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/euclid_VI.lean | rescale_triangle_of_base_inductive | [53, 1] | [111, 73] | have ed : e ≠ d := fun ed => by
have := He.2.1; rw [ed, hlen] at this; simp at this
exact bc (length_eq_zero_iff.mp this) | case neg.intro
i : incidence_geometry
a b c : point
L : line
bL : online b L
cL : online c L
aL : ¬online a L
n : ℕ
hn : ∀ (d : point), online d L → length b d = ↑n * length b c → B b c d → area a b d = ↑n * area a b c
d : point
dL : online d L
hlen : length b d = ↑(Nat.succ n) * length b c
Bbcd : B b c d
bc : b ≠ c
bd : b ≠ d
cd : c ≠ d
n0 : ¬n = 0
n1 : ¬n = 1
e : point
He : online e L ∧ length b e = ↑n * length b c ∧ (n ≥ 2 ∧ b ≠ c → B b c e)
Bbce : B b c e
be : b ≠ e
⊢ area a b d = area a b e + area a b c | case neg.intro
i : incidence_geometry
a b c : point
L : line
bL : online b L
cL : online c L
aL : ¬online a L
n : ℕ
hn : ∀ (d : point), online d L → length b d = ↑n * length b c → B b c d → area a b d = ↑n * area a b c
d : point
dL : online d L
hlen : length b d = ↑(Nat.succ n) * length b c
Bbcd : B b c d
bc : b ≠ c
bd : b ≠ d
cd : c ≠ d
n0 : ¬n = 0
n1 : ¬n = 1
e : point
He : online e L ∧ length b e = ↑n * length b c ∧ (n ≥ 2 ∧ b ≠ c → B b c e)
Bbce : B b c e
be : b ≠ e
ed : e ≠ d
⊢ area a b d = area a b e + area a b c | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro
i : incidence_geometry
a b c : point
L : line
bL : online b L
cL : online c L
aL : ¬online a L
n : ℕ
hn : ∀ (d : point), online d L → length b d = ↑n * length b c → B b c d → area a b d = ↑n * area a b c
d : point
dL : online d L
hlen : length b d = ↑(Nat.succ n) * length b c
Bbcd : B b c d
bc : b ≠ c
bd : b ≠ d
cd : c ≠ d
n0 : ¬n = 0
n1 : ¬n = 1
e : point
He : online e L ∧ length b e = ↑n * length b c ∧ (n ≥ 2 ∧ b ≠ c → B b c e)
Bbce : B b c e
be : b ≠ e
⊢ area a b d = area a b e + area a b c
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_triangle_of_base_inductive | [53, 1] | [111, 73] | have Bbed : B b e d := by
have: length b e < length b d := by rw [He.2.1, hlen]; simp; linarith [len_pos_of_nq bc]
have := B_of_three_online_ne_short this be bd ed bL He.1 dL
cases this with
| inl h => exact h
| inr h =>
exfalso
exact not_B324_of_B123_B124 Bbce Bbcd (B124_of_B134_B123 h (B_symm Bbce)) | case neg.intro
i : incidence_geometry
a b c : point
L : line
bL : online b L
cL : online c L
aL : ¬online a L
n : ℕ
hn : ∀ (d : point), online d L → length b d = ↑n * length b c → B b c d → area a b d = ↑n * area a b c
d : point
dL : online d L
hlen : length b d = ↑(Nat.succ n) * length b c
Bbcd : B b c d
bc : b ≠ c
bd : b ≠ d
cd : c ≠ d
n0 : ¬n = 0
n1 : ¬n = 1
e : point
He : online e L ∧ length b e = ↑n * length b c ∧ (n ≥ 2 ∧ b ≠ c → B b c e)
Bbce : B b c e
be : b ≠ e
ed : e ≠ d
⊢ area a b d = area a b e + area a b c | case neg.intro
i : incidence_geometry
a b c : point
L : line
bL : online b L
cL : online c L
aL : ¬online a L
n : ℕ
hn : ∀ (d : point), online d L → length b d = ↑n * length b c → B b c d → area a b d = ↑n * area a b c
d : point
dL : online d L
hlen : length b d = ↑(Nat.succ n) * length b c
Bbcd : B b c d
bc : b ≠ c
bd : b ≠ d
cd : c ≠ d
n0 : ¬n = 0
n1 : ¬n = 1
e : point
He : online e L ∧ length b e = ↑n * length b c ∧ (n ≥ 2 ∧ b ≠ c → B b c e)
Bbce : B b c e
be : b ≠ e
ed : e ≠ d
Bbed : B b e d
⊢ area a b d = area a b e + area a b c | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro
i : incidence_geometry
a b c : point
L : line
bL : online b L
cL : online c L
aL : ¬online a L
n : ℕ
hn : ∀ (d : point), online d L → length b d = ↑n * length b c → B b c d → area a b d = ↑n * area a b c
d : point
dL : online d L
hlen : length b d = ↑(Nat.succ n) * length b c
Bbcd : B b c d
bc : b ≠ c
bd : b ≠ d
cd : c ≠ d
n0 : ¬n = 0
n1 : ¬n = 1
e : point
He : online e L ∧ length b e = ↑n * length b c ∧ (n ≥ 2 ∧ b ≠ c → B b c e)
Bbce : B b c e
be : b ≠ e
ed : e ≠ d
⊢ area a b d = area a b e + area a b c
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_triangle_of_base_inductive | [53, 1] | [111, 73] | rw [((area_add_iff_B be ed bd.symm bL He.1 dL aL).mp Bbed).symm] | case neg.intro
i : incidence_geometry
a b c : point
L : line
bL : online b L
cL : online c L
aL : ¬online a L
n : ℕ
hn : ∀ (d : point), online d L → length b d = ↑n * length b c → B b c d → area a b d = ↑n * area a b c
d : point
dL : online d L
hlen : length b d = ↑(Nat.succ n) * length b c
Bbcd : B b c d
bc : b ≠ c
bd : b ≠ d
cd : c ≠ d
n0 : ¬n = 0
n1 : ¬n = 1
e : point
He : online e L ∧ length b e = ↑n * length b c ∧ (n ≥ 2 ∧ b ≠ c → B b c e)
Bbce : B b c e
be : b ≠ e
ed : e ≠ d
Bbed : B b e d
⊢ area a b d = area a b e + area a b c | case neg.intro
i : incidence_geometry
a b c : point
L : line
bL : online b L
cL : online c L
aL : ¬online a L
n : ℕ
hn : ∀ (d : point), online d L → length b d = ↑n * length b c → B b c d → area a b d = ↑n * area a b c
d : point
dL : online d L
hlen : length b d = ↑(Nat.succ n) * length b c
Bbcd : B b c d
bc : b ≠ c
bd : b ≠ d
cd : c ≠ d
n0 : ¬n = 0
n1 : ¬n = 1
e : point
He : online e L ∧ length b e = ↑n * length b c ∧ (n ≥ 2 ∧ b ≠ c → B b c e)
Bbce : B b c e
be : b ≠ e
ed : e ≠ d
Bbed : B b e d
⊢ area a b e + area a d e = area a b e + area a b c | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro
i : incidence_geometry
a b c : point
L : line
bL : online b L
cL : online c L
aL : ¬online a L
n : ℕ
hn : ∀ (d : point), online d L → length b d = ↑n * length b c → B b c d → area a b d = ↑n * area a b c
d : point
dL : online d L
hlen : length b d = ↑(Nat.succ n) * length b c
Bbcd : B b c d
bc : b ≠ c
bd : b ≠ d
cd : c ≠ d
n0 : ¬n = 0
n1 : ¬n = 1
e : point
He : online e L ∧ length b e = ↑n * length b c ∧ (n ≥ 2 ∧ b ≠ c → B b c e)
Bbce : B b c e
be : b ≠ e
ed : e ≠ d
Bbed : B b e d
⊢ area a b d = area a b e + area a b c
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_triangle_of_base_inductive | [53, 1] | [111, 73] | have: length b c = length e d := by
have := length_sum_of_B Bbed
rw [hlen, He.2.1] at this; simp [one_mul] at this; linarith | case neg.intro
i : incidence_geometry
a b c : point
L : line
bL : online b L
cL : online c L
aL : ¬online a L
n : ℕ
hn : ∀ (d : point), online d L → length b d = ↑n * length b c → B b c d → area a b d = ↑n * area a b c
d : point
dL : online d L
hlen : length b d = ↑(Nat.succ n) * length b c
Bbcd : B b c d
bc : b ≠ c
bd : b ≠ d
cd : c ≠ d
n0 : ¬n = 0
n1 : ¬n = 1
e : point
He : online e L ∧ length b e = ↑n * length b c ∧ (n ≥ 2 ∧ b ≠ c → B b c e)
Bbce : B b c e
be : b ≠ e
ed : e ≠ d
Bbed : B b e d
⊢ area a b e + area a d e = area a b e + area a b c | case neg.intro
i : incidence_geometry
a b c : point
L : line
bL : online b L
cL : online c L
aL : ¬online a L
n : ℕ
hn : ∀ (d : point), online d L → length b d = ↑n * length b c → B b c d → area a b d = ↑n * area a b c
d : point
dL : online d L
hlen : length b d = ↑(Nat.succ n) * length b c
Bbcd : B b c d
bc : b ≠ c
bd : b ≠ d
cd : c ≠ d
n0 : ¬n = 0
n1 : ¬n = 1
e : point
He : online e L ∧ length b e = ↑n * length b c ∧ (n ≥ 2 ∧ b ≠ c → B b c e)
Bbce : B b c e
be : b ≠ e
ed : e ≠ d
Bbed : B b e d
this : length b c = length e d
⊢ area a b e + area a d e = area a b e + area a b c | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro
i : incidence_geometry
a b c : point
L : line
bL : online b L
cL : online c L
aL : ¬online a L
n : ℕ
hn : ∀ (d : point), online d L → length b d = ↑n * length b c → B b c d → area a b d = ↑n * area a b c
d : point
dL : online d L
hlen : length b d = ↑(Nat.succ n) * length b c
Bbcd : B b c d
bc : b ≠ c
bd : b ≠ d
cd : c ≠ d
n0 : ¬n = 0
n1 : ¬n = 1
e : point
He : online e L ∧ length b e = ↑n * length b c ∧ (n ≥ 2 ∧ b ≠ c → B b c e)
Bbce : B b c e
be : b ≠ e
ed : e ≠ d
Bbed : B b e d
⊢ area a b e + area a d e = area a b e + area a b c
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_triangle_of_base_inductive | [53, 1] | [111, 73] | perm [eq_area_of_eq_base_samevertex a bL cL He.1 dL this] | case neg.intro
i : incidence_geometry
a b c : point
L : line
bL : online b L
cL : online c L
aL : ¬online a L
n : ℕ
hn : ∀ (d : point), online d L → length b d = ↑n * length b c → B b c d → area a b d = ↑n * area a b c
d : point
dL : online d L
hlen : length b d = ↑(Nat.succ n) * length b c
Bbcd : B b c d
bc : b ≠ c
bd : b ≠ d
cd : c ≠ d
n0 : ¬n = 0
n1 : ¬n = 1
e : point
He : online e L ∧ length b e = ↑n * length b c ∧ (n ≥ 2 ∧ b ≠ c → B b c e)
Bbce : B b c e
be : b ≠ e
ed : e ≠ d
Bbed : B b e d
this : length b c = length e d
⊢ area a b e + area a d e = area a b e + area a b c | case neg.intro
i : incidence_geometry
a b c : point
L : line
bL : online b L
cL : online c L
aL : ¬online a L
n : ℕ
hn : ∀ (d : point), online d L → length b d = ↑n * length b c → B b c d → area a b d = ↑n * area a b c
d : point
dL : online d L
hlen : length b d = ↑(Nat.succ n) * length b c
Bbcd : B b c d
bc : b ≠ c
bd : b ≠ d
cd : c ≠ d
n0 : ¬n = 0
n1 : ¬n = 1
e : point
He : online e L ∧ length b e = ↑n * length b c ∧ (n ≥ 2 ∧ b ≠ c → B b c e)
Bbce : B b c e
be : b ≠ e
ed : e ≠ d
Bbed : B b e d
this✝ : length b c = length e d
this : area a b c = area a d e
⊢ area a b e + area a d e = area a b e + area a b c | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro
i : incidence_geometry
a b c : point
L : line
bL : online b L
cL : online c L
aL : ¬online a L
n : ℕ
hn : ∀ (d : point), online d L → length b d = ↑n * length b c → B b c d → area a b d = ↑n * area a b c
d : point
dL : online d L
hlen : length b d = ↑(Nat.succ n) * length b c
Bbcd : B b c d
bc : b ≠ c
bd : b ≠ d
cd : c ≠ d
n0 : ¬n = 0
n1 : ¬n = 1
e : point
He : online e L ∧ length b e = ↑n * length b c ∧ (n ≥ 2 ∧ b ≠ c → B b c e)
Bbce : B b c e
be : b ≠ e
ed : e ≠ d
Bbed : B b e d
this : length b c = length e d
⊢ area a b e + area a d e = area a b e + area a b c
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_triangle_of_base_inductive | [53, 1] | [111, 73] | rw [this] | case neg.intro
i : incidence_geometry
a b c : point
L : line
bL : online b L
cL : online c L
aL : ¬online a L
n : ℕ
hn : ∀ (d : point), online d L → length b d = ↑n * length b c → B b c d → area a b d = ↑n * area a b c
d : point
dL : online d L
hlen : length b d = ↑(Nat.succ n) * length b c
Bbcd : B b c d
bc : b ≠ c
bd : b ≠ d
cd : c ≠ d
n0 : ¬n = 0
n1 : ¬n = 1
e : point
He : online e L ∧ length b e = ↑n * length b c ∧ (n ≥ 2 ∧ b ≠ c → B b c e)
Bbce : B b c e
be : b ≠ e
ed : e ≠ d
Bbed : B b e d
this✝ : length b c = length e d
this : area a b c = area a d e
⊢ area a b e + area a d e = area a b e + area a b c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro
i : incidence_geometry
a b c : point
L : line
bL : online b L
cL : online c L
aL : ¬online a L
n : ℕ
hn : ∀ (d : point), online d L → length b d = ↑n * length b c → B b c d → area a b d = ↑n * area a b c
d : point
dL : online d L
hlen : length b d = ↑(Nat.succ n) * length b c
Bbcd : B b c d
bc : b ≠ c
bd : b ≠ d
cd : c ≠ d
n0 : ¬n = 0
n1 : ¬n = 1
e : point
He : online e L ∧ length b e = ↑n * length b c ∧ (n ≥ 2 ∧ b ≠ c → B b c e)
Bbce : B b c e
be : b ≠ e
ed : e ≠ d
Bbed : B b e d
this✝ : length b c = length e d
this : area a b c = area a d e
⊢ area a b e + area a d e = area a b e + area a b c
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_triangle_of_base_inductive | [53, 1] | [111, 73] | rw [← length_sum_of_B Bbcd, n0] at hlen | case pos
i : incidence_geometry
a b c : point
L : line
bL : online b L
cL : online c L
aL : ¬online a L
n : ℕ
hn : ∀ (d : point), online d L → length b d = ↑n * length b c → B b c d → area a b d = ↑n * area a b c
d : point
dL : online d L
hlen : length b d = ↑(Nat.succ n) * length b c
Bbcd : B b c d
bc : b ≠ c
bd : b ≠ d
cd : c ≠ d
n0 : n = 0
⊢ area a b d = ↑(Nat.succ n) * area a b c | case pos
i : incidence_geometry
a b c : point
L : line
bL : online b L
cL : online c L
aL : ¬online a L
