url stringclasses 147
values | commit stringclasses 147
values | file_path stringlengths 7 101 | full_name stringlengths 1 94 | start stringlengths 6 10 | end stringlengths 6 11 | tactic stringlengths 1 11.2k | state_before stringlengths 3 2.09M | state_after stringlengths 6 2.09M | input stringlengths 73 2.09M |
|---|---|---|---|---|---|---|---|---|---|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/proportion.lean | proportion_symm' | [133, 1] | [145, 38] | have t_ne := (zero_of_proportion_iff inv).not.mp r_ne | case neg
r s t u : ℝ
h : proportion r s t u
s_ne : s = 0
u_ne : u = 0
inv : proportion s r u t
r_ne : ¬r = 0
⊢ proportion t u r s | case neg
r s t u : ℝ
h : proportion r s t u
s_ne : s = 0
u_ne : u = 0
inv : proportion s r u t
r_ne : ¬r = 0
t_ne : ¬t = 0
⊢ proportion t u r s | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
r s t u : ℝ
h : proportion r s t u
s_ne : s = 0
u_ne : u = 0
inv : proportion s r u t
r_ne : ¬r = 0
⊢ proportion t u r s
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/proportion.lean | proportion_symm' | [133, 1] | [145, 38] | rw [proportion_inv_iff] | case neg
r s t u : ℝ
h : proportion r s t u
s_ne : s = 0
u_ne : u = 0
inv : proportion s r u t
r_ne : ¬r = 0
t_ne : ¬t = 0
⊢ proportion t u r s | case neg
r s t u : ℝ
h : proportion r s t u
s_ne : s = 0
u_ne : u = 0
inv : proportion s r u t
r_ne : ¬r = 0
t_ne : ¬t = 0
⊢ proportion u t s r | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
r s t u : ℝ
h : proportion r s t u
s_ne : s = 0
u_ne : u = 0
inv : proportion s r u t
r_ne : ¬r = 0
t_ne : ¬t = 0
⊢ proportion t u r s
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/proportion.lean | proportion_symm' | [133, 1] | [145, 38] | exact proportion_symm r_ne t_ne inv | case neg
r s t u : ℝ
h : proportion r s t u
s_ne : s = 0
u_ne : u = 0
inv : proportion s r u t
r_ne : ¬r = 0
t_ne : ¬t = 0
⊢ proportion u t s r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
r s t u : ℝ
h : proportion r s t u
s_ne : s = 0
u_ne : u = 0
inv : proportion s r u t
r_ne : ¬r = 0
t_ne : ¬t = 0
⊢ proportion u t s r
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/proportion.lean | proportion_symm' | [133, 1] | [145, 38] | have u_ne := (zero_of_proportion_iff h).not.mp s_ne | case neg
r s t u : ℝ
h : proportion r s t u
s_ne : ¬s = 0
⊢ proportion t u r s | case neg
r s t u : ℝ
h : proportion r s t u
s_ne : ¬s = 0
u_ne : ¬u = 0
⊢ proportion t u r s | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
r s t u : ℝ
h : proportion r s t u
s_ne : ¬s = 0
⊢ proportion t u r s
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/proportion.lean | proportion_symm' | [133, 1] | [145, 38] | exact proportion_symm s_ne u_ne h | case neg
r s t u : ℝ
h : proportion r s t u
s_ne : ¬s = 0
u_ne : ¬u = 0
⊢ proportion t u r s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
r s t u : ℝ
h : proportion r s t u
s_ne : ¬s = 0
u_ne : ¬u = 0
⊢ proportion t u r s
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/proportion.lean | proportion_eq | [150, 1] | [151, 63] | rw [(proportion_iff hr hr hs hs r_ne s_ne).symm] | r s : ℝ
hr : 0 ≤ r
hs : 0 ≤ s
r_ne : r ≠ 0
s_ne : s ≠ 0
⊢ proportion r r s s | r s : ℝ
hr : 0 ≤ r
hs : 0 ≤ s
r_ne : r ≠ 0
s_ne : s ≠ 0
⊢ r / r = s / s | Please generate a tactic in lean4 to solve the state.
STATE:
r s : ℝ
hr : 0 ≤ r
hs : 0 ≤ s
r_ne : r ≠ 0
s_ne : s ≠ 0
⊢ proportion r r s s
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/proportion.lean | proportion_eq | [150, 1] | [151, 63] | field_simp | r s : ℝ
hr : 0 ≤ r
hs : 0 ≤ s
r_ne : r ≠ 0
s_ne : s ≠ 0
⊢ r / r = s / s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
r s : ℝ
hr : 0 ≤ r
hs : 0 ≤ s
r_ne : r ≠ 0
s_ne : s ≠ 0
⊢ r / r = s / s
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/pythagoras_2.lean | aux | [5, 1] | [11, 49] | obtain ⟨_, bL, cL⟩ := line_of_pts b c | i : incidence_geometry
a b c : point
Tabc : ¬colinear a b c
ang_a : rightangle ≤ angle b a c
⊢ ∃ d, B b d c ∧ angle b d a = rightangle ∧ angle c d a = rightangle | case intro.intro
i : incidence_geometry
a b c : point
Tabc : ¬colinear a b c
ang_a : rightangle ≤ angle b a c
w✝ : line
bL : online b w✝
cL : online c w✝
⊢ ∃ d, B b d c ∧ angle b d a = rightangle ∧ angle c d a = rightangle | Please generate a tactic in lean4 to solve the state.
STATE:
i : incidence_geometry
a b c : point
Tabc : ¬colinear a b c
ang_a : rightangle ≤ angle b a c
⊢ ∃ d, B b d c ∧ angle b d a = rightangle ∧ angle c d a = rightangle
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/pythagoras_2.lean | aux | [5, 1] | [11, 49] | have aL := online_1_of_triangle bL cL Tabc | case intro.intro
i : incidence_geometry
a b c : point
Tabc : ¬colinear a b c
ang_a : rightangle ≤ angle b a c
w✝ : line
bL : online b w✝
cL : online c w✝
⊢ ∃ d, B b d c ∧ angle b d a = rightangle ∧ angle c d a = rightangle | case intro.intro
i : incidence_geometry
a b c : point
Tabc : ¬colinear a b c
ang_a : rightangle ≤ angle b a c
w✝ : line
bL : online b w✝
cL : online c w✝
aL : ¬online a w✝
⊢ ∃ d, B b d c ∧ angle b d a = rightangle ∧ angle c d a = rightangle | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
i : incidence_geometry
a b c : point
Tabc : ¬colinear a b c
ang_a : rightangle ≤ angle b a c
w✝ : line
bL : online b w✝
cL : online c w✝
⊢ ∃ d, B b d c ∧ angle b d a = rightangle ∧ angle c d a = rightangle
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/pythagoras_2.lean | aux | [5, 1] | [11, 49] | obtain ⟨d, adc, adb, Bcdb⟩ := pythlem (ne_23_of_tri Tabc).symm cL bL aL ang_a | case intro.intro
i : incidence_geometry
a b c : point
Tabc : ¬colinear a b c
ang_a : rightangle ≤ angle b a c
w✝ : line
bL : online b w✝
cL : online c w✝
aL : ¬online a w✝
⊢ ∃ d, B b d c ∧ angle b d a = rightangle ∧ angle c d a = rightangle | case intro.intro.intro.intro.intro
i : incidence_geometry
a b c : point
Tabc : ¬colinear a b c
ang_a : rightangle ≤ angle b a c
w✝ : line
bL : online b w✝
cL : online c w✝
aL : ¬online a w✝
d : point
adc : angle a d c = rightangle
adb : angle a d b = rightangle
Bcdb : B c d b
⊢ ∃ d, B b d c ∧ angle b d a = rightangle ∧ angle c d a = rightangle | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
i : incidence_geometry
a b c : point
Tabc : ¬colinear a b c
ang_a : rightangle ≤ angle b a c
w✝ : line
bL : online b w✝
cL : online c w✝
aL : ¬online a w✝
⊢ ∃ d, B b d c ∧ angle b d a = rightangle ∧ angle c d a = rightangle
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/pythagoras_2.lean | aux | [5, 1] | [11, 49] | exact ⟨d, B_symm Bcdb, (by perma), (by perma)⟩ | case intro.intro.intro.intro.intro
i : incidence_geometry
a b c : point
Tabc : ¬colinear a b c
ang_a : rightangle ≤ angle b a c
w✝ : line
bL : online b w✝
cL : online c w✝
aL : ¬online a w✝
d : point
adc : angle a d c = rightangle
adb : angle a d b = rightangle
Bcdb : B c d b
⊢ ∃ d, B b d c ∧ angle b d a = rightangle ∧ angle c d a = rightangle | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro
i : incidence_geometry
a b c : point
Tabc : ¬colinear a b c
ang_a : rightangle ≤ angle b a c
w✝ : line
bL : online b w✝
cL : online c w✝
aL : ¬online a w✝
d : point
adc : angle a d c = rightangle
adb : angle a d b = rightangle
Bcdb : B c d b
⊢ ∃ d, B b d c ∧ angle b d a = rightangle ∧ angle c d a = rightangle
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/pythagoras_2.lean | aux | [5, 1] | [11, 49] | perma | i : incidence_geometry
a b c : point
Tabc : ¬colinear a b c
ang_a : rightangle ≤ angle b a c
w✝ : line
bL : online b w✝
cL : online c w✝
aL : ¬online a w✝
d : point
adc : angle a d c = rightangle
adb : angle a d b = rightangle
Bcdb : B c d b
⊢ angle b d a = rightangle | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
i : incidence_geometry
a b c : point
Tabc : ¬colinear a b c
ang_a : rightangle ≤ angle b a c
w✝ : line
bL : online b w✝
cL : online c w✝
aL : ¬online a w✝
d : point
adc : angle a d c = rightangle
adb : angle a d b = rightangle
Bcdb : B c d b
⊢ angle b d a = rightangle
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/pythagoras_2.lean | aux | [5, 1] | [11, 49] | perma | i : incidence_geometry
a b c : point
Tabc : ¬colinear a b c
ang_a : rightangle ≤ angle b a c
w✝ : line
bL : online b w✝
cL : online c w✝
aL : ¬online a w✝
d : point
adc : angle a d c = rightangle
adb : angle a d b = rightangle
Bcdb : B c d b
⊢ angle c d a = rightangle | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
i : incidence_geometry
a b c : point
Tabc : ¬colinear a b c
ang_a : rightangle ≤ angle b a c
w✝ : line
bL : online b w✝
cL : online c w✝
aL : ¬online a w✝
d : point
adc : angle a d c = rightangle
adb : angle a d b = rightangle
Bcdb : B c d b
⊢ angle c d a = rightangle
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/pythagoras_2.lean | aux2 | [13, 1] | [19, 27] | have ba := length_eq_zero_iff.not.mpr $ ne_21_of_tri Tabc | i : incidence_geometry
a b c : point
Tabc : triangle a b c
⊢ length b a ≠ 0 ∧ length c a ≠ 0 ∧ length b c ≠ 0 ∧ length c b ≠ 0 | i : incidence_geometry
a b c : point
Tabc : triangle a b c
ba : ¬length b a = 0
⊢ length b a ≠ 0 ∧ length c a ≠ 0 ∧ length b c ≠ 0 ∧ length c b ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
i : incidence_geometry
a b c : point
Tabc : triangle a b c
⊢ length b a ≠ 0 ∧ length c a ≠ 0 ∧ length b c ≠ 0 ∧ length c b ≠ 0
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/pythagoras_2.lean | aux2 | [13, 1] | [19, 27] | have ca := length_eq_zero_iff.not.mpr $ ne_31_of_tri Tabc | i : incidence_geometry
a b c : point
Tabc : triangle a b c
ba : ¬length b a = 0
⊢ length b a ≠ 0 ∧ length c a ≠ 0 ∧ length b c ≠ 0 ∧ length c b ≠ 0 | i : incidence_geometry
a b c : point
Tabc : triangle a b c
ba : ¬length b a = 0
ca : ¬length c a = 0
⊢ length b a ≠ 0 ∧ length c a ≠ 0 ∧ length b c ≠ 0 ∧ length c b ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
i : incidence_geometry
a b c : point
Tabc : triangle a b c
ba : ¬length b a = 0
⊢ length b a ≠ 0 ∧ length c a ≠ 0 ∧ length b c ≠ 0 ∧ length c b ≠ 0
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/pythagoras_2.lean | aux2 | [13, 1] | [19, 27] | have bc := length_eq_zero_iff.not.mpr $ ne_23_of_tri Tabc | i : incidence_geometry
a b c : point
Tabc : triangle a b c
ba : ¬length b a = 0
ca : ¬length c a = 0
⊢ length b a ≠ 0 ∧ length c a ≠ 0 ∧ length b c ≠ 0 ∧ length c b ≠ 0 | i : incidence_geometry
a b c : point
Tabc : triangle a b c
ba : ¬length b a = 0
ca : ¬length c a = 0
bc : ¬length b c = 0
⊢ length b a ≠ 0 ∧ length c a ≠ 0 ∧ length b c ≠ 0 ∧ length c b ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
i : incidence_geometry
a b c : point
Tabc : triangle a b c
ba : ¬length b a = 0
ca : ¬length c a = 0
⊢ length b a ≠ 0 ∧ length c a ≠ 0 ∧ length b c ≠ 0 ∧ length c b ≠ 0
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/pythagoras_2.lean | aux2 | [13, 1] | [19, 27] | have cb := fun h => bc ((length_symm b c).trans h) | i : incidence_geometry
a b c : point
Tabc : triangle a b c
ba : ¬length b a = 0
ca : ¬length c a = 0
bc : ¬length b c = 0
⊢ length b a ≠ 0 ∧ length c a ≠ 0 ∧ length b c ≠ 0 ∧ length c b ≠ 0 | i : incidence_geometry
a b c : point
Tabc : triangle a b c
ba : ¬length b a = 0
ca : ¬length c a = 0
bc : ¬length b c = 0
cb : length c b = 0 → False
⊢ length b a ≠ 0 ∧ length c a ≠ 0 ∧ length b c ≠ 0 ∧ length c b ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
i : incidence_geometry
a b c : point
Tabc : triangle a b c
ba : ¬length b a = 0
ca : ¬length c a = 0
bc : ¬length b c = 0
⊢ length b a ≠ 0 ∧ length c a ≠ 0 ∧ length b c ≠ 0 ∧ length c b ≠ 0
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/pythagoras_2.lean | aux2 | [13, 1] | [19, 27] | exact ⟨ ba, ca, bc, cb ⟩ | i : incidence_geometry
a b c : point
Tabc : triangle a b c
ba : ¬length b a = 0
ca : ¬length c a = 0
bc : ¬length b c = 0
cb : length c b = 0 → False
⊢ length b a ≠ 0 ∧ length c a ≠ 0 ∧ length b c ≠ 0 ∧ length c b ≠ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
i : incidence_geometry
a b c : point
Tabc : triangle a b c
ba : ¬length b a = 0
ca : ¬length c a = 0
bc : ¬length b c = 0
cb : length c b = 0 → False
⊢ length b a ≠ 0 ∧ length c a ≠ 0 ∧ length b c ≠ 0 ∧ length c b ≠ 0
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/pythagoras_2.lean | pythagorean_proof_two | [31, 1] | [44, 65] | obtain ⟨ d, Bbdc, ang_bda, ang_cda ⟩ := aux Tabc ang_a.symm.le | i : incidence_geometry
a b c : point
Tabc : triangle a b c
ang_a : angle b a c = rightangle
⊢ length a b ^ 2 + length a c ^ 2 = length b c ^ 2 | case intro.intro.intro
i : incidence_geometry
a b c : point
Tabc : triangle a b c
ang_a : angle b a c = rightangle
d : point
Bbdc : B b d c
ang_bda : angle b d a = rightangle
ang_cda : angle c d a = rightangle
⊢ length a b ^ 2 + length a c ^ 2 = length b c ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
i : incidence_geometry
a b c : point
Tabc : triangle a b c
ang_a : angle b a c = rightangle
⊢ length a b ^ 2 + length a c ^ 2 = length b c ^ 2
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/pythagoras_2.lean | pythagorean_proof_two | [31, 1] | [44, 65] | obtain ⟨ ba, ca, bc, cb ⟩ := aux2 Tabc | case intro.intro.intro
i : incidence_geometry
a b c : point
Tabc : triangle a b c
ang_a : angle b a c = rightangle
d : point
Bbdc : B b d c
ang_bda : angle b d a = rightangle
ang_cda : angle c d a = rightangle
⊢ length a b ^ 2 + length a c ^ 2 = length b c ^ 2 | case intro.intro.intro.intro.intro.intro
i : incidence_geometry
a b c : point
Tabc : triangle a b c
ang_a : angle b a c = rightangle
d : point
Bbdc : B b d c
ang_bda : angle b d a = rightangle
ang_cda : angle c d a = rightangle
ba : length b a ≠ 0
ca : length c a ≠ 0
bc : length b c ≠ 0
cb : length c b ≠ 0
⊢ length a b ^ 2 + length a c ^ 2 = length b c ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
i : incidence_geometry
a b c : point
Tabc : triangle a b c
ang_a : angle b a c = rightangle
d : point
Bbdc : B b d c
ang_bda : angle b d a = rightangle
ang_cda : angle c d a = rightangle
⊢ length a b ^ 2 + length a c ^ 2 = length b c ^ 2
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/pythagoras_2.lean | pythagorean_proof_two | [31, 1] | [44, 65] | obtain ⟨ Tdba, Tdca ⟩ := aux3 Tabc Bbdc | case intro.intro.intro.intro.intro.intro
i : incidence_geometry
a b c : point
Tabc : triangle a b c
ang_a : angle b a c = rightangle
d : point
Bbdc : B b d c
ang_bda : angle b d a = rightangle
ang_cda : angle c d a = rightangle
ba : length b a ≠ 0
ca : length c a ≠ 0
bc : length b c ≠ 0
cb : length c b ≠ 0
⊢ length a b ^ 2 + length a c ^ 2 = length b c ^ 2 | case intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
a b c : point
Tabc : triangle a b c
ang_a : angle b a c = rightangle
d : point
Bbdc : B b d c
ang_bda : angle b d a = rightangle
ang_cda : angle c d a = rightangle
ba : length b a ≠ 0
ca : length c a ≠ 0
bc : length b c ≠ 0
cb : length c b ≠ 0
Tdba : triangle d b a
Tdca : triangle d c a
⊢ length a b ^ 2 + length a c ^ 2 = length b c ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro
i : incidence_geometry
a b c : point
Tabc : triangle a b c
ang_a : angle b a c = rightangle
d : point
Bbdc : B b d c
ang_bda : angle b d a = rightangle
ang_cda : angle c d a = rightangle
ba : length b a ≠ 0
ca : length c a ≠ 0
bc : length b c ≠ 0
cb : length c b ≠ 0
⊢ length a b ^ 2 + length a c ^ 2 = length b c ^ 2
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/pythagoras_2.lean | pythagorean_proof_two | [31, 1] | [44, 65] | have abc_dba : angle a b c = angle d b a := by linperm [aux4 ba Bbdc] | case intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
a b c : point
Tabc : triangle a b c
ang_a : angle b a c = rightangle
d : point
Bbdc : B b d c
ang_bda : angle b d a = rightangle
ang_cda : angle c d a = rightangle
ba : length b a ≠ 0
ca : length c a ≠ 0
bc : length b c ≠ 0
cb : length c b ≠ 0
Tdba : triangle d b a
Tdca : triangle d c a
⊢ length a b ^ 2 + length a c ^ 2 = length b c ^ 2 | case intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
a b c : point
Tabc : triangle a b c
ang_a : angle b a c = rightangle
d : point
Bbdc : B b d c
ang_bda : angle b d a = rightangle
ang_cda : angle c d a = rightangle
ba : length b a ≠ 0
ca : length c a ≠ 0
bc : length b c ≠ 0
cb : length c b ≠ 0
Tdba : triangle d b a
Tdca : triangle d c a
abc_dba : angle a b c = angle d b a
⊢ length a b ^ 2 + length a c ^ 2 = length b c ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
a b c : point
Tabc : triangle a b c
ang_a : angle b a c = rightangle
