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
|
|---|---|---|---|---|---|---|---|---|
https://github.com/loewenheim/projective-plane.git | 7ade52678ac3df537f38d04de46db9aca752f3ca | ProjectivePlane.lean | ProjectivePlane.dual_quadrangle_part | [159, 1] | [232, 13] | . apply And.intro
. simp
. simp | case right
P L : Type
p q r s : P
inst✝ : ProjectivePlane P
triangle_pqr : ¬collinear p q r
left✝¹ : ¬collinear p q s
left✝ : ¬collinear p r s
triangle_qrs : ¬collinear q r s
a : line P := p ⊔ q
b : line P := q ⊔ r
c : line P := r ⊔ s
t : P := a ⊓ c
t_is_not_q : q ≠ t
h : r = t
⊢ q ∈ a ∧ t ∈ a | no goals |
https://github.com/loewenheim/projective-plane.git | 7ade52678ac3df537f38d04de46db9aca752f3ca | ProjectivePlane.lean | ProjectivePlane.dual_quadrangle_part | [159, 1] | [232, 13] | apply connectingLine_uniq | P L : Type
p q r s : P
inst✝ : ProjectivePlane P
triangle_pqr : ¬collinear p q r
left✝¹ : ¬collinear p q s
left✝ : ¬collinear p r s
triangle_qrs : ¬collinear q r s
a : line P := p ⊔ q
b : line P := q ⊔ r
c : line P := r ⊔ s
t : P := a ⊓ c
t_is_not_q : q ≠ t
t_is_not_r : r ≠ t
⊢ a = q ⊔ t | case a
P L : Type
p q r s : P
inst✝ : ProjectivePlane P
triangle_pqr : ¬collinear p q r
left✝¹ : ¬collinear p q s
left✝ : ¬collinear p r s
triangle_qrs : ¬collinear q r s
a : line P := p ⊔ q
b : line P := q ⊔ r
c : line P := r ⊔ s
t : P := a ⊓ c
t_is_not_q : q ≠ t
t_is_not_r : r ≠ t
⊢ q ≠ t
case a
P L : Type
p q r s : P
inst✝ : ProjectivePlane P
triangle_pqr : ¬collinear p q r
left✝¹ : ¬collinear p q s
left✝ : ¬collinear p r s
triangle_qrs : ¬collinear q r s
a : line P := p ⊔ q
b : line P := q ⊔ r
c : line P := r ⊔ s
t : P := a ⊓ c
t_is_not_q : q ≠ t
t_is_not_r : r ≠ t
⊢ q ∈ a
case a
P L : Type
p q r s : P
inst✝ : ProjectivePlane P
triangle_pqr : ¬collinear p q r
left✝¹ : ¬collinear p q s
left✝ : ¬collinear p r s
triangle_qrs : ¬collinear q r s
a : line P := p ⊔ q
b : line P := q ⊔ r
c : line P := r ⊔ s
t : P := a ⊓ c
t_is_not_q : q ≠ t
t_is_not_r : r ≠ t
⊢ t ∈ a |
https://github.com/loewenheim/projective-plane.git | 7ade52678ac3df537f38d04de46db9aca752f3ca | ProjectivePlane.lean | ProjectivePlane.dual_quadrangle_part | [159, 1] | [232, 13] | . assumption | case a
P L : Type
p q r s : P
inst✝ : ProjectivePlane P
triangle_pqr : ¬collinear p q r
left✝¹ : ¬collinear p q s
left✝ : ¬collinear p r s
triangle_qrs : ¬collinear q r s
a : line P := p ⊔ q
b : line P := q ⊔ r
c : line P := r ⊔ s
t : P := a ⊓ c
t_is_not_q : q ≠ t
t_is_not_r : r ≠ t
⊢ q ≠ t
case a
P L : Type
p q r s : P
inst✝ : ProjectivePlane P
triangle_pqr : ¬collinear p q r
left✝¹ : ¬collinear p q s
left✝ : ¬collinear p r s
triangle_qrs : ¬collinear q r s
a : line P := p ⊔ q
b : line P := q ⊔ r
c : line P := r ⊔ s
t : P := a ⊓ c
t_is_not_q : q ≠ t
t_is_not_r : r ≠ t
⊢ q ∈ a
case a
P L : Type
p q r s : P
inst✝ : ProjectivePlane P
triangle_pqr : ¬collinear p q r
left✝¹ : ¬collinear p q s
left✝ : ¬collinear p r s
triangle_qrs : ¬collinear q r s
a : line P := p ⊔ q
b : line P := q ⊔ r
c : line P := r ⊔ s
t : P := a ⊓ c
t_is_not_q : q ≠ t
t_is_not_r : r ≠ t
⊢ t ∈ a | case a
P L : Type
p q r s : P
inst✝ : ProjectivePlane P
triangle_pqr : ¬collinear p q r
left✝¹ : ¬collinear p q s
left✝ : ¬collinear p r s
triangle_qrs : ¬collinear q r s
a : line P := p ⊔ q
b : line P := q ⊔ r
c : line P := r ⊔ s
t : P := a ⊓ c
t_is_not_q : q ≠ t
t_is_not_r : r ≠ t
⊢ q ∈ a
case a
P L : Type
p q r s : P
inst✝ : ProjectivePlane P
triangle_pqr : ¬collinear p q r
left✝¹ : ¬collinear p q s
left✝ : ¬collinear p r s
triangle_qrs : ¬collinear q r s
a : line P := p ⊔ q
b : line P := q ⊔ r
c : line P := r ⊔ s
t : P := a ⊓ c
t_is_not_q : q ≠ t
t_is_not_r : r ≠ t
⊢ t ∈ a |
https://github.com/loewenheim/projective-plane.git | 7ade52678ac3df537f38d04de46db9aca752f3ca | ProjectivePlane.lean | ProjectivePlane.dual_quadrangle_part | [159, 1] | [232, 13] | . simp | case a
P L : Type
p q r s : P
inst✝ : ProjectivePlane P
triangle_pqr : ¬collinear p q r
left✝¹ : ¬collinear p q s
left✝ : ¬collinear p r s
triangle_qrs : ¬collinear q r s
a : line P := p ⊔ q
b : line P := q ⊔ r
c : line P := r ⊔ s
t : P := a ⊓ c
t_is_not_q : q ≠ t
t_is_not_r : r ≠ t
⊢ q ∈ a
case a
P L : Type
p q r s : P
inst✝ : ProjectivePlane P
triangle_pqr : ¬collinear p q r
left✝¹ : ¬collinear p q s
left✝ : ¬collinear p r s
triangle_qrs : ¬collinear q r s
a : line P := p ⊔ q
b : line P := q ⊔ r
c : line P := r ⊔ s
t : P := a ⊓ c
t_is_not_q : q ≠ t
t_is_not_r : r ≠ t
⊢ t ∈ a | case a
P L : Type
p q r s : P
inst✝ : ProjectivePlane P
triangle_pqr : ¬collinear p q r
left✝¹ : ¬collinear p q s
left✝ : ¬collinear p r s
triangle_qrs : ¬collinear q r s
a : line P := p ⊔ q
b : line P := q ⊔ r
c : line P := r ⊔ s
t : P := a ⊓ c
t_is_not_q : q ≠ t
t_is_not_r : r ≠ t
⊢ t ∈ a |
https://github.com/loewenheim/projective-plane.git | 7ade52678ac3df537f38d04de46db9aca752f3ca | ProjectivePlane.lean | ProjectivePlane.dual_quadrangle_part | [159, 1] | [232, 13] | . simp | case a
P L : Type
p q r s : P
inst✝ : ProjectivePlane P
triangle_pqr : ¬collinear p q r
left✝¹ : ¬collinear p q s
left✝ : ¬collinear p r s
triangle_qrs : ¬collinear q r s
a : line P := p ⊔ q
b : line P := q ⊔ r
c : line P := r ⊔ s
t : P := a ⊓ c
t_is_not_q : q ≠ t
t_is_not_r : r ≠ t
⊢ t ∈ a | no goals |
https://github.com/loewenheim/projective-plane.git | 7ade52678ac3df537f38d04de46db9aca752f3ca | ProjectivePlane.lean | ProjectivePlane.dual_quadrangle_part | [159, 1] | [232, 13] | apply connectingLine_uniq | P L : Type
p q r s : P
inst✝ : ProjectivePlane P
triangle_pqr : ¬collinear p q r
left✝¹ : ¬collinear p q s
left✝ : ¬collinear p r s
triangle_qrs : ¬collinear q r s
a : line P := p ⊔ q
b : line P := q ⊔ r
c : line P := r ⊔ s
t : P := a ⊓ c
t_is_not_q : q ≠ t
t_is_not_r : r ≠ t
a_is_q_t : a = q ⊔ t
⊢ c = r ⊔ t | case a
P L : Type
p q r s : P
inst✝ : ProjectivePlane P
triangle_pqr : ¬collinear p q r
left✝¹ : ¬collinear p q s
left✝ : ¬collinear p r s
triangle_qrs : ¬collinear q r s
a : line P := p ⊔ q
b : line P := q ⊔ r
c : line P := r ⊔ s
t : P := a ⊓ c
t_is_not_q : q ≠ t
t_is_not_r : r ≠ t
a_is_q_t : a = q ⊔ t
⊢ r ≠ t
case a
P L : Type
p q r s : P
inst✝ : ProjectivePlane P
triangle_pqr : ¬collinear p q r
left✝¹ : ¬collinear p q s
left✝ : ¬collinear p r s
triangle_qrs : ¬collinear q r s
a : line P := p ⊔ q
b : line P := q ⊔ r
c : line P := r ⊔ s
t : P := a ⊓ c
t_is_not_q : q ≠ t
t_is_not_r : r ≠ t
a_is_q_t : a = q ⊔ t
⊢ r ∈ c
case a
P L : Type
p q r s : P
inst✝ : ProjectivePlane P
triangle_pqr : ¬collinear p q r
left✝¹ : ¬collinear p q s
left✝ : ¬collinear p r s
triangle_qrs : ¬collinear q r s
a : line P := p ⊔ q
b : line P := q ⊔ r
c : line P := r ⊔ s
t : P := a ⊓ c
t_is_not_q : q ≠ t
t_is_not_r : r ≠ t
a_is_q_t : a = q ⊔ t
⊢ t ∈ c |
https://github.com/loewenheim/projective-plane.git | 7ade52678ac3df537f38d04de46db9aca752f3ca | ProjectivePlane.lean | ProjectivePlane.dual_quadrangle_part | [159, 1] | [232, 13] | . assumption | case a
P L : Type
p q r s : P
inst✝ : ProjectivePlane P
triangle_pqr : ¬collinear p q r
left✝¹ : ¬collinear p q s
left✝ : ¬collinear p r s
triangle_qrs : ¬collinear q r s
a : line P := p ⊔ q
b : line P := q ⊔ r
c : line P := r ⊔ s
t : P := a ⊓ c
t_is_not_q : q ≠ t
t_is_not_r : r ≠ t
a_is_q_t : a = q ⊔ t
⊢ r ≠ t
case a
P L : Type
p q r s : P
inst✝ : ProjectivePlane P
triangle_pqr : ¬collinear p q r
left✝¹ : ¬collinear p q s
left✝ : ¬collinear p r s
triangle_qrs : ¬collinear q r s
a : line P := p ⊔ q
b : line P := q ⊔ r
c : line P := r ⊔ s
t : P := a ⊓ c
t_is_not_q : q ≠ t
t_is_not_r : r ≠ t
a_is_q_t : a = q ⊔ t
⊢ r ∈ c
case a
P L : Type
p q r s : P
inst✝ : ProjectivePlane P
triangle_pqr : ¬collinear p q r
left✝¹ : ¬collinear p q s
left✝ : ¬collinear p r s
triangle_qrs : ¬collinear q r s
a : line P := p ⊔ q
b : line P := q ⊔ r
c : line P := r ⊔ s
t : P := a ⊓ c
t_is_not_q : q ≠ t
t_is_not_r : r ≠ t
a_is_q_t : a = q ⊔ t
⊢ t ∈ c | case a
P L : Type
p q r s : P
inst✝ : ProjectivePlane P
triangle_pqr : ¬collinear p q r
left✝¹ : ¬collinear p q s
left✝ : ¬collinear p r s
triangle_qrs : ¬collinear q r s
a : line P := p ⊔ q
b : line P := q ⊔ r
c : line P := r ⊔ s
t : P := a ⊓ c
t_is_not_q : q ≠ t
t_is_not_r : r ≠ t
a_is_q_t : a = q ⊔ t
⊢ r ∈ c
case a
P L : Type
p q r s : P
inst✝ : ProjectivePlane P
triangle_pqr : ¬collinear p q r
left✝¹ : ¬collinear p q s
left✝ : ¬collinear p r s
triangle_qrs : ¬collinear q r s
a : line P := p ⊔ q
b : line P := q ⊔ r
c : line P := r ⊔ s
t : P := a ⊓ c
t_is_not_q : q ≠ t
t_is_not_r : r ≠ t
a_is_q_t : a = q ⊔ t
⊢ t ∈ c |
https://github.com/loewenheim/projective-plane.git | 7ade52678ac3df537f38d04de46db9aca752f3ca | ProjectivePlane.lean | ProjectivePlane.dual_quadrangle_part | [159, 1] | [232, 13] | . simp | case a
P L : Type
p q r s : P
inst✝ : ProjectivePlane P
triangle_pqr : ¬collinear p q r
left✝¹ : ¬collinear p q s
left✝ : ¬collinear p r s
triangle_qrs : ¬collinear q r s
a : line P := p ⊔ q
b : line P := q ⊔ r
c : line P := r ⊔ s
t : P := a ⊓ c
t_is_not_q : q ≠ t
t_is_not_r : r ≠ t
a_is_q_t : a = q ⊔ t
⊢ r ∈ c
case a
P L : Type
p q r s : P
inst✝ : ProjectivePlane P
triangle_pqr : ¬collinear p q r
left✝¹ : ¬collinear p q s
left✝ : ¬collinear p r s