n : ℕ
hn : ∀ (d : point), online d L → length b d = ↑n * length b c → B b c d → area a b d = ↑n * area a b c
d : point
dL : online d L
hlen : length b c + length c d = ↑(Nat.succ 0) * length b c
Bbcd : B b c d
bc : b ≠ c
bd : b ≠ d
cd : c ≠ d
n0 : n = 0
⊢ area a b d = ↑(Nat.succ n) * area a b c | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
i : incidence_geometry
a b c : point
L : line
bL : online b L
cL : online c L
aL : ¬online a L
n : ℕ
hn : ∀ (d : point), online d L → length b d = ↑n * length b c → B b c d → area a b d = ↑n * area a b c
d : point
dL : online d L
hlen : length b d = ↑(Nat.succ n) * length b c
Bbcd : B b c d
bc : b ≠ c
bd : b ≠ d
cd : c ≠ d
n0 : n = 0
⊢ area a b d = ↑(Nat.succ n) * area a b c
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_triangle_of_base_inductive | [53, 1] | [111, 73] | simp at hlen | case pos
i : incidence_geometry
a b c : point
L : line
bL : online b L
cL : online c L
aL : ¬online a L
n : ℕ
hn : ∀ (d : point), online d L → length b d = ↑n * length b c → B b c d → area a b d = ↑n * area a b c
d : point
dL : online d L
hlen : length b c + length c d = ↑(Nat.succ 0) * length b c
Bbcd : B b c d
bc : b ≠ c
bd : b ≠ d
cd : c ≠ d
n0 : n = 0
⊢ area a b d = ↑(Nat.succ n) * area a b c | case pos
i : incidence_geometry
a b c : point
L : line
bL : online b L
cL : online c L
aL : ¬online a L
n : ℕ
hn : ∀ (d : point), online d L → length b d = ↑n * length b c → B b c d → area a b d = ↑n * area a b c
d : point
dL : online d L
Bbcd : B b c d
bc : b ≠ c
bd : b ≠ d
cd : c ≠ d
n0 : n = 0
hlen : length c d = 0
⊢ area a b d = ↑(Nat.succ n) * area a b c | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
i : incidence_geometry
a b c : point
L : line
bL : online b L
cL : online c L
aL : ¬online a L
n : ℕ
hn : ∀ (d : point), online d L → length b d = ↑n * length b c → B b c d → area a b d = ↑n * area a b c
d : point
dL : online d L
hlen : length b c + length c d = ↑(Nat.succ 0) * length b c
Bbcd : B b c d
bc : b ≠ c
bd : b ≠ d
cd : c ≠ d
n0 : n = 0
⊢ area a b d = ↑(Nat.succ n) * area a b c
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_triangle_of_base_inductive | [53, 1] | [111, 73] | exfalso | case pos
i : incidence_geometry
a b c : point
L : line
bL : online b L
cL : online c L
aL : ¬online a L
n : ℕ
hn : ∀ (d : point), online d L → length b d = ↑n * length b c → B b c d → area a b d = ↑n * area a b c
d : point
dL : online d L
Bbcd : B b c d
bc : b ≠ c
bd : b ≠ d
cd : c ≠ d
n0 : n = 0
hlen : length c d = 0
⊢ area a b d = ↑(Nat.succ n) * area a b c | case pos.h
i : incidence_geometry
a b c : point
L : line
bL : online b L
cL : online c L
aL : ¬online a L
n : ℕ
hn : ∀ (d : point), online d L → length b d = ↑n * length b c → B b c d → area a b d = ↑n * area a b c
d : point
dL : online d L
Bbcd : B b c d
bc : b ≠ c
bd : b ≠ d
cd : c ≠ d
n0 : n = 0
hlen : length c d = 0
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
i : incidence_geometry
a b c : point
L : line
bL : online b L
cL : online c L
aL : ¬online a L
n : ℕ
hn : ∀ (d : point), online d L → length b d = ↑n * length b c → B b c d → area a b d = ↑n * area a b c
d : point
dL : online d L
Bbcd : B b c d
bc : b ≠ c
bd : b ≠ d
cd : c ≠ d
n0 : n = 0
hlen : length c d = 0
⊢ area a b d = ↑(Nat.succ n) * area a b c
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_triangle_of_base_inductive | [53, 1] | [111, 73] | exact cd (length_eq_zero_iff.mp hlen) | case pos.h
i : incidence_geometry
a b c : point
L : line
bL : online b L
cL : online c L
aL : ¬online a L
n : ℕ
hn : ∀ (d : point), online d L → length b d = ↑n * length b c → B b c d → area a b d = ↑n * area a b c
d : point
dL : online d L
Bbcd : B b c d
bc : b ≠ c
bd : b ≠ d
cd : c ≠ d
n0 : n = 0
hlen : length c d = 0
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.h
i : incidence_geometry
a b c : point
L : line
bL : online b L
cL : online c L
aL : ¬online a L
n : ℕ
hn : ∀ (d : point), online d L → length b d = ↑n * length b c → B b c d → area a b d = ↑n * area a b c
d : point
dL : online d L
Bbcd : B b c d
bc : b ≠ c
bd : b ≠ d
cd : c ≠ d
n0 : n = 0
hlen : length c d = 0
⊢ False
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_triangle_of_base_inductive | [53, 1] | [111, 73] | rw [← length_sum_of_B Bbcd, n1,] at hlen | case pos
i : incidence_geometry
a b c : point
L : line
bL : online b L
cL : online c L
aL : ¬online a L
n : ℕ
hn : ∀ (d : point), online d L → length b d = ↑n * length b c → B b c d → area a b d = ↑n * area a b c
d : point
dL : online d L
hlen : length b d = ↑(Nat.succ n) * length b c
Bbcd : B b c d
bc : b ≠ c
bd : b ≠ d
cd : c ≠ d
n0 : ¬n = 0
n1 : n = 1
⊢ area a b d = ↑(Nat.succ n) * area a b c | case pos
i : incidence_geometry
a b c : point
L : line
bL : online b L
cL : online c L
aL : ¬online a L
n : ℕ
hn : ∀ (d : point), online d L → length b d = ↑n * length b c → B b c d → area a b d = ↑n * area a b c
d : point
dL : online d L
hlen : length b c + length c d = ↑(Nat.succ 1) * length b c
Bbcd : B b c d
bc : b ≠ c
bd : b ≠ d
cd : c ≠ d
n0 : ¬n = 0
n1 : n = 1
⊢ area a b d = ↑(Nat.succ n) * area a b c | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
i : incidence_geometry
a b c : point
L : line
bL : online b L
cL : online c L
aL : ¬online a L
n : ℕ
hn : ∀ (d : point), online d L → length b d = ↑n * length b c → B b c d → area a b d = ↑n * area a b c
d : point
dL : online d L
hlen : length b d = ↑(Nat.succ n) * length b c
Bbcd : B b c d
bc : b ≠ c
bd : b ≠ d
cd : c ≠ d
n0 : ¬n = 0
n1 : n = 1
⊢ area a b d = ↑(Nat.succ n) * area a b c
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_triangle_of_base_inductive | [53, 1] | [111, 73] | simp at hlen | case pos
i : incidence_geometry
a b c : point
L : line
bL : online b L
cL : online c L
aL : ¬online a L
n : ℕ
hn : ∀ (d : point), online d L → length b d = ↑n * length b c → B b c d → area a b d = ↑n * area a b c
d : point
dL : online d L
hlen : length b c + length c d = ↑(Nat.succ 1) * length b c
Bbcd : B b c d
bc : b ≠ c
bd : b ≠ d
cd : c ≠ d
n0 : ¬n = 0
n1 : n = 1
⊢ area a b d = ↑(Nat.succ n) * area a b c | case pos
i : incidence_geometry
a b c : point
L : line
bL : online b L
cL : online c L
aL : ¬online a L
n : ℕ
hn : ∀ (d : point), online d L → length b d = ↑n * length b c → B b c d → area a b d = ↑n * area a b c
d : point
dL : online d L
Bbcd : B b c d
bc : b ≠ c
bd : b ≠ d
cd : c ≠ d
n0 : ¬n = 0
n1 : n = 1
hlen : length b c + length c d = (1 + 1) * length b c
⊢ area a b d = ↑(Nat.succ n) * area a b c | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
i : incidence_geometry
a b c : point
L : line
bL : online b L
cL : online c L
aL : ¬online a L
n : ℕ
hn : ∀ (d : point), online d L → length b d = ↑n * length b c → B b c d → area a b d = ↑n * area a b c
d : point
dL : online d L
hlen : length b c + length c d = ↑(Nat.succ 1) * length b c
Bbcd : B b c d
bc : b ≠ c
bd : b ≠ d
cd : c ≠ d
n0 : ¬n = 0
n1 : n = 1
⊢ area a b d = ↑(Nat.succ n) * area a b c
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_triangle_of_base_inductive | [53, 1] | [111, 73] | rw [← (area_add_iff_B bc cd bd.symm bL cL dL aL).mp Bbcd, n1] | case pos
i : incidence_geometry
a b c : point
L : line
bL : online b L
cL : online c L
aL : ¬online a L
n : ℕ
hn : ∀ (d : point), online d L → length b d = ↑n * length b c → B b c d → area a b d = ↑n * area a b c
d : point
dL : online d L
Bbcd : B b c d
bc : b ≠ c
bd : b ≠ d
cd : c ≠ d
n0 : ¬n = 0
n1 : n = 1
hlen : length b c + length c d = (1 + 1) * length b c
⊢ area a b d = ↑(Nat.succ n) * area a b c | case pos
i : incidence_geometry
a b c : point
L : line
bL : online b L
cL : online c L
aL : ¬online a L
n : ℕ
hn : ∀ (d : point), online d L → length b d = ↑n * length b c → B b c d → area a b d = ↑n * area a b c
d : point
dL : online d L
Bbcd : B b c d
bc : b ≠ c
bd : b ≠ d
cd : c ≠ d
n0 : ¬n = 0
n1 : n = 1
hlen : length b c + length c d = (1 + 1) * length b c
⊢ area a b c + area a d c = ↑(Nat.succ 1) * area a b c | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
i : incidence_geometry
a b c : point
L : line
bL : online b L
cL : online c L
aL : ¬online a L
n : ℕ
hn : ∀ (d : point), online d L → length b d = ↑n * length b c → B b c d → area a b d = ↑n * area a b c
d : point
dL : online d L
Bbcd : B b c d
bc : b ≠ c
bd : b ≠ d
cd : c ≠ d
n0 : ¬n = 0
n1 : n = 1
hlen : length b c + length c d = (1 + 1) * length b c
⊢ area a b d = ↑(Nat.succ n) * area a b c
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_triangle_of_base_inductive | [53, 1] | [111, 73] | perm | case pos
i : incidence_geometry
a b c : point
L : line
bL : online b L
cL : online c L
aL : ¬online a L
n : ℕ
hn : ∀ (d : point), online d L → length b d = ↑n * length b c → B b c d → area a b d = ↑n * area a b c
d : point
dL : online d L
Bbcd : B b c d
bc : b ≠ c
bd : b ≠ d
cd : c ≠ d
n0 : ¬n = 0
n1 : n = 1
hlen : length b c + length c d = (1 + 1) * length b c
⊢ area a b c + area a d c = ↑(Nat.succ 1) * area a b c | case pos
i : incidence_geometry
a b c : point
L : line
bL : online b L
cL : online c L
aL : ¬online a L
n : ℕ
hn : ∀ (d : point), online d L → length b d = ↑n * length b c → B b c d → area a b d = ↑n * area a b c
d : point
dL : online d L
Bbcd : B b c d
bc : b ≠ c
bd : b ≠ d
cd : c ≠ d
n0 : ¬n = 0
n1 : n = 1
hlen : length b c + length c d = (1 + 1) * length b c
⊢ area a b c + area a c d = ↑(Nat.succ 1) * area a b c | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
i : incidence_geometry
a b c : point
L : line
bL : online b L
cL : online c L
aL : ¬online a L
n : ℕ
hn : ∀ (d : point), online d L → length b d = ↑n * length b c → B b c d → area a b d = ↑n * area a b c
d : point
dL : online d L
Bbcd : B b c d
bc : b ≠ c
bd : b ≠ d
cd : c ≠ d
n0 : ¬n = 0
n1 : n = 1
hlen : length b c + length c d = (1 + 1) * length b c
⊢ area a b c + area a d c = ↑(Nat.succ 1) * area a b c
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_triangle_of_base_inductive | [53, 1] | [111, 73] | simp | case pos
i : incidence_geometry
a b c : point
L : line
bL : online b L
cL : online c L
aL : ¬online a L
n : ℕ
hn : ∀ (d : point), online d L → length b d = ↑n * length b c → B b c d → area a b d = ↑n * area a b c
d : point
dL : online d L
Bbcd : B b c d
bc : b ≠ c
bd : b ≠ d
cd : c ≠ d
n0 : ¬n = 0
n1 : n = 1
hlen : length b c + length c d = (1 + 1) * length b c
⊢ area a b c + area a c d = ↑(Nat.succ 1) * area a b c | case pos
i : incidence_geometry
a b c : point
L : line
bL : online b L
cL : online c L
aL : ¬online a L
n : ℕ
hn : ∀ (d : point), online d L → length b d = ↑n * length b c → B b c d → area a b d = ↑n * area a b c
d : point
dL : online d L
Bbcd : B b c d
bc : b ≠ c
bd : b ≠ d
cd : c ≠ d
n0 : ¬n = 0
n1 : n = 1
hlen : length b c + length c d = (1 + 1) * length b c
⊢ area a b c + area a c d = (1 + 1) * area a b c | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
i : incidence_geometry
a b c : point
L : line
bL : online b L
cL : online c L
aL : ¬online a L
n : ℕ
hn : ∀ (d : point), online d L → length b d = ↑n * length b c → B b c d → area a b d = ↑n * area a b c
d : point
dL : online d L
Bbcd : B b c d
bc : b ≠ c
bd : b ≠ d
cd : c ≠ d
n0 : ¬n = 0
n1 : n = 1
hlen : length b c + length c d = (1 + 1) * length b c
⊢ area a b c + area a c d = ↑(Nat.succ 1) * area a b c
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_triangle_of_base_inductive | [53, 1] | [111, 73] | ring_nf | case pos
i : incidence_geometry
a b c : point
L : line
bL : online b L
cL : online c L
aL : ¬online a L
n : ℕ
hn : ∀ (d : point), online d L → length b d = ↑n * length b c → B b c d → area a b d = ↑n * area a b c
d : point
dL : online d L
Bbcd : B b c d
bc : b ≠ c
bd : b ≠ d
cd : c ≠ d
n0 : ¬n = 0
n1 : n = 1
hlen : length b c + length c d = (1 + 1) * length b c
⊢ area a b c + area a c d = (1 + 1) * area a b c | case pos
