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_I_extras.lean | eq_area_of_eq_base | [339, 1] | [374, 11] | perma | i : incidence_geometry
a b c d e f : point
L M : line
aM : online a M
bL : online b L
cL : online c L
dM : online d M
eL : online e L
fL : online f L
pLM : para L M
hlen : length b c = length e f
bc : ¬b = c
ef : e ≠ f
K : line
aK : online a K
cK : online c K
N : line
dN : online d N
eN : online e N
aL : ¬online a L
this✝ : ¬online b K
g : point
O : line
gM : online g M
hgO : online g O
bO : online b O
pKO : para K O
dL : ¬online d L
this : ¬online f N
h : point
P : line
hhM : online h M
hP : online h P
hfP : online f P
pNP : para N P
⊢ para M L | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
i : incidence_geometry
a b c d e f : point
L M : line
aM : online a M
bL : online b L
cL : online c L
dM : online d M
eL : online e L
fL : online f L
pLM : para L M
hlen : length b c = length e f
bc : ¬b = c
ef : e ≠ f
K : line
aK : online a K
cK : online c K
N : line
dN : online d N
eN : online e N
aL : ¬online a L
this✝ : ¬online b K
g : point
O : line
gM : online g M
hgO : online g O
bO : online b O
pKO : para K O
dL : ¬online d L
this : ¬online f N
h : point
P : line
hhM : online h M
hP : online h P
hfP : online f P
pNP : para N P
⊢ para M L
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_I_extras.lean | eq_area_of_eq_base | [339, 1] | [374, 11] | perma | i : incidence_geometry
a b c d e f : point
L M : line
aM : online a M
bL : online b L
cL : online c L
dM : online d M
eL : online e L
fL : online f L
pLM : para L M
hlen : length b c = length e f
bc : ¬b = c
ef : e ≠ f
K : line
aK : online a K
cK : online c K
N : line
dN : online d N
eN : online e N
aL : ¬online a L
this✝ : ¬online b K
g : point
O : line
gM : online g M
hgO : online g O
bO : online b O
pKO : para K O
dL : ¬online d L
this : ¬online f N
h : point
P : line
hhM : online h M
hP : online h P
hfP : online f P
pNP : para N P
⊢ para O K | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
i : incidence_geometry
a b c d e f : point
L M : line
aM : online a M
bL : online b L
cL : online c L
dM : online d M
eL : online e L
fL : online f L
pLM : para L M
hlen : length b c = length e f
bc : ¬b = c
ef : e ≠ f
K : line
aK : online a K
cK : online c K
N : line
dN : online d N
eN : online e N
aL : ¬online a L
this✝ : ¬online b K
g : point
O : line
gM : online g M
hgO : online g O
bO : online b O
pKO : para K O
dL : ¬online d L
this : ¬online f N
h : point
P : line
hhM : online h M
hP : online h P
hfP : online f P
pNP : para N P
⊢ para O K
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_I_extras.lean | eq_area_of_eq_base | [339, 1] | [374, 11] | splitAll <;> perma only [para] | i : incidence_geometry
a b c d e f : point
L M : line
aM : online a M
bL : online b L
cL : online c L
dM : online d M
eL : online e L
fL : online f L
pLM : para L M
hlen : length b c = length e f
bc : ¬b = c
ef : e ≠ f
K : line
aK : online a K
cK : online c K
N : line
dN : online d N
eN : online e N
aL : ¬online a L
this✝¹ : ¬online b K
g : point
O : line
gM : online g M
hgO : online g O
bO : online b O
pKO : para K O
dL : ¬online d L
this✝ : ¬online f N
h : point
P : line
hhM : online h M
hP : online h P
hfP : online f P
pNP : para N P
this : area b c g + area a c g = area d e f + area d f h
⊢ paragram b c a g L K M O | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
i : incidence_geometry
a b c d e f : point
L M : line
aM : online a M
bL : online b L
cL : online c L
dM : online d M
eL : online e L
fL : online f L
pLM : para L M
hlen : length b c = length e f
bc : ¬b = c
ef : e ≠ f
K : line
aK : online a K
cK : online c K
N : line
dN : online d N
eN : online e N
aL : ¬online a L
this✝¹ : ¬online b K
g : point
O : line
gM : online g M
hgO : online g O
bO : online b O
pKO : para K O
dL : ¬online d L
this✝ : ¬online f N
h : point
P : line
hhM : online h M
hP : online h P
hfP : online f P
pNP : para N P
this : area b c g + area a c g = area d e f + area d f h
⊢ paragram b c a g L K M O
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_I_extras.lean | eq_area_of_eq_base | [339, 1] | [374, 11] | splitAll <;> perma only [para] | i : incidence_geometry
a b c d e f : point
L M : line
aM : online a M
bL : online b L
cL : online c L
dM : online d M
eL : online e L
fL : online f L
pLM : para L M
hlen : length b c = length e f
bc : ¬b = c
ef : e ≠ f
K : line
aK : online a K
cK : online c K
N : line
dN : online d N
eN : online e N
aL : ¬online a L
this✝¹ : ¬online b K
g : point
O : line
gM : online g M
hgO : online g O
bO : online b O
pKO : para K O
dL : ¬online d L
this✝ : ¬online f N
h : point
P : line
hhM : online h M
hP : online h P
hfP : online f P
pNP : para N P
this : area b c g + area a c g = area d e f + area d f h
bcag : paragram b c a g L K M O
⊢ paragram f e d h L N M P | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
i : incidence_geometry
a b c d e f : point
L M : line
aM : online a M
bL : online b L
cL : online c L
dM : online d M
eL : online e L
fL : online f L
pLM : para L M
hlen : length b c = length e f
bc : ¬b = c
ef : e ≠ f
K : line
aK : online a K
cK : online c K
N : line
dN : online d N
eN : online e N
aL : ¬online a L
this✝¹ : ¬online b K
g : point
O : line
gM : online g M
hgO : online g O
bO : online b O
pKO : para K O
dL : ¬online d L
this✝ : ¬online f N
h : point
P : line
hhM : online h M
hP : online h P
hfP : online f P
pNP : para N P
this : area b c g + area a c g = area d e f + area d f h
bcag : paragram b c a g L K M O
⊢ paragram f e d h L N M P
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_I_extras.lean | eq_area_of_eq_base | [339, 1] | [374, 11] | perma [(area_of_parallelogram bcag).2] | i : incidence_geometry
a b c d e f : point
L M : line
aM : online a M
bL : online b L
cL : online c L
dM : online d M
eL : online e L
fL : online f L
pLM : para L M
hlen : length b c = length e f
bc : ¬b = c
ef : e ≠ f
K : line
aK : online a K
cK : online c K
N : line
dN : online d N
eN : online e N
aL : ¬online a L
this✝¹ : ¬online b K
g : point
O : line
gM : online g M
hgO : online g O
bO : online b O
pKO : para K O
dL : ¬online d L
this✝ : ¬online f N
h : point
P : line
hhM : online h M
hP : online h P
hfP : online f P
pNP : para N P
this : area b c g + area a c g = area d e f + area d f h
bcag : paragram b c a g L K M O
fedh : paragram f e d h L N M P
⊢ area b c g + area a c g = 2 * area a b c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
i : incidence_geometry
a b c d e f : point
L M : line
aM : online a M
bL : online b L
cL : online c L
dM : online d M
eL : online e L
fL : online f L
pLM : para L M
hlen : length b c = length e f
bc : ¬b = c
ef : e ≠ f
K : line
aK : online a K
cK : online c K
N : line
dN : online d N
eN : online e N
aL : ¬online a L
this✝¹ : ¬online b K
g : point
O : line
gM : online g M
hgO : online g O
bO : online b O
pKO : para K O
dL : ¬online d L
this✝ : ¬online f N
h : point
P : line
hhM : online h M
hP : online h P
hfP : online f P
pNP : para N P
this : area b c g + area a c g = area d e f + area d f h
bcag : paragram b c a g L K M O
fedh : paragram f e d h L N M P
⊢ area b c g + area a c g = 2 * area a b c
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_I_extras.lean | eq_area_of_eq_base | [339, 1] | [374, 11] | linperm [(area_of_parallelogram fedh).1] | i : incidence_geometry
a b c d e f : point
L M : line
aM : online a M
bL : online b L
cL : online c L
dM : online d M
eL : online e L
fL : online f L
pLM : para L M
hlen : length b c = length e f
bc : ¬b = c
ef : e ≠ f
K : line
aK : online a K
cK : online c K
N : line
dN : online d N
eN : online e N
aL : ¬online a L
this✝² : ¬online b K
g : point
O : line
gM : online g M
hgO : online g O
bO : online b O
pKO : para K O
dL : ¬online d L
this✝¹ : ¬online f N
h : point
P : line
hhM : online h M
hP : online h P
hfP : online f P
pNP : para N P
this✝ : area b c g + area a c g = area d e f + area d f h
bcag : paragram b c a g L K M O
fedh : paragram f e d h L N M P
this : area b c g + area a c g = 2 * area a b c
⊢ area d e f = area d f h | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
i : incidence_geometry
a b c d e f : point
L M : line
aM : online a M
bL : online b L
cL : online c L
dM : online d M
eL : online e L
fL : online f L
pLM : para L M
hlen : length b c = length e f
bc : ¬b = c
ef : e ≠ f
K : line
aK : online a K
cK : online c K
N : line
dN : online d N
eN : online e N
aL : ¬online a L
this✝² : ¬online b K
g : point
O : line
gM : online g M
hgO : online g O
bO : online b O
pKO : para K O
dL : ¬online d L
this✝¹ : ¬online f N
h : point
P : line
hhM : online h M
hP : online h P
hfP : online f P
pNP : para N P
this✝ : area b c g + area a c g = area d e f + area d f h
bcag : paragram b c a g L K M O
fedh : paragram f e d h L N M P
this : area b c g + area a c g = 2 * area a b c
⊢ area d e f = area d f h
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_I_extras.lean | eq_area_of_eq_base_samevertex | [379, 1] | [398, 59] | by_cases bc : b = c | i : incidence_geometry
a b c e f : point
L : line
bL : online b L
cL : online c L
eL : online e L
fL : online f L
hlen : length b c = length e f
⊢ area a b c = area a e f | case pos
i : incidence_geometry
a b c e f : point
L : line
bL : online b L
cL : online c L
eL : online e L
fL : online f L
hlen : length b c = length e f
bc : b = c
⊢ area a b c = area a e f
case neg
i : incidence_geometry
a b c e f : point
L : line
bL : online b L
cL : online c L
eL : online e L
fL : online f L
hlen : length b c = length e f
bc : ¬b = c
⊢ area a b c = area a e f | Please generate a tactic in lean4 to solve the state.
