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|---|---|---|---|---|---|---|---|---|---|
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) | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ ∀ (a b c : Rotation), a + b + c = a + (b + c)
TACTIC:
|
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) | Please generate a tactic in lean4 to solve the state.
STATE:
a b c : Rotation
⊢ a + b + c = a + (b + c)
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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)
TACTIC:
|
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) | Please generate a tactic in lean4 to solve the state.
STATE:
case r240
b c : Rotation
⊢ r240 + b + c = r240 + (b + c)
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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)
TACTIC:
|
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) | Please generate a tactic in lean4 to solve the state.
STATE:
case r240.r240
c : Rotation
⊢ r240 + r240 + c = r240 + (r240 + c)
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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)
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
case r240.r240.r240
⊢ r240 + r240 + r240 = r240 + (r240 + r240)
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ ∀ (a : Rotation), 0 + a = a
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
a : Rotation
⊢ 0 + a = a
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
case r0
⊢ 0 + r0 = r0
case r120
⊢ 0 + r120 = r120
case r240
⊢ 0 + r240 = r240
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
case r240
⊢ 0 + r240 = r240
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ ∀ (a : Rotation), a + 0 = a
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
a : Rotation
⊢ a + 0 = a
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
case r0
⊢ r0 + 0 = r0
case r120
⊢ r120 + 0 = r120
case r240
⊢ r240 + 0 = r240
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
case r240
⊢ r240 + 0 = r240
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ ∀ (p : State), 0 +ᵥ p = p
TACTIC:
|
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✝ | Please generate a tactic in lean4 to solve the state.
STATE:
p : State
⊢ 0 +ᵥ p = p
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
case heading
h✝ : K
unit✝ : ℕ
⊢ 0 +ᵥ heading h✝ unit✝ = heading h✝ unit✝
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
case heading
h✝ : K
unit✝ : ℕ
⊢ 0 +ᵥ heading h✝ unit✝ = heading h✝ unit✝
TACTIC:
|
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) | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ ∀ (g₁ g₂ : Rotation) (p : State), g₁ + g₂ +ᵥ p = g₁ +ᵥ (g₂ +ᵥ p)
TACTIC:
|
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) | Please generate a tactic in lean4 to solve the state.
STATE:
g₁ g₂ : Rotation
p : State
⊢ g₁ + g₂ +ᵥ p = g₁ +ᵥ (g₂ +ᵥ p)
TACTIC:
|
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) | Please generate a tactic in lean4 to solve the state.
STATE:
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)
TACTIC:
|
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) | Please generate a tactic in lean4 to solve the state.
STATE:
case r120
g₂ : Rotation
p : State
⊢ r120 + g₂ +ᵥ p = r120 +ᵥ (g₂ +ᵥ p)
TACTIC:
|
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) | Please generate a tactic in lean4 to solve the state.
STATE:
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)
TACTIC:
|
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✝) | Please generate a tactic in lean4 to solve the state.
STATE:
case r120.r240
p : State
⊢ r120 + r240 +ᵥ p = r120 +ᵥ (r240 +ᵥ p)
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
case r120.r120.heading
h✝ : K
unit✝ : ℕ
⊢ r120 + r120 +ᵥ heading h✝ unit✝ = r120 +ᵥ (r120 +ᵥ heading h✝ unit✝)
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
case r120.r120.heading
h✝ : K
unit✝ : ℕ
⊢ r120 + r120 +ᵥ heading h✝ unit✝ = r120 +ᵥ (r120 +ᵥ heading h✝ unit✝)
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ ∀ (a : Rotation), -a + a = 0
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
a : Rotation
⊢ -a + a = 0
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
case r0
⊢ -r0 + r0 = 0
case r120
⊢ -r120 + r120 = 0
case r240
⊢ -r240 + r240 = 0
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
case r240
⊢ -r240 + r240 = 0
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ ∀ (p1 p2 : State), p1 -ᵥ p2 +ᵥ p2 = p1
TACTIC:
|
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✝ | Please generate a tactic in lean4 to solve the state.