triangle_qrs : ¬collinear q r s
a : line P := p ⊔ q
b : line P := q ⊔ r
c : line P := r ⊔ s
t : P := a ⊓ c
t_is_not_q : q ≠ t
t_is_not_r : r ≠ t
a_is_q_t : a = q ⊔ t
⊢ t ∈ c | case a
P L : Type
p q r s : P
inst✝ : ProjectivePlane P
triangle_pqr : ¬collinear p q r
left✝¹ : ¬collinear p q s
left✝ : ¬collinear p r s
triangle_qrs : ¬collinear q r s
a : line P := p ⊔ q
b : line P := q ⊔ r
c : line P := r ⊔ s
t : P := a ⊓ c
t_is_not_q : q ≠ t
t_is_not_r : r ≠ t
a_is_q_t : a = q ⊔ t
⊢ t ∈ c |
https://github.com/loewenheim/projective-plane.git | 7ade52678ac3df537f38d04de46db9aca752f3ca | ProjectivePlane.lean | ProjectivePlane.dual_quadrangle_part | [159, 1] | [232, 13] | . simp | case a
P L : Type
p q r s : P
inst✝ : ProjectivePlane P
triangle_pqr : ¬collinear p q r
left✝¹ : ¬collinear p q s
left✝ : ¬collinear p r s
triangle_qrs : ¬collinear q r s
a : line P := p ⊔ q
b : line P := q ⊔ r
c : line P := r ⊔ s
t : P := a ⊓ c
t_is_not_q : q ≠ t
t_is_not_r : r ≠ t
a_is_q_t : a = q ⊔ t
⊢ t ∈ c | no goals |
https://github.com/loewenheim/projective-plane.git | 7ade52678ac3df537f38d04de46db9aca752f3ca | ProjectivePlane.lean | ProjectivePlane.dual_quadrangle_part | [159, 1] | [232, 13] | simp[collinear] | P L : Type
p q r s : P
inst✝ : ProjectivePlane P
triangle_pqr : ¬collinear p q r
left✝¹ : ¬collinear p q s
left✝ : ¬collinear p r s
triangle_qrs : ¬collinear q r s
a : line P := p ⊔ q
b : line P := q ⊔ r
c : line P := r ⊔ s
t : P := a ⊓ c
t_is_not_q : q ≠ t
t_is_not_r : r ≠ t
a_is_q_t : a = q ⊔ t
c_is_r_t : c = r ⊔ t
⊢ ¬collinear t q r | P L : Type
p q r s : P
inst✝ : ProjectivePlane P
triangle_pqr : ¬collinear p q r
left✝¹ : ¬collinear p q s
left✝ : ¬collinear p r s
triangle_qrs : ¬collinear q r s
a : line P := p ⊔ q
b : line P := q ⊔ r
c : line P := r ⊔ s
t : P := a ⊓ c
t_is_not_q : q ≠ t
t_is_not_r : r ≠ t
a_is_q_t : a = q ⊔ t
c_is_r_t : c = r ⊔ t
⊢ ∀ (x : line P), (p ⊔ q) ⊓ (r ⊔ s) ∈ x → q ∈ x → ¬r ∈ x |
https://github.com/loewenheim/projective-plane.git | 7ade52678ac3df537f38d04de46db9aca752f3ca | ProjectivePlane.lean | ProjectivePlane.dual_quadrangle_part | [159, 1] | [232, 13] | intro l htl hql hrl | P L : Type
p q r s : P
inst✝ : ProjectivePlane P
triangle_pqr : ¬collinear p q r
left✝¹ : ¬collinear p q s
left✝ : ¬collinear p r s
triangle_qrs : ¬collinear q r s
a : line P := p ⊔ q
b : line P := q ⊔ r
c : line P := r ⊔ s
t : P := a ⊓ c
t_is_not_q : q ≠ t
t_is_not_r : r ≠ t
a_is_q_t : a = q ⊔ t
c_is_r_t : c = r ⊔ t
⊢ ∀ (x : line P), (p ⊔ q) ⊓ (r ⊔ s) ∈ x → q ∈ x → ¬r ∈ x | P L : Type
p q r s : P
inst✝ : ProjectivePlane P
triangle_pqr : ¬collinear p q r
left✝¹ : ¬collinear p q s
left✝ : ¬collinear p r s
triangle_qrs : ¬collinear q r s
a : line P := p ⊔ q
b : line P := q ⊔ r
c : line P := r ⊔ s
t : P := a ⊓ c
t_is_not_q : q ≠ t
t_is_not_r : r ≠ t
a_is_q_t : a = q ⊔ t
c_is_r_t : c = r ⊔ t
l : line P
htl : (p ⊔ q) ⊓ (r ⊔ s) ∈ l
hql : q ∈ l
hrl : r ∈ l
⊢ False |
https://github.com/loewenheim/projective-plane.git | 7ade52678ac3df537f38d04de46db9aca752f3ca | ProjectivePlane.lean | ProjectivePlane.dual_quadrangle_part | [159, 1] | [232, 13] | have l_is_a : l = a := by
rw[a_is_q_t]
apply connectingLine_uniq <;> assumption | P L : Type
p q r s : P
inst✝ : ProjectivePlane P
triangle_pqr : ¬collinear p q r
left✝¹ : ¬collinear p q s
left✝ : ¬collinear p r s
triangle_qrs : ¬collinear q r s
a : line P := p ⊔ q
b : line P := q ⊔ r
c : line P := r ⊔ s
t : P := a ⊓ c
t_is_not_q : q ≠ t
t_is_not_r : r ≠ t
a_is_q_t : a = q ⊔ t
c_is_r_t : c = r ⊔ t
l : line P
htl : (p ⊔ q) ⊓ (r ⊔ s) ∈ l
hql : q ∈ l
hrl : r ∈ l
⊢ False | P L : Type
p q r s : P
inst✝ : ProjectivePlane P
triangle_pqr : ¬collinear p q r
left✝¹ : ¬collinear p q s
left✝ : ¬collinear p r s
triangle_qrs : ¬collinear q r s
a : line P := p ⊔ q
b : line P := q ⊔ r
c : line P := r ⊔ s
t : P := a ⊓ c
t_is_not_q : q ≠ t
t_is_not_r : r ≠ t
a_is_q_t : a = q ⊔ t
c_is_r_t : c = r ⊔ t
l : line P
htl : (p ⊔ q) ⊓ (r ⊔ s) ∈ l
hql : q ∈ l
hrl : r ∈ l
l_is_a : l = a
⊢ False |
https://github.com/loewenheim/projective-plane.git | 7ade52678ac3df537f38d04de46db9aca752f3ca | ProjectivePlane.lean | ProjectivePlane.dual_quadrangle_part | [159, 1] | [232, 13] | rw[l_is_a] at hrl | P L : Type
p q r s : P
inst✝ : ProjectivePlane P
triangle_pqr : ¬collinear p q r
left✝¹ : ¬collinear p q s
left✝ : ¬collinear p r s
triangle_qrs : ¬collinear q r s
a : line P := p ⊔ q
b : line P := q ⊔ r
c : line P := r ⊔ s
t : P := a ⊓ c
t_is_not_q : q ≠ t
t_is_not_r : r ≠ t
a_is_q_t : a = q ⊔ t
c_is_r_t : c = r ⊔ t
l : line P
htl : (p ⊔ q) ⊓ (r ⊔ s) ∈ l
hql : q ∈ l
hrl : r ∈ l
l_is_a : l = a
⊢ False | P L : Type
p q r s : P
inst✝ : ProjectivePlane P
triangle_pqr : ¬collinear p q r
left✝¹ : ¬collinear p q s
left✝ : ¬collinear p r s
triangle_qrs : ¬collinear q r s
a : line P := p ⊔ q
b : line P := q ⊔ r
c : line P := r ⊔ s
t : P := a ⊓ c
t_is_not_q : q ≠ t
t_is_not_r : r ≠ t
a_is_q_t : a = q ⊔ t
c_is_r_t : c = r ⊔ t
l : line P
htl : (p ⊔ q) ⊓ (r ⊔ s) ∈ l
hql : q ∈ l
hrl : r ∈ a
l_is_a : l = a
⊢ False |
https://github.com/loewenheim/projective-plane.git | 7ade52678ac3df537f38d04de46db9aca752f3ca | ProjectivePlane.lean | ProjectivePlane.dual_quadrangle_part | [159, 1] | [232, 13] | have : collinear p q r := by
simp[collinear]
exists a
apply And.intro
. simp
. apply And.intro
. simp
. assumption | P L : Type
p q r s : P
inst✝ : ProjectivePlane P
triangle_pqr : ¬collinear p q r
left✝¹ : ¬collinear p q s
left✝ : ¬collinear p r s
triangle_qrs : ¬collinear q r s
a : line P := p ⊔ q
b : line P := q ⊔ r
c : line P := r ⊔ s
t : P := a ⊓ c
t_is_not_q : q ≠ t
t_is_not_r : r ≠ t
a_is_q_t : a = q ⊔ t
c_is_r_t : c = r ⊔ t
l : line P
htl : (p ⊔ q) ⊓ (r ⊔ s) ∈ l
hql : q ∈ l
hrl : r ∈ a
l_is_a : l = a
⊢ False | P L : Type
p q r s : P
inst✝ : ProjectivePlane P
triangle_pqr : ¬collinear p q r
left✝¹ : ¬collinear p q s
left✝ : ¬collinear p r s
triangle_qrs : ¬collinear q r s
a : line P := p ⊔ q
b : line P := q ⊔ r
c : line P := r ⊔ s
t : P := a ⊓ c
t_is_not_q : q ≠ t
t_is_not_r : r ≠ t
a_is_q_t : a = q ⊔ t
c_is_r_t : c = r ⊔ t
l : line P
htl : (p ⊔ q) ⊓ (r ⊔ s) ∈ l
hql : q ∈ l
hrl : r ∈ a
l_is_a : l = a
this : collinear p q r
⊢ False |
https://github.com/loewenheim/projective-plane.git | 7ade52678ac3df537f38d04de46db9aca752f3ca | ProjectivePlane.lean | ProjectivePlane.dual_quadrangle_part | [159, 1] | [232, 13] | contradiction | P L : Type
p q r s : P
inst✝ : ProjectivePlane P
triangle_pqr : ¬collinear p q r
left✝¹ : ¬collinear p q s
left✝ : ¬collinear p r s
triangle_qrs : ¬collinear q r s
a : line P := p ⊔ q
b : line P := q ⊔ r
c : line P := r ⊔ s
t : P := a ⊓ c
t_is_not_q : q ≠ t
t_is_not_r : r ≠ t
a_is_q_t : a = q ⊔ t
c_is_r_t : c = r ⊔ t
l : line P
htl : (p ⊔ q) ⊓ (r ⊔ s) ∈ l
hql : q ∈ l
hrl : r ∈ a
l_is_a : l = a
this : collinear p q r
⊢ False | no goals |
https://github.com/loewenheim/projective-plane.git | 7ade52678ac3df537f38d04de46db9aca752f3ca | ProjectivePlane.lean | ProjectivePlane.dual_quadrangle_part | [159, 1] | [232, 13] | rw[a_is_q_t] | P L : Type
p q r s : P
inst✝ : ProjectivePlane P
triangle_pqr : ¬collinear p q r
left✝¹ : ¬collinear p q s
left✝ : ¬collinear p r s
triangle_qrs : ¬collinear q r s
a : line P := p ⊔ q
b : line P := q ⊔ r
c : line P := r ⊔ s
t : P := a ⊓ c
t_is_not_q : q ≠ t
t_is_not_r : r ≠ t
a_is_q_t : a = q ⊔ t
c_is_r_t : c = r ⊔ t
l : line P
htl : (p ⊔ q) ⊓ (r ⊔ s) ∈ l
hql : q ∈ l
hrl : r ∈ l
⊢ l = a | P L : Type
p q r s : P
inst✝ : ProjectivePlane P
triangle_pqr : ¬collinear p q r
left✝¹ : ¬collinear p q s
left✝ : ¬collinear p r s
triangle_qrs : ¬collinear q r s
a : line P := p ⊔ q
b : line P := q ⊔ r
c : line P := r ⊔ s
t : P := a ⊓ c
t_is_not_q : q ≠ t
t_is_not_r : r ≠ t
a_is_q_t : a = q ⊔ t
c_is_r_t : c = r ⊔ t
l : line P
htl : (p ⊔ q) ⊓ (r ⊔ s) ∈ l
hql : q ∈ l
hrl : r ∈ l
⊢ l = q ⊔ t |
https://github.com/loewenheim/projective-plane.git | 7ade52678ac3df537f38d04de46db9aca752f3ca | ProjectivePlane.lean | ProjectivePlane.dual_quadrangle_part | [159, 1] | [232, 13] | apply connectingLine_uniq <;> assumption | P L : Type
p q r s : P
inst✝ : ProjectivePlane P
triangle_pqr : ¬collinear p q r
left✝¹ : ¬collinear p q s
left✝ : ¬collinear p r s
triangle_qrs : ¬collinear q r s
a : line P := p ⊔ q
b : line P := q ⊔ r
c : line P := r ⊔ s
t : P := a ⊓ c
t_is_not_q : q ≠ t
t_is_not_r : r ≠ t
a_is_q_t : a = q ⊔ t
c_is_r_t : c = r ⊔ t
l : line P
htl : (p ⊔ q) ⊓ (r ⊔ s) ∈ l
hql : q ∈ l
hrl : r ∈ l
⊢ l = q ⊔ t | no goals |
https://github.com/loewenheim/projective-plane.git | 7ade52678ac3df537f38d04de46db9aca752f3ca | ProjectivePlane.lean | ProjectivePlane.dual_quadrangle_part | [159, 1] | [232, 13] | simp[collinear] | P L : Type
p q r s : P
inst✝ : ProjectivePlane P
triangle_pqr : ¬collinear p q r
left✝¹ : ¬collinear p q s
left✝ : ¬collinear p r s
triangle_qrs : ¬collinear q r s
a : line P := p ⊔ q
b : line P := q ⊔ r
c : line P := r ⊔ s
t : P := a ⊓ c
t_is_not_q : q ≠ t
t_is_not_r : r ≠ t
a_is_q_t : a = q ⊔ t
c_is_r_t : c = r ⊔ t
l : line P
htl : (p ⊔ q) ⊓ (r ⊔ s) ∈ l
hql : q ∈ l
hrl : r ∈ a
l_is_a : l = a
⊢ collinear p q r | P L : Type
p q r s : P
inst✝ : ProjectivePlane P
triangle_pqr : ¬collinear p q r