STATE:
i : incidence_geometry
a b c e f : point
L : line
bL : online b L
cL : online c L
eL : online e L
fL : online f L
hlen : length b c = length e f
⊢ area a b c = area a e f
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_I_extras.lean | eq_area_of_eq_base_samevertex | [379, 1] | [398, 59] | have ef : e ≠ f := ne_of_ne_len bc hlen | case neg
i : incidence_geometry
a b c e f : point
L : line
bL : online b L
cL : online c L
eL : online e L
fL : online f L
hlen : length b c = length e f
bc : ¬b = c
⊢ area a b c = area a e f | case neg
i : incidence_geometry
a b c e f : point
L : line
bL : online b L
cL : online c L
eL : online e L
fL : online f L
hlen : length b c = length e f
bc : ¬b = c
ef : e ≠ f
⊢ area a b c = area a e f | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
i : incidence_geometry
a b c e f : point
L : line
bL : online b L
cL : online c L
eL : online e L
fL : online f L
hlen : length b c = length e f
bc : ¬b = c
⊢ area a b c = area a e f
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_I_extras.lean | eq_area_of_eq_base_samevertex | [379, 1] | [398, 59] | by_cases aL : online a L | case neg
i : incidence_geometry
a b c e f : point
L : line
bL : online b L
cL : online c L
eL : online e L
fL : online f L
hlen : length b c = length e f
bc : ¬b = c
ef : e ≠ f
⊢ area a b c = area a e f | case pos
i : incidence_geometry
a b c e f : point
L : line
bL : online b L
cL : online c L
eL : online e L
fL : online f L
hlen : length b c = length e f
bc : ¬b = c
ef : e ≠ f
aL : online a L
⊢ area a b c = area a e f
case neg
i : incidence_geometry
a b c e f : point
L : line
bL : online b L
cL : online c L
eL : online e L
fL : online f L
hlen : length b c = length e f
bc : ¬b = c
ef : e ≠ f
aL : ¬online a L
⊢ area a b c = area a e f | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
i : incidence_geometry
a b c e f : point
L : line
bL : online b L
cL : online c L
eL : online e L
fL : online f L
hlen : length b c = length e f
bc : ¬b = c
ef : e ≠ f
⊢ area a b c = area a e f
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_I_extras.lean | eq_area_of_eq_base_samevertex | [379, 1] | [398, 59] | obtain ⟨ _, hM ⟩ := parallel_of_line_pt aL | case neg
i : incidence_geometry
a b c e f : point
L : line
bL : online b L
cL : online c L
eL : online e L
fL : online f L
hlen : length b c = length e f
bc : ¬b = c
ef : e ≠ f
aL : ¬online a L
⊢ area a b c = area a e f | case neg.intro
i : incidence_geometry
a b c e f : point
L : line
bL : online b L
cL : online c L
eL : online e L
fL : online f L
hlen : length b c = length e f
bc : ¬b = c
ef : e ≠ f
aL : ¬online a L
w✝ : line
hM : online a w✝ ∧ para L w✝
⊢ area a b c = area a e f | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
i : incidence_geometry
a b c e f : point
L : line
bL : online b L
cL : online c L
eL : online e L
fL : online f L
hlen : length b c = length e f
bc : ¬b = c
ef : e ≠ f
aL : ¬online a L
⊢ area a b c = area a e f
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_I_extras.lean | eq_area_of_eq_base_samevertex | [379, 1] | [398, 59] | exact eq_area_of_eq_base hM.1 bL cL hM.1 eL fL hM.2 hlen | case neg.intro
i : incidence_geometry
a b c e f : point
L : line
bL : online b L
cL : online c L
eL : online e L
fL : online f L
hlen : length b c = length e f
bc : ¬b = c
ef : e ≠ f
aL : ¬online a L
w✝ : line
hM : online a w✝ ∧ para L w✝
⊢ area a b c = area a e f | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro
i : incidence_geometry
a b c e f : point
L : line
bL : online b L
cL : online c L
eL : online e L
fL : online f L
hlen : length b c = length e f
bc : ¬b = c
ef : e ≠ f
aL : ¬online a L
w✝ : line
hM : online a w✝ ∧ para L w✝
⊢ area a b c = area a e f
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_I_extras.lean | eq_area_of_eq_base_samevertex | [379, 1] | [398, 59] | have ef: e = f := (same_len_pts_coincide_iff hlen).mp bc | case pos
i : incidence_geometry
a b c e f : point
L : line
bL : online b L
cL : online c L
eL : online e L
fL : online f L
hlen : length b c = length e f
bc : b = c
⊢ area a b c = area a e f | case pos
i : incidence_geometry
a b c e f : point
L : line
bL : online b L
cL : online c L
eL : online e L
fL : online f L
hlen : length b c = length e f
bc : b = c
ef : e = f
⊢ area a b c = area a e f | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
i : incidence_geometry
a b c e f : point
L : line
bL : online b L
cL : online c L
eL : online e L
fL : online f L
hlen : length b c = length e f
bc : b = c
⊢ area a b c = area a e f
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_I_extras.lean | eq_area_of_eq_base_samevertex | [379, 1] | [398, 59] | rw [area_of_eq a b c (by tauto), area_of_eq a e f (by tauto)] | case pos
i : incidence_geometry
a b c e f : point
L : line
bL : online b L
cL : online c L
eL : online e L
fL : online f L
hlen : length b c = length e f
bc : b = c
ef : e = f
⊢ area a b c = area a e f | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
i : incidence_geometry
a b c e f : point
L : line
bL : online b L
cL : online c L
eL : online e L
fL : online f L
hlen : length b c = length e f
bc : b = c
ef : e = f
⊢ area a b c = area a e f
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_I_extras.lean | eq_area_of_eq_base_samevertex | [379, 1] | [398, 59] | tauto | i : incidence_geometry
a b c e f : point
L : line
bL : online b L
cL : online c L
eL : online e L
fL : online f L
hlen : length b c = length e f
bc : b = c
ef : e = f
⊢ 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 e f : point
L : line
bL : online b L
cL : online c L
eL : online e L
fL : online f L
hlen : length b c = length e f
bc : b = c
ef : e = f
⊢ a = b ∨ a = c ∨ b = c
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_I_extras.lean | eq_area_of_eq_base_samevertex | [379, 1] | [398, 59] | tauto | i : incidence_geometry
a b c e f : point
L : line
bL : online b L
cL : online c L
eL : online e L
fL : online f L
hlen : length b c = length e f
bc : b = c
ef : e = f
⊢ a = e ∨ a = f ∨ e = f | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
i : incidence_geometry
a b c e f : point
L : line
bL : online b L
cL : online c L
eL : online e L
fL : online f L
hlen : length b c = length e f
bc : b = c
ef : e = f
⊢ a = e ∨ a = f ∨ e = f
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_I_extras.lean | eq_area_of_eq_base_samevertex | [379, 1] | [398, 59] | perm [(area_zero_iff_online bc bL cL).mpr aL] | case pos
i : incidence_geometry
a b c e f : point
L : line
bL : online b L
cL : online c L
eL : online e L
fL : online f L
hlen : length b c = length e f
bc : ¬b = c
ef : e ≠ f
aL : online a L
⊢ area a b c = area a e f | case pos
i : incidence_geometry
a b c e f : point
L : line
bL : online b L
cL : online c L
eL : online e L
fL : online f L
hlen : length b c = length e f
bc : ¬b = c
ef : e ≠ f
aL : online a L
this : area a b c = 0
⊢ area a b c = area a e f | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
i : incidence_geometry
a b c e f : point
L : line
bL : online b L
cL : online c L
eL : online e L
fL : online f L
hlen : length b c = length e f
bc : ¬b = c
ef : e ≠ f
aL : online a L
⊢ area a b c = area a e f
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_I_extras.lean | eq_area_of_eq_base_samevertex | [379, 1] | [398, 59] | rw [this] | case pos
i : incidence_geometry
a b c e f : point
L : line
bL : online b L
cL : online c L
eL : online e L
fL : online f L
hlen : length b c = length e f
bc : ¬b = c
ef : e ≠ f
aL : online a L
this : area a b c = 0
⊢ area a b c = area a e f | case pos
i : incidence_geometry
a b c e f : point
L : line
bL : online b L
cL : online c L
eL : online e L
fL : online f L
hlen : length b c = length e f
bc : ¬b = c
ef : e ≠ f
aL : online a L
this : area a b c = 0
⊢ 0 = area a e f | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
i : incidence_geometry
a b c e f : point
L : line
bL : online b L
cL : online c L
eL : online e L
fL : online f L
hlen : length b c = length e f
bc : ¬b = c
ef : e ≠ f
aL : online a L
this : area a b c = 0
⊢ area a b c = area a e f
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_I_extras.lean | eq_area_of_eq_base_samevertex | [379, 1] | [398, 59] | linperm [(area_zero_iff_online ef eL fL).mpr aL] | case pos
i : incidence_geometry
a b c e f : point
L : line
bL : online b L
cL : online c L
eL : online e L
fL : online f L
hlen : length b c = length e f
bc : ¬b = c
ef : e ≠ f
aL : online a L
this : area a b c = 0
⊢ 0 = area a e f | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
i : incidence_geometry
a b c e f : point
L : line
bL : online b L
cL : online c L
eL : online e L
fL : online f L
hlen : length b c = length e f
bc : ¬b = c
ef : e ≠ f
aL : online a L
this : area a b c = 0
⊢ 0 = area a e f
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_I_extras.lean | para_implies_eq_area_of_same_base | [403, 1] | [407, 85] | exact eq_area_of_eq_base aM bL cL dM bL cL pLM rfl | i : incidence_geometry
a b c d : point
L M : line
aM : online a M
dM : online d M
bL : online b L
cL : online c L
pLM : para L M
⊢ area a b c = area d b c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
i : incidence_geometry
a b c d : point
L M : line
aM : online a M
dM : online d M
bL : online b L
cL : online c L
pLM : para L M
⊢ area a b c = area d b c
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_I_extras.lean | tri_sum_contra | [409, 1] | [420, 73] | apply cO | i : incidence_geometry
b c d e : point
O : line
bO : online b O
dO : online d O
eO : online e O
cO : ¬online c O
bd : b ≠ d
de : d ≠ e
eb : e ≠ b
Bbed : B b e d
ar : area b c d = area b c e
⊢ False | i : incidence_geometry
b c d e : point
O : line
bO : online b O
dO : online d O
eO : online e O
cO : ¬online c O
bd : b ≠ d
de : d ≠ e
eb : e ≠ b
Bbed : B b e d
ar : area b c d = area b c e
⊢ online c O | Please generate a tactic in lean4 to solve the state.
STATE:
i : incidence_geometry
b c d e : point
O : line
bO : online b O
dO : online d O
eO : online e O
cO : ¬online c O
bd : b ≠ d
de : d ≠ e
eb : e ≠ b
Bbed : B b e d
ar : area b c d = area b c e
⊢ False
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_I_extras.lean | tri_sum_contra | [409, 1] | [420, 73] | apply (area_zero_iff_online de dO eO).mp | i : incidence_geometry
b c d e : point
O : line
bO : online b O
dO : online d O
eO : online e O
cO : ¬online c O
bd : b ≠ d
de : d ≠ e
eb : e ≠ b
Bbed : B b e d
ar : area b c d = area b c e
⊢ online c O | i : incidence_geometry
b c d e : point
O : line
bO : online b O
dO : online d O
eO : online e O
cO : ¬online c O
bd : b ≠ d
de : d ≠ e
eb : e ≠ b
Bbed : B b e d
ar : area b c d = area b c e
⊢ area d e c = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
i : incidence_geometry
b c d e : point
O : line
bO : online b O
dO : online d O
eO : online e O
cO : ¬online c O
bd : b ≠ d
de : d ≠ e
eb : e ≠ b
Bbed : B b e d
ar : area b c d = area b c e
⊢ online c O
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_I_extras.lean | tri_sum_contra | [409, 1] | [420, 73] | linperm [(area_add_iff_B eb.symm de.symm bd.symm bO eO dO cO).mp Bbed] | i : incidence_geometry
b c d e : point
O : line
bO : online b O
dO : online d O
eO : online e O
cO : ¬online c O
bd : b ≠ d
de : d ≠ e
eb : e ≠ b
Bbed : B b e d
ar : area b c d = area b c e
⊢ area d e c = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
i : incidence_geometry
b c d e : point
O : line
bO : online b O
dO : online d O
eO : online e O
cO : ¬online c O
bd : b ≠ d
de : d ≠ e
eb : e ≠ b
Bbed : B b e d
ar : area b c d = area b c e
⊢ area d e c = 0
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_I_extras.lean | eq_area_of_same_base_implies_para | [425, 3] | [461, 83] | obtain ⟨N, aN, pNL⟩ := para_of_offline aL | i : incidence_geometry
a b c d : point
L M O : line
bL : online b L
cL : online c L
aL : ¬online a L
aM : online a M
dM : online d M
bO : online b O
dO : online d O
cO : ¬online c O
ad : a ≠ d
bd : b ≠ d
ssadL : sameside a d L
ar : area a b c = area d b c
⊢ para L M | case intro.intro
i : incidence_geometry
a b c d : point
L M O : line
bL : online b L
cL : online c L
aL : ¬online a L
aM : online a M
dM : online d M
bO : online b O
dO : online d O
cO : ¬online c O
ad : a ≠ d
bd : b ≠ d
ssadL : sameside a d L
ar : area a b c = area d b c
N : line
