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/kevinsullivan/cs2120f23.git
|
c2da7c4a6be769c6fca375a54e8c1fa31ba2854b
|
Instructor/Lectures/grad/vector_spaces.lean
|
rot_add_assoc
|
[134, 1]
|
[144, 4]
|
intros a b c
|
⊢ ∀ (a b c : Rotation), a + b + c = a + (b + c)
|
a b c : Rotation
⊢ a + b + c = a + (b + c)
|
https://github.com/kevinsullivan/cs2120f23.git
|
c2da7c4a6be769c6fca375a54e8c1fa31ba2854b
|
Instructor/Lectures/grad/vector_spaces.lean
|
rot_add_assoc
|
[134, 1]
|
[144, 4]
|
cases a
|
a b c : Rotation
⊢ a + b + c = a + (b + c)
|
case r0
b c : Rotation
⊢ r0 + b + c = r0 + (b + c)
case r120
b c : Rotation
⊢ r120 + b + c = r120 + (b + c)
case r240
b c : Rotation
⊢ r240 + b + c = r240 + (b + c)
|
https://github.com/kevinsullivan/cs2120f23.git
|
c2da7c4a6be769c6fca375a54e8c1fa31ba2854b
|
Instructor/Lectures/grad/vector_spaces.lean
|
rot_add_assoc
|
[134, 1]
|
[144, 4]
|
repeat {
cases b
repeat {
cases c
repeat { rfl }
}
}
|
case r0
b c : Rotation
⊢ r0 + b + c = r0 + (b + c)
case r120
b c : Rotation
⊢ r120 + b + c = r120 + (b + c)
case r240
b c : Rotation
⊢ r240 + b + c = r240 + (b + c)
|
no goals
|
https://github.com/kevinsullivan/cs2120f23.git
|
c2da7c4a6be769c6fca375a54e8c1fa31ba2854b
|
Instructor/Lectures/grad/vector_spaces.lean
|
rot_add_assoc
|
[134, 1]
|
[144, 4]
|
cases b
|
case r240
b c : Rotation
⊢ r240 + b + c = r240 + (b + c)
|
case r240.r0
c : Rotation
⊢ r240 + r0 + c = r240 + (r0 + c)
case r240.r120
c : Rotation
⊢ r240 + r120 + c = r240 + (r120 + c)
case r240.r240
c : Rotation
⊢ r240 + r240 + c = r240 + (r240 + c)
|
https://github.com/kevinsullivan/cs2120f23.git
|
c2da7c4a6be769c6fca375a54e8c1fa31ba2854b
|
Instructor/Lectures/grad/vector_spaces.lean
|
rot_add_assoc
|
[134, 1]
|
[144, 4]
|
repeat {
cases c
repeat { rfl }
}
|
case r240.r0
c : Rotation
⊢ r240 + r0 + c = r240 + (r0 + c)
case r240.r120
c : Rotation
⊢ r240 + r120 + c = r240 + (r120 + c)
case r240.r240
c : Rotation
⊢ r240 + r240 + c = r240 + (r240 + c)
|
no goals
|
https://github.com/kevinsullivan/cs2120f23.git
|
c2da7c4a6be769c6fca375a54e8c1fa31ba2854b
|
Instructor/Lectures/grad/vector_spaces.lean
|
rot_add_assoc
|
[134, 1]
|
[144, 4]
|
cases c
|
case r240.r240
c : Rotation
⊢ r240 + r240 + c = r240 + (r240 + c)
|
case r240.r240.r0
⊢ r240 + r240 + r0 = r240 + (r240 + r0)
case r240.r240.r120
⊢ r240 + r240 + r120 = r240 + (r240 + r120)
case r240.r240.r240
⊢ r240 + r240 + r240 = r240 + (r240 + r240)
|
https://github.com/kevinsullivan/cs2120f23.git
|
c2da7c4a6be769c6fca375a54e8c1fa31ba2854b
|
Instructor/Lectures/grad/vector_spaces.lean
|
rot_add_assoc
|
[134, 1]
|
[144, 4]
|
repeat { rfl }
|
case r240.r240.r0
⊢ r240 + r240 + r0 = r240 + (r240 + r0)
case r240.r240.r120
⊢ r240 + r240 + r120 = r240 + (r240 + r120)
case r240.r240.r240
⊢ r240 + r240 + r240 = r240 + (r240 + r240)
|
no goals
|
https://github.com/kevinsullivan/cs2120f23.git
|
c2da7c4a6be769c6fca375a54e8c1fa31ba2854b
|
Instructor/Lectures/grad/vector_spaces.lean
|
rot_add_assoc
|
[134, 1]
|
[144, 4]
|
rfl
|
case r240.r240.r240
⊢ r240 + r240 + r240 = r240 + (r240 + r240)
|
no goals
|
https://github.com/kevinsullivan/cs2120f23.git
|
c2da7c4a6be769c6fca375a54e8c1fa31ba2854b
|
Instructor/Lectures/grad/vector_spaces.lean
|
rot_zero_add
|
[157, 1]
|
[161, 17]
|
intro a
|
⊢ ∀ (a : Rotation), 0 + a = a
|
a : Rotation
⊢ 0 + a = a
|
https://github.com/kevinsullivan/cs2120f23.git
|
c2da7c4a6be769c6fca375a54e8c1fa31ba2854b
|
Instructor/Lectures/grad/vector_spaces.lean
|
rot_zero_add
|
[157, 1]
|
[161, 17]
|
cases a
|
a : Rotation
⊢ 0 + a = a
|
case r0
⊢ 0 + r0 = r0
case r120
⊢ 0 + r120 = r120
case r240
⊢ 0 + r240 = r240
|
https://github.com/kevinsullivan/cs2120f23.git
|
c2da7c4a6be769c6fca375a54e8c1fa31ba2854b
|