d : point
Bbdc : B b d c
ang_bda : angle b d a = rightangle
ang_cda : angle c d a = rightangle
ba : length b a ≠ 0
ca : length c a ≠ 0
bc : length b c ≠ 0
cb : length c b ≠ 0
Tdba : triangle d b a
Tdca : triangle d c a
⊢ length a b ^ 2 + length a c ^ 2 = length b c ^ 2
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/pythagoras_2.lean | pythagorean_proof_two | [31, 1] | [44, 65] | have acb_dca : angle a c b = angle d c a := by linperm [aux4 ca $ B_symm Bbdc] | case intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
a b c : point
Tabc : triangle a b c
ang_a : angle b a c = rightangle
d : point
Bbdc : B b d c
ang_bda : angle b d a = rightangle
ang_cda : angle c d a = rightangle
ba : length b a ≠ 0
ca : length c a ≠ 0
bc : length b c ≠ 0
cb : length c b ≠ 0
Tdba : triangle d b a
Tdca : triangle d c a
abc_dba : angle a b c = angle d b a
⊢ length a b ^ 2 + length a c ^ 2 = length b c ^ 2 | case intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
a b c : point
Tabc : triangle a b c
ang_a : angle b a c = rightangle
d : point
Bbdc : B b d c
ang_bda : angle b d a = rightangle
ang_cda : angle c d a = rightangle
ba : length b a ≠ 0
ca : length c a ≠ 0
bc : length b c ≠ 0
cb : length c b ≠ 0
Tdba : triangle d b a
Tdca : triangle d c a
abc_dba : angle a b c = angle d b a
acb_dca : angle a c b = angle d c a
⊢ length a b ^ 2 + length a c ^ 2 = length b c ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
a b c : point
Tabc : triangle a b c
ang_a : angle b a c = rightangle
d : point
Bbdc : B b d c
ang_bda : angle b d a = rightangle
ang_cda : angle c d a = rightangle
ba : length b a ≠ 0
ca : length c a ≠ 0
bc : length b c ≠ 0
cb : length c b ≠ 0
Tdba : triangle d b a
Tdca : triangle d c a
abc_dba : angle a b c = angle d b a
⊢ length a b ^ 2 + length a c ^ 2 = length b c ^ 2
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/pythagoras_2.lean | pythagorean_proof_two | [31, 1] | [44, 65] | have prop1 := similar_of_AA (tri213 Tdba) (tri213 Tabc) abc_dba.symm (by linarith) | case intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
a b c : point
Tabc : triangle a b c
ang_a : angle b a c = rightangle
d : point
Bbdc : B b d c
ang_bda : angle b d a = rightangle
ang_cda : angle c d a = rightangle
ba : length b a ≠ 0
ca : length c a ≠ 0
bc : length b c ≠ 0
cb : length c b ≠ 0
Tdba : triangle d b a
Tdca : triangle d c a
abc_dba : angle a b c = angle d b a
acb_dca : angle a c b = angle d c a
⊢ length a b ^ 2 + length a c ^ 2 = length b c ^ 2 | case intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
a b c : point
Tabc : triangle a b c
ang_a : angle b a c = rightangle
d : point
Bbdc : B b d c
ang_bda : angle b d a = rightangle
ang_cda : angle c d a = rightangle
ba : length b a ≠ 0
ca : length c a ≠ 0
bc : length b c ≠ 0
cb : length c b ≠ 0
Tdba : triangle d b a
Tdca : triangle d c a
abc_dba : angle a b c = angle d b a
acb_dca : angle a c b = angle d c a
prop1 : proportion (length b d) (length b a) (length b a) (length b c)
⊢ length a b ^ 2 + length a c ^ 2 = length b c ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
a b c : point
Tabc : triangle a b c
ang_a : angle b a c = rightangle
d : point
Bbdc : B b d c
ang_bda : angle b d a = rightangle
ang_cda : angle c d a = rightangle
ba : length b a ≠ 0
ca : length c a ≠ 0
bc : length b c ≠ 0
cb : length c b ≠ 0
Tdba : triangle d b a
Tdca : triangle d c a
abc_dba : angle a b c = angle d b a
acb_dca : angle a c b = angle d c a
⊢ length a b ^ 2 + length a c ^ 2 = length b c ^ 2
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/pythagoras_2.lean | pythagorean_proof_two | [31, 1] | [44, 65] | have rat1 := (proportion_len_iff _ _ _ _ _ _ _ _ ba bc).mpr prop1 | case intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
a b c : point
Tabc : triangle a b c
ang_a : angle b a c = rightangle
d : point
Bbdc : B b d c
ang_bda : angle b d a = rightangle
ang_cda : angle c d a = rightangle
ba : length b a ≠ 0
ca : length c a ≠ 0
bc : length b c ≠ 0
cb : length c b ≠ 0
Tdba : triangle d b a
Tdca : triangle d c a
abc_dba : angle a b c = angle d b a
acb_dca : angle a c b = angle d c a
prop1 : proportion (length b d) (length b a) (length b a) (length b c)
⊢ length a b ^ 2 + length a c ^ 2 = length b c ^ 2 | case intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
a b c : point
Tabc : triangle a b c
ang_a : angle b a c = rightangle
d : point
Bbdc : B b d c
ang_bda : angle b d a = rightangle
ang_cda : angle c d a = rightangle
ba : length b a ≠ 0
ca : length c a ≠ 0
bc : length b c ≠ 0
cb : length c b ≠ 0
Tdba : triangle d b a
Tdca : triangle d c a
abc_dba : angle a b c = angle d b a
acb_dca : angle a c b = angle d c a
prop1 : proportion (length b d) (length b a) (length b a) (length b c)
rat1 : length b d / length b a = length b a / length b c
⊢ length a b ^ 2 + length a c ^ 2 = length b c ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
a b c : point
Tabc : triangle a b c
ang_a : angle b a c = rightangle
d : point
Bbdc : B b d c
ang_bda : angle b d a = rightangle
ang_cda : angle c d a = rightangle
ba : length b a ≠ 0
ca : length c a ≠ 0
bc : length b c ≠ 0
cb : length c b ≠ 0
Tdba : triangle d b a
Tdca : triangle d c a
abc_dba : angle a b c = angle d b a
acb_dca : angle a c b = angle d c a
prop1 : proportion (length b d) (length b a) (length b a) (length b c)
⊢ length a b ^ 2 + length a c ^ 2 = length b c ^ 2
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/pythagoras_2.lean | pythagorean_proof_two | [31, 1] | [44, 65] | field_simp at rat1 | case intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
a b c : point
Tabc : triangle a b c
ang_a : angle b a c = rightangle
d : point
Bbdc : B b d c
ang_bda : angle b d a = rightangle
ang_cda : angle c d a = rightangle
ba : length b a ≠ 0
ca : length c a ≠ 0
bc : length b c ≠ 0
cb : length c b ≠ 0
Tdba : triangle d b a
Tdca : triangle d c a
abc_dba : angle a b c = angle d b a
acb_dca : angle a c b = angle d c a
prop1 : proportion (length b d) (length b a) (length b a) (length b c)
rat1 : length b d / length b a = length b a / length b c
⊢ length a b ^ 2 + length a c ^ 2 = length b c ^ 2 | case intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
a b c : point
Tabc : triangle a b c
ang_a : angle b a c = rightangle
d : point
Bbdc : B b d c
ang_bda : angle b d a = rightangle
ang_cda : angle c d a = rightangle
ba : length b a ≠ 0
ca : length c a ≠ 0
bc : length b c ≠ 0
cb : length c b ≠ 0
Tdba : triangle d b a
Tdca : triangle d c a
abc_dba : angle a b c = angle d b a
acb_dca : angle a c b = angle d c a
prop1 : proportion (length b d) (length b a) (length b a) (length b c)
rat1 : length b d * length b c = length b a * length b a
⊢ length a b ^ 2 + length a c ^ 2 = length b c ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
a b c : point
Tabc : triangle a b c
ang_a : angle b a c = rightangle
d : point
Bbdc : B b d c
ang_bda : angle b d a = rightangle
ang_cda : angle c d a = rightangle
ba : length b a ≠ 0
ca : length c a ≠ 0
bc : length b c ≠ 0
cb : length c b ≠ 0
Tdba : triangle d b a
Tdca : triangle d c a
abc_dba : angle a b c = angle d b a
acb_dca : angle a c b = angle d c a
prop1 : proportion (length b d) (length b a) (length b a) (length b c)
rat1 : length b d / length b a = length b a / length b c
⊢ length a b ^ 2 + length a c ^ 2 = length b c ^ 2
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/pythagoras_2.lean | pythagorean_proof_two | [31, 1] | [44, 65] | have prop2 := similar_of_AA (tri213 Tdca) (tri312 Tabc) acb_dca.symm (by linperm) | case intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
a b c : point
Tabc : triangle a b c
ang_a : angle b a c = rightangle
d : point
Bbdc : B b d c
ang_bda : angle b d a = rightangle
ang_cda : angle c d a = rightangle
ba : length b a ≠ 0
ca : length c a ≠ 0
bc : length b c ≠ 0
cb : length c b ≠ 0
Tdba : triangle d b a
Tdca : triangle d c a
abc_dba : angle a b c = angle d b a
acb_dca : angle a c b = angle d c a
prop1 : proportion (length b d) (length b a) (length b a) (length b c)
rat1 : length b d * length b c = length b a * length b a
⊢ length a b ^ 2 + length a c ^ 2 = length b c ^ 2 | case intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
a b c : point
Tabc : triangle a b c
ang_a : angle b a c = rightangle
d : point
Bbdc : B b d c
ang_bda : angle b d a = rightangle
ang_cda : angle c d a = rightangle
ba : length b a ≠ 0
ca : length c a ≠ 0
bc : length b c ≠ 0
cb : length c b ≠ 0
Tdba : triangle d b a
Tdca : triangle d c a
abc_dba : angle a b c = angle d b a
acb_dca : angle a c b = angle d c a
prop1 : proportion (length b d) (length b a) (length b a) (length b c)
rat1 : length b d * length b c = length b a * length b a
prop2 : proportion (length c d) (length c a) (length c a) (length c b)
⊢ length a b ^ 2 + length a c ^ 2 = length b c ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
a b c : point
Tabc : triangle a b c
ang_a : angle b a c = rightangle
d : point
Bbdc : B b d c
ang_bda : angle b d a = rightangle
ang_cda : angle c d a = rightangle
ba : length b a ≠ 0
ca : length c a ≠ 0
bc : length b c ≠ 0
cb : length c b ≠ 0
Tdba : triangle d b a
Tdca : triangle d c a
abc_dba : angle a b c = angle d b a
acb_dca : angle a c b = angle d c a
prop1 : proportion (length b d) (length b a) (length b a) (length b c)
rat1 : length b d * length b c = length b a * length b a
⊢ length a b ^ 2 + length a c ^ 2 = length b c ^ 2
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/pythagoras_2.lean | pythagorean_proof_two | [31, 1] | [44, 65] | have rat2 := (proportion_len_iff _ _ _ _ _ _ _ _ ca cb).mpr prop2 | case intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
a b c : point
Tabc : triangle a b c
ang_a : angle b a c = rightangle
d : point
Bbdc : B b d c
ang_bda : angle b d a = rightangle
ang_cda : angle c d a = rightangle
ba : length b a ≠ 0
ca : length c a ≠ 0
bc : length b c ≠ 0
cb : length c b ≠ 0
Tdba : triangle d b a
Tdca : triangle d c a
abc_dba : angle a b c = angle d b a
acb_dca : angle a c b = angle d c a
prop1 : proportion (length b d) (length b a) (length b a) (length b c)
rat1 : length b d * length b c = length b a * length b a
prop2 : proportion (length c d) (length c a) (length c a) (length c b)
⊢ length a b ^ 2 + length a c ^ 2 = length b c ^ 2 | case intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
a b c : point
Tabc : triangle a b c
ang_a : angle b a c = rightangle
d : point
Bbdc : B b d c
ang_bda : angle b d a = rightangle
ang_cda : angle c d a = rightangle
ba : length b a ≠ 0
ca : length c a ≠ 0
bc : length b c ≠ 0
cb : length c b ≠ 0
Tdba : triangle d b a
Tdca : triangle d c a
abc_dba : angle a b c = angle d b a
acb_dca : angle a c b = angle d c a
prop1 : proportion (length b d) (length b a) (length b a) (length b c)
rat1 : length b d * length b c = length b a * length b a
prop2 : proportion (length c d) (length c a) (length c a) (length c b)
rat2 : length c d / length c a = length c a / length c b
⊢ length a b ^ 2 + length a c ^ 2 = length b c ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
a b c : point
Tabc : triangle a b c
ang_a : angle b a c = rightangle
d : point
Bbdc : B b d c
ang_bda : angle b d a = rightangle
ang_cda : angle c d a = rightangle
ba : length b a ≠ 0
ca : length c a ≠ 0
bc : length b c ≠ 0
cb : length c b ≠ 0
Tdba : triangle d b a
Tdca : triangle d c a
abc_dba : angle a b c = angle d b a
acb_dca : angle a c b = angle d c a
prop1 : proportion (length b d) (length b a) (length b a) (length b c)
rat1 : length b d * length b c = length b a * length b a
prop2 : proportion (length c d) (length c a) (length c a) (length c b)
⊢ length a b ^ 2 + length a c ^ 2 = length b c ^ 2
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/pythagoras_2.lean | pythagorean_proof_two | [31, 1] | [44, 65] | field_simp at rat2 | case intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
a b c : point
Tabc : triangle a b c
ang_a : angle b a c = rightangle
d : point
Bbdc : B b d c
ang_bda : angle b d a = rightangle
ang_cda : angle c d a = rightangle
ba : length b a ≠ 0
ca : length c a ≠ 0
bc : length b c ≠ 0
cb : length c b ≠ 0
Tdba : triangle d b a
Tdca : triangle d c a
abc_dba : angle a b c = angle d b a
acb_dca : angle a c b = angle d c a
prop1 : proportion (length b d) (length b a) (length b a) (length b c)
rat1 : length b d * length b c = length b a * length b a
prop2 : proportion (length c d) (length c a) (length c a) (length c b)
rat2 : length c d / length c a = length c a / length c b
⊢ length a b ^ 2 + length a c ^ 2 = length b c ^ 2 | case intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
a b c : point
Tabc : triangle a b c
ang_a : angle b a c = rightangle
d : point
Bbdc : B b d c
ang_bda : angle b d a = rightangle
ang_cda : angle c d a = rightangle
ba : length b a ≠ 0
ca : length c a ≠ 0
bc : length b c ≠ 0
cb : length c b ≠ 0
Tdba : triangle d b a
Tdca : triangle d c a
abc_dba : angle a b c = angle d b a
acb_dca : angle a c b = angle d c a
prop1 : proportion (length b d) (length b a) (length b a) (length b c)
rat1 : length b d * length b c = length b a * length b a
prop2 : proportion (length c d) (length c a) (length c a) (length c b)
rat2 : length c d * length c b = length c a * length c a
⊢ length a b ^ 2 + length a c ^ 2 = length b c ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
a b c : point
Tabc : triangle a b c
ang_a : angle b a c = rightangle
d : point
Bbdc : B b d c
ang_bda : angle b d a = rightangle
ang_cda : angle c d a = rightangle
ba : length b a ≠ 0
ca : length c a ≠ 0
bc : length b c ≠ 0
cb : length c b ≠ 0
Tdba : triangle d b a
Tdca : triangle d c a
abc_dba : angle a b c = angle d b a
acb_dca : angle a c b = angle d c a
prop1 : proportion (length b d) (length b a) (length b a) (length b c)
rat1 : length b d * length b c = length b a * length b a
prop2 : proportion (length c d) (length c a) (length c a) (length c b)
rat2 : length c d / length c a = length c a / length c b
⊢ length a b ^ 2 + length a c ^ 2 = length b c ^ 2
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/pythagoras_2.lean | pythagorean_proof_two | [31, 1] | [44, 65] | perm [length_sum_of_B Bbdc] | case intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
a b c : point
Tabc : triangle a b c
ang_a : angle b a c = rightangle
d : point
Bbdc : B b d c
ang_bda : angle b d a = rightangle
ang_cda : angle c d a = rightangle
ba : length b a ≠ 0
ca : length c a ≠ 0
bc : length b c ≠ 0
cb : length c b ≠ 0
Tdba : triangle d b a
Tdca : triangle d c a
abc_dba : angle a b c = angle d b a
acb_dca : angle a c b = angle d c a
prop1 : proportion (length b d) (length b a) (length b a) (length b c)
rat1 : length b d * length b c = length b a * length b a
prop2 : proportion (length c d) (length c a) (length c a) (length c b)
rat2 : length c d * length c b = length c a * length c a
⊢ length a b ^ 2 + length a c ^ 2 = length b c ^ 2 | case intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
a b c : point
Tabc : triangle a b c
ang_a : angle b a c = rightangle
d : point
Bbdc : B b d c
ang_bda : angle b d a = rightangle
ang_cda : angle c d a = rightangle
ba : length b a ≠ 0
ca : length c a ≠ 0
bc : length b c ≠ 0
cb : length c b ≠ 0
Tdba : triangle d b a
Tdca : triangle d c a
abc_dba : angle a b c = angle d b a
acb_dca : angle a c b = angle d c a
prop1 : proportion (length b d) (length b a) (length b a) (length b c)
rat1 : length b d * length b c = length b a * length b a
prop2 : proportion (length c d) (length c a) (length c a) (length c b)
rat2 : length c d * length c b = length c a * length c a
this : length b d + length c d = length b c
⊢ length a b ^ 2 + length a c ^ 2 = length b c ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
a b c : point
Tabc : triangle a b c
ang_a : angle b a c = rightangle
d : point
Bbdc : B b d c
ang_bda : angle b d a = rightangle
ang_cda : angle c d a = rightangle
ba : length b a ≠ 0
ca : length c a ≠ 0
bc : length b c ≠ 0
cb : length c b ≠ 0
Tdba : triangle d b a
Tdca : triangle d c a
abc_dba : angle a b c = angle d b a
acb_dca : angle a c b = angle d c a
prop1 : proportion (length b d) (length b a) (length b a) (length b c)
rat1 : length b d * length b c = length b a * length b a
prop2 : proportion (length c d) (length c a) (length c a) (length c b)
rat2 : length c d * length c b = length c a * length c a
⊢ length a b ^ 2 + length a c ^ 2 = length b c ^ 2