i : incidence_geometry
a b c : point
L : line
bL : online b L
cL : online c L
aL : ¬online a L
n : ℕ
hn : ∀ (d : point), online d L → length b d = ↑n * length b c → B b c d → area a b d = ↑n * area a b c
d : point
dL : online d L
Bbcd : B b c d
bc : b ≠ c
bd : b ≠ d
cd : c ≠ d
n0 : ¬n = 0
n1 : n = 1
hlen : length b c + length c d = (1 + 1) * length b c
⊢ area a b c + area a c d = area a b c * 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
i : incidence_geometry
a b c : point
L : line
bL : online b L
cL : online c L
aL : ¬online a L
n : ℕ
hn : ∀ (d : point), online d L → length b d = ↑n * length b c → B b c d → area a b d = ↑n * area a b c
d : point
dL : online d L
Bbcd : B b c d
bc : b ≠ c
bd : b ≠ d
cd : c ≠ d
n0 : ¬n = 0
n1 : n = 1
hlen : length b c + length c d = (1 + 1) * length b c
⊢ area a b c + area a c d = (1 + 1) * area a b c
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_triangle_of_base_inductive | [53, 1] | [111, 73] | simp [mul_two] | case pos
i : incidence_geometry
a b c : point
L : line
bL : online b L
cL : online c L
aL : ¬online a L
n : ℕ
hn : ∀ (d : point), online d L → length b d = ↑n * length b c → B b c d → area a b d = ↑n * area a b c
d : point
dL : online d L
Bbcd : B b c d
bc : b ≠ c
bd : b ≠ d
cd : c ≠ d
n0 : ¬n = 0
n1 : n = 1
hlen : length b c + length c d = (1 + 1) * length b c
⊢ area a b c + area a c d = area a b c * 2 | case pos
i : incidence_geometry
a b c : point
L : line
bL : online b L
cL : online c L
aL : ¬online a L
n : ℕ
hn : ∀ (d : point), online d L → length b d = ↑n * length b c → B b c d → area a b d = ↑n * area a b c
d : point
dL : online d L
Bbcd : B b c d
bc : b ≠ c
bd : b ≠ d
cd : c ≠ d
n0 : ¬n = 0
n1 : n = 1
hlen : length b c + length c d = (1 + 1) * length b c
⊢ area a c d = area a b c | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
i : incidence_geometry
a b c : point
L : line
bL : online b L
cL : online c L
aL : ¬online a L
n : ℕ
hn : ∀ (d : point), online d L → length b d = ↑n * length b c → B b c d → area a b d = ↑n * area a b c
d : point
dL : online d L
Bbcd : B b c d
bc : b ≠ c
bd : b ≠ d
cd : c ≠ d
n0 : ¬n = 0
n1 : n = 1
hlen : length b c + length c d = (1 + 1) * length b c
⊢ area a b c + area a c d = area a b c * 2
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_triangle_of_base_inductive | [53, 1] | [111, 73] | apply (eq_area_of_eq_base_samevertex a bL cL cL dL (by linarith)).symm | case pos
i : incidence_geometry
a b c : point
L : line
bL : online b L
cL : online c L
aL : ¬online a L
n : ℕ
hn : ∀ (d : point), online d L → length b d = ↑n * length b c → B b c d → area a b d = ↑n * area a b c
d : point
dL : online d L
Bbcd : B b c d
bc : b ≠ c
bd : b ≠ d
cd : c ≠ d
n0 : ¬n = 0
n1 : n = 1
hlen : length b c + length c d = (1 + 1) * length b c
⊢ area a c d = area a b c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
i : incidence_geometry
a b c : point
L : line
bL : online b L
cL : online c L
aL : ¬online a L
n : ℕ
hn : ∀ (d : point), online d L → length b d = ↑n * length b c → B b c d → area a b d = ↑n * area a b c
d : point
dL : online d L
Bbcd : B b c d
bc : b ≠ c
bd : b ≠ d
cd : c ≠ d
n0 : ¬n = 0
n1 : n = 1
hlen : length b c + length c d = (1 + 1) * length b c
⊢ area a c d = area a b c
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_triangle_of_base_inductive | [53, 1] | [111, 73] | linarith | i : incidence_geometry
a b c : point
L : line
bL : online b L
cL : online c L
aL : ¬online a L
n : ℕ
hn : ∀ (d : point), online d L → length b d = ↑n * length b c → B b c d → area a b d = ↑n * area a b c
d : point
dL : online d L
Bbcd : B b c d
bc : b ≠ c
bd : b ≠ d
cd : c ≠ d
n0 : ¬n = 0
n1 : n = 1
hlen : length b c + length c d = (1 + 1) * length b c
⊢ length b c = length c d | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
i : incidence_geometry
a b c : point
L : line
bL : online b L
cL : online c L
aL : ¬online a L
n : ℕ
hn : ∀ (d : point), online d L → length b d = ↑n * length b c → B b c d → area a b d = ↑n * area a b c
d : point
dL : online d L
Bbcd : B b c d
bc : b ≠ c
bd : b ≠ d
cd : c ≠ d
n0 : ¬n = 0
n1 : n = 1
hlen : length b c + length c d = (1 + 1) * length b c
⊢ length b c = length c d
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_triangle_of_base_inductive | [53, 1] | [111, 73] | have := He.2.1 | i : incidence_geometry
a b c : point
L : line
bL : online b L
cL : online c L
aL : ¬online a L
n : ℕ
hn : ∀ (d : point), online d L → length b d = ↑n * length b c → B b c d → area a b d = ↑n * area a b c
d : point
dL : online d L
hlen : length b d = ↑(Nat.succ n) * length b c
Bbcd : B b c d
bc : b ≠ c
bd : b ≠ d
cd : c ≠ d
n0 : ¬n = 0
n1 : ¬n = 1
e : point
He : online e L ∧ length b e = ↑n * length b c ∧ (n ≥ 2 ∧ b ≠ c → B b c e)
Bbce : B b c e
be : b ≠ e
ed : e = d
⊢ False | i : incidence_geometry
a b c : point
L : line
bL : online b L
cL : online c L
aL : ¬online a L
n : ℕ
hn : ∀ (d : point), online d L → length b d = ↑n * length b c → B b c d → area a b d = ↑n * area a b c
d : point
dL : online d L
hlen : length b d = ↑(Nat.succ n) * length b c
Bbcd : B b c d
bc : b ≠ c
bd : b ≠ d
cd : c ≠ d
n0 : ¬n = 0
n1 : ¬n = 1
e : point
He : online e L ∧ length b e = ↑n * length b c ∧ (n ≥ 2 ∧ b ≠ c → B b c e)
Bbce : B b c e
be : b ≠ e
ed : e = d
this : length b e = ↑n * length b c
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
i : incidence_geometry
a b c : point
L : line
bL : online b L
cL : online c L
aL : ¬online a L
n : ℕ
hn : ∀ (d : point), online d L → length b d = ↑n * length b c → B b c d → area a b d = ↑n * area a b c
d : point
dL : online d L
hlen : length b d = ↑(Nat.succ n) * length b c
Bbcd : B b c d
bc : b ≠ c
bd : b ≠ d
cd : c ≠ d
n0 : ¬n = 0
n1 : ¬n = 1
e : point
He : online e L ∧ length b e = ↑n * length b c ∧ (n ≥ 2 ∧ b ≠ c → B b c e)
Bbce : B b c e
be : b ≠ e
ed : e = d
⊢ False
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_triangle_of_base_inductive | [53, 1] | [111, 73] | rw [ed, hlen] at this | i : incidence_geometry
a b c : point
L : line
bL : online b L
cL : online c L
aL : ¬online a L
n : ℕ
hn : ∀ (d : point), online d L → length b d = ↑n * length b c → B b c d → area a b d = ↑n * area a b c
d : point
dL : online d L
hlen : length b d = ↑(Nat.succ n) * length b c
Bbcd : B b c d
bc : b ≠ c
bd : b ≠ d
cd : c ≠ d
n0 : ¬n = 0
n1 : ¬n = 1
e : point
He : online e L ∧ length b e = ↑n * length b c ∧ (n ≥ 2 ∧ b ≠ c → B b c e)
Bbce : B b c e
be : b ≠ e
ed : e = d
this : length b e = ↑n * length b c
⊢ False | i : incidence_geometry
a b c : point
L : line
bL : online b L
cL : online c L
aL : ¬online a L
n : ℕ
hn : ∀ (d : point), online d L → length b d = ↑n * length b c → B b c d → area a b d = ↑n * area a b c
d : point
dL : online d L
hlen : length b d = ↑(Nat.succ n) * length b c
Bbcd : B b c d
bc : b ≠ c
bd : b ≠ d
cd : c ≠ d
n0 : ¬n = 0
n1 : ¬n = 1
e : point
He : online e L ∧ length b e = ↑n * length b c ∧ (n ≥ 2 ∧ b ≠ c → B b c e)
Bbce : B b c e
be : b ≠ e
ed : e = d
this : ↑(Nat.succ n) * length b c = ↑n * length b c
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
i : incidence_geometry
a b c : point
L : line
bL : online b L
cL : online c L
aL : ¬online a L
n : ℕ
hn : ∀ (d : point), online d L → length b d = ↑n * length b c → B b c d → area a b d = ↑n * area a b c
d : point
dL : online d L
hlen : length b d = ↑(Nat.succ n) * length b c
Bbcd : B b c d
bc : b ≠ c
bd : b ≠ d
cd : c ≠ d
n0 : ¬n = 0
n1 : ¬n = 1
e : point
He : online e L ∧ length b e = ↑n * length b c ∧ (n ≥ 2 ∧ b ≠ c → B b c e)
Bbce : B b c e
be : b ≠ e
ed : e = d
this : length b e = ↑n * length b c
⊢ False
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_triangle_of_base_inductive | [53, 1] | [111, 73] | simp at this | i : incidence_geometry
a b c : point
L : line
bL : online b L
cL : online c L
aL : ¬online a L
n : ℕ
hn : ∀ (d : point), online d L → length b d = ↑n * length b c → B b c d → area a b d = ↑n * area a b c
d : point
dL : online d L
hlen : length b d = ↑(Nat.succ n) * length b c
Bbcd : B b c d
bc : b ≠ c
bd : b ≠ d
cd : c ≠ d
n0 : ¬n = 0
n1 : ¬n = 1
e : point
He : online e L ∧ length b e = ↑n * length b c ∧ (n ≥ 2 ∧ b ≠ c → B b c e)
Bbce : B b c e
be : b ≠ e
ed : e = d
this : ↑(Nat.succ n) * length b c = ↑n * length b c
⊢ False | i : incidence_geometry
a b c : point
L : line
bL : online b L
cL : online c L
aL : ¬online a L
n : ℕ
hn : ∀ (d : point), online d L → length b d = ↑n * length b c → B b c d → area a b d = ↑n * area a b c
d : point
dL : online d L
hlen : length b d = ↑(Nat.succ n) * length b c
Bbcd : B b c d
bc : b ≠ c
bd : b ≠ d
cd : c ≠ d
n0 : ¬n = 0
n1 : ¬n = 1
e : point
He : online e L ∧ length b e = ↑n * length b c ∧ (n ≥ 2 ∧ b ≠ c → B b c e)
Bbce : B b c e
be : b ≠ e
ed : e = d
this : length b c = 0
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
i : incidence_geometry
a b c : point
L : line
bL : online b L
cL : online c L
aL : ¬online a L
n : ℕ
hn : ∀ (d : point), online d L → length b d = ↑n * length b c → B b c d → area a b d = ↑n * area a b c
d : point
dL : online d L
hlen : length b d = ↑(Nat.succ n) * length b c
Bbcd : B b c d
bc : b ≠ c
bd : b ≠ d
cd : c ≠ d
n0 : ¬n = 0
n1 : ¬n = 1
e : point
He : online e L ∧ length b e = ↑n * length b c ∧ (n ≥ 2 ∧ b ≠ c → B b c e)
Bbce : B b c e
be : b ≠ e
ed : e = d
this : ↑(Nat.succ n) * length b c = ↑n * length b c
⊢ False
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_triangle_of_base_inductive | [53, 1] | [111, 73] | exact bc (length_eq_zero_iff.mp this) | i : incidence_geometry
a b c : point
L : line
bL : online b L
cL : online c L
aL : ¬online a L
n : ℕ
hn : ∀ (d : point), online d L → length b d = ↑n * length b c → B b c d → area a b d = ↑n * area a b c
d : point
dL : online d L
hlen : length b d = ↑(Nat.succ n) * length b c
Bbcd : B b c d
bc : b ≠ c
bd : b ≠ d
cd : c ≠ d
n0 : ¬n = 0
n1 : ¬n = 1
e : point
He : online e L ∧ length b e = ↑n * length b c ∧ (n ≥ 2 ∧ b ≠ c → B b c e)
Bbce : B b c e
be : b ≠ e
ed : e = d
this : length b c = 0
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
i : incidence_geometry
a b c : point
L : line
bL : online b L
cL : online c L
aL : ¬online a L
n : ℕ
hn : ∀ (d : point), online d L → length b d = ↑n * length b c → B b c d → area a b d = ↑n * area a b c
d : point
dL : online d L
hlen : length b d = ↑(Nat.succ n) * length b c
Bbcd : B b c d
bc : b ≠ c
bd : b ≠ d
cd : c ≠ d
n0 : ¬n = 0
n1 : ¬n = 1
e : point
He : online e L ∧ length b e = ↑n * length b c ∧ (n ≥ 2 ∧ b ≠ c → B b c e)
Bbce : B b c e
be : b ≠ e
ed : e = d
this : length b c = 0
⊢ False
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_triangle_of_base_inductive | [53, 1] | [111, 73] | have: length b e < length b d := by rw [He.2.1, hlen]; simp; linarith [len_pos_of_nq bc] | i : incidence_geometry
a b c : point
L : line
bL : online b L
cL : online c L
aL : ¬online a L
n : ℕ
hn : ∀ (d : point), online d L → length b d = ↑n * length b c → B b c d → area a b d = ↑n * area a b c
d : point
dL : online d L
hlen : length b d = ↑(Nat.succ n) * length b c
Bbcd : B b c d
bc : b ≠ c
bd : b ≠ d
cd : c ≠ d
n0 : ¬n = 0
n1 : ¬n = 1
e : point
He : online e L ∧ length b e = ↑n * length b c ∧ (n ≥ 2 ∧ b ≠ c → B b c e)
Bbce : B b c e
be : b ≠ e
ed : e ≠ d
⊢ B b e d | i : incidence_geometry
a b c : point
L : line
bL : online b L
cL : online c L
aL : ¬online a L
n : ℕ
hn : ∀ (d : point), online d L → length b d = ↑n * length b c → B b c d → area a b d = ↑n * area a b c
d : point
dL : online d L
hlen : length b d = ↑(Nat.succ n) * length b c
Bbcd : B b c d
bc : b ≠ c
bd : b ≠ d
cd : c ≠ d
n0 : ¬n = 0
n1 : ¬n = 1
e : point
He : online e L ∧ length b e = ↑n * length b c ∧ (n ≥ 2 ∧ b ≠ c → B b c e)
Bbce : B b c e
be : b ≠ e
ed : e ≠ d
this : length b e < length b d
⊢ B b e d | Please generate a tactic in lean4 to solve the state.