Instructor/Lectures/grad/vector_spaces.lean
|
rot_zero_add
|
[157, 1]
|
[161, 17]
|
repeat { rfl }
|
case r0
⊢ 0 + r0 = r0
case r120
⊢ 0 + r120 = r120
case r240
⊢ 0 + r240 = r240
|
no goals
|
https://github.com/kevinsullivan/cs2120f23.git
|
c2da7c4a6be769c6fca375a54e8c1fa31ba2854b
|
Instructor/Lectures/grad/vector_spaces.lean
|
rot_zero_add
|
[157, 1]
|
[161, 17]
|
rfl
|
case r240
⊢ 0 + r240 = r240
|
no goals
|
https://github.com/kevinsullivan/cs2120f23.git
|
c2da7c4a6be769c6fca375a54e8c1fa31ba2854b
|
Instructor/Lectures/grad/vector_spaces.lean
|
rot_add_zero
|
[163, 1]
|
[167, 17]
|
intro a
|
⊢ ∀ (a : Rotation), a + 0 = a
|
a : Rotation
⊢ a + 0 = a
|
https://github.com/kevinsullivan/cs2120f23.git
|
c2da7c4a6be769c6fca375a54e8c1fa31ba2854b
|
Instructor/Lectures/grad/vector_spaces.lean
|
rot_add_zero
|
[163, 1]
|
[167, 17]
|
cases a
|
a : Rotation
⊢ a + 0 = a
|
case r0
⊢ r0 + 0 = r0
case r120
⊢ r120 + 0 = r120
case r240
⊢ r240 + 0 = r240
|
https://github.com/kevinsullivan/cs2120f23.git
|
c2da7c4a6be769c6fca375a54e8c1fa31ba2854b
|
Instructor/Lectures/grad/vector_spaces.lean
|
rot_add_zero
|
[163, 1]
|
[167, 17]
|
repeat { rfl }
|
case r0
⊢ r0 + 0 = r0
case r120
⊢ r120 + 0 = r120
case r240
⊢ r240 + 0 = r240
|
no goals
|
https://github.com/kevinsullivan/cs2120f23.git
|
c2da7c4a6be769c6fca375a54e8c1fa31ba2854b
|
Instructor/Lectures/grad/vector_spaces.lean
|
rot_add_zero
|
[163, 1]
|
[167, 17]
|
rfl
|
case r240
⊢ r240 + 0 = r240
|
no goals
|
https://github.com/kevinsullivan/cs2120f23.git
|
c2da7c4a6be769c6fca375a54e8c1fa31ba2854b
|
Instructor/Lectures/grad/vector_spaces.lean
|
rot_zero_vadd
|
[219, 1]
|
[223, 17]
|
intro p
|
⊢ ∀ (p : State), 0 +ᵥ p = p
|
p : State
⊢ 0 +ᵥ p = p
|
https://github.com/kevinsullivan/cs2120f23.git
|
c2da7c4a6be769c6fca375a54e8c1fa31ba2854b
|
Instructor/Lectures/grad/vector_spaces.lean
|
rot_zero_vadd
|
[219, 1]
|
[223, 17]
|
cases p
|
p : State
⊢ 0 +ᵥ p = p
|
case heading
h✝ : K
unit✝ : ℕ
⊢ 0 +ᵥ heading h✝ unit✝ = heading h✝ unit✝
|
https://github.com/kevinsullivan/cs2120f23.git
|
c2da7c4a6be769c6fca375a54e8c1fa31ba2854b
|
Instructor/Lectures/grad/vector_spaces.lean
|
rot_zero_vadd
|
[219, 1]
|
[223, 17]
|
repeat { rfl }
|
case heading
h✝ : K
unit✝ : ℕ
⊢ 0 +ᵥ heading h✝ unit✝ = heading h✝ unit✝
|
no goals
|
https://github.com/kevinsullivan/cs2120f23.git
|
c2da7c4a6be769c6fca375a54e8c1fa31ba2854b
|
Instructor/Lectures/grad/vector_spaces.lean
|
rot_zero_vadd
|
[219, 1]
|
[223, 17]
|
rfl
|
case heading
h✝ : K
unit✝ : ℕ
⊢ 0 +ᵥ heading h✝ unit✝ = heading h✝ unit✝
|
no goals
|
https://github.com/kevinsullivan/cs2120f23.git
|
c2da7c4a6be769c6fca375a54e8c1fa31ba2854b
|
Instructor/Lectures/grad/vector_spaces.lean
|
rot_add_vadd
|
[225, 1]
|
[235, 4]
|
intros g₁ g₂ p
|
⊢ ∀ (g₁ g₂ : Rotation) (p : State), g₁ + g₂ +ᵥ p = g₁ +ᵥ (g₂ +ᵥ p)
|
g₁ g₂ : Rotation
p : State
⊢ g₁ + g₂ +ᵥ p = g₁ +ᵥ (g₂ +ᵥ p)
|
https://github.com/kevinsullivan/cs2120f23.git
|
c2da7c4a6be769c6fca375a54e8c1fa31ba2854b
|
Instructor/Lectures/grad/vector_spaces.lean
|
rot_add_vadd
|
[225, 1]
|
[235, 4]
|
cases g₁
|
g₁ g₂ : Rotation
p : State
⊢ g₁ + g₂ +ᵥ p = g₁ +ᵥ (g₂ +ᵥ p)
|
case r0
g₂ : Rotation
p : State
⊢ r0 + g₂ +ᵥ p = r0 +ᵥ (g₂ +ᵥ p)
case r120
g₂ : Rotation
p : State
⊢ r120 + g₂ +ᵥ p = r120 +ᵥ (g₂ +ᵥ p)
case r240
g₂ : Rotation
p : State
⊢ r240 + g₂ +ᵥ p = r240 +ᵥ (g₂ +ᵥ p)
|
https://github.com/kevinsullivan/cs2120f23.git
|
c2da7c4a6be769c6fca375a54e8c1fa31ba2854b
|
Instructor/Lectures/grad/vector_spaces.lean
|
rot_add_vadd
|
[225, 1]
|
[235, 4]
|
repeat {
cases g₂
repeat {
cases p
repeat rfl
}
}
|
case r0
g₂ : Rotation
p : State
⊢ r0 + g₂ +ᵥ p = r0 +ᵥ (g₂ +ᵥ p)
case r120
g₂ : Rotation
p : State
⊢ r120 + g₂ +ᵥ p = r120 +ᵥ (g₂ +ᵥ p)
case r240
g₂ : Rotation
p : State
⊢ r240 + g₂ +ᵥ p = r240 +ᵥ (g₂ +ᵥ p)