left✝¹ : ¬collinear p q s
left✝ : ¬collinear p r s
triangle_qrs : ¬collinear q r s
a : line P := p ⊔ q
b : line P := q ⊔ r
c : line P := r ⊔ s
t : P := a ⊓ c
t_is_not_q : q ≠ t
t_is_not_r : r ≠ t
a_is_q_t : a = q ⊔ t
c_is_r_t : c = r ⊔ t
l : line P
htl : (p ⊔ q) ⊓ (r ⊔ s) ∈ l
hql : q ∈ l
hrl : r ∈ a
l_is_a : l = a
⊢ ∃ l, p ∈ l ∧ q ∈ l ∧ r ∈ l |
https://github.com/loewenheim/projective-plane.git | 7ade52678ac3df537f38d04de46db9aca752f3ca | ProjectivePlane.lean | ProjectivePlane.dual_quadrangle_part | [159, 1] | [232, 13] | exists a | P L : Type
p q r s : P
inst✝ : ProjectivePlane P
triangle_pqr : ¬collinear p q r
left✝¹ : ¬collinear p q s
left✝ : ¬collinear p r s
triangle_qrs : ¬collinear q r s
a : line P := p ⊔ q
b : line P := q ⊔ r
c : line P := r ⊔ s
t : P := a ⊓ c
t_is_not_q : q ≠ t
t_is_not_r : r ≠ t
a_is_q_t : a = q ⊔ t
c_is_r_t : c = r ⊔ t
l : line P
htl : (p ⊔ q) ⊓ (r ⊔ s) ∈ l
hql : q ∈ l
hrl : r ∈ a
l_is_a : l = a
⊢ ∃ l, p ∈ l ∧ q ∈ l ∧ r ∈ l | P L : Type
p q r s : P
inst✝ : ProjectivePlane P
triangle_pqr : ¬collinear p q r
left✝¹ : ¬collinear p q s
left✝ : ¬collinear p r s
triangle_qrs : ¬collinear q r s
a : line P := p ⊔ q
b : line P := q ⊔ r
c : line P := r ⊔ s
t : P := a ⊓ c
t_is_not_q : q ≠ t
t_is_not_r : r ≠ t
a_is_q_t : a = q ⊔ t
c_is_r_t : c = r ⊔ t
l : line P
htl : (p ⊔ q) ⊓ (r ⊔ s) ∈ l
hql : q ∈ l
hrl : r ∈ a
l_is_a : l = a
⊢ p ∈ a ∧ q ∈ a ∧ r ∈ a |
https://github.com/loewenheim/projective-plane.git | 7ade52678ac3df537f38d04de46db9aca752f3ca | ProjectivePlane.lean | ProjectivePlane.dual_quadrangle_part | [159, 1] | [232, 13] | apply And.intro | P L : Type
p q r s : P
inst✝ : ProjectivePlane P
triangle_pqr : ¬collinear p q r
left✝¹ : ¬collinear p q s
left✝ : ¬collinear p r s
triangle_qrs : ¬collinear q r s
a : line P := p ⊔ q
b : line P := q ⊔ r
c : line P := r ⊔ s
t : P := a ⊓ c
t_is_not_q : q ≠ t
t_is_not_r : r ≠ t
a_is_q_t : a = q ⊔ t
c_is_r_t : c = r ⊔ t
l : line P
htl : (p ⊔ q) ⊓ (r ⊔ s) ∈ l
hql : q ∈ l
hrl : r ∈ a
l_is_a : l = a
⊢ p ∈ a ∧ q ∈ a ∧ r ∈ a | case left
P L : Type
p q r s : P
inst✝ : ProjectivePlane P
triangle_pqr : ¬collinear p q r
left✝¹ : ¬collinear p q s
left✝ : ¬collinear p r s
triangle_qrs : ¬collinear q r s
a : line P := p ⊔ q
b : line P := q ⊔ r
c : line P := r ⊔ s
t : P := a ⊓ c
t_is_not_q : q ≠ t
t_is_not_r : r ≠ t
a_is_q_t : a = q ⊔ t
c_is_r_t : c = r ⊔ t
l : line P
htl : (p ⊔ q) ⊓ (r ⊔ s) ∈ l
hql : q ∈ l
hrl : r ∈ a
l_is_a : l = a
⊢ p ∈ a
case right
P L : Type
p q r s : P
inst✝ : ProjectivePlane P
triangle_pqr : ¬collinear p q r
left✝¹ : ¬collinear p q s
left✝ : ¬collinear p r s
triangle_qrs : ¬collinear q r s
a : line P := p ⊔ q
b : line P := q ⊔ r
c : line P := r ⊔ s
t : P := a ⊓ c
t_is_not_q : q ≠ t
t_is_not_r : r ≠ t
a_is_q_t : a = q ⊔ t
c_is_r_t : c = r ⊔ t
l : line P
htl : (p ⊔ q) ⊓ (r ⊔ s) ∈ l
hql : q ∈ l
hrl : r ∈ a
l_is_a : l = a
⊢ q ∈ a ∧ r ∈ a |
https://github.com/loewenheim/projective-plane.git | 7ade52678ac3df537f38d04de46db9aca752f3ca | ProjectivePlane.lean | ProjectivePlane.dual_quadrangle_part | [159, 1] | [232, 13] | . simp | case left
P L : Type
p q r s : P
inst✝ : ProjectivePlane P
triangle_pqr : ¬collinear p q r
left✝¹ : ¬collinear p q s
left✝ : ¬collinear p r s
triangle_qrs : ¬collinear q r s
a : line P := p ⊔ q
b : line P := q ⊔ r
c : line P := r ⊔ s
t : P := a ⊓ c
t_is_not_q : q ≠ t
t_is_not_r : r ≠ t
a_is_q_t : a = q ⊔ t
c_is_r_t : c = r ⊔ t
l : line P
htl : (p ⊔ q) ⊓ (r ⊔ s) ∈ l
hql : q ∈ l
hrl : r ∈ a
l_is_a : l = a
⊢ p ∈ a
case right
P L : Type
p q r s : P
inst✝ : ProjectivePlane P
triangle_pqr : ¬collinear p q r
left✝¹ : ¬collinear p q s
left✝ : ¬collinear p r s
triangle_qrs : ¬collinear q r s
a : line P := p ⊔ q
b : line P := q ⊔ r
c : line P := r ⊔ s
t : P := a ⊓ c
t_is_not_q : q ≠ t
t_is_not_r : r ≠ t
a_is_q_t : a = q ⊔ t
c_is_r_t : c = r ⊔ t
l : line P
htl : (p ⊔ q) ⊓ (r ⊔ s) ∈ l
hql : q ∈ l
hrl : r ∈ a
l_is_a : l = a
⊢ q ∈ a ∧ r ∈ a | case right
P L : Type
p q r s : P
inst✝ : ProjectivePlane P
triangle_pqr : ¬collinear p q r
left✝¹ : ¬collinear p q s
left✝ : ¬collinear p r s
triangle_qrs : ¬collinear q r s
a : line P := p ⊔ q
b : line P := q ⊔ r
c : line P := r ⊔ s
t : P := a ⊓ c
t_is_not_q : q ≠ t
t_is_not_r : r ≠ t
a_is_q_t : a = q ⊔ t
c_is_r_t : c = r ⊔ t
l : line P
htl : (p ⊔ q) ⊓ (r ⊔ s) ∈ l
hql : q ∈ l
hrl : r ∈ a
l_is_a : l = a
⊢ q ∈ a ∧ r ∈ a |
https://github.com/loewenheim/projective-plane.git | 7ade52678ac3df537f38d04de46db9aca752f3ca | ProjectivePlane.lean | ProjectivePlane.dual_quadrangle_part | [159, 1] | [232, 13] | . apply And.intro
. simp
. assumption | case right
P L : Type
p q r s : P
inst✝ : ProjectivePlane P
triangle_pqr : ¬collinear p q r
left✝¹ : ¬collinear p q s
left✝ : ¬collinear p r s
triangle_qrs : ¬collinear q r s
a : line P := p ⊔ q
b : line P := q ⊔ r
c : line P := r ⊔ s
t : P := a ⊓ c
t_is_not_q : q ≠ t
t_is_not_r : r ≠ t
a_is_q_t : a = q ⊔ t
c_is_r_t : c = r ⊔ t
l : line P
htl : (p ⊔ q) ⊓ (r ⊔ s) ∈ l
hql : q ∈ l
hrl : r ∈ a
l_is_a : l = a
⊢ q ∈ a ∧ r ∈ a | no goals |
https://github.com/loewenheim/projective-plane.git | 7ade52678ac3df537f38d04de46db9aca752f3ca | ProjectivePlane.lean | ProjectivePlane.dual_quadrangle_part | [159, 1] | [232, 13] | rw[a_is_q_t, c_is_r_t, connectingLine_comm] | P L : Type
p q r s : P
inst✝ : ProjectivePlane P
triangle_pqr : ¬collinear p q r
left✝¹ : ¬collinear p q s
left✝ : ¬collinear p r s
triangle_qrs : ¬collinear q r s
a : line P := p ⊔ q
b : line P := q ⊔ r
c : line P := r ⊔ s
t : P := a ⊓ c
t_is_not_q : q ≠ t
t_is_not_r : r ≠ t
a_is_q_t : a = q ⊔ t
c_is_r_t : c = r ⊔ t
triangle_tqr : ¬collinear t q r
⊢ ¬collinear a b c | P L : Type
p q r s : P
inst✝ : ProjectivePlane P
triangle_pqr : ¬collinear p q r
left✝¹ : ¬collinear p q s
left✝ : ¬collinear p r s
triangle_qrs : ¬collinear q r s
a : line P := p ⊔ q
b : line P := q ⊔ r
c : line P := r ⊔ s
t : P := a ⊓ c
t_is_not_q : q ≠ t
t_is_not_r : r ≠ t
a_is_q_t : a = q ⊔ t
c_is_r_t : c = r ⊔ t
triangle_tqr : ¬collinear t q r
⊢ ¬collinear (t ⊔ q) b (r ⊔ t) |
https://github.com/loewenheim/projective-plane.git | 7ade52678ac3df537f38d04de46db9aca752f3ca | ProjectivePlane.lean | ProjectivePlane.dual_quadrangle_part | [159, 1] | [232, 13] | apply dual_triangle | P L : Type
p q r s : P
inst✝ : ProjectivePlane P
triangle_pqr : ¬collinear p q r
left✝¹ : ¬collinear p q s
left✝ : ¬collinear p r s
triangle_qrs : ¬collinear q r s
a : line P := p ⊔ q
b : line P := q ⊔ r
c : line P := r ⊔ s
t : P := a ⊓ c
t_is_not_q : q ≠ t
t_is_not_r : r ≠ t
a_is_q_t : a = q ⊔ t
c_is_r_t : c = r ⊔ t
triangle_tqr : ¬collinear t q r
⊢ ¬collinear (t ⊔ q) b (r ⊔ t) | case a
P L : Type
p q r s : P
inst✝ : ProjectivePlane P
triangle_pqr : ¬collinear p q r
left✝¹ : ¬collinear p q s
left✝ : ¬collinear p r s
triangle_qrs : ¬collinear q r s
a : line P := p ⊔ q
b : line P := q ⊔ r
c : line P := r ⊔ s
t : P := a ⊓ c
t_is_not_q : q ≠ t
t_is_not_r : r ≠ t
a_is_q_t : a = q ⊔ t
c_is_r_t : c = r ⊔ t
triangle_tqr : ¬collinear t q r
⊢ ¬collinear t q r |
https://github.com/loewenheim/projective-plane.git | 7ade52678ac3df537f38d04de46db9aca752f3ca | ProjectivePlane.lean | ProjectivePlane.dual_quadrangle_part | [159, 1] | [232, 13] | assumption | case a
P L : Type
p q r s : P
inst✝ : ProjectivePlane P
triangle_pqr : ¬collinear p q r
left✝¹ : ¬collinear p q s
left✝ : ¬collinear p r s
triangle_qrs : ¬collinear q r s
a : line P := p ⊔ q
b : line P := q ⊔ r
c : line P := r ⊔ s
t : P := a ⊓ c
t_is_not_q : q ≠ t
t_is_not_r : r ≠ t
a_is_q_t : a = q ⊔ t
c_is_r_t : c = r ⊔ t
triangle_tqr : ¬collinear t q r
⊢ ¬collinear t q r | no goals |
https://github.com/loewenheim/projective-plane.git | 7ade52678ac3df537f38d04de46db9aca752f3ca | ProjectivePlane.lean | ProjectivePlane.dual_quadrangle | [234, 1] | [266, 65] | intro ⟨triangle_pqr, triangle_pqs, triangle_prs, triangle_qrs⟩ | P L : Type
p q r s : P
inst✝ : ProjectivePlane P
⊢ isQuadrangle p q r s → isQuadrangle (p ⊔ q) (q ⊔ r) (r ⊔ s) (s ⊔ p) | P L : Type
p q r s : P
inst✝ : ProjectivePlane P
triangle_pqr : ¬collinear p q r
triangle_pqs : ¬collinear p q s
triangle_prs : ¬collinear p r s
triangle_qrs : ¬collinear q r s
⊢ isQuadrangle (p ⊔ q) (q ⊔ r) (r ⊔ s) (s ⊔ p) |
https://github.com/loewenheim/projective-plane.git | 7ade52678ac3df537f38d04de46db9aca752f3ca | ProjectivePlane.lean | ProjectivePlane.dual_quadrangle | [234, 1] | [266, 65] | let a := p ⊔ q | P L : Type
p q r s : P
inst✝ : ProjectivePlane P
triangle_pqr : ¬collinear p q r
triangle_pqs : ¬collinear p q s
triangle_prs : ¬collinear p r s
triangle_qrs : ¬collinear q r s
⊢ isQuadrangle (p ⊔ q) (q ⊔ r) (r ⊔ s) (s ⊔ p) | P L : Type
p q r s : P
inst✝ : ProjectivePlane P
triangle_pqr : ¬collinear p q r
triangle_pqs : ¬collinear p q s
triangle_prs : ¬collinear p r s
triangle_qrs : ¬collinear q r s
a : line P := p ⊔ q
⊢ isQuadrangle (p ⊔ q) (q ⊔ r) (r ⊔ s) (s ⊔ p) |
https://github.com/loewenheim/projective-plane.git | 7ade52678ac3df537f38d04de46db9aca752f3ca | ProjectivePlane.lean | ProjectivePlane.dual_quadrangle | [234, 1] | [266, 65] | let b := q ⊔ r | P L : Type
p q r s : P
inst✝ : ProjectivePlane P
triangle_pqr : ¬collinear p q r
triangle_pqs : ¬collinear p q s
triangle_prs : ¬collinear p r s