aN : online a N
pNL : para L N
⊢ para L M | Please generate a tactic in lean4 to solve the state.
STATE:
i : incidence_geometry
a b c d : point
L M O : line
bL : online b L
cL : online c L
aL : ¬online a L
aM : online a M
dM : online d M
bO : online b O
dO : online d O
cO : ¬online c O
ad : a ≠ d
bd : b ≠ d
ssadL : sameside a d L
ar : area a b c = area d b c
⊢ para L M
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_I_extras.lean | eq_area_of_same_base_implies_para | [425, 3] | [461, 83] | have pNO: ¬ para N O := fun pNO => by
have LO_or_pLO := para_trans (para_symm pNL) pNO
cases LO_or_pLO with
| inl LO => rw [← LO] at cO; exact cO cL
| inr pLO => exact neq_of_para bL bO pLO rfl | case intro.intro
i : incidence_geometry
a b c d : point
L M O : line
bL : online b L
cL : online c L
aL : ¬online a L
aM : online a M
dM : online d M
bO : online b O
dO : online d O
cO : ¬online c O
ad : a ≠ d
bd : b ≠ d
ssadL : sameside a d L
ar : area a b c = area d b c
N : line
aN : online a N
pNL : para L N
⊢ para L M | case intro.intro
i : incidence_geometry
a b c d : point
L M O : line
bL : online b L
cL : online c L
aL : ¬online a L
aM : online a M
dM : online d M
bO : online b O
dO : online d O
cO : ¬online c O
ad : a ≠ d
bd : b ≠ d
ssadL : sameside a d L
ar : area a b c = area d b c
N : line
aN : online a N
pNL : para L N
pNO : ¬para N O
⊢ para L M | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
i : incidence_geometry
a b c d : point
L M O : line
bL : online b L
cL : online c L
aL : ¬online a L
aM : online a M
dM : online d M
bO : online b O
dO : online d O
cO : ¬online c O
ad : a ≠ d
bd : b ≠ d
ssadL : sameside a d L
ar : area a b c = area d b c
N : line
aN : online a N
pNL : para L N
⊢ para L M
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_I_extras.lean | eq_area_of_same_base_implies_para | [425, 3] | [461, 83] | rcases pt_of_line_line pNO with ⟨e, eN, eO⟩ | case intro.intro
i : incidence_geometry
a b c d : point
L M O : line
bL : online b L
cL : online c L
aL : ¬online a L
aM : online a M
dM : online d M
bO : online b O
dO : online d O
cO : ¬online c O
ad : a ≠ d
bd : b ≠ d
ssadL : sameside a d L
ar : area a b c = area d b c
N : line
aN : online a N
pNL : para L N
pNO : ¬para N O
⊢ para L M | case intro.intro.intro.intro
i : incidence_geometry
a b c d : point
L M O : line
bL : online b L
cL : online c L
aL : ¬online a L
aM : online a M
dM : online d M
bO : online b O
dO : online d O
cO : ¬online c O
ad : a ≠ d
bd : b ≠ d
ssadL : sameside a d L
ar : area a b c = area d b c
N : line
aN : online a N
pNL : para L N
pNO : ¬para N O
e : point
eN : online e N
eO : online e O
⊢ para L M | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
i : incidence_geometry
a b c d : point
L M O : line
bL : online b L
cL : online c L
aL : ¬online a L
aM : online a M
dM : online d M
bO : online b O
dO : online d O
cO : ¬online c O
ad : a ≠ d
bd : b ≠ d
ssadL : sameside a d L
ar : area a b c = area d b c
N : line
aN : online a N
pNL : para L N
pNO : ¬para N O
⊢ para L M
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_I_extras.lean | eq_area_of_same_base_implies_para | [425, 3] | [461, 83] | have ar2 := eq_area_of_eq_base aN bL cL eN bL cL pNL rfl | case intro.intro.intro.intro
i : incidence_geometry
a b c d : point
L M O : line
bL : online b L
cL : online c L
aL : ¬online a L
aM : online a M
dM : online d M
bO : online b O
dO : online d O
cO : ¬online c O
ad : a ≠ d
bd : b ≠ d
ssadL : sameside a d L
ar : area a b c = area d b c
N : line
aN : online a N
pNL : para L N
pNO : ¬para N O
e : point
eN : online e N
eO : online e O
⊢ para L M | case intro.intro.intro.intro
i : incidence_geometry
a b c d : point
L M O : line
bL : online b L
cL : online c L
aL : ¬online a L
aM : online a M
dM : online d M
bO : online b O
dO : online d O
cO : ¬online c O
ad : a ≠ d
bd : b ≠ d
ssadL : sameside a d L
ar : area a b c = area d b c
N : line
aN : online a N
pNL : para L N
pNO : ¬para N O
e : point
eN : online e N
eO : online e O
ar2 : area a b c = area e b c
⊢ para L M | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
i : incidence_geometry
a b c d : point
L M O : line
bL : online b L
cL : online c L
aL : ¬online a L
aM : online a M
dM : online d M
bO : online b O
dO : online d O
cO : ¬online c O
ad : a ≠ d
bd : b ≠ d
ssadL : sameside a d L
ar : area a b c = area d b c
N : line
aN : online a N
pNL : para L N
pNO : ¬para N O
e : point
eN : online e N
eO : online e O
⊢ para L M
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_I_extras.lean | eq_area_of_same_base_implies_para | [425, 3] | [461, 83] | rw [ar] at ar2 | case intro.intro.intro.intro
i : incidence_geometry
a b c d : point
L M O : line
bL : online b L
cL : online c L
aL : ¬online a L
aM : online a M
dM : online d M
bO : online b O
dO : online d O
cO : ¬online c O
ad : a ≠ d
bd : b ≠ d
ssadL : sameside a d L
ar : area a b c = area d b c
N : line
aN : online a N
pNL : para L N
pNO : ¬para N O
e : point
eN : online e N
eO : online e O
ar2 : area a b c = area e b c
⊢ para L M | case intro.intro.intro.intro
i : incidence_geometry
a b c d : point
L M O : line
bL : online b L
cL : online c L
aL : ¬online a L
aM : online a M
dM : online d M
bO : online b O
dO : online d O
cO : ¬online c O
ad : a ≠ d
bd : b ≠ d
ssadL : sameside a d L
ar : area a b c = area d b c
N : line
aN : online a N
pNL : para L N
pNO : ¬para N O
e : point
eN : online e N
eO : online e O
ar2 : area d b c = area e b c
⊢ para L M | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
i : incidence_geometry
a b c d : point
L M O : line
bL : online b L
cL : online c L
aL : ¬online a L
aM : online a M
dM : online d M
bO : online b O
dO : online d O
cO : ¬online c O
ad : a ≠ d
bd : b ≠ d
ssadL : sameside a d L
ar : area a b c = area d b c
N : line
aN : online a N
pNL : para L N
pNO : ¬para N O
e : point
eN : online e N
eO : online e O
ar2 : area a b c = area e b c
⊢ para L M
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_I_extras.lean | eq_area_of_same_base_implies_para | [425, 3] | [461, 83] | have be := neq_of_para bL eN pNL | case intro.intro.intro.intro
i : incidence_geometry
a b c d : point
L M O : line
bL : online b L
cL : online c L
aL : ¬online a L
aM : online a M
dM : online d M
bO : online b O
dO : online d O
cO : ¬online c O
ad : a ≠ d
bd : b ≠ d
ssadL : sameside a d L
ar : area a b c = area d b c
N : line
aN : online a N
pNL : para L N
pNO : ¬para N O
e : point
eN : online e N
eO : online e O
ar2 : area d b c = area e b c
⊢ para L M | case intro.intro.intro.intro
i : incidence_geometry
a b c d : point
L M O : line
bL : online b L
cL : online c L
aL : ¬online a L
aM : online a M
dM : online d M
bO : online b O
dO : online d O
cO : ¬online c O
ad : a ≠ d
bd : b ≠ d
ssadL : sameside a d L
ar : area a b c = area d b c
N : line
aN : online a N
pNL : para L N
pNO : ¬para N O
e : point
eN : online e N
eO : online e O
ar2 : area d b c = area e b c
be : b ≠ e
⊢ para L M | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
i : incidence_geometry
a b c d : point
L M O : line
bL : online b L
cL : online c L
aL : ¬online a L
aM : online a M
dM : online d M
bO : online b O
dO : online d O
cO : ¬online c O
ad : a ≠ d
bd : b ≠ d
ssadL : sameside a d L
ar : area a b c = area d b c
N : line
aN : online a N
pNL : para L N
pNO : ¬para N O
e : point
eN : online e N
eO : online e O
ar2 : area d b c = area e b c
⊢ para L M
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_I_extras.lean | eq_area_of_same_base_implies_para | [425, 3] | [461, 83] | by_cases de: d = e | case intro.intro.intro.intro
i : incidence_geometry
a b c d : point
L M O : line
bL : online b L
cL : online c L
aL : ¬online a L
aM : online a M
dM : online d M
bO : online b O
dO : online d O
cO : ¬online c O
ad : a ≠ d
bd : b ≠ d
ssadL : sameside a d L
ar : area a b c = area d b c
N : line
aN : online a N
pNL : para L N
pNO : ¬para N O
e : point
eN : online e N
eO : online e O
ar2 : area d b c = area e b c
be : b ≠ e
⊢ para L M | case pos
i : incidence_geometry
a b c d : point
L M O : line
bL : online b L
cL : online c L
aL : ¬online a L
aM : online a M
dM : online d M
bO : online b O
dO : online d O
cO : ¬online c O
ad : a ≠ d
bd : b ≠ d
ssadL : sameside a d L
ar : area a b c = area d b c
N : line
aN : online a N
pNL : para L N
pNO : ¬para N O
e : point
eN : online e N
eO : online e O
ar2 : area d b c = area e b c
be : b ≠ e
de : d = e
⊢ para L M
case neg
i : incidence_geometry
a b c d : point
L M O : line
bL : online b L
cL : online c L
aL : ¬online a L
aM : online a M
dM : online d M
bO : online b O
dO : online d O
cO : ¬online c O
ad : a ≠ d
bd : b ≠ d
ssadL : sameside a d L
ar : area a b c = area d b c
N : line
aN : online a N
pNL : para L N
pNO : ¬para N O
e : point
eN : online e N
eO : online e O
ar2 : area d b c = area e b c
be : b ≠ e
de : ¬d = e
⊢ para L M | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
i : incidence_geometry
a b c d : point
L M O : line
bL : online b L
cL : online c L
aL : ¬online a L
aM : online a M
dM : online d M
bO : online b O
dO : online d O
cO : ¬online c O
ad : a ≠ d
bd : b ≠ d
ssadL : sameside a d L
ar : area a b c = area d b c
N : line
aN : online a N
pNL : para L N
pNO : ¬para N O
e : point
eN : online e N
eO : online e O
ar2 : area d b c = area e b c
be : b ≠ e
⊢ para L M
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_I_extras.lean | eq_area_of_same_base_implies_para | [425, 3] | [461, 83] | have LO_or_pLO := para_trans (para_symm pNL) pNO | i : incidence_geometry
a b c d : point
L M O : line
bL : online b L
cL : online c L
aL : ¬online a L
aM : online a M
dM : online d M
bO : online b O
dO : online d O
cO : ¬online c O
ad : a ≠ d
bd : b ≠ d
ssadL : sameside a d L
ar : area a b c = area d b c
N : line
aN : online a N
pNL : para L N
pNO : para N O
⊢ False | i : incidence_geometry
a b c d : point
L M O : line
bL : online b L
cL : online c L
aL : ¬online a L
aM : online a M
dM : online d M
bO : online b O
dO : online d O
cO : ¬online c O
ad : a ≠ d
bd : b ≠ d
ssadL : sameside a d L
ar : area a b c = area d b c
N : line
aN : online a N
pNL : para L N
pNO : para N O
LO_or_pLO : L = O ∨ para L O
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
i : incidence_geometry
a b c d : point
L M O : line
bL : online b L
cL : online c L
aL : ¬online a L
aM : online a M
dM : online d M
bO : online b O
dO : online d O
cO : ¬online c O
ad : a ≠ d
bd : b ≠ d
ssadL : sameside a d L
ar : area a b c = area d b c
N : line
aN : online a N
pNL : para L N
pNO : para N O
⊢ False
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_I_extras.lean | eq_area_of_same_base_implies_para | [425, 3] | [461, 83] | cases LO_or_pLO with
| inl LO => rw [← LO] at cO; exact cO cL
| inr pLO => exact neq_of_para bL bO pLO rfl | i : incidence_geometry
a b c d : point
L M O : line
bL : online b L
cL : online c L
aL : ¬online a L
aM : online a M
dM : online d M
bO : online b O
dO : online d O
cO : ¬online c O
ad : a ≠ d
bd : b ≠ d
ssadL : sameside a d L
ar : area a b c = area d b c
N : line
aN : online a N
pNL : para L N
pNO : para N O
LO_or_pLO : L = O ∨ para L O
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
i : incidence_geometry
a b c d : point
L M O : line
bL : online b L
cL : online c L
aL : ¬online a L
aM : online a M
dM : online d M
bO : online b O
dO : online d O
cO : ¬online c O
ad : a ≠ d
bd : b ≠ d
ssadL : sameside a d L
ar : area a b c = area d b c
N : line
aN : online a N
pNL : para L N
pNO : para N O
LO_or_pLO : L = O ∨ para L O
⊢ False
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_I_extras.lean | eq_area_of_same_base_implies_para | [425, 3] | [461, 83] | rw [← LO] at cO | case inl
i : incidence_geometry
a b c d : point
L M O : line
bL : online b L
cL : online c L
aL : ¬online a L
aM : online a M
dM : online d M
bO : online b O
dO : online d O
cO : ¬online c O
ad : a ≠ d
bd : b ≠ d
ssadL : sameside a d L
ar : area a b c = area d b c
N : line
aN : online a N
pNL : para L N
pNO : para N O
LO : L = O
⊢ False | case inl
i : incidence_geometry
a b c d : point
L M O : line
bL : online b L
cL : online c L
aL : ¬online a L
aM : online a M
dM : online d M
bO : online b O
dO : online d O
cO : ¬online c L
ad : a ≠ d
bd : b ≠ d
ssadL : sameside a d L
ar : area a b c = area d b c
N : line
aN : online a N
pNL : para L N