|
case r120
g₂ : Rotation
p : State
⊢ r120 + g₂ +ᵥ p = r120 +ᵥ (g₂ +ᵥ p)
case r240
g₂ : Rotation
p : State
⊢ r240 + g₂ +ᵥ p = r240 +ᵥ (g₂ +ᵥ p)
|
https://github.com/kevinsullivan/cs2120f23.git
|
c2da7c4a6be769c6fca375a54e8c1fa31ba2854b
|
Instructor/Lectures/grad/vector_spaces.lean
|
rot_add_vadd
|
[225, 1]
|
[235, 4]
|
cases g₂
|
case r120
g₂ : Rotation
p : State
⊢ r120 + g₂ +ᵥ p = r120 +ᵥ (g₂ +ᵥ p)
|
case r120.r0
p : State
⊢ r120 + r0 +ᵥ p = r120 +ᵥ (r0 +ᵥ p)
case r120.r120
p : State
⊢ r120 + r120 +ᵥ p = r120 +ᵥ (r120 +ᵥ p)
case r120.r240
p : State
⊢ r120 + r240 +ᵥ p = r120 +ᵥ (r240 +ᵥ p)
|
https://github.com/kevinsullivan/cs2120f23.git
|
c2da7c4a6be769c6fca375a54e8c1fa31ba2854b
|
Instructor/Lectures/grad/vector_spaces.lean
|
rot_add_vadd
|
[225, 1]
|
[235, 4]
|
repeat {
cases p
repeat rfl
}
|
case r120.r0
p : State
⊢ r120 + r0 +ᵥ p = r120 +ᵥ (r0 +ᵥ p)
case r120.r120
p : State
⊢ r120 + r120 +ᵥ p = r120 +ᵥ (r120 +ᵥ p)
case r120.r240
p : State
⊢ r120 + r240 +ᵥ p = r120 +ᵥ (r240 +ᵥ p)
|
case r120.r240
p : State
⊢ r120 + r240 +ᵥ p = r120 +ᵥ (r240 +ᵥ p)
|
https://github.com/kevinsullivan/cs2120f23.git
|
c2da7c4a6be769c6fca375a54e8c1fa31ba2854b
|
Instructor/Lectures/grad/vector_spaces.lean
|
rot_add_vadd
|
[225, 1]
|
[235, 4]
|
cases p
|
case r120.r240
p : State
⊢ r120 + r240 +ᵥ p = r120 +ᵥ (r240 +ᵥ p)
|
case r120.r240.heading
h✝ : K
unit✝ : ℕ
⊢ r120 + r240 +ᵥ heading h✝ unit✝ = r120 +ᵥ (r240 +ᵥ heading h✝ unit✝)
|
https://github.com/kevinsullivan/cs2120f23.git
|
c2da7c4a6be769c6fca375a54e8c1fa31ba2854b
|
Instructor/Lectures/grad/vector_spaces.lean
|
rot_add_vadd
|
[225, 1]
|
[235, 4]
|
repeat rfl
|
case r120.r120.heading
h✝ : K
unit✝ : ℕ
⊢ r120 + r120 +ᵥ heading h✝ unit✝ = r120 +ᵥ (r120 +ᵥ heading h✝ unit✝)
|
no goals
|
https://github.com/kevinsullivan/cs2120f23.git
|
c2da7c4a6be769c6fca375a54e8c1fa31ba2854b
|
Instructor/Lectures/grad/vector_spaces.lean
|
rot_add_vadd
|
[225, 1]
|
[235, 4]
|
rfl
|
case r120.r120.heading
h✝ : K
unit✝ : ℕ
⊢ r120 + r120 +ᵥ heading h✝ unit✝ = r120 +ᵥ (r120 +ᵥ heading h✝ unit✝)
|
no goals
|
https://github.com/kevinsullivan/cs2120f23.git
|
c2da7c4a6be769c6fca375a54e8c1fa31ba2854b
|
Instructor/Lectures/grad/vector_spaces.lean
|
rot_add_neg_left
|
[349, 1]
|
[353, 17]
|
intro a
|
⊢ ∀ (a : Rotation), -a + a = 0
|
a : Rotation
⊢ -a + a = 0
|
https://github.com/kevinsullivan/cs2120f23.git
|
c2da7c4a6be769c6fca375a54e8c1fa31ba2854b
|
Instructor/Lectures/grad/vector_spaces.lean
|
rot_add_neg_left
|
[349, 1]
|
[353, 17]
|
cases a
|
a : Rotation
⊢ -a + a = 0
|
case r0
⊢ -r0 + r0 = 0
case r120
⊢ -r120 + r120 = 0
case r240
⊢ -r240 + r240 = 0
|
https://github.com/kevinsullivan/cs2120f23.git
|
c2da7c4a6be769c6fca375a54e8c1fa31ba2854b
|
Instructor/Lectures/grad/vector_spaces.lean
|
rot_add_neg_left
|
[349, 1]
|
[353, 17]
|
repeat { rfl }
|
case r0
⊢ -r0 + r0 = 0
case r120
⊢ -r120 + r120 = 0
case r240
⊢ -r240 + r240 = 0
|
no goals
|
https://github.com/kevinsullivan/cs2120f23.git
|
c2da7c4a6be769c6fca375a54e8c1fa31ba2854b
|
Instructor/Lectures/grad/vector_spaces.lean
|
rot_add_neg_left
|
[349, 1]
|
[353, 17]
|
rfl
|
case r240
⊢ -r240 + r240 = 0
|
no goals
|
https://github.com/kevinsullivan/cs2120f23.git
|
c2da7c4a6be769c6fca375a54e8c1fa31ba2854b
|
Instructor/Lectures/grad/vector_spaces.lean
|
rot_state_vsub_vadd'
|
[405, 1]
|
[412, 4]
|
intro p1 p2
|
⊢ ∀ (p1 p2 : State), p1 -ᵥ p2 +ᵥ p2 = p1
|
p1 p2 : State
⊢ p1 -ᵥ p2 +ᵥ p2 = p1
|
https://github.com/kevinsullivan/cs2120f23.git
|
c2da7c4a6be769c6fca375a54e8c1fa31ba2854b
|
Instructor/Lectures/grad/vector_spaces.lean
|
rot_state_vsub_vadd'
|
[405, 1]
|
[412, 4]
|
cases p1
|
p1 p2 : State
⊢ p1 -ᵥ p2 +ᵥ p2 = p1
|
case heading
p2 : State
h✝ : K
unit✝ : ℕ
⊢ heading h✝ unit✝ -ᵥ p2 +ᵥ p2 = heading h✝ unit✝
|
