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/Lemmas.lean | Nat.le_of_mul_le_mul_of_pos_right | [18, 11] | [25, 29] | exact Nat.le_of_not_gt h | x y z : Nat
hx : 0 < x
hxyz : y * x ≤ z * x
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 : y * x ≤ z * x
h : ¬z < y
⊢ y ≤ z
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Lemmas.lean | Nat.pow_lt_pow_of_lt_left | [27, 11] | [32, 55] | have hpos : 0 < b ^ n := Nat.pos_pow_of_pos n (Nat.zero_lt_of_lt h) | a b : Nat
h : a < b
n : Nat
x✝ : 0 < n + 1
⊢ a ^ (n + 1) < b ^ (n + 1) | a b : Nat
h : a < b
n : Nat
x✝ : 0 < n + 1
hpos : 0 < b ^ n
⊢ a ^ (n + 1) < b ^ (n + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
a b : Nat
h : a < b
n : Nat
x✝ : 0 < n + 1
⊢ a ^ (n + 1) < b ^ (n + 1)
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Lemmas.lean | Nat.pow_lt_pow_of_lt_left | [27, 11] | [32, 55] | simp [Nat.pow_succ] | a b : Nat
h : a < b
n : Nat
x✝ : 0 < n + 1
hpos : 0 < b ^ n
⊢ a ^ (n + 1) < b ^ (n + 1) | a b : Nat
h : a < b
n : Nat
x✝ : 0 < n + 1
hpos : 0 < b ^ n
⊢ a ^ n * a < b ^ n * b | Please generate a tactic in lean4 to solve the state.
STATE:
a b : Nat
h : a < b
n : Nat
x✝ : 0 < n + 1
hpos : 0 < b ^ n
⊢ a ^ (n + 1) < b ^ (n + 1)
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Lemmas.lean | Nat.pow_lt_pow_of_lt_left | [27, 11] | [32, 55] | apply Nat.mul_lt_mul_of_le_of_lt _ h hpos | a b : Nat
h : a < b
n : Nat
x✝ : 0 < n + 1
hpos : 0 < b ^ n
⊢ a ^ n * a < b ^ n * b | a b : Nat
h : a < b
n : Nat
x✝ : 0 < n + 1
hpos : 0 < b ^ n
⊢ a ^ n ≤ b ^ n | Please generate a tactic in lean4 to solve the state.
STATE:
a b : Nat
h : a < b
n : Nat
x✝ : 0 < n + 1
hpos : 0 < b ^ n
⊢ a ^ n * a < b ^ n * b
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Lemmas.lean | Nat.pow_lt_pow_of_lt_left | [27, 11] | [32, 55] | exact Nat.pow_le_pow_of_le_left (Nat.le_of_lt h) n | a b : Nat
h : a < b
n : Nat
x✝ : 0 < n + 1
hpos : 0 < b ^ n
⊢ a ^ n ≤ b ^ n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b : Nat
h : a < b
n : Nat
x✝ : 0 < n + 1
hpos : 0 < b ^ n
⊢ a ^ n ≤ b ^ n
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Lemmas.lean | Nat.pow_lt_pow_of_lt_right | [34, 11] | [43, 40] | intro ha h | m n a : Nat
⊢ 1 < a → m < n → a ^ m < a ^ n | m n a : Nat
ha : 1 < a
h : m < n
⊢ a ^ m < a ^ n | Please generate a tactic in lean4 to solve the state.
