url stringclasses 147
values | commit stringclasses 147
values | file_path stringlengths 7 101 | full_name stringlengths 1 94 | start stringlengths 6 10 | end stringlengths 6 11 | tactic stringlengths 1 11.2k | state_before stringlengths 3 2.09M | state_after stringlengths 6 2.09M | input stringlengths 73 2.09M |
|---|---|---|---|---|---|---|---|---|---|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Bitwise.lean | Nat.bit0_bitwise_bit0 | [61, 1] | [65, 35] | simp [Nat.mul_eq_zero] at h | x y : Nat
f : Bool → Bool → Bool
hx : x ≠ 0
hy : y ≠ 0
hx2 : 2 * x ≠ 0
h : 2 * y = 0
⊢ False | x y : Nat
f : Bool → Bool → Bool
hx : x ≠ 0
hy : y ≠ 0
hx2 : 2 * x ≠ 0
h : y = 0
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Nat
f : Bool → Bool → Bool
hx : x ≠ 0
hy : y ≠ 0
hx2 : 2 * x ≠ 0
h : 2 * y = 0
⊢ False
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Bitwise.lean | Nat.bit0_bitwise_bit0 | [61, 1] | [65, 35] | contradiction | x y : Nat
f : Bool → Bool → Bool
hx : x ≠ 0
hy : y ≠ 0
hx2 : 2 * x ≠ 0
h : y = 0
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Nat
f : Bool → Bool → Bool
hx : x ≠ 0
hy : y ≠ 0
hx2 : 2 * x ≠ 0
h : y = 0
⊢ False
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Bitwise.lean | Nat.bit1_bitwise_bit0 | [67, 1] | [74, 34] | if y = 0 then
have : 2 * y = 0 := by simp [*]
simp [*]; cases (f true false) <;> simp [*]
else
have hy : 2 * y ≠ 0 := by intro h; simp [Nat.mul_eq_zero] at h; contradiction
simp [pos_bitwise_pos _ _ hy] | f : Bool → Bool → Bool
x y : Nat
⊢ bitwise f (2 * x + 1) (2 * y) = 2 * bitwise f x y + bif f true false then 1 else 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : Bool → Bool → Bool
x y : Nat
⊢ bitwise f (2 * x + 1) (2 * y) = 2 * bitwise f x y + bif f true false then 1 else 0
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Bitwise.lean | Nat.bit1_bitwise_bit0 | [67, 1] | [74, 34] | have : 2 * y = 0 := by simp [*] | f : Bool → Bool → Bool
x y : Nat
h✝ : y = 0
⊢ bitwise f (2 * x + 1) (2 * y) = 2 * bitwise f x y + bif f true false then 1 else 0 | f : Bool → Bool → Bool
x y : Nat
h✝ : y = 0
this : 2 * y = 0
⊢ bitwise f (2 * x + 1) (2 * y) = 2 * bitwise f x y + bif f true false then 1 else 0 | Please generate a tactic in lean4 to solve the state.
STATE:
f : Bool → Bool → Bool
x y : Nat
h✝ : y = 0
⊢ bitwise f (2 * x + 1) (2 * y) = 2 * bitwise f x y + bif f true false then 1 else 0
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Bitwise.lean | Nat.bit1_bitwise_bit0 | [67, 1] | [74, 34] | simp [*] | f : Bool → Bool → Bool
x y : Nat
h✝ : y = 0
this : 2 * y = 0
⊢ bitwise f (2 * x + 1) (2 * y) = 2 * bitwise f x y + bif f true false then 1 else 0 | f : Bool → Bool → Bool
x y : Nat
h✝ : y = 0
this : 2 * y = 0
⊢ (bif f true false then 2 * x + 1 else 0) = (2 * bif f true false then x else 0) + bif f true false then 1 else 0 | Please generate a tactic in lean4 to solve the state.
STATE:
f : Bool → Bool → Bool
x y : Nat
h✝ : y = 0
this : 2 * y = 0
⊢ bitwise f (2 * x + 1) (2 * y) = 2 * bitwise f x y + bif f true false then 1 else 0
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Bitwise.lean | Nat.bit1_bitwise_bit0 | [67, 1] | [74, 34] | cases (f true false) <;> simp [*] | f : Bool → Bool → Bool
x y : Nat
h✝ : y = 0
this : 2 * y = 0
⊢ (bif f true false then 2 * x + 1 else 0) = (2 * bif f true false then x else 0) + bif f true false then 1 else 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : Bool → Bool → Bool
x y : Nat
h✝ : y = 0
this : 2 * y = 0
⊢ (bif f true false then 2 * x + 1 else 0) = (2 * bif f true false then x else 0) + bif f true false then 1 else 0
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Bitwise.lean | Nat.bit1_bitwise_bit0 | [67, 1] | [74, 34] | simp [*] | f : Bool → Bool → Bool
x y : Nat
h✝ : y = 0
⊢ 2 * y = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : Bool → Bool → Bool
x y : Nat
h✝ : y = 0
⊢ 2 * y = 0
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Bitwise.lean | Nat.bit1_bitwise_bit0 | [67, 1] | [74, 34] | have hy : 2 * y ≠ 0 := by intro h; simp [Nat.mul_eq_zero] at h; contradiction | f : Bool → Bool → Bool
x y : Nat
h✝ : ¬y = 0
⊢ bitwise f (2 * x + 1) (2 * y) = 2 * bitwise f x y + bif f true false then 1 else 0 | f : Bool → Bool → Bool
x y : Nat
h✝ : ¬y = 0
hy : 2 * y ≠ 0
⊢ bitwise f (2 * x + 1) (2 * y) = 2 * bitwise f x y + bif f true false then 1 else 0 | Please generate a tactic in lean4 to solve the state.