https://github.com/kevinsullivan/cs2120f23.git
|
c2da7c4a6be769c6fca375a54e8c1fa31ba2854b
|
Instructor/Lectures/grad/vector_spaces.lean
|
rot_state_vsub_vadd'
|
[405, 1]
|
[412, 4]
|
cases p2
|
case heading
p2 : State
h✝ : K
unit✝ : ℕ
⊢ heading h✝ unit✝ -ᵥ p2 +ᵥ p2 = heading h✝ unit✝
|
case heading.heading
h✝¹ : K
unit✝¹ : ℕ
h✝ : K
unit✝ : ℕ
⊢ heading h✝¹ unit✝¹ -ᵥ heading h✝ unit✝ +ᵥ heading h✝ unit✝ = heading h✝¹ unit✝¹
|
https://github.com/kevinsullivan/cs2120f23.git
|
c2da7c4a6be769c6fca375a54e8c1fa31ba2854b
|
Instructor/Lectures/grad/vector_spaces.lean
|
rot_state_vadd_vsub'
|
[414, 1]
|
[421, 4]
|
intros g p
|
⊢ ∀ (g : Rotation) (p : State), g +ᵥ p -ᵥ p = g
|
g : Rotation
p : State
⊢ g +ᵥ p -ᵥ p = g
|
https://github.com/kevinsullivan/cs2120f23.git
|
c2da7c4a6be769c6fca375a54e8c1fa31ba2854b
|
Instructor/Lectures/grad/vector_spaces.lean
|
rot_state_vadd_vsub'
|
[414, 1]
|
[421, 4]
|
cases g
|
g : Rotation
p : State
⊢ g +ᵥ p -ᵥ p = g
|
case r0
p : State
⊢ r0 +ᵥ p -ᵥ p = r0
case r120
p : State
⊢ r120 +ᵥ p -ᵥ p = r120
case r240
p : State
⊢ r240 +ᵥ p -ᵥ p = r240
|
https://github.com/kevinsullivan/cs2120f23.git
|
c2da7c4a6be769c6fca375a54e8c1fa31ba2854b
|
Instructor/Lectures/grad/vector_spaces.lean
|
rot_state_vadd_vsub'
|
[414, 1]
|
[421, 4]
|
cases p
|
case r0
p : State
⊢ r0 +ᵥ p -ᵥ p = r0
|
case r0.heading
h✝ : K
unit✝ : ℕ
⊢ r0 +ᵥ heading h✝ unit✝ -ᵥ heading h✝ unit✝ = r0
|
https://github.com/kevinsullivan/cs2120f23.git
|
c2da7c4a6be769c6fca375a54e8c1fa31ba2854b
|
Instructor/Lectures/grad/sets_relations.lean
|
eq_nat_is_refl
|
[68, 1]
|
[71, 12]
|
unfold Reflexive
|
⊢ Reflexive Eq
|
⊢ ∀ (x : ℕ), x = x
|
https://github.com/kevinsullivan/cs2120f23.git
|
c2da7c4a6be769c6fca375a54e8c1fa31ba2854b
|
Instructor/Lectures/grad/sets_relations.lean
|
eq_nat_is_refl
|
[68, 1]
|
[71, 12]
|
intro x
|
⊢ ∀ (x : ℕ), x = x
|
x : ℕ
⊢ x = x
|
https://github.com/kevinsullivan/cs2120f23.git
|
c2da7c4a6be769c6fca375a54e8c1fa31ba2854b
|
Instructor/Lectures/grad/sets_relations.lean
|
eq_nat_is_refl
|
[68, 1]
|
[71, 12]
|
exact rfl
|
x : ℕ
⊢ x = x
|
no goals
|
https://github.com/kevinsullivan/cs2120f23.git
|
c2da7c4a6be769c6fca375a54e8c1fa31ba2854b
|
Instructor/Lectures/grad/sets_relations.lean
|
eq_nat_is_symm
|
[73, 1]
|
[77, 11]
|
unfold Symmetric
|
⊢ Symmetric Eq
|
⊢ ∀ ⦃x y : ℕ⦄, x = y → y = x
|
https://github.com/kevinsullivan/cs2120f23.git
|
c2da7c4a6be769c6fca375a54e8c1fa31ba2854b
|
Instructor/Lectures/grad/sets_relations.lean
|
eq_nat_is_symm
|
[73, 1]
|
[77, 11]
|
intro x y
|
⊢ ∀ ⦃x y : ℕ⦄, x = y → y = x
|
x y : ℕ
⊢ x = y → y = x
|
https://github.com/kevinsullivan/cs2120f23.git
|
c2da7c4a6be769c6fca375a54e8c1fa31ba2854b
|
Instructor/Lectures/grad/sets_relations.lean
|
eq_nat_is_symm
|
[73, 1]
|
[77, 11]
|
intro hxy
|
x y : ℕ
⊢ x = y → y = x
|
x y : ℕ
hxy : x = y
⊢ y = x
|
https://github.com/kevinsullivan/cs2120f23.git
|
c2da7c4a6be769c6fca375a54e8c1fa31ba2854b
|
Instructor/Lectures/grad/sets_relations.lean
|
eq_nat_is_symm
|
[73, 1]
|
[77, 11]
|
rw [hxy]
|
x y : ℕ
hxy : x = y
⊢ y = x
|
no goals
|
https://github.com/kevinsullivan/cs2120f23.git
|
c2da7c4a6be769c6fca375a54e8c1fa31ba2854b
|
Instructor/Lectures/grad/sets_relations.lean
|
eq_nat_is_trans
|
[82, 1]
|
[87, 9]
|
unfold Transitive
|
⊢ Transitive Eq
|
⊢ ∀ ⦃x y z : ℕ⦄, x = y → y = z → x = z
|
https://github.com/kevinsullivan/cs2120f23.git
|
c2da7c4a6be769c6fca375a54e8c1fa31ba2854b
|
Instructor/Lectures/grad/sets_relations.lean
|
eq_nat_is_trans
|
[82, 1]
|
[87, 9]
|
intros x y z
|
⊢ ∀ ⦃x y z : ℕ⦄, x = y → y = z → x = z
|
x y z : ℕ
⊢ x = y → y = z → x = z
|
https://github.com/kevinsullivan/cs2120f23.git
|
c2da7c4a6be769c6fca375a54e8c1fa31ba2854b
|
Instructor/Lectures/grad/sets_relations.lean
|
eq_nat_is_trans
|
[82, 1]
|