STATE:
m n a : Nat
⊢ 1 < a → m < n → a ^ m < a ^ n
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Lemmas.lean | Nat.pow_lt_pow_of_lt_right | [34, 11] | [43, 40] | induction m, n using Nat.recDiag with try contradiction
| zero_succ n =>
have hpos := Nat.pos_pow_of_pos n (Nat.le_of_lt ha)
apply Nat.lt_of_le_of_lt hpos
have : a ^ n * 1 < a ^ n * a := Nat.mul_lt_mul_of_pos_left ha hpos
rwa [Nat.mul_one] at this
| succ_succ m n ih =>
apply Nat.mul_lt_mul_of_pos_right _ (Nat.le_of_lt ... | m n a : Nat
ha : 1 < a
h : m < n
⊢ a ^ m < a ^ n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
m n a : Nat
ha : 1 < a
h : m < n
⊢ a ^ m < a ^ n
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Lemmas.lean | Nat.pow_lt_pow_of_lt_right | [34, 11] | [43, 40] | contradiction | case succ_zero
a : Nat
ha : 1 < a
m✝ : Nat
a✝ : m✝ < 0 → a ^ m✝ < a ^ 0
h : m✝ + 1 < 0
⊢ a ^ (m✝ + 1) < a ^ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ_zero
a : Nat
ha : 1 < a
m✝ : Nat
a✝ : m✝ < 0 → a ^ m✝ < a ^ 0
h : m✝ + 1 < 0
⊢ a ^ (m✝ + 1) < a ^ 0
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Lemmas.lean | Nat.pow_lt_pow_of_lt_right | [34, 11] | [43, 40] | have hpos := Nat.pos_pow_of_pos n (Nat.le_of_lt ha) | case zero_succ
a : Nat
ha : 1 < a
n : Nat
a✝ : 0 < n → a ^ 0 < a ^ n
h : 0 < n + 1
⊢ a ^ 0 < a ^ (n + 1) | case zero_succ
a : Nat
ha : 1 < a
n : Nat
a✝ : 0 < n → a ^ 0 < a ^ n
h : 0 < n + 1
hpos : 0 < a ^ n
⊢ a ^ 0 < a ^ (n + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case zero_succ
a : Nat
ha : 1 < a
n : Nat
a✝ : 0 < n → a ^ 0 < a ^ n
h : 0 < n + 1
⊢ a ^ 0 < a ^ (n + 1)
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Lemmas.lean | Nat.pow_lt_pow_of_lt_right | [34, 11] | [43, 40] | apply Nat.lt_of_le_of_lt hpos | case zero_succ
a : Nat
ha : 1 < a
n : Nat
a✝ : 0 < n → a ^ 0 < a ^ n
h : 0 < n + 1
hpos : 0 < a ^ n
⊢ a ^ 0 < a ^ (n + 1) | case zero_succ
a : Nat
ha : 1 < a
n : Nat
a✝ : 0 < n → a ^ 0 < a ^ n
h : 0 < n + 1
hpos : 0 < a ^ n
⊢ a ^ n < a ^ (n + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case zero_succ
a : Nat
ha : 1 < a
n : Nat
a✝ : 0 < n → a ^ 0 < a ^ n
h : 0 < n + 1
hpos : 0 < a ^ n
⊢ a ^ 0 < a ^ (n + 1)
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Lemmas.lean | Nat.pow_lt_pow_of_lt_right | [34, 11] | [43, 40] | have : a ^ n * 1 < a ^ n * a := Nat.mul_lt_mul_of_pos_left ha hpos | case zero_succ
a : Nat
ha : 1 < a
n : Nat
a✝ : 0 < n → a ^ 0 < a ^ n
h : 0 < n + 1
hpos : 0 < a ^ n
⊢ a ^ n < a ^ (n + 1) | case zero_succ
a : Nat
ha : 1 < a
n : Nat
a✝ : 0 < n → a ^ 0 < a ^ n
h : 0 < n + 1
hpos : 0 < a ^ n
this : a ^ n * 1 < a ^ n * a
⊢ a ^ n < a ^ (n + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case zero_succ
a : Nat
ha : 1 < a
n : Nat
a✝ : 0 < n → a ^ 0 < a ^ n
h : 0 < n + 1
hpos : 0 < a ^ n
⊢ a ^ n < a ^ (n + 1)
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Lemmas.lean | Nat.pow_lt_pow_of_lt_right | [34, 11] | [43, 40] | rwa [Nat.mul_one] at this | case zero_succ
a : Nat
ha : 1 < a
n : Nat
a✝ : 0 < n → a ^ 0 < a ^ n
h : 0 < n + 1
hpos : 0 < a ^ n
this : a ^ n * 1 < a ^ n * a
⊢ a ^ n < a ^ (n + 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case zero_succ
a : Nat
ha : 1 < a
n : Nat
a✝ : 0 < n → a ^ 0 < a ^ n
h : 0 < n + 1
hpos : 0 < a ^ n
this : a ^ n * 1 < a ^ n * a
⊢ a ^ n < a ^ (n + 1)
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Lemmas.lean | Nat.pow_lt_pow_of_lt_right | [34, 11] | [43, 40] | apply Nat.mul_lt_mul_of_pos_right _ (Nat.le_of_lt ha) | case succ_succ
a : Nat
ha : 1 < a
m n : Nat
ih : m < n → a ^ m < a ^ n
h : m + 1 < n + 1
⊢ a ^ (m + 1) < a ^ (n + 1) | a : Nat
ha : 1 < a
m n : Nat
ih : m < n → a ^ m < a ^ n
h : m + 1 < n + 1
⊢ a.pow m < a.pow n | Please generate a tactic in lean4 to solve the state.