STATE:
f : Bool → Bool → Bool
x y : Nat
h✝ : ¬y = 0
⊢ bitwise f (2 * x + 1) (2 * y) = 2 * bitwise f x y + bif f true false then 1 else 0
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Bitwise.lean | Nat.bit1_bitwise_bit0 | [67, 1] | [74, 34] | simp [pos_bitwise_pos _ _ hy] | f : Bool → Bool → Bool
x y : Nat
h✝ : ¬y = 0
hy : 2 * y ≠ 0
⊢ bitwise f (2 * x + 1) (2 * y) = 2 * bitwise f x y + bif f true false then 1 else 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : Bool → Bool → Bool
x y : Nat
h✝ : ¬y = 0
hy : 2 * y ≠ 0
⊢ bitwise f (2 * x + 1) (2 * y) = 2 * bitwise f x y + bif f true false then 1 else 0
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Bitwise.lean | Nat.bit1_bitwise_bit0 | [67, 1] | [74, 34] | intro h | f : Bool → Bool → Bool
x y : Nat
h✝ : ¬y = 0
⊢ 2 * y ≠ 0 | f : Bool → Bool → Bool
x y : Nat
h✝ : ¬y = 0
h : 2 * y = 0
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
f : Bool → Bool → Bool
x y : Nat
h✝ : ¬y = 0
⊢ 2 * y ≠ 0
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Bitwise.lean | Nat.bit1_bitwise_bit0 | [67, 1] | [74, 34] | simp [Nat.mul_eq_zero] at h | f : Bool → Bool → Bool
x y : Nat
h✝ : ¬y = 0
h : 2 * y = 0
⊢ False | f : Bool → Bool → Bool
x y : Nat
h✝ : ¬y = 0
h : y = 0
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
f : Bool → Bool → Bool
x y : Nat
h✝ : ¬y = 0
h : 2 * y = 0
⊢ False
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Bitwise.lean | Nat.bit1_bitwise_bit0 | [67, 1] | [74, 34] | contradiction | f : Bool → Bool → Bool
x y : Nat
h✝ : ¬y = 0
h : y = 0
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : Bool → Bool → Bool
x y : Nat
h✝ : ¬y = 0
h : y = 0
⊢ False
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Bitwise.lean | Nat.bit0_bitwise_bit1 | [76, 1] | [83, 32] | if x = 0 then
have : 2 * x = 0 := by simp [*]
simp [*]; cases (f false true) <;> simp [*]
else
have hx : 2 * x ≠ 0 := by intro h; simp [Nat.mul_eq_zero] at h; contradiction
simp [pos_bitwise_pos _ hx] | f : Bool → Bool → Bool
x y : Nat
⊢ bitwise f (2 * x) (2 * y + 1) = 2 * bitwise f x y + bif f false true then 1 else 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : Bool → Bool → Bool
x y : Nat
⊢ bitwise f (2 * x) (2 * y + 1) = 2 * bitwise f x y + bif f false true then 1 else 0
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Bitwise.lean | Nat.bit0_bitwise_bit1 | [76, 1] | [83, 32] | have : 2 * x = 0 := by simp [*] | f : Bool → Bool → Bool
x y : Nat
h✝ : x = 0
⊢ bitwise f (2 * x) (2 * y + 1) = 2 * bitwise f x y + bif f false true then 1 else 0 | f : Bool → Bool → Bool
x y : Nat
h✝ : x = 0
this : 2 * x = 0
⊢ bitwise f (2 * x) (2 * y + 1) = 2 * bitwise f x y + bif f false true then 1 else 0 | Please generate a tactic in lean4 to solve the state.
STATE:
f : Bool → Bool → Bool
x y : Nat
h✝ : x = 0
⊢ bitwise f (2 * x) (2 * y + 1) = 2 * bitwise f x y + bif f false true then 1 else 0
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Bitwise.lean | Nat.bit0_bitwise_bit1 | [76, 1] | [83, 32] | simp [*] | f : Bool → Bool → Bool
x y : Nat
h✝ : x = 0
this : 2 * x = 0
⊢ bitwise f (2 * x) (2 * y + 1) = 2 * bitwise f x y + bif f false true then 1 else 0 | f : Bool → Bool → Bool
x y : Nat
h✝ : x = 0
this : 2 * x = 0
⊢ (bif f false true then 2 * y + 1 else 0) = (2 * bif f false true then y else 0) + bif f false true then 1 else 0 | Please generate a tactic in lean4 to solve the state.
STATE:
f : Bool → Bool → Bool
x y : Nat
h✝ : x = 0
this : 2 * x = 0
⊢ bitwise f (2 * x) (2 * y + 1) = 2 * bitwise f x y + bif f false true then 1 else 0
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Bitwise.lean | Nat.bit0_bitwise_bit1 | [76, 1] | [83, 32] | cases (f false true) <;> simp [*] | f : Bool → Bool → Bool
x y : Nat
h✝ : x = 0
this : 2 * x = 0
⊢ (bif f false true then 2 * y + 1 else 0) = (2 * bif f false true then y else 0) + bif f false true then 1 else 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : Bool → Bool → Bool
x y : Nat
h✝ : x = 0
this : 2 * x = 0
⊢ (bif f false true then 2 * y + 1 else 0) = (2 * bif f false true then y else 0) + bif f false true then 1 else 0
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Bitwise.lean | Nat.bit0_bitwise_bit1 | [76, 1] | [83, 32] | simp [*] | f : Bool → Bool → Bool
x y : Nat
h✝ : x = 0
⊢ 2 * x = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : Bool → Bool → Bool
x y : Nat
h✝ : x = 0
⊢ 2 * x = 0
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Bitwise.lean | Nat.bit0_bitwise_bit1 | [76, 1] | [83, 32] | have hx : 2 * x ≠ 0 := by intro h; simp [Nat.mul_eq_zero] at h; contradiction | f : Bool → Bool → Bool
x y : Nat
h✝ : ¬x = 0
⊢ bitwise f (2 * x) (2 * y + 1) = 2 * bitwise f x y + bif f false true then 1 else 0 | f : Bool → Bool → Bool
x y : Nat
h✝ : ¬x = 0
hx : 2 * x ≠ 0
⊢ bitwise f (2 * x) (2 * y + 1) = 2 * bitwise f x y + bif f false true then 1 else 0 | Please generate a tactic in lean4 to solve the state.