[87, 9]
|
intros hxy hyz
|
x y z : ℕ
⊢ x = y → y = z → x = z
|
x y z : ℕ
hxy : x = y
hyz : y = z
⊢ x = z
|
https://github.com/kevinsullivan/cs2120f23.git
|
c2da7c4a6be769c6fca375a54e8c1fa31ba2854b
|
Instructor/Lectures/grad/sets_relations.lean
|
eq_nat_is_trans
|
[82, 1]
|
[87, 9]
|
rw [hxy]
|
x y z : ℕ
hxy : x = y
hyz : y = z
⊢ x = z
|
x y z : ℕ
hxy : x = y
hyz : y = z
⊢ y = z
|
https://github.com/kevinsullivan/cs2120f23.git
|
c2da7c4a6be769c6fca375a54e8c1fa31ba2854b
|
Instructor/Lectures/grad/sets_relations.lean
|
eq_nat_is_trans
|
[82, 1]
|
[87, 9]
|
rw [hyz]
|
x y z : ℕ
hxy : x = y
hyz : y = z
⊢ y = z
|
no goals
|
https://github.com/kevinsullivan/cs2120f23.git
|
c2da7c4a6be769c6fca375a54e8c1fa31ba2854b
|
Instructor/Lectures/grad/sets_relations.lean
|
cong_mod_n_equiv'
|
[99, 1]
|
[102, 31]
|
intro n
|
⊢ ∀ (n : ℕ), Equivalence (cong_mod_n n)
|
n : ℕ
⊢ Equivalence (cong_mod_n n)
|
https://github.com/kevinsullivan/cs2120f23.git
|
c2da7c4a6be769c6fca375a54e8c1fa31ba2854b
|
Instructor/Lectures/grad/sets_relations.lean
|
cong_mod_n_rfl
|
[106, 1]
|
[111, 12]
|
intro n
|
⊢ ∀ (n : ℕ), Reflexive (cong_mod_n n)
|
n : ℕ
⊢ Reflexive (cong_mod_n n)
|
https://github.com/kevinsullivan/cs2120f23.git
|
c2da7c4a6be769c6fca375a54e8c1fa31ba2854b
|
Instructor/Lectures/grad/sets_relations.lean
|
cong_mod_n_rfl
|
[106, 1]
|
[111, 12]
|
unfold cong_mod_n
|
n : ℕ
⊢ Reflexive (cong_mod_n n)
|
n : ℕ
⊢ Reflexive fun a b => a % n = b % n
|
https://github.com/kevinsullivan/cs2120f23.git
|
c2da7c4a6be769c6fca375a54e8c1fa31ba2854b
|
Instructor/Lectures/grad/sets_relations.lean
|
cong_mod_n_rfl
|
[106, 1]
|
[111, 12]
|
unfold Reflexive
|
n : ℕ
⊢ Reflexive fun a b => a % n = b % n
|
n : ℕ
⊢ ∀ (x : ℕ), (fun a b => a % n = b % n) x x
|
https://github.com/kevinsullivan/cs2120f23.git
|
c2da7c4a6be769c6fca375a54e8c1fa31ba2854b
|
Instructor/Lectures/grad/sets_relations.lean
|
cong_mod_n_rfl
|
[106, 1]
|
[111, 12]
|
intro a
|
n : ℕ
⊢ ∀ (x : ℕ), (fun a b => a % n = b % n) x x
|
n a : ℕ
⊢ a % n = a % n
|
https://github.com/kevinsullivan/cs2120f23.git
|
c2da7c4a6be769c6fca375a54e8c1fa31ba2854b
|
Instructor/Lectures/grad/sets_relations.lean
|
cong_mod_n_rfl
|
[106, 1]
|
[111, 12]
|
exact rfl
|
n a : ℕ
⊢ a % n = a % n
|
no goals
|
https://github.com/kevinsullivan/cs2120f23.git
|
c2da7c4a6be769c6fca375a54e8c1fa31ba2854b
|
Instructor/Lectures/grad/sets_relations.lean
|
cong_mod_n_symm
|
[113, 1]
|
[120, 11]
|
intro n
|
⊢ ∀ (n : ℕ), Symmetric (cong_mod_n n)
|
n : ℕ
⊢ Symmetric (cong_mod_n n)
|
https://github.com/kevinsullivan/cs2120f23.git
|
c2da7c4a6be769c6fca375a54e8c1fa31ba2854b
|
Instructor/Lectures/grad/sets_relations.lean
|
cong_mod_n_symm
|
[113, 1]
|
[120, 11]
|
unfold Symmetric
|
n : ℕ
⊢ Symmetric (cong_mod_n n)
|
n : ℕ
⊢ ∀ ⦃x y : ℕ⦄, cong_mod_n n x y → cong_mod_n n y x
|
https://github.com/kevinsullivan/cs2120f23.git
|
c2da7c4a6be769c6fca375a54e8c1fa31ba2854b
|
Instructor/Lectures/grad/sets_relations.lean
|
cong_mod_n_symm
|
[113, 1]
|
[120, 11]
|
intros x y
|
n : ℕ
⊢ ∀ ⦃x y : ℕ⦄, cong_mod_n n x y → cong_mod_n n y x
|
n x y : ℕ
⊢ cong_mod_n n x y → cong_mod_n n y x
|
https://github.com/kevinsullivan/cs2120f23.git
|
c2da7c4a6be769c6fca375a54e8c1fa31ba2854b
|
Instructor/Lectures/grad/sets_relations.lean
|
cong_mod_n_symm
|
[113, 1]
|
[120, 11]
|
intro hxy
|
n x y : ℕ
⊢ cong_mod_n n x y → cong_mod_n n y x
|
n x y : ℕ
hxy : cong_mod_n n x y
⊢ cong_mod_n n y x
|
https://github.com/kevinsullivan/cs2120f23.git
|
c2da7c4a6be769c6fca375a54e8c1fa31ba2854b
|
Instructor/Lectures/grad/sets_relations.lean
|
cong_mod_n_symm
|
[113, 1]
|
[120, 11]
|
unfold cong_mod_n
|
n x y : ℕ
hxy : cong_mod_n n x y
⊢ cong_mod_n n y x
|
n x y : ℕ
hxy : cong_mod_n n x y
⊢ y % n = x % n
|
https://github.com/kevinsullivan/cs2120f23.git