STATE:
i : incidence_geometry
a b c : point
L : line
bL : online b L
cL : online c L
aL : ¬online a L
n : ℕ
hn : ∀ (d : point), online d L → length b d = ↑n * length b c → B b c d → area a b d = ↑n * area a b c
d : point
dL : online d L
hlen : length b d = ↑(Nat.succ n) * length b c
Bbcd : B b c d
bc : b ≠ c
bd : b ≠ d
cd : c ≠ d
n0 : ¬n = 0
n1 : ¬n = 1
e : point
He : online e L ∧ length b e = ↑n * length b c ∧ (n ≥ 2 ∧ b ≠ c → B b c e)
Bbce : B b c e
be : b ≠ e
ed : e ≠ d
⊢ B b e d
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_triangle_of_base_inductive | [53, 1] | [111, 73] | have := B_of_three_online_ne_short this be bd ed bL He.1 dL | i : incidence_geometry
a b c : point
L : line
bL : online b L
cL : online c L
aL : ¬online a L
n : ℕ
hn : ∀ (d : point), online d L → length b d = ↑n * length b c → B b c d → area a b d = ↑n * area a b c
d : point
dL : online d L
hlen : length b d = ↑(Nat.succ n) * length b c
Bbcd : B b c d
bc : b ≠ c
bd : b ≠ d
cd : c ≠ d
n0 : ¬n = 0
n1 : ¬n = 1
e : point
He : online e L ∧ length b e = ↑n * length b c ∧ (n ≥ 2 ∧ b ≠ c → B b c e)
Bbce : B b c e
be : b ≠ e
ed : e ≠ d
this : length b e < length b d
⊢ B b e d | i : incidence_geometry
a b c : point
L : line
bL : online b L
cL : online c L
aL : ¬online a L
n : ℕ
hn : ∀ (d : point), online d L → length b d = ↑n * length b c → B b c d → area a b d = ↑n * area a b c
d : point
dL : online d L
hlen : length b d = ↑(Nat.succ n) * length b c
Bbcd : B b c d
bc : b ≠ c
bd : b ≠ d
cd : c ≠ d
n0 : ¬n = 0
n1 : ¬n = 1
e : point
He : online e L ∧ length b e = ↑n * length b c ∧ (n ≥ 2 ∧ b ≠ c → B b c e)
Bbce : B b c e
be : b ≠ e
ed : e ≠ d
this✝ : length b e < length b d
this : B b e d ∨ B e b d
⊢ B b e d | Please generate a tactic in lean4 to solve the state.
STATE:
i : incidence_geometry
a b c : point
L : line
bL : online b L
cL : online c L
aL : ¬online a L
n : ℕ
hn : ∀ (d : point), online d L → length b d = ↑n * length b c → B b c d → area a b d = ↑n * area a b c
d : point
dL : online d L
hlen : length b d = ↑(Nat.succ n) * length b c
Bbcd : B b c d
bc : b ≠ c
bd : b ≠ d
cd : c ≠ d
n0 : ¬n = 0
n1 : ¬n = 1
e : point
He : online e L ∧ length b e = ↑n * length b c ∧ (n ≥ 2 ∧ b ≠ c → B b c e)
Bbce : B b c e
be : b ≠ e
ed : e ≠ d
this : length b e < length b d
⊢ B b e d
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_triangle_of_base_inductive | [53, 1] | [111, 73] | cases this with
| inl h => exact h
| inr h =>
exfalso
exact not_B324_of_B123_B124 Bbce Bbcd (B124_of_B134_B123 h (B_symm Bbce)) | i : incidence_geometry
a b c : point
L : line
bL : online b L
cL : online c L
aL : ¬online a L
n : ℕ
hn : ∀ (d : point), online d L → length b d = ↑n * length b c → B b c d → area a b d = ↑n * area a b c
d : point
dL : online d L
hlen : length b d = ↑(Nat.succ n) * length b c
Bbcd : B b c d
bc : b ≠ c
bd : b ≠ d
cd : c ≠ d
n0 : ¬n = 0
n1 : ¬n = 1
e : point
He : online e L ∧ length b e = ↑n * length b c ∧ (n ≥ 2 ∧ b ≠ c → B b c e)
Bbce : B b c e
be : b ≠ e
ed : e ≠ d
this✝ : length b e < length b d
this : B b e d ∨ B e b d
⊢ B b e d | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
i : incidence_geometry
a b c : point
L : line
bL : online b L
cL : online c L
aL : ¬online a L
n : ℕ
hn : ∀ (d : point), online d L → length b d = ↑n * length b c → B b c d → area a b d = ↑n * area a b c
d : point
dL : online d L
hlen : length b d = ↑(Nat.succ n) * length b c
Bbcd : B b c d
bc : b ≠ c
bd : b ≠ d
cd : c ≠ d
n0 : ¬n = 0
n1 : ¬n = 1
e : point
He : online e L ∧ length b e = ↑n * length b c ∧ (n ≥ 2 ∧ b ≠ c → B b c e)
Bbce : B b c e
be : b ≠ e
ed : e ≠ d
this✝ : length b e < length b d
this : B b e d ∨ B e b d
⊢ B b e d
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_triangle_of_base_inductive | [53, 1] | [111, 73] | rw [He.2.1, hlen] | i : incidence_geometry
a b c : point
L : line
bL : online b L
cL : online c L
aL : ¬online a L
n : ℕ
hn : ∀ (d : point), online d L → length b d = ↑n * length b c → B b c d → area a b d = ↑n * area a b c
d : point
dL : online d L
hlen : length b d = ↑(Nat.succ n) * length b c
Bbcd : B b c d
bc : b ≠ c
bd : b ≠ d
cd : c ≠ d
n0 : ¬n = 0
n1 : ¬n = 1
e : point
He : online e L ∧ length b e = ↑n * length b c ∧ (n ≥ 2 ∧ b ≠ c → B b c e)
Bbce : B b c e
be : b ≠ e
ed : e ≠ d
⊢ length b e < length b d | i : incidence_geometry
a b c : point
L : line
bL : online b L
cL : online c L
aL : ¬online a L
n : ℕ
hn : ∀ (d : point), online d L → length b d = ↑n * length b c → B b c d → area a b d = ↑n * area a b c
d : point
dL : online d L
hlen : length b d = ↑(Nat.succ n) * length b c
Bbcd : B b c d
bc : b ≠ c
bd : b ≠ d
cd : c ≠ d
n0 : ¬n = 0
n1 : ¬n = 1
e : point
He : online e L ∧ length b e = ↑n * length b c ∧ (n ≥ 2 ∧ b ≠ c → B b c e)
Bbce : B b c e
be : b ≠ e
ed : e ≠ d
⊢ ↑n * length b c < ↑(Nat.succ n) * length b c | Please generate a tactic in lean4 to solve the state.
STATE:
i : incidence_geometry
a b c : point
L : line
bL : online b L
cL : online c L
aL : ¬online a L
n : ℕ
hn : ∀ (d : point), online d L → length b d = ↑n * length b c → B b c d → area a b d = ↑n * area a b c
d : point
dL : online d L
hlen : length b d = ↑(Nat.succ n) * length b c
Bbcd : B b c d
bc : b ≠ c
bd : b ≠ d
cd : c ≠ d
n0 : ¬n = 0
n1 : ¬n = 1
e : point
He : online e L ∧ length b e = ↑n * length b c ∧ (n ≥ 2 ∧ b ≠ c → B b c e)
Bbce : B b c e
be : b ≠ e
ed : e ≠ d
⊢ length b e < length b d
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_triangle_of_base_inductive | [53, 1] | [111, 73] | simp | i : incidence_geometry
a b c : point
L : line
bL : online b L
cL : online c L
aL : ¬online a L
n : ℕ
hn : ∀ (d : point), online d L → length b d = ↑n * length b c → B b c d → area a b d = ↑n * area a b c
d : point
dL : online d L
hlen : length b d = ↑(Nat.succ n) * length b c
Bbcd : B b c d
bc : b ≠ c
bd : b ≠ d
cd : c ≠ d
n0 : ¬n = 0
n1 : ¬n = 1
e : point
He : online e L ∧ length b e = ↑n * length b c ∧ (n ≥ 2 ∧ b ≠ c → B b c e)
Bbce : B b c e
be : b ≠ e
ed : e ≠ d
⊢ ↑n * length b c < ↑(Nat.succ n) * length b c | i : incidence_geometry
a b c : point
L : line
bL : online b L
cL : online c L
aL : ¬online a L
n : ℕ
hn : ∀ (d : point), online d L → length b d = ↑n * length b c → B b c d → area a b d = ↑n * area a b c
d : point
dL : online d L
hlen : length b d = ↑(Nat.succ n) * length b c
Bbcd : B b c d
bc : b ≠ c
bd : b ≠ d
cd : c ≠ d
n0 : ¬n = 0
n1 : ¬n = 1
e : point
He : online e L ∧ length b e = ↑n * length b c ∧ (n ≥ 2 ∧ b ≠ c → B b c e)
Bbce : B b c e
be : b ≠ e
ed : e ≠ d
⊢ ↑n * length b c < (↑n + 1) * length b c | Please generate a tactic in lean4 to solve the state.
STATE:
i : incidence_geometry
a b c : point
L : line
bL : online b L
cL : online c L
aL : ¬online a L
n : ℕ
hn : ∀ (d : point), online d L → length b d = ↑n * length b c → B b c d → area a b d = ↑n * area a b c
d : point
dL : online d L
hlen : length b d = ↑(Nat.succ n) * length b c
Bbcd : B b c d
bc : b ≠ c
bd : b ≠ d
cd : c ≠ d
n0 : ¬n = 0
n1 : ¬n = 1
e : point
He : online e L ∧ length b e = ↑n * length b c ∧ (n ≥ 2 ∧ b ≠ c → B b c e)
Bbce : B b c e
be : b ≠ e
ed : e ≠ d
⊢ ↑n * length b c < ↑(Nat.succ n) * length b c
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_triangle_of_base_inductive | [53, 1] | [111, 73] | linarith [len_pos_of_nq bc] | i : incidence_geometry
a b c : point
L : line
bL : online b L
cL : online c L
aL : ¬online a L
n : ℕ
hn : ∀ (d : point), online d L → length b d = ↑n * length b c → B b c d → area a b d = ↑n * area a b c
d : point
dL : online d L
hlen : length b d = ↑(Nat.succ n) * length b c
Bbcd : B b c d
bc : b ≠ c
bd : b ≠ d
cd : c ≠ d
n0 : ¬n = 0
n1 : ¬n = 1
e : point
He : online e L ∧ length b e = ↑n * length b c ∧ (n ≥ 2 ∧ b ≠ c → B b c e)
Bbce : B b c e
be : b ≠ e
ed : e ≠ d
⊢ ↑n * length b c < (↑n + 1) * length b c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
i : incidence_geometry
a b c : point
L : line
bL : online b L
cL : online c L
aL : ¬online a L
n : ℕ
hn : ∀ (d : point), online d L → length b d = ↑n * length b c → B b c d → area a b d = ↑n * area a b c
d : point
dL : online d L
hlen : length b d = ↑(Nat.succ n) * length b c
Bbcd : B b c d
bc : b ≠ c
bd : b ≠ d
cd : c ≠ d
n0 : ¬n = 0
n1 : ¬n = 1
e : point
He : online e L ∧ length b e = ↑n * length b c ∧ (n ≥ 2 ∧ b ≠ c → B b c e)
Bbce : B b c e
be : b ≠ e
ed : e ≠ d
⊢ ↑n * length b c < (↑n + 1) * length b c
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_triangle_of_base_inductive | [53, 1] | [111, 73] | exact h | case inl
i : incidence_geometry
a b c : point
L : line
bL : online b L
cL : online c L
aL : ¬online a L
n : ℕ
hn : ∀ (d : point), online d L → length b d = ↑n * length b c → B b c d → area a b d = ↑n * area a b c
d : point
dL : online d L
hlen : length b d = ↑(Nat.succ n) * length b c
Bbcd : B b c d
bc : b ≠ c
bd : b ≠ d
cd : c ≠ d
n0 : ¬n = 0
n1 : ¬n = 1
e : point
He : online e L ∧ length b e = ↑n * length b c ∧ (n ≥ 2 ∧ b ≠ c → B b c e)
Bbce : B b c e
be : b ≠ e
ed : e ≠ d
this : length b e < length b d
h : B b e d
⊢ B b e d | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
i : incidence_geometry
a b c : point
L : line
bL : online b L
cL : online c L
aL : ¬online a L
n : ℕ
hn : ∀ (d : point), online d L → length b d = ↑n * length b c → B b c d → area a b d = ↑n * area a b c
d : point
dL : online d L
hlen : length b d = ↑(Nat.succ n) * length b c
Bbcd : B b c d
bc : b ≠ c
bd : b ≠ d
cd : c ≠ d
n0 : ¬n = 0
n1 : ¬n = 1
e : point
He : online e L ∧ length b e = ↑n * length b c ∧ (n ≥ 2 ∧ b ≠ c → B b c e)
Bbce : B b c e
be : b ≠ e
ed : e ≠ d
this : length b e < length b d
h : B b e d
⊢ B b e d
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_triangle_of_base_inductive | [53, 1] | [111, 73] | exfalso | case inr
i : incidence_geometry
a b c : point
L : line
bL : online b L
cL : online c L
aL : ¬online a L
n : ℕ
hn : ∀ (d : point), online d L → length b d = ↑n * length b c → B b c d → area a b d = ↑n * area a b c
d : point
dL : online d L
hlen : length b d = ↑(Nat.succ n) * length b c
Bbcd : B b c d
bc : b ≠ c
bd : b ≠ d
cd : c ≠ d
n0 : ¬n = 0
n1 : ¬n = 1
e : point
He : online e L ∧ length b e = ↑n * length b c ∧ (n ≥ 2 ∧ b ≠ c → B b c e)
Bbce : B b c e
be : b ≠ e
ed : e ≠ d
this : length b e < length b d
h : B e b d
⊢ B b e d | case inr.h
i : incidence_geometry
a b c : point
L : line
bL : online b L
cL : online c L
aL : ¬online a L
n : ℕ
hn : ∀ (d : point), online d L → length b d = ↑n * length b c → B b c d → area a b d = ↑n * area a b c
d : point
dL : online d L
hlen : length b d = ↑(Nat.succ n) * length b c
Bbcd : B b c d
bc : b ≠ c
bd : b ≠ d
cd : c ≠ d
n0 : ¬n = 0
n1 : ¬n = 1
e : point
He : online e L ∧ length b e = ↑n * length b c ∧ (n ≥ 2 ∧ b ≠ c → B b c e)
Bbce : B b c e
be : b ≠ e
ed : e ≠ d
this : length b e < length b d
h : B e b d
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
i : incidence_geometry
a b c : point
L : line
bL : online b L
cL : online c L
aL : ¬online a L
n : ℕ
hn : ∀ (d : point), online d L → length b d = ↑n * length b c → B b c d → area a b d = ↑n * area a b c
d : point
dL : online d L
hlen : length b d = ↑(Nat.succ n) * length b c
Bbcd : B b c d
bc : b ≠ c
bd : b ≠ d
cd : c ≠ d
n0 : ¬n = 0
n1 : ¬n = 1
e : point
He : online e L ∧ length b e = ↑n * length b c ∧ (n ≥ 2 ∧ b ≠ c → B b c e)
Bbce : B b c e
be : b ≠ e
ed : e ≠ d
this : length b e < length b d
h : B e b d
⊢ B b e d
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_triangle_of_base_inductive | [53, 1] | [111, 73] | exact not_B324_of_B123_B124 Bbce Bbcd (B124_of_B134_B123 h (B_symm Bbce)) | case inr.h
i : incidence_geometry
a b c : point
L : line
bL : online b L
cL : online c L
aL : ¬online a L
n : ℕ
hn : ∀ (d : point), online d L → length b d = ↑n * length b c → B b c d → area a b d = ↑n * area a b c
d : point
dL : online d L
hlen : length b d = ↑(Nat.succ n) * length b c
Bbcd : B b c d
bc : b ≠ c
bd : b ≠ d
cd : c ≠ d
n0 : ¬n = 0
n1 : ¬n = 1
e : point
He : online e L ∧ length b e = ↑n * length b c ∧ (n ≥ 2 ∧ b ≠ c → B b c e)
Bbce : B b c e
be : b ≠ e
ed : e ≠ d
this : length b e < length b d
h : B e b d
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.h
i : incidence_geometry
a b c : point
L : line
bL : online b L
cL : online c L
aL : ¬online a L
n : ℕ
hn : ∀ (d : point), online d L → length b d = ↑n * length b c → B b c d → area a b d = ↑n * area a b c
d : point
dL : online d L
hlen : length b d = ↑(Nat.succ n) * length b c
Bbcd : B b c d
bc : b ≠ c
bd : b ≠ d
cd : c ≠ d
n0 : ¬n = 0
n1 : ¬n = 1
e : point
He : online e L ∧ length b e = ↑n * length b c ∧ (n ≥ 2 ∧ b ≠ c → B b c e)
Bbce : B b c e
be : b ≠ e
ed : e ≠ d
this : length b e < length b d
h : B e b d
⊢ False
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_triangle_of_base_inductive | [53, 1] | [111, 73] | have := length_sum_of_B Bbed | i : incidence_geometry
a b c : point
L : line
bL : online b L
cL : online c L
aL : ¬online a L
n : ℕ
hn : ∀ (d : point), online d L → length b d = ↑n * length b c → B b c d → area a b d = ↑n * area a b c
d : point
dL : online d L
hlen : length b d = ↑(Nat.succ n) * length b c
Bbcd : B b c d
bc : b ≠ c
bd : b ≠ d
cd : c ≠ d
n0 : ¬n = 0
n1 : ¬n = 1
e : point
He : online e L ∧ length b e = ↑n * length b c ∧ (n ≥ 2 ∧ b ≠ c → B b c e)
Bbce : B b c e
be : b ≠ e
ed : e ≠ d
Bbed : B b e d
⊢ length b c = length e d | i : incidence_geometry
a b c : point
L : line
bL : online b L
cL : online c L
aL : ¬online a L
n : ℕ
hn : ∀ (d : point), online d L → length b d = ↑n * length b c → B b c d → area a b d = ↑n * area a b c
d : point
dL : online d L
hlen : length b d = ↑(Nat.succ n) * length b c
Bbcd : B b c d
bc : b ≠ c
bd : b ≠ d
cd : c ≠ d
n0 : ¬n = 0
n1 : ¬n = 1
e : point
He : online e L ∧ length b e = ↑n * length b c ∧ (n ≥ 2 ∧ b ≠ c → B b c e)
Bbce : B b c e
be : b ≠ e
ed : e ≠ d
Bbed : B b e d
this : length b e + length e d = length b d
⊢ length b c = length e d | Please generate a tactic in lean4 to solve the state.