triangle_qrs : ¬collinear q r s
a : line P := p ⊔ q
⊢ isQuadrangle (p ⊔ q) (q ⊔ r) (r ⊔ s) (s ⊔ p) | P L : Type
p q r s : P
inst✝ : ProjectivePlane P
triangle_pqr : ¬collinear p q r
triangle_pqs : ¬collinear p q s
triangle_prs : ¬collinear p r s
triangle_qrs : ¬collinear q r s
a : line P := p ⊔ q
b : line P := q ⊔ r
⊢ isQuadrangle (p ⊔ q) (q ⊔ r) (r ⊔ s) (s ⊔ p) |
https://github.com/loewenheim/projective-plane.git | 7ade52678ac3df537f38d04de46db9aca752f3ca | ProjectivePlane.lean | ProjectivePlane.dual_quadrangle | [234, 1] | [266, 65] | let c := r ⊔ s | P L : Type
p q r s : P
inst✝ : ProjectivePlane P
triangle_pqr : ¬collinear p q r
triangle_pqs : ¬collinear p q s
triangle_prs : ¬collinear p r s
triangle_qrs : ¬collinear q r s
a : line P := p ⊔ q
b : line P := q ⊔ r
⊢ isQuadrangle (p ⊔ q) (q ⊔ r) (r ⊔ s) (s ⊔ p) | P L : Type
p q r s : P
inst✝ : ProjectivePlane P
triangle_pqr : ¬collinear p q r
triangle_pqs : ¬collinear p q s
triangle_prs : ¬collinear p r s
triangle_qrs : ¬collinear q r s
a : line P := p ⊔ q
b : line P := q ⊔ r
c : line P := r ⊔ s
⊢ isQuadrangle (p ⊔ q) (q ⊔ r) (r ⊔ s) (s ⊔ p) |
https://github.com/loewenheim/projective-plane.git | 7ade52678ac3df537f38d04de46db9aca752f3ca | ProjectivePlane.lean | ProjectivePlane.dual_quadrangle | [234, 1] | [266, 65] | let d := s ⊔ p | P L : Type
p q r s : P
inst✝ : ProjectivePlane P
triangle_pqr : ¬collinear p q r
triangle_pqs : ¬collinear p q s
triangle_prs : ¬collinear p r s
triangle_qrs : ¬collinear q r s
a : line P := p ⊔ q
b : line P := q ⊔ r
c : line P := r ⊔ s
⊢ isQuadrangle (p ⊔ q) (q ⊔ r) (r ⊔ s) (s ⊔ p) | P L : Type
p q r s : P
inst✝ : ProjectivePlane P
triangle_pqr : ¬collinear p q r
triangle_pqs : ¬collinear p q s
triangle_prs : ¬collinear p r s
triangle_qrs : ¬collinear q r s
a : line P := p ⊔ q
b : line P := q ⊔ r
c : line P := r ⊔ s
d : line P := s ⊔ p
⊢ isQuadrangle (p ⊔ q) (q ⊔ r) (r ⊔ s) (s ⊔ p) |
https://github.com/loewenheim/projective-plane.git | 7ade52678ac3df537f38d04de46db9aca752f3ca | ProjectivePlane.lean | ProjectivePlane.dual_quadrangle | [234, 1] | [266, 65] | have triangle_abc : ¬ collinear a b c := by
apply dual_quadrangle_part
exact ⟨triangle_pqr, triangle_pqs, triangle_prs, triangle_qrs⟩ | P L : Type
p q r s : P
inst✝ : ProjectivePlane P
triangle_pqr : ¬collinear p q r
triangle_pqs : ¬collinear p q s
triangle_prs : ¬collinear p r s
triangle_qrs : ¬collinear q r s
a : line P := p ⊔ q
b : line P := q ⊔ r
c : line P := r ⊔ s
d : line P := s ⊔ p
⊢ isQuadrangle (p ⊔ q) (q ⊔ r) (r ⊔ s) (s ⊔ p) | P L : Type
p q r s : P
inst✝ : ProjectivePlane P
triangle_pqr : ¬collinear p q r
triangle_pqs : ¬collinear p q s
triangle_prs : ¬collinear p r s
triangle_qrs : ¬collinear q r s
a : line P := p ⊔ q
b : line P := q ⊔ r
c : line P := r ⊔ s
d : line P := s ⊔ p
triangle_abc : ¬collinear a b c
⊢ isQuadrangle (p ⊔ q) (q ⊔ r) (r ⊔ s) (s ⊔ p) |
https://github.com/loewenheim/projective-plane.git | 7ade52678ac3df537f38d04de46db9aca752f3ca | ProjectivePlane.lean | ProjectivePlane.dual_quadrangle | [234, 1] | [266, 65] | have triangle_abd : ¬ collinear a b d := by
apply triangle_rotate
apply dual_quadrangle_part
apply quadrangle_rotate
apply quadrangle_rotate
apply quadrangle_rotate
exact ⟨triangle_pqr, triangle_pqs, triangle_prs, triangle_qrs⟩ | P L : Type
p q r s : P
inst✝ : ProjectivePlane P
triangle_pqr : ¬collinear p q r
triangle_pqs : ¬collinear p q s
triangle_prs : ¬collinear p r s
triangle_qrs : ¬collinear q r s
a : line P := p ⊔ q
b : line P := q ⊔ r
c : line P := r ⊔ s
d : line P := s ⊔ p
triangle_abc : ¬collinear a b c
⊢ isQuadrangle (p ⊔ q) (q ⊔ r) (r ⊔ s) (s ⊔ p) | P L : Type
p q r s : P
inst✝ : ProjectivePlane P
triangle_pqr : ¬collinear p q r
triangle_pqs : ¬collinear p q s
triangle_prs : ¬collinear p r s
triangle_qrs : ¬collinear q r s
a : line P := p ⊔ q
b : line P := q ⊔ r
c : line P := r ⊔ s
d : line P := s ⊔ p
triangle_abc : ¬collinear a b c
triangle_abd : ¬collinear a b d
⊢ isQuadrangle (p ⊔ q) (q ⊔ r) (r ⊔ s) (s ⊔ p) |
https://github.com/loewenheim/projective-plane.git | 7ade52678ac3df537f38d04de46db9aca752f3ca | ProjectivePlane.lean | ProjectivePlane.dual_quadrangle | [234, 1] | [266, 65] | have triangle_acd : ¬ collinear a c d := by
apply triangle_rotate
apply triangle_rotate
apply dual_quadrangle_part
apply quadrangle_rotate
apply quadrangle_rotate
exact ⟨triangle_pqr, triangle_pqs, triangle_prs, triangle_qrs⟩ | P L : Type
p q r s : P
inst✝ : ProjectivePlane P
triangle_pqr : ¬collinear p q r
triangle_pqs : ¬collinear p q s
triangle_prs : ¬collinear p r s
triangle_qrs : ¬collinear q r s
a : line P := p ⊔ q
b : line P := q ⊔ r
c : line P := r ⊔ s
d : line P := s ⊔ p
triangle_abc : ¬collinear a b c
triangle_abd : ¬collinear a b d
⊢ isQuadrangle (p ⊔ q) (q ⊔ r) (r ⊔ s) (s ⊔ p) | P L : Type
p q r s : P
inst✝ : ProjectivePlane P
triangle_pqr : ¬collinear p q r
triangle_pqs : ¬collinear p q s
triangle_prs : ¬collinear p r s
triangle_qrs : ¬collinear q r s
a : line P := p ⊔ q
b : line P := q ⊔ r
c : line P := r ⊔ s
d : line P := s ⊔ p
triangle_abc : ¬collinear a b c
triangle_abd : ¬collinear a b d
triangle_acd : ¬collinear a c d
⊢ isQuadrangle (p ⊔ q) (q ⊔ r) (r ⊔ s) (s ⊔ p) |
https://github.com/loewenheim/projective-plane.git | 7ade52678ac3df537f38d04de46db9aca752f3ca | ProjectivePlane.lean | ProjectivePlane.dual_quadrangle | [234, 1] | [266, 65] | have triangle_bcd : ¬ collinear b c d := by
apply dual_quadrangle_part
apply quadrangle_rotate
exact ⟨triangle_pqr, triangle_pqs, triangle_prs, triangle_qrs⟩ | P L : Type
p q r s : P
inst✝ : ProjectivePlane P
triangle_pqr : ¬collinear p q r
triangle_pqs : ¬collinear p q s
triangle_prs : ¬collinear p r s
triangle_qrs : ¬collinear q r s
a : line P := p ⊔ q
b : line P := q ⊔ r
c : line P := r ⊔ s
d : line P := s ⊔ p
triangle_abc : ¬collinear a b c
triangle_abd : ¬collinear a b d
triangle_acd : ¬collinear a c d
⊢ isQuadrangle (p ⊔ q) (q ⊔ r) (r ⊔ s) (s ⊔ p) | P L : Type
p q r s : P
inst✝ : ProjectivePlane P
triangle_pqr : ¬collinear p q r
triangle_pqs : ¬collinear p q s
triangle_prs : ¬collinear p r s
triangle_qrs : ¬collinear q r s
a : line P := p ⊔ q
b : line P := q ⊔ r
c : line P := r ⊔ s
d : line P := s ⊔ p
triangle_abc : ¬collinear a b c
triangle_abd : ¬collinear a b d
triangle_acd : ¬collinear a c d
triangle_bcd : ¬collinear b c d
⊢ isQuadrangle (p ⊔ q) (q ⊔ r) (r ⊔ s) (s ⊔ p) |
https://github.com/loewenheim/projective-plane.git | 7ade52678ac3df537f38d04de46db9aca752f3ca | ProjectivePlane.lean | ProjectivePlane.dual_quadrangle | [234, 1] | [266, 65] | exact ⟨triangle_abc, triangle_abd, triangle_acd, triangle_bcd⟩ | P L : Type
p q r s : P
inst✝ : ProjectivePlane P
triangle_pqr : ¬collinear p q r
triangle_pqs : ¬collinear p q s
triangle_prs : ¬collinear p r s
triangle_qrs : ¬collinear q r s
a : line P := p ⊔ q
b : line P := q ⊔ r
c : line P := r ⊔ s
d : line P := s ⊔ p
triangle_abc : ¬collinear a b c
triangle_abd : ¬collinear a b d
triangle_acd : ¬collinear a c d
triangle_bcd : ¬collinear b c d
⊢ isQuadrangle (p ⊔ q) (q ⊔ r) (r ⊔ s) (s ⊔ p) | no goals |
https://github.com/loewenheim/projective-plane.git | 7ade52678ac3df537f38d04de46db9aca752f3ca | ProjectivePlane.lean | ProjectivePlane.dual_quadrangle | [234, 1] | [266, 65] | apply dual_quadrangle_part | P L : Type
p q r s : P
inst✝ : ProjectivePlane P
triangle_pqr : ¬collinear p q r
triangle_pqs : ¬collinear p q s
triangle_prs : ¬collinear p r s
triangle_qrs : ¬collinear q r s
a : line P := p ⊔ q
b : line P := q ⊔ r
c : line P := r ⊔ s
d : line P := s ⊔ p
⊢ ¬collinear a b c | case a
P L : Type
p q r s : P
inst✝ : ProjectivePlane P
triangle_pqr : ¬collinear p q r
triangle_pqs : ¬collinear p q s
triangle_prs : ¬collinear p r s
triangle_qrs : ¬collinear q r s
a : line P := p ⊔ q
b : line P := q ⊔ r
c : line P := r ⊔ s
d : line P := s ⊔ p
⊢ isQuadrangle p q r s |
https://github.com/loewenheim/projective-plane.git | 7ade52678ac3df537f38d04de46db9aca752f3ca | ProjectivePlane.lean | ProjectivePlane.dual_quadrangle | [234, 1] | [266, 65] | exact ⟨triangle_pqr, triangle_pqs, triangle_prs, triangle_qrs⟩ | case a
P L : Type
p q r s : P
inst✝ : ProjectivePlane P
triangle_pqr : ¬collinear p q r
triangle_pqs : ¬collinear p q s
triangle_prs : ¬collinear p r s
triangle_qrs : ¬collinear q r s
a : line P := p ⊔ q
b : line P := q ⊔ r
c : line P := r ⊔ s
d : line P := s ⊔ p
⊢ isQuadrangle p q r s | no goals |
https://github.com/loewenheim/projective-plane.git | 7ade52678ac3df537f38d04de46db9aca752f3ca | ProjectivePlane.lean | ProjectivePlane.dual_quadrangle | [234, 1] | [266, 65] | apply triangle_rotate | P L : Type
p q r s : P
inst✝ : ProjectivePlane P
triangle_pqr : ¬collinear p q r
triangle_pqs : ¬collinear p q s
triangle_prs : ¬collinear p r s
triangle_qrs : ¬collinear q r s
a : line P := p ⊔ q
b : line P := q ⊔ r
c : line P := r ⊔ s
d : line P := s ⊔ p
triangle_abc : ¬collinear a b c
⊢ ¬collinear a b d | case a
P L : Type
p q r s : P
inst✝ : ProjectivePlane P
triangle_pqr : ¬collinear p q r
triangle_pqs : ¬collinear p q s
triangle_prs : ¬collinear p r s
triangle_qrs : ¬collinear q r s
a : line P := p ⊔ q
b : line P := q ⊔ r
c : line P := r ⊔ s