STATE:
p1 p2 : State
⊢ p1 -ᵥ p2 +ᵥ p2 = p1
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
case heading
p2 : State
h✝ : K
unit✝ : ℕ
⊢ heading h✝ unit✝ -ᵥ p2 +ᵥ p2 = heading h✝ unit✝
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ ∀ (g : Rotation) (p : State), g +ᵥ p -ᵥ p = g
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
g : Rotation
p : State
⊢ g +ᵥ p -ᵥ p = g
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
case r0
p : State
⊢ r0 +ᵥ p -ᵥ p = r0
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ Reflexive Eq
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ ∀ (x : ℕ), x = x
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℕ
⊢ x = x
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ Symmetric Eq
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ ∀ ⦃x y : ℕ⦄, x = y → y = x
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
x y : ℕ
⊢ x = y → y = x
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
x y : ℕ
hxy : x = y
⊢ y = x
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ Transitive Eq
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ ∀ ⦃x y z : ℕ⦄, x = y → y = z → x = z
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
x y z : ℕ
⊢ x = y → y = z → x = z
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
x y z : ℕ
hxy : x = y
hyz : y = z
⊢ x = z
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
x y z : ℕ
hxy : x = y
hyz : y = z
⊢ y = z
TACTIC:
|
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) | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ ∀ (n : ℕ), Equivalence (cong_mod_n n)
TACTIC:
|
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) | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ ∀ (n : ℕ), Reflexive (cong_mod_n n)
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
⊢ Reflexive (cong_mod_n n)
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
⊢ Reflexive fun a b => a % n = b % n
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
⊢ ∀ (x : ℕ), (fun a b => a % n = b % n) x x
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
n a : ℕ
⊢ a % n = a % n
TACTIC:
|
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) | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ ∀ (n : ℕ), Symmetric (cong_mod_n n)
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
⊢ Symmetric (cong_mod_n n)
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
⊢ ∀ ⦃x y : ℕ⦄, cong_mod_n n x y → cong_mod_n n y x
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
n x y : ℕ
⊢ cong_mod_n n x y → cong_mod_n n y x
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
n x y : ℕ
hxy : cong_mod_n n x y
⊢ cong_mod_n n y x
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
n x y : ℕ
hxy : cong_mod_n n x y
⊢ y % n = x % n
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
n x y : ℕ
hxy : x % n = y % n
⊢ y % n = x % n
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ ∀ (n : ℕ), Transitive (cong_mod_n n)
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
n a b c : ℕ
hab : cong_mod_n n a b
hbc : cong_mod_n n b c
⊢ a % n = c % n
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
n a b c : ℕ
hab : a % n = b % n
hbc : b % n = c % n
⊢ a % n = c % n
TACTIC:
|
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) | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ ∀ (n : ℕ), Equivalence (cong_mod_n n)
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
⊢ Equivalence (cong_mod_n n)
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
⊢ Equivalence fun a b => a % n = b % n
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
⊢ ∀ (x : ℕ), x % n = x % n
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
n x : ℕ
⊢ x % n = x % n
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
⊢ ∀ {x y : ℕ}, x % n = y % n → y % n = x % n
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
n x y : ℕ
h : x % n = y % n
⊢ y % n = x % n
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
⊢ ∀ {x y z : ℕ}, x % n = y % n → y % n = z % n → x % n = z % n
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
n x y z : ℕ
hxy : x % n = y % n
hyz : y % n = z % n
⊢ x % n = z % n
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ Injective fun n => n + 1
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ ∀ ⦃a₁ a₂ : ℕ⦄, (fun n => n + 1) a₁ = (fun n => n + 1) a₂ → a₁ = a₂
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
a1 a2 : ℕ
h : (fun n => n + 1) a1 = (fun n => n + 1) a2
⊢ a1 = a2
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
a1 a2 : ℕ
h : a1 = a2
⊢ a1 = a2
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ ¬Surjective fun n => n + 1
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ ¬∀ (b : ℕ), ∃ a, (fun n => n + 1) a = b
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
h : ∀ (b : ℕ), ∃ a, (fun n => n + 1) a = b
⊢ False
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
h : ∀ (b : ℕ), ∃ a, (fun n => n + 1) a = b
contra : ∃ a, (fun n => n + 1) a = 0 := h 0
⊢ False
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ Injective Bool.not
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ ∀ ⦃a₁ a₂ : Bool⦄, Bool.not a₁ = Bool.not a₂ → a₁ = a₂
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
a1 a2 : Bool
h : Bool.not a1 = Bool.not a2
⊢ a1 = a2
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
case true
a2 : Bool
h : Bool.not true = Bool.not a2
⊢ true = a2
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
case true.true
h : Bool.not true = Bool.not true
⊢ true = true
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
case true.false
h : Bool.not true = Bool.not false
⊢ true = false
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
case false
a2 : Bool
h : Bool.not false = Bool.not a2
⊢ false = a2
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
case false.true
h : Bool.not false = Bool.not true
⊢ false = true
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
case false.false
h : Bool.not false = Bool.not false
⊢ false = false
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ Surjective Bool.not
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ ∀ (b : Bool), ∃ a, Bool.not a = b
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
b : Bool
⊢ ∃ a, Bool.not a = b
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
case true
⊢ ∃ a, Bool.not a = true
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
case false
⊢ ∃ a, Bool.not a = false
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ Bijective Bool.not
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ Injective Bool.not ∧ Surjective Bool.not
TACTIC:
|