pNO : para N O
LO : L = O
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
i : incidence_geometry
a b c d : point
L M O : line
bL : online b L
cL : online c L
aL : ¬online a L
aM : online a M
dM : online d M
bO : online b O
dO : online d O
cO : ¬online c O
ad : a ≠ d
bd : b ≠ d
ssadL : sameside a d L
ar : area a b c = area d b c
N : line
aN : online a N
pNL : para L N
pNO : para N O
LO : L = O
⊢ False
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_I_extras.lean | eq_area_of_same_base_implies_para | [425, 3] | [461, 83] | exact cO cL | case inl
i : incidence_geometry
a b c d : point
L M O : line
bL : online b L
cL : online c L
aL : ¬online a L
aM : online a M
dM : online d M
bO : online b O
dO : online d O
cO : ¬online c L
ad : a ≠ d
bd : b ≠ d
ssadL : sameside a d L
ar : area a b c = area d b c
N : line
aN : online a N
pNL : para L N
pNO : para N O
LO : L = O
⊢ 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 M O : line
bL : online b L
cL : online c L
aL : ¬online a L
aM : online a M
dM : online d M
bO : online b O
dO : online d O
cO : ¬online c L
ad : a ≠ d
bd : b ≠ d
ssadL : sameside a d L
ar : area a b c = area d b c
N : line
aN : online a N
pNL : para L N
pNO : para N O
LO : L = O
⊢ False
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_I_extras.lean | eq_area_of_same_base_implies_para | [425, 3] | [461, 83] | exact neq_of_para bL bO pLO rfl | case inr
i : incidence_geometry
a b c d : point
L M O : line
bL : online b L
cL : online c L
aL : ¬online a L
aM : online a M
dM : online d M
bO : online b O
dO : online d O
cO : ¬online c O
ad : a ≠ d
bd : b ≠ d
ssadL : sameside a d L
ar : area a b c = area d b c
N : line
aN : online a N
pNL : para L N
pNO : para N O
pLO : para L O
⊢ 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 M O : line
bL : online b L
cL : online c L
aL : ¬online a L
aM : online a M
dM : online d M
bO : online b O
dO : online d O
cO : ¬online c O
ad : a ≠ d
bd : b ≠ d
ssadL : sameside a d L
ar : area a b c = area d b c
N : line
aN : online a N
pNL : para L N
pNO : para N O
pLO : para L O
⊢ False
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_I_extras.lean | eq_area_of_same_base_implies_para | [425, 3] | [461, 83] | rw [← de] at eN | case pos
i : incidence_geometry
a b c d : point
L M O : line
bL : online b L
cL : online c L
aL : ¬online a L
aM : online a M
dM : online d M
bO : online b O
dO : online d O
cO : ¬online c O
ad : a ≠ d
bd : b ≠ d
ssadL : sameside a d L
ar : area a b c = area d b c
N : line
aN : online a N
pNL : para L N
pNO : ¬para N O
e : point
eN : online e N
eO : online e O
ar2 : area d b c = area e b c
be : b ≠ e
de : d = e
⊢ para L M | case pos
i : incidence_geometry
a b c d : point
L M O : line
bL : online b L
cL : online c L
aL : ¬online a L
aM : online a M
dM : online d M
bO : online b O
dO : online d O
cO : ¬online c O
ad : a ≠ d
bd : b ≠ d
ssadL : sameside a d L
ar : area a b c = area d b c
N : line
aN : online a N
pNL : para L N
pNO : ¬para N O
e : point
eN : online d N
eO : online e O
ar2 : area d b c = area e b c
be : b ≠ e
de : d = e
⊢ para L M | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
i : incidence_geometry
a b c d : point
L M O : line
bL : online b L
cL : online c L
aL : ¬online a L
aM : online a M
dM : online d M
bO : online b O
dO : online d O
cO : ¬online c O
ad : a ≠ d
bd : b ≠ d
ssadL : sameside a d L
ar : area a b c = area d b c
N : line
aN : online a N
pNL : para L N
pNO : ¬para N O
e : point
eN : online e N
eO : online e O
ar2 : area d b c = area e b c
be : b ≠ e
de : d = e
⊢ para L M
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_I_extras.lean | eq_area_of_same_base_implies_para | [425, 3] | [461, 83] | rwa [line_unique_of_pts ad aM dM aN eN] | case pos
i : incidence_geometry
a b c d : point
L M O : line
bL : online b L
cL : online c L
aL : ¬online a L
aM : online a M
dM : online d M
bO : online b O
dO : online d O
cO : ¬online c O
ad : a ≠ d
bd : b ≠ d
ssadL : sameside a d L
ar : area a b c = area d b c
N : line
aN : online a N
pNL : para L N
pNO : ¬para N O
e : point
eN : online d N
eO : online e O
ar2 : area d b c = area e b c
be : b ≠ e
de : d = e
⊢ para L M | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
i : incidence_geometry
a b c d : point
L M O : line
bL : online b L
cL : online c L
aL : ¬online a L
aM : online a M
dM : online d M
bO : online b O
dO : online d O
cO : ¬online c O
ad : a ≠ d
bd : b ≠ d
ssadL : sameside a d L
ar : area a b c = area d b c
N : line
aN : online a N
pNL : para L N
pNO : ¬para N O
e : point
eN : online d N
eO : online e O
ar2 : area d b c = area e b c
be : b ≠ e
de : d = e
⊢ para L M
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_I_extras.lean | eq_area_of_same_base_implies_para | [425, 3] | [461, 83] | rw [← Ne.def d e] at de | case neg
i : incidence_geometry
a b c d : point
L M O : line
bL : online b L
cL : online c L
aL : ¬online a L
aM : online a M
dM : online d M
bO : online b O
dO : online d O
cO : ¬online c O
ad : a ≠ d
bd : b ≠ d
ssadL : sameside a d L
ar : area a b c = area d b c
N : line
aN : online a N
pNL : para L N
pNO : ¬para N O
e : point
eN : online e N
eO : online e O
ar2 : area d b c = area e b c
be : b ≠ e
de : ¬d = e
⊢ para L M | case neg
i : incidence_geometry
a b c d : point
L M O : line
bL : online b L
cL : online c L
aL : ¬online a L
aM : online a M
dM : online d M
bO : online b O
dO : online d O
cO : ¬online c O
ad : a ≠ d
bd : b ≠ d
ssadL : sameside a d L
ar : area a b c = area d b c
N : line
aN : online a N
pNL : para L N
pNO : ¬para N O
e : point
eN : online e N
eO : online e O
ar2 : area d b c = area e b c
be : b ≠ e
de : d ≠ e
⊢ para L M | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
i : incidence_geometry
a b c d : point
L M O : line
bL : online b L
cL : online c L
aL : ¬online a L
aM : online a M
dM : online d M
bO : online b O
dO : online d O
cO : ¬online c O
ad : a ≠ d
bd : b ≠ d
ssadL : sameside a d L
ar : area a b c = area d b c
N : line
aN : online a N
pNL : para L N
pNO : ¬para N O
e : point
eN : online e N
eO : online e O
ar2 : area d b c = area e b c
be : b ≠ e
de : ¬d = e
⊢ para L M
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_I_extras.lean | eq_area_of_same_base_implies_para | [425, 3] | [461, 83] | exfalso | case neg
i : incidence_geometry
a b c d : point
L M O : line
bL : online b L
cL : online c L
aL : ¬online a L
aM : online a M
dM : online d M
bO : online b O
dO : online d O
cO : ¬online c O
ad : a ≠ d
bd : b ≠ d
ssadL : sameside a d L
ar : area a b c = area d b c
N : line
aN : online a N
pNL : para L N
pNO : ¬para N O
e : point
eN : online e N
eO : online e O
ar2 : area d b c = area e b c
be : b ≠ e
de : d ≠ e
⊢ para L M | case neg.h
i : incidence_geometry
a b c d : point
L M O : line
bL : online b L
cL : online c L
aL : ¬online a L
aM : online a M
dM : online d M
bO : online b O
dO : online d O
cO : ¬online c O
ad : a ≠ d
bd : b ≠ d
ssadL : sameside a d L
ar : area a b c = area d b c
N : line
aN : online a N
pNL : para L N
pNO : ¬para N O
e : point
eN : online e N
eO : online e O
ar2 : area d b c = area e b c
be : b ≠ e
de : d ≠ e
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
i : incidence_geometry
a b c d : point
L M O : line
bL : online b L
cL : online c L
aL : ¬online a L
aM : online a M
dM : online d M
bO : online b O
dO : online d O
cO : ¬online c O
ad : a ≠ d
bd : b ≠ d
ssadL : sameside a d L
ar : area a b c = area d b c
N : line
aN : online a N
pNL : para L N
pNO : ¬para N O
e : point
eN : online e N
eO : online e O
ar2 : area d b c = area e b c
be : b ≠ e
de : d ≠ e
⊢ para L M
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_I_extras.lean | eq_area_of_same_base_implies_para | [425, 3] | [461, 83] | cases B_of_three_online_ne be bd de.symm bO eO dO with
| inl Bbed =>
exact tri_sum_contra bO dO eO cO bd de be.symm Bbed (by perma at ar2)
| inr HB =>
cases HB with
| inl Bebd =>
have ssaeL := sameside_of_para_online aN eN (para_symm pNL)
have ssedL := sameside_symm (sameside_trans ssadL ssaeL)
have dsedL:= not_sameside13_of_B123_online2 Bebd bL
exact dsedL ssedL
| inr Bbde =>
exact tri_sum_contra bO eO dO cO be de.symm bd.symm Bbde (by perma at ar2) | case neg.h
i : incidence_geometry
a b c d : point
L M O : line
bL : online b L
cL : online c L
aL : ¬online a L
aM : online a M
dM : online d M
bO : online b O
dO : online d O
cO : ¬online c O
ad : a ≠ d
bd : b ≠ d
ssadL : sameside a d L
ar : area a b c = area d b c
N : line
aN : online a N
pNL : para L N
pNO : ¬para N O
e : point
eN : online e N
eO : online e O
ar2 : area d b c = area e b c
be : b ≠ e
de : d ≠ e
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.h
i : incidence_geometry
a b c d : point
L M O : line
bL : online b L
cL : online c L
aL : ¬online a L
aM : online a M
dM : online d M
bO : online b O
dO : online d O
cO : ¬online c O
ad : a ≠ d
bd : b ≠ d
ssadL : sameside a d L
ar : area a b c = area d b c
N : line
aN : online a N
pNL : para L N
pNO : ¬para N O
e : point
eN : online e N
eO : online e O
ar2 : area d b c = area e b c
be : b ≠ e
de : d ≠ e
⊢ False
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_I_extras.lean | eq_area_of_same_base_implies_para | [425, 3] | [461, 83] | exact tri_sum_contra bO dO eO cO bd de be.symm Bbed (by perma at ar2) | case neg.h.inl
i : incidence_geometry
a b c d : point
L M O : line
bL : online b L
cL : online c L
aL : ¬online a L
aM : online a M
dM : online d M
bO : online b O
dO : online d O
cO : ¬online c O
ad : a ≠ d
bd : b ≠ d
ssadL : sameside a d L
ar : area a b c = area d b c
N : line
aN : online a N
pNL : para L N
pNO : ¬para N O
e : point
eN : online e N
eO : online e O
ar2 : area d b c = area e b c
be : b ≠ e
de : d ≠ e
Bbed : B b e d
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.h.inl
i : incidence_geometry
a b c d : point
L M O : line
bL : online b L
cL : online c L
aL : ¬online a L
aM : online a M
dM : online d M
bO : online b O
dO : online d O
cO : ¬online c O
ad : a ≠ d
bd : b ≠ d
ssadL : sameside a d L
ar : area a b c = area d b c
N : line
aN : online a N
pNL : para L N
pNO : ¬para N O
e : point
eN : online e N
eO : online e O
ar2 : area d b c = area e b c
be : b ≠ e
de : d ≠ e
Bbed : B b e d
⊢ False
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_I_extras.lean | eq_area_of_same_base_implies_para | [425, 3] | [461, 83] | perma at ar2 | i : incidence_geometry
a b c d : point
L M O : line
bL : online b L
cL : online c L
aL : ¬online a L
aM : online a M
dM : online d M
bO : online b O
dO : online d O
cO : ¬online c O
ad : a ≠ d
bd : b ≠ d
ssadL : sameside a d L
ar : area a b c = area d b c
N : line
aN : online a N
pNL : para L N
pNO : ¬para N O
e : point
eN : online e N
eO : online e O
ar2 : area d b c = area e b c
be : b ≠ e
de : d ≠ e
Bbed : B b e d
⊢ area b c d = area b c e | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
i : incidence_geometry
a b c d : point
L M O : line
bL : online b L
cL : online c L
aL : ¬online a L
aM : online a M
dM : online d M
bO : online b O
dO : online d O
cO : ¬online c O
ad : a ≠ d
bd : b ≠ d
ssadL : sameside a d L
ar : area a b c = area d b c
N : line
aN : online a N
pNL : para L N
pNO : ¬para N O
e : point
eN : online e N
eO : online e O
ar2 : area d b c = area e b c
be : b ≠ e
de : d ≠ e
Bbed : B b e d
⊢ area b c d = area b c e
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_I_extras.lean | eq_area_of_same_base_implies_para | [425, 3] | [461, 83] | cases HB with
| inl Bebd =>
have ssaeL := sameside_of_para_online aN eN (para_symm pNL)
have ssedL := sameside_symm (sameside_trans ssadL ssaeL)
have dsedL:= not_sameside13_of_B123_online2 Bebd bL
exact dsedL ssedL
| inr Bbde =>
exact tri_sum_contra bO eO dO cO be de.symm bd.symm Bbde (by perma at ar2) | case neg.h.inr
i : incidence_geometry
a b c d : point
L M O : line
bL : online b L
cL : online c L
aL : ¬online a L
aM : online a M
dM : online d M
bO : online b O
dO : online d O
cO : ¬online c O
ad : a ≠ d
bd : b ≠ d
ssadL : sameside a d L
ar : area a b c = area d b c
N : line
aN : online a N
pNL : para L N
pNO : ¬para N O
e : point
eN : online e N
eO : online e O
ar2 : area d b c = area e b c
be : b ≠ e
de : d ≠ e
HB : B e b d ∨ B b d e
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.h.inr
i : incidence_geometry
a b c d : point
L M O : line
bL : online b L