STATE:
f : Bool → Bool → Bool
x y : Nat
h✝ : ¬x = 0
⊢ bitwise f (2 * x) (2 * y + 1) = 2 * bitwise f x y + bif f false true then 1 else 0
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Bitwise.lean | Nat.bit0_bitwise_bit1 | [76, 1] | [83, 32] | simp [pos_bitwise_pos _ hx] | f : Bool → Bool → Bool
x y : Nat
h✝ : ¬x = 0
hx : 2 * x ≠ 0
⊢ bitwise f (2 * x) (2 * y + 1) = 2 * bitwise f x y + bif f false true then 1 else 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : Bool → Bool → Bool
x y : Nat
h✝ : ¬x = 0
hx : 2 * x ≠ 0
⊢ bitwise f (2 * x) (2 * y + 1) = 2 * bitwise f x y + bif f false true then 1 else 0
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Bitwise.lean | Nat.bit0_bitwise_bit1 | [76, 1] | [83, 32] | intro h | f : Bool → Bool → Bool
x y : Nat
h✝ : ¬x = 0
⊢ 2 * x ≠ 0 | f : Bool → Bool → Bool
x y : Nat
h✝ : ¬x = 0
h : 2 * x = 0
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
f : Bool → Bool → Bool
x y : Nat
h✝ : ¬x = 0
⊢ 2 * x ≠ 0
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Bitwise.lean | Nat.bit0_bitwise_bit1 | [76, 1] | [83, 32] | simp [Nat.mul_eq_zero] at h | f : Bool → Bool → Bool
x y : Nat
h✝ : ¬x = 0
h : 2 * x = 0
⊢ False | f : Bool → Bool → Bool
x y : Nat
h✝ : ¬x = 0
h : x = 0
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
f : Bool → Bool → Bool
x y : Nat
h✝ : ¬x = 0
h : 2 * x = 0
⊢ False
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Bitwise.lean | Nat.bit0_bitwise_bit1 | [76, 1] | [83, 32] | contradiction | f : Bool → Bool → Bool
x y : Nat
h✝ : ¬x = 0
h : x = 0
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : Bool → Bool → Bool
x y : Nat
h✝ : ¬x = 0
h : x = 0
⊢ False
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Bitwise.lean | Nat.bit1_bitwise_bit1 | [85, 1] | [87, 25] | simp [pos_bitwise_pos] | f : Bool → Bool → Bool
x y : Nat
⊢ bitwise f (2 * x + 1) (2 * y + 1) = 2 * bitwise f x y + bif f true true then 1 else 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : Bool → Bool → Bool
x y : Nat
⊢ bitwise f (2 * x + 1) (2 * y + 1) = 2 * bitwise f x y + bif f true true then 1 else 0
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Bitwise.lean | Nat.bit0_and_bit0 | [93, 15] | [99, 41] | if x = 0 then
simp [*, and_def]
else if y = 0 then
simp [*, and_def]
else
simp [*, and_def, bit0_bitwise_bit0] | x y : Nat
⊢ 2 * x &&& 2 * y = 2 * (x &&& y) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Nat
⊢ 2 * x &&& 2 * y = 2 * (x &&& y)
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Bitwise.lean | Nat.bit0_and_bit0 | [93, 15] | [99, 41] | simp [*, and_def] | x y : Nat
h✝ : x = 0
⊢ 2 * x &&& 2 * y = 2 * (x &&& y) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Nat
h✝ : x = 0
⊢ 2 * x &&& 2 * y = 2 * (x &&& y)
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Bitwise.lean | Nat.bit0_and_bit0 | [93, 15] | [99, 41] | if y = 0 then
simp [*, and_def]
else
simp [*, and_def, bit0_bitwise_bit0] | x y : Nat
h✝ : ¬x = 0
⊢ 2 * x &&& 2 * y = 2 * (x &&& y) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Nat
h✝ : ¬x = 0
⊢ 2 * x &&& 2 * y = 2 * (x &&& y)
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Bitwise.lean | Nat.bit0_and_bit0 | [93, 15] | [99, 41] | simp [*, and_def] | x y : Nat
h✝¹ : ¬x = 0
h✝ : y = 0
⊢ 2 * x &&& 2 * y = 2 * (x &&& y) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Nat
h✝¹ : ¬x = 0
h✝ : y = 0
⊢ 2 * x &&& 2 * y = 2 * (x &&& y)
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Bitwise.lean | Nat.bit0_and_bit0 | [93, 15] | [99, 41] | simp [*, and_def, bit0_bitwise_bit0] | x y : Nat
h✝¹ : ¬x = 0
h✝ : ¬y = 0
⊢ 2 * x &&& 2 * y = 2 * (x &&& y) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Nat
h✝¹ : ¬x = 0
h✝ : ¬y = 0
⊢ 2 * x &&& 2 * y = 2 * (x &&& y)
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Bitwise.lean | Nat.bit0_and_bit1 | [101, 15] | [102, 36] | simp [and_def, bit0_bitwise_bit1] | x y : Nat
⊢ 2 * x &&& 2 * y + 1 = 2 * (x &&& y) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Nat
⊢ 2 * x &&& 2 * y + 1 = 2 * (x &&& y)
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Bitwise.lean | Nat.bit1_and_bit0 | [104, 15] | [105, 36] | simp [and_def, bit1_bitwise_bit0] | x y : Nat
⊢ 2 * x + 1 &&& 2 * y = 2 * (x &&& y) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Nat
⊢ 2 * x + 1 &&& 2 * y = 2 * (x &&& y)
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Bitwise.lean | Nat.bit1_and_bit1 | [107, 15] | [108, 36] | simp [and_def, bit1_bitwise_bit1] | x y : Nat
⊢ 2 * x + 1 &&& 2 * y + 1 = 2 * (x &&& y) + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Nat
⊢ 2 * x + 1 &&& 2 * y + 1 = 2 * (x &&& y) + 1
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Bitwise.lean | Nat.and_self | [110, 9] | [111, 44] | induction x using Nat.recBit <;> simp [*] | x : Nat
⊢ x &&& x = x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x : Nat
⊢ x &&& x = x
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Bitwise.lean | Nat.and_comm | [113, 1] | [114, 92] | induction x using Nat.recBit generalizing y <;> cases y using Nat.casesBitOn <;> simp [*] | x y : Nat
⊢ x &&& y = y &&& x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Nat
⊢ x &&& y = y &&& x
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Bitwise.lean | Nat.and_assoc | [116, 1] | [118, 79] | induction x using Nat.recBit generalizing y z <;>
cases y using Nat.casesBitOn <;> cases z using Nat.casesBitOn <;> simp [*] | x y z : Nat
⊢ x &&& y &&& z = x &&& (y &&& z) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y z : Nat