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/pythagoras_2.lean | pythagorean_proof_two | [31, 1] | [44, 65] | conv in (occs := *) length _ _ ^ 2 => all_goals rw [sq] | case intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
a b c : point
Tabc : triangle a b c
ang_a : angle b a c = rightangle
d : point
Bbdc : B b d c
ang_bda : angle b d a = rightangle
ang_cda : angle c d a = rightangle
ba : length b a ≠ 0
ca : length c a ≠ 0
bc : length b c ≠ 0
cb : length c b ≠ 0
Tdba : triangle d b a
Tdca : triangle d c a
abc_dba : angle a b c = angle d b a
acb_dca : angle a c b = angle d c a
prop1 : proportion (length b d) (length b a) (length b a) (length b c)
rat1 : length b d * length b c = length b a * length b a
prop2 : proportion (length c d) (length c a) (length c a) (length c b)
rat2 : length c d * length c b = length c a * length c a
this : length b d + length c d = length b c
⊢ length a b ^ 2 + length a c ^ 2 = length b c ^ 2 | case intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
a b c : point
Tabc : triangle a b c
ang_a : angle b a c = rightangle
d : point
Bbdc : B b d c
ang_bda : angle b d a = rightangle
ang_cda : angle c d a = rightangle
ba : length b a ≠ 0
ca : length c a ≠ 0
bc : length b c ≠ 0
cb : length c b ≠ 0
Tdba : triangle d b a
Tdca : triangle d c a
abc_dba : angle a b c = angle d b a
acb_dca : angle a c b = angle d c a
prop1 : proportion (length b d) (length b a) (length b a) (length b c)
rat1 : length b d * length b c = length b a * length b a
prop2 : proportion (length c d) (length c a) (length c a) (length c b)
rat2 : length c d * length c b = length c a * length c a
this : length b d + length c d = length b c
⊢ length a b * length a b + length a c * length a c = length b c * length b c | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
a b c : point
Tabc : triangle a b c
ang_a : angle b a c = rightangle
d : point
Bbdc : B b d c
ang_bda : angle b d a = rightangle
ang_cda : angle c d a = rightangle
ba : length b a ≠ 0
ca : length c a ≠ 0
bc : length b c ≠ 0
cb : length c b ≠ 0
Tdba : triangle d b a
Tdca : triangle d c a
abc_dba : angle a b c = angle d b a
acb_dca : angle a c b = angle d c a
prop1 : proportion (length b d) (length b a) (length b a) (length b c)
rat1 : length b d * length b c = length b a * length b a
prop2 : proportion (length c d) (length c a) (length c a) (length c b)
rat2 : length c d * length c b = length c a * length c a
this : length b d + length c d = length b c
⊢ length a b ^ 2 + length a c ^ 2 = length b c ^ 2
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/pythagoras_2.lean | pythagorean_proof_two | [31, 1] | [44, 65] | perm at rat1, rat2 | case intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
a b c : point
Tabc : triangle a b c
ang_a : angle b a c = rightangle
d : point
Bbdc : B b d c
ang_bda : angle b d a = rightangle
ang_cda : angle c d a = rightangle
ba : length b a ≠ 0
ca : length c a ≠ 0
bc : length b c ≠ 0
cb : length c b ≠ 0
Tdba : triangle d b a
Tdca : triangle d c a
abc_dba : angle a b c = angle d b a
acb_dca : angle a c b = angle d c a
prop1 : proportion (length b d) (length b a) (length b a) (length b c)
rat1 : length b d * length b c = length b a * length b a
prop2 : proportion (length c d) (length c a) (length c a) (length c b)
rat2 : length c d * length c b = length c a * length c a
this : length b d + length c d = length b c
⊢ length a b * length a b + length a c * length a c = length b c * length b c | case intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
a b c : point
Tabc : triangle a b c
ang_a : angle b a c = rightangle
d : point
Bbdc : B b d c
ang_bda : angle b d a = rightangle
ang_cda : angle c d a = rightangle
ba : length b a ≠ 0
ca : length c a ≠ 0
bc : length b c ≠ 0
cb : length c b ≠ 0
Tdba : triangle d b a
Tdca : triangle d c a
abc_dba : angle a b c = angle d b a
acb_dca : angle a c b = angle d c a
prop1 : proportion (length b d) (length b a) (length b a) (length b c)
rat1 : length b d * length b c = length a b * length a b
prop2 : proportion (length c d) (length c a) (length c a) (length c b)
rat2 : length c d * length b c = length a c * length a c
this : length b d + length c d = length b c
⊢ length a b * length a b + length a c * length a c = length b c * length b c | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
a b c : point
Tabc : triangle a b c
ang_a : angle b a c = rightangle
d : point
Bbdc : B b d c
ang_bda : angle b d a = rightangle
ang_cda : angle c d a = rightangle
ba : length b a ≠ 0
ca : length c a ≠ 0
bc : length b c ≠ 0
cb : length c b ≠ 0
Tdba : triangle d b a
Tdca : triangle d c a
abc_dba : angle a b c = angle d b a
acb_dca : angle a c b = angle d c a
prop1 : proportion (length b d) (length b a) (length b a) (length b c)
rat1 : length b d * length b c = length b a * length b a
prop2 : proportion (length c d) (length c a) (length c a) (length c b)
rat2 : length c d * length c b = length c a * length c a
this : length b d + length c d = length b c
⊢ length a b * length a b + length a c * length a c = length b c * length b c
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/pythagoras_2.lean | pythagorean_proof_two | [31, 1] | [44, 65] | rw [← rat1, ← rat2, ← right_distrib, this] | case intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
a b c : point
Tabc : triangle a b c
ang_a : angle b a c = rightangle
d : point
Bbdc : B b d c
ang_bda : angle b d a = rightangle
ang_cda : angle c d a = rightangle
ba : length b a ≠ 0
ca : length c a ≠ 0
bc : length b c ≠ 0
cb : length c b ≠ 0
Tdba : triangle d b a
Tdca : triangle d c a
abc_dba : angle a b c = angle d b a
acb_dca : angle a c b = angle d c a
prop1 : proportion (length b d) (length b a) (length b a) (length b c)
rat1 : length b d * length b c = length a b * length a b
prop2 : proportion (length c d) (length c a) (length c a) (length c b)
rat2 : length c d * length b c = length a c * length a c
this : length b d + length c d = length b c
⊢ length a b * length a b + length a c * length a c = length b c * length b c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
a b c : point
Tabc : triangle a b c
ang_a : angle b a c = rightangle
d : point
Bbdc : B b d c
ang_bda : angle b d a = rightangle
ang_cda : angle c d a = rightangle
ba : length b a ≠ 0
ca : length c a ≠ 0
bc : length b c ≠ 0
cb : length c b ≠ 0
Tdba : triangle d b a
Tdca : triangle d c a
abc_dba : angle a b c = angle d b a
acb_dca : angle a c b = angle d c a
prop1 : proportion (length b d) (length b a) (length b a) (length b c)
rat1 : length b d * length b c = length a b * length a b
prop2 : proportion (length c d) (length c a) (length c a) (length c b)
rat2 : length c d * length b c = length a c * length a c
this : length b d + length c d = length b c
⊢ length a b * length a b + length a c * length a c = length b c * length b c
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/pythagoras_2.lean | pythagorean_proof_two | [31, 1] | [44, 65] | linperm [aux4 ba Bbdc] | i : incidence_geometry
a b c : point
Tabc : triangle a b c
ang_a : angle b a c = rightangle
d : point
Bbdc : B b d c
ang_bda : angle b d a = rightangle
ang_cda : angle c d a = rightangle
ba : length b a ≠ 0
ca : length c a ≠ 0
bc : length b c ≠ 0
cb : length c b ≠ 0
Tdba : triangle d b a
Tdca : triangle d c a
⊢ angle a b c = angle d b a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
i : incidence_geometry
a b c : point
Tabc : triangle a b c
ang_a : angle b a c = rightangle
d : point
Bbdc : B b d c
ang_bda : angle b d a = rightangle
ang_cda : angle c d a = rightangle
ba : length b a ≠ 0
ca : length c a ≠ 0
bc : length b c ≠ 0
cb : length c b ≠ 0
Tdba : triangle d b a
Tdca : triangle d c a
⊢ angle a b c = angle d b a
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/pythagoras_2.lean | pythagorean_proof_two | [31, 1] | [44, 65] | linperm [aux4 ca $ B_symm Bbdc] | i : incidence_geometry
a b c : point
Tabc : triangle a b c
ang_a : angle b a c = rightangle
d : point
Bbdc : B b d c
ang_bda : angle b d a = rightangle
ang_cda : angle c d a = rightangle
ba : length b a ≠ 0
ca : length c a ≠ 0
bc : length b c ≠ 0
cb : length c b ≠ 0
Tdba : triangle d b a
Tdca : triangle d c a
abc_dba : angle a b c = angle d b a
⊢ angle a c b = angle d c a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
i : incidence_geometry
a b c : point
Tabc : triangle a b c
ang_a : angle b a c = rightangle
d : point
Bbdc : B b d c
ang_bda : angle b d a = rightangle
ang_cda : angle c d a = rightangle
ba : length b a ≠ 0
ca : length c a ≠ 0
bc : length b c ≠ 0
cb : length c b ≠ 0
Tdba : triangle d b a
Tdca : triangle d c a
abc_dba : angle a b c = angle d b a
⊢ angle a c b = angle d c a
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/pythagoras_2.lean | pythagorean_proof_two | [31, 1] | [44, 65] | linarith | i : incidence_geometry
a b c : point
Tabc : triangle a b c
ang_a : angle b a c = rightangle
d : point
Bbdc : B b d c
ang_bda : angle b d a = rightangle
ang_cda : angle c d a = rightangle
ba : length b a ≠ 0
ca : length c a ≠ 0
bc : length b c ≠ 0
cb : length c b ≠ 0
Tdba : triangle d b a
Tdca : triangle d c a
abc_dba : angle a b c = angle d b a
acb_dca : angle a c b = angle d c a
⊢ angle b d a = angle b a c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
i : incidence_geometry
a b c : point
Tabc : triangle a b c
ang_a : angle b a c = rightangle
d : point
Bbdc : B b d c
ang_bda : angle b d a = rightangle
ang_cda : angle c d a = rightangle
ba : length b a ≠ 0
ca : length c a ≠ 0
bc : length b c ≠ 0
cb : length c b ≠ 0
Tdba : triangle d b a
Tdca : triangle d c a
abc_dba : angle a b c = angle d b a
acb_dca : angle a c b = angle d c a
⊢ angle b d a = angle b a c
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/pythagoras_2.lean | pythagorean_proof_two | [31, 1] | [44, 65] | linperm | i : incidence_geometry
a b c : point
Tabc : triangle a b c
ang_a : angle b a c = rightangle
d : point
Bbdc : B b d c
ang_bda : angle b d a = rightangle
ang_cda : angle c d a = rightangle
ba : length b a ≠ 0
ca : length c a ≠ 0
bc : length b c ≠ 0
cb : length c b ≠ 0
Tdba : triangle d b a
Tdca : triangle d c a
abc_dba : angle a b c = angle d b a
acb_dca : angle a c b = angle d c a
prop1 : proportion (length b d) (length b a) (length b a) (length b c)
rat1 : length b d * length b c = length b a * length b a
⊢ angle c d a = angle c a b | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
i : incidence_geometry
a b c : point
Tabc : triangle a b c
ang_a : angle b a c = rightangle
d : point
Bbdc : B b d c
ang_bda : angle b d a = rightangle
ang_cda : angle c d a = rightangle
ba : length b a ≠ 0
ca : length c a ≠ 0
bc : length b c ≠ 0
cb : length c b ≠ 0
Tdba : triangle d b a
Tdca : triangle d c a
abc_dba : angle a b c = angle d b a
acb_dca : angle a c b = angle d c a
prop1 : proportion (length b d) (length b a) (length b a) (length b c)
rat1 : length b d * length b c = length b a * length b a
⊢ angle c d a = angle c a b
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/pythagoras_2.lean | perpendicular_of_online'' | [47, 1] | [52, 116] | rcases length_eq_B_of_ne ab ab.symm with ⟨d, Babd, _⟩ | i : incidence_geometry
a b : point
L : line
f : point
ab : a ≠ b
aL : online a L
bL : online b L
fL : ¬online f L
⊢ ∃ c, diffside c f L ∧ angle a b c = rightangle | case intro.intro
i : incidence_geometry
a b : point
L : line
f : point
ab : a ≠ b
aL : online a L
bL : online b L
fL : ¬online f L
d : point
Babd : B a b d
right✝ : length b a = length b d
⊢ ∃ c, diffside c f L ∧ angle a b c = rightangle | Please generate a tactic in lean4 to solve the state.
STATE:
i : incidence_geometry
a b : point
L : line
f : point
ab : a ≠ b
aL : online a L
bL : online b L
fL : ¬online f L
⊢ ∃ c, diffside c f L ∧ angle a b c = rightangle
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/pythagoras_2.lean | perpendicular_of_online'' | [47, 1] | [52, 116] | rcases diffside_of_not_online fL with ⟨f', f'L, ff'L⟩ | case intro.intro
i : incidence_geometry
a b : point
L : line
f : point
ab : a ≠ b
aL : online a L
bL : online b L
fL : ¬online f L
d : point
Babd : B a b d
right✝ : length b a = length b d
⊢ ∃ c, diffside c f L ∧ angle a b c = rightangle | case intro.intro.intro.intro
i : incidence_geometry
a b : point
L : line
f : point
ab : a ≠ b
aL : online a L
bL : online b L
fL : ¬online f L
d : point
Babd : B a b d
right✝ : length b a = length b d
f' : point
f'L : ¬online f' L
ff'L : ¬sameside f f' L
⊢ ∃ c, diffside c f L ∧ angle a b c = rightangle | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
i : incidence_geometry
a b : point
L : line
f : point
ab : a ≠ b
aL : online a L
bL : online b L
fL : ¬online f L
d : point
Babd : B a b d
right✝ : length b a = length b d
⊢ ∃ c, diffside c f L ∧ angle a b c = rightangle
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/pythagoras_2.lean | perpendicular_of_online'' | [47, 1] | [52, 116] | rcases perpendicular_of_online Babd aL bL f'L with ⟨c, cf'L, right, _⟩ | case intro.intro.intro.intro
i : incidence_geometry
a b : point
L : line
f : point
ab : a ≠ b
aL : online a L
bL : online b L
fL : ¬online f L
d : point
Babd : B a b d
right✝ : length b a = length b d
f' : point
f'L : ¬online f' L
ff'L : ¬sameside f f' L
⊢ ∃ c, diffside c f L ∧ angle a b c = rightangle | case intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
a b : point
L : line
f : point
ab : a ≠ b
aL : online a L
bL : online b L
fL : ¬online f L
d : point
Babd : B a b d
right✝¹ : length b a = length b d
f' : point
f'L : ¬online f' L
ff'L : ¬sameside f f' L
c : point
cf'L : sameside c f' L
right : angle c b a = rightangle
right✝ : angle c b d = rightangle
⊢ ∃ c, diffside c f L ∧ angle a b c = rightangle | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
i : incidence_geometry
a b : point
L : line
f : point
ab : a ≠ b
aL : online a L
bL : online b L
fL : ¬online f L
d : point
Babd : B a b d
right✝ : length b a = length b d
f' : point
f'L : ¬online f' L
ff'L : ¬sameside f f' L
⊢ ∃ c, diffside c f L ∧ angle a b c = rightangle
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/pythagoras_2.lean | perpendicular_of_online'' | [47, 1] | [52, 116] | exact ⟨c, diffside_of_sameside_diffside (sameside_symm cf'L) ⟨f'L, fL, not_sameside_symm ff'L⟩ , by perma[right]⟩ | case intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
a b : point
L : line
f : point
ab : a ≠ b
aL : online a L
bL : online b L
fL : ¬online f L
d : point
Babd : B a b d
right✝¹ : length b a = length b d
f' : point
f'L : ¬online f' L
ff'L : ¬sameside f f' L
c : point
cf'L : sameside c f' L
right : angle c b a = rightangle
right✝ : angle c b d = rightangle
⊢ ∃ c, diffside c f L ∧ angle a b c = rightangle | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
a b : point
L : line
f : point
ab : a ≠ b
aL : online a L
bL : online b L
fL : ¬online f L
d : point
Babd : B a b d
right✝¹ : length b a = length b d
f' : point
f'L : ¬online f' L
ff'L : ¬sameside f f' L
c : point
cf'L : sameside c f' L
right : angle c b a = rightangle
right✝ : angle c b d = rightangle
⊢ ∃ c, diffside c f L ∧ angle a b c = rightangle
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/pythagoras_2.lean | perpendicular_of_online'' | [47, 1] | [52, 116] | perma[right] | i : incidence_geometry
a b : point
L : line
f : point
ab : a ≠ b
aL : online a L
bL : online b L
fL : ¬online f L
d : point
Babd : B a b d
right✝¹ : length b a = length b d
f' : point
f'L : ¬online f' L
ff'L : ¬sameside f f' L
c : point
cf'L : sameside c f' L
right : angle c b a = rightangle
right✝ : angle c b d = rightangle
⊢ angle a b c = rightangle | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
i : incidence_geometry
a b : point
L : line
f : point
ab : a ≠ b
aL : online a L
bL : online b L
fL : ¬online f L
d : point
Babd : B a b d
right✝¹ : length b a = length b d
f' : point
f'L : ¬online f' L
ff'L : ¬sameside f f' L
c : point
cf'L : sameside c f' L
right : angle c b a = rightangle
right✝ : angle c b d = rightangle
⊢ angle a b c = rightangle
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/pythagoras_2.lean | pythagoras_converse | [55, 1] | [71, 101] | rcases line_of_pts a c with ⟨L, aL, cL⟩ | i : incidence_geometry
a b c : point
tri_abc : triangle a b c
sq_sum : length a b ^ 2 + length a c ^ 2 = length b c ^ 2
⊢ angle b a c = rightangle | case intro.intro
i : incidence_geometry
a b c : point
tri_abc : triangle a b c
sq_sum : length a b ^ 2 + length a c ^ 2 = length b c ^ 2
L : line
aL : online a L
cL : online c L
⊢ angle b a c = rightangle | Please generate a tactic in lean4 to solve the state.