cL : online c L
aL : ¬online a L
aM : online a M
dM : online d M
bO : online b O
dO : online d O
cO : ¬online c O
ad : a ≠ d
bd : b ≠ d
ssadL : sameside a d L
ar : area a b c = area d b c
N : line
aN : online a N
pNL : para L N
pNO : ¬para N O
e : point
eN : online e N
eO : online e O
ar2 : area d b c = area e b c
be : b ≠ e
de : d ≠ e
HB : B e b d ∨ B b d e
⊢ False
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_I_extras.lean | eq_area_of_same_base_implies_para | [425, 3] | [461, 83] | have ssaeL := sameside_of_para_online aN eN (para_symm pNL) | case neg.h.inr.inl
i : incidence_geometry
a b c d : point
L M O : line
bL : online b L
cL : online c L
aL : ¬online a L
aM : online a M
dM : online d M
bO : online b O
dO : online d O
cO : ¬online c O
ad : a ≠ d
bd : b ≠ d
ssadL : sameside a d L
ar : area a b c = area d b c
N : line
aN : online a N
pNL : para L N
pNO : ¬para N O
e : point
eN : online e N
eO : online e O
ar2 : area d b c = area e b c
be : b ≠ e
de : d ≠ e
Bebd : B e b d
⊢ False | case neg.h.inr.inl
i : incidence_geometry
a b c d : point
L M O : line
bL : online b L
cL : online c L
aL : ¬online a L
aM : online a M
dM : online d M
bO : online b O
dO : online d O
cO : ¬online c O
ad : a ≠ d
bd : b ≠ d
ssadL : sameside a d L
ar : area a b c = area d b c
N : line
aN : online a N
pNL : para L N
pNO : ¬para N O
e : point
eN : online e N
eO : online e O
ar2 : area d b c = area e b c
be : b ≠ e
de : d ≠ e
Bebd : B e b d
ssaeL : sameside a e L
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.h.inr.inl
i : incidence_geometry
a b c d : point
L M O : line
bL : online b L
cL : online c L
aL : ¬online a L
aM : online a M
dM : online d M
bO : online b O
dO : online d O
cO : ¬online c O
ad : a ≠ d
bd : b ≠ d
ssadL : sameside a d L
ar : area a b c = area d b c
N : line
aN : online a N
pNL : para L N
pNO : ¬para N O
e : point
eN : online e N
eO : online e O
ar2 : area d b c = area e b c
be : b ≠ e
de : d ≠ e
Bebd : B e b d
⊢ False
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_I_extras.lean | eq_area_of_same_base_implies_para | [425, 3] | [461, 83] | have ssedL := sameside_symm (sameside_trans ssadL ssaeL) | case neg.h.inr.inl
i : incidence_geometry
a b c d : point
L M O : line
bL : online b L
cL : online c L
aL : ¬online a L
aM : online a M
dM : online d M
bO : online b O
dO : online d O
cO : ¬online c O
ad : a ≠ d
bd : b ≠ d
ssadL : sameside a d L
ar : area a b c = area d b c
N : line
aN : online a N
pNL : para L N
pNO : ¬para N O
e : point
eN : online e N
eO : online e O
ar2 : area d b c = area e b c
be : b ≠ e
de : d ≠ e
Bebd : B e b d
ssaeL : sameside a e L
⊢ False | case neg.h.inr.inl
i : incidence_geometry
a b c d : point
L M O : line
bL : online b L
cL : online c L
aL : ¬online a L
aM : online a M
dM : online d M
bO : online b O
dO : online d O
cO : ¬online c O
ad : a ≠ d
bd : b ≠ d
ssadL : sameside a d L
ar : area a b c = area d b c
N : line
aN : online a N
pNL : para L N
pNO : ¬para N O
e : point
eN : online e N
eO : online e O
ar2 : area d b c = area e b c
be : b ≠ e
de : d ≠ e
Bebd : B e b d
ssaeL : sameside a e L
ssedL : sameside e d L
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.h.inr.inl
i : incidence_geometry
a b c d : point
L M O : line
bL : online b L
cL : online c L
aL : ¬online a L
aM : online a M
dM : online d M
bO : online b O
dO : online d O
cO : ¬online c O
ad : a ≠ d
bd : b ≠ d
ssadL : sameside a d L
ar : area a b c = area d b c
N : line
aN : online a N
pNL : para L N
pNO : ¬para N O
e : point
eN : online e N
eO : online e O
ar2 : area d b c = area e b c
be : b ≠ e
de : d ≠ e
Bebd : B e b d
ssaeL : sameside a e L
⊢ False
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_I_extras.lean | eq_area_of_same_base_implies_para | [425, 3] | [461, 83] | have dsedL:= not_sameside13_of_B123_online2 Bebd bL | case neg.h.inr.inl
i : incidence_geometry
a b c d : point
L M O : line
bL : online b L
cL : online c L
aL : ¬online a L
aM : online a M
dM : online d M
bO : online b O
dO : online d O
cO : ¬online c O
ad : a ≠ d
bd : b ≠ d
ssadL : sameside a d L
ar : area a b c = area d b c
N : line
aN : online a N
pNL : para L N
pNO : ¬para N O
e : point
eN : online e N
eO : online e O
ar2 : area d b c = area e b c
be : b ≠ e
de : d ≠ e
Bebd : B e b d
ssaeL : sameside a e L
ssedL : sameside e d L
⊢ False | case neg.h.inr.inl
i : incidence_geometry
a b c d : point
L M O : line
bL : online b L
cL : online c L
aL : ¬online a L
aM : online a M
dM : online d M
bO : online b O
dO : online d O
cO : ¬online c O
ad : a ≠ d
bd : b ≠ d
ssadL : sameside a d L
ar : area a b c = area d b c
N : line
aN : online a N
pNL : para L N
pNO : ¬para N O
e : point
eN : online e N
eO : online e O
ar2 : area d b c = area e b c
be : b ≠ e
de : d ≠ e
Bebd : B e b d
ssaeL : sameside a e L
ssedL : sameside e d L
dsedL : ¬sameside e d L
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.h.inr.inl
i : incidence_geometry
a b c d : point
L M O : line
bL : online b L
cL : online c L
aL : ¬online a L
aM : online a M
dM : online d M
bO : online b O
dO : online d O
cO : ¬online c O
ad : a ≠ d
bd : b ≠ d
ssadL : sameside a d L
ar : area a b c = area d b c
N : line
aN : online a N
pNL : para L N
pNO : ¬para N O
e : point
eN : online e N
eO : online e O
ar2 : area d b c = area e b c
be : b ≠ e
de : d ≠ e
Bebd : B e b d
ssaeL : sameside a e L
ssedL : sameside e d L
⊢ False
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_I_extras.lean | eq_area_of_same_base_implies_para | [425, 3] | [461, 83] | exact dsedL ssedL | case neg.h.inr.inl
i : incidence_geometry
a b c d : point
L M O : line
bL : online b L
cL : online c L
aL : ¬online a L
aM : online a M
dM : online d M
bO : online b O
dO : online d O
cO : ¬online c O
ad : a ≠ d
bd : b ≠ d
ssadL : sameside a d L
ar : area a b c = area d b c
N : line
aN : online a N
pNL : para L N
pNO : ¬para N O
e : point
eN : online e N
eO : online e O
ar2 : area d b c = area e b c
be : b ≠ e
de : d ≠ e
Bebd : B e b d
ssaeL : sameside a e L
ssedL : sameside e d L
dsedL : ¬sameside e d L
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.h.inr.inl
i : incidence_geometry
a b c d : point
L M O : line
bL : online b L
cL : online c L
aL : ¬online a L
aM : online a M
dM : online d M
bO : online b O
dO : online d O
cO : ¬online c O
ad : a ≠ d
bd : b ≠ d
ssadL : sameside a d L
ar : area a b c = area d b c
N : line
aN : online a N
pNL : para L N
pNO : ¬para N O
e : point
eN : online e N
eO : online e O
ar2 : area d b c = area e b c
be : b ≠ e
de : d ≠ e
Bebd : B e b d
ssaeL : sameside a e L
ssedL : sameside e d L
dsedL : ¬sameside e d L
⊢ False
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_I_extras.lean | eq_area_of_same_base_implies_para | [425, 3] | [461, 83] | exact tri_sum_contra bO eO dO cO be de.symm bd.symm Bbde (by perma at ar2) | case neg.h.inr.inr
i : incidence_geometry
a b c d : point
L M O : line
bL : online b L
cL : online c L
aL : ¬online a L
aM : online a M
dM : online d M
bO : online b O
dO : online d O
cO : ¬online c O
ad : a ≠ d
bd : b ≠ d
ssadL : sameside a d L
ar : area a b c = area d b c
N : line
aN : online a N
pNL : para L N
pNO : ¬para N O
e : point
eN : online e N
eO : online e O
ar2 : area d b c = area e b c
be : b ≠ e
de : d ≠ e
Bbde : B b d e
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.h.inr.inr
i : incidence_geometry
a b c d : point
L M O : line
bL : online b L
cL : online c L
aL : ¬online a L
aM : online a M
dM : online d M
bO : online b O
dO : online d O
cO : ¬online c O
ad : a ≠ d
bd : b ≠ d
ssadL : sameside a d L
ar : area a b c = area d b c
N : line
aN : online a N
pNL : para L N
pNO : ¬para N O
e : point
eN : online e N
eO : online e O
ar2 : area d b c = area e b c
be : b ≠ e
de : d ≠ e
Bbde : B b d e
⊢ False
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_I_extras.lean | eq_area_of_same_base_implies_para | [425, 3] | [461, 83] | perma at ar2 | i : incidence_geometry
a b c d : point
L M O : line
bL : online b L
cL : online c L
aL : ¬online a L
aM : online a M
dM : online d M
bO : online b O
dO : online d O
cO : ¬online c O
ad : a ≠ d
bd : b ≠ d
ssadL : sameside a d L
ar : area a b c = area d b c
N : line
aN : online a N
pNL : para L N
pNO : ¬para N O
e : point
eN : online e N
eO : online e O
ar2 : area d b c = area e b c
be : b ≠ e
de : d ≠ e
Bbde : B b d e
⊢ area b c e = area b c d | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
i : incidence_geometry
a b c d : point
L M O : line
bL : online b L
cL : online c L
aL : ¬online a L
aM : online a M
dM : online d M
bO : online b O
dO : online d O
cO : ¬online c O
ad : a ≠ d
bd : b ≠ d
ssadL : sameside a d L
ar : area a b c = area d b c
N : line
aN : online a N
pNL : para L N
pNO : ¬para N O
e : point
eN : online e N
eO : online e O
ar2 : area d b c = area e b c
be : b ≠ e
de : d ≠ e
Bbde : B b d e
⊢ area b c e = area b c d
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_length | [17, 1] | [51, 92] | by_cases ab: a = b | i : incidence_geometry
a b : point
L : line
n : ℕ
aL : online a L
bL : online b L
⊢ ∃ c, online c L ∧ length a c = ↑n * length a b ∧ (n ≥ 2 ∧ a ≠ b → B a b c) | case pos
i : incidence_geometry
a b : point
L : line
n : ℕ
aL : online a L
bL : online b L
ab : a = b
⊢ ∃ c, online c L ∧ length a c = ↑n * length a b ∧ (n ≥ 2 ∧ a ≠ b → B a b c)
case neg
i : incidence_geometry
a b : point
L : line
n : ℕ
aL : online a L
bL : online b L
ab : ¬a = b
⊢ ∃ c, online c L ∧ length a c = ↑n * length a b ∧ (n ≥ 2 ∧ a ≠ b → B a b c) | Please generate a tactic in lean4 to solve the state.
STATE:
i : incidence_geometry
a b : point
L : line
n : ℕ
aL : online a L
bL : online b L
⊢ ∃ c, online c L ∧ length a c = ↑n * length a b ∧ (n ≥ 2 ∧ a ≠ b → B a b c)
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_length | [17, 1] | [51, 92] | use a | case pos
i : incidence_geometry
a b : point
L : line
n : ℕ
aL : online a L
bL : online b L
ab : a = b
⊢ ∃ c, online c L ∧ length a c = ↑n * length a b ∧ (n ≥ 2 ∧ a ≠ b → B a b c) | case pos
i : incidence_geometry
a b : point
L : line
n : ℕ
aL : online a L
bL : online b L
ab : a = b
⊢ online a L ∧ length a a = ↑n * length a b ∧ (n ≥ 2 ∧ a ≠ b → B a b a) | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
i : incidence_geometry
a b : point
L : line
n : ℕ
aL : online a L
bL : online b L
ab : a = b
⊢ ∃ c, online c L ∧ length a c = ↑n * length a b ∧ (n ≥ 2 ∧ a ≠ b → B a b c)
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_length | [17, 1] | [51, 92] | rw [← ab, length_eq_zero_iff.mpr rfl] | case pos
i : incidence_geometry
a b : point
L : line
n : ℕ
aL : online a L
bL : online b L
ab : a = b
⊢ online a L ∧ length a a = ↑n * length a b ∧ (n ≥ 2 ∧ a ≠ b → B a b a) | case pos
i : incidence_geometry
a b : point
L : line
n : ℕ
aL : online a L
bL : online b L
ab : a = b
⊢ online a L ∧ 0 = ↑n * 0 ∧ (n ≥ 2 ∧ a ≠ a → B a a a) | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
i : incidence_geometry
a b : point
L : line
n : ℕ
aL : online a L
bL : online b L
ab : a = b
⊢ online a L ∧ length a a = ↑n * length a b ∧ (n ≥ 2 ∧ a ≠ b → B a b a)
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_length | [17, 1] | [51, 92] | simp | case pos
i : incidence_geometry
a b : point
L : line
n : ℕ
aL : online a L
bL : online b L
ab : a = b
⊢ online a L ∧ 0 = ↑n * 0 ∧ (n ≥ 2 ∧ a ≠ a → B a a a) | case pos
i : incidence_geometry
a b : point
L : line
n : ℕ
aL : online a L
bL : online b L
ab : a = b
⊢ online a L | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
i : incidence_geometry
a b : point
L : line
n : ℕ
aL : online a L
bL : online b L
ab : a = b
⊢ online a L ∧ 0 = ↑n * 0 ∧ (n ≥ 2 ∧ a ≠ a → B a a a)
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_length | [17, 1] | [51, 92] | exact aL | case pos
i : incidence_geometry
a b : point
L : line
n : ℕ
aL : online a L
bL : online b L
ab : a = b
⊢ online a L | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
i : incidence_geometry
a b : point
L : line
n : ℕ
aL : online a L
bL : online b L
ab : a = b
⊢ online a L
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_length | [17, 1] | [51, 92] | use a | case neg.zero
i : incidence_geometry
a b : point
L : line
aL : online a L
bL : online b L
ab : ¬a = b
⊢ ∃ c, online c L ∧ length a c = ↑Nat.zero * length a b ∧ (Nat.zero ≥ 2 ∧ a ≠ b → B a b c) | case neg.zero
i : incidence_geometry