|
c2da7c4a6be769c6fca375a54e8c1fa31ba2854b
|
Instructor/Lectures/grad/sets_relations.lean
|
cong_mod_n_symm
|
[113, 1]
|
[120, 11]
|
unfold cong_mod_n at hxy
|
n x y : ℕ
hxy : cong_mod_n n x y
⊢ y % n = x % n
|
n x y : ℕ
hxy : x % n = y % n
⊢ y % n = x % n
|
https://github.com/kevinsullivan/cs2120f23.git
|
c2da7c4a6be769c6fca375a54e8c1fa31ba2854b
|
Instructor/Lectures/grad/sets_relations.lean
|
cong_mod_n_symm
|
[113, 1]
|
[120, 11]
|
rw [hxy]
|
n x y : ℕ
hxy : x % n = y % n
⊢ y % n = x % n
|
no goals
|
https://github.com/kevinsullivan/cs2120f23.git
|
c2da7c4a6be769c6fca375a54e8c1fa31ba2854b
|
Instructor/Lectures/grad/sets_relations.lean
|
cong_mod_trans
|
[122, 1]
|
[126, 16]
|
intro n a b c hab hbc
|
⊢ ∀ (n : ℕ), Transitive (cong_mod_n n)
|
n a b c : ℕ
hab : cong_mod_n n a b
hbc : cong_mod_n n b c
⊢ cong_mod_n n a c
|
https://github.com/kevinsullivan/cs2120f23.git
|
c2da7c4a6be769c6fca375a54e8c1fa31ba2854b
|
Instructor/Lectures/grad/sets_relations.lean
|
cong_mod_trans
|
[122, 1]
|
[126, 16]
|
unfold cong_mod_n
|
n a b c : ℕ
hab : cong_mod_n n a b
hbc : cong_mod_n n b c
⊢ cong_mod_n n a c
|
n a b c : ℕ
hab : cong_mod_n n a b
hbc : cong_mod_n n b c
⊢ a % n = c % n
|
https://github.com/kevinsullivan/cs2120f23.git
|
c2da7c4a6be769c6fca375a54e8c1fa31ba2854b
|
Instructor/Lectures/grad/sets_relations.lean
|
cong_mod_trans
|
[122, 1]
|
[126, 16]
|
unfold cong_mod_n at hab hbc
|
n a b c : ℕ
hab : cong_mod_n n a b
hbc : cong_mod_n n b c
⊢ a % n = c % n
|
n a b c : ℕ
hab : a % n = b % n
hbc : b % n = c % n
⊢ a % n = c % n
|
https://github.com/kevinsullivan/cs2120f23.git
|
c2da7c4a6be769c6fca375a54e8c1fa31ba2854b
|
Instructor/Lectures/grad/sets_relations.lean
|
cong_mod_trans
|
[122, 1]
|
[126, 16]
|
rw [hab, hbc]
|
n a b c : ℕ
hab : a % n = b % n
hbc : b % n = c % n
⊢ a % n = c % n
|
no goals
|
https://github.com/kevinsullivan/cs2120f23.git
|
c2da7c4a6be769c6fca375a54e8c1fa31ba2854b
|
Instructor/Lectures/grad/sets_relations.lean
|
cong_mod_n_equiv
|
[128, 1]
|
[135, 47]
|
intro n
|
⊢ ∀ (n : ℕ), Equivalence (cong_mod_n n)
|
n : ℕ
⊢ Equivalence (cong_mod_n n)
|
https://github.com/kevinsullivan/cs2120f23.git
|
c2da7c4a6be769c6fca375a54e8c1fa31ba2854b
|
Instructor/Lectures/grad/sets_relations.lean
|
cong_mod_n_equiv
|
[128, 1]
|
[135, 47]
|
unfold cong_mod_n
|
n : ℕ
⊢ Equivalence (cong_mod_n n)
|
n : ℕ
⊢ Equivalence fun a b => a % n = b % n
|
https://github.com/kevinsullivan/cs2120f23.git
|
c2da7c4a6be769c6fca375a54e8c1fa31ba2854b
|
Instructor/Lectures/grad/sets_relations.lean
|
cong_mod_n_equiv
|
[128, 1]
|
[135, 47]
|
exact Equivalence.mk
(by intro x; rfl)
(by intros x y h; rw [h])
(by intros x y z hxy hyz; rw [hxy, hyz])
|
n : ℕ
⊢ Equivalence fun a b => a % n = b % n
|
no goals
|
https://github.com/kevinsullivan/cs2120f23.git
|
c2da7c4a6be769c6fca375a54e8c1fa31ba2854b
|
Instructor/Lectures/grad/sets_relations.lean
|
cong_mod_n_equiv
|
[128, 1]
|
[135, 47]
|
intro x
|
n : ℕ
⊢ ∀ (x : ℕ), x % n = x % n
|
n x : ℕ
⊢ x % n = x % n
|
https://github.com/kevinsullivan/cs2120f23.git
|
c2da7c4a6be769c6fca375a54e8c1fa31ba2854b
|
Instructor/Lectures/grad/sets_relations.lean
|
cong_mod_n_equiv
|
[128, 1]
|
[135, 47]
|
rfl
|
n x : ℕ
⊢ x % n = x % n
|
no goals
|
https://github.com/kevinsullivan/cs2120f23.git
|
c2da7c4a6be769c6fca375a54e8c1fa31ba2854b
|
Instructor/Lectures/grad/sets_relations.lean
|
cong_mod_n_equiv
|
[128, 1]
|
[135, 47]
|
intros x y h
|
n : ℕ
⊢ ∀ {x y : ℕ}, x % n = y % n → y % n = x % n
|
n x y : ℕ
h : x % n = y % n
⊢ y % n = x % n
|
https://github.com/kevinsullivan/cs2120f23.git
|
c2da7c4a6be769c6fca375a54e8c1fa31ba2854b
|
Instructor/Lectures/grad/sets_relations.lean
|
cong_mod_n_equiv
|
[128, 1]
|
[135, 47]
|
rw [h]
|
n x y : ℕ
h : x % n = y % n
⊢ y % n = x % n