STATE:
i : incidence_geometry
a b c : point
tri_abc : triangle a b c
sq_sum : length a b ^ 2 + length a c ^ 2 = length b c ^ 2
⊢ angle b a c = rightangle
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/pythagoras_2.lean | pythagoras_converse | [55, 1] | [71, 101] | rcases line_of_pts a b with ⟨N, aN, bN⟩ | case intro.intro
i : incidence_geometry
a b c : point
tri_abc : triangle a b c
sq_sum : length a b ^ 2 + length a c ^ 2 = length b c ^ 2
L : line
aL : online a L
cL : online c L
⊢ angle b a c = rightangle | case intro.intro.intro.intro
i : incidence_geometry
a b c : point
tri_abc : triangle a b c
sq_sum : length a b ^ 2 + length a c ^ 2 = length b c ^ 2
L : line
aL : online a L
cL : online c L
N : line
aN : online a N
bN : online b N
⊢ angle b a c = rightangle | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
i : incidence_geometry
a b c : point
tri_abc : triangle a b c
sq_sum : length a b ^ 2 + length a c ^ 2 = length b c ^ 2
L : line
aL : online a L
cL : online c L
⊢ angle b a c = rightangle
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/pythagoras_2.lean | pythagoras_converse | [55, 1] | [71, 101] | rcases length_eq_B_of_ne_four (ne_21_of_tri tri_abc) (ne_21_of_tri tri_abc) with ⟨e, Bbae, ba_ae⟩ | case intro.intro.intro.intro
i : incidence_geometry
a b c : point
tri_abc : triangle a b c
sq_sum : length a b ^ 2 + length a c ^ 2 = length b c ^ 2
L : line
aL : online a L
cL : online c L
N : line
aN : online a N
bN : online b N
⊢ angle b a c = rightangle | case intro.intro.intro.intro.intro.intro
i : incidence_geometry
a b c : point
tri_abc : triangle a b c
sq_sum : length a b ^ 2 + length a c ^ 2 = length b c ^ 2
L : line
aL : online a L
cL : online c L
N : line
aN : online a N
bN : online b N
e : point
Bbae : B b a e
ba_ae : length b a = length a e
⊢ angle b a c = rightangle | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
i : incidence_geometry
a b c : point
tri_abc : triangle a b c
sq_sum : length a b ^ 2 + length a c ^ 2 = length b c ^ 2
L : line
aL : online a L
cL : online c L
N : line
aN : online a N
bN : online b N
⊢ angle b a c = rightangle
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/pythagoras_2.lean | pythagoras_converse | [55, 1] | [71, 101] | rcases perpendicular_of_online'' (ne_31_of_tri tri_abc) cL aL (offline_of_B_offline (B_symm Bbae)
(online_3_of_B Bbae bN aN) aN aL cL (online_3_of_triangle aN bN tri_abc)) with ⟨d', d'eL, cad'_ra⟩ | case intro.intro.intro.intro.intro.intro
i : incidence_geometry
a b c : point
tri_abc : triangle a b c
sq_sum : length a b ^ 2 + length a c ^ 2 = length b c ^ 2
L : line
aL : online a L
cL : online c L
N : line
aN : online a N
bN : online b N
e : point
Bbae : B b a e
ba_ae : length b a = length a e
⊢ angle b a c = rightangle | case intro.intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
a b c : point
tri_abc : triangle a b c
sq_sum : length a b ^ 2 + length a c ^ 2 = length b c ^ 2
L : line
aL : online a L
cL : online c L
N : line
aN : online a N
bN : online b N
e : point
Bbae : B b a e
ba_ae : length b a = length a e
d' : point
d'eL : diffside d' e L
cad'_ra : angle c a d' = rightangle
⊢ angle b a c = rightangle | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro
i : incidence_geometry
a b c : point
tri_abc : triangle a b c
sq_sum : length a b ^ 2 + length a c ^ 2 = length b c ^ 2
L : line
aL : online a L
cL : online c L
N : line
aN : online a N
bN : online b N
e : point
Bbae : B b a e
ba_ae : length b a = length a e
⊢ angle b a c = rightangle
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/pythagoras_2.lean | pythagoras_converse | [55, 1] | [71, 101] | rcases length_eq_B_of_ne_four (ne_of_online aL d'eL.1).symm (ne_21_of_tri tri_abc)
with ⟨d, Bd'ad, ba_ad⟩ | case intro.intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
a b c : point
tri_abc : triangle a b c
sq_sum : length a b ^ 2 + length a c ^ 2 = length b c ^ 2
L : line
aL : online a L
cL : online c L
N : line
aN : online a N
bN : online b N
e : point
Bbae : B b a e
ba_ae : length b a = length a e
d' : point
d'eL : diffside d' e L
cad'_ra : angle c a d' = rightangle
⊢ angle b a c = rightangle | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
a b c : point
tri_abc : triangle a b c
sq_sum : length a b ^ 2 + length a c ^ 2 = length b c ^ 2
L : line
aL : online a L
cL : online c L
N : line
aN : online a N
bN : online b N
e : point
Bbae : B b a e
ba_ae : length b a = length a e
d' : point
d'eL : diffside d' e L
cad'_ra : angle c a d' = rightangle
d : point
Bd'ad : B d' a d
ba_ad : length b a = length a d
⊢ angle b a c = rightangle | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
a b c : point
tri_abc : triangle a b c
sq_sum : length a b ^ 2 + length a c ^ 2 = length b c ^ 2
L : line
aL : online a L
cL : online c L
N : line
aN : online a N
bN : online b N
e : point
Bbae : B b a e
ba_ae : length b a = length a e
d' : point
d'eL : diffside d' e L
cad'_ra : angle c a d' = rightangle
⊢ angle b a c = rightangle
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/pythagoras_2.lean | pythagoras_converse | [55, 1] | [71, 101] | rcases line_of_pts a d' with ⟨M, aM, d'M⟩ | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
a b c : point
tri_abc : triangle a b c
sq_sum : length a b ^ 2 + length a c ^ 2 = length b c ^ 2
L : line
aL : online a L
cL : online c L
N : line
aN : online a N
bN : online b N
e : point
Bbae : B b a e
ba_ae : length b a = length a e
d' : point
d'eL : diffside d' e L
cad'_ra : angle c a d' = rightangle
d : point
Bd'ad : B d' a d
ba_ad : length b a = length a d
⊢ angle b a c = rightangle | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
a b c : point
tri_abc : triangle a b c
sq_sum : length a b ^ 2 + length a c ^ 2 = length b c ^ 2
L : line
aL : online a L
cL : online c L
N : line
aN : online a N
bN : online b N
e : point
Bbae : B b a e
ba_ae : length b a = length a e
d' : point
d'eL : diffside d' e L
cad'_ra : angle c a d' = rightangle
d : point
Bd'ad : B d' a d
ba_ad : length b a = length a d
M : line
aM : online a M
d'M : online d' M
⊢ angle b a c = rightangle | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
a b c : point
tri_abc : triangle a b c
sq_sum : length a b ^ 2 + length a c ^ 2 = length b c ^ 2
L : line
aL : online a L
cL : online c L
N : line
aN : online a N
bN : online b N
e : point
Bbae : B b a e
ba_ae : length b a = length a e
d' : point
d'eL : diffside d' e L
cad'_ra : angle c a d' = rightangle
d : point
Bd'ad : B d' a d
ba_ad : length b a = length a d
⊢ angle b a c = rightangle
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/pythagoras_2.lean | pythagoras_converse | [55, 1] | [71, 101] | have cM := online_2_of_triangle aM d'M $ triangle_of_ne_online (ne_13_of_tri tri_abc) aL cL d'eL.1 | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
a b c : point
tri_abc : triangle a b c
sq_sum : length a b ^ 2 + length a c ^ 2 = length b c ^ 2
L : line
aL : online a L
cL : online c L
N : line
aN : online a N
bN : online b N
e : point
Bbae : B b a e
ba_ae : length b a = length a e
d' : point
d'eL : diffside d' e L
cad'_ra : angle c a d' = rightangle
d : point
Bd'ad : B d' a d
ba_ad : length b a = length a d
M : line
aM : online a M
d'M : online d' M
⊢ angle b a c = rightangle | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
a b c : point
tri_abc : triangle a b c
sq_sum : length a b ^ 2 + length a c ^ 2 = length b c ^ 2
L : line
aL : online a L
cL : online c L
N : line
aN : online a N
bN : online b N
e : point
Bbae : B b a e
ba_ae : length b a = length a e
d' : point
d'eL : diffside d' e L
cad'_ra : angle c a d' = rightangle
d : point
Bd'ad : B d' a d
ba_ad : length b a = length a d
M : line
aM : online a M
d'M : online d' M
cM : ¬online c M
⊢ angle b a c = rightangle | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
a b c : point
tri_abc : triangle a b c
sq_sum : length a b ^ 2 + length a c ^ 2 = length b c ^ 2
L : line
aL : online a L
cL : online c L
N : line
aN : online a N
bN : online b N
e : point
Bbae : B b a e
ba_ae : length b a = length a e
d' : point
d'eL : diffside d' e L
cad'_ra : angle c a d' = rightangle
d : point
Bd'ad : B d' a d
ba_ad : length b a = length a d
M : line
aM : online a M
d'M : online d' M
⊢ angle b a c = rightangle
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/pythagoras_2.lean | pythagoras_converse | [55, 1] | [71, 101] | have ang_split := two_right_of_flat_angle Bd'ad d'M aM cM | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
a b c : point
tri_abc : triangle a b c
sq_sum : length a b ^ 2 + length a c ^ 2 = length b c ^ 2
L : line
aL : online a L
cL : online c L
N : line
aN : online a N
bN : online b N
e : point
Bbae : B b a e
ba_ae : length b a = length a e
d' : point
d'eL : diffside d' e L
cad'_ra : angle c a d' = rightangle
d : point
Bd'ad : B d' a d
ba_ad : length b a = length a d
M : line
aM : online a M
d'M : online d' M
cM : ¬online c M
⊢ angle b a c = rightangle | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
a b c : point
tri_abc : triangle a b c
sq_sum : length a b ^ 2 + length a c ^ 2 = length b c ^ 2
L : line
aL : online a L
cL : online c L
N : line
aN : online a N
bN : online b N
e : point
Bbae : B b a e
ba_ae : length b a = length a e
d' : point
d'eL : diffside d' e L
cad'_ra : angle c a d' = rightangle
d : point
Bd'ad : B d' a d
ba_ad : length b a = length a d
M : line
aM : online a M
d'M : online d' M
cM : ¬online c M
ang_split : angle c a d' + angle c a d = 2 * rightangle
⊢ angle b a c = rightangle | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
a b c : point
tri_abc : triangle a b c
sq_sum : length a b ^ 2 + length a c ^ 2 = length b c ^ 2
L : line
aL : online a L
cL : online c L
N : line
aN : online a N
bN : online b N
e : point
Bbae : B b a e
ba_ae : length b a = length a e
d' : point
d'eL : diffside d' e L
cad'_ra : angle c a d' = rightangle
d : point
Bd'ad : B d' a d
ba_ad : length b a = length a d
M : line
aM : online a M
d'M : online d' M
cM : ¬online c M
⊢ angle b a c = rightangle
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/pythagoras_2.lean | pythagoras_converse | [55, 1] | [71, 101] | have pyth_2 := pythagorean_proof_two (triangle_of_ne_online (ne_23_of_B Bd'ad) aM
(online_3_of_B Bd'ad d'M aM) cM) (by linperm) | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
a b c : point
tri_abc : triangle a b c
sq_sum : length a b ^ 2 + length a c ^ 2 = length b c ^ 2
L : line
aL : online a L
cL : online c L
N : line
aN : online a N
bN : online b N
e : point
Bbae : B b a e
ba_ae : length b a = length a e
d' : point
d'eL : diffside d' e L
cad'_ra : angle c a d' = rightangle
d : point
Bd'ad : B d' a d
ba_ad : length b a = length a d
M : line
aM : online a M
d'M : online d' M
cM : ¬online c M
ang_split : angle c a d' + angle c a d = 2 * rightangle
⊢ angle b a c = rightangle | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
a b c : point
tri_abc : triangle a b c
sq_sum : length a b ^ 2 + length a c ^ 2 = length b c ^ 2
L : line
aL : online a L
cL : online c L
N : line
aN : online a N
bN : online b N
e : point
Bbae : B b a e
ba_ae : length b a = length a e
d' : point
d'eL : diffside d' e L
cad'_ra : angle c a d' = rightangle
d : point
Bd'ad : B d' a d
ba_ad : length b a = length a d
M : line
aM : online a M
d'M : online d' M
cM : ¬online c M
ang_split : angle c a d' + angle c a d = 2 * rightangle
pyth_2 : length a d ^ 2 + length a c ^ 2 = length d c ^ 2
⊢ angle b a c = rightangle | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
a b c : point
tri_abc : triangle a b c
sq_sum : length a b ^ 2 + length a c ^ 2 = length b c ^ 2
L : line
aL : online a L
cL : online c L
N : line
aN : online a N
bN : online b N
e : point
Bbae : B b a e
ba_ae : length b a = length a e
d' : point
d'eL : diffside d' e L
cad'_ra : angle c a d' = rightangle
d : point
Bd'ad : B d' a d
ba_ad : length b a = length a d
M : line
aM : online a M
d'M : online d' M
cM : ¬online c M
ang_split : angle c a d' + angle c a d = 2 * rightangle
⊢ angle b a c = rightangle
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/pythagoras_2.lean | pythagoras_converse | [55, 1] | [71, 101] | rw[←ba_ad] at pyth_2 | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
a b c : point
tri_abc : triangle a b c
sq_sum : length a b ^ 2 + length a c ^ 2 = length b c ^ 2
L : line
aL : online a L
cL : online c L
N : line
aN : online a N
bN : online b N
e : point
Bbae : B b a e
ba_ae : length b a = length a e
d' : point
d'eL : diffside d' e L
cad'_ra : angle c a d' = rightangle
d : point
Bd'ad : B d' a d
ba_ad : length b a = length a d
M : line
aM : online a M
d'M : online d' M
cM : ¬online c M
ang_split : angle c a d' + angle c a d = 2 * rightangle
pyth_2 : length a d ^ 2 + length a c ^ 2 = length d c ^ 2
⊢ angle b a c = rightangle | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
a b c : point
tri_abc : triangle a b c
sq_sum : length a b ^ 2 + length a c ^ 2 = length b c ^ 2
L : line
aL : online a L
cL : online c L
N : line
aN : online a N
bN : online b N
e : point
Bbae : B b a e
ba_ae : length b a = length a e
d' : point
d'eL : diffside d' e L
cad'_ra : angle c a d' = rightangle
d : point
Bd'ad : B d' a d
ba_ad : length b a = length a d
M : line
aM : online a M
d'M : online d' M
cM : ¬online c M
ang_split : angle c a d' + angle c a d = 2 * rightangle
pyth_2 : length b a ^ 2 + length a c ^ 2 = length d c ^ 2
⊢ angle b a c = rightangle | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
a b c : point
tri_abc : triangle a b c
sq_sum : length a b ^ 2 + length a c ^ 2 = length b c ^ 2
L : line
aL : online a L
cL : online c L
N : line
aN : online a N
bN : online b N
e : point
Bbae : B b a e
ba_ae : length b a = length a e
d' : point
d'eL : diffside d' e L
cad'_ra : angle c a d' = rightangle
d : point
Bd'ad : B d' a d
ba_ad : length b a = length a d
M : line
aM : online a M
d'M : online d' M
cM : ¬online c M
ang_split : angle c a d' + angle c a d = 2 * rightangle
pyth_2 : length a d ^ 2 + length a c ^ 2 = length d c ^ 2
⊢ angle b a c = rightangle
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/pythagoras_2.lean | pythagoras_converse | [55, 1] | [71, 101] | have sq_eq : (length c d)^2 = (length b c)^2 := by linperm | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
a b c : point
tri_abc : triangle a b c
sq_sum : length a b ^ 2 + length a c ^ 2 = length b c ^ 2
L : line
aL : online a L
cL : online c L
N : line
aN : online a N
bN : online b N
e : point
Bbae : B b a e
ba_ae : length b a = length a e
d' : point
d'eL : diffside d' e L
cad'_ra : angle c a d' = rightangle
d : point
Bd'ad : B d' a d
ba_ad : length b a = length a d
M : line
aM : online a M
d'M : online d' M
cM : ¬online c M
ang_split : angle c a d' + angle c a d = 2 * rightangle
pyth_2 : length b a ^ 2 + length a c ^ 2 = length d c ^ 2
⊢ angle b a c = rightangle | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
a b c : point
tri_abc : triangle a b c
sq_sum : length a b ^ 2 + length a c ^ 2 = length b c ^ 2
L : line
aL : online a L
cL : online c L
N : line
aN : online a N
bN : online b N
e : point
Bbae : B b a e
ba_ae : length b a = length a e
d' : point
d'eL : diffside d' e L
cad'_ra : angle c a d' = rightangle
d : point
Bd'ad : B d' a d
ba_ad : length b a = length a d
M : line
aM : online a M
d'M : online d' M
cM : ¬online c M
ang_split : angle c a d' + angle c a d = 2 * rightangle
pyth_2 : length b a ^ 2 + length a c ^ 2 = length d c ^ 2
sq_eq : length c d ^ 2 = length b c ^ 2
⊢ angle b a c = rightangle | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
a b c : point
tri_abc : triangle a b c
sq_sum : length a b ^ 2 + length a c ^ 2 = length b c ^ 2
L : line
aL : online a L
cL : online c L
N : line
aN : online a N
bN : online b N
e : point
Bbae : B b a e
ba_ae : length b a = length a e
d' : point
d'eL : diffside d' e L
cad'_ra : angle c a d' = rightangle
d : point
Bd'ad : B d' a d
ba_ad : length b a = length a d
M : line
aM : online a M
d'M : online d' M
cM : ¬online c M
ang_split : angle c a d' + angle c a d = 2 * rightangle
pyth_2 : length b a ^ 2 + length a c ^ 2 = length d c ^ 2
⊢ angle b a c = rightangle
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/pythagoras_2.lean | pythagoras_converse | [55, 1] | [71, 101] | rw[sq_eq_sq_iff_abs_eq_abs] at sq_eq | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
a b c : point
tri_abc : triangle a b c
sq_sum : length a b ^ 2 + length a c ^ 2 = length b c ^ 2
L : line
aL : online a L