a b : point
L : line
aL : online a L
bL : online b L
ab : ¬a = b
⊢ online a L ∧ length a a = ↑Nat.zero * length a b ∧ (Nat.zero ≥ 2 ∧ a ≠ b → B a b a) | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.zero
i : incidence_geometry
a b : point
L : line
aL : online a L
bL : online b L
ab : ¬a = b
⊢ ∃ c, online c L ∧ length a c = ↑Nat.zero * length a b ∧ (Nat.zero ≥ 2 ∧ a ≠ b → B a b c)
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_length | [17, 1] | [51, 92] | rw [length_eq_zero_iff.mpr rfl] | case neg.zero
i : incidence_geometry
a b : point
L : line
aL : online a L
bL : online b L
ab : ¬a = b
⊢ online a L ∧ length a a = ↑Nat.zero * length a b ∧ (Nat.zero ≥ 2 ∧ a ≠ b → B a b a) | case neg.zero
i : incidence_geometry
a b : point
L : line
aL : online a L
bL : online b L
ab : ¬a = b
⊢ online a L ∧ 0 = ↑Nat.zero * length a b ∧ (Nat.zero ≥ 2 ∧ a ≠ b → B a b a) | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.zero
i : incidence_geometry
a b : point
L : line
aL : online a L
bL : online b L
ab : ¬a = b
⊢ online a L ∧ length a a = ↑Nat.zero * length a b ∧ (Nat.zero ≥ 2 ∧ a ≠ b → B a b a)
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_length | [17, 1] | [51, 92] | simp | case neg.zero
i : incidence_geometry
a b : point
L : line
aL : online a L
bL : online b L
ab : ¬a = b
⊢ online a L ∧ 0 = ↑Nat.zero * length a b ∧ (Nat.zero ≥ 2 ∧ a ≠ b → B a b a) | case neg.zero
i : incidence_geometry
a b : point
L : line
aL : online a L
bL : online b L
ab : ¬a = b
⊢ online a L | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.zero
i : incidence_geometry
a b : point
L : line
aL : online a L
bL : online b L
ab : ¬a = b
⊢ online a L ∧ 0 = ↑Nat.zero * length a b ∧ (Nat.zero ≥ 2 ∧ a ≠ b → B a b a)
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_length | [17, 1] | [51, 92] | exact aL | case neg.zero
i : incidence_geometry
a b : point
L : line
aL : online a L
bL : online b L
ab : ¬a = b
⊢ online a L | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.zero
i : incidence_geometry
a b : point
L : line
aL : online a L
bL : online b L
ab : ¬a = b
⊢ online a L
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_length | [17, 1] | [51, 92] | by_cases n0: n = 0 | case neg.succ
i : incidence_geometry
a b : point
L : line
aL : online a L
bL : online b L
ab : ¬a = b
n : ℕ
hn : ∃ c, online c L ∧ length a c = ↑n * length a b ∧ (n ≥ 2 ∧ a ≠ b → B a b c)
⊢ ∃ c, online c L ∧ length a c = ↑(Nat.succ n) * length a b ∧ (Nat.succ n ≥ 2 ∧ a ≠ b → B a b c) | case pos
i : incidence_geometry
a b : point
L : line
aL : online a L
bL : online b L
ab : ¬a = b
n : ℕ
hn : ∃ c, online c L ∧ length a c = ↑n * length a b ∧ (n ≥ 2 ∧ a ≠ b → B a b c)
n0 : n = 0
⊢ ∃ c, online c L ∧ length a c = ↑(Nat.succ n) * length a b ∧ (Nat.succ n ≥ 2 ∧ a ≠ b → B a b c)
case neg
i : incidence_geometry
a b : point
L : line
aL : online a L
bL : online b L
ab : ¬a = b
n : ℕ
hn : ∃ c, online c L ∧ length a c = ↑n * length a b ∧ (n ≥ 2 ∧ a ≠ b → B a b c)
n0 : ¬n = 0
⊢ ∃ c, online c L ∧ length a c = ↑(Nat.succ n) * length a b ∧ (Nat.succ n ≥ 2 ∧ a ≠ b → B a b c) | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.succ
i : incidence_geometry
a b : point
L : line
aL : online a L
bL : online b L
ab : ¬a = b
n : ℕ
hn : ∃ c, online c L ∧ length a c = ↑n * length a b ∧ (n ≥ 2 ∧ a ≠ b → B a b c)
⊢ ∃ c, online c L ∧ length a c = ↑(Nat.succ n) * length a b ∧ (Nat.succ n ≥ 2 ∧ a ≠ b → B a b c)
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_length | [17, 1] | [51, 92] | by_cases n1 : n = 1 | case neg
i : incidence_geometry
a b : point
L : line
aL : online a L
bL : online b L
ab : ¬a = b
n : ℕ
hn : ∃ c, online c L ∧ length a c = ↑n * length a b ∧ (n ≥ 2 ∧ a ≠ b → B a b c)
n0 : ¬n = 0
⊢ ∃ c, online c L ∧ length a c = ↑(Nat.succ n) * length a b ∧ (Nat.succ n ≥ 2 ∧ a ≠ b → B a b c) | case pos
i : incidence_geometry
a b : point
L : line
aL : online a L
bL : online b L
ab : ¬a = b
n : ℕ
hn : ∃ c, online c L ∧ length a c = ↑n * length a b ∧ (n ≥ 2 ∧ a ≠ b → B a b c)
n0 : ¬n = 0
n1 : n = 1
⊢ ∃ c, online c L ∧ length a c = ↑(Nat.succ n) * length a b ∧ (Nat.succ n ≥ 2 ∧ a ≠ b → B a b c)
case neg
i : incidence_geometry
a b : point
L : line
aL : online a L
bL : online b L
ab : ¬a = b
n : ℕ
hn : ∃ c, online c L ∧ length a c = ↑n * length a b ∧ (n ≥ 2 ∧ a ≠ b → B a b c)
n0 : ¬n = 0
n1 : ¬n = 1
⊢ ∃ c, online c L ∧ length a c = ↑(Nat.succ n) * length a b ∧ (Nat.succ n ≥ 2 ∧ a ≠ b → B a b c) | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
i : incidence_geometry
a b : point
L : line
aL : online a L
bL : online b L
ab : ¬a = b
n : ℕ
hn : ∃ c, online c L ∧ length a c = ↑n * length a b ∧ (n ≥ 2 ∧ a ≠ b → B a b c)
n0 : ¬n = 0
⊢ ∃ c, online c L ∧ length a c = ↑(Nat.succ n) * length a b ∧ (Nat.succ n ≥ 2 ∧ a ≠ b → B a b c)
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_length | [17, 1] | [51, 92] | obtain ⟨ d, Hd ⟩ := hn | case neg
i : incidence_geometry
a b : point
L : line
aL : online a L
bL : online b L
ab : ¬a = b
n : ℕ
hn : ∃ c, online c L ∧ length a c = ↑n * length a b ∧ (n ≥ 2 ∧ a ≠ b → B a b c)
n0 : ¬n = 0
n1 : ¬n = 1
⊢ ∃ c, online c L ∧ length a c = ↑(Nat.succ n) * length a b ∧ (Nat.succ n ≥ 2 ∧ a ≠ b → B a b c) | case neg.intro
i : incidence_geometry
a b : point
L : line
aL : online a L
bL : online b L
ab : ¬a = b
n : ℕ
n0 : ¬n = 0
n1 : ¬n = 1
d : point
Hd : online d L ∧ length a d = ↑n * length a b ∧ (n ≥ 2 ∧ a ≠ b → B a b d)
⊢ ∃ c, online c L ∧ length a c = ↑(Nat.succ n) * length a b ∧ (Nat.succ n ≥ 2 ∧ a ≠ b → B a b c) | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
i : incidence_geometry
a b : point
L : line
aL : online a L
bL : online b L
ab : ¬a = b
n : ℕ
hn : ∃ c, online c L ∧ length a c = ↑n * length a b ∧ (n ≥ 2 ∧ a ≠ b → B a b c)
n0 : ¬n = 0
n1 : ¬n = 1
⊢ ∃ c, online c L ∧ length a c = ↑(Nat.succ n) * length a b ∧ (Nat.succ n ≥ 2 ∧ a ≠ b → B a b c)
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_length | [17, 1] | [51, 92] | have ad : a ≠ d := by
apply nq_of_len_pos
rw [Hd.2.1]; simp [Nat.pos_of_ne_zero n0, zero_lt_mul_left, Nat.cast_pos]
exact len_pos_of_nq ab | case neg.intro
i : incidence_geometry
a b : point
L : line
aL : online a L
bL : online b L
ab : ¬a = b
n : ℕ
n0 : ¬n = 0
n1 : ¬n = 1
d : point
Hd : online d L ∧ length a d = ↑n * length a b ∧ (n ≥ 2 ∧ a ≠ b → B a b d)
⊢ ∃ c, online c L ∧ length a c = ↑(Nat.succ n) * length a b ∧ (Nat.succ n ≥ 2 ∧ a ≠ b → B a b c) | case neg.intro
i : incidence_geometry
a b : point
L : line
aL : online a L
bL : online b L
ab : ¬a = b
n : ℕ
n0 : ¬n = 0
n1 : ¬n = 1
d : point
Hd : online d L ∧ length a d = ↑n * length a b ∧ (n ≥ 2 ∧ a ≠ b → B a b d)
ad : a ≠ d
⊢ ∃ c, online c L ∧ length a c = ↑(Nat.succ n) * length a b ∧ (Nat.succ n ≥ 2 ∧ a ≠ b → B a b c) | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro
i : incidence_geometry
a b : point
L : line
aL : online a L
bL : online b L
ab : ¬a = b
n : ℕ
n0 : ¬n = 0
n1 : ¬n = 1
d : point
Hd : online d L ∧ length a d = ↑n * length a b ∧ (n ≥ 2 ∧ a ≠ b → B a b d)
⊢ ∃ c, online c L ∧ length a c = ↑(Nat.succ n) * length a b ∧ (Nat.succ n ≥ 2 ∧ a ≠ b → B a b c)
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_length | [17, 1] | [51, 92] | obtain ⟨ e, He ⟩ := length_eq_B_of_ne_four ad ab | case neg.intro
i : incidence_geometry
a b : point
L : line
aL : online a L
bL : online b L
ab : ¬a = b
n : ℕ
n0 : ¬n = 0
n1 : ¬n = 1
d : point
Hd : online d L ∧ length a d = ↑n * length a b ∧ (n ≥ 2 ∧ a ≠ b → B a b d)
ad : a ≠ d
⊢ ∃ c, online c L ∧ length a c = ↑(Nat.succ n) * length a b ∧ (Nat.succ n ≥ 2 ∧ a ≠ b → B a b c) | case neg.intro.intro
i : incidence_geometry
a b : point
L : line
aL : online a L
bL : online b L
ab : ¬a = b
n : ℕ
n0 : ¬n = 0
n1 : ¬n = 1
d : point
Hd : online d L ∧ length a d = ↑n * length a b ∧ (n ≥ 2 ∧ a ≠ b → B a b d)
ad : a ≠ d
e : point
He : B a d e ∧ length a b = length d e
⊢ ∃ c, online c L ∧ length a c = ↑(Nat.succ n) * length a b ∧ (Nat.succ n ≥ 2 ∧ a ≠ b → B a b c) | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro
i : incidence_geometry
a b : point
L : line
aL : online a L
bL : online b L
ab : ¬a = b
n : ℕ
n0 : ¬n = 0
n1 : ¬n = 1
d : point
Hd : online d L ∧ length a d = ↑n * length a b ∧ (n ≥ 2 ∧ a ≠ b → B a b d)
ad : a ≠ d
⊢ ∃ c, online c L ∧ length a c = ↑(Nat.succ n) * length a b ∧ (Nat.succ n ≥ 2 ∧ a ≠ b → B a b c)
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_length | [17, 1] | [51, 92] | use e | case neg.intro.intro
i : incidence_geometry
a b : point
L : line
aL : online a L
bL : online b L
ab : ¬a = b
n : ℕ
n0 : ¬n = 0
n1 : ¬n = 1
d : point
Hd : online d L ∧ length a d = ↑n * length a b ∧ (n ≥ 2 ∧ a ≠ b → B a b d)
ad : a ≠ d
e : point
He : B a d e ∧ length a b = length d e
⊢ ∃ c, online c L ∧ length a c = ↑(Nat.succ n) * length a b ∧ (Nat.succ n ≥ 2 ∧ a ≠ b → B a b c) | case neg.intro.intro
i : incidence_geometry
a b : point
L : line
aL : online a L
bL : online b L
ab : ¬a = b
n : ℕ
n0 : ¬n = 0
n1 : ¬n = 1
d : point
Hd : online d L ∧ length a d = ↑n * length a b ∧ (n ≥ 2 ∧ a ≠ b → B a b d)
ad : a ≠ d
e : point
He : B a d e ∧ length a b = length d e
⊢ online e L ∧ length a e = ↑(Nat.succ n) * length a b ∧ (Nat.succ n ≥ 2 ∧ a ≠ b → B a b e) | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro
i : incidence_geometry
a b : point
L : line
aL : online a L
bL : online b L
ab : ¬a = b
n : ℕ
n0 : ¬n = 0
n1 : ¬n = 1
d : point
Hd : online d L ∧ length a d = ↑n * length a b ∧ (n ≥ 2 ∧ a ≠ b → B a b d)
ad : a ≠ d
e : point
He : B a d e ∧ length a b = length d e
⊢ ∃ c, online c L ∧ length a c = ↑(Nat.succ n) * length a b ∧ (Nat.succ n ≥ 2 ∧ a ≠ b → B a b c)
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_length | [17, 1] | [51, 92] | rw [← length_sum_of_B He.1, Hd.2.1, He.2] | case neg.intro.intro
i : incidence_geometry
a b : point
L : line
aL : online a L
bL : online b L
ab : ¬a = b
n : ℕ
n0 : ¬n = 0
n1 : ¬n = 1
d : point
Hd : online d L ∧ length a d = ↑n * length a b ∧ (n ≥ 2 ∧ a ≠ b → B a b d)
ad : a ≠ d
e : point
He : B a d e ∧ length a b = length d e
⊢ online e L ∧ length a e = ↑(Nat.succ n) * length a b ∧ (Nat.succ n ≥ 2 ∧ a ≠ b → B a b e) | case neg.intro.intro
i : incidence_geometry
a b : point
L : line
aL : online a L
bL : online b L
ab : ¬a = b
n : ℕ
n0 : ¬n = 0
n1 : ¬n = 1
d : point
Hd : online d L ∧ length a d = ↑n * length a b ∧ (n ≥ 2 ∧ a ≠ b → B a b d)
ad : a ≠ d
e : point
He : B a d e ∧ length a b = length d e
⊢ online e L ∧ ↑n * length d e + length d e = ↑(Nat.succ n) * length d e ∧ (Nat.succ n ≥ 2 ∧ a ≠ b → B a b e) | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro
i : incidence_geometry
a b : point
L : line
aL : online a L
bL : online b L
ab : ¬a = b
n : ℕ
n0 : ¬n = 0
n1 : ¬n = 1
d : point
Hd : online d L ∧ length a d = ↑n * length a b ∧ (n ≥ 2 ∧ a ≠ b → B a b d)
ad : a ≠ d
e : point
He : B a d e ∧ length a b = length d e
⊢ online e L ∧ length a e = ↑(Nat.succ n) * length a b ∧ (Nat.succ n ≥ 2 ∧ a ≠ b → B a b e)
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_length | [17, 1] | [51, 92] | simp | case neg.intro.intro
i : incidence_geometry
a b : point
L : line
aL : online a L
bL : online b L
ab : ¬a = b
n : ℕ
n0 : ¬n = 0
n1 : ¬n = 1
d : point
Hd : online d L ∧ length a d = ↑n * length a b ∧ (n ≥ 2 ∧ a ≠ b → B a b d)
ad : a ≠ d
e : point
He : B a d e ∧ length a b = length d e
⊢ online e L ∧ ↑n * length d e + length d e = ↑(Nat.succ n) * length d e ∧ (Nat.succ n ≥ 2 ∧ a ≠ b → B a b e) | case neg.intro.intro
i : incidence_geometry
a b : point
L : line
aL : online a L
bL : online b L
ab : ¬a = b
n : ℕ
n0 : ¬n = 0
n1 : ¬n = 1
d : point
Hd : online d L ∧ length a d = ↑n * length a b ∧ (n ≥ 2 ∧ a ≠ b → B a b d)
ad : a ≠ d
e : point
He : B a d e ∧ length a b = length d e
⊢ online e L ∧ ↑n * length d e + length d e = (↑n + 1) * length d e ∧ (2 ≤ Nat.succ n → ¬a = b → B a b e) | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro
i : incidence_geometry
a b : point
L : line
aL : online a L
bL : online b L
ab : ¬a = b
n : ℕ
n0 : ¬n = 0
n1 : ¬n = 1
d : point
Hd : online d L ∧ length a d = ↑n * length a b ∧ (n ≥ 2 ∧ a ≠ b → B a b d)
ad : a ≠ d
e : point
He : B a d e ∧ length a b = length d e
⊢ online e L ∧ ↑n * length d e + length d e = ↑(Nat.succ n) * length d e ∧ (Nat.succ n ≥ 2 ∧ a ≠ b → B a b e)
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_length | [17, 1] | [51, 92] | ring_nf | case neg.intro.intro