⊢ x &&& y &&& z = x &&& (y &&& z)
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Bitwise.lean | Nat.bit0_or_bit0 | [133, 15] | [139, 40] | if x = 0 then
simp [*, or_def]
else if y = 0 then
simp [*, or_def]
else
simp [*, or_def, bit0_bitwise_bit0] | x y : Nat
⊢ 2 * x ||| 2 * y = 2 * (x ||| y) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Nat
⊢ 2 * x ||| 2 * y = 2 * (x ||| y)
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Bitwise.lean | Nat.bit0_or_bit0 | [133, 15] | [139, 40] | simp [*, or_def] | x y : Nat
h✝ : x = 0
⊢ 2 * x ||| 2 * y = 2 * (x ||| y) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Nat
h✝ : x = 0
⊢ 2 * x ||| 2 * y = 2 * (x ||| y)
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Bitwise.lean | Nat.bit0_or_bit0 | [133, 15] | [139, 40] | if y = 0 then
simp [*, or_def]
else
simp [*, or_def, bit0_bitwise_bit0] | x y : Nat
h✝ : ¬x = 0
⊢ 2 * x ||| 2 * y = 2 * (x ||| y) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Nat
h✝ : ¬x = 0
⊢ 2 * x ||| 2 * y = 2 * (x ||| y)
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Bitwise.lean | Nat.bit0_or_bit0 | [133, 15] | [139, 40] | simp [*, or_def] | x y : Nat
h✝¹ : ¬x = 0
h✝ : y = 0
⊢ 2 * x ||| 2 * y = 2 * (x ||| y) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Nat
h✝¹ : ¬x = 0
h✝ : y = 0
⊢ 2 * x ||| 2 * y = 2 * (x ||| y)
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Bitwise.lean | Nat.bit0_or_bit0 | [133, 15] | [139, 40] | simp [*, or_def, bit0_bitwise_bit0] | x y : Nat
h✝¹ : ¬x = 0
h✝ : ¬y = 0
⊢ 2 * x ||| 2 * y = 2 * (x ||| y) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Nat
h✝¹ : ¬x = 0
h✝ : ¬y = 0
⊢ 2 * x ||| 2 * y = 2 * (x ||| y)
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Bitwise.lean | Nat.bit0_or_bit1 | [141, 15] | [142, 35] | simp [or_def, bit0_bitwise_bit1] | x y : Nat
⊢ 2 * x ||| 2 * y + 1 = 2 * (x ||| y) + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Nat
⊢ 2 * x ||| 2 * y + 1 = 2 * (x ||| y) + 1
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Bitwise.lean | Nat.bit1_or_bit0 | [144, 15] | [145, 35] | simp [or_def, bit1_bitwise_bit0] | x y : Nat
⊢ 2 * x + 1 ||| 2 * y = 2 * (x ||| y) + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Nat
⊢ 2 * x + 1 ||| 2 * y = 2 * (x ||| y) + 1
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Bitwise.lean | Nat.bit1_or_bit1 | [147, 15] | [148, 35] | simp [or_def, bit1_bitwise_bit1] | x y : Nat
⊢ 2 * x + 1 ||| 2 * y + 1 = 2 * (x ||| y) + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Nat
⊢ 2 * x + 1 ||| 2 * y + 1 = 2 * (x ||| y) + 1
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Bitwise.lean | Nat.or_self | [150, 9] | [151, 44] | induction x using Nat.recBit <;> simp [*] | x : Nat
⊢ x ||| x = x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x : Nat
⊢ x ||| x = x
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Bitwise.lean | Nat.or_comm | [153, 1] | [154, 92] | induction x using Nat.recBit generalizing y <;> cases y using Nat.casesBitOn <;> simp [*] | x y : Nat
⊢ x ||| y = y ||| x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Nat
⊢ x ||| y = y ||| x
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Bitwise.lean | Nat.or_assoc | [156, 1] | [158, 79] | induction x using Nat.recBit generalizing y z <;>
cases y using Nat.casesBitOn <;> cases z using Nat.casesBitOn <;> simp [*] | x y z : Nat
⊢ x ||| y ||| z = x ||| (y ||| z) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y z : Nat
⊢ x ||| y ||| z = x ||| (y ||| z)
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Bitwise.lean | Nat.and_or_distrib_left | [160, 1] | [162, 79] | induction x using Nat.recBit generalizing y z <;>
cases y using Nat.casesBitOn <;> cases z using Nat.casesBitOn <;> simp [*] | x y z : Nat
⊢ x &&& (y ||| z) = x &&& y ||| x &&& z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y z : Nat
⊢ x &&& (y ||| z) = x &&& y ||| x &&& z
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Bitwise.lean | Nat.and_distrib_right | [164, 1] | [166, 79] | induction z using Nat.recBit generalizing x y <;>
cases y using Nat.casesBitOn <;> cases x using Nat.casesBitOn <;> simp [*] | x y z : Nat
⊢ (x ||| y) &&& z = x &&& z ||| y &&& z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y z : Nat
⊢ (x ||| y) &&& z = x &&& z ||| y &&& z
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Bitwise.lean | Nat.or_and_distrib_left | [168, 1] | [170, 79] | induction x using Nat.recBit generalizing y z <;>
cases y using Nat.casesBitOn <;> cases z using Nat.casesBitOn <;> simp [*] | x y z : Nat
⊢ x ||| y &&& z = (x ||| y) &&& (x ||| z) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y z : Nat
⊢ x ||| y &&& z = (x ||| y) &&& (x ||| z)
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Bitwise.lean | Nat.or_and_distrib_right | [172, 1] | [174, 79] | induction z using Nat.recBit generalizing x y <;>
cases y using Nat.casesBitOn <;> cases x using Nat.casesBitOn <;> simp [*] | x y z : Nat
⊢ x &&& y ||| z = (x ||| z) &&& (y ||| z) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y z : Nat
⊢ x &&& y ||| z = (x ||| z) &&& (y ||| z)
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Bitwise.lean | Nat.bit0_xor_bit0 | [193, 15] | [199, 41] | if x = 0 then
simp [*, xor_def]
else if y = 0 then
simp [*, xor_def]
else
simp [*, xor_def, bit0_bitwise_bit0] | x y : Nat
⊢ 2 * x ^^^ 2 * y = 2 * (x ^^^ y) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Nat
⊢ 2 * x ^^^ 2 * y = 2 * (x ^^^ y)