STATE:
case succ_succ
a : Nat
ha : 1 < a
m n : Nat
ih : m < n → a ^ m < a ^ n
h : m + 1 < n + 1
⊢ a ^ (m + 1) < a ^ (n + 1)
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Lemmas.lean | Nat.pow_lt_pow_of_lt_right | [34, 11] | [43, 40] | exact ih (Nat.lt_of_succ_lt_succ h) | a : Nat
ha : 1 < a
m n : Nat
ih : m < n → a ^ m < a ^ n
h : m + 1 < n + 1
⊢ a.pow m < a.pow n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a : Nat
ha : 1 < a
m n : Nat
ih : m < n → a ^ m < a ^ n
h : m + 1 < n + 1
⊢ a.pow m < a.pow n
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Lemmas.lean | Nat.pow_lt_pow_of_pos_right | [48, 11] | [62, 16] | intro | case zero
x y : Nat
h : x < y
⊢ 0 > 0 → x ^ 0 < y ^ 0 | case zero
x y : Nat
h : x < y
a✝ : 0 > 0
⊢ x ^ 0 < y ^ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
x y : Nat
h : x < y
⊢ 0 > 0 → x ^ 0 < y ^ 0
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Lemmas.lean | Nat.pow_lt_pow_of_pos_right | [48, 11] | [62, 16] | contradiction | case zero
x y : Nat
h : x < y
a✝ : 0 > 0
⊢ x ^ 0 < y ^ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
x y : Nat
h : x < y
a✝ : 0 > 0
⊢ x ^ 0 < y ^ 0
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Lemmas.lean | Nat.pow_lt_pow_of_pos_right | [48, 11] | [62, 16] | intro | case succ
x y : Nat
h : x < y
z : Nat
⊢ z + 1 > 0 → x ^ (z + 1) < y ^ (z + 1) | case succ
x y : Nat
h : x < y
z : Nat
a✝ : z + 1 > 0
⊢ x ^ (z + 1) < y ^ (z + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
x y : Nat
h : x < y
z : Nat
⊢ z + 1 > 0 → x ^ (z + 1) < y ^ (z + 1)
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Lemmas.lean | Nat.pow_lt_pow_of_pos_right | [48, 11] | [62, 16] | rw [Nat.pow_succ, Nat.pow_succ] | case succ
x y : Nat
h : x < y
z : Nat
a✝ : z + 1 > 0
⊢ x ^ (z + 1) < y ^ (z + 1) | case succ
x y : Nat
h : x < y
z : Nat
a✝ : z + 1 > 0
⊢ x ^ z * x < y ^ z * y | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
x y : Nat
h : x < y
z : Nat
a✝ : z + 1 > 0
⊢ x ^ (z + 1) < y ^ (z + 1)
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Lemmas.lean | Nat.pow_lt_pow_of_pos_right | [48, 11] | [62, 16] | apply Nat.mul_lt_mul_of_le_of_lt | case succ
x y : Nat
h : x < y
z : Nat
a✝ : z + 1 > 0
⊢ x ^ z * x < y ^ z * y | case succ.hac
x y : Nat
h : x < y
z : Nat
a✝ : z + 1 > 0
⊢ x ^ z ≤ y ^ z
case succ.hbd
x y : Nat
h : x < y
z : Nat
a✝ : z + 1 > 0
⊢ x < y
case succ.hc
x y : Nat
h : x < y
z : Nat
a✝ : z + 1 > 0
⊢ 0 < y ^ z | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
x y : Nat
h : x < y
z : Nat
a✝ : z + 1 > 0
⊢ x ^ z * x < y ^ z * y
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Lemmas.lean | Nat.pow_lt_pow_of_pos_right | [48, 11] | [62, 16] | apply Nat.pow_le_pow_left | case succ.hac
x y : Nat
h : x < y
z : Nat
a✝ : z + 1 > 0
⊢ x ^ z ≤ y ^ z | case succ.hac.h
x y : Nat
h : x < y
z : Nat
a✝ : z + 1 > 0
⊢ x ≤ y | Please generate a tactic in lean4 to solve the state.