i : incidence_geometry
a b : point
L : line
aL : online a L
bL : online b L
ab : ¬a = b
n : ℕ
n0 : ¬n = 0
n1 : ¬n = 1
d : point
Hd : online d L ∧ length a d = ↑n * length a b ∧ (n ≥ 2 ∧ a ≠ b → B a b d)
ad : a ≠ d
e : point
He : B a d e ∧ length a b = length d e
⊢ online e L ∧ ↑n * length d e + length d e = (↑n + 1) * length d e ∧ (2 ≤ Nat.succ n → ¬a = b → B a b e) | case neg.intro.intro
i : incidence_geometry
a b : point
L : line
aL : online a L
bL : online b L
ab : ¬a = b
n : ℕ
n0 : ¬n = 0
n1 : ¬n = 1
d : point
Hd : online d L ∧ length a d = ↑n * length a b ∧ (n ≥ 2 ∧ a ≠ b → B a b d)
ad : a ≠ d
e : point
He : B a d e ∧ length a b = length d e
⊢ online e L ∧ True ∧ (2 ≤ Nat.succ n → ¬a = b → B a b e) | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro
i : incidence_geometry
a b : point
L : line
aL : online a L
bL : online b L
ab : ¬a = b
n : ℕ
n0 : ¬n = 0
n1 : ¬n = 1
d : point
Hd : online d L ∧ length a d = ↑n * length a b ∧ (n ≥ 2 ∧ a ≠ b → B a b d)
ad : a ≠ d
e : point
He : B a d e ∧ length a b = length d e
⊢ online e L ∧ ↑n * length d e + length d e = (↑n + 1) * length d e ∧ (2 ≤ Nat.succ n → ¬a = b → B a b e)
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_length | [17, 1] | [51, 92] | have := Hd.2.2 ⟨ ge2_of_n1_n0 n0 n1, ab ⟩ | case neg.intro.intro
i : incidence_geometry
a b : point
L : line
aL : online a L
bL : online b L
ab : ¬a = b
n : ℕ
n0 : ¬n = 0
n1 : ¬n = 1
d : point
Hd : online d L ∧ length a d = ↑n * length a b ∧ (n ≥ 2 ∧ a ≠ b → B a b d)
ad : a ≠ d
e : point
He : B a d e ∧ length a b = length d e
⊢ online e L ∧ True ∧ (2 ≤ Nat.succ n → ¬a = b → B a b e) | case neg.intro.intro
i : incidence_geometry
a b : point
L : line
aL : online a L
bL : online b L
ab : ¬a = b
n : ℕ
n0 : ¬n = 0
n1 : ¬n = 1
d : point
Hd : online d L ∧ length a d = ↑n * length a b ∧ (n ≥ 2 ∧ a ≠ b → B a b d)
ad : a ≠ d
e : point
He : B a d e ∧ length a b = length d e
this : B a b d
⊢ online e L ∧ True ∧ (2 ≤ Nat.succ n → ¬a = b → B a b e) | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro
i : incidence_geometry
a b : point
L : line
aL : online a L
bL : online b L
ab : ¬a = b
n : ℕ
n0 : ¬n = 0
n1 : ¬n = 1
d : point
Hd : online d L ∧ length a d = ↑n * length a b ∧ (n ≥ 2 ∧ a ≠ b → B a b d)
ad : a ≠ d
e : point
He : B a d e ∧ length a b = length d e
⊢ online e L ∧ True ∧ (2 ≤ Nat.succ n → ¬a = b → B a b e)
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_length | [17, 1] | [51, 92] | exact ⟨ online_3_of_B He.1 aL Hd.1, (by tauto), fun _ _ => B124_of_B134_B123 He.1 this⟩ | case neg.intro.intro
i : incidence_geometry
a b : point
L : line
aL : online a L
bL : online b L
ab : ¬a = b
n : ℕ
n0 : ¬n = 0
n1 : ¬n = 1
d : point
Hd : online d L ∧ length a d = ↑n * length a b ∧ (n ≥ 2 ∧ a ≠ b → B a b d)
ad : a ≠ d
e : point
He : B a d e ∧ length a b = length d e
this : B a b d
⊢ online e L ∧ True ∧ (2 ≤ Nat.succ n → ¬a = b → B a b e) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro
i : incidence_geometry
a b : point
L : line
aL : online a L
bL : online b L
ab : ¬a = b
n : ℕ
n0 : ¬n = 0
n1 : ¬n = 1
d : point
Hd : online d L ∧ length a d = ↑n * length a b ∧ (n ≥ 2 ∧ a ≠ b → B a b d)
ad : a ≠ d
e : point
He : B a d e ∧ length a b = length d e
this : B a b d
⊢ online e L ∧ True ∧ (2 ≤ Nat.succ n → ¬a = b → B a b e)
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_length | [17, 1] | [51, 92] | exact ⟨ b, bL, (by rw [n0]; simp) ⟩ | case pos
i : incidence_geometry
a b : point
L : line
aL : online a L
bL : online b L
ab : ¬a = b
n : ℕ
hn : ∃ c, online c L ∧ length a c = ↑n * length a b ∧ (n ≥ 2 ∧ a ≠ b → B a b c)
n0 : n = 0
⊢ ∃ c, online c L ∧ length a c = ↑(Nat.succ n) * length a b ∧ (Nat.succ n ≥ 2 ∧ a ≠ b → B a b c) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
i : incidence_geometry
a b : point
L : line
aL : online a L
bL : online b L
ab : ¬a = b
n : ℕ
hn : ∃ c, online c L ∧ length a c = ↑n * length a b ∧ (n ≥ 2 ∧ a ≠ b → B a b c)
n0 : n = 0
⊢ ∃ c, online c L ∧ length a c = ↑(Nat.succ n) * length a b ∧ (Nat.succ n ≥ 2 ∧ a ≠ b → B a b c)
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_length | [17, 1] | [51, 92] | rw [n0] | i : incidence_geometry
a b : point
L : line
aL : online a L
bL : online b L
ab : ¬a = b
n : ℕ
hn : ∃ c, online c L ∧ length a c = ↑n * length a b ∧ (n ≥ 2 ∧ a ≠ b → B a b c)
n0 : n = 0
⊢ length a b = ↑(Nat.succ n) * length a b ∧ (Nat.succ n ≥ 2 ∧ a ≠ b → B a b b) | i : incidence_geometry
a b : point
L : line
aL : online a L
bL : online b L
ab : ¬a = b
n : ℕ
hn : ∃ c, online c L ∧ length a c = ↑n * length a b ∧ (n ≥ 2 ∧ a ≠ b → B a b c)
n0 : n = 0
⊢ length a b = ↑(Nat.succ 0) * length a b ∧ (Nat.succ 0 ≥ 2 ∧ a ≠ b → B a b b) | Please generate a tactic in lean4 to solve the state.
STATE:
i : incidence_geometry
a b : point
L : line
aL : online a L
bL : online b L
ab : ¬a = b
n : ℕ
hn : ∃ c, online c L ∧ length a c = ↑n * length a b ∧ (n ≥ 2 ∧ a ≠ b → B a b c)
n0 : n = 0
⊢ length a b = ↑(Nat.succ n) * length a b ∧ (Nat.succ n ≥ 2 ∧ a ≠ b → B a b b)
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_length | [17, 1] | [51, 92] | simp | i : incidence_geometry
a b : point
L : line
aL : online a L
bL : online b L
ab : ¬a = b
n : ℕ
hn : ∃ c, online c L ∧ length a c = ↑n * length a b ∧ (n ≥ 2 ∧ a ≠ b → B a b c)
n0 : n = 0
⊢ length a b = ↑(Nat.succ 0) * length a b ∧ (Nat.succ 0 ≥ 2 ∧ a ≠ b → B a b b) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
i : incidence_geometry
a b : point
L : line
aL : online a L
bL : online b L
ab : ¬a = b
n : ℕ
hn : ∃ c, online c L ∧ length a c = ↑n * length a b ∧ (n ≥ 2 ∧ a ≠ b → B a b c)
n0 : n = 0
⊢ length a b = ↑(Nat.succ 0) * length a b ∧ (Nat.succ 0 ≥ 2 ∧ a ≠ b → B a b b)
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_length | [17, 1] | [51, 92] | obtain ⟨ e, He ⟩ := length_eq_B_of_ne_four ab ab | case pos
i : incidence_geometry
a b : point
L : line
aL : online a L
bL : online b L
ab : ¬a = b
n : ℕ
hn : ∃ c, online c L ∧ length a c = ↑n * length a b ∧ (n ≥ 2 ∧ a ≠ b → B a b c)
n0 : ¬n = 0
n1 : n = 1
⊢ ∃ c, online c L ∧ length a c = ↑(Nat.succ n) * length a b ∧ (Nat.succ n ≥ 2 ∧ a ≠ b → B a b c) | case pos.intro
i : incidence_geometry
a b : point
L : line
aL : online a L
bL : online b L
ab : ¬a = b
n : ℕ
hn : ∃ c, online c L ∧ length a c = ↑n * length a b ∧ (n ≥ 2 ∧ a ≠ b → B a b c)
n0 : ¬n = 0
n1 : n = 1
e : point
He : B a b e ∧ length a b = length b e
⊢ ∃ c, online c L ∧ length a c = ↑(Nat.succ n) * length a b ∧ (Nat.succ n ≥ 2 ∧ a ≠ b → B a b c) | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
i : incidence_geometry
a b : point
L : line
aL : online a L
bL : online b L
ab : ¬a = b
n : ℕ
hn : ∃ c, online c L ∧ length a c = ↑n * length a b ∧ (n ≥ 2 ∧ a ≠ b → B a b c)
n0 : ¬n = 0
n1 : n = 1
⊢ ∃ c, online c L ∧ length a c = ↑(Nat.succ n) * length a b ∧ (Nat.succ n ≥ 2 ∧ a ≠ b → B a b c)
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_length | [17, 1] | [51, 92] | use e | case pos.intro
i : incidence_geometry
a b : point
L : line
aL : online a L
bL : online b L
ab : ¬a = b
n : ℕ
hn : ∃ c, online c L ∧ length a c = ↑n * length a b ∧ (n ≥ 2 ∧ a ≠ b → B a b c)
n0 : ¬n = 0
n1 : n = 1
e : point
He : B a b e ∧ length a b = length b e
⊢ ∃ c, online c L ∧ length a c = ↑(Nat.succ n) * length a b ∧ (Nat.succ n ≥ 2 ∧ a ≠ b → B a b c) | case pos.intro
i : incidence_geometry
a b : point
L : line
aL : online a L
bL : online b L
ab : ¬a = b
n : ℕ
hn : ∃ c, online c L ∧ length a c = ↑n * length a b ∧ (n ≥ 2 ∧ a ≠ b → B a b c)
n0 : ¬n = 0
n1 : n = 1
e : point
He : B a b e ∧ length a b = length b e
⊢ online e L ∧ length a e = ↑(Nat.succ n) * length a b ∧ (Nat.succ n ≥ 2 ∧ a ≠ b → B a b e) | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.intro
i : incidence_geometry
a b : point
L : line
aL : online a L
bL : online b L
ab : ¬a = b
n : ℕ
hn : ∃ c, online c L ∧ length a c = ↑n * length a b ∧ (n ≥ 2 ∧ a ≠ b → B a b c)
n0 : ¬n = 0
n1 : n = 1
e : point
He : B a b e ∧ length a b = length b e
⊢ ∃ c, online c L ∧ length a c = ↑(Nat.succ n) * length a b ∧ (Nat.succ n ≥ 2 ∧ a ≠ b → B a b c)
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_length | [17, 1] | [51, 92] | simp_rw [(length_sum_of_B He.1).symm, He.2, n1] | case pos.intro
i : incidence_geometry
a b : point
L : line
aL : online a L
bL : online b L
ab : ¬a = b
n : ℕ
hn : ∃ c, online c L ∧ length a c = ↑n * length a b ∧ (n ≥ 2 ∧ a ≠ b → B a b c)
n0 : ¬n = 0
n1 : n = 1
e : point
He : B a b e ∧ length a b = length b e
⊢ online e L ∧ length a e = ↑(Nat.succ n) * length a b ∧ (Nat.succ n ≥ 2 ∧ a ≠ b → B a b e) | case pos.intro
i : incidence_geometry
a b : point
L : line
aL : online a L
bL : online b L
ab : ¬a = b
n : ℕ
hn : ∃ c, online c L ∧ length a c = ↑n * length a b ∧ (n ≥ 2 ∧ a ≠ b → B a b c)
n0 : ¬n = 0
n1 : n = 1
e : point
He : B a b e ∧ length a b = length b e
⊢ online e L ∧ length b e + length b e = ↑(Nat.succ 1) * length b e ∧ (True ∧ a ≠ b → B a b e) | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.intro
i : incidence_geometry
a b : point
L : line
aL : online a L
bL : online b L
ab : ¬a = b
n : ℕ
hn : ∃ c, online c L ∧ length a c = ↑n * length a b ∧ (n ≥ 2 ∧ a ≠ b → B a b c)
n0 : ¬n = 0
n1 : n = 1
e : point
He : B a b e ∧ length a b = length b e
⊢ online e L ∧ length a e = ↑(Nat.succ n) * length a b ∧ (Nat.succ n ≥ 2 ∧ a ≠ b → B a b e)
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_length | [17, 1] | [51, 92] | simp | case pos.intro
i : incidence_geometry
a b : point
L : line
aL : online a L
bL : online b L
ab : ¬a = b
n : ℕ
hn : ∃ c, online c L ∧ length a c = ↑n * length a b ∧ (n ≥ 2 ∧ a ≠ b → B a b c)
n0 : ¬n = 0
n1 : n = 1
e : point
He : B a b e ∧ length a b = length b e
⊢ online e L ∧ length b e + length b e = ↑(Nat.succ 1) * length b e ∧ (True ∧ a ≠ b → B a b e) | case pos.intro
i : incidence_geometry
a b : point
L : line
aL : online a L
bL : online b L
ab : ¬a = b
n : ℕ
hn : ∃ c, online c L ∧ length a c = ↑n * length a b ∧ (n ≥ 2 ∧ a ≠ b → B a b c)
n0 : ¬n = 0
n1 : n = 1
e : point
He : B a b e ∧ length a b = length b e
⊢ online e L ∧ length b e + length b e = (1 + 1) * length b e ∧ (¬a = b → B a b e) | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.intro
i : incidence_geometry
a b : point
L : line
aL : online a L
bL : online b L
ab : ¬a = b
n : ℕ
hn : ∃ c, online c L ∧ length a c = ↑n * length a b ∧ (n ≥ 2 ∧ a ≠ b → B a b c)
n0 : ¬n = 0
n1 : n = 1
e : point
He : B a b e ∧ length a b = length b e
⊢ online e L ∧ length b e + length b e = ↑(Nat.succ 1) * length b e ∧ (True ∧ a ≠ b → B a b e)
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_length | [17, 1] | [51, 92] | ring_nf | case pos.intro
i : incidence_geometry
a b : point
L : line
aL : online a L
bL : online b L
ab : ¬a = b
n : ℕ
hn : ∃ c, online c L ∧ length a c = ↑n * length a b ∧ (n ≥ 2 ∧ a ≠ b → B a b c)
n0 : ¬n = 0
n1 : n = 1
e : point
He : B a b e ∧ length a b = length b e
⊢ online e L ∧ length b e + length b e = (1 + 1) * length b e ∧ (¬a = b → B a b e) | case pos.intro
i : incidence_geometry
a b : point
L : line
aL : online a L
bL : online b L
ab : ¬a = b
n : ℕ
hn : ∃ c, online c L ∧ length a c = ↑n * length a b ∧ (n ≥ 2 ∧ a ≠ b → B a b c)
n0 : ¬n = 0
n1 : n = 1
e : point
He : B a b e ∧ length a b = length b e
⊢ online e L ∧ True ∧ (¬a = b → B a b e) | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.intro
i : incidence_geometry
a b : point
L : line
aL : online a L
bL : online b L
ab : ¬a = b
n : ℕ
hn : ∃ c, online c L ∧ length a c = ↑n * length a b ∧ (n ≥ 2 ∧ a ≠ b → B a b c)
n0 : ¬n = 0
n1 : n = 1
e : point
He : B a b e ∧ length a b = length b e
⊢ online e L ∧ length b e + length b e = (1 + 1) * length b e ∧ (¬a = b → B a b e)
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_length | [17, 1] | [51, 92] | refine' ⟨ online_3_of_B He.1 aL bL, (by tauto), fun _ => He.1 ⟩ | case pos.intro
i : incidence_geometry
a b : point
L : line
aL : online a L
bL : online b L
ab : ¬a = b
n : ℕ
hn : ∃ c, online c L ∧ length a c = ↑n * length a b ∧ (n ≥ 2 ∧ a ≠ b → B a b c)
n0 : ¬n = 0
n1 : n = 1
e : point
He : B a b e ∧ length a b = length b e
⊢ online e L ∧ True ∧ (¬a = b → B a b e) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.intro
i : incidence_geometry
a b : point
L : line
aL : online a L
bL : online b L
ab : ¬a = b
n : ℕ
hn : ∃ c, online c L ∧ length a c = ↑n * length a b ∧ (n ≥ 2 ∧ a ≠ b → B a b c)
