A \ne B$.
+instance A_ne_B {Plane : Type _} [EuclideanPlane Plane] {e : Setting1 Plane} : PtNe e.A e.B := ⟨(ne_of_not_colinear not_colinear_BCA).2.1.symm⟩
+-- Claim : $E \ne F$.
+instance E_ne_F {Plane : Type _} [EuclideanPlane Plane] {e : Setting1 Plane} : PtNe e.E e.F := ⟨(ne_of_not_colinear not_colinear_EFD).2.2.symm⟩
+-- Claim : $E \ne D$.
+instance E_ne_D {Plane : Type _} [EuclideanPlane Plane] {e : Setting1 Plane} : PtNe e.E e.D := ⟨(ne_of_not_colinear not_colinear_EFD).2.1⟩
+
+structure Setting2 (Plane : Type _) [EuclideanPlane Plane] extends Setting1 Plane where
+  not_colinear_BCA : ¬ colinear B C A := not_colinear_BCA
+  not_colinear_EFD : ¬ colinear E F D := not_colinear_EFD
+-- and that $CB \parallel FE$,
+  CB_parallel_FE : (LIN C B) ∥ (LIN F E)
+-- If $\angle BAC = - \angle EDF$,
+  angle_BAC_eq_angle_EDF : (∠ B A C) = (∠ E D F)
+-- and $AF = DC$,
+  AF_eq_DC : (SEG A F).length = (SEG D C).length
+-- Prove that $AB = DE$.
+
+theorem result {Plane : Type _} [EuclideanPlane Plane] (e : Setting2 Plane) : (SEG e.A e.B).length = (SEG e.D e.E).length := by
+  have F_int_AC : e.F LiesInt (SEG e.A e.C) := by
+    apply DirLine.lies_int_of_lt_and_lt e.A_on_l e.F_on_l e.C_on_l
+    simp only [e.A_lt_F_on_l]
+    simp only [e.F_lt_C_on_l]
+  have C_int_DF : e.C LiesInt (SEG e.D e.F) := by
+    apply DirLine.lies_int_of_gt_and_gt e.D_on_l e.C_on_l e.F_on_l
+    simp only [e.C_lt_D_on_l]
+    simp only [e.F_lt_C_on_l]
+  have C_ne_F' : e.C ≠ e.F := by
+    apply (DirLine.ne_iff_ne_as_line_elem e.C_on_l e.F_on_l).mpr; apply ne_of_gt
+    simp only [e.F_lt_C_on_l]
+  haveI C_ne_F : PtNe e.C e.F := ⟨C_ne_F'⟩
+  have CA_eq_FD : (SEG e.C e.A).length = (SEG e.F e.D).length := by
+    calc
+    _= (SEG e.A e.C).length := by apply length_of_rev_eq_length'
+    _= (SEG e.A e.F).length + (SEG e.F e.C).length := by
+      apply length_eq_length_add_length
+      apply Seg.lies_on_of_lies_int
+      exact F_int_AC
+    _= (SEG e.D e.C).length + (SEG e.C e.F).length := by congr 1; exact e.AF_eq_DC; exact length_of_rev_eq_length'
+    _= (SEG e.D e.F).length := by
+      symm; apply length_eq_length_add_length
+      apply Seg.lies_on_of_lies_int
+      exact C_int_DF
+    _=_ := by apply length_of_rev_eq_length'
+  have angle_ACB_eq_angle_DFE : (∠ e.A e.C e.B) = (∠ e.D e.F e.E) := by
+    have dir_CA_eq_neg_dir_FD : (RAY e.C e.A).toDir = - (RAY e.F e.D).toDir := by
+      calc
+      _= - e.l.toDir := by
+        apply DirLine.neg_toDir_of_gt e.C_on_l e.A_on_l
+        calc
+        (⟨e.A, e.A_on_l⟩ : e.l.carrier.Elem)
+        _< ⟨e.F, e.F_on_l⟩ := e.A_lt_F_on_l
+        _< ⟨e.C, e.C_on_l⟩ := e.F_lt_C_on_l
+      _=_ := by
+        simp only [neg_inj]; symm
+        apply DirLine.eq_toDir_of_lt e.F_on_l e.D_on_l
+        calc
+        (⟨e.F, e.F_on_l⟩ : e.l.carrier.Elem)
+        _< ⟨e.C, e.C_on_l⟩ := e.F_lt_C_on_l
+        _< ⟨e.D, e.D_on_l⟩ := e.C_lt_D_on_l
+    have dir_CB_eq_neg_dir_FE : (RAY e.C e.B).toDir = - (RAY e.F e.E).toDir := by
+      have h : IsOnOppositeSide e.B e.E (SEG_nd e.C e.F) := by
+        have h1 : (SEG_nd e.C e.F).toLine = e.l := by
+          calc
+          _= (LIN e.C e.F) := by
+            symm;
+            apply Line.eq_line_of_pt_pt_of_ne
+            apply SegND.lies_on_toLine_of_lie_on; exact SegND.source_lies_on
+            apply SegND.lies_on_toLine_of_lie_on; exact SegND.target_lies_on
+          _=_ := by
+            apply Line.eq_line_of_pt_pt_of_ne
+            exact e.C_on_l
+            exact e.F_on_l
+        have h2 : IsOnOppositeSide e.B e.E (SEG_nd e.C e.F) = IsOnOppositeSide e.B e.E e.l := by
+          calc
+          _= IsOnOppositeSide e.B e.E (SEG_nd e.C e.F).toLine := by rfl
+          _=_ := by congr
+        simp only [h2]; exact e.B_E_IsOnOppositeSide_l
+      apply neg_toDir_of_parallel_and_IsOnOppositeSide
+      · exact e.CB_parallel_FE
+      · exact h
+    apply ang_eq_ang_of_toDir_rev_toDir
+    · exact dir_CA_eq_neg_dir_FD
+    · exact dir_CB_eq_neg_dir_FE
+  have triangle_BCA_congr_triangle_EFD : (TRI_nd e.B e.C e.A e.not_colinear_BCA) ≅ (TRI_nd e.E e.F e.D e.not_colinear_EFD) := by
+    apply TriangleND.congr_of_ASA
+    · exact angle_ACB_eq_angle_DFE
+    · exact CA_eq_FD
+    · exact e.angle_BAC_eq_angle_EDF
+  exact triangle_BCA_congr_triangle_EFD.edge₂
+end Problem_5
+
+end EuclidGeom