cL : online c L
N : line
aN : online a N
bN : online b N
e : point
Bbae : B b a e
ba_ae : length b a = length a e
d' : point
d'eL : diffside d' e L
cad'_ra : angle c a d' = rightangle
d : point
Bd'ad : B d' a d
ba_ad : length b a = length a d
M : line
aM : online a M
d'M : online d' M
cM : ¬online c M
ang_split : angle c a d' + angle c a d = 2 * rightangle
pyth_2 : length b a ^ 2 + length a c ^ 2 = length d c ^ 2
sq_eq : length c d ^ 2 = length b c ^ 2
⊢ angle b a c = rightangle | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
a b c : point
tri_abc : triangle a b c
sq_sum : length a b ^ 2 + length a c ^ 2 = length b c ^ 2
L : line
aL : online a L
cL : online c L
N : line
aN : online a N
bN : online b N
e : point
Bbae : B b a e
ba_ae : length b a = length a e
d' : point
d'eL : diffside d' e L
cad'_ra : angle c a d' = rightangle
d : point
Bd'ad : B d' a d
ba_ad : length b a = length a d
M : line
aM : online a M
d'M : online d' M
cM : ¬online c M
ang_split : angle c a d' + angle c a d = 2 * rightangle
pyth_2 : length b a ^ 2 + length a c ^ 2 = length d c ^ 2
sq_eq : abs (length c d) = abs (length b c)
⊢ angle b a c = rightangle | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
a b c : point
tri_abc : triangle a b c
sq_sum : length a b ^ 2 + length a c ^ 2 = length b c ^ 2
L : line
aL : online a L
cL : online c L
N : line
aN : online a N
bN : online b N
e : point
Bbae : B b a e
ba_ae : length b a = length a e
d' : point
d'eL : diffside d' e L
cad'_ra : angle c a d' = rightangle
d : point
Bd'ad : B d' a d
ba_ad : length b a = length a d
M : line
aM : online a M
d'M : online d' M
cM : ¬online c M
ang_split : angle c a d' + angle c a d = 2 * rightangle
pyth_2 : length b a ^ 2 + length a c ^ 2 = length d c ^ 2
sq_eq : length c d ^ 2 = length b c ^ 2
⊢ angle b a c = rightangle
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/pythagoras_2.lean | pythagoras_converse | [55, 1] | [71, 101] | have sq_eq' : (length c d) = (length b c) :=
by linperm[abs_of_nonneg (length_nonneg b c), abs_of_nonneg (length_nonneg c d)] | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
a b c : point
tri_abc : triangle a b c
sq_sum : length a b ^ 2 + length a c ^ 2 = length b c ^ 2
L : line
aL : online a L
cL : online c L
N : line
aN : online a N
bN : online b N
e : point
Bbae : B b a e
ba_ae : length b a = length a e
d' : point
d'eL : diffside d' e L
cad'_ra : angle c a d' = rightangle
d : point
Bd'ad : B d' a d
ba_ad : length b a = length a d
M : line
aM : online a M
d'M : online d' M
cM : ¬online c M
ang_split : angle c a d' + angle c a d = 2 * rightangle
pyth_2 : length b a ^ 2 + length a c ^ 2 = length d c ^ 2
sq_eq : abs (length c d) = abs (length b c)
⊢ angle b a c = rightangle | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
a b c : point
tri_abc : triangle a b c
sq_sum : length a b ^ 2 + length a c ^ 2 = length b c ^ 2
L : line
aL : online a L
cL : online c L
N : line
aN : online a N
bN : online b N
e : point
Bbae : B b a e
ba_ae : length b a = length a e
d' : point
d'eL : diffside d' e L
cad'_ra : angle c a d' = rightangle
d : point
Bd'ad : B d' a d
ba_ad : length b a = length a d
M : line
aM : online a M
d'M : online d' M
cM : ¬online c M
ang_split : angle c a d' + angle c a d = 2 * rightangle
pyth_2 : length b a ^ 2 + length a c ^ 2 = length d c ^ 2
sq_eq : abs (length c d) = abs (length b c)
sq_eq' : length c d = length b c
⊢ angle b a c = rightangle | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
a b c : point
tri_abc : triangle a b c
sq_sum : length a b ^ 2 + length a c ^ 2 = length b c ^ 2
L : line
aL : online a L
cL : online c L
N : line
aN : online a N
bN : online b N
e : point
Bbae : B b a e
ba_ae : length b a = length a e
d' : point
d'eL : diffside d' e L
cad'_ra : angle c a d' = rightangle
d : point
Bd'ad : B d' a d
ba_ad : length b a = length a d
M : line
aM : online a M
d'M : online d' M
cM : ¬online c M
ang_split : angle c a d' + angle c a d = 2 * rightangle
pyth_2 : length b a ^ 2 + length a c ^ 2 = length d c ^ 2
sq_eq : abs (length c d) = abs (length b c)
⊢ angle b a c = rightangle
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/pythagoras_2.lean | pythagoras_converse | [55, 1] | [71, 101] | linperm[(sss (by linperm : length a d = length a b) (by linperm : length d c = length b c) rfl).2.1] | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
a b c : point
tri_abc : triangle a b c
sq_sum : length a b ^ 2 + length a c ^ 2 = length b c ^ 2
L : line
aL : online a L
cL : online c L
N : line
aN : online a N
bN : online b N
e : point
Bbae : B b a e
ba_ae : length b a = length a e
d' : point
d'eL : diffside d' e L
cad'_ra : angle c a d' = rightangle
d : point
Bd'ad : B d' a d
ba_ad : length b a = length a d
M : line
aM : online a M
d'M : online d' M
cM : ¬online c M
ang_split : angle c a d' + angle c a d = 2 * rightangle
pyth_2 : length b a ^ 2 + length a c ^ 2 = length d c ^ 2
sq_eq : abs (length c d) = abs (length b c)
sq_eq' : length c d = length b c
⊢ angle b a c = rightangle | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
a b c : point
tri_abc : triangle a b c
sq_sum : length a b ^ 2 + length a c ^ 2 = length b c ^ 2
L : line
aL : online a L
cL : online c L
N : line
aN : online a N
bN : online b N
e : point
Bbae : B b a e
ba_ae : length b a = length a e
d' : point
d'eL : diffside d' e L
cad'_ra : angle c a d' = rightangle
d : point
Bd'ad : B d' a d
ba_ad : length b a = length a d
M : line
aM : online a M
d'M : online d' M
cM : ¬online c M
ang_split : angle c a d' + angle c a d = 2 * rightangle
pyth_2 : length b a ^ 2 + length a c ^ 2 = length d c ^ 2
sq_eq : abs (length c d) = abs (length b c)
sq_eq' : length c d = length b c
⊢ angle b a c = rightangle
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/pythagoras_2.lean | pythagoras_converse | [55, 1] | [71, 101] | linperm | i : incidence_geometry
a b c : point
tri_abc : triangle a b c
sq_sum : length a b ^ 2 + length a c ^ 2 = length b c ^ 2
L : line
aL : online a L
cL : online c L
N : line
aN : online a N
bN : online b N
e : point
Bbae : B b a e
ba_ae : length b a = length a e
d' : point
d'eL : diffside d' e L
cad'_ra : angle c a d' = rightangle
d : point
Bd'ad : B d' a d
ba_ad : length b a = length a d
M : line
aM : online a M
d'M : online d' M
cM : ¬online c M
ang_split : angle c a d' + angle c a d = 2 * rightangle
⊢ angle d a c = rightangle | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
i : incidence_geometry
a b c : point
tri_abc : triangle a b c
sq_sum : length a b ^ 2 + length a c ^ 2 = length b c ^ 2
L : line
aL : online a L
cL : online c L
N : line
aN : online a N
bN : online b N
e : point
Bbae : B b a e
ba_ae : length b a = length a e
d' : point
d'eL : diffside d' e L
cad'_ra : angle c a d' = rightangle
d : point
Bd'ad : B d' a d
ba_ad : length b a = length a d
M : line
aM : online a M
d'M : online d' M
cM : ¬online c M
ang_split : angle c a d' + angle c a d = 2 * rightangle
⊢ angle d a c = rightangle
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/pythagoras_2.lean | pythagoras_converse | [55, 1] | [71, 101] | linperm | i : incidence_geometry
a b c : point
tri_abc : triangle a b c
sq_sum : length a b ^ 2 + length a c ^ 2 = length b c ^ 2
L : line
aL : online a L
cL : online c L
N : line
aN : online a N
bN : online b N
e : point
Bbae : B b a e
ba_ae : length b a = length a e
d' : point
d'eL : diffside d' e L
cad'_ra : angle c a d' = rightangle
d : point
Bd'ad : B d' a d
ba_ad : length b a = length a d
M : line
aM : online a M
d'M : online d' M
cM : ¬online c M
ang_split : angle c a d' + angle c a d = 2 * rightangle
pyth_2 : length b a ^ 2 + length a c ^ 2 = length d c ^ 2
⊢ length c d ^ 2 = length b c ^ 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
i : incidence_geometry
a b c : point
tri_abc : triangle a b c
sq_sum : length a b ^ 2 + length a c ^ 2 = length b c ^ 2
L : line
aL : online a L
cL : online c L
N : line
aN : online a N
bN : online b N
e : point
Bbae : B b a e
ba_ae : length b a = length a e
d' : point
d'eL : diffside d' e L
cad'_ra : angle c a d' = rightangle
d : point
Bd'ad : B d' a d
ba_ad : length b a = length a d
M : line
aM : online a M
d'M : online d' M
cM : ¬online c M
ang_split : angle c a d' + angle c a d = 2 * rightangle
pyth_2 : length b a ^ 2 + length a c ^ 2 = length d c ^ 2
⊢ length c d ^ 2 = length b c ^ 2
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/pythagoras_2.lean | pythagoras_converse | [55, 1] | [71, 101] | linperm[abs_of_nonneg (length_nonneg b c), abs_of_nonneg (length_nonneg c d)] | i : incidence_geometry
a b c : point
tri_abc : triangle a b c
sq_sum : length a b ^ 2 + length a c ^ 2 = length b c ^ 2
L : line
aL : online a L
cL : online c L
N : line
aN : online a N
bN : online b N
e : point
Bbae : B b a e
ba_ae : length b a = length a e
d' : point
d'eL : diffside d' e L
cad'_ra : angle c a d' = rightangle
d : point
Bd'ad : B d' a d
ba_ad : length b a = length a d
M : line
aM : online a M
d'M : online d' M
cM : ¬online c M
ang_split : angle c a d' + angle c a d = 2 * rightangle
pyth_2 : length b a ^ 2 + length a c ^ 2 = length d c ^ 2
sq_eq : abs (length c d) = abs (length b c)
⊢ length c d = length b c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
i : incidence_geometry
a b c : point
tri_abc : triangle a b c
sq_sum : length a b ^ 2 + length a c ^ 2 = length b c ^ 2
L : line
aL : online a L
cL : online c L
N : line
aN : online a N
bN : online b N
e : point
Bbae : B b a e
ba_ae : length b a = length a e
d' : point
d'eL : diffside d' e L
cad'_ra : angle c a d' = rightangle
d : point
Bd'ad : B d' a d
ba_ad : length b a = length a d
M : line
aM : online a M
d'M : online d' M
cM : ¬online c M
ang_split : angle c a d' + angle c a d = 2 * rightangle
pyth_2 : length b a ^ 2 + length a c ^ 2 = length d c ^ 2
sq_eq : abs (length c d) = abs (length b c)
⊢ length c d = length b c
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/pythagoras_2.lean | pythagoras_converse | [55, 1] | [71, 101] | linperm | i : incidence_geometry
a b c : point
tri_abc : triangle a b c
sq_sum : length a b ^ 2 + length a c ^ 2 = length b c ^ 2
L : line
aL : online a L
cL : online c L
N : line
aN : online a N
bN : online b N
e : point
Bbae : B b a e
ba_ae : length b a = length a e
d' : point
d'eL : diffside d' e L
cad'_ra : angle c a d' = rightangle
d : point
Bd'ad : B d' a d
ba_ad : length b a = length a d
M : line
aM : online a M
d'M : online d' M
cM : ¬online c M
ang_split : angle c a d' + angle c a d = 2 * rightangle
pyth_2 : length b a ^ 2 + length a c ^ 2 = length d c ^ 2
sq_eq : abs (length c d) = abs (length b c)
sq_eq' : length c d = length b c
⊢ length a d = length a b | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
i : incidence_geometry
a b c : point
tri_abc : triangle a b c
sq_sum : length a b ^ 2 + length a c ^ 2 = length b c ^ 2
L : line
aL : online a L
cL : online c L
N : line
aN : online a N
bN : online b N
e : point
Bbae : B b a e
ba_ae : length b a = length a e
d' : point
d'eL : diffside d' e L
cad'_ra : angle c a d' = rightangle
d : point
Bd'ad : B d' a d
ba_ad : length b a = length a d
M : line
aM : online a M
d'M : online d' M
cM : ¬online c M
ang_split : angle c a d' + angle c a d = 2 * rightangle
pyth_2 : length b a ^ 2 + length a c ^ 2 = length d c ^ 2
sq_eq : abs (length c d) = abs (length b c)
sq_eq' : length c d = length b c
⊢ length a d = length a b
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/pythagoras_2.lean | pythagoras_converse | [55, 1] | [71, 101] | linperm | i : incidence_geometry
a b c : point
tri_abc : triangle a b c
sq_sum : length a b ^ 2 + length a c ^ 2 = length b c ^ 2
L : line
aL : online a L
cL : online c L
N : line
aN : online a N
bN : online b N
e : point
Bbae : B b a e
ba_ae : length b a = length a e
d' : point
d'eL : diffside d' e L
cad'_ra : angle c a d' = rightangle
d : point
Bd'ad : B d' a d
ba_ad : length b a = length a d
M : line
aM : online a M
d'M : online d' M
cM : ¬online c M
ang_split : angle c a d' + angle c a d = 2 * rightangle
pyth_2 : length b a ^ 2 + length a c ^ 2 = length d c ^ 2
sq_eq : abs (length c d) = abs (length b c)
sq_eq' : length c d = length b c
⊢ length d c = length b c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
i : incidence_geometry
a b c : point
tri_abc : triangle a b c
sq_sum : length a b ^ 2 + length a c ^ 2 = length b c ^ 2
L : line
aL : online a L
cL : online c L
N : line
aN : online a N
bN : online b N
e : point
Bbae : B b a e
ba_ae : length b a = length a e
d' : point
d'eL : diffside d' e L
cad'_ra : angle c a d' = rightangle
d : point
Bd'ad : B d' a d
ba_ad : length b a = length a d
M : line
aM : online a M
d'M : online d' M
cM : ¬online c M
ang_split : angle c a d' + angle c a d = 2 * rightangle
pyth_2 : length b a ^ 2 + length a c ^ 2 = length d c ^ 2
sq_eq : abs (length c d) = abs (length b c)
sq_eq' : length c d = length b c
⊢ length d c = length b c
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_I_extras.lean | same_len_pts_coincide_iff | [13, 1] | [15, 72] | rw [← @length_eq_zero_iff i a b, ← @length_eq_zero_iff i c d, hlen] | i : incidence_geometry
a b c d : point
hlen : length a b = length c d
⊢ a = b ↔ c = d | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
i : incidence_geometry
a b c d : point
hlen : length a b = length c d
⊢ a = b ↔ c = d
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_I_extras.lean | pt_of_line_ne_pt | [17, 1] | [23, 28] | obtain ⟨ b, c, Hbc ⟩ := online_ne_of_line L | i : incidence_geometry
a : point
L : line
⊢ ∃ b, b ≠ a ∧ online b L | case intro.intro
i : incidence_geometry
a : point
L : line
b c : point
Hbc : b ≠ c ∧ online b L ∧ online c L
⊢ ∃ b, b ≠ a ∧ online b L | Please generate a tactic in lean4 to solve the state.
STATE:
i : incidence_geometry
a : point
L : line
⊢ ∃ b, b ≠ a ∧ online b L
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_I_extras.lean | pt_of_line_ne_pt | [17, 1] | [23, 28] | by_cases b = a | case intro.intro
i : incidence_geometry
a : point
L : line
b c : point
Hbc : b ≠ c ∧ online b L ∧ online c L
⊢ ∃ b, b ≠ a ∧ online b L | case pos
i : incidence_geometry
a : point
L : line
b c : point
Hbc : b ≠ c ∧ online b L ∧ online c L
h : b = a
⊢ ∃ b, b ≠ a ∧ online b L
case neg
i : incidence_geometry
a : point
L : line
b c : point
Hbc : b ≠ c ∧ online b L ∧ online c L
h : ¬b = a
⊢ ∃ b, b ≠ a ∧ online b L | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
i : incidence_geometry
a : point
L : line
b c : point
Hbc : b ≠ c ∧ online b L ∧ online c L
⊢ ∃ b, b ≠ a ∧ online b L
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_I_extras.lean | pt_of_line_ne_pt | [17, 1] | [23, 28] | simp_rw [h] at Hbc | case pos
i : incidence_geometry
a : point
L : line
b c : point
Hbc : b ≠ c ∧ online b L ∧ online c L
h : b = a
⊢ ∃ b, b ≠ a ∧ online b L | case pos
i : incidence_geometry
a : point
L : line
b c : point
h : b = a
Hbc : a ≠ c ∧ online a L ∧ online c L
⊢ ∃ b, b ≠ a ∧ online b L | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
i : incidence_geometry
a : point
L : line
b c : point
Hbc : b ≠ c ∧ online b L ∧ online c L
h : b = a
⊢ ∃ b, b ≠ a ∧ online b L
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_I_extras.lean | pt_of_line_ne_pt | [17, 1] | [23, 28] | exact ⟨c, Hbc.1.symm, Hbc.2.2 ⟩ | case pos
i : incidence_geometry
a : point
L : line
b c : point
h : b = a
Hbc : a ≠ c ∧ online a L ∧ online c L
⊢ ∃ b, b ≠ a ∧ online b L | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
i : incidence_geometry
a : point
L : line
b c : point
h : b = a
Hbc : a ≠ c ∧ online a L ∧ online c L
⊢ ∃ b, b ≠ a ∧ online b L
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_I_extras.lean | pt_of_line_ne_pt | [17, 1] | [23, 28] | exact ⟨ b, h, Hbc.2.1 ⟩ | case neg
i : incidence_geometry
a : point
L : line
b c : point
Hbc : b ≠ c ∧ online b L ∧ online c L
h : ¬b = a
⊢ ∃ b, b ≠ a ∧ online b L | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
i : incidence_geometry
a : point
L : line
b c : point
Hbc : b ≠ c ∧ online b L ∧ online c L
h : ¬b = a
⊢ ∃ b, b ≠ a ∧ online b L
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_I_extras.lean | pt_of_line_line | [25, 1] | [28, 62] | simp_rw [para, not_forall, not_or, not_not] at hp | i : incidence_geometry
L M : line
hp : ¬para L M
⊢ ∃ a, online a L ∧ online a M | i : incidence_geometry
L M : line
hp : ∃ x, online x L ∧ online x M
⊢ ∃ a, online a L ∧ online a M | Please generate a tactic in lean4 to solve the state.