n0 : ¬n = 0
n1 : n = 1
e : point
He : B a b e ∧ length a b = length b e
⊢ online e L ∧ True ∧ (¬a = b → B a b e)
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_length | [17, 1] | [51, 92] | tauto | i : incidence_geometry
a b : point
L : line
aL : online a L
bL : online b L
ab : ¬a = b
n : ℕ
hn : ∃ c, online c L ∧ length a c = ↑n * length a b ∧ (n ≥ 2 ∧ a ≠ b → B a b c)
n0 : ¬n = 0
n1 : n = 1
e : point
He : B a b e ∧ length a b = length b e
⊢ True | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
i : incidence_geometry
a b : point
L : line
aL : online a L
bL : online b L
ab : ¬a = b
n : ℕ
hn : ∃ c, online c L ∧ length a c = ↑n * length a b ∧ (n ≥ 2 ∧ a ≠ b → B a b c)
n0 : ¬n = 0
n1 : n = 1
e : point
He : B a b e ∧ length a b = length b e
⊢ True
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_length | [17, 1] | [51, 92] | apply nq_of_len_pos | i : incidence_geometry
a b : point
L : line
aL : online a L
bL : online b L
ab : ¬a = b
n : ℕ
n0 : ¬n = 0
n1 : ¬n = 1
d : point
Hd : online d L ∧ length a d = ↑n * length a b ∧ (n ≥ 2 ∧ a ≠ b → B a b d)
⊢ a ≠ d | case length
i : incidence_geometry
a b : point
L : line
aL : online a L
bL : online b L
ab : ¬a = b
n : ℕ
n0 : ¬n = 0
n1 : ¬n = 1
d : point
Hd : online d L ∧ length a d = ↑n * length a b ∧ (n ≥ 2 ∧ a ≠ b → B a b d)
⊢ 0 < length a d | Please generate a tactic in lean4 to solve the state.
STATE:
i : incidence_geometry
a b : point
L : line
aL : online a L
bL : online b L
ab : ¬a = b
n : ℕ
n0 : ¬n = 0
n1 : ¬n = 1
d : point
Hd : online d L ∧ length a d = ↑n * length a b ∧ (n ≥ 2 ∧ a ≠ b → B a b d)
⊢ a ≠ d
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_length | [17, 1] | [51, 92] | rw [Hd.2.1] | case length
i : incidence_geometry
a b : point
L : line
aL : online a L
bL : online b L
ab : ¬a = b
n : ℕ
n0 : ¬n = 0
n1 : ¬n = 1
d : point
Hd : online d L ∧ length a d = ↑n * length a b ∧ (n ≥ 2 ∧ a ≠ b → B a b d)
⊢ 0 < length a d | case length
i : incidence_geometry
a b : point
L : line
aL : online a L
bL : online b L
ab : ¬a = b
n : ℕ
n0 : ¬n = 0
n1 : ¬n = 1
d : point
Hd : online d L ∧ length a d = ↑n * length a b ∧ (n ≥ 2 ∧ a ≠ b → B a b d)
⊢ 0 < ↑n * length a b | Please generate a tactic in lean4 to solve the state.
STATE:
case length
i : incidence_geometry
a b : point
L : line
aL : online a L
bL : online b L
ab : ¬a = b
n : ℕ
n0 : ¬n = 0
n1 : ¬n = 1
d : point
Hd : online d L ∧ length a d = ↑n * length a b ∧ (n ≥ 2 ∧ a ≠ b → B a b d)
⊢ 0 < length a d
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_length | [17, 1] | [51, 92] | simp [Nat.pos_of_ne_zero n0, zero_lt_mul_left, Nat.cast_pos] | case length
i : incidence_geometry
a b : point
L : line
aL : online a L
bL : online b L
ab : ¬a = b
n : ℕ
n0 : ¬n = 0
n1 : ¬n = 1
d : point
Hd : online d L ∧ length a d = ↑n * length a b ∧ (n ≥ 2 ∧ a ≠ b → B a b d)
⊢ 0 < ↑n * length a b | case length
i : incidence_geometry
a b : point
L : line
aL : online a L
bL : online b L
ab : ¬a = b
n : ℕ
n0 : ¬n = 0
n1 : ¬n = 1
d : point
Hd : online d L ∧ length a d = ↑n * length a b ∧ (n ≥ 2 ∧ a ≠ b → B a b d)
⊢ 0 < length a b | Please generate a tactic in lean4 to solve the state.
STATE:
case length
i : incidence_geometry
a b : point
L : line
aL : online a L
bL : online b L
ab : ¬a = b
n : ℕ
n0 : ¬n = 0
n1 : ¬n = 1
d : point
Hd : online d L ∧ length a d = ↑n * length a b ∧ (n ≥ 2 ∧ a ≠ b → B a b d)
⊢ 0 < ↑n * length a b
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_length | [17, 1] | [51, 92] | exact len_pos_of_nq ab | case length
i : incidence_geometry
a b : point
L : line
aL : online a L
bL : online b L
ab : ¬a = b
n : ℕ
n0 : ¬n = 0
n1 : ¬n = 1
d : point
Hd : online d L ∧ length a d = ↑n * length a b ∧ (n ≥ 2 ∧ a ≠ b → B a b d)
⊢ 0 < length a b | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case length
i : incidence_geometry
a b : point
L : line
aL : online a L
bL : online b L
ab : ¬a = b
n : ℕ
n0 : ¬n = 0
n1 : ¬n = 1
d : point
Hd : online d L ∧ length a d = ↑n * length a b ∧ (n ≥ 2 ∧ a ≠ b → B a b d)
⊢ 0 < length a b
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_length | [17, 1] | [51, 92] | tauto | i : incidence_geometry
a b : point
L : line
aL : online a L
bL : online b L
ab : ¬a = b
n : ℕ
n0 : ¬n = 0
n1 : ¬n = 1
d : point
Hd : online d L ∧ length a d = ↑n * length a b ∧ (n ≥ 2 ∧ a ≠ b → B a b d)
ad : a ≠ d
e : point
He : B a d e ∧ length a b = length d e
this : B a b d
⊢ True | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
i : incidence_geometry
a b : point
L : line
aL : online a L
bL : online b L
ab : ¬a = b
n : ℕ
n0 : ¬n = 0
n1 : ¬n = 1
d : point
Hd : online d L ∧ length a d = ↑n * length a b ∧ (n ≥ 2 ∧ a ≠ b → B a b d)
ad : a ≠ d
e : point
He : B a d e ∧ length a b = length d e
this : B a b d
⊢ True
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_triangle_of_base_inductive | [53, 1] | [111, 73] | simp | case zero
i : incidence_geometry
a b c : point
L : line
bL : online b L
cL : online c L
aL : ¬online a L
⊢ ∀ (d : point), online d L → length b d = ↑Nat.zero * length b c → B b c d → area a b d = ↑Nat.zero * area a b c | case zero
i : incidence_geometry
a b c : point
L : line
bL : online b L
cL : online c L
aL : ¬online a L
⊢ ∀ (d : point), online d L → length b d = 0 → B b c d → area a b d = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
i : incidence_geometry
a b c : point
L : line
bL : online b L
cL : online c L
aL : ¬online a L
⊢ ∀ (d : point), online d L → length b d = ↑Nat.zero * length b c → B b c d → area a b d = ↑Nat.zero * 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] | intros d _ hlen _ | case zero
i : incidence_geometry
a b c : point
L : line
bL : online b L
cL : online c L
aL : ¬online a L
⊢ ∀ (d : point), online d L → length b d = 0 → B b c d → area a b d = 0 | case zero
i : incidence_geometry
a b c : point
L : line
bL : online b L
cL : online c L
aL : ¬online a L
d : point
a✝¹ : online d L
hlen : length b d = 0
a✝ : B b c d
⊢ area a b d = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
i : incidence_geometry
a b c : point
L : line
bL : online b L
cL : online c L
aL : ¬online a L
⊢ ∀ (d : point), online d L → length b d = 0 → B b c d → area a b d = 0
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_triangle_of_base_inductive | [53, 1] | [111, 73] | rw [length_eq_zero_iff.mp hlen, area_of_eq a d d (by tauto)] | case zero
i : incidence_geometry
a b c : point
L : line
bL : online b L
cL : online c L
aL : ¬online a L
d : point
a✝¹ : online d L
hlen : length b d = 0
a✝ : B b c d
⊢ area a b d = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
i : incidence_geometry
a b c : point
L : line
bL : online b L
cL : online c L
aL : ¬online a L
d : point
a✝¹ : online d L
hlen : length b d = 0
a✝ : B b c d
⊢ area a b d = 0
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_triangle_of_base_inductive | [53, 1] | [111, 73] | tauto | i : incidence_geometry
a b c : point
L : line
bL : online b L
cL : online c L
aL : ¬online a L
d : point
a✝¹ : online d L
hlen : length b d = 0
a✝ : B b c d
⊢ a = d ∨ a = d ∨ d = 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
d : point
a✝¹ : online d L
hlen : length b d = 0
a✝ : B b c d
⊢ a = d ∨ a = d ∨ d = d
TACTIC:
|
https://github.com/ianjauslin-rutgers/pythagoras4.git | f97e5d2375d2e350b15d79e541520f8ba81600ec | Pythagoras/euclid_VI.lean | rescale_triangle_of_base_inductive | [53, 1] | [111, 73] | intros d dL hlen Bbcd | case succ
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),
online d L → length b d = ↑(Nat.succ n) * length b c → B b c d → area a b d = ↑(Nat.succ n) * area a b c | case succ
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
⊢ area a b d = ↑(Nat.succ n) * area a b c | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
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),
online d L → length b d = ↑(Nat.succ n) * length b c → B b c d → 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] | have bc := ne_12_of_B Bbcd | case succ
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
⊢ area a b d = ↑(Nat.succ n) * area a b c | case succ
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
⊢ area a b d = ↑(Nat.succ n) * area a b c | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
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
⊢ 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] | have bd := ne_13_of_B Bbcd | case succ
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
⊢ area a b d = ↑(Nat.succ n) * area a b c | case succ
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
⊢ area a b d = ↑(Nat.succ n) * area a b c | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
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
⊢ 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] | have cd := ne_23_of_B Bbcd | case succ
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
⊢ area a b d = ↑(Nat.succ n) * area a b c | case succ
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
⊢ area a b d = ↑(Nat.succ n) * area a b c | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
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
⊢ 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] | by_cases n0 : n = 0 | case succ
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
⊢ 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 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 neg
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 | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
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
⊢ 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] | by_cases n1 : n = 1 | case neg
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 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 neg
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 | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
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 [Nat.cast_succ] | case neg
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 neg
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 = (↑n + 1) * area a b c | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
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] | rw [add_mul] | case neg
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 = (↑n + 1) * area a b c | case neg
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 = ↑n * area a b c + 1 * area a b c | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
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 = (↑n + 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 [one_mul] | case neg
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 = ↑n * area a b c + 1 * area a b c | case neg
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 = ↑n * area a b c + area a b c | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
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 = ↑n * area a b c + 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] | obtain ⟨ e, He ⟩ := rescale_length n bL cL | case neg
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 = ↑n * area a b c + 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)
⊢ area a b d = ↑n * area a b c + area a b c | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
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 = ↑n * area a b c + 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 Bbce := He.2.2 ⟨ ge2_of_n1_n0 n0 n1, bc ⟩ | 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)
⊢ area a b d = ↑n * area a b c + 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
⊢ area a b d = ↑n * area a b c + 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)
⊢ area a b d = ↑n * area a b c + 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 [← hn e He.1 He.2.1 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
⊢ area a b d = ↑n * area a b c + 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
⊢ 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
⊢ area a b d = ↑n * area a b c + 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 be := ne_13_of_B 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
⊢ 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
⊢ 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
⊢ area a b d = area a b e + area a b c
TACTIC:
|
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