d : line P := s ⊔ p
triangle_abc : ¬collinear a b c
⊢ ¬collinear d a b |
https://github.com/loewenheim/projective-plane.git | 7ade52678ac3df537f38d04de46db9aca752f3ca | ProjectivePlane.lean | ProjectivePlane.dual_quadrangle | [234, 1] | [266, 65] | apply dual_quadrangle_part | case a
P L : Type
p q r s : P
inst✝ : ProjectivePlane P
triangle_pqr : ¬collinear p q r
triangle_pqs : ¬collinear p q s
triangle_prs : ¬collinear p r s
triangle_qrs : ¬collinear q r s
a : line P := p ⊔ q
b : line P := q ⊔ r
c : line P := r ⊔ s
d : line P := s ⊔ p
triangle_abc : ¬collinear a b c
⊢ ¬collinear d a b | case a.a
P L : Type
p q r s : P
inst✝ : ProjectivePlane P
triangle_pqr : ¬collinear p q r
triangle_pqs : ¬collinear p q s
triangle_prs : ¬collinear p r s
triangle_qrs : ¬collinear q r s
a : line P := p ⊔ q
b : line P := q ⊔ r
c : line P := r ⊔ s
d : line P := s ⊔ p
triangle_abc : ¬collinear a b c
⊢ isQuadrangle s p q r |
https://github.com/loewenheim/projective-plane.git | 7ade52678ac3df537f38d04de46db9aca752f3ca | ProjectivePlane.lean | ProjectivePlane.dual_quadrangle | [234, 1] | [266, 65] | apply quadrangle_rotate | case a.a
P L : Type
p q r s : P
inst✝ : ProjectivePlane P
triangle_pqr : ¬collinear p q r
triangle_pqs : ¬collinear p q s
triangle_prs : ¬collinear p r s
triangle_qrs : ¬collinear q r s
a : line P := p ⊔ q
b : line P := q ⊔ r
c : line P := r ⊔ s
d : line P := s ⊔ p
triangle_abc : ¬collinear a b c
⊢ isQuadrangle s p q r | case a.a.a
P L : Type
p q r s : P
inst✝ : ProjectivePlane P
triangle_pqr : ¬collinear p q r
triangle_pqs : ¬collinear p q s
triangle_prs : ¬collinear p r s
triangle_qrs : ¬collinear q r s
a : line P := p ⊔ q
b : line P := q ⊔ r
c : line P := r ⊔ s
d : line P := s ⊔ p
triangle_abc : ¬collinear a b c
⊢ isQuadrangle r s p q |
https://github.com/loewenheim/projective-plane.git | 7ade52678ac3df537f38d04de46db9aca752f3ca | ProjectivePlane.lean | ProjectivePlane.dual_quadrangle | [234, 1] | [266, 65] | apply quadrangle_rotate | case a.a.a
P L : Type
p q r s : P
inst✝ : ProjectivePlane P
triangle_pqr : ¬collinear p q r
triangle_pqs : ¬collinear p q s
triangle_prs : ¬collinear p r s
triangle_qrs : ¬collinear q r s
a : line P := p ⊔ q
b : line P := q ⊔ r
c : line P := r ⊔ s
d : line P := s ⊔ p
triangle_abc : ¬collinear a b c
⊢ isQuadrangle r s p q | case a.a.a.a
P L : Type
p q r s : P
inst✝ : ProjectivePlane P
triangle_pqr : ¬collinear p q r
triangle_pqs : ¬collinear p q s
triangle_prs : ¬collinear p r s
triangle_qrs : ¬collinear q r s
a : line P := p ⊔ q
b : line P := q ⊔ r
c : line P := r ⊔ s
d : line P := s ⊔ p
triangle_abc : ¬collinear a b c
⊢ isQuadrangle q r s p |
https://github.com/loewenheim/projective-plane.git | 7ade52678ac3df537f38d04de46db9aca752f3ca | ProjectivePlane.lean | ProjectivePlane.dual_quadrangle | [234, 1] | [266, 65] | apply quadrangle_rotate | case a.a.a.a
P L : Type
p q r s : P
inst✝ : ProjectivePlane P
triangle_pqr : ¬collinear p q r
triangle_pqs : ¬collinear p q s
triangle_prs : ¬collinear p r s
triangle_qrs : ¬collinear q r s
a : line P := p ⊔ q
b : line P := q ⊔ r
c : line P := r ⊔ s
d : line P := s ⊔ p
triangle_abc : ¬collinear a b c
⊢ isQuadrangle q r s p | case a.a.a.a.a
P L : Type
p q r s : P
inst✝ : ProjectivePlane P
triangle_pqr : ¬collinear p q r
triangle_pqs : ¬collinear p q s
triangle_prs : ¬collinear p r s
triangle_qrs : ¬collinear q r s
a : line P := p ⊔ q
b : line P := q ⊔ r
c : line P := r ⊔ s
d : line P := s ⊔ p
triangle_abc : ¬collinear a b c
⊢ isQuadrangle p q r s |
https://github.com/loewenheim/projective-plane.git | 7ade52678ac3df537f38d04de46db9aca752f3ca | ProjectivePlane.lean | ProjectivePlane.dual_quadrangle | [234, 1] | [266, 65] | exact ⟨triangle_pqr, triangle_pqs, triangle_prs, triangle_qrs⟩ | case a.a.a.a.a
P L : Type
p q r s : P
inst✝ : ProjectivePlane P
triangle_pqr : ¬collinear p q r
triangle_pqs : ¬collinear p q s
triangle_prs : ¬collinear p r s
triangle_qrs : ¬collinear q r s
a : line P := p ⊔ q
b : line P := q ⊔ r
c : line P := r ⊔ s
d : line P := s ⊔ p
triangle_abc : ¬collinear a b c
⊢ isQuadrangle p q r s | no goals |
https://github.com/loewenheim/projective-plane.git | 7ade52678ac3df537f38d04de46db9aca752f3ca | ProjectivePlane.lean | ProjectivePlane.dual_quadrangle | [234, 1] | [266, 65] | apply triangle_rotate | P L : Type
p q r s : P
inst✝ : ProjectivePlane P
triangle_pqr : ¬collinear p q r
triangle_pqs : ¬collinear p q s
triangle_prs : ¬collinear p r s
triangle_qrs : ¬collinear q r s
a : line P := p ⊔ q
b : line P := q ⊔ r
c : line P := r ⊔ s
d : line P := s ⊔ p
triangle_abc : ¬collinear a b c
triangle_abd : ¬collinear a b d
⊢ ¬collinear a c d | case a
P L : Type
p q r s : P
inst✝ : ProjectivePlane P
triangle_pqr : ¬collinear p q r
triangle_pqs : ¬collinear p q s
triangle_prs : ¬collinear p r s
triangle_qrs : ¬collinear q r s
a : line P := p ⊔ q
b : line P := q ⊔ r
c : line P := r ⊔ s
d : line P := s ⊔ p
triangle_abc : ¬collinear a b c
triangle_abd : ¬collinear a b d
⊢ ¬collinear d a c |
https://github.com/loewenheim/projective-plane.git | 7ade52678ac3df537f38d04de46db9aca752f3ca | ProjectivePlane.lean | ProjectivePlane.dual_quadrangle | [234, 1] | [266, 65] | apply triangle_rotate | case a
P L : Type
p q r s : P
inst✝ : ProjectivePlane P
triangle_pqr : ¬collinear p q r
triangle_pqs : ¬collinear p q s
triangle_prs : ¬collinear p r s
triangle_qrs : ¬collinear q r s
a : line P := p ⊔ q
b : line P := q ⊔ r
c : line P := r ⊔ s
d : line P := s ⊔ p
triangle_abc : ¬collinear a b c
triangle_abd : ¬collinear a b d
⊢ ¬collinear d a c | case a.a
P L : Type
p q r s : P
inst✝ : ProjectivePlane P
triangle_pqr : ¬collinear p q r
triangle_pqs : ¬collinear p q s
triangle_prs : ¬collinear p r s
triangle_qrs : ¬collinear q r s
a : line P := p ⊔ q
b : line P := q ⊔ r
c : line P := r ⊔ s
d : line P := s ⊔ p
triangle_abc : ¬collinear a b c
triangle_abd : ¬collinear a b d
⊢ ¬collinear c d a |
https://github.com/loewenheim/projective-plane.git | 7ade52678ac3df537f38d04de46db9aca752f3ca | ProjectivePlane.lean | ProjectivePlane.dual_quadrangle | [234, 1] | [266, 65] | apply dual_quadrangle_part | case a.a
P L : Type
p q r s : P
inst✝ : ProjectivePlane P
triangle_pqr : ¬collinear p q r
triangle_pqs : ¬collinear p q s
triangle_prs : ¬collinear p r s
triangle_qrs : ¬collinear q r s
a : line P := p ⊔ q
b : line P := q ⊔ r
c : line P := r ⊔ s
d : line P := s ⊔ p
triangle_abc : ¬collinear a b c
triangle_abd : ¬collinear a b d
⊢ ¬collinear c d a | case a.a.a
P L : Type
p q r s : P
inst✝ : ProjectivePlane P
triangle_pqr : ¬collinear p q r
triangle_pqs : ¬collinear p q s
triangle_prs : ¬collinear p r s
triangle_qrs : ¬collinear q r s
a : line P := p ⊔ q
b : line P := q ⊔ r
c : line P := r ⊔ s
d : line P := s ⊔ p
triangle_abc : ¬collinear a b c
triangle_abd : ¬collinear a b d
⊢ isQuadrangle r s p q |
https://github.com/loewenheim/projective-plane.git | 7ade52678ac3df537f38d04de46db9aca752f3ca | ProjectivePlane.lean | ProjectivePlane.dual_quadrangle | [234, 1] | [266, 65] | apply quadrangle_rotate | case a.a.a
P L : Type
p q r s : P
inst✝ : ProjectivePlane P
triangle_pqr : ¬collinear p q r
triangle_pqs : ¬collinear p q s
triangle_prs : ¬collinear p r s
triangle_qrs : ¬collinear q r s
a : line P := p ⊔ q
b : line P := q ⊔ r
c : line P := r ⊔ s
d : line P := s ⊔ p
triangle_abc : ¬collinear a b c
triangle_abd : ¬collinear a b d
⊢ isQuadrangle r s p q | case a.a.a.a
P L : Type
p q r s : P
inst✝ : ProjectivePlane P
triangle_pqr : ¬collinear p q r
triangle_pqs : ¬collinear p q s
triangle_prs : ¬collinear p r s
triangle_qrs : ¬collinear q r s
a : line P := p ⊔ q
b : line P := q ⊔ r
c : line P := r ⊔ s
d : line P := s ⊔ p
triangle_abc : ¬collinear a b c
triangle_abd : ¬collinear a b d
⊢ isQuadrangle q r s p |
https://github.com/loewenheim/projective-plane.git | 7ade52678ac3df537f38d04de46db9aca752f3ca | ProjectivePlane.lean | ProjectivePlane.dual_quadrangle | [234, 1] | [266, 65] | apply quadrangle_rotate | case a.a.a.a
P L : Type
p q r s : P
inst✝ : ProjectivePlane P
triangle_pqr : ¬collinear p q r
triangle_pqs : ¬collinear p q s
triangle_prs : ¬collinear p r s
triangle_qrs : ¬collinear q r s
a : line P := p ⊔ q
b : line P := q ⊔ r
c : line P := r ⊔ s
d : line P := s ⊔ p
triangle_abc : ¬collinear a b c
triangle_abd : ¬collinear a b d
⊢ isQuadrangle q r s p | case a.a.a.a.a
P L : Type
p q r s : P
inst✝ : ProjectivePlane P
triangle_pqr : ¬collinear p q r
triangle_pqs : ¬collinear p q s
triangle_prs : ¬collinear p r s
triangle_qrs : ¬collinear q r s
a : line P := p ⊔ q
b : line P := q ⊔ r
c : line P := r ⊔ s
d : line P := s ⊔ p
triangle_abc : ¬collinear a b c
triangle_abd : ¬collinear a b d
⊢ isQuadrangle p q r s |
https://github.com/loewenheim/projective-plane.git | 7ade52678ac3df537f38d04de46db9aca752f3ca | ProjectivePlane.lean | ProjectivePlane.dual_quadrangle | [234, 1] | [266, 65] | exact ⟨triangle_pqr, triangle_pqs, triangle_prs, triangle_qrs⟩ | case a.a.a.a.a
P L : Type
p q r s : P
inst✝ : ProjectivePlane P
triangle_pqr : ¬collinear p q r
triangle_pqs : ¬collinear p q s
triangle_prs : ¬collinear p r s
triangle_qrs : ¬collinear q r s
a : line P := p ⊔ q
b : line P := q ⊔ r
c : line P := r ⊔ s