STATE:
i : incidence_geometry
L M : line
hp : ¬para L M
⊢ ∃ a, online a L ∧ online a M
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_I_extras.lean | pt_of_line_line | [25, 1] | [28, 62] | exact hp | i : incidence_geometry
L M : line
hp : ∃ x, online x L ∧ online x M
⊢ ∃ a, online a L ∧ online a M | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
i : incidence_geometry
L M : line
hp : ∃ x, online x L ∧ online x M
⊢ ∃ a, online a L ∧ online a M
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_I_extras.lean | neq_of_para | [30, 1] | [36, 65] | have := offline_of_para aL pLM | i : incidence_geometry
a b : point
L M : line
aL : online a L
bM : online b M
pLM : para L M
ab : a = b
⊢ False | i : incidence_geometry
a b : point
L M : line
aL : online a L
bM : online b M
pLM : para L M
ab : a = b
this : ¬online a M
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
i : incidence_geometry
a b : point
L M : line
aL : online a L
bM : online b M
pLM : para L M
ab : a = b
⊢ False
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_I_extras.lean | neq_of_para | [30, 1] | [36, 65] | rw [ab] at this | i : incidence_geometry
a b : point
L M : line
aL : online a L
bM : online b M
pLM : para L M
ab : a = b
this : ¬online a M
⊢ False | i : incidence_geometry
a b : point
L M : line
aL : online a L
bM : online b M
pLM : para L M
ab : a = b
this : ¬online b M
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
i : incidence_geometry
a b : point
L M : line
aL : online a L
bM : online b M
pLM : para L M
ab : a = b
this : ¬online a M
⊢ False
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_I_extras.lean | neq_of_para | [30, 1] | [36, 65] | exact this bM | i : incidence_geometry
a b : point
L M : line
aL : online a L
bM : online b M
pLM : para L M
ab : a = b
this : ¬online b M
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
i : incidence_geometry
a b : point
L M : line
aL : online a L
bM : online b M
pLM : para L M
ab : a = b
this : ¬online b M
⊢ False
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_I_extras.lean | para_trans | [40, 1] | [119, 73] | by_cases MN : M = N | i : incidence_geometry
L M N : line
pLM : para L M
pLN : para L N
⊢ M = N ∨ para M N | case pos
i : incidence_geometry
L M N : line
pLM : para L M
pLN : para L N
MN : M = N
⊢ M = N ∨ para M N
case neg
i : incidence_geometry
L M N : line
pLM : para L M
pLN : para L N
MN : ¬M = N
⊢ M = N ∨ para M N | Please generate a tactic in lean4 to solve the state.
STATE:
i : incidence_geometry
L M N : line
pLM : para L M
pLN : para L N
⊢ M = N ∨ para M N
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_I_extras.lean | para_trans | [40, 1] | [119, 73] | by_contra npMN | case neg
i : incidence_geometry
L M N : line
pLM : para L M
pLN : para L N
MN : ¬M = N
⊢ M = N ∨ para M N | case neg
i : incidence_geometry
L M N : line
pLM : para L M
pLN : para L N
MN : ¬M = N
npMN : ¬(M = N ∨ para M N)
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
i : incidence_geometry
L M N : line
pLM : para L M
pLN : para L N
MN : ¬M = N
⊢ M = N ∨ para M N
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_I_extras.lean | para_trans | [40, 1] | [119, 73] | simp [not_or] at npMN | case neg
i : incidence_geometry
L M N : line
pLM : para L M
pLN : para L N
MN : ¬M = N
npMN : ¬(M = N ∨ para M N)
⊢ False | case neg
i : incidence_geometry
L M N : line
pLM : para L M
pLN : para L N
MN : ¬M = N
npMN : ¬M = N ∧ ¬para M N
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
i : incidence_geometry
L M N : line
pLM : para L M
pLN : para L N
MN : ¬M = N
npMN : ¬(M = N ∨ para M N)
⊢ False
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_I_extras.lean | para_trans | [40, 1] | [119, 73] | rcases pt_of_line_line npMN.2 with ⟨c, cM, cN⟩ | case neg
i : incidence_geometry
L M N : line
pLM : para L M
pLN : para L N
MN : ¬M = N
npMN : ¬M = N ∧ ¬para M N
⊢ False | case neg.intro.intro
i : incidence_geometry
L M N : line
pLM : para L M
pLN : para L N
MN : ¬M = N
npMN : ¬M = N ∧ ¬para M N
c : point
cM : online c M
cN : online c N
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
i : incidence_geometry
L M N : line
pLM : para L M
pLN : para L N
MN : ¬M = N
npMN : ¬M = N ∧ ¬para M N
⊢ False
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_I_extras.lean | para_trans | [40, 1] | [119, 73] | have cL := offline_of_para cN (para_symm pLN) | case neg.intro.intro
i : incidence_geometry
L M N : line
pLM : para L M
pLN : para L N
MN : ¬M = N
npMN : ¬M = N ∧ ¬para M N
c : point
cM : online c M
cN : online c N
⊢ False | case neg.intro.intro
i : incidence_geometry
L M N : line
pLM : para L M
pLN : para L N
MN : ¬M = N
npMN : ¬M = N ∧ ¬para M N
c : point
cM : online c M
cN : online c N
cL : ¬online c L
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro
i : incidence_geometry
L M N : line
pLM : para L M
pLN : para L N
MN : ¬M = N
npMN : ¬M = N ∧ ¬para M N
c : point
cM : online c M
cN : online c N
⊢ False
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_I_extras.lean | para_trans | [40, 1] | [119, 73] | rcases perpendicular_of_not_online cL with ⟨-, -, d, -, -, -, dL, -, -⟩ | case neg.intro.intro
i : incidence_geometry
L M N : line
pLM : para L M
pLN : para L N
MN : ¬M = N
npMN : ¬M = N ∧ ¬para M N
c : point
cM : online c M
cN : online c N
cL : ¬online c L
⊢ False | case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
L M N : line
pLM : para L M
pLN : para L N
MN : ¬M = N
npMN : ¬M = N ∧ ¬para M N
c : point
cM : online c M
cN : online c N
cL : ¬online c L
d : point
dL : online d L
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro
i : incidence_geometry
L M N : line
pLM : para L M
pLN : para L N
MN : ¬M = N
npMN : ¬M = N ∧ ¬para M N
c : point
cM : online c M
cN : online c N
cL : ¬online c L
⊢ False
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_I_extras.lean | para_trans | [40, 1] | [119, 73] | obtain ⟨O, cO, dO⟩ := line_of_pts c d | case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
L M N : line
pLM : para L M
pLN : para L N
MN : ¬M = N
npMN : ¬M = N ∧ ¬para M N
c : point
cM : online c M
cN : online c N
cL : ¬online c L
d : point
dL : online d L
⊢ False | case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
L M N : line
pLM : para L M
pLN : para L N
MN : ¬M = N
npMN : ¬M = N ∧ ¬para M N
c : point
cM : online c M
cN : online c N
cL : ¬online c L
d : point
dL : online d L
O : line
cO : online c O
dO : online d O
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
L M N : line
pLM : para L M
pLN : para L N
MN : ¬M = N
npMN : ¬M = N ∧ ¬para M N
c : point
cM : online c M
cN : online c N
cL : ¬online c L
d : point
dL : online d L
⊢ False
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_I_extras.lean | para_trans | [40, 1] | [119, 73] | have cd : c ≠ d := neq_of_para cM dL (para_symm pLM) | case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
L M N : line
pLM : para L M
pLN : para L N
MN : ¬M = N
npMN : ¬M = N ∧ ¬para M N
c : point
cM : online c M
cN : online c N
cL : ¬online c L
d : point
dL : online d L
O : line
cO : online c O
dO : online d O
⊢ False | case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
L M N : line
pLM : para L M
pLN : para L N
MN : ¬M = N
npMN : ¬M = N ∧ ¬para M N
c : point
cM : online c M
cN : online c N
cL : ¬online c L
d : point
dL : online d L
O : line
cO : online c O
dO : online d O
cd : c ≠ d
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
L M N : line
pLM : para L M
pLN : para L N
MN : ¬M = N
npMN : ¬M = N ∧ ¬para M N
c : point
cM : online c M
cN : online c N
cL : ¬online c L
d : point
dL : online d L
O : line
cO : online c O
dO : online d O
⊢ False
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_I_extras.lean | para_trans | [40, 1] | [119, 73] | have LO : L ≠ O := fun LO => cL (by rwa [← LO] at cO) | case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
L M N : line
pLM : para L M
pLN : para L N
MN : ¬M = N
npMN : ¬M = N ∧ ¬para M N
c : point
cM : online c M
cN : online c N
cL : ¬online c L
d : point
dL : online d L
O : line
cO : online c O
dO : online d O
cd : c ≠ d
⊢ False | case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
L M N : line
pLM : para L M
pLN : para L N
MN : ¬M = N
npMN : ¬M = N ∧ ¬para M N
c : point
cM : online c M
cN : online c N
cL : ¬online c L
d : point
dL : online d L
O : line
cO : online c O
dO : online d O
cd : c ≠ d
LO : L ≠ O
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
L M N : line
pLM : para L M
pLN : para L N
MN : ¬M = N
npMN : ¬M = N ∧ ¬para M N
c : point
cM : online c M
cN : online c N
cL : ¬online c L
d : point
dL : online d L
O : line
cO : online c O
dO : online d O
cd : c ≠ d
⊢ False
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_I_extras.lean | para_trans | [40, 1] | [119, 73] | obtain ⟨α, Hα⟩ := circle_of_ne cd | case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
L M N : line
pLM : para L M
pLN : para L N
MN : ¬M = N
npMN : ¬M = N ∧ ¬para M N
c : point
cM : online c M
cN : online c N
cL : ¬online c L
d : point
dL : online d L
O : line
cO : online c O
dO : online d O
cd : c ≠ d
LO : L ≠ O
⊢ False | case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
L M N : line
pLM : para L M
pLN : para L N
MN : ¬M = N
npMN : ¬M = N ∧ ¬para M N
c : point
cM : online c M
cN : online c N
cL : ¬online c L
d : point
dL : online d L
O : line
cO : online c O
dO : online d O
cd : c ≠ d
LO : L ≠ O
α : circle
Hα : center_circle c α ∧ on_circle d α
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
L M N : line
pLM : para L M
pLN : para L N
MN : ¬M = N
npMN : ¬M = N ∧ ¬para M N
c : point
cM : online c M
cN : online c N
cL : ¬online c L
d : point
dL : online d L
O : line
cO : online c O
dO : online d O
cd : c ≠ d
LO : L ≠ O
⊢ False
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_I_extras.lean | para_trans | [40, 1] | [119, 73] | have cα := inside_circle_of_center Hα.1 | case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
L M N : line
pLM : para L M
pLN : para L N
MN : ¬M = N
npMN : ¬M = N ∧ ¬para M N
c : point
cM : online c M
cN : online c N
cL : ¬online c L
d : point
dL : online d L
O : line
cO : online c O
dO : online d O
cd : c ≠ d
LO : L ≠ O
α : circle
Hα : center_circle c α ∧ on_circle d α
⊢ False | case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
L M N : line
pLM : para L M
pLN : para L N
MN : ¬M = N
npMN : ¬M = N ∧ ¬para M N
c : point
cM : online c M
cN : online c N
cL : ¬online c L
d : point
dL : online d L
O : line
cO : online c O
dO : online d O
cd : c ≠ d
LO : L ≠ O
α : circle
Hα : center_circle c α ∧ on_circle d α
cα : in_circle c α
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
L M N : line
pLM : para L M
pLN : para L N
MN : ¬M = N
npMN : ¬M = N ∧ ¬para M N
c : point
cM : online c M
cN : online c N
cL : ¬online c L
d : point
dL : online d L
O : line
cO : online c O
dO : online d O
cd : c ≠ d
LO : L ≠ O
α : circle
Hα : center_circle c α ∧ on_circle d α
⊢ False
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_I_extras.lean | para_trans | [40, 1] | [119, 73] | have αM := line_circle_inter_of_inside_online cM cα | case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
L M N : line
pLM : para L M
pLN : para L N
MN : ¬M = N
npMN : ¬M = N ∧ ¬para M N
c : point
cM : online c M
cN : online c N
cL : ¬online c L
d : point
dL : online d L
O : line
cO : online c O
dO : online d O
cd : c ≠ d
LO : L ≠ O
α : circle
Hα : center_circle c α ∧ on_circle d α
cα : in_circle c α
⊢ False | case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
L M N : line
pLM : para L M
pLN : para L N
MN : ¬M = N
npMN : ¬M = N ∧ ¬para M N
c : point
cM : online c M
cN : online c N
cL : ¬online c L
d : point
dL : online d L
O : line
cO : online c O
dO : online d O
cd : c ≠ d
LO : L ≠ O
α : circle
Hα : center_circle c α ∧ on_circle d α
cα : in_circle c α
αM : line_circle_inter M α
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
L M N : line
pLM : para L M
pLN : para L N
MN : ¬M = N
npMN : ¬M = N ∧ ¬para M N
c : point
cM : online c M
cN : online c N
cL : ¬online c L
d : point
dL : online d L
O : line
cO : online c O
dO : online d O
cd : c ≠ d
LO : L ≠ O
α : circle
Hα : center_circle c α ∧ on_circle d α
cα : in_circle c α
⊢ False
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_I_extras.lean | para_trans | [40, 1] | [119, 73] | have αN := line_circle_inter_of_inside_online cN cα | case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
L M N : line
pLM : para L M
pLN : para L N
MN : ¬M = N
npMN : ¬M = N ∧ ¬para M N
c : point
cM : online c M
cN : online c N
cL : ¬online c L
d : point
dL : online d L
O : line
cO : online c O
dO : online d O
cd : c ≠ d
LO : L ≠ O
α : circle
Hα : center_circle c α ∧ on_circle d α
cα : in_circle c α
αM : line_circle_inter M α
⊢ False | case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
L M N : line
pLM : para L M
pLN : para L N
MN : ¬M = N
npMN : ¬M = N ∧ ¬para M N
c : point
cM : online c M
cN : online c N
cL : ¬online c L
d : point
dL : online d L
O : line
cO : online c O
dO : online d O
cd : c ≠ d
LO : L ≠ O
α : circle
Hα : center_circle c α ∧ on_circle d α
cα : in_circle c α
αM : line_circle_inter M α
αN : line_circle_inter N α
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
L M N : line
pLM : para L M
pLN : para L N
MN : ¬M = N
npMN : ¬M = N ∧ ¬para M N
c : point
cM : online c M
cN : online c N
cL : ¬online c L
d : point
dL : online d L
O : line
cO : online c O
dO : online d O
cd : c ≠ d
LO : L ≠ O
α : circle
Hα : center_circle c α ∧ on_circle d α
cα : in_circle c α
αM : line_circle_inter M α
⊢ False
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_I_extras.lean | para_trans | [40, 1] | [119, 73] | obtain ⟨a, a', aa', aM, a'M, aα⟩ := pts_of_line_circle_inter αM | case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
L M N : line
pLM : para L M
pLN : para L N
MN : ¬M = N
npMN : ¬M = N ∧ ¬para M N
c : point
cM : online c M
cN : online c N
cL : ¬online c L
d : point
dL : online d L
O : line
cO : online c O
dO : online d O
cd : c ≠ d
LO : L ≠ O
α : circle
Hα : center_circle c α ∧ on_circle d α
cα : in_circle c α
αM : line_circle_inter M α
αN : line_circle_inter N α
⊢ False | case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
L M N : line
pLM : para L M
pLN : para L N
MN : ¬M = N
npMN : ¬M = N ∧ ¬para M N
c : point
cM : online c M
cN : online c N
cL : ¬online c L
d : point
dL : online d L
O : line
cO : online c O
dO : online d O
cd : c ≠ d
LO : L ≠ O
α : circle
Hα : center_circle c α ∧ on_circle d α
cα : in_circle c α
αM : line_circle_inter M α
αN : line_circle_inter N α
a a' : point
aa' : a ≠ a'
aM : online a M
a'M : online a' M
aα : on_circle a α ∧ on_circle a' α
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
L M N : line
pLM : para L M
pLN : para L N
MN : ¬M = N
npMN : ¬M = N ∧ ¬para M N
c : point
cM : online c M
cN : online c N
cL : ¬online c L
d : point
dL : online d L
O : line
cO : online c O
dO : online d O
cd : c ≠ d
LO : L ≠ O
α : circle
Hα : center_circle c α ∧ on_circle d α
cα : in_circle c α
αM : line_circle_inter M α
αN : line_circle_inter N α
⊢ False
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_I_extras.lean | para_trans | [40, 1] | [119, 73] | obtain ⟨b, b', bb', bN, b'N, bα⟩ := pts_of_line_circle_inter αN | case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
L M N : line
pLM : para L M
pLN : para L N
MN : ¬M = N
npMN : ¬M = N ∧ ¬para M N
c : point
cM : online c M
cN : online c N
cL : ¬online c L
d : point
dL : online d L
O : line
cO : online c O
dO : online d O
cd : c ≠ d
LO : L ≠ O
α : circle
Hα : center_circle c α ∧ on_circle d α
cα : in_circle c α