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Bitwise.lean | Nat.bit0_xor_bit0 | [193, 15] | [199, 41] | simp [*, xor_def] | x y : Nat
h✝ : x = 0
⊢ 2 * x ^^^ 2 * y = 2 * (x ^^^ y) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Nat
h✝ : x = 0
⊢ 2 * x ^^^ 2 * y = 2 * (x ^^^ y)
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Bitwise.lean | Nat.bit0_xor_bit0 | [193, 15] | [199, 41] | if y = 0 then
simp [*, xor_def]
else
simp [*, xor_def, bit0_bitwise_bit0] | x y : Nat
h✝ : ¬x = 0
⊢ 2 * x ^^^ 2 * y = 2 * (x ^^^ y) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Nat
h✝ : ¬x = 0
⊢ 2 * x ^^^ 2 * y = 2 * (x ^^^ y)
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Bitwise.lean | Nat.bit0_xor_bit0 | [193, 15] | [199, 41] | simp [*, xor_def] | x y : Nat
h✝¹ : ¬x = 0
h✝ : y = 0
⊢ 2 * x ^^^ 2 * y = 2 * (x ^^^ y) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Nat
h✝¹ : ¬x = 0
h✝ : y = 0
⊢ 2 * x ^^^ 2 * y = 2 * (x ^^^ y)
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Bitwise.lean | Nat.bit0_xor_bit0 | [193, 15] | [199, 41] | simp [*, xor_def, bit0_bitwise_bit0] | x y : Nat
h✝¹ : ¬x = 0
h✝ : ¬y = 0
⊢ 2 * x ^^^ 2 * y = 2 * (x ^^^ y) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Nat
h✝¹ : ¬x = 0
h✝ : ¬y = 0
⊢ 2 * x ^^^ 2 * y = 2 * (x ^^^ y)
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Bitwise.lean | Nat.bit0_xor_bit1 | [201, 15] | [202, 36] | simp [xor_def, bit0_bitwise_bit1] | x y : Nat
⊢ 2 * x ^^^ 2 * y + 1 = 2 * (x ^^^ y) + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Nat
⊢ 2 * x ^^^ 2 * y + 1 = 2 * (x ^^^ y) + 1
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Bitwise.lean | Nat.bit1_xor_bit0 | [204, 15] | [205, 36] | simp [xor_def, bit1_bitwise_bit0] | x y : Nat
⊢ 2 * x + 1 ^^^ 2 * y = 2 * (x ^^^ y) + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Nat
⊢ 2 * x + 1 ^^^ 2 * y = 2 * (x ^^^ y) + 1
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Bitwise.lean | Nat.bit1_xor_bit1 | [207, 15] | [208, 36] | simp [xor_def, bit1_bitwise_bit1] | x y : Nat
⊢ 2 * x + 1 ^^^ 2 * y + 1 = 2 * (x ^^^ y) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Nat
⊢ 2 * x + 1 ^^^ 2 * y + 1 = 2 * (x ^^^ y)
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Bitwise.lean | Nat.zero_xor | [210, 9] | [211, 17] | simp [xor_def] | x : Nat
⊢ 0 ^^^ x = x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x : Nat
⊢ 0 ^^^ x = x
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Bitwise.lean | Nat.xor_zero | [213, 9] | [214, 17] | simp [xor_def] | x : Nat
⊢ x ^^^ 0 = x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x : Nat
⊢ x ^^^ 0 = x
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Bitwise.lean | Nat.xor_self | [216, 9] | [217, 44] | induction x using Nat.recBit <;> simp [*] | x : Nat
⊢ x ^^^ x = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x : Nat
⊢ x ^^^ x = 0
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Bitwise.lean | Nat.xor_comm | [219, 1] | [220, 92] | induction x using Nat.recBit generalizing y <;> cases y using Nat.casesBitOn <;> simp [*] | x y : Nat
⊢ x ^^^ y = y ^^^ x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Nat
⊢ x ^^^ y = y ^^^ x
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Bitwise.lean | Nat.xor_assoc | [222, 1] | [224, 79] | induction x using Nat.recBit generalizing y z <;>
cases y using Nat.casesBitOn <;> cases z using Nat.casesBitOn <;> simp [*] | x y z : Nat
⊢ x ^^^ y ^^^ z = x ^^^ (y ^^^ z) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y z : Nat
⊢ x ^^^ y ^^^ z = x ^^^ (y ^^^ z)
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Bitwise.lean | Nat.and_xor_distrib_left | [226, 1] | [228, 79] | induction x using Nat.recBit generalizing y z <;>
cases y using Nat.casesBitOn <;> cases z using Nat.casesBitOn <;> simp [*] | x y z : Nat
⊢ x &&& (y ^^^ z) = x &&& y ^^^ x &&& z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y z : Nat
⊢ x &&& (y ^^^ z) = x &&& y ^^^ x &&& z
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Bitwise.lean | Nat.and_xor_distrib_right | [230, 1] | [232, 79] | induction x using Nat.recBit generalizing y z <;>
cases y using Nat.casesBitOn <;> cases z using Nat.casesBitOn <;> simp [*] | x y z : Nat
⊢ (x ^^^ y) &&& z = x &&& z ^^^ y &&& z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y z : Nat
⊢ (x ^^^ y) &&& z = x &&& z ^^^ y &&& z
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Find.lean | Find.find_is_some_iff_exists_true | [11, 1] | [28, 22] | constructor | α : Type u_1
inst✝ : Find α
p : α → Bool
⊢ (find? p).isSome = true ↔ ∃ x, p x = true | case mp
α : Type u_1
inst✝ : Find α
p : α → Bool
⊢ (find? p).isSome = true → ∃ x, p x = true
case mpr
α : Type u_1
inst✝ : Find α
p : α → Bool
⊢ (∃ x, p x = true) → (find? p).isSome = true | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : Find α
p : α → Bool
⊢ (find? p).isSome = true ↔ ∃ x, p x = true
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Find.lean | Find.find_is_some_iff_exists_true | [11, 1] | [28, 22] | match hp : find? p with
| some x =>
intro
exists x
exact find?_eq_some hp
| none =>
intro
contradiction | case mp
α : Type u_1
inst✝ : Find α
p : α → Bool
⊢ (find? p).isSome = true → ∃ x, p x = true | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
α : Type u_1
inst✝ : Find α
p : α → Bool
⊢ (find? p).isSome = true → ∃ x, p x = true
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Find.lean | Find.find_is_some_iff_exists_true | [11, 1] | [28, 22] | intro | α : Type u_1
inst✝ : Find α
p : α → Bool