d : line P := s ⊔ p
triangle_abc : ¬collinear a b c
triangle_abd : ¬collinear a b d
⊢ isQuadrangle p q r s | no goals |
https://github.com/loewenheim/projective-plane.git | 7ade52678ac3df537f38d04de46db9aca752f3ca | ProjectivePlane.lean | ProjectivePlane.dual_quadrangle | [234, 1] | [266, 65] | apply dual_quadrangle_part | P L : Type
p q r s : P
inst✝ : ProjectivePlane P
triangle_pqr : ¬collinear p q r
triangle_pqs : ¬collinear p q s
triangle_prs : ¬collinear p r s
triangle_qrs : ¬collinear q r s
a : line P := p ⊔ q
b : line P := q ⊔ r
c : line P := r ⊔ s
d : line P := s ⊔ p
triangle_abc : ¬collinear a b c
triangle_abd : ¬collinear a b d
triangle_acd : ¬collinear a c d
⊢ ¬collinear b c d | case a
P L : Type
p q r s : P
inst✝ : ProjectivePlane P
triangle_pqr : ¬collinear p q r
triangle_pqs : ¬collinear p q s
triangle_prs : ¬collinear p r s
triangle_qrs : ¬collinear q r s
a : line P := p ⊔ q
b : line P := q ⊔ r
c : line P := r ⊔ s
d : line P := s ⊔ p
triangle_abc : ¬collinear a b c
triangle_abd : ¬collinear a b d
triangle_acd : ¬collinear a c d
⊢ isQuadrangle q r s p |
https://github.com/loewenheim/projective-plane.git | 7ade52678ac3df537f38d04de46db9aca752f3ca | ProjectivePlane.lean | ProjectivePlane.dual_quadrangle | [234, 1] | [266, 65] | apply quadrangle_rotate | case a
P L : Type
p q r s : P
inst✝ : ProjectivePlane P
triangle_pqr : ¬collinear p q r
triangle_pqs : ¬collinear p q s
triangle_prs : ¬collinear p r s
triangle_qrs : ¬collinear q r s
a : line P := p ⊔ q
b : line P := q ⊔ r
c : line P := r ⊔ s
d : line P := s ⊔ p
triangle_abc : ¬collinear a b c
triangle_abd : ¬collinear a b d
triangle_acd : ¬collinear a c d
⊢ isQuadrangle q r s p | case a.a
P L : Type
p q r s : P
inst✝ : ProjectivePlane P
triangle_pqr : ¬collinear p q r
triangle_pqs : ¬collinear p q s
triangle_prs : ¬collinear p r s
triangle_qrs : ¬collinear q r s
a : line P := p ⊔ q
b : line P := q ⊔ r
c : line P := r ⊔ s
d : line P := s ⊔ p
triangle_abc : ¬collinear a b c
triangle_abd : ¬collinear a b d
triangle_acd : ¬collinear a c d
⊢ isQuadrangle p q r s |
https://github.com/loewenheim/projective-plane.git | 7ade52678ac3df537f38d04de46db9aca752f3ca | ProjectivePlane.lean | ProjectivePlane.dual_quadrangle | [234, 1] | [266, 65] | exact ⟨triangle_pqr, triangle_pqs, triangle_prs, triangle_qrs⟩ | case a.a
P L : Type
p q r s : P
inst✝ : ProjectivePlane P
triangle_pqr : ¬collinear p q r
triangle_pqs : ¬collinear p q s
triangle_prs : ¬collinear p r s
triangle_qrs : ¬collinear q r s
a : line P := p ⊔ q
b : line P := q ⊔ r
c : line P := r ⊔ s
d : line P := s ⊔ p
triangle_abc : ¬collinear a b c
triangle_abd : ¬collinear a b d
triangle_acd : ¬collinear a c d
⊢ isQuadrangle p q r s | no goals |
https://github.com/loewenheim/projective-plane.git | 7ade52678ac3df537f38d04de46db9aca752f3ca | ProjectivePlane.lean | ProjectivePlane.dual_involution | [281, 1] | [283, 19] | rfl | P L : Type
p q r s : P
h : ProjectivePlane P
⊢ let h' := inferInstance;
h' = h | no goals |
https://github.com/loewenheim/projective-plane.git | 7ade52678ac3df537f38d04de46db9aca752f3ca | ProjectivePlane.lean | Fano.Fano.quadrangle1236 | [383, 1] | [384, 83] | simp[collinear] | P L : Type
⊢ ¬collinear Points.p1 Points.p2 Points.p3 | no goals |
https://github.com/loewenheim/projective-plane.git | 7ade52678ac3df537f38d04de46db9aca752f3ca | ProjectivePlane.lean | Fano.Fano.quadrangle1236 | [383, 1] | [384, 83] | simp[collinear] | P L : Type
⊢ ¬collinear Points.p1 Points.p2 Points.p6 | no goals |
https://github.com/loewenheim/projective-plane.git | 7ade52678ac3df537f38d04de46db9aca752f3ca | ProjectivePlane.lean | Fano.Fano.quadrangle1236 | [383, 1] | [384, 83] | simp[collinear] | P L : Type
⊢ ¬collinear Points.p1 Points.p3 Points.p6 | no goals |
https://github.com/loewenheim/projective-plane.git | 7ade52678ac3df537f38d04de46db9aca752f3ca | ProjectivePlane.lean | Fano.Fano.quadrangle1236 | [383, 1] | [384, 83] | simp[collinear] | P L : Type
⊢ ¬collinear Points.p2 Points.p3 Points.p6 | no goals |
https://github.com/loewenheim/projective-plane.git | 7ade52678ac3df537f38d04de46db9aca752f3ca | ProjectivePlane.lean | Fano.Fano.point_line_uniq | [386, 1] | [387, 9] | decide | P L : Type
⊢ ∀ {p q : Points} {l m : Lines}, p ∈ l → q ∈ l → p ∈ m → q ∈ m → p = q ∨ l = m | no goals |
https://github.com/benjaminfjones/reckonlean.git | 8768f7342ba226cfc2d7b92e47432f1da66eff25 | ReckonLean/Fol.lean | max_var_name_mem | [399, 1] | [418, 83] | intro v vars | ⊢ ∀ (v : String) (vars : List String), v ∈ vars → String.length v ≤ max_var_name vars | v : String
vars : List String
⊢ v ∈ vars → String.length v ≤ max_var_name vars |
https://github.com/benjaminfjones/reckonlean.git | 8768f7342ba226cfc2d7b92e47432f1da66eff25 | ReckonLean/Fol.lean | max_var_name_mem | [399, 1] | [418, 83] | induction vars with
| nil =>
intro hm
exact absurd hm (List.not_mem_nil v)
| cons x xs ih =>
intro h
rw [List.mem_cons] at h
cases h with
| inl hvx =>
unfold max_var_name
conv => rhs; rw [← hvx]
apply Nat.le_max_left
| inr hvxs =>
unfold max_var_name
calc
v.length ≤ max_var_name xs := ih hvxs
_ ≤ Nat.max x.length (max_var_name xs) := by apply Nat.le_max_right | v : String
vars : List String
⊢ v ∈ vars → String.length v ≤ max_var_name vars | no goals |
https://github.com/benjaminfjones/reckonlean.git | 8768f7342ba226cfc2d7b92e47432f1da66eff25 | ReckonLean/Fol.lean | max_var_name_mem | [399, 1] | [418, 83] | intro hm | case nil
v : String
⊢ v ∈ [] → String.length v ≤ max_var_name [] | case nil
v : String
hm : v ∈ []
⊢ String.length v ≤ max_var_name [] |
https://github.com/benjaminfjones/reckonlean.git | 8768f7342ba226cfc2d7b92e47432f1da66eff25 | ReckonLean/Fol.lean | max_var_name_mem | [399, 1] | [418, 83] | exact absurd hm (List.not_mem_nil v) | case nil
v : String
hm : v ∈ []
⊢ String.length v ≤ max_var_name [] | no goals |
https://github.com/benjaminfjones/reckonlean.git | 8768f7342ba226cfc2d7b92e47432f1da66eff25 | ReckonLean/Fol.lean | max_var_name_mem | [399, 1] | [418, 83] | intro h | case cons
v x : String
xs : List String
ih : v ∈ xs → String.length v ≤ max_var_name xs
⊢ v ∈ x :: xs → String.length v ≤ max_var_name (x :: xs) | case cons
v x : String
xs : List String
ih : v ∈ xs → String.length v ≤ max_var_name xs
h : v ∈ x :: xs
⊢ String.length v ≤ max_var_name (x :: xs) |
https://github.com/benjaminfjones/reckonlean.git | 8768f7342ba226cfc2d7b92e47432f1da66eff25 | ReckonLean/Fol.lean | max_var_name_mem | [399, 1] | [418, 83] | rw [List.mem_cons] at h | case cons
v x : String
xs : List String
ih : v ∈ xs → String.length v ≤ max_var_name xs
h : v ∈ x :: xs
⊢ String.length v ≤ max_var_name (x :: xs) | case cons
v x : String
xs : List String
ih : v ∈ xs → String.length v ≤ max_var_name xs
h : v = x ∨ v ∈ xs
⊢ String.length v ≤ max_var_name (x :: xs) |
https://github.com/benjaminfjones/reckonlean.git | 8768f7342ba226cfc2d7b92e47432f1da66eff25 | ReckonLean/Fol.lean | max_var_name_mem | [399, 1] | [418, 83] | cases h with
| inl hvx =>
unfold max_var_name
conv => rhs; rw [← hvx]
apply Nat.le_max_left
| inr hvxs =>
unfold max_var_name
calc
v.length ≤ max_var_name xs := ih hvxs
_ ≤ Nat.max x.length (max_var_name xs) := by apply Nat.le_max_right | case cons
v x : String
xs : List String
ih : v ∈ xs → String.length v ≤ max_var_name xs
h : v = x ∨ v ∈ xs
⊢ String.length v ≤ max_var_name (x :: xs) | no goals |
https://github.com/benjaminfjones/reckonlean.git | 8768f7342ba226cfc2d7b92e47432f1da66eff25 | ReckonLean/Fol.lean | max_var_name_mem | [399, 1] | [418, 83] | unfold max_var_name | case cons.inl
v x : String
xs : List String
ih : v ∈ xs → String.length v ≤ max_var_name xs
hvx : v = x
⊢ String.length v ≤ max_var_name (x :: xs) | case cons.inl
v x : String
xs : List String
ih : v ∈ xs → String.length v ≤ max_var_name xs
hvx : v = x
⊢ String.length v ≤ Nat.max (String.length x) (max_var_name xs) |
https://github.com/benjaminfjones/reckonlean.git | 8768f7342ba226cfc2d7b92e47432f1da66eff25 | ReckonLean/Fol.lean | max_var_name_mem | [399, 1] | [418, 83] | conv => rhs; rw [← hvx] | case cons.inl
v x : String
xs : List String
ih : v ∈ xs → String.length v ≤ max_var_name xs
hvx : v = x
⊢ String.length v ≤ Nat.max (String.length x) (max_var_name xs) | case cons.inl
v x : String
xs : List String
ih : v ∈ xs → String.length v ≤ max_var_name xs
hvx : v = x
⊢ String.length v ≤ Nat.max (String.length v) (max_var_name xs) |
https://github.com/benjaminfjones/reckonlean.git | 8768f7342ba226cfc2d7b92e47432f1da66eff25 | ReckonLean/Fol.lean | max_var_name_mem | [399, 1] | [418, 83] | apply Nat.le_max_left | case cons.inl
v x : String
xs : List String
ih : v ∈ xs → String.length v ≤ max_var_name xs
hvx : v = x
⊢ String.length v ≤ Nat.max (String.length v) (max_var_name xs) | no goals |
https://github.com/benjaminfjones/reckonlean.git | 8768f7342ba226cfc2d7b92e47432f1da66eff25 | ReckonLean/Fol.lean | max_var_name_mem | [399, 1] | [418, 83] | unfold max_var_name | case cons.inr
v x : String
xs : List String
ih : v ∈ xs → String.length v ≤ max_var_name xs
hvxs : v ∈ xs
⊢ String.length v ≤ max_var_name (x :: xs) | case cons.inr
v x : String