STATE:
case succ.hac
x y : Nat
h : x < y
z : Nat
a✝ : z + 1 > 0
⊢ x ^ z ≤ y ^ z
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Lemmas.lean | Nat.pow_lt_pow_of_pos_right | [48, 11] | [62, 16] | apply Nat.le_of_lt | case succ.hac.h
x y : Nat
h : x < y
z : Nat
a✝ : z + 1 > 0
⊢ x ≤ y | case succ.hac.h.a
x y : Nat
h : x < y
z : Nat
a✝ : z + 1 > 0
⊢ x < y | Please generate a tactic in lean4 to solve the state.
STATE:
case succ.hac.h
x y : Nat
h : x < y
z : Nat
a✝ : z + 1 > 0
⊢ x ≤ y
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Lemmas.lean | Nat.pow_lt_pow_of_pos_right | [48, 11] | [62, 16] | exact h | case succ.hac.h.a
x y : Nat
h : x < y
z : Nat
a✝ : z + 1 > 0
⊢ x < y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ.hac.h.a
x y : Nat
h : x < y
z : Nat
a✝ : z + 1 > 0
⊢ x < y
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Lemmas.lean | Nat.pow_lt_pow_of_pos_right | [48, 11] | [62, 16] | exact h | case succ.hbd
x y : Nat
h : x < y
z : Nat
a✝ : z + 1 > 0
⊢ x < y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ.hbd
x y : Nat
h : x < y
z : Nat
a✝ : z + 1 > 0
⊢ x < y
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Lemmas.lean | Nat.pow_lt_pow_of_pos_right | [48, 11] | [62, 16] | apply Nat.pos_pow_of_pos | case succ.hc
x y : Nat
h : x < y
z : Nat
a✝ : z + 1 > 0
⊢ 0 < y ^ z | case succ.hc.h
x y : Nat
h : x < y
z : Nat
a✝ : z + 1 > 0
⊢ 0 < y | Please generate a tactic in lean4 to solve the state.
STATE:
case succ.hc
x y : Nat
h : x < y
z : Nat
a✝ : z + 1 > 0
⊢ 0 < y ^ z
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Lemmas.lean | Nat.pow_lt_pow_of_pos_right | [48, 11] | [62, 16] | apply Nat.lt_of_le_of_lt | case succ.hc.h
x y : Nat
h : x < y
z : Nat
a✝ : z + 1 > 0
⊢ 0 < y | case succ.hc.h.h₁
x y : Nat
h : x < y
z : Nat
a✝ : z + 1 > 0
⊢ 0 ≤ ?succ.hc.h.m
case succ.hc.h.h₂
x y : Nat
h : x < y
z : Nat
a✝ : z + 1 > 0
⊢ ?succ.hc.h.m < y
case succ.hc.h.m
x y : Nat
h : x < y
z : Nat
a✝ : z + 1 > 0
⊢ Nat | Please generate a tactic in lean4 to solve the state.