STATE:
i : incidence_geometry
a b c : point
L : line
bL : online b L
cL : online c L
aL : ¬online a L
n : ℕ
hn : ∀ (d : point), online d L → length b d = ↑n * length b c → B b c d → area a b d = ↑n * area a b c
d : point
dL : online d L
hlen : length b d = ↑(Nat.succ n) * length b c
Bbcd : B b c d
bc : b ≠ c
bd : b ≠ d
cd : c ≠ d
n0 : ¬n = 0
n1 : ¬n = 1
e : point
He : online e L ∧ length b e = ↑n * length b c ∧ (n ≥ 2 ∧ b ≠ c → B b c e)
Bbce : B b c e
be : b ≠ e
ed : e ≠ d
Bbed : B b e d
⊢ length b c = length e d
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_triangle_of_base_inductive | [53, 1] | [111, 73] | rw [hlen, He.2.1] at this | i : incidence_geometry
a b c : point
L : line
bL : online b L
cL : online c L
aL : ¬online a L
n : ℕ
hn : ∀ (d : point), online d L → length b d = ↑n * length b c → B b c d → area a b d = ↑n * area a b c
d : point
dL : online d L
hlen : length b d = ↑(Nat.succ n) * length b c
Bbcd : B b c d
bc : b ≠ c
bd : b ≠ d
cd : c ≠ d
n0 : ¬n = 0
n1 : ¬n = 1
e : point
He : online e L ∧ length b e = ↑n * length b c ∧ (n ≥ 2 ∧ b ≠ c → B b c e)
Bbce : B b c e
be : b ≠ e
ed : e ≠ d
Bbed : B b e d
this : length b e + length e d = length b d
⊢ length b c = length e d | i : incidence_geometry
a b c : point
L : line
bL : online b L
cL : online c L
aL : ¬online a L
n : ℕ
hn : ∀ (d : point), online d L → length b d = ↑n * length b c → B b c d → area a b d = ↑n * area a b c
d : point
dL : online d L
hlen : length b d = ↑(Nat.succ n) * length b c
Bbcd : B b c d
bc : b ≠ c
bd : b ≠ d
cd : c ≠ d
n0 : ¬n = 0
n1 : ¬n = 1
e : point
He : online e L ∧ length b e = ↑n * length b c ∧ (n ≥ 2 ∧ b ≠ c → B b c e)
Bbce : B b c e
be : b ≠ e
ed : e ≠ d
Bbed : B b e d
this : ↑n * length b c + length e d = ↑(Nat.succ n) * length b c
⊢ length b c = length e d | Please generate a tactic in lean4 to solve the state.
STATE:
i : incidence_geometry
a b c : point
L : line
bL : online b L
cL : online c L
aL : ¬online a L
n : ℕ
hn : ∀ (d : point), online d L → length b d = ↑n * length b c → B b c d → area a b d = ↑n * area a b c
d : point
dL : online d L
hlen : length b d = ↑(Nat.succ n) * length b c
Bbcd : B b c d
bc : b ≠ c
bd : b ≠ d
cd : c ≠ d
n0 : ¬n = 0
n1 : ¬n = 1
e : point
He : online e L ∧ length b e = ↑n * length b c ∧ (n ≥ 2 ∧ b ≠ c → B b c e)
Bbce : B b c e
be : b ≠ e
ed : e ≠ d
Bbed : B b e d
this : length b e + length e d = length b d
⊢ length b c = length e d
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_triangle_of_base_inductive | [53, 1] | [111, 73] | simp [one_mul] at this | i : incidence_geometry
a b c : point
L : line
bL : online b L
cL : online c L
aL : ¬online a L
n : ℕ
hn : ∀ (d : point), online d L → length b d = ↑n * length b c → B b c d → area a b d = ↑n * area a b c
d : point
dL : online d L
hlen : length b d = ↑(Nat.succ n) * length b c
Bbcd : B b c d
bc : b ≠ c
bd : b ≠ d
cd : c ≠ d
n0 : ¬n = 0
n1 : ¬n = 1
e : point
He : online e L ∧ length b e = ↑n * length b c ∧ (n ≥ 2 ∧ b ≠ c → B b c e)
Bbce : B b c e
be : b ≠ e
ed : e ≠ d
Bbed : B b e d
this : ↑n * length b c + length e d = ↑(Nat.succ n) * length b c
⊢ length b c = length e d | i : incidence_geometry
a b c : point
L : line
bL : online b L
cL : online c L
aL : ¬online a L
n : ℕ
hn : ∀ (d : point), online d L → length b d = ↑n * length b c → B b c d → area a b d = ↑n * area a b c
d : point
dL : online d L
hlen : length b d = ↑(Nat.succ n) * length b c
Bbcd : B b c d
bc : b ≠ c
bd : b ≠ d
cd : c ≠ d
n0 : ¬n = 0
n1 : ¬n = 1
e : point
He : online e L ∧ length b e = ↑n * length b c ∧ (n ≥ 2 ∧ b ≠ c → B b c e)
Bbce : B b c e
be : b ≠ e
ed : e ≠ d
Bbed : B b e d
this : ↑n * length b c + length e d = (↑n + 1) * length b c
⊢ length b c = length e d | Please generate a tactic in lean4 to solve the state.
STATE:
i : incidence_geometry
a b c : point
L : line
bL : online b L
cL : online c L
aL : ¬online a L
n : ℕ
hn : ∀ (d : point), online d L → length b d = ↑n * length b c → B b c d → area a b d = ↑n * area a b c
d : point
dL : online d L
hlen : length b d = ↑(Nat.succ n) * length b c
Bbcd : B b c d
bc : b ≠ c
bd : b ≠ d
cd : c ≠ d
n0 : ¬n = 0
n1 : ¬n = 1
e : point
He : online e L ∧ length b e = ↑n * length b c ∧ (n ≥ 2 ∧ b ≠ c → B b c e)
Bbce : B b c e
be : b ≠ e
ed : e ≠ d
Bbed : B b e d
this : ↑n * length b c + length e d = ↑(Nat.succ n) * length b c
⊢ length b c = length e d
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_triangle_of_base_inductive | [53, 1] | [111, 73] | linarith | i : incidence_geometry
a b c : point
L : line
bL : online b L
cL : online c L
aL : ¬online a L
n : ℕ
hn : ∀ (d : point), online d L → length b d = ↑n * length b c → B b c d → area a b d = ↑n * area a b c
d : point
dL : online d L
hlen : length b d = ↑(Nat.succ n) * length b c
Bbcd : B b c d
bc : b ≠ c
bd : b ≠ d
cd : c ≠ d
n0 : ¬n = 0
n1 : ¬n = 1
e : point
He : online e L ∧ length b e = ↑n * length b c ∧ (n ≥ 2 ∧ b ≠ c → B b c e)
Bbce : B b c e
be : b ≠ e
ed : e ≠ d
Bbed : B b e d
this : ↑n * length b c + length e d = (↑n + 1) * length b c
⊢ length b c = length e d | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
i : incidence_geometry
a b c : point
L : line
bL : online b L
cL : online c L
aL : ¬online a L
n : ℕ
hn : ∀ (d : point), online d L → length b d = ↑n * length b c → B b c d → area a b d = ↑n * area a b c
d : point
dL : online d L
hlen : length b d = ↑(Nat.succ n) * length b c
Bbcd : B b c d
bc : b ≠ c
bd : b ≠ d
cd : c ≠ d
n0 : ¬n = 0
n1 : ¬n = 1
e : point
He : online e L ∧ length b e = ↑n * length b c ∧ (n ≥ 2 ∧ b ≠ c → B b c e)
Bbce : B b c e
be : b ≠ e
ed : e ≠ d
Bbed : B b e d
this : ↑n * length b c + length e d = (↑n + 1) * length b c
⊢ length b c = length e d
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_triangle_of_base_B | [114, 1] | [122, 69] | exact rescale_triangle_of_base_inductive a bL cL aL d dL hlen Bbcd | i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = ↑n * length b c
Bbcd : B b c d
aL : ¬online a L
⊢ area a b d = ↑n * area a b c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = ↑n * length b c
Bbcd : B b c d
aL : ¬online a L
⊢ area a b d = ↑n * area a b c
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_triangle_of_base_notcbd | [125, 1] | [153, 37] | by_cases n1 : n =1 | i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = ↑n * length b c
hB : ¬B c b d
bc : b ≠ c
n0 : n ≠ 0
bd : b ≠ d
aL : ¬online a L
⊢ area a b d = ↑n * area a b c | case pos
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = ↑n * length b c
hB : ¬B c b d
bc : b ≠ c
n0 : n ≠ 0
bd : b ≠ d
aL : ¬online a L
n1 : n = 1
⊢ area a b d = ↑n * area a b c
case neg
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = ↑n * length b c
hB : ¬B c b d
bc : b ≠ c
n0 : n ≠ 0
bd : b ≠ d
aL : ¬online a L
n1 : ¬n = 1
⊢ area a b d = ↑n * area a b c | Please generate a tactic in lean4 to solve the state.