xs : List String
ih : v ∈ xs → String.length v ≤ max_var_name xs
hvxs : v ∈ xs
⊢ String.length v ≤ Nat.max (String.length x) (max_var_name xs) |
https://github.com/benjaminfjones/reckonlean.git | 8768f7342ba226cfc2d7b92e47432f1da66eff25 | ReckonLean/Fol.lean | max_var_name_mem | [399, 1] | [418, 83] | calc
v.length ≤ max_var_name xs := ih hvxs
_ ≤ Nat.max x.length (max_var_name xs) := by apply Nat.le_max_right | case cons.inr
v x : String
xs : List String
ih : v ∈ xs → String.length v ≤ max_var_name xs
hvxs : v ∈ xs
⊢ String.length v ≤ Nat.max (String.length x) (max_var_name xs) | no goals |
https://github.com/benjaminfjones/reckonlean.git | 8768f7342ba226cfc2d7b92e47432f1da66eff25 | ReckonLean/Fol.lean | max_var_name_mem | [399, 1] | [418, 83] | apply Nat.le_max_right | v x : String
xs : List String
ih : v ∈ xs → String.length v ≤ max_var_name xs
hvxs : v ∈ xs
⊢ max_var_name xs ≤ Nat.max (String.length x) (max_var_name xs) | no goals |
https://github.com/benjaminfjones/reckonlean.git | 8768f7342ba226cfc2d7b92e47432f1da66eff25 | ReckonLean/Parser.lean | lexwhile_monotonic | [35, 1] | [42, 31] | intro p inp | ⊢ ∀ (p : Char → Bool) (inp : List Char), List.length (lexwhile p inp).snd ≤ List.length inp | p : Char → Bool
inp : List Char
⊢ List.length (lexwhile p inp).snd ≤ List.length inp |
https://github.com/benjaminfjones/reckonlean.git | 8768f7342ba226cfc2d7b92e47432f1da66eff25 | ReckonLean/Parser.lean | lexwhile_monotonic | [35, 1] | [42, 31] | induction inp with
| nil => simp [lexwhile]
| cons c cs ih =>
simp [lexwhile]
cases p c <;> simp
exact Nat.le_succ_of_le ih | p : Char → Bool
inp : List Char
⊢ List.length (lexwhile p inp).snd ≤ List.length inp | no goals |
https://github.com/benjaminfjones/reckonlean.git | 8768f7342ba226cfc2d7b92e47432f1da66eff25 | ReckonLean/Parser.lean | lexwhile_monotonic | [35, 1] | [42, 31] | simp [lexwhile] | case nil
p : Char → Bool
⊢ List.length (lexwhile p []).snd ≤ List.length [] | no goals |
https://github.com/benjaminfjones/reckonlean.git | 8768f7342ba226cfc2d7b92e47432f1da66eff25 | ReckonLean/Parser.lean | lexwhile_monotonic | [35, 1] | [42, 31] | simp [lexwhile] | case cons
p : Char → Bool
c : Char
cs : List Char
ih : List.length (lexwhile p cs).snd ≤ List.length cs
⊢ List.length (lexwhile p (c :: cs)).snd ≤ List.length (c :: cs) | case cons
p : Char → Bool
c : Char
cs : List Char
ih : List.length (lexwhile p cs).snd ≤ List.length cs
⊢ List.length
(if p c = true then (String.append (Char.toString c) (lexwhile p cs).fst, (lexwhile p cs).snd)
else ("", c :: cs)).snd ≤
Nat.succ (List.length cs) |
https://github.com/benjaminfjones/reckonlean.git | 8768f7342ba226cfc2d7b92e47432f1da66eff25 | ReckonLean/Parser.lean | lexwhile_monotonic | [35, 1] | [42, 31] | cases p c <;> simp | case cons
p : Char → Bool
c : Char
cs : List Char
ih : List.length (lexwhile p cs).snd ≤ List.length cs
⊢ List.length
(if p c = true then (String.append (Char.toString c) (lexwhile p cs).fst, (lexwhile p cs).snd)
else ("", c :: cs)).snd ≤
Nat.succ (List.length cs) | case cons.true
p : Char → Bool
c : Char
cs : List Char
ih : List.length (lexwhile p cs).snd ≤ List.length cs
⊢ List.length (lexwhile p cs).snd ≤ Nat.succ (List.length cs) |
https://github.com/benjaminfjones/reckonlean.git | 8768f7342ba226cfc2d7b92e47432f1da66eff25 | ReckonLean/Parser.lean | lexwhile_monotonic | [35, 1] | [42, 31] | exact Nat.le_succ_of_le ih | case cons.true
p : Char → Bool
c : Char
cs : List Char
ih : List.length (lexwhile p cs).snd ≤ List.length cs
⊢ List.length (lexwhile p cs).snd ≤ Nat.succ (List.length cs) | no goals |
https://github.com/benjaminfjones/reckonlean.git | 8768f7342ba226cfc2d7b92e47432f1da66eff25 | ReckonLean/Common.lean | Set.merge_length | [105, 1] | [121, 14] | intro l1 l2 | α : Type
inst✝¹ : Ord α
inst✝ : BEq α
⊢ ∀ (l1 l2 : List α), List.length (merge l1 l2) = List.length l1 + List.length l2 | α : Type
inst✝¹ : Ord α
inst✝ : BEq α
l1 l2 : List α
⊢ List.length (merge l1 l2) = List.length l1 + List.length l2 |
https://github.com/benjaminfjones/reckonlean.git | 8768f7342ba226cfc2d7b92e47432f1da66eff25 | ReckonLean/Common.lean | Set.merge_length | [105, 1] | [121, 14] | match l1 with
| .nil => unfold merge; rw [List.length_nil, Nat.zero_add]
| .cons h1 t1 =>
unfold merge
match l2 with
| .nil => simp only [List.length_cons, List.length_nil, Nat.add_zero]
| .cons h2 t2 =>
simp only [List.length_cons]
split
. rw [List.length_cons, merge_length t1 (h2 :: t2)]
simp only [List.length_cons]
omega
. rw [List.length_cons, merge_length (h1 :: t1) t2]
simp only [List.length_cons]
omega | α : Type
inst✝¹ : Ord α
inst✝ : BEq α
l1 l2 : List α
⊢ List.length (merge l1 l2) = List.length l1 + List.length l2 | no goals |
https://github.com/benjaminfjones/reckonlean.git | 8768f7342ba226cfc2d7b92e47432f1da66eff25 | ReckonLean/Common.lean | Set.merge_length | [105, 1] | [121, 14] | unfold merge | α : Type
inst✝¹ : Ord α
inst✝ : BEq α
l1 l2 : List α
⊢ List.length (merge [] l2) = List.length [] + List.length l2 | α : Type
inst✝¹ : Ord α
inst✝ : BEq α
l1 l2 : List α
⊢ List.length l2 = List.length [] + List.length l2 |
https://github.com/benjaminfjones/reckonlean.git | 8768f7342ba226cfc2d7b92e47432f1da66eff25 | ReckonLean/Common.lean | Set.merge_length | [105, 1] | [121, 14] | rw [List.length_nil, Nat.zero_add] | α : Type
inst✝¹ : Ord α
inst✝ : BEq α
l1 l2 : List α
⊢ List.length l2 = List.length [] + List.length l2 | no goals |
https://github.com/benjaminfjones/reckonlean.git | 8768f7342ba226cfc2d7b92e47432f1da66eff25 | ReckonLean/Common.lean | Set.merge_length | [105, 1] | [121, 14] | unfold merge | α : Type
inst✝¹ : Ord α
inst✝ : BEq α
l1 l2 : List α
h1 : α
t1 : List α
⊢ List.length (merge (h1 :: t1) l2) = List.length (h1 :: t1) + List.length l2 | α : Type
inst✝¹ : Ord α
inst✝ : BEq α
l1 l2 : List α
h1 : α
t1 : List α
⊢ List.length
(match l2 with
| [] => h1 :: t1
| l2@h:(h2 :: t2) =>
let_fun x := (_ : List.length t1 + List.length l2 < List.length (h1 :: t1) + List.length l2);
let_fun x := (_ : List.length (h1 :: t1) + List.length t2 < List.length (h1 :: t1) + List.length l2);
if isLE (compare h1 h2) = true then h1 :: merge t1 l2 else h2 :: merge (h1 :: t1) t2) =
List.length (h1 :: t1) + List.length l2 |
https://github.com/benjaminfjones/reckonlean.git | 8768f7342ba226cfc2d7b92e47432f1da66eff25 | ReckonLean/Common.lean | Set.merge_length | [105, 1] | [121, 14] | match l2 with
| .nil => simp only [List.length_cons, List.length_nil, Nat.add_zero]
| .cons h2 t2 =>
simp only [List.length_cons]
split
. rw [List.length_cons, merge_length t1 (h2 :: t2)]
simp only [List.length_cons]
omega
. rw [List.length_cons, merge_length (h1 :: t1) t2]
simp only [List.length_cons]
omega | α : Type
inst✝¹ : Ord α
inst✝ : BEq α
l1 l2 : List α
h1 : α
t1 : List α
⊢ List.length
(match l2 with
| [] => h1 :: t1
| l2@h:(h2 :: t2) =>
let_fun x := (_ : List.length t1 + List.length l2 < List.length (h1 :: t1) + List.length l2);
let_fun x := (_ : List.length (h1 :: t1) + List.length t2 < List.length (h1 :: t1) + List.length l2);
if isLE (compare h1 h2) = true then h1 :: merge t1 l2 else h2 :: merge (h1 :: t1) t2) =
List.length (h1 :: t1) + List.length l2 | no goals |
https://github.com/benjaminfjones/reckonlean.git | 8768f7342ba226cfc2d7b92e47432f1da66eff25 | ReckonLean/Common.lean | Set.merge_length | [105, 1] | [121, 14] | simp only [List.length_cons, List.length_nil, Nat.add_zero] | α : Type
inst✝¹ : Ord α
inst✝ : BEq α
l1 l2 : List α
h1 : α
t1 : List α
⊢ List.length
(match [] with
| [] => h1 :: t1
| l2@h:(h2 :: t2) =>
let_fun x := (_ : List.length t1 + List.length l2 < List.length (h1 :: t1) + List.length l2);
let_fun x := (_ : List.length (h1 :: t1) + List.length t2 < List.length (h1 :: t1) + List.length l2);
if isLE (compare h1 h2) = true then h1 :: merge t1 l2 else h2 :: merge (h1 :: t1) t2) =
List.length (h1 :: t1) + List.length [] | no goals |
https://github.com/benjaminfjones/reckonlean.git | 8768f7342ba226cfc2d7b92e47432f1da66eff25 | ReckonLean/Common.lean | Set.merge_length | [105, 1] | [121, 14] | simp only [List.length_cons] | α : Type
inst✝¹ : Ord α
inst✝ : BEq α
l1 l2 : List α
h1 : α
t1 : List α
h2 : α
t2 : List α
⊢ List.length
(match h2 :: t2 with
| [] => h1 :: t1
| l2@h:(h2 :: t2) =>
let_fun x := (_ : List.length t1 + List.length l2 < List.length (h1 :: t1) + List.length l2);
let_fun x := (_ : List.length (h1 :: t1) + List.length t2 < List.length (h1 :: t1) + List.length l2);
if isLE (compare h1 h2) = true then h1 :: merge t1 l2 else h2 :: merge (h1 :: t1) t2) =
List.length (h1 :: t1) + List.length (h2 :: t2) | α : Type
inst✝¹ : Ord α
inst✝ : BEq α
l1 l2 : List α
h1 : α
t1 : List α
h2 : α
t2 : List α
⊢ List.length (if isLE (compare h1 h2) = true then h1 :: merge t1 (h2 :: t2) else h2 :: merge (h1 :: t1) t2) =
Nat.succ (List.length t1) + Nat.succ (List.length t2) |
https://github.com/benjaminfjones/reckonlean.git | 8768f7342ba226cfc2d7b92e47432f1da66eff25 | ReckonLean/Common.lean | Set.merge_length | [105, 1] | [121, 14] | split | α : Type
inst✝¹ : Ord α
inst✝ : BEq α
l1 l2 : List α
h1 : α
t1 : List α
h2 : α
t2 : List α