x : α
hp : find? p = some x
⊢ (some x).isSome = true → ∃ x, p x = true | α : Type u_1
inst✝ : Find α
p : α → Bool
x : α
hp : find? p = some x
a✝ : (some x).isSome = true
⊢ ∃ x, p x = true | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : Find α
p : α → Bool
x : α
hp : find? p = some x
⊢ (some x).isSome = true → ∃ x, p x = true
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Find.lean | Find.find_is_some_iff_exists_true | [11, 1] | [28, 22] | exists x | α : Type u_1
inst✝ : Find α
p : α → Bool
x : α
hp : find? p = some x
a✝ : (some x).isSome = true
⊢ ∃ x, p x = true | α : Type u_1
inst✝ : Find α
p : α → Bool
x : α
hp : find? p = some x
a✝ : (some x).isSome = true
⊢ p x = true | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : Find α
p : α → Bool
x : α
hp : find? p = some x
a✝ : (some x).isSome = true
⊢ ∃ x, p x = true
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Find.lean | Find.find_is_some_iff_exists_true | [11, 1] | [28, 22] | exact find?_eq_some hp | α : Type u_1
inst✝ : Find α
p : α → Bool
x : α
hp : find? p = some x
a✝ : (some x).isSome = true
⊢ p x = true | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : Find α
p : α → Bool
x : α
hp : find? p = some x
a✝ : (some x).isSome = true
⊢ p x = true
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Find.lean | Find.find_is_some_iff_exists_true | [11, 1] | [28, 22] | intro | α : Type u_1
inst✝ : Find α
p : α → Bool
hp : find? p = none
⊢ none.isSome = true → ∃ x, p x = true | α : Type u_1
inst✝ : Find α
p : α → Bool
hp : find? p = none
a✝ : none.isSome = true
⊢ ∃ x, p x = true | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : Find α
p : α → Bool
hp : find? p = none
⊢ none.isSome = true → ∃ x, p x = true
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Find.lean | Find.find_is_some_iff_exists_true | [11, 1] | [28, 22] | contradiction | α : Type u_1
inst✝ : Find α
p : α → Bool
hp : find? p = none
a✝ : none.isSome = true
⊢ ∃ x, p x = true | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : Find α
p : α → Bool
hp : find? p = none
a✝ : none.isSome = true
⊢ ∃ x, p x = true
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Find.lean | Find.find_is_some_iff_exists_true | [11, 1] | [28, 22] | intro
| ⟨x, hx⟩ =>
match hp : find? p with
| some _ =>
rfl
| none =>
rw [find?_eq_none hp] at hx
contradiction | case mpr
α : Type u_1
inst✝ : Find α
p : α → Bool
⊢ (∃ x, p x = true) → (find? p).isSome = true | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
α : Type u_1
inst✝ : Find α
p : α → Bool
⊢ (∃ x, p x = true) → (find? p).isSome = true
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Find.lean | Find.find_is_some_iff_exists_true | [11, 1] | [28, 22] | match hp : find? p with
| some _ =>
rfl
| none =>
rw [find?_eq_none hp] at hx
contradiction | α : Type u_1
inst✝ : Find α
p : α → Bool
x✝ : ∃ x, p x = true
x : α
hx : p x = true
⊢ (find? p).isSome = true | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : Find α
p : α → Bool
x✝ : ∃ x, p x = true
x : α
hx : p x = true
⊢ (find? p).isSome = true
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Find.lean | Find.find_is_some_iff_exists_true | [11, 1] | [28, 22] | rfl | α : Type u_1
inst✝ : Find α
p : α → Bool
x✝ : ∃ x, p x = true
x : α
hx : p x = true
val✝ : α
hp : find? p = some val✝
⊢ (some val✝).isSome = true | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : Find α
p : α → Bool
x✝ : ∃ x, p x = true
x : α
hx : p x = true
val✝ : α
hp : find? p = some val✝
⊢ (some val✝).isSome = true
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Find.lean | Find.find_is_some_iff_exists_true | [11, 1] | [28, 22] | rw [find?_eq_none hp] at hx | α : Type u_1
inst✝ : Find α
p : α → Bool
x✝ : ∃ x, p x = true
x : α
hx : p x = true
hp : find? p = none
⊢ none.isSome = true | α : Type u_1
inst✝ : Find α
p : α → Bool
x✝ : ∃ x, p x = true
x : α
hx : false = true
hp : find? p = none
⊢ none.isSome = true | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : Find α
p : α → Bool
x✝ : ∃ x, p x = true
x : α
hx : p x = true
hp : find? p = none
⊢ none.isSome = true
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Find.lean | Find.find_is_some_iff_exists_true | [11, 1] | [28, 22] | contradiction | α : Type u_1
inst✝ : Find α
p : α → Bool
x✝ : ∃ x, p x = true
x : α
hx : false = true
hp : find? p = none
⊢ none.isSome = true | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : Find α
p : α → Bool
x✝ : ∃ x, p x = true
x : α
hx : false = true
hp : find? p = none
⊢ none.isSome = true
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Find.lean | Find.find_is_none_iff_forall_false | [30, 1] | [47, 10] | constructor | α : Type u_1
inst✝ : Find α
p : α → Bool
⊢ (find? p).isNone = true ↔ ∀ (x : α), p x = false | case mp
α : Type u_1
inst✝ : Find α
p : α → Bool
⊢ (find? p).isNone = true → ∀ (x : α), p x = false
case mpr
α : Type u_1
inst✝ : Find α
p : α → Bool
⊢ (∀ (x : α), p x = false) → (find? p).isNone = true | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : Find α
p : α → Bool
⊢ (find? p).isNone = true ↔ ∀ (x : α), p x = false
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Find.lean | Find.find_is_none_iff_forall_false | [30, 1] | [47, 10] | match hp : find? p with
| some x =>
intro _
contradiction
| none =>
intro _ x
exact find?_eq_none hp | case mp
α : Type u_1
inst✝ : Find α
p : α → Bool
⊢ (find? p).isNone = true → ∀ (x : α), p x = false | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
α : Type u_1
inst✝ : Find α
p : α → Bool