STATE:
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = ↑n * length b c
hB : ¬B c b d
bc : b ≠ c
n0 : n ≠ 0
bd : b ≠ d
aL : ¬online a L
⊢ area a b d = ↑n * area a b c
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_triangle_of_base_notcbd | [125, 1] | [153, 37] | have cd: c ≠ d := fun cd => by
have n0 := length_eq_zero_iff.not.mpr bd
rw [cd] at hlen; field_simp at hlen; exact n1 hlen | case neg
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = ↑n * length b c
hB : ¬B c b d
bc : b ≠ c
n0 : n ≠ 0
bd : b ≠ d
aL : ¬online a L
n1 : ¬n = 1
⊢ area a b d = ↑n * area a b c | case neg
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = ↑n * length b c
hB : ¬B c b d
bc : b ≠ c
n0 : n ≠ 0
bd : b ≠ d
aL : ¬online a L
n1 : ¬n = 1
cd : c ≠ d
⊢ area a b d = ↑n * area a b c | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = ↑n * length b c
hB : ¬B c b d
bc : b ≠ c
n0 : n ≠ 0
bd : b ≠ d
aL : ¬online a L
n1 : ¬n = 1
⊢ area a b d = ↑n * area a b c
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_triangle_of_base_notcbd | [125, 1] | [153, 37] | have: length b c < length b d := by
have := len_pos_of_nq bc
have n_ge1 := gt1_of_n1_n0 n0 n1; simp [*] | case neg
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = ↑n * length b c
hB : ¬B c b d
bc : b ≠ c
n0 : n ≠ 0
bd : b ≠ d
aL : ¬online a L
n1 : ¬n = 1
cd : c ≠ d
⊢ area a b d = ↑n * area a b c | case neg
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = ↑n * length b c
hB : ¬B c b d
bc : b ≠ c
n0 : n ≠ 0
bd : b ≠ d
aL : ¬online a L
n1 : ¬n = 1
cd : c ≠ d
this : length b c < length b d
⊢ area a b d = ↑n * area a b c | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = ↑n * length b c
hB : ¬B c b d
bc : b ≠ c
n0 : n ≠ 0
bd : b ≠ d
aL : ¬online a L
n1 : ¬n = 1
cd : c ≠ d
⊢ area a b d = ↑n * area a b c
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_triangle_of_base_notcbd | [125, 1] | [153, 37] | have hBs := B_of_three_online_ne_short this bc bd cd bL cL dL | case neg
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = ↑n * length b c
hB : ¬B c b d
bc : b ≠ c
n0 : n ≠ 0
bd : b ≠ d
aL : ¬online a L
n1 : ¬n = 1
cd : c ≠ d
this : length b c < length b d
⊢ area a b d = ↑n * area a b c | case neg
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = ↑n * length b c
hB : ¬B c b d
bc : b ≠ c
n0 : n ≠ 0
bd : b ≠ d
aL : ¬online a L
n1 : ¬n = 1
cd : c ≠ d
this : length b c < length b d
hBs : B b c d ∨ B c b d
⊢ area a b d = ↑n * area a b c | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = ↑n * length b c
hB : ¬B c b d
bc : b ≠ c
n0 : n ≠ 0
bd : b ≠ d
aL : ¬online a L
n1 : ¬n = 1
cd : c ≠ d
this : length b c < length b d
⊢ area a b d = ↑n * area a b c
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_triangle_of_base_notcbd | [125, 1] | [153, 37] | cases hBs with
| inl hBs => exact rescale_triangle_of_base_B a bL cL dL hlen hBs aL
| inr hBs => exfalso; exact hB hBs | case neg
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = ↑n * length b c
hB : ¬B c b d
bc : b ≠ c
n0 : n ≠ 0
bd : b ≠ d
aL : ¬online a L
n1 : ¬n = 1
cd : c ≠ d
this : length b c < length b d
hBs : B b c d ∨ B c b d
⊢ area a b d = ↑n * area a b c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = ↑n * length b c
hB : ¬B c b d
bc : b ≠ c
n0 : n ≠ 0
bd : b ≠ d
aL : ¬online a L
n1 : ¬n = 1
cd : c ≠ d
this : length b c < length b d
hBs : B b c d ∨ B c b d
⊢ area a b d = ↑n * area a b c
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_triangle_of_base_notcbd | [125, 1] | [153, 37] | rw [n1] at hlen | case pos
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = ↑n * length b c
hB : ¬B c b d
bc : b ≠ c
n0 : n ≠ 0
bd : b ≠ d
aL : ¬online a L
n1 : n = 1
⊢ area a b d = ↑n * area a b c | case pos
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = ↑1 * length b c
hB : ¬B c b d
bc : b ≠ c
n0 : n ≠ 0
bd : b ≠ d
aL : ¬online a L
n1 : n = 1
⊢ area a b d = ↑n * area a b c | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = ↑n * length b c
hB : ¬B c b d
bc : b ≠ c
n0 : n ≠ 0
bd : b ≠ d
aL : ¬online a L
n1 : n = 1
⊢ area a b d = ↑n * area a b c
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_triangle_of_base_notcbd | [125, 1] | [153, 37] | rw [n1] | case pos
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = ↑1 * length b c
hB : ¬B c b d
bc : b ≠ c
n0 : n ≠ 0
bd : b ≠ d
aL : ¬online a L
n1 : n = 1
⊢ area a b d = ↑n * area a b c | case pos
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = ↑1 * length b c
hB : ¬B c b d
bc : b ≠ c
n0 : n ≠ 0
bd : b ≠ d
aL : ¬online a L
n1 : n = 1
⊢ area a b d = ↑1 * area a b c | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = ↑1 * length b c
hB : ¬B c b d
bc : b ≠ c
n0 : n ≠ 0
bd : b ≠ d
aL : ¬online a L
n1 : n = 1
⊢ area a b d = ↑n * area a b c
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_triangle_of_base_notcbd | [125, 1] | [153, 37] | simp [Nat.cast_one] at hlen | case pos
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = ↑1 * length b c
hB : ¬B c b d
bc : b ≠ c
n0 : n ≠ 0
bd : b ≠ d
aL : ¬online a L
n1 : n = 1
⊢ area a b d = ↑1 * area a b c | case pos
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hB : ¬B c b d
bc : b ≠ c
n0 : n ≠ 0
bd : b ≠ d
aL : ¬online a L
n1 : n = 1
hlen : length b d = length b c
⊢ area a b d = ↑1 * area a b c | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = ↑1 * length b c
hB : ¬B c b d
bc : b ≠ c
n0 : n ≠ 0
bd : b ≠ d
aL : ¬online a L
n1 : n = 1
⊢ area a b d = ↑1 * area a b c
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_triangle_of_base_notcbd | [125, 1] | [153, 37] | simp [Nat.cast_one] | case pos
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hB : ¬B c b d
bc : b ≠ c
n0 : n ≠ 0
bd : b ≠ d
aL : ¬online a L
n1 : n = 1
hlen : length b d = length b c
⊢ area a b d = ↑1 * area a b c | case pos
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hB : ¬B c b d
bc : b ≠ c
n0 : n ≠ 0
bd : b ≠ d
aL : ¬online a L
n1 : n = 1
hlen : length b d = length b c
⊢ area a b d = area a b c | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hB : ¬B c b d
bc : b ≠ c
n0 : n ≠ 0
bd : b ≠ d
aL : ¬online a L
n1 : n = 1
hlen : length b d = length b c
⊢ area a b d = ↑1 * area a b c
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_triangle_of_base_notcbd | [125, 1] | [153, 37] | exact (eq_area_of_eq_base_samevertex a bL cL bL dL hlen.symm).symm | case pos
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hB : ¬B c b d
bc : b ≠ c
n0 : n ≠ 0
bd : b ≠ d
aL : ¬online a L
n1 : n = 1
hlen : length b d = length b c
⊢ area a b d = area a b c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hB : ¬B c b d
bc : b ≠ c
n0 : n ≠ 0
bd : b ≠ d
aL : ¬online a L
n1 : n = 1
hlen : length b d = length b c
⊢ area a b d = area a b c
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_triangle_of_base_notcbd | [125, 1] | [153, 37] | have n0 := length_eq_zero_iff.not.mpr bd | i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = ↑n * length b c
hB : ¬B c b d
bc : b ≠ c
n0 : n ≠ 0
bd : b ≠ d
aL : ¬online a L
n1 : ¬n = 1
cd : c = d
⊢ False | i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = ↑n * length b c
hB : ¬B c b d
bc : b ≠ c
n0✝ : n ≠ 0
bd : b ≠ d
aL : ¬online a L
n1 : ¬n = 1
cd : c = d
n0 : ¬length b d = 0
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = ↑n * length b c
hB : ¬B c b d
bc : b ≠ c
n0 : n ≠ 0
bd : b ≠ d
aL : ¬online a L
n1 : ¬n = 1
cd : c = d
⊢ False
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_triangle_of_base_notcbd | [125, 1] | [153, 37] | rw [cd] at hlen | i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = ↑n * length b c
hB : ¬B c b d
bc : b ≠ c
n0✝ : n ≠ 0
bd : b ≠ d
aL : ¬online a L
n1 : ¬n = 1
cd : c = d
n0 : ¬length b d = 0
⊢ False | i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = ↑n * length b d
hB : ¬B c b d
bc : b ≠ c
n0✝ : n ≠ 0
bd : b ≠ d
aL : ¬online a L
n1 : ¬n = 1
cd : c = d
n0 : ¬length b d = 0
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = ↑n * length b c
hB : ¬B c b d
bc : b ≠ c
n0✝ : n ≠ 0
bd : b ≠ d
aL : ¬online a L
n1 : ¬n = 1
cd : c = d
n0 : ¬length b d = 0
⊢ False
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_triangle_of_base_notcbd | [125, 1] | [153, 37] | field_simp at hlen | i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = ↑n * length b d
hB : ¬B c b d
bc : b ≠ c
n0✝ : n ≠ 0
bd : b ≠ d
aL : ¬online a L
n1 : ¬n = 1
cd : c = d
n0 : ¬length b d = 0
⊢ False | i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hB : ¬B c b d
bc : b ≠ c
n0✝ : n ≠ 0
bd : b ≠ d
aL : ¬online a L
n1 : ¬n = 1
cd : c = d
n0 : ¬length b d = 0
hlen : n = 1
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = ↑n * length b d
hB : ¬B c b d
bc : b ≠ c
n0✝ : n ≠ 0
bd : b ≠ d
aL : ¬online a L
n1 : ¬n = 1
cd : c = d
n0 : ¬length b d = 0
⊢ False
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_triangle_of_base_notcbd | [125, 1] | [153, 37] | exact n1 hlen | i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hB : ¬B c b d
bc : b ≠ c
n0✝ : n ≠ 0
bd : b ≠ d
aL : ¬online a L
n1 : ¬n = 1
cd : c = d
n0 : ¬length b d = 0
hlen : n = 1
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hB : ¬B c b d
bc : b ≠ c
n0✝ : n ≠ 0
bd : b ≠ d
aL : ¬online a L
n1 : ¬n = 1
cd : c = d
n0 : ¬length b d = 0
hlen : n = 1
⊢ False
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_triangle_of_base_notcbd | [125, 1] | [153, 37] | have := len_pos_of_nq bc | i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = ↑n * length b c
hB : ¬B c b d
bc : b ≠ c
n0 : n ≠ 0
bd : b ≠ d
aL : ¬online a L
n1 : ¬n = 1
cd : c ≠ d
⊢ length b c < length b d | i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = ↑n * length b c
hB : ¬B c b d
bc : b ≠ c
n0 : n ≠ 0
bd : b ≠ d
aL : ¬online a L
n1 : ¬n = 1
cd : c ≠ d
this : 0 < length b c
⊢ length b c < length b d | Please generate a tactic in lean4 to solve the state.
STATE:
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = ↑n * length b c
hB : ¬B c b d
bc : b ≠ c
n0 : n ≠ 0
bd : b ≠ d
aL : ¬online a L
n1 : ¬n = 1
cd : c ≠ d
⊢ length b c < length b d
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_triangle_of_base_notcbd | [125, 1] | [153, 37] | have n_ge1 := gt1_of_n1_n0 n0 n1 | i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = ↑n * length b c
hB : ¬B c b d
bc : b ≠ c
n0 : n ≠ 0
bd : b ≠ d
aL : ¬online a L
n1 : ¬n = 1
cd : c ≠ d
this : 0 < length b c
⊢ length b c < length b d | i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = ↑n * length b c
hB : ¬B c b d
bc : b ≠ c
n0 : n ≠ 0
bd : b ≠ d
aL : ¬online a L
n1 : ¬n = 1
cd : c ≠ d
this : 0 < length b c
n_ge1 : n > 1
⊢ length b c < length b d | Please generate a tactic in lean4 to solve the state.
STATE:
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = ↑n * length b c
hB : ¬B c b d
bc : b ≠ c
n0 : n ≠ 0
bd : b ≠ d
aL : ¬online a L
n1 : ¬n = 1
cd : c ≠ d
this : 0 < length b c
⊢ length b c < length b d
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_triangle_of_base_notcbd | [125, 1] | [153, 37] | simp [*] | i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = ↑n * length b c
hB : ¬B c b d
bc : b ≠ c
n0 : n ≠ 0
bd : b ≠ d
aL : ¬online a L
n1 : ¬n = 1
cd : c ≠ d
this : 0 < length b c
n_ge1 : n > 1
⊢ length b c < length b d | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = ↑n * length b c
hB : ¬B c b d
bc : b ≠ c
n0 : n ≠ 0
bd : b ≠ d
aL : ¬online a L
n1 : ¬n = 1
cd : c ≠ d
this : 0 < length b c
n_ge1 : n > 1
⊢ length b c < length b d
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_triangle_of_base_notcbd | [125, 1] | [153, 37] | exact rescale_triangle_of_base_B a bL cL dL hlen hBs aL | case neg.inl
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = ↑n * length b c
hB : ¬B c b d
bc : b ≠ c
n0 : n ≠ 0
bd : b ≠ d
aL : ¬online a L
n1 : ¬n = 1
cd : c ≠ d
this : length b c < length b d
hBs : B b c d
⊢ area a b d = ↑n * area a b c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.inl
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = ↑n * length b c
hB : ¬B c b d
bc : b ≠ c
n0 : n ≠ 0
bd : b ≠ d
aL : ¬online a L
n1 : ¬n = 1
cd : c ≠ d
this : length b c < length b d
hBs : B b c d
⊢ area a b d = ↑n * area a b c
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_triangle_of_base_notcbd | [125, 1] | [153, 37] | exfalso | case neg.inr
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = ↑n * length b c
hB : ¬B c b d
bc : b ≠ c
n0 : n ≠ 0
bd : b ≠ d
aL : ¬online a L
n1 : ¬n = 1
cd : c ≠ d
this : length b c < length b d
hBs : B c b d
⊢ area a b d = ↑n * area a b c | case neg.inr.h
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = ↑n * length b c
hB : ¬B c b d
bc : b ≠ c
n0 : n ≠ 0
bd : b ≠ d
aL : ¬online a L
n1 : ¬n = 1
cd : c ≠ d
this : length b c < length b d
hBs : B c b d
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.inr
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = ↑n * length b c
hB : ¬B c b d
bc : b ≠ c
n0 : n ≠ 0
bd : b ≠ d
aL : ¬online a L
n1 : ¬n = 1
cd : c ≠ d
this : length b c < length b d
hBs : B c b d
⊢ area a b d = ↑n * area a b c
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_triangle_of_base_notcbd | [125, 1] | [153, 37] | exact hB hBs | case neg.inr.h
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = ↑n * length b c
hB : ¬B c b d
bc : b ≠ c
n0 : n ≠ 0
bd : b ≠ d
aL : ¬online a L
n1 : ¬n = 1
cd : c ≠ d
this : length b c < length b d
hBs : B c b d
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.inr.h
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = ↑n * length b c
hB : ¬B c b d
bc : b ≠ c
n0 : n ≠ 0
bd : b ≠ d
aL : ¬online a L
n1 : ¬n = 1
cd : c ≠ d
this : length b c < length b d
hBs : B c b d
⊢ False
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_triangle_of_base | [156, 1] | [199, 76] | by_cases bc : b = c | i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = ↑n * length b c
⊢ area a b d = ↑n * area a b c | case pos
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = ↑n * length b c
bc : b = c
⊢ area a b d = ↑n * area a b c
case neg
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = ↑n * length b c
bc : ¬b = c
⊢ area a b d = ↑n * area a b c | Please generate a tactic in lean4 to solve the state.