⊢ List.length (if isLE (compare h1 h2) = true then h1 :: merge t1 (h2 :: t2) else h2 :: merge (h1 :: t1) t2) =
Nat.succ (List.length t1) + Nat.succ (List.length t2) | case inl
α : Type
inst✝¹ : Ord α
inst✝ : BEq α
l1 l2 : List α
h1 : α
t1 : List α
h2 : α
t2 : List α
h✝ : isLE (compare h1 h2) = true
⊢ List.length (h1 :: merge t1 (h2 :: t2)) = Nat.succ (List.length t1) + Nat.succ (List.length t2)
case inr
α : Type
inst✝¹ : Ord α
inst✝ : BEq α
l1 l2 : List α
h1 : α
t1 : List α
h2 : α
t2 : List α
h✝ : ¬isLE (compare h1 h2) = true
⊢ List.length (h2 :: merge (h1 :: t1) t2) = Nat.succ (List.length t1) + Nat.succ (List.length t2) |
https://github.com/benjaminfjones/reckonlean.git | 8768f7342ba226cfc2d7b92e47432f1da66eff25 | ReckonLean/Common.lean | Set.merge_length | [105, 1] | [121, 14] | . rw [List.length_cons, merge_length t1 (h2 :: t2)]
simp only [List.length_cons]
omega | case inl
α : Type
inst✝¹ : Ord α
inst✝ : BEq α
l1 l2 : List α
h1 : α
t1 : List α
h2 : α
t2 : List α
h✝ : isLE (compare h1 h2) = true
⊢ List.length (h1 :: merge t1 (h2 :: t2)) = Nat.succ (List.length t1) + Nat.succ (List.length t2)
case inr
α : Type
inst✝¹ : Ord α
inst✝ : BEq α
l1 l2 : List α
h1 : α
t1 : List α
h2 : α
t2 : List α
h✝ : ¬isLE (compare h1 h2) = true
⊢ List.length (h2 :: merge (h1 :: t1) t2) = Nat.succ (List.length t1) + Nat.succ (List.length t2) | case inr
α : Type
inst✝¹ : Ord α
inst✝ : BEq α
l1 l2 : List α
h1 : α
t1 : List α
h2 : α
t2 : List α
h✝ : ¬isLE (compare h1 h2) = true
⊢ List.length (h2 :: merge (h1 :: t1) t2) = Nat.succ (List.length t1) + Nat.succ (List.length t2) |
https://github.com/benjaminfjones/reckonlean.git | 8768f7342ba226cfc2d7b92e47432f1da66eff25 | ReckonLean/Common.lean | Set.merge_length | [105, 1] | [121, 14] | . rw [List.length_cons, merge_length (h1 :: t1) t2]
simp only [List.length_cons]
omega | case inr
α : Type
inst✝¹ : Ord α
inst✝ : BEq α
l1 l2 : List α
h1 : α
t1 : List α
h2 : α
t2 : List α
h✝ : ¬isLE (compare h1 h2) = true
⊢ List.length (h2 :: merge (h1 :: t1) t2) = Nat.succ (List.length t1) + Nat.succ (List.length t2) | no goals |
https://github.com/benjaminfjones/reckonlean.git | 8768f7342ba226cfc2d7b92e47432f1da66eff25 | ReckonLean/Common.lean | Set.merge_length | [105, 1] | [121, 14] | rw [List.length_cons, merge_length t1 (h2 :: t2)] | case inl
α : Type
inst✝¹ : Ord α
inst✝ : BEq α
l1 l2 : List α
h1 : α
t1 : List α
h2 : α
t2 : List α
h✝ : isLE (compare h1 h2) = true
⊢ List.length (h1 :: merge t1 (h2 :: t2)) = Nat.succ (List.length t1) + Nat.succ (List.length t2) | case inl
α : Type
inst✝¹ : Ord α
inst✝ : BEq α
l1 l2 : List α
h1 : α
t1 : List α
h2 : α
t2 : List α
h✝ : isLE (compare h1 h2) = true
⊢ Nat.succ (List.length t1 + List.length (h2 :: t2)) = Nat.succ (List.length t1) + Nat.succ (List.length t2) |
https://github.com/benjaminfjones/reckonlean.git | 8768f7342ba226cfc2d7b92e47432f1da66eff25 | ReckonLean/Common.lean | Set.merge_length | [105, 1] | [121, 14] | simp only [List.length_cons] | case inl
α : Type
inst✝¹ : Ord α
inst✝ : BEq α
l1 l2 : List α
h1 : α
t1 : List α
h2 : α
t2 : List α
h✝ : isLE (compare h1 h2) = true
⊢ Nat.succ (List.length t1 + List.length (h2 :: t2)) = Nat.succ (List.length t1) + Nat.succ (List.length t2) | case inl
α : Type
inst✝¹ : Ord α
inst✝ : BEq α
l1 l2 : List α
h1 : α
t1 : List α
h2 : α
t2 : List α
h✝ : isLE (compare h1 h2) = true
⊢ Nat.succ (List.length t1 + Nat.succ (List.length t2)) = Nat.succ (List.length t1) + Nat.succ (List.length t2) |
https://github.com/benjaminfjones/reckonlean.git | 8768f7342ba226cfc2d7b92e47432f1da66eff25 | ReckonLean/Common.lean | Set.merge_length | [105, 1] | [121, 14] | omega | case inl
α : Type
inst✝¹ : Ord α
inst✝ : BEq α
l1 l2 : List α
h1 : α
t1 : List α
h2 : α
t2 : List α
h✝ : isLE (compare h1 h2) = true
⊢ Nat.succ (List.length t1 + Nat.succ (List.length t2)) = Nat.succ (List.length t1) + Nat.succ (List.length t2) | no goals |
https://github.com/benjaminfjones/reckonlean.git | 8768f7342ba226cfc2d7b92e47432f1da66eff25 | ReckonLean/Common.lean | Set.merge_length | [105, 1] | [121, 14] | rw [List.length_cons, merge_length (h1 :: t1) t2] | case inr
α : Type
inst✝¹ : Ord α
inst✝ : BEq α
l1 l2 : List α
h1 : α
t1 : List α
h2 : α
t2 : List α
h✝ : ¬isLE (compare h1 h2) = true
⊢ List.length (h2 :: merge (h1 :: t1) t2) = Nat.succ (List.length t1) + Nat.succ (List.length t2) | case inr
α : Type
inst✝¹ : Ord α
inst✝ : BEq α
l1 l2 : List α
h1 : α
t1 : List α
h2 : α
t2 : List α
h✝ : ¬isLE (compare h1 h2) = true
⊢ Nat.succ (List.length (h1 :: t1) + List.length t2) = Nat.succ (List.length t1) + Nat.succ (List.length t2) |
https://github.com/benjaminfjones/reckonlean.git | 8768f7342ba226cfc2d7b92e47432f1da66eff25 | ReckonLean/Common.lean | Set.merge_length | [105, 1] | [121, 14] | simp only [List.length_cons] | case inr
α : Type
inst✝¹ : Ord α
inst✝ : BEq α
l1 l2 : List α
h1 : α
t1 : List α
h2 : α
t2 : List α
h✝ : ¬isLE (compare h1 h2) = true
⊢ Nat.succ (List.length (h1 :: t1) + List.length t2) = Nat.succ (List.length t1) + Nat.succ (List.length t2) | case inr
α : Type
inst✝¹ : Ord α
inst✝ : BEq α
l1 l2 : List α
h1 : α
t1 : List α
h2 : α
t2 : List α
h✝ : ¬isLE (compare h1 h2) = true
⊢ Nat.succ (Nat.succ (List.length t1) + List.length t2) = Nat.succ (List.length t1) + Nat.succ (List.length t2) |
https://github.com/benjaminfjones/reckonlean.git | 8768f7342ba226cfc2d7b92e47432f1da66eff25 | ReckonLean/Common.lean | Set.merge_length | [105, 1] | [121, 14] | omega | case inr
α : Type
inst✝¹ : Ord α
inst✝ : BEq α
l1 l2 : List α
h1 : α
t1 : List α
h2 : α
t2 : List α
h✝ : ¬isLE (compare h1 h2) = true
⊢ Nat.succ (Nat.succ (List.length t1) + List.length t2) = Nat.succ (List.length t1) + Nat.succ (List.length t2) | no goals |
https://github.com/benjaminfjones/reckonlean.git | 8768f7342ba226cfc2d7b92e47432f1da66eff25 | ReckonLean/Dpll.lean | length_backtrack | [195, 1] | [204, 50] | intro t | α : Type
inst✝³ : BEq α
inst✝² : Ord α
inst✝¹ : Repr α
inst✝ : Hashable α
⊢ ∀ (tr : Trail α), List.length (backtrack tr) ≤ List.length tr | α : Type
inst✝³ : BEq α
inst✝² : Ord α
inst✝¹ : Repr α
inst✝ : Hashable α
t : Trail α
⊢ List.length (backtrack t) ≤ List.length t |
https://github.com/benjaminfjones/reckonlean.git | 8768f7342ba226cfc2d7b92e47432f1da66eff25 | ReckonLean/Dpll.lean | length_backtrack | [195, 1] | [204, 50] | induction t with
| nil => unfold backtrack; apply Nat.le_refl
| cons d ds ih =>
simp [backtrack]
cases d.decision == .Deduced
. simp
. simp; apply Nat.le_succ_of_le; assumption | α : Type
inst✝³ : BEq α
inst✝² : Ord α
inst✝¹ : Repr α
inst✝ : Hashable α
t : Trail α
⊢ List.length (backtrack t) ≤ List.length t | no goals |
https://github.com/benjaminfjones/reckonlean.git | 8768f7342ba226cfc2d7b92e47432f1da66eff25 | ReckonLean/Dpll.lean | length_backtrack | [195, 1] | [204, 50] | unfold backtrack | case nil
α : Type
inst✝³ : BEq α
inst✝² : Ord α
inst✝¹ : Repr α
inst✝ : Hashable α
⊢ List.length (backtrack []) ≤ List.length [] | case nil
α : Type
inst✝³ : BEq α
inst✝² : Ord α
inst✝¹ : Repr α
inst✝ : Hashable α
⊢ List.length [] ≤ List.length [] |
https://github.com/benjaminfjones/reckonlean.git | 8768f7342ba226cfc2d7b92e47432f1da66eff25 | ReckonLean/Dpll.lean | length_backtrack | [195, 1] | [204, 50] | apply Nat.le_refl | case nil
α : Type
inst✝³ : BEq α
inst✝² : Ord α
inst✝¹ : Repr α
inst✝ : Hashable α
⊢ List.length [] ≤ List.length [] | no goals |
https://github.com/benjaminfjones/reckonlean.git | 8768f7342ba226cfc2d7b92e47432f1da66eff25 | ReckonLean/Dpll.lean | length_backtrack | [195, 1] | [204, 50] | simp [backtrack] | case cons
α : Type
inst✝³ : BEq α
inst✝² : Ord α
inst✝¹ : Repr α
inst✝ : Hashable α
d : Decision α
ds : List (Decision α)
ih : List.length (backtrack ds) ≤ List.length ds
⊢ List.length (backtrack (d :: ds)) ≤ List.length (d :: ds) | case cons
α : Type
inst✝³ : BEq α
inst✝² : Ord α
inst✝¹ : Repr α
inst✝ : Hashable α
d : Decision α
ds : List (Decision α)
ih : List.length (backtrack ds) ≤ List.length ds
⊢ List.length (if (d.decision == LitState.Deduced) = true then backtrack ds else d :: ds) ≤ Nat.succ (List.length ds) |
https://github.com/benjaminfjones/reckonlean.git | 8768f7342ba226cfc2d7b92e47432f1da66eff25 | ReckonLean/Dpll.lean | length_backtrack | [195, 1] | [204, 50] | cases d.decision == .Deduced | case cons
α : Type
inst✝³ : BEq α
inst✝² : Ord α
inst✝¹ : Repr α
inst✝ : Hashable α
d : Decision α
ds : List (Decision α)
ih : List.length (backtrack ds) ≤ List.length ds
⊢ List.length (if (d.decision == LitState.Deduced) = true then backtrack ds else d :: ds) ≤ Nat.succ (List.length ds) | case cons.false
α : Type
inst✝³ : BEq α
inst✝² : Ord α
inst✝¹ : Repr α
inst✝ : Hashable α
d : Decision α
ds : List (Decision α)
ih : List.length (backtrack ds) ≤ List.length ds
⊢ List.length (if false = true then backtrack ds else d :: ds) ≤ Nat.succ (List.length ds)
case cons.true
α : Type
inst✝³ : BEq α
inst✝² : Ord α
inst✝¹ : Repr α
inst✝ : Hashable α
d : Decision α
ds : List (Decision α)
ih : List.length (backtrack ds) ≤ List.length ds
⊢ List.length (if true = true then backtrack ds else d :: ds) ≤ Nat.succ (List.length ds) |
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