⊢ (find? p).isNone = true → ∀ (x : α), p x = false
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Find.lean | Find.find_is_none_iff_forall_false | [30, 1] | [47, 10] | intro _ | α : Type u_1
inst✝ : Find α
p : α → Bool
x : α
hp : find? p = some x
⊢ (some x).isNone = true → ∀ (x : α), p x = false | α : Type u_1
inst✝ : Find α
p : α → Bool
x : α
hp : find? p = some x
a✝ : (some x).isNone = true
⊢ ∀ (x : α), p x = false | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : Find α
p : α → Bool
x : α
hp : find? p = some x
⊢ (some x).isNone = true → ∀ (x : α), p x = false
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Find.lean | Find.find_is_none_iff_forall_false | [30, 1] | [47, 10] | contradiction | α : Type u_1
inst✝ : Find α
p : α → Bool
x : α
hp : find? p = some x
a✝ : (some x).isNone = true
⊢ ∀ (x : α), p x = false | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : Find α
p : α → Bool
x : α
hp : find? p = some x
a✝ : (some x).isNone = true
⊢ ∀ (x : α), p x = false
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Find.lean | Find.find_is_none_iff_forall_false | [30, 1] | [47, 10] | intro _ x | α : Type u_1
inst✝ : Find α
p : α → Bool
hp : find? p = none
⊢ none.isNone = true → ∀ (x : α), p x = false | α : Type u_1
inst✝ : Find α
p : α → Bool
hp : find? p = none
a✝ : none.isNone = true
x : α
⊢ p x = false | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : Find α
p : α → Bool
hp : find? p = none
⊢ none.isNone = true → ∀ (x : α), p x = false
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Find.lean | Find.find_is_none_iff_forall_false | [30, 1] | [47, 10] | exact find?_eq_none hp | α : Type u_1
inst✝ : Find α
p : α → Bool
hp : find? p = none
a✝ : none.isNone = true
x : α
⊢ p x = false | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : Find α
p : α → Bool
hp : find? p = none
a✝ : none.isNone = true
x : α
⊢ p x = false
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Find.lean | Find.find_is_none_iff_forall_false | [30, 1] | [47, 10] | intro h | case mpr
α : Type u_1
inst✝ : Find α
p : α → Bool
⊢ (∀ (x : α), p x = false) → (find? p).isNone = true | case mpr
α : Type u_1
inst✝ : Find α
p : α → Bool
h : ∀ (x : α), p x = false
⊢ (find? p).isNone = true | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
α : Type u_1
inst✝ : Find α
p : α → Bool
⊢ (∀ (x : α), p x = false) → (find? p).isNone = true
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Find.lean | Find.find_is_none_iff_forall_false | [30, 1] | [47, 10] | match hp : find? p with
| some x =>
absurd find?_eq_some hp
rw [h]
intro
contradiction
| none =>
rfl | case mpr
α : Type u_1
inst✝ : Find α
p : α → Bool
h : ∀ (x : α), p x = false
⊢ (find? p).isNone = true | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
α : Type u_1
inst✝ : Find α
p : α → Bool
h : ∀ (x : α), p x = false
⊢ (find? p).isNone = true
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Find.lean | Find.find_is_none_iff_forall_false | [30, 1] | [47, 10] | absurd find?_eq_some hp | α : Type u_1
inst✝ : Find α
p : α → Bool
h : ∀ (x : α), p x = false
x : α
hp : find? p = some x
⊢ (some x).isNone = true | α : Type u_1
inst✝ : Find α
p : α → Bool
h : ∀ (x : α), p x = false
x : α
hp : find? p = some x
⊢ ¬p x = true | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : Find α
p : α → Bool
h : ∀ (x : α), p x = false
x : α
hp : find? p = some x
⊢ (some x).isNone = true
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Find.lean | Find.find_is_none_iff_forall_false | [30, 1] | [47, 10] | rw [h] | α : Type u_1
inst✝ : Find α
p : α → Bool
h : ∀ (x : α), p x = false
x : α
hp : find? p = some x
⊢ ¬p x = true | α : Type u_1
inst✝ : Find α
p : α → Bool
h : ∀ (x : α), p x = false
x : α
hp : find? p = some x
⊢ ¬false = true | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : Find α
p : α → Bool
h : ∀ (x : α), p x = false
x : α
hp : find? p = some x
⊢ ¬p x = true
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Find.lean | Find.find_is_none_iff_forall_false | [30, 1] | [47, 10] | intro | α : Type u_1
inst✝ : Find α
p : α → Bool
h : ∀ (x : α), p x = false
x : α
hp : find? p = some x
⊢ ¬false = true | α : Type u_1
inst✝ : Find α
p : α → Bool
h : ∀ (x : α), p x = false
x : α
hp : find? p = some x
a✝ : false = true
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : Find α
p : α → Bool
h : ∀ (x : α), p x = false
x : α
hp : find? p = some x
⊢ ¬false = true
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Find.lean | Find.find_is_none_iff_forall_false | [30, 1] | [47, 10] | contradiction | α : Type u_1
inst✝ : Find α
p : α → Bool
h : ∀ (x : α), p x = false
x : α
hp : find? p = some x
a✝ : false = true
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : Find α
p : α → Bool
h : ∀ (x : α), p x = false
x : α
hp : find? p = some x
a✝ : false = true
⊢ False
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Find.lean | Find.find_is_none_iff_forall_false | [30, 1] | [47, 10] | rfl | α : Type u_1
inst✝ : Find α
p : α → Bool
h : ∀ (x : α), p x = false
hp : find? p = none
⊢ none.isNone = true | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : Find α
p : α → Bool
h : ∀ (x : α), p x = false
hp : find? p = none
⊢ none.isNone = true
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Lemmas.lean | Nat.le_of_mul_le_mul_of_pos_left | [9, 11] | [16, 29] | intro hx hxyz | x y z : Nat
⊢ 0 < x → x * y ≤ x * z → y ≤ z | x y z : Nat
hx : 0 < x
hxyz : x * y ≤ x * z
⊢ y ≤ z | Please generate a tactic in lean4 to solve the state.