|
no goals
|
https://github.com/kevinsullivan/cs2120f23.git
|
c2da7c4a6be769c6fca375a54e8c1fa31ba2854b
|
Instructor/Lectures/grad/sets_relations.lean
|
cong_mod_n_equiv
|
[128, 1]
|
[135, 47]
|
intros x y z hxy hyz
|
n : ℕ
⊢ ∀ {x y z : ℕ}, x % n = y % n → y % n = z % n → x % n = z % n
|
n x y z : ℕ
hxy : x % n = y % n
hyz : y % n = z % n
⊢ x % n = z % n
|
https://github.com/kevinsullivan/cs2120f23.git
|
c2da7c4a6be769c6fca375a54e8c1fa31ba2854b
|
Instructor/Lectures/grad/sets_relations.lean
|
cong_mod_n_equiv
|
[128, 1]
|
[135, 47]
|
rw [hxy, hyz]
|
n x y z : ℕ
hxy : x % n = y % n
hyz : y % n = z % n
⊢ x % n = z % n
|
no goals
|
https://github.com/kevinsullivan/cs2120f23.git
|
c2da7c4a6be769c6fca375a54e8c1fa31ba2854b
|
Instructor/Lectures/grad/sets_relations.lean
|
succ_inj
|
[189, 1]
|
[193, 13]
|
unfold Injective
|
⊢ Injective fun n => n + 1
|
⊢ ∀ ⦃a₁ a₂ : ℕ⦄, (fun n => n + 1) a₁ = (fun n => n + 1) a₂ → a₁ = a₂
|
https://github.com/kevinsullivan/cs2120f23.git
|
c2da7c4a6be769c6fca375a54e8c1fa31ba2854b
|
Instructor/Lectures/grad/sets_relations.lean
|
succ_inj
|
[189, 1]
|
[193, 13]
|
intros a1 a2 h
|
⊢ ∀ ⦃a₁ a₂ : ℕ⦄, (fun n => n + 1) a₁ = (fun n => n + 1) a₂ → a₁ = a₂
|
a1 a2 : ℕ
h : (fun n => n + 1) a1 = (fun n => n + 1) a2
⊢ a1 = a2
|
https://github.com/kevinsullivan/cs2120f23.git
|
c2da7c4a6be769c6fca375a54e8c1fa31ba2854b
|
Instructor/Lectures/grad/sets_relations.lean
|
succ_inj
|
[189, 1]
|
[193, 13]
|
simp [Nat.add] at h
|
a1 a2 : ℕ
h : (fun n => n + 1) a1 = (fun n => n + 1) a2
⊢ a1 = a2
|
a1 a2 : ℕ
h : a1 = a2
⊢ a1 = a2
|
https://github.com/kevinsullivan/cs2120f23.git
|
c2da7c4a6be769c6fca375a54e8c1fa31ba2854b
|
Instructor/Lectures/grad/sets_relations.lean
|
succ_inj
|
[189, 1]
|
[193, 13]
|
assumption
|
a1 a2 : ℕ
h : a1 = a2
⊢ a1 = a2
|
no goals
|
https://github.com/kevinsullivan/cs2120f23.git
|
c2da7c4a6be769c6fca375a54e8c1fa31ba2854b
|
Instructor/Lectures/grad/sets_relations.lean
|
not_succ_surj
|
[196, 1]
|
[200, 27]
|
unfold Surjective
|
⊢ ¬Surjective fun n => n + 1
|
⊢ ¬∀ (b : ℕ), ∃ a, (fun n => n + 1) a = b
|
https://github.com/kevinsullivan/cs2120f23.git
|
c2da7c4a6be769c6fca375a54e8c1fa31ba2854b
|
Instructor/Lectures/grad/sets_relations.lean
|
not_succ_surj
|
[196, 1]
|
[200, 27]
|
intro h
|
⊢ ¬∀ (b : ℕ), ∃ a, (fun n => n + 1) a = b
|
h : ∀ (b : ℕ), ∃ a, (fun n => n + 1) a = b
⊢ False
|
https://github.com/kevinsullivan/cs2120f23.git
|
c2da7c4a6be769c6fca375a54e8c1fa31ba2854b
|
Instructor/Lectures/grad/sets_relations.lean
|
not_succ_surj
|
[196, 1]
|
[200, 27]
|
let contra := h 0
|
h : ∀ (b : ℕ), ∃ a, (fun n => n + 1) a = b
⊢ False
|
h : ∀ (b : ℕ), ∃ a, (fun n => n + 1) a = b
contra : ∃ a, (fun n => n + 1) a = 0 := h 0
⊢ False
|
https://github.com/kevinsullivan/cs2120f23.git
|
c2da7c4a6be769c6fca375a54e8c1fa31ba2854b
|
Instructor/Lectures/grad/sets_relations.lean
|
not_succ_surj
|
[196, 1]
|
[200, 27]
|
simp [Nat.add] at contra
|
h : ∀ (b : ℕ), ∃ a, (fun n => n + 1) a = b
contra : ∃ a, (fun n => n + 1) a = 0 := h 0
⊢ False
|
no goals
|
https://github.com/kevinsullivan/cs2120f23.git
|
c2da7c4a6be769c6fca375a54e8c1fa31ba2854b
|
Instructor/Lectures/grad/sets_relations.lean
|
inj_not
|
[204, 1]
|
[215, 25]
|
unfold Injective
|
⊢ Injective Bool.not
|
⊢ ∀ ⦃a₁ a₂ : Bool⦄, Bool.not a₁ = Bool.not a₂ → a₁ = a₂
|
https://github.com/kevinsullivan/cs2120f23.git
|
c2da7c4a6be769c6fca375a54e8c1fa31ba2854b
|
Instructor/Lectures/grad/sets_relations.lean
|
inj_not
|
[204, 1]
|
[215, 25]
|
intros a1 a2 h
|
⊢ ∀ ⦃a₁ a₂ : Bool⦄, Bool.not a₁ = Bool.not a₂ → a₁ = a₂
|
a1 a2 : Bool
h : Bool.not a1 = Bool.not a2
⊢ a1 = a2
|
https://github.com/kevinsullivan/cs2120f23.git
|
c2da7c4a6be769c6fca375a54e8c1fa31ba2854b
|
Instructor/Lectures/grad/sets_relations.lean
|
inj_not
|