αM : line_circle_inter M α
αN : line_circle_inter N α
a a' : point
aa' : a ≠ a'
aM : online a M
a'M : online a' M
aα : on_circle a α ∧ on_circle a' α
⊢ False | case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
L M N : line
pLM : para L M
pLN : para L N
MN : ¬M = N
npMN : ¬M = N ∧ ¬para M N
c : point
cM : online c M
cN : online c N
cL : ¬online c L
d : point
dL : online d L
O : line
cO : online c O
dO : online d O
cd : c ≠ d
LO : L ≠ O
α : circle
Hα : center_circle c α ∧ on_circle d α
cα : in_circle c α
αM : line_circle_inter M α
αN : line_circle_inter N α
a a' : point
aa' : a ≠ a'
aM : online a M
a'M : online a' M
aα : on_circle a α ∧ on_circle a' α
b b' : point
bb' : b ≠ b'
bN : online b N
b'N : online b' N
bα : on_circle b α ∧ on_circle b' α
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
L M N : line
pLM : para L M
pLN : para L N
MN : ¬M = N
npMN : ¬M = N ∧ ¬para M N
c : point
cM : online c M
cN : online c N
cL : ¬online c L
d : point
dL : online d L
O : line
cO : online c O
dO : online d O
cd : c ≠ d
LO : L ≠ O
α : circle
Hα : center_circle c α ∧ on_circle d α
cα : in_circle c α
αM : line_circle_inter M α
αN : line_circle_inter N α
a a' : point
aa' : a ≠ a'
aM : online a M
a'M : online a' M
aα : on_circle a α ∧ on_circle a' α
⊢ False
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_I_extras.lean | para_trans | [40, 1] | [119, 73] | have Baca' := B_of_line_circle_inter aa' cM aM a'M aα.1 aα.2 cα | case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
L M N : line
pLM : para L M
pLN : para L N
MN : ¬M = N
npMN : ¬M = N ∧ ¬para M N
c : point
cM : online c M
cN : online c N
cL : ¬online c L
d : point
dL : online d L
O : line
cO : online c O
dO : online d O
cd : c ≠ d
LO : L ≠ O
α : circle
Hα : center_circle c α ∧ on_circle d α
cα : in_circle c α
αM : line_circle_inter M α
αN : line_circle_inter N α
a a' : point
aa' : a ≠ a'
aM : online a M
a'M : online a' M
aα : on_circle a α ∧ on_circle a' α
b b' : point
bb' : b ≠ b'
bN : online b N
b'N : online b' N
bα : on_circle b α ∧ on_circle b' α
⊢ False | case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
L M N : line
pLM : para L M
pLN : para L N
MN : ¬M = N
npMN : ¬M = N ∧ ¬para M N
c : point
cM : online c M
cN : online c N
cL : ¬online c L
d : point
dL : online d L
O : line
cO : online c O
dO : online d O
cd : c ≠ d
LO : L ≠ O
α : circle
Hα : center_circle c α ∧ on_circle d α
cα : in_circle c α
αM : line_circle_inter M α
αN : line_circle_inter N α
a a' : point
aa' : a ≠ a'
aM : online a M
a'M : online a' M
aα : on_circle a α ∧ on_circle a' α
b b' : point
bb' : b ≠ b'
bN : online b N
b'N : online b' N
bα : on_circle b α ∧ on_circle b' α
Baca' : B a c a'
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
L M N : line
pLM : para L M
pLN : para L N
MN : ¬M = N
npMN : ¬M = N ∧ ¬para M N
c : point
cM : online c M
cN : online c N
cL : ¬online c L
d : point
dL : online d L
O : line
cO : online c O
dO : online d O
cd : c ≠ d
LO : L ≠ O
α : circle
Hα : center_circle c α ∧ on_circle d α
cα : in_circle c α
αM : line_circle_inter M α
αN : line_circle_inter N α
a a' : point
aa' : a ≠ a'
aM : online a M
a'M : online a' M
aα : on_circle a α ∧ on_circle a' α
b b' : point
bb' : b ≠ b'
bN : online b N
b'N : online b' N
bα : on_circle b α ∧ on_circle b' α
⊢ False
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_I_extras.lean | para_trans | [40, 1] | [119, 73] | have Bbcb' := B_of_line_circle_inter bb' cN bN b'N bα.1 bα.2 cα | case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
L M N : line
pLM : para L M
pLN : para L N
MN : ¬M = N
npMN : ¬M = N ∧ ¬para M N
c : point
cM : online c M
cN : online c N
cL : ¬online c L
d : point
dL : online d L
O : line
cO : online c O
dO : online d O
cd : c ≠ d
LO : L ≠ O
α : circle
Hα : center_circle c α ∧ on_circle d α
cα : in_circle c α
αM : line_circle_inter M α
αN : line_circle_inter N α
a a' : point
aa' : a ≠ a'
aM : online a M
a'M : online a' M
aα : on_circle a α ∧ on_circle a' α
b b' : point
bb' : b ≠ b'
bN : online b N
b'N : online b' N
bα : on_circle b α ∧ on_circle b' α
Baca' : B a c a'
⊢ False | case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
L M N : line
pLM : para L M
pLN : para L N
MN : ¬M = N
npMN : ¬M = N ∧ ¬para M N
c : point
cM : online c M
cN : online c N
cL : ¬online c L
d : point
dL : online d L
O : line
cO : online c O
dO : online d O
cd : c ≠ d
LO : L ≠ O
α : circle
Hα : center_circle c α ∧ on_circle d α
cα : in_circle c α
αM : line_circle_inter M α
αN : line_circle_inter N α
a a' : point
aa' : a ≠ a'
aM : online a M
a'M : online a' M
aα : on_circle a α ∧ on_circle a' α
b b' : point
bb' : b ≠ b'
bN : online b N
b'N : online b' N
bα : on_circle b α ∧ on_circle b' α
Baca' : B a c a'
Bbcb' : B b c b'
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
L M N : line
pLM : para L M
pLN : para L N
MN : ¬M = N
npMN : ¬M = N ∧ ¬para M N
c : point
cM : online c M
cN : online c N
cL : ¬online c L
d : point
dL : online d L
O : line
cO : online c O
dO : online d O
cd : c ≠ d
LO : L ≠ O
α : circle
Hα : center_circle c α ∧ on_circle d α
cα : in_circle c α
αM : line_circle_inter M α
αN : line_circle_inter N α
a a' : point
aa' : a ≠ a'
aM : online a M
a'M : online a' M
aα : on_circle a α ∧ on_circle a' α
b b' : point
bb' : b ≠ b'
bN : online b N
b'N : online b' N
bα : on_circle b α ∧ on_circle b' α
Baca' : B a c a'
⊢ False
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_I_extras.lean | para_trans | [40, 1] | [119, 73] | have ac := ne_12_of_B Baca' | case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
L M N : line
pLM : para L M
pLN : para L N
MN : ¬M = N
npMN : ¬M = N ∧ ¬para M N
c : point
cM : online c M
cN : online c N
cL : ¬online c L
d : point
dL : online d L
O : line
cO : online c O
dO : online d O
cd : c ≠ d
LO : L ≠ O
α : circle
Hα : center_circle c α ∧ on_circle d α
cα : in_circle c α
αM : line_circle_inter M α
αN : line_circle_inter N α
a a' : point
aa' : a ≠ a'
aM : online a M
a'M : online a' M
aα : on_circle a α ∧ on_circle a' α
b b' : point
bb' : b ≠ b'
bN : online b N
b'N : online b' N
bα : on_circle b α ∧ on_circle b' α
Baca' : B a c a'
Bbcb' : B b c b'
⊢ False | case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
L M N : line
pLM : para L M
pLN : para L N
MN : ¬M = N
npMN : ¬M = N ∧ ¬para M N
c : point
cM : online c M
cN : online c N
cL : ¬online c L
d : point
dL : online d L
O : line
cO : online c O
dO : online d O
cd : c ≠ d
LO : L ≠ O
α : circle
Hα : center_circle c α ∧ on_circle d α
cα : in_circle c α
αM : line_circle_inter M α
αN : line_circle_inter N α
a a' : point
aa' : a ≠ a'
aM : online a M
a'M : online a' M
aα : on_circle a α ∧ on_circle a' α
b b' : point
bb' : b ≠ b'
bN : online b N
b'N : online b' N
bα : on_circle b α ∧ on_circle b' α
Baca' : B a c a'
Bbcb' : B b c b'
ac : a ≠ c
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
L M N : line
pLM : para L M
pLN : para L N
MN : ¬M = N
npMN : ¬M = N ∧ ¬para M N
c : point
cM : online c M
cN : online c N
cL : ¬online c L
d : point
dL : online d L
O : line
cO : online c O
dO : online d O
cd : c ≠ d
LO : L ≠ O
α : circle
Hα : center_circle c α ∧ on_circle d α
cα : in_circle c α
αM : line_circle_inter M α
αN : line_circle_inter N α
a a' : point
aa' : a ≠ a'
aM : online a M
a'M : online a' M
aα : on_circle a α ∧ on_circle a' α
b b' : point
bb' : b ≠ b'
bN : online b N
b'N : online b' N
bα : on_circle b α ∧ on_circle b' α
Baca' : B a c a'
Bbcb' : B b c b'
⊢ False
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_I_extras.lean | para_trans | [40, 1] | [119, 73] | have bc := ne_12_of_B Bbcb' | case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
L M N : line
pLM : para L M
pLN : para L N
MN : ¬M = N
npMN : ¬M = N ∧ ¬para M N
c : point
cM : online c M
cN : online c N
cL : ¬online c L
d : point
dL : online d L
O : line
cO : online c O
dO : online d O
cd : c ≠ d
LO : L ≠ O
α : circle
Hα : center_circle c α ∧ on_circle d α
cα : in_circle c α
αM : line_circle_inter M α
αN : line_circle_inter N α
a a' : point
aa' : a ≠ a'
aM : online a M
a'M : online a' M
aα : on_circle a α ∧ on_circle a' α
b b' : point
bb' : b ≠ b'
bN : online b N
b'N : online b' N
bα : on_circle b α ∧ on_circle b' α
Baca' : B a c a'
Bbcb' : B b c b'
ac : a ≠ c
⊢ False | case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
L M N : line
pLM : para L M
pLN : para L N
MN : ¬M = N
npMN : ¬M = N ∧ ¬para M N
c : point
cM : online c M
cN : online c N
cL : ¬online c L
d : point
dL : online d L
O : line
cO : online c O
dO : online d O
cd : c ≠ d
LO : L ≠ O
α : circle
Hα : center_circle c α ∧ on_circle d α
cα : in_circle c α
αM : line_circle_inter M α
αN : line_circle_inter N α
a a' : point
aa' : a ≠ a'
aM : online a M
a'M : online a' M
aα : on_circle a α ∧ on_circle a' α
b b' : point
bb' : b ≠ b'
bN : online b N
b'N : online b' N
bα : on_circle b α ∧ on_circle b' α
Baca' : B a c a'
Bbcb' : B b c b'
ac : a ≠ c
bc : b ≠ c
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
L M N : line
pLM : para L M
pLN : para L N
MN : ¬M = N
npMN : ¬M = N ∧ ¬para M N
c : point
cM : online c M
cN : online c N
cL : ¬online c L
d : point
dL : online d L
O : line
cO : online c O
dO : online d O
cd : c ≠ d
LO : L ≠ O
α : circle
Hα : center_circle c α ∧ on_circle d α
cα : in_circle c α
αM : line_circle_inter M α
αN : line_circle_inter N α
a a' : point
aa' : a ≠ a'
aM : online a M
a'M : online a' M
aα : on_circle a α ∧ on_circle a' α
b b' : point
bb' : b ≠ b'
bN : online b N
b'N : online b' N
bα : on_circle b α ∧ on_circle b' α
Baca' : B a c a'
Bbcb' : B b c b'
ac : a ≠ c
⊢ False
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_I_extras.lean | para_trans | [40, 1] | [119, 73] | have b'c := (ne_23_of_B Bbcb').symm | case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
L M N : line
pLM : para L M
pLN : para L N
MN : ¬M = N
npMN : ¬M = N ∧ ¬para M N
c : point
cM : online c M
cN : online c N
cL : ¬online c L
d : point
dL : online d L
O : line
cO : online c O
dO : online d O
cd : c ≠ d
LO : L ≠ O
α : circle
Hα : center_circle c α ∧ on_circle d α
cα : in_circle c α
αM : line_circle_inter M α
αN : line_circle_inter N α
a a' : point
aa' : a ≠ a'
aM : online a M
a'M : online a' M
aα : on_circle a α ∧ on_circle a' α
b b' : point
bb' : b ≠ b'
bN : online b N
b'N : online b' N
bα : on_circle b α ∧ on_circle b' α
Baca' : B a c a'
Bbcb' : B b c b'
ac : a ≠ c
bc : b ≠ c
⊢ False | case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
L M N : line
pLM : para L M
pLN : para L N
MN : ¬M = N
npMN : ¬M = N ∧ ¬para M N
c : point
cM : online c M
cN : online c N
cL : ¬online c L
d : point
dL : online d L
O : line
cO : online c O
dO : online d O
cd : c ≠ d
LO : L ≠ O
α : circle
Hα : center_circle c α ∧ on_circle d α
cα : in_circle c α
αM : line_circle_inter M α
αN : line_circle_inter N α
a a' : point
aa' : a ≠ a'
aM : online a M
a'M : online a' M
aα : on_circle a α ∧ on_circle a' α
b b' : point
bb' : b ≠ b'
bN : online b N
b'N : online b' N
bα : on_circle b α ∧ on_circle b' α
Baca' : B a c a'
Bbcb' : B b c b'
ac : a ≠ c
bc : b ≠ c
b'c : b' ≠ c
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
L M N : line
pLM : para L M
pLN : para L N
MN : ¬M = N
npMN : ¬M = N ∧ ¬para M N
c : point
cM : online c M
cN : online c N
cL : ¬online c L
d : point
dL : online d L
O : line
cO : online c O
dO : online d O
cd : c ≠ d
LO : L ≠ O
α : circle
Hα : center_circle c α ∧ on_circle d α
cα : in_circle c α
αM : line_circle_inter M α
αN : line_circle_inter N α
a a' : point
aa' : a ≠ a'
aM : online a M
a'M : online a' M
aα : on_circle a α ∧ on_circle a' α
b b' : point
bb' : b ≠ b'
bN : online b N
b'N : online b' N
bα : on_circle b α ∧ on_circle b' α
Baca' : B a c a'
Bbcb' : B b c b'
ac : a ≠ c
bc : b ≠ c
⊢ False
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_I_extras.lean | para_trans | [40, 1] | [119, 73] | have aO: ¬ online a O := fun aO => neq_of_para dL dO (by rwa [← line_unique_of_pts ac aO cO aM cM] at pLM) rfl | case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
L M N : line
pLM : para L M
pLN : para L N
MN : ¬M = N
npMN : ¬M = N ∧ ¬para M N
c : point
cM : online c M
cN : online c N
cL : ¬online c L
d : point
dL : online d L
O : line
cO : online c O
dO : online d O
cd : c ≠ d
LO : L ≠ O
α : circle
Hα : center_circle c α ∧ on_circle d α
cα : in_circle c α
αM : line_circle_inter M α
αN : line_circle_inter N α
a a' : point
aa' : a ≠ a'
aM : online a M
a'M : online a' M
aα : on_circle a α ∧ on_circle a' α
b b' : point
bb' : b ≠ b'
bN : online b N
b'N : online b' N
bα : on_circle b α ∧ on_circle b' α
Baca' : B a c a'
Bbcb' : B b c b'
ac : a ≠ c
bc : b ≠ c
b'c : b' ≠ c
⊢ False | case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
L M N : line
pLM : para L M
pLN : para L N
MN : ¬M = N
npMN : ¬M = N ∧ ¬para M N
c : point
cM : online c M
cN : online c N
cL : ¬online c L
d : point
dL : online d L
O : line
cO : online c O
dO : online d O
cd : c ≠ d
LO : L ≠ O
α : circle
Hα : center_circle c α ∧ on_circle d α
cα : in_circle c α
αM : line_circle_inter M α
αN : line_circle_inter N α
a a' : point
aa' : a ≠ a'
aM : online a M
a'M : online a' M
aα : on_circle a α ∧ on_circle a' α
b b' : point
bb' : b ≠ b'
bN : online b N
b'N : online b' N
bα : on_circle b α ∧ on_circle b' α
Baca' : B a c a'
Bbcb' : B b c b'
ac : a ≠ c
bc : b ≠ c
b'c : b' ≠ c
aO : ¬online a O
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
L M N : line
pLM : para L M
pLN : para L N
MN : ¬M = N
npMN : ¬M = N ∧ ¬para M N
c : point
cM : online c M
cN : online c N
cL : ¬online c L
d : point
dL : online d L
O : line
cO : online c O
dO : online d O
cd : c ≠ d
LO : L ≠ O
α : circle
Hα : center_circle c α ∧ on_circle d α
cα : in_circle c α
αM : line_circle_inter M α
αN : line_circle_inter N α
a a' : point
aa' : a ≠ a'
aM : online a M
a'M : online a' M
aα : on_circle a α ∧ on_circle a' α
b b' : point
bb' : b ≠ b'
bN : online b N
b'N : online b' N
bα : on_circle b α ∧ on_circle b' α
Baca' : B a c a'
Bbcb' : B b c b'
ac : a ≠ c
bc : b ≠ c
b'c : b' ≠ c
⊢ False
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_I_extras.lean | para_trans | [40, 1] | [119, 73] | have bO : ¬ online b O := fun bO => neq_of_para dL dO (by rwa [← line_unique_of_pts bc bO cO bN cN] at pLN) rfl | case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
L M N : line
pLM : para L M
pLN : para L N
MN : ¬M = N
npMN : ¬M = N ∧ ¬para M N
c : point
cM : online c M
cN : online c N
cL : ¬online c L
d : point
dL : online d L
O : line
cO : online c O
dO : online d O
cd : c ≠ d
LO : L ≠ O
α : circle
Hα : center_circle c α ∧ on_circle d α
cα : in_circle c α
αM : line_circle_inter M α
αN : line_circle_inter N α
a a' : point
aa' : a ≠ a'
aM : online a M
a'M : online a' M
aα : on_circle a α ∧ on_circle a' α
b b' : point
bb' : b ≠ b'
bN : online b N
b'N : online b' N
bα : on_circle b α ∧ on_circle b' α
Baca' : B a c a'
Bbcb' : B b c b'
ac : a ≠ c
bc : b ≠ c
b'c : b' ≠ c
aO : ¬online a O
⊢ False | case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
L M N : line
pLM : para L M
pLN : para L N
MN : ¬M = N
npMN : ¬M = N ∧ ¬para M N
c : point
cM : online c M
cN : online c N
cL : ¬online c L
d : point
dL : online d L
O : line
cO : online c O
dO : online d O
cd : c ≠ d
LO : L ≠ O
α : circle
Hα : center_circle c α ∧ on_circle d α
cα : in_circle c α
αM : line_circle_inter M α
αN : line_circle_inter N α
a a' : point
aa' : a ≠ a'
aM : online a M
a'M : online a' M
aα : on_circle a α ∧ on_circle a' α
b b' : point
bb' : b ≠ b'
bN : online b N
b'N : online b' N
bα : on_circle b α ∧ on_circle b' α
Baca' : B a c a'
Bbcb' : B b c b'
ac : a ≠ c
bc : b ≠ c
b'c : b' ≠ c
aO : ¬online a O
bO : ¬online b O