STATE:
x y z : Nat
⊢ 0 < x → x * y ≤ x * z → y ≤ z
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Lemmas.lean | Nat.le_of_mul_le_mul_of_pos_left | [9, 11] | [16, 29] | if h : z < y then
absurd hxyz
apply Nat.not_le_of_gt
exact Nat.mul_lt_mul_of_pos_left h hx
else
exact Nat.le_of_not_gt h | x y z : Nat
hx : 0 < x
hxyz : x * y ≤ x * z
⊢ y ≤ z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y z : Nat
hx : 0 < x
hxyz : x * y ≤ x * z
⊢ y ≤ z
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Lemmas.lean | Nat.le_of_mul_le_mul_of_pos_left | [9, 11] | [16, 29] | absurd hxyz | x y z : Nat
hx : 0 < x
hxyz : x * y ≤ x * z
h : z < y
⊢ y ≤ z | x y z : Nat
hx : 0 < x
hxyz : x * y ≤ x * z
h : z < y
⊢ ¬x * y ≤ x * z | Please generate a tactic in lean4 to solve the state.
STATE:
x y z : Nat
hx : 0 < x
hxyz : x * y ≤ x * z
h : z < y
⊢ y ≤ z
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Lemmas.lean | Nat.le_of_mul_le_mul_of_pos_left | [9, 11] | [16, 29] | apply Nat.not_le_of_gt | x y z : Nat
hx : 0 < x
hxyz : x * y ≤ x * z
h : z < y
⊢ ¬x * y ≤ x * z | case h
x y z : Nat
hx : 0 < x
hxyz : x * y ≤ x * z
h : z < y
⊢ x * y > x * z | Please generate a tactic in lean4 to solve the state.
STATE:
x y z : Nat
hx : 0 < x
hxyz : x * y ≤ x * z
h : z < y
⊢ ¬x * y ≤ x * z
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Lemmas.lean | Nat.le_of_mul_le_mul_of_pos_left | [9, 11] | [16, 29] | exact Nat.mul_lt_mul_of_pos_left h hx | case h
x y z : Nat
hx : 0 < x
hxyz : x * y ≤ x * z
h : z < y
⊢ x * y > x * z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
x y z : Nat
hx : 0 < x
hxyz : x * y ≤ x * z
h : z < y
⊢ x * y > x * z
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Lemmas.lean | Nat.le_of_mul_le_mul_of_pos_left | [9, 11] | [16, 29] | exact Nat.le_of_not_gt h | x y z : Nat
hx : 0 < x
hxyz : x * y ≤ x * z
h : ¬z < y
⊢ y ≤ z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y z : Nat
hx : 0 < x
hxyz : x * y ≤ x * z
h : ¬z < y
⊢ y ≤ z
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Lemmas.lean | Nat.le_of_mul_le_mul_of_pos_right | [18, 11] | [25, 29] | intro hx hxyz | x y z : Nat
⊢ 0 < x → y * x ≤ z * x → y ≤ z | x y z : Nat
hx : 0 < x
hxyz : y * x ≤ z * x
⊢ y ≤ z | Please generate a tactic in lean4 to solve the state.
STATE:
x y z : Nat
⊢ 0 < x → y * x ≤ z * x → y ≤ z
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Lemmas.lean | Nat.le_of_mul_le_mul_of_pos_right | [18, 11] | [25, 29] | if h : z < y then
absurd hxyz
apply Nat.not_le_of_gt
exact Nat.mul_lt_mul_of_pos_right h hx
else
exact Nat.le_of_not_gt h | x y z : Nat
hx : 0 < x
hxyz : y * x ≤ z * x
⊢ y ≤ z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y z : Nat
hx : 0 < x
hxyz : y * x ≤ z * x
⊢ y ≤ z
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Lemmas.lean | Nat.le_of_mul_le_mul_of_pos_right | [18, 11] | [25, 29] | absurd hxyz | x y z : Nat
hx : 0 < x
hxyz : y * x ≤ z * x
h : z < y
⊢ y ≤ z | x y z : Nat
hx : 0 < x
hxyz : y * x ≤ z * x
h : z < y
⊢ ¬y * x ≤ z * x | Please generate a tactic in lean4 to solve the state.
STATE:
x y z : Nat
hx : 0 < x
hxyz : y * x ≤ z * x
h : z < y
⊢ y ≤ z
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Lemmas.lean | Nat.le_of_mul_le_mul_of_pos_right | [18, 11] | [25, 29] | apply Nat.not_le_of_gt | x y z : Nat
hx : 0 < x
hxyz : y * x ≤ z * x
h : z < y
⊢ ¬y * x ≤ z * x | case h
x y z : Nat
hx : 0 < x
hxyz : y * x ≤ z * x
h : z < y
⊢ y * x > z * x | Please generate a tactic in lean4 to solve the state.
STATE:
x y z : Nat
hx : 0 < x
hxyz : y * x ≤ z * x
h : z < y
⊢ ¬y * x ≤ z * x
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Lemmas.lean | Nat.le_of_mul_le_mul_of_pos_right | [18, 11] | [25, 29] | exact Nat.mul_lt_mul_of_pos_right h hx | case h
x y z : Nat
hx : 0 < x
hxyz : y * x ≤ z * x
h : z < y
⊢ y * x > z * x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
x y z : Nat
hx : 0 < x
hxyz : y * x ≤ z * x
h : z < y
⊢ y * x > z * x
TACTIC:
|
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