STATE:
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = ↑n * length b c
⊢ area a b d = ↑n * area a b c
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_triangle_of_base | [156, 1] | [199, 76] | by_cases n0 : n = 0 | case neg
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = ↑n * length b c
bc : ¬b = c
⊢ area a b d = ↑n * area a b c | case pos
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = ↑n * length b c
bc : ¬b = c
n0 : n = 0
⊢ area a b d = ↑n * area a b c
case neg
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = ↑n * length b c
bc : ¬b = c
n0 : ¬n = 0
⊢ area a b d = ↑n * area a b c | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = ↑n * length b c
bc : ¬b = c
⊢ area a b d = ↑n * area a b c
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_triangle_of_base | [156, 1] | [199, 76] | have bd: b ≠ d := fun bd => by
rw [length_eq_zero_iff.mpr bd, zero_eq_mul, Nat.cast_eq_zero] at hlen
cases hlen with
| inl hlen => exact n0 hlen
| inr hlen => exact bc (length_eq_zero_iff.mp hlen) | case neg
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = ↑n * length b c
bc : ¬b = c
n0 : ¬n = 0
⊢ area a b d = ↑n * area a b c | case neg
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = ↑n * length b c
bc : ¬b = c
n0 : ¬n = 0
bd : b ≠ d
⊢ area a b d = ↑n * area a b c | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = ↑n * length b c
bc : ¬b = c
n0 : ¬n = 0
⊢ area a b d = ↑n * area a b c
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_triangle_of_base | [156, 1] | [199, 76] | by_cases aL : online a L | case neg
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = ↑n * length b c
bc : ¬b = c
n0 : ¬n = 0
bd : b ≠ d
⊢ area a b d = ↑n * area a b c | case pos
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = ↑n * length b c
bc : ¬b = c
n0 : ¬n = 0
bd : b ≠ d
aL : online a L
⊢ area a b d = ↑n * area a b c
case neg
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = ↑n * length b c
bc : ¬b = c
n0 : ¬n = 0
bd : b ≠ d
aL : ¬online a L
⊢ area a b d = ↑n * area a b c | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = ↑n * length b c
bc : ¬b = c
n0 : ¬n = 0
bd : b ≠ d
⊢ area a b d = ↑n * area a b c
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_triangle_of_base | [156, 1] | [199, 76] | by_cases h_B_cbd : B c b d | case neg
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = ↑n * length b c
bc : ¬b = c
n0 : ¬n = 0
bd : b ≠ d
aL : ¬online a L
⊢ area a b d = ↑n * area a b c | case pos
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = ↑n * length b c
bc : ¬b = c
n0 : ¬n = 0
bd : b ≠ d
aL : ¬online a L
h_B_cbd : B c b d
⊢ area a b d = ↑n * area a b c
case neg
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = ↑n * length b c
bc : ¬b = c
n0 : ¬n = 0
bd : b ≠ d
aL : ¬online a L
h_B_cbd : ¬B c b d
⊢ area a b d = ↑n * area a b c | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = ↑n * length b c
bc : ¬b = c
n0 : ¬n = 0
bd : b ≠ d
aL : ¬online a L
⊢ area a b d = ↑n * area a b c
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_triangle_of_base | [156, 1] | [199, 76] | exact rescale_triangle_of_base_notcbd a bL cL dL hlen h_B_cbd bc n0 bd aL | case neg
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = ↑n * length b c
bc : ¬b = c
n0 : ¬n = 0
bd : b ≠ d
aL : ¬online a L
h_B_cbd : ¬B c b d
⊢ area a b d = ↑n * area a b c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = ↑n * length b c
bc : ¬b = c
n0 : ¬n = 0
bd : b ≠ d
aL : ¬online a L
h_B_cbd : ¬B c b d
⊢ area a b d = ↑n * area a b c
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_triangle_of_base | [156, 1] | [199, 76] | rw [length_eq_zero_iff.mpr bc, mul_zero] at hlen | case pos
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = ↑n * length b c
bc : b = c
⊢ area a b d = ↑n * area a b c | case pos
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = 0
bc : b = c
⊢ area a b d = ↑n * area a b c | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = ↑n * length b c
bc : b = c
⊢ area a b d = ↑n * area a b c
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_triangle_of_base | [156, 1] | [199, 76] | have := length_eq_zero_iff.mp hlen | case pos
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = 0
bc : b = c
⊢ area a b d = ↑n * area a b c | case pos
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = 0
bc : b = c
this : b = d
⊢ area a b d = ↑n * area a b c | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = 0
bc : b = c
⊢ area a b d = ↑n * area a b c
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_triangle_of_base | [156, 1] | [199, 76] | rw [area_of_eq a b d (by tauto), area_of_eq a b c (by tauto)] | case pos
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = 0
bc : b = c
this : b = d
⊢ area a b d = ↑n * area a b c | case pos
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = 0
bc : b = c
this : b = d
⊢ 0 = ↑n * 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = 0
bc : b = c
this : b = d
⊢ area a b d = ↑n * area a b c
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_triangle_of_base | [156, 1] | [199, 76] | simp [mul_zero] | case pos
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = 0
bc : b = c
this : b = d
⊢ 0 = ↑n * 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = 0
bc : b = c
this : b = d
⊢ 0 = ↑n * 0
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_triangle_of_base | [156, 1] | [199, 76] | tauto | i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = 0
bc : b = c
this : b = d
⊢ a = b ∨ a = d ∨ b = d | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = 0
bc : b = c
this : b = d
⊢ a = b ∨ a = d ∨ b = d
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_triangle_of_base | [156, 1] | [199, 76] | tauto | i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = 0
bc : b = c
this : b = d
⊢ a = b ∨ a = c ∨ b = c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = 0
bc : b = c
this : b = d
⊢ a = b ∨ a = c ∨ b = c
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_triangle_of_base | [156, 1] | [199, 76] | rw [n0] at hlen | case pos
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = ↑n * length b c
bc : ¬b = c
n0 : n = 0
⊢ area a b d = ↑n * area a b c | case pos
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = ↑0 * length b c
bc : ¬b = c
n0 : n = 0
⊢ area a b d = ↑n * area a b c | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = ↑n * length b c
bc : ¬b = c
n0 : n = 0
⊢ area a b d = ↑n * area a b c
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_triangle_of_base | [156, 1] | [199, 76] | simp [Nat.cast_zero] at hlen | case pos
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = ↑0 * length b c
bc : ¬b = c
n0 : n = 0
⊢ area a b d = ↑n * area a b c | case pos
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
bc : ¬b = c
n0 : n = 0
hlen : length b d = 0
⊢ area a b d = ↑n * area a b c | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = ↑0 * length b c
bc : ¬b = c
n0 : n = 0
⊢ area a b d = ↑n * area a b c
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_triangle_of_base | [156, 1] | [199, 76] | have := length_eq_zero_iff.mp hlen | case pos
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
bc : ¬b = c
n0 : n = 0
hlen : length b d = 0
⊢ area a b d = ↑n * area a b c | case pos
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
bc : ¬b = c
n0 : n = 0
hlen : length b d = 0
this : b = d
⊢ area a b d = ↑n * area a b c | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
bc : ¬b = c
n0 : n = 0
hlen : length b d = 0
⊢ area a b d = ↑n * area a b c
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_triangle_of_base | [156, 1] | [199, 76] | rw [n0, area_of_eq a b d (by tauto)] | case pos
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
bc : ¬b = c
n0 : n = 0
hlen : length b d = 0
this : b = d
⊢ area a b d = ↑n * area a b c | case pos
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
bc : ¬b = c
n0 : n = 0
hlen : length b d = 0
this : b = d
⊢ 0 = ↑0 * area a b c | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
bc : ¬b = c
n0 : n = 0
hlen : length b d = 0
this : b = d
⊢ area a b d = ↑n * area a b c
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_triangle_of_base | [156, 1] | [199, 76] | simp [Nat.cast_zero] | case pos
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
bc : ¬b = c
n0 : n = 0
hlen : length b d = 0
this : b = d
⊢ 0 = ↑0 * area a b c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
bc : ¬b = c
n0 : n = 0
hlen : length b d = 0
this : b = d
⊢ 0 = ↑0 * area a b c
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_triangle_of_base | [156, 1] | [199, 76] | tauto | i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
bc : ¬b = c
n0 : n = 0
hlen : length b d = 0
this : b = d
⊢ a = b ∨ a = d ∨ b = d | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
bc : ¬b = c
n0 : n = 0
hlen : length b d = 0
this : b = d
⊢ a = b ∨ a = d ∨ b = d
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_triangle_of_base | [156, 1] | [199, 76] | rw [length_eq_zero_iff.mpr bd, zero_eq_mul, Nat.cast_eq_zero] at hlen | i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = ↑n * length b c
bc : ¬b = c
n0 : ¬n = 0
bd : b = d
⊢ False | i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : n = 0 ∨ length b c = 0
bc : ¬b = c
n0 : ¬n = 0
bd : b = d
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = ↑n * length b c
bc : ¬b = c
n0 : ¬n = 0
bd : b = d
⊢ False
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_triangle_of_base | [156, 1] | [199, 76] | cases hlen with
| inl hlen => exact n0 hlen
| inr hlen => exact bc (length_eq_zero_iff.mp hlen) | i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : n = 0 ∨ length b c = 0
bc : ¬b = c
n0 : ¬n = 0
bd : b = d
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : n = 0 ∨ length b c = 0
bc : ¬b = c
n0 : ¬n = 0
bd : b = d
⊢ False
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_triangle_of_base | [156, 1] | [199, 76] | exact n0 hlen | case inl
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
bc : ¬b = c
n0 : ¬n = 0
bd : b = d
hlen : n = 0
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
bc : ¬b = c
n0 : ¬n = 0
bd : b = d
hlen : n = 0
⊢ False
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_triangle_of_base | [156, 1] | [199, 76] | exact bc (length_eq_zero_iff.mp hlen) | case inr
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
bc : ¬b = c
n0 : ¬n = 0
bd : b = d
hlen : length b c = 0
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
bc : ¬b = c
n0 : ¬n = 0
bd : b = d
hlen : length b c = 0
⊢ False
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_triangle_of_base | [156, 1] | [199, 76] | perm [(area_zero_iff_online bc bL cL).mpr aL] | case pos
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = ↑n * length b c
bc : ¬b = c
n0 : ¬n = 0
bd : b ≠ d
aL : online a L
⊢ area a b d = ↑n * area a b c | case pos
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = ↑n * length b c
bc : ¬b = c
n0 : ¬n = 0
bd : b ≠ d
aL : online a L
this : area a b c = 0
⊢ area a b d = ↑n * area a b c | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = ↑n * length b c
bc : ¬b = c
n0 : ¬n = 0
bd : b ≠ d
aL : online a L
⊢ area a b d = ↑n * area a b c
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_triangle_of_base | [156, 1] | [199, 76] | rw [this] | case pos
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = ↑n * length b c
bc : ¬b = c
n0 : ¬n = 0
bd : b ≠ d
aL : online a L
this : area a b c = 0
⊢ area a b d = ↑n * area a b c | case pos
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = ↑n * length b c
bc : ¬b = c
n0 : ¬n = 0
bd : b ≠ d
aL : online a L
this : area a b c = 0
⊢ area a b d = ↑n * 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = ↑n * length b c
bc : ¬b = c
n0 : ¬n = 0
bd : b ≠ d
aL : online a L
this : area a b c = 0
⊢ area a b d = ↑n * area a b c
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_triangle_of_base | [156, 1] | [199, 76] | linperm [(area_zero_iff_online bd bL dL).mpr aL] | case pos
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = ↑n * length b c
bc : ¬b = c
n0 : ¬n = 0
bd : b ≠ d
aL : online a L
this : area a b c = 0
⊢ area a b d = ↑n * 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = ↑n * length b c
bc : ¬b = c
n0 : ¬n = 0
bd : b ≠ d
aL : online a L
this : area a b c = 0
⊢ area a b d = ↑n * 0
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_triangle_of_base | [156, 1] | [199, 76] | obtain ⟨ e, He ⟩ := rescale_length 2 cL bL | case pos
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = ↑n * length b c
bc : ¬b = c
n0 : ¬n = 0
bd : b ≠ d
aL : ¬online a L
h_B_cbd : B c b d
⊢ area a b d = ↑n * area a b c | case pos.intro
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = ↑n * length b c
bc : ¬b = c
n0 : ¬n = 0
bd : b ≠ d
aL : ¬online a L
h_B_cbd : B c b d
e : point
He : online e L ∧ length c e = ↑2 * length c b ∧ (2 ≥ 2 ∧ c ≠ b → B c b e)
⊢ area a b d = ↑n * area a b c | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = ↑n * length b c
bc : ¬b = c
n0 : ¬n = 0
bd : b ≠ d
aL : ¬online a L
h_B_cbd : B c b d
⊢ area a b d = ↑n * area a b c
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_triangle_of_base | [156, 1] | [199, 76] | have Bcbe := He.2.2 ⟨ by simp [ge_iff_le], fun cb => bc cb.symm⟩ | case pos.intro
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = ↑n * length b c
bc : ¬b = c
n0 : ¬n = 0
bd : b ≠ d
aL : ¬online a L
h_B_cbd : B c b d
e : point
He : online e L ∧ length c e = ↑2 * length c b ∧ (2 ≥ 2 ∧ c ≠ b → B c b e)
⊢ area a b d = ↑n * area a b c | case pos.intro
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = ↑n * length b c
bc : ¬b = c
n0 : ¬n = 0
bd : b ≠ d
aL : ¬online a L
h_B_cbd : B c b d
e : point
He : online e L ∧ length c e = ↑2 * length c b ∧ (2 ≥ 2 ∧ c ≠ b → B c b e)
Bcbe : B c b e
⊢ area a b d = ↑n * area a b c | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.intro
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = ↑n * length b c
bc : ¬b = c
n0 : ¬n = 0
bd : b ≠ d
aL : ¬online a L
h_B_cbd : B c b d
e : point
He : online e L ∧ length c e = ↑2 * length c b ∧ (2 ≥ 2 ∧ c ≠ b → B c b e)
⊢ area a b d = ↑n * area a b c
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_triangle_of_base | [156, 1] | [199, 76] | have Lbe : length b e = length b c := by
perm [length_sum_of_B Bcbe]
rw [He.2.1] at this; norm_cast at this; linperm | case pos.intro
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = ↑n * length b c
bc : ¬b = c
n0 : ¬n = 0
bd : b ≠ d
aL : ¬online a L
h_B_cbd : B c b d
e : point
He : online e L ∧ length c e = ↑2 * length c b ∧ (2 ≥ 2 ∧ c ≠ b → B c b e)
Bcbe : B c b e
⊢ area a b d = ↑n * area a b c | case pos.intro
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = ↑n * length b c
bc : ¬b = c
n0 : ¬n = 0
bd : b ≠ d
aL : ¬online a L
h_B_cbd : B c b d
e : point
He : online e L ∧ length c e = ↑2 * length c b ∧ (2 ≥ 2 ∧ c ≠ b → B c b e)
Bcbe : B c b e
Lbe : length b e = length b c
⊢ area a b d = ↑n * area a b c | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.intro
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = ↑n * length b c
bc : ¬b = c
n0 : ¬n = 0
bd : b ≠ d
aL : ¬online a L
h_B_cbd : B c b d
e : point
He : online e L ∧ length c e = ↑2 * length c b ∧ (2 ≥ 2 ∧ c ≠ b → B c b e)
Bcbe : B c b e
⊢ area a b d = ↑n * area a b c
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_triangle_of_base | [156, 1] | [199, 76] | rw [← Lbe] at hlen | case pos.intro
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = ↑n * length b c
bc : ¬b = c
n0 : ¬n = 0
bd : b ≠ d
aL : ¬online a L
h_B_cbd : B c b d
e : point
He : online e L ∧ length c e = ↑2 * length c b ∧ (2 ≥ 2 ∧ c ≠ b → B c b e)
Bcbe : B c b e
Lbe : length b e = length b c
⊢ area a b d = ↑n * area a b c | case pos.intro
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
bc : ¬b = c
n0 : ¬n = 0
bd : b ≠ d
aL : ¬online a L
h_B_cbd : B c b d
e : point
hlen : length b d = ↑n * length b e
He : online e L ∧ length c e = ↑2 * length c b ∧ (2 ≥ 2 ∧ c ≠ b → B c b e)
Bcbe : B c b e
Lbe : length b e = length b c
⊢ area a b d = ↑n * area a b c | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.intro
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = ↑n * length b c
bc : ¬b = c
n0 : ¬n = 0
bd : b ≠ d
aL : ¬online a L
h_B_cbd : B c b d
e : point
He : online e L ∧ length c e = ↑2 * length c b ∧ (2 ≥ 2 ∧ c ≠ b → B c b e)
Bcbe : B c b e
Lbe : length b e = length b c
⊢ area a b d = ↑n * area a b c
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_triangle_of_base | [156, 1] | [199, 76] | have := not_B324_of_B123_B124 Bcbe h_B_cbd | case pos.intro
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
bc : ¬b = c
n0 : ¬n = 0
bd : b ≠ d
aL : ¬online a L
h_B_cbd : B c b d
e : point
hlen : length b d = ↑n * length b e
He : online e L ∧ length c e = ↑2 * length c b ∧ (2 ≥ 2 ∧ c ≠ b → B c b e)
Bcbe : B c b e
Lbe : length b e = length b c
⊢ area a b d = ↑n * area a b c | case pos.intro
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
bc : ¬b = c
n0 : ¬n = 0
bd : b ≠ d
aL : ¬online a L
h_B_cbd : B c b d
e : point
hlen : length b d = ↑n * length b e
He : online e L ∧ length c e = ↑2 * length c b ∧ (2 ≥ 2 ∧ c ≠ b → B c b e)
Bcbe : B c b e
Lbe : length b e = length b c
this : ¬B e b d
⊢ area a b d = ↑n * area a b c | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.intro
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
bc : ¬b = c
n0 : ¬n = 0
bd : b ≠ d
aL : ¬online a L
h_B_cbd : B c b d
e : point
hlen : length b d = ↑n * length b e
He : online e L ∧ length c e = ↑2 * length c b ∧ (2 ≥ 2 ∧ c ≠ b → B c b e)
Bcbe : B c b e
Lbe : length b e = length b c
⊢ area a b d = ↑n * area a b c
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_triangle_of_base | [156, 1] | [199, 76] | rw [rescale_triangle_of_base_notcbd a bL He.1 dL hlen this (ne_23_of_B Bcbe) n0 bd aL] | case pos.intro
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
bc : ¬b = c
n0 : ¬n = 0
bd : b ≠ d
aL : ¬online a L
h_B_cbd : B c b d
e : point
hlen : length b d = ↑n * length b e
He : online e L ∧ length c e = ↑2 * length c b ∧ (2 ≥ 2 ∧ c ≠ b → B c b e)
Bcbe : B c b e
Lbe : length b e = length b c
this : ¬B e b d
⊢ area a b d = ↑n * area a b c | case pos.intro
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
bc : ¬b = c
n0 : ¬n = 0
bd : b ≠ d
aL : ¬online a L
h_B_cbd : B c b d
e : point
hlen : length b d = ↑n * length b e
He : online e L ∧ length c e = ↑2 * length c b ∧ (2 ≥ 2 ∧ c ≠ b → B c b e)
Bcbe : B c b e
Lbe : length b e = length b c
this : ¬B e b d
⊢ ↑n * area a b e = ↑n * area a b c | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.intro
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
bc : ¬b = c
n0 : ¬n = 0
bd : b ≠ d
aL : ¬online a L
h_B_cbd : B c b d
e : point
hlen : length b d = ↑n * length b e
He : online e L ∧ length c e = ↑2 * length c b ∧ (2 ≥ 2 ∧ c ≠ b → B c b e)
Bcbe : B c b e
Lbe : length b e = length b c
this : ¬B e b d
⊢ area a b d = ↑n * area a b c
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_triangle_of_base | [156, 1] | [199, 76] | rw [eq_area_of_eq_base_samevertex a bL He.1 bL cL Lbe] | case pos.intro
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
bc : ¬b = c
n0 : ¬n = 0
bd : b ≠ d
aL : ¬online a L
h_B_cbd : B c b d
e : point
hlen : length b d = ↑n * length b e
He : online e L ∧ length c e = ↑2 * length c b ∧ (2 ≥ 2 ∧ c ≠ b → B c b e)
Bcbe : B c b e
Lbe : length b e = length b c
this : ¬B e b d
⊢ ↑n * area a b e = ↑n * area a b c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.intro
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
bc : ¬b = c
n0 : ¬n = 0
bd : b ≠ d
aL : ¬online a L
h_B_cbd : B c b d
e : point
hlen : length b d = ↑n * length b e
He : online e L ∧ length c e = ↑2 * length c b ∧ (2 ≥ 2 ∧ c ≠ b → B c b e)
Bcbe : B c b e
Lbe : length b e = length b c
this : ¬B e b d
⊢ ↑n * area a b e = ↑n * area a b c
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_triangle_of_base | [156, 1] | [199, 76] | simp [ge_iff_le] | i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = ↑n * length b c
bc : ¬b = c
n0 : ¬n = 0
bd : b ≠ d
aL : ¬online a L
h_B_cbd : B c b d
e : point
He : online e L ∧ length c e = ↑2 * length c b ∧ (2 ≥ 2 ∧ c ≠ b → B c b e)
⊢ 2 ≥ 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = ↑n * length b c
bc : ¬b = c
n0 : ¬n = 0
bd : b ≠ d
aL : ¬online a L
h_B_cbd : B c b d
e : point
He : online e L ∧ length c e = ↑2 * length c b ∧ (2 ≥ 2 ∧ c ≠ b → B c b e)
⊢ 2 ≥ 2
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_triangle_of_base | [156, 1] | [199, 76] | perm [length_sum_of_B Bcbe] | i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = ↑n * length b c
bc : ¬b = c
n0 : ¬n = 0
bd : b ≠ d
aL : ¬online a L
h_B_cbd : B c b d
e : point
He : online e L ∧ length c e = ↑2 * length c b ∧ (2 ≥ 2 ∧ c ≠ b → B c b e)
Bcbe : B c b e
⊢ length b e = length b c | i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = ↑n * length b c
bc : ¬b = c
n0 : ¬n = 0
bd : b ≠ d
aL : ¬online a L
h_B_cbd : B c b d
e : point
He : online e L ∧ length c e = ↑2 * length c b ∧ (2 ≥ 2 ∧ c ≠ b → B c b e)
Bcbe : B c b e
this : length b c + length b e = length c e
⊢ length b e = length b c | Please generate a tactic in lean4 to solve the state.