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
L M N : line
pLM : para L M
pLN : para L N
MN : ¬M = N
npMN : ¬M = N ∧ ¬para M N
c : point
cM : online c M
cN : online c N
cL : ¬online c L
d : point
dL : online d L
O : line
cO : online c O
dO : online d O
cd : c ≠ d
LO : L ≠ O
α : circle
Hα : center_circle c α ∧ on_circle d α
cα : in_circle c α
αM : line_circle_inter M α
αN : line_circle_inter N α
a a' : point
aa' : a ≠ a'
aM : online a M
a'M : online a' M
aα : on_circle a α ∧ on_circle a' α
b b' : point
bb' : b ≠ b'
bN : online b N
b'N : online b' N
bα : on_circle b α ∧ on_circle b' α
Baca' : B a c a'
Bbcb' : B b c b'
ac : a ≠ c
bc : b ≠ c
b'c : b' ≠ c
aO : ¬online a O
⊢ False
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_I_extras.lean | para_trans | [40, 1] | [119, 73] | have b'O : ¬ online b' O := fun hb'O => neq_of_para dL dO (by rwa [← line_unique_of_pts b'c hb'O cO b'N cN] at pLN) rfl | case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
L M N : line
pLM : para L M
pLN : para L N
MN : ¬M = N
npMN : ¬M = N ∧ ¬para M N
c : point
cM : online c M
cN : online c N
cL : ¬online c L
d : point
dL : online d L
O : line
cO : online c O
dO : online d O
cd : c ≠ d
LO : L ≠ O
α : circle
Hα : center_circle c α ∧ on_circle d α
cα : in_circle c α
αM : line_circle_inter M α
αN : line_circle_inter N α
a a' : point
aa' : a ≠ a'
aM : online a M
a'M : online a' M
aα : on_circle a α ∧ on_circle a' α
b b' : point
bb' : b ≠ b'
bN : online b N
b'N : online b' N
bα : on_circle b α ∧ on_circle b' α
Baca' : B a c a'
Bbcb' : B b c b'
ac : a ≠ c
bc : b ≠ c
b'c : b' ≠ c
aO : ¬online a O
bO : ¬online b O
⊢ False | case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
L M N : line
pLM : para L M
pLN : para L N
MN : ¬M = N
npMN : ¬M = N ∧ ¬para M N
c : point
cM : online c M
cN : online c N
cL : ¬online c L
d : point
dL : online d L
O : line
cO : online c O
dO : online d O
cd : c ≠ d
LO : L ≠ O
α : circle
Hα : center_circle c α ∧ on_circle d α
cα : in_circle c α
αM : line_circle_inter M α
αN : line_circle_inter N α
a a' : point
aa' : a ≠ a'
aM : online a M
a'M : online a' M
aα : on_circle a α ∧ on_circle a' α
b b' : point
bb' : b ≠ b'
bN : online b N
b'N : online b' N
bα : on_circle b α ∧ on_circle b' α
Baca' : B a c a'
Bbcb' : B b c b'
ac : a ≠ c
bc : b ≠ c
b'c : b' ≠ c
aO : ¬online a O
bO : ¬online b O
b'O : ¬online b' O
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
L M N : line
pLM : para L M
pLN : para L N
MN : ¬M = N
npMN : ¬M = N ∧ ¬para M N
c : point
cM : online c M
cN : online c N
cL : ¬online c L
d : point
dL : online d L
O : line
cO : online c O
dO : online d O
cd : c ≠ d
LO : L ≠ O
α : circle
Hα : center_circle c α ∧ on_circle d α
cα : in_circle c α
αM : line_circle_inter M α
αN : line_circle_inter N α
a a' : point
aa' : a ≠ a'
aM : online a M
a'M : online a' M
aα : on_circle a α ∧ on_circle a' α
b b' : point
bb' : b ≠ b'
bN : online b N
b'N : online b' N
bα : on_circle b α ∧ on_circle b' α
Baca' : B a c a'
Bbcb' : B b c b'
ac : a ≠ c
bc : b ≠ c
b'c : b' ≠ c
aO : ¬online a O
bO : ¬online b O
⊢ False
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_I_extras.lean | para_trans | [40, 1] | [119, 73] | have aN: ¬ online a N := fun aN => MN (line_unique_of_pts ac aM cM aN cN) | case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
L M N : line
pLM : para L M
pLN : para L N
MN : ¬M = N
npMN : ¬M = N ∧ ¬para M N
c : point
cM : online c M
cN : online c N
cL : ¬online c L
d : point
dL : online d L
O : line
cO : online c O
dO : online d O
cd : c ≠ d
LO : L ≠ O
α : circle
Hα : center_circle c α ∧ on_circle d α
cα : in_circle c α
αM : line_circle_inter M α
αN : line_circle_inter N α
a a' : point
aa' : a ≠ a'
aM : online a M
a'M : online a' M
aα : on_circle a α ∧ on_circle a' α
b b' : point
bb' : b ≠ b'
bN : online b N
b'N : online b' N
bα : on_circle b α ∧ on_circle b' α
Baca' : B a c a'
Bbcb' : B b c b'
ac : a ≠ c
bc : b ≠ c
b'c : b' ≠ c
aO : ¬online a O
bO : ¬online b O
b'O : ¬online b' O
⊢ False | case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
L M N : line
pLM : para L M
pLN : para L N
MN : ¬M = N
npMN : ¬M = N ∧ ¬para M N
c : point
cM : online c M
cN : online c N
cL : ¬online c L
d : point
dL : online d L
O : line
cO : online c O
dO : online d O
cd : c ≠ d
LO : L ≠ O
α : circle
Hα : center_circle c α ∧ on_circle d α
cα : in_circle c α
αM : line_circle_inter M α
αN : line_circle_inter N α
a a' : point
aa' : a ≠ a'
aM : online a M
a'M : online a' M
aα : on_circle a α ∧ on_circle a' α
b b' : point
bb' : b ≠ b'
bN : online b N
b'N : online b' N
bα : on_circle b α ∧ on_circle b' α
Baca' : B a c a'
Bbcb' : B b c b'
ac : a ≠ c
bc : b ≠ c
b'c : b' ≠ c
aO : ¬online a O
bO : ¬online b O
b'O : ¬online b' O
aN : ¬online a N
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
L M N : line
pLM : para L M
pLN : para L N
MN : ¬M = N
npMN : ¬M = N ∧ ¬para M N
c : point
cM : online c M
cN : online c N
cL : ¬online c L
d : point
dL : online d L
O : line
cO : online c O
dO : online d O
cd : c ≠ d
LO : L ≠ O
α : circle
Hα : center_circle c α ∧ on_circle d α
cα : in_circle c α
αM : line_circle_inter M α
αN : line_circle_inter N α
a a' : point
aa' : a ≠ a'
aM : online a M
a'M : online a' M
aα : on_circle a α ∧ on_circle a' α
b b' : point
bb' : b ≠ b'
bN : online b N
b'N : online b' N
bα : on_circle b α ∧ on_circle b' α
Baca' : B a c a'
Bbcb' : B b c b'
ac : a ≠ c
bc : b ≠ c
b'c : b' ≠ c
aO : ¬online a O
bO : ¬online b O
b'O : ¬online b' O
⊢ False
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_I_extras.lean | para_trans | [40, 1] | [119, 73] | have bM: ¬ online b M := fun bM => MN (line_unique_of_pts bc bM cM bN cN) | case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
L M N : line
pLM : para L M
pLN : para L N
MN : ¬M = N
npMN : ¬M = N ∧ ¬para M N
c : point
cM : online c M
cN : online c N
cL : ¬online c L
d : point
dL : online d L
O : line
cO : online c O
dO : online d O
cd : c ≠ d
LO : L ≠ O
α : circle
Hα : center_circle c α ∧ on_circle d α
cα : in_circle c α
αM : line_circle_inter M α
αN : line_circle_inter N α
a a' : point
aa' : a ≠ a'
aM : online a M
a'M : online a' M
aα : on_circle a α ∧ on_circle a' α
b b' : point
bb' : b ≠ b'
bN : online b N
b'N : online b' N
bα : on_circle b α ∧ on_circle b' α
Baca' : B a c a'
Bbcb' : B b c b'
ac : a ≠ c
bc : b ≠ c
b'c : b' ≠ c
aO : ¬online a O
bO : ¬online b O
b'O : ¬online b' O
aN : ¬online a N
⊢ False | case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
L M N : line
pLM : para L M
pLN : para L N
MN : ¬M = N
npMN : ¬M = N ∧ ¬para M N
c : point
cM : online c M
cN : online c N
cL : ¬online c L
d : point
dL : online d L
O : line
cO : online c O
dO : online d O
cd : c ≠ d
LO : L ≠ O
α : circle
Hα : center_circle c α ∧ on_circle d α
cα : in_circle c α
αM : line_circle_inter M α
αN : line_circle_inter N α
a a' : point
aa' : a ≠ a'
aM : online a M
a'M : online a' M
aα : on_circle a α ∧ on_circle a' α
b b' : point
bb' : b ≠ b'
bN : online b N
b'N : online b' N
bα : on_circle b α ∧ on_circle b' α
Baca' : B a c a'
Bbcb' : B b c b'
ac : a ≠ c
bc : b ≠ c
b'c : b' ≠ c
aO : ¬online a O
bO : ¬online b O
b'O : ¬online b' O
aN : ¬online a N
bM : ¬online b M
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
L M N : line
pLM : para L M
pLN : para L N
MN : ¬M = N
npMN : ¬M = N ∧ ¬para M N
c : point
cM : online c M
cN : online c N
cL : ¬online c L
d : point
dL : online d L
O : line
cO : online c O
dO : online d O
cd : c ≠ d
LO : L ≠ O
α : circle
Hα : center_circle c α ∧ on_circle d α
cα : in_circle c α
αM : line_circle_inter M α
αN : line_circle_inter N α
a a' : point
aa' : a ≠ a'
aM : online a M
a'M : online a' M
aα : on_circle a α ∧ on_circle a' α
b b' : point
bb' : b ≠ b'
bN : online b N
b'N : online b' N
bα : on_circle b α ∧ on_circle b' α
Baca' : B a c a'
Bbcb' : B b c b'
ac : a ≠ c
bc : b ≠ c
b'c : b' ≠ c
aO : ¬online a O
bO : ¬online b O
b'O : ¬online b' O
aN : ¬online a N
⊢ False
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_I_extras.lean | para_trans | [40, 1] | [119, 73] | have b'M: ¬ online b' M := fun hb'M => MN (line_unique_of_pts b'c hb'M cM b'N cN) | case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
L M N : line
pLM : para L M
pLN : para L N
MN : ¬M = N
npMN : ¬M = N ∧ ¬para M N
c : point
cM : online c M
cN : online c N
cL : ¬online c L
d : point
dL : online d L
O : line
cO : online c O
dO : online d O
cd : c ≠ d
LO : L ≠ O
α : circle
Hα : center_circle c α ∧ on_circle d α
cα : in_circle c α
αM : line_circle_inter M α
αN : line_circle_inter N α
a a' : point
aa' : a ≠ a'
aM : online a M
a'M : online a' M
aα : on_circle a α ∧ on_circle a' α
b b' : point
bb' : b ≠ b'
bN : online b N
b'N : online b' N
bα : on_circle b α ∧ on_circle b' α
Baca' : B a c a'
Bbcb' : B b c b'
ac : a ≠ c
bc : b ≠ c
b'c : b' ≠ c
aO : ¬online a O
bO : ¬online b O
b'O : ¬online b' O
aN : ¬online a N
bM : ¬online b M
⊢ False | case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
L M N : line
pLM : para L M
pLN : para L N
MN : ¬M = N
npMN : ¬M = N ∧ ¬para M N
c : point
cM : online c M
cN : online c N
cL : ¬online c L
d : point
dL : online d L
O : line
cO : online c O
dO : online d O
cd : c ≠ d
LO : L ≠ O
α : circle
Hα : center_circle c α ∧ on_circle d α
cα : in_circle c α
αM : line_circle_inter M α
αN : line_circle_inter N α
a a' : point
aa' : a ≠ a'
aM : online a M
a'M : online a' M
aα : on_circle a α ∧ on_circle a' α
b b' : point
bb' : b ≠ b'
bN : online b N
b'N : online b' N
bα : on_circle b α ∧ on_circle b' α
Baca' : B a c a'
Bbcb' : B b c b'
ac : a ≠ c
bc : b ≠ c
b'c : b' ≠ c
aO : ¬online a O
bO : ¬online b O
b'O : ¬online b' O
aN : ¬online a N
bM : ¬online b M
b'M : ¬online b' M
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
L M N : line
pLM : para L M
pLN : para L N
MN : ¬M = N
npMN : ¬M = N ∧ ¬para M N
c : point
cM : online c M
cN : online c N
cL : ¬online c L
d : point
dL : online d L
O : line
cO : online c O
dO : online d O
cd : c ≠ d
LO : L ≠ O
α : circle
Hα : center_circle c α ∧ on_circle d α
cα : in_circle c α
αM : line_circle_inter M α
αN : line_circle_inter N α
a a' : point
aa' : a ≠ a'
aM : online a M
a'M : online a' M
aα : on_circle a α ∧ on_circle a' α
b b' : point
bb' : b ≠ b'
bN : online b N
b'N : online b' N
bα : on_circle b α ∧ on_circle b' α
Baca' : B a c a'
Bbcb' : B b c b'
ac : a ≠ c
bc : b ≠ c
b'c : b' ≠ c
aO : ¬online a O
bO : ¬online b O
b'O : ¬online b' O
aN : ¬online a N
bM : ¬online b M
⊢ False
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_I_extras.lean | para_trans | [40, 1] | [119, 73] | have NO : N ≠ O := fun NO => bO (by rwa [NO] at bN) | case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
L M N : line
pLM : para L M
pLN : para L N
MN : ¬M = N
npMN : ¬M = N ∧ ¬para M N
c : point
cM : online c M
cN : online c N
cL : ¬online c L
d : point
dL : online d L
O : line
cO : online c O
dO : online d O
cd : c ≠ d
LO : L ≠ O
α : circle
Hα : center_circle c α ∧ on_circle d α
cα : in_circle c α
αM : line_circle_inter M α
αN : line_circle_inter N α
a a' : point
aa' : a ≠ a'
aM : online a M
a'M : online a' M
aα : on_circle a α ∧ on_circle a' α
b b' : point
bb' : b ≠ b'
bN : online b N
b'N : online b' N
bα : on_circle b α ∧ on_circle b' α
Baca' : B a c a'
Bbcb' : B b c b'
ac : a ≠ c
bc : b ≠ c
b'c : b' ≠ c
aO : ¬online a O
bO : ¬online b O
b'O : ¬online b' O
aN : ¬online a N
bM : ¬online b M
b'M : ¬online b' M
⊢ False | case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
L M N : line
pLM : para L M
pLN : para L N
MN : ¬M = N
npMN : ¬M = N ∧ ¬para M N
c : point
cM : online c M
cN : online c N
cL : ¬online c L
d : point
dL : online d L
O : line
cO : online c O
dO : online d O
cd : c ≠ d
LO : L ≠ O
α : circle
Hα : center_circle c α ∧ on_circle d α
cα : in_circle c α
αM : line_circle_inter M α
αN : line_circle_inter N α
a a' : point
aa' : a ≠ a'
aM : online a M
a'M : online a' M
aα : on_circle a α ∧ on_circle a' α
b b' : point
bb' : b ≠ b'
bN : online b N
b'N : online b' N
bα : on_circle b α ∧ on_circle b' α
Baca' : B a c a'
Bbcb' : B b c b'
ac : a ≠ c
bc : b ≠ c
b'c : b' ≠ c
aO : ¬online a O
bO : ¬online b O
b'O : ¬online b' O
aN : ¬online a N
bM : ¬online b M
b'M : ¬online b' M
NO : N ≠ O
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
L M N : line
pLM : para L M
pLN : para L N
MN : ¬M = N
npMN : ¬M = N ∧ ¬para M N
c : point
cM : online c M
cN : online c N
cL : ¬online c L
d : point
dL : online d L
O : line
cO : online c O
dO : online d O
cd : c ≠ d
LO : L ≠ O
α : circle
Hα : center_circle c α ∧ on_circle d α
cα : in_circle c α
αM : line_circle_inter M α
αN : line_circle_inter N α
a a' : point
aa' : a ≠ a'
aM : online a M
a'M : online a' M
aα : on_circle a α ∧ on_circle a' α
b b' : point
bb' : b ≠ b'
bN : online b N
b'N : online b' N
bα : on_circle b α ∧ on_circle b' α
Baca' : B a c a'
Bbcb' : B b c b'
ac : a ≠ c
bc : b ≠ c
b'c : b' ≠ c
aO : ¬online a O
bO : ¬online b O
b'O : ¬online b' O
aN : ¬online a N
bM : ¬online b M
b'M : ¬online b' M
⊢ False
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_I_extras.lean | para_trans | [40, 1] | [119, 73] | have MO : M ≠ O := fun MO => aO (by rwa [MO] at aM) | case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
L M N : line
pLM : para L M
pLN : para L N
MN : ¬M = N
npMN : ¬M = N ∧ ¬para M N
c : point
cM : online c M
cN : online c N
cL : ¬online c L
d : point
dL : online d L
O : line
cO : online c O
dO : online d O
cd : c ≠ d
LO : L ≠ O
α : circle
Hα : center_circle c α ∧ on_circle d α
cα : in_circle c α
αM : line_circle_inter M α
αN : line_circle_inter N α
a a' : point
aa' : a ≠ a'
aM : online a M
a'M : online a' M
aα : on_circle a α ∧ on_circle a' α
b b' : point
bb' : b ≠ b'
bN : online b N
b'N : online b' N
bα : on_circle b α ∧ on_circle b' α
Baca' : B a c a'
Bbcb' : B b c b'
ac : a ≠ c
bc : b ≠ c
b'c : b' ≠ c
aO : ¬online a O
bO : ¬online b O
b'O : ¬online b' O
aN : ¬online a N
bM : ¬online b M
b'M : ¬online b' M
NO : N ≠ O
⊢ False | case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
L M N : line
pLM : para L M
pLN : para L N
MN : ¬M = N
npMN : ¬M = N ∧ ¬para M N
c : point
cM : online c M
cN : online c N
cL : ¬online c L
d : point
dL : online d L
O : line
cO : online c O
dO : online d O
cd : c ≠ d
LO : L ≠ O
α : circle
Hα : center_circle c α ∧ on_circle d α
cα : in_circle c α
αM : line_circle_inter M α
αN : line_circle_inter N α
a a' : point
aa' : a ≠ a'
aM : online a M
a'M : online a' M
aα : on_circle a α ∧ on_circle a' α
b b' : point
bb' : b ≠ b'
bN : online b N
b'N : online b' N
bα : on_circle b α ∧ on_circle b' α
Baca' : B a c a'
Bbcb' : B b c b'
ac : a ≠ c
bc : b ≠ c
b'c : b' ≠ c
aO : ¬online a O
bO : ¬online b O
b'O : ¬online b' O
aN : ¬online a N
bM : ¬online b M
b'M : ¬online b' M
NO : N ≠ O
MO : M ≠ O
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
i : incidence_geometry
L M N : line
pLM : para L M
pLN : para L N
MN : ¬M = N
npMN : ¬M = N ∧ ¬para M N
c : point
cM : online c M
cN : online c N
cL : ¬online c L
d : point
dL : online d L
O : line
cO : online c O
dO : online d O
cd : c ≠ d
LO : L ≠ O
α : circle
Hα : center_circle c α ∧ on_circle d α
cα : in_circle c α
αM : line_circle_inter M α
αN : line_circle_inter N α
a a' : point
aa' : a ≠ a'
aM : online a M
a'M : online a' M
aα : on_circle a α ∧ on_circle a' α
b b' : point
bb' : b ≠ b'
bN : online b N
b'N : online b' N
bα : on_circle b α ∧ on_circle b' α
Baca' : B a c a'
Bbcb' : B b c b'
ac : a ≠ c
bc : b ≠ c
b'c : b' ≠ c
aO : ¬online a O
bO : ¬online b O
b'O : ¬online b' O
aN : ¬online a N
bM : ¬online b M
b'M : ¬online b' M
NO : N ≠ O
⊢ False
TACTIC:
|
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