[204, 1]
|
[215, 25]
|
cases a1 with
| true =>
cases a2 with
| true => rfl
| false => nomatch h
| false =>
cases a2 with
| true => nomatch h
| false => rfl
|
a1 a2 : Bool
h : Bool.not a1 = Bool.not a2
⊢ a1 = a2
|
no goals
|
https://github.com/kevinsullivan/cs2120f23.git
|
c2da7c4a6be769c6fca375a54e8c1fa31ba2854b
|
Instructor/Lectures/grad/sets_relations.lean
|
inj_not
|
[204, 1]
|
[215, 25]
|
cases a2 with
| true => rfl
| false => nomatch h
|
case true
a2 : Bool
h : Bool.not true = Bool.not a2
⊢ true = a2
|
no goals
|
https://github.com/kevinsullivan/cs2120f23.git
|
c2da7c4a6be769c6fca375a54e8c1fa31ba2854b
|
Instructor/Lectures/grad/sets_relations.lean
|
inj_not
|
[204, 1]
|
[215, 25]
|
rfl
|
case true.true
h : Bool.not true = Bool.not true
⊢ true = true
|
no goals
|
https://github.com/kevinsullivan/cs2120f23.git
|
c2da7c4a6be769c6fca375a54e8c1fa31ba2854b
|
Instructor/Lectures/grad/sets_relations.lean
|
inj_not
|
[204, 1]
|
[215, 25]
|
nomatch h
|
case true.false
h : Bool.not true = Bool.not false
⊢ true = false
|
no goals
|
https://github.com/kevinsullivan/cs2120f23.git
|
c2da7c4a6be769c6fca375a54e8c1fa31ba2854b
|
Instructor/Lectures/grad/sets_relations.lean
|
inj_not
|
[204, 1]
|
[215, 25]
|
cases a2 with
| true => nomatch h
| false => rfl
|
case false
a2 : Bool
h : Bool.not false = Bool.not a2
⊢ false = a2
|
no goals
|
https://github.com/kevinsullivan/cs2120f23.git
|
c2da7c4a6be769c6fca375a54e8c1fa31ba2854b
|
Instructor/Lectures/grad/sets_relations.lean
|
inj_not
|
[204, 1]
|
[215, 25]
|
nomatch h
|
case false.true
h : Bool.not false = Bool.not true
⊢ false = true
|
no goals
|
https://github.com/kevinsullivan/cs2120f23.git
|
c2da7c4a6be769c6fca375a54e8c1fa31ba2854b
|
Instructor/Lectures/grad/sets_relations.lean
|
inj_not
|
[204, 1]
|
[215, 25]
|
rfl
|
case false.false
h : Bool.not false = Bool.not false
⊢ false = false
|
no goals
|
https://github.com/kevinsullivan/cs2120f23.git
|
c2da7c4a6be769c6fca375a54e8c1fa31ba2854b
|
Instructor/Lectures/grad/sets_relations.lean
|
surj_not
|
[217, 1]
|
[222, 33]
|
unfold Surjective
|
⊢ Surjective Bool.not
|
⊢ ∀ (b : Bool), ∃ a, Bool.not a = b
|
https://github.com/kevinsullivan/cs2120f23.git
|
c2da7c4a6be769c6fca375a54e8c1fa31ba2854b
|
Instructor/Lectures/grad/sets_relations.lean
|
surj_not
|
[217, 1]
|
[222, 33]
|
intro b
|
⊢ ∀ (b : Bool), ∃ a, Bool.not a = b
|
b : Bool
⊢ ∃ a, Bool.not a = b
|
https://github.com/kevinsullivan/cs2120f23.git
|
c2da7c4a6be769c6fca375a54e8c1fa31ba2854b
|
Instructor/Lectures/grad/sets_relations.lean
|
surj_not
|
[217, 1]
|
[222, 33]
|
cases b with
| true => exact ⟨ false, rfl ⟩
| false => exact ⟨ true, rfl ⟩
|
b : Bool
⊢ ∃ a, Bool.not a = b
|
no goals
|
https://github.com/kevinsullivan/cs2120f23.git
|
c2da7c4a6be769c6fca375a54e8c1fa31ba2854b
|
Instructor/Lectures/grad/sets_relations.lean
|
surj_not
|
[217, 1]
|
[222, 33]
|
exact ⟨ false, rfl ⟩
|
case true
⊢ ∃ a, Bool.not a = true
|
no goals
|
https://github.com/kevinsullivan/cs2120f23.git
|
c2da7c4a6be769c6fca375a54e8c1fa31ba2854b
|
Instructor/Lectures/grad/sets_relations.lean
|
surj_not
|
[217, 1]
|
[222, 33]
|
exact ⟨ true, rfl ⟩
|
case false
⊢ ∃ a, Bool.not a = false
|
no goals
|
https://github.com/kevinsullivan/cs2120f23.git
|
c2da7c4a6be769c6fca375a54e8c1fa31ba2854b
|
Instructor/Lectures/grad/sets_relations.lean
|
not_bij
|
[224, 1]
|
[226, 30]
|
unfold Bijective
|
⊢ Bijective Bool.not
|
⊢ Injective Bool.not ∧ Surjective Bool.not
|
https://github.com/kevinsullivan/cs2120f23.git
|
c2da7c4a6be769c6fca375a54e8c1fa31ba2854b
|
Instructor/Lectures/grad/sets_relations.lean
|
not_bij
|
[224, 1]
|
[226, 30]
|
exact ⟨ inj_not, surj_not ⟩
|
⊢ Injective Bool.not ∧ Surjective Bool.not
|
no goals
|
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