STATE:
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = ↑n * length b c
bc : ¬b = c
n0 : ¬n = 0
bd : b ≠ d
aL : ¬online a L
h_B_cbd : B c b d
e : point
He : online e L ∧ length c e = ↑2 * length c b ∧ (2 ≥ 2 ∧ c ≠ b → B c b e)
Bcbe : B c b e
⊢ length b e = length b c
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_triangle_of_base | [156, 1] | [199, 76] | rw [He.2.1] at this | i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = ↑n * length b c
bc : ¬b = c
n0 : ¬n = 0
bd : b ≠ d
aL : ¬online a L
h_B_cbd : B c b d
e : point
He : online e L ∧ length c e = ↑2 * length c b ∧ (2 ≥ 2 ∧ c ≠ b → B c b e)
Bcbe : B c b e
this : length b c + length b e = length c e
⊢ length b e = length b c | i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = ↑n * length b c
bc : ¬b = c
n0 : ¬n = 0
bd : b ≠ d
aL : ¬online a L
h_B_cbd : B c b d
e : point
He : online e L ∧ length c e = ↑2 * length c b ∧ (2 ≥ 2 ∧ c ≠ b → B c b e)
Bcbe : B c b e
this : length b c + length b e = ↑2 * length c b
⊢ length b e = length b c | Please generate a tactic in lean4 to solve the state.
STATE:
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = ↑n * length b c
bc : ¬b = c
n0 : ¬n = 0
bd : b ≠ d
aL : ¬online a L
h_B_cbd : B c b d
e : point
He : online e L ∧ length c e = ↑2 * length c b ∧ (2 ≥ 2 ∧ c ≠ b → B c b e)
Bcbe : B c b e
this : length b c + length b e = length c e
⊢ length b e = length b c
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_triangle_of_base | [156, 1] | [199, 76] | norm_cast at this | i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = ↑n * length b c
bc : ¬b = c
n0 : ¬n = 0
bd : b ≠ d
aL : ¬online a L
h_B_cbd : B c b d
e : point
He : online e L ∧ length c e = ↑2 * length c b ∧ (2 ≥ 2 ∧ c ≠ b → B c b e)
Bcbe : B c b e
this : length b c + length b e = ↑2 * length c b
⊢ length b e = length b c | i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = ↑n * length b c
bc : ¬b = c
n0 : ¬n = 0
bd : b ≠ d
aL : ¬online a L
h_B_cbd : B c b d
e : point
He : online e L ∧ length c e = ↑2 * length c b ∧ (2 ≥ 2 ∧ c ≠ b → B c b e)
Bcbe : B c b e
this : length b c + length b e = 2 * length c b
⊢ length b e = length b c | Please generate a tactic in lean4 to solve the state.
STATE:
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = ↑n * length b c
bc : ¬b = c
n0 : ¬n = 0
bd : b ≠ d
aL : ¬online a L
h_B_cbd : B c b d
e : point
He : online e L ∧ length c e = ↑2 * length c b ∧ (2 ≥ 2 ∧ c ≠ b → B c b e)
Bcbe : B c b e
this : length b c + length b e = ↑2 * length c b
⊢ length b e = length b c
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_triangle_of_base | [156, 1] | [199, 76] | linperm | i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = ↑n * length b c
bc : ¬b = c
n0 : ¬n = 0
bd : b ≠ d
aL : ¬online a L
h_B_cbd : B c b d
e : point
He : online e L ∧ length c e = ↑2 * length c b ∧ (2 ≥ 2 ∧ c ≠ b → B c b e)
Bcbe : B c b e
this : length b c + length b e = 2 * length c b
⊢ length b e = length b c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
i : incidence_geometry
a b c d : point
L : line
n : ℕ
bL : online b L
cL : online c L
dL : online d L
hlen : length b d = ↑n * length b c
bc : ¬b = c
n0 : ¬n = 0
bd : b ≠ d
aL : ¬online a L
h_B_cbd : B c b d
e : point
He : online e L ∧ length c e = ↑2 * length c b ∧ (2 ≥ 2 ∧ c ≠ b → B c b e)
Bcbe : B c b e
this : length b c + length b e = 2 * length c b
⊢ length b e = length b c
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | lt_area_of_lt_base_sameedge_Bbfc | [201, 1] | [216, 52] | have := (area_add_iff_B bf cf.symm bc.symm bL fL cL aL).mp hB | i : incidence_geometry
a b c f : point
L : line
bL : online b L
cL : online c L
fL : online f L
hB : B b f c
bc : b ≠ c
bf : b ≠ f
cf : c ≠ f
aL : ¬online a L
⊢ length b c > length b f → area a b c > area a b f | i : incidence_geometry
a b c f : point
L : line
bL : online b L
cL : online c L
fL : online f L
hB : B b f c
bc : b ≠ c
bf : b ≠ f
cf : c ≠ f
aL : ¬online a L
this : area a b f + area a c f = area a b c
⊢ length b c > length b f → area a b c > area a b f | Please generate a tactic in lean4 to solve the state.
STATE:
i : incidence_geometry
a b c f : point
L : line
bL : online b L
cL : online c L
fL : online f L
hB : B b f c
bc : b ≠ c
bf : b ≠ f
cf : c ≠ f
aL : ¬online a L
⊢ length b c > length b f → area a b c > area a b f
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | lt_area_of_lt_base_sameedge_Bbfc | [201, 1] | [216, 52] | intro | i : incidence_geometry
a b c f : point
L : line
bL : online b L
cL : online c L
fL : online f L
hB : B b f c
bc : b ≠ c
bf : b ≠ f
cf : c ≠ f
aL : ¬online a L
this : area a b f + area a c f = area a b c
⊢ length b c > length b f → area a b c > area a b f | i : incidence_geometry
a b c f : point
L : line
bL : online b L
cL : online c L
fL : online f L
hB : B b f c
bc : b ≠ c
bf : b ≠ f
cf : c ≠ f
aL : ¬online a L
this : area a b f + area a c f = area a b c
a✝ : length b c > length b f
⊢ area a b c > area a b f | Please generate a tactic in lean4 to solve the state.
STATE:
i : incidence_geometry
a b c f : point
L : line
bL : online b L
cL : online c L
fL : online f L
hB : B b f c
bc : b ≠ c
bf : b ≠ f
cf : c ≠ f
aL : ¬online a L
this : area a b f + area a c f = area a b c
⊢ length b c > length b f → area a b c > area a b f
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | lt_area_of_lt_base_sameedge_Bbfc | [201, 1] | [216, 52] | rw [← this] | i : incidence_geometry
a b c f : point
L : line
bL : online b L
cL : online c L
fL : online f L
hB : B b f c
bc : b ≠ c
bf : b ≠ f
cf : c ≠ f
aL : ¬online a L
this : area a b f + area a c f = area a b c
a✝ : length b c > length b f
⊢ area a b c > area a b f | i : incidence_geometry
a b c f : point
L : line
bL : online b L
cL : online c L
fL : online f L
hB : B b f c
bc : b ≠ c
bf : b ≠ f
cf : c ≠ f
aL : ¬online a L
this : area a b f + area a c f = area a b c
a✝ : length b c > length b f
⊢ area a b f + area a c f > area a b f | Please generate a tactic in lean4 to solve the state.
STATE:
i : incidence_geometry
a b c f : point
L : line
bL : online b L
cL : online c L
fL : online f L
hB : B b f c
bc : b ≠ c
bf : b ≠ f
cf : c ≠ f
aL : ¬online a L
this : area a b f + area a c f = area a b c
a✝ : length b c > length b f
⊢ area a b c > area a b f
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | lt_area_of_lt_base_sameedge_Bbfc | [201, 1] | [216, 52] | simp [gt_iff_lt, lt_add_iff_pos_right] | i : incidence_geometry
a b c f : point
L : line
bL : online b L
cL : online c L
fL : online f L
hB : B b f c
bc : b ≠ c
bf : b ≠ f
cf : c ≠ f
aL : ¬online a L
this : area a b f + area a c f = area a b c
a✝ : length b c > length b f
⊢ area a b f + area a c f > area a b f | i : incidence_geometry
a b c f : point
L : line
bL : online b L
cL : online c L
fL : online f L
hB : B b f c
bc : b ≠ c
bf : b ≠ f
cf : c ≠ f
aL : ¬online a L
this : area a b f + area a c f = area a b c
a✝ : length b c > length b f
⊢ 0 < area a c f | Please generate a tactic in lean4 to solve the state.
STATE:
i : incidence_geometry
a b c f : point
L : line
bL : online b L
cL : online c L
fL : online f L
hB : B b f c
bc : b ≠ c
bf : b ≠ f
cf : c ≠ f
aL : ¬online a L
this : area a b f + area a c f = area a b c
a✝ : length b c > length b f
⊢ area a b f + area a c f > area a b f
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | lt_area_of_lt_base_sameedge_Bbfc | [201, 1] | [216, 52] | have : area a c f ≠ 0 := by perma [(area_zero_iff_online cf cL fL).not.mpr aL] | i : incidence_geometry
a b c f : point
L : line
bL : online b L
cL : online c L
fL : online f L
hB : B b f c
bc : b ≠ c
bf : b ≠ f
cf : c ≠ f
aL : ¬online a L
this : area a b f + area a c f = area a b c
a✝ : length b c > length b f
⊢ 0 < area a c f | i : incidence_geometry
a b c f : point
L : line
bL : online b L
cL : online c L
fL : online f L
hB : B b f c
bc : b ≠ c
bf : b ≠ f
cf : c ≠ f
aL : ¬online a L
this✝ : area a b f + area a c f = area a b c
a✝ : length b c > length b f
this : area a c f ≠ 0
⊢ 0 < area a c f | Please generate a tactic in lean4 to solve the state.
STATE:
i : incidence_geometry
a b c f : point
L : line
bL : online b L
cL : online c L
fL : online f L
hB : B b f c
bc : b ≠ c
bf : b ≠ f
cf : c ≠ f
aL : ¬online a L
this : area a b f + area a c f = area a b c
a✝ : length b c > length b f
⊢ 0 < area a c f
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | lt_area_of_lt_base_sameedge_Bbfc | [201, 1] | [216, 52] | exact (Ne.symm this).lt_of_le (area_nonneg a c f) | i : incidence_geometry
a b c f : point
L : line
bL : online b L
cL : online c L
fL : online f L
hB : B b f c
bc : b ≠ c
bf : b ≠ f
cf : c ≠ f
aL : ¬online a L
this✝ : area a b f + area a c f = area a b c
a✝ : length b c > length b f
this : area a c f ≠ 0
⊢ 0 < area a c f | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
i : incidence_geometry
a b c f : point
L : line
bL : online b L
cL : online c L
fL : online f L
hB : B b f c
bc : b ≠ c
bf : b ≠ f
cf : c ≠ f
aL : ¬online a L
this✝ : area a b f + area a c f = area a b c
a✝ : length b c > length b f
this : area a c f ≠ 0
⊢ 0 < area a c f
TACTIC:
|
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