STATE:
case succ.hc.h
x y : Nat
h : x < y
z : Nat
a✝ : z + 1 > 0
⊢ 0 < y
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Lemmas.lean | Nat.pow_lt_pow_of_pos_right | [48, 11] | [62, 16] | apply Nat.zero_le | case succ.hc.h.h₁
x y : Nat
h : x < y
z : Nat
a✝ : z + 1 > 0
⊢ 0 ≤ ?succ.hc.h.m | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ.hc.h.h₁
x y : Nat
h : x < y
z : Nat
a✝ : z + 1 > 0
⊢ 0 ≤ ?succ.hc.h.m
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Lemmas.lean | Nat.pow_lt_pow_of_pos_right | [48, 11] | [62, 16] | exact h | case succ.hc.h.h₂
x y : Nat
h : x < y
z : Nat
a✝ : z + 1 > 0
⊢ ?succ.hc.h.m < y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ.hc.h.h₂
x y : Nat
h : x < y
z : Nat
a✝ : z + 1 > 0
⊢ ?succ.hc.h.m < y
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Lemmas.lean | Nat.eq_one_of_mul_eq_one_left | [64, 1] | [71, 57] | contradiction | m n : Nat
h : m * 0 = 1
⊢ 0 = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
m n : Nat
h : m * 0 = 1
⊢ 0 = 1
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Lemmas.lean | Nat.eq_one_of_mul_eq_one_left | [64, 1] | [71, 57] | rfl | m n : Nat
h : m * 1 = 1
⊢ 1 = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
m n : Nat
h : m * 1 = 1
⊢ 1 = 1
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Lemmas.lean | Nat.eq_one_of_mul_eq_one_left | [64, 1] | [71, 57] | cases m | m n n✝ : Nat
h : m * (n✝ + 2) = 1
⊢ n✝ + 2 = 1 | case zero
n n✝ : Nat
h : 0 * (n✝ + 2) = 1
⊢ n✝ + 2 = 1
case succ
n n✝¹ n✝ : Nat
h : (n✝ + 1) * (n✝¹ + 2) = 1
⊢ n✝¹ + 2 = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
m n n✝ : Nat
h : m * (n✝ + 2) = 1
⊢ n✝ + 2 = 1
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Lemmas.lean | Nat.eq_one_of_mul_eq_one_left | [64, 1] | [71, 57] | rw [Nat.zero_mul] at h | case zero
n n✝ : Nat
h : 0 * (n✝ + 2) = 1
⊢ n✝ + 2 = 1 | case zero
n n✝ : Nat
h : 0 = 1
⊢ n✝ + 2 = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
n n✝ : Nat
h : 0 * (n✝ + 2) = 1
⊢ n✝ + 2 = 1
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Lemmas.lean | Nat.eq_one_of_mul_eq_one_left | [64, 1] | [71, 57] | contradiction | case zero
n n✝ : Nat
h : 0 = 1
⊢ n✝ + 2 = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
n n✝ : Nat
h : 0 = 1
⊢ n✝ + 2 = 1
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Lemmas.lean | Nat.eq_one_of_mul_eq_one_left | [64, 1] | [71, 57] | rw [Nat.succ_mul] at h | case succ
n n✝¹ n✝ : Nat
h : (n✝ + 1) * (n✝¹ + 2) = 1
⊢ n✝¹ + 2 = 1 | case succ
n n✝¹ n✝ : Nat
h : n✝ * (n✝¹ + 2) + (n✝¹ + 2) = 1
⊢ n✝¹ + 2 = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
n n✝¹ n✝ : Nat
h : (n✝ + 1) * (n✝¹ + 2) = 1
⊢ n✝¹ + 2 = 1
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Lemmas.lean | Nat.eq_one_of_mul_eq_one_left | [64, 1] | [71, 57] | injection h | case succ
n n✝¹ n✝ : Nat
h : n✝ * (n✝¹ + 2) + (n✝¹ + 2) = 1
⊢ n✝¹ + 2 = 1 | case succ
n n✝¹ n✝ : Nat
n_eq✝ : (n✝ * (n✝¹ + 2)).add (n✝¹ + 1) = 0
⊢ n✝¹ + 2 = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
n n✝¹ n✝ : Nat
h : n✝ * (n✝¹ + 2) + (n✝¹ + 2) = 1
⊢ n✝¹ + 2 = 1
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Lemmas.lean | Nat.eq_one_of_mul_eq_one_left | [64, 1] | [71, 57] | contradiction | case succ
n n✝¹ n✝ : Nat
n_eq✝ : (n✝ * (n✝¹ + 2)).add (n✝¹ + 1) = 0
⊢ n✝¹ + 2 = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
n n✝¹ n✝ : Nat
n_eq✝ : (n✝ * (n✝¹ + 2)).add (n✝¹ + 1) = 0
⊢ n✝¹ + 2 = 1
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Lemmas.lean | Nat.eq_one_of_mul_eq_one_right | [73, 1] | [74, 60] | rw [Nat.mul_comm] at h | m n : Nat
h : m * n = 1
⊢ m = 1 | m n : Nat
h : n * m = 1
⊢ m = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
m n : Nat
h : m * n = 1
⊢ m = 1
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Lemmas.lean | Nat.eq_one_of_mul_eq_one_right | [73, 1] | [74, 60] | exact eq_one_of_mul_eq_one_left h | m n : Nat
h : n * m = 1
⊢ m = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
m n : Nat
h : n * m = 1
⊢ m = 1
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Lemmas.lean | Nat.mul_eq_one | [76, 1] | [79, 32] | constructor | m n : Nat
⊢ m * n = 1 ↔ m = 1 ∧ n = 1 | case mp
m n : Nat
⊢ m * n = 1 → m = 1 ∧ n = 1
case mpr
m n : Nat
⊢ m = 1 ∧ n = 1 → m * n = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
m n : Nat
⊢ m * n = 1 ↔ m = 1 ∧ n = 1
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Lemmas.lean | Nat.mul_eq_one | [76, 1] | [79, 32] | intro h | case mp
m n : Nat
⊢ m * n = 1 → m = 1 ∧ n = 1 | case mp
m n : Nat
h : m * n = 1
⊢ m = 1 ∧ n = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
m n : Nat
⊢ m * n = 1 → m = 1 ∧ n = 1
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Lemmas.lean | Nat.mul_eq_one | [76, 1] | [79, 32] | rw [eq_one_of_mul_eq_one_right h, eq_one_of_mul_eq_one_left h] | case mp
m n : Nat
h : m * n = 1
⊢ m = 1 ∧ n = 1 | case mp
m n : Nat
h : m * n = 1
⊢ 1 = 1 ∧ 1 = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
m n : Nat
h : m * n = 1
⊢ m = 1 ∧ n = 1
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Lemmas.lean | Nat.mul_eq_one | [76, 1] | [79, 32] | trivial | case mp
m n : Nat
h : m * n = 1
⊢ 1 = 1 ∧ 1 = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
m n : Nat
h : m * n = 1
⊢ 1 = 1 ∧ 1 = 1
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Lemmas.lean | Nat.mul_eq_one | [76, 1] | [79, 32] | intro ⟨hm, hn⟩ | case mpr
m n : Nat
⊢ m = 1 ∧ n = 1 → m * n = 1 | case mpr
m n : Nat
hm : m = 1
hn : n = 1
⊢ m * n = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
m n : Nat
⊢ m = 1 ∧ n = 1 → m * n = 1
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Lemmas.lean | Nat.mul_eq_one | [76, 1] | [79, 32] | rw [hm, hn] | case mpr
m n : Nat
hm : m = 1
hn : n = 1
⊢ m * n = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
m n : Nat
hm : m = 1
hn : n = 1
⊢ m * n = 1
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Lemmas.lean | Nat.pow_eq_zero | [81, 1] | [97, 33] | simp | case zero
m : Nat
⊢ m ^ 0 = 0 ↔ m = 0 ∧ 0 ≠ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
m : Nat
⊢ m ^ 0 = 0 ↔ m = 0 ∧ 0 ≠ 0
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Lemmas.lean | Nat.pow_eq_zero | [81, 1] | [97, 33] | rw [Nat.pow_succ] | case succ
m n : Nat
ih : m ^ n = 0 ↔ m = 0 ∧ n ≠ 0
⊢ m ^ (n + 1) = 0 ↔ m = 0 ∧ n + 1 ≠ 0 | case succ
m n : Nat
ih : m ^ n = 0 ↔ m = 0 ∧ n ≠ 0
⊢ m ^ n * m = 0 ↔ m = 0 ∧ n + 1 ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
m n : Nat
ih : m ^ n = 0 ↔ m = 0 ∧ n ≠ 0
⊢ m ^ (n + 1) = 0 ↔ m = 0 ∧ n + 1 ≠ 0
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Lemmas.lean | Nat.pow_eq_zero | [81, 1] | [97, 33] | rw [Nat.mul_eq_zero] | case succ
m n : Nat
ih : m ^ n = 0 ↔ m = 0 ∧ n ≠ 0
⊢ m ^ n * m = 0 ↔ m = 0 ∧ n + 1 ≠ 0 | case succ
m n : Nat
ih : m ^ n = 0 ↔ m = 0 ∧ n ≠ 0
⊢ m ^ n = 0 ∨ m = 0 ↔ m = 0 ∧ n + 1 ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
m n : Nat
ih : m ^ n = 0 ↔ m = 0 ∧ n ≠ 0
⊢ m ^ n * m = 0 ↔ m = 0 ∧ n + 1 ≠ 0
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Lemmas.lean | Nat.pow_eq_zero | [81, 1] | [97, 33] | constructor | case succ
m n : Nat
ih : m ^ n = 0 ↔ m = 0 ∧ n ≠ 0
⊢ m ^ n = 0 ∨ m = 0 ↔ m = 0 ∧ n + 1 ≠ 0 | case succ.mp
m n : Nat
ih : m ^ n = 0 ↔ m = 0 ∧ n ≠ 0
⊢ m ^ n = 0 ∨ m = 0 → m = 0 ∧ n + 1 ≠ 0
case succ.mpr
m n : Nat
ih : m ^ n = 0 ↔ m = 0 ∧ n ≠ 0
⊢ m = 0 ∧ n + 1 ≠ 0 → m ^ n = 0 ∨ m = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
m n : Nat
ih : m ^ n = 0 ↔ m = 0 ∧ n ≠ 0
⊢ m ^ n = 0 ∨ m = 0 ↔ m = 0 ∧ n + 1 ≠ 0
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Lemmas.lean | Nat.pow_eq_zero | [81, 1] | [97, 33] | constructor | m n : Nat
ih : m ^ n = 0 ↔ m = 0 ∧ n ≠ 0
x✝ : m ^ n = 0 ∨ m = 0
h : m ^ n = 0
⊢ m = 0 ∧ n + 1 ≠ 0 | case left
m n : Nat
ih : m ^ n = 0 ↔ m = 0 ∧ n ≠ 0
x✝ : m ^ n = 0 ∨ m = 0
h : m ^ n = 0
⊢ m = 0
case right
m n : Nat
ih : m ^ n = 0 ↔ m = 0 ∧ n ≠ 0
x✝ : m ^ n = 0 ∨ m = 0
h : m ^ n = 0
⊢ n + 1 ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
m n : Nat
ih : m ^ n = 0 ↔ m = 0 ∧ n ≠ 0
x✝ : m ^ n = 0 ∨ m = 0
h : m ^ n = 0
⊢ m = 0 ∧ n + 1 ≠ 0
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Lemmas.lean | Nat.pow_eq_zero | [81, 1] | [97, 33] | rw [ih] at h | case left
m n : Nat
ih : m ^ n = 0 ↔ m = 0 ∧ n ≠ 0
x✝ : m ^ n = 0 ∨ m = 0
h : m ^ n = 0
⊢ m = 0 | case left
m n : Nat
ih : m ^ n = 0 ↔ m = 0 ∧ n ≠ 0
x✝ : m ^ n = 0 ∨ m = 0
h : m = 0 ∧ n ≠ 0
⊢ m = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case left
m n : Nat
ih : m ^ n = 0 ↔ m = 0 ∧ n ≠ 0
x✝ : m ^ n = 0 ∨ m = 0
h : m ^ n = 0
⊢ m = 0
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Lemmas.lean | Nat.pow_eq_zero | [81, 1] | [97, 33] | exact h.1 | case left
m n : Nat
ih : m ^ n = 0 ↔ m = 0 ∧ n ≠ 0
x✝ : m ^ n = 0 ∨ m = 0
h : m = 0 ∧ n ≠ 0
⊢ m = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case left
m n : Nat
ih : m ^ n = 0 ↔ m = 0 ∧ n ≠ 0
x✝ : m ^ n = 0 ∨ m = 0
h : m = 0 ∧ n ≠ 0
⊢ m = 0
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Lemmas.lean | Nat.pow_eq_zero | [81, 1] | [97, 33] | exact Nat.succ_ne_zero _ | case right
m n : Nat
ih : m ^ n = 0 ↔ m = 0 ∧ n ≠ 0
x✝ : m ^ n = 0 ∨ m = 0
h : m ^ n = 0
⊢ n + 1 ≠ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right
m n : Nat
ih : m ^ n = 0 ↔ m = 0 ∧ n ≠ 0
x✝ : m ^ n = 0 ∨ m = 0
h : m ^ n = 0
⊢ n + 1 ≠ 0
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Lemmas.lean | Nat.pow_eq_zero | [81, 1] | [97, 33] | constructor | m n : Nat
ih : m ^ n = 0 ↔ m = 0 ∧ n ≠ 0
x✝ : m ^ n = 0 ∨ m = 0
h : m = 0
⊢ m = 0 ∧ n + 1 ≠ 0 | case left
m n : Nat
ih : m ^ n = 0 ↔ m = 0 ∧ n ≠ 0
x✝ : m ^ n = 0 ∨ m = 0
h : m = 0
⊢ m = 0
case right
m n : Nat
ih : m ^ n = 0 ↔ m = 0 ∧ n ≠ 0
x✝ : m ^ n = 0 ∨ m = 0
h : m = 0
⊢ n + 1 ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
m n : Nat
ih : m ^ n = 0 ↔ m = 0 ∧ n ≠ 0
x✝ : m ^ n = 0 ∨ m = 0
h : m = 0
⊢ m = 0 ∧ n + 1 ≠ 0
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Lemmas.lean | Nat.pow_eq_zero | [81, 1] | [97, 33] | exact h | case left
m n : Nat
ih : m ^ n = 0 ↔ m = 0 ∧ n ≠ 0
x✝ : m ^ n = 0 ∨ m = 0
h : m = 0
⊢ m = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case left
m n : Nat
ih : m ^ n = 0 ↔ m = 0 ∧ n ≠ 0
x✝ : m ^ n = 0 ∨ m = 0
h : m = 0
⊢ m = 0
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Lemmas.lean | Nat.pow_eq_zero | [81, 1] | [97, 33] | exact Nat.succ_ne_zero _ | case right
m n : Nat
ih : m ^ n = 0 ↔ m = 0 ∧ n ≠ 0
x✝ : m ^ n = 0 ∨ m = 0
h : m = 0
⊢ n + 1 ≠ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right
m n : Nat
ih : m ^ n = 0 ↔ m = 0 ∧ n ≠ 0
x✝ : m ^ n = 0 ∨ m = 0
h : m = 0
⊢ n + 1 ≠ 0
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Lemmas.lean | Nat.pow_eq_zero | [81, 1] | [97, 33] | intro ⟨h, _⟩ | case succ.mpr
m n : Nat
ih : m ^ n = 0 ↔ m = 0 ∧ n ≠ 0
⊢ m = 0 ∧ n + 1 ≠ 0 → m ^ n = 0 ∨ m = 0 | case succ.mpr
m n : Nat
ih : m ^ n = 0 ↔ m = 0 ∧ n ≠ 0
h : m = 0
right✝ : n + 1 ≠ 0
⊢ m ^ n = 0 ∨ m = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case succ.mpr
m n : Nat
ih : m ^ n = 0 ↔ m = 0 ∧ n ≠ 0
⊢ m = 0 ∧ n + 1 ≠ 0 → m ^ n = 0 ∨ m = 0
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Lemmas.lean | Nat.pow_eq_zero | [81, 1] | [97, 33] | exact .inr h | case succ.mpr
m n : Nat
ih : m ^ n = 0 ↔ m = 0 ∧ n ≠ 0
h : m = 0
right✝ : n + 1 ≠ 0
⊢ m ^ n = 0 ∨ m = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ.mpr
m n : Nat
ih : m ^ n = 0 ↔ m = 0 ∧ n ≠ 0
h : m = 0
right✝ : n + 1 ≠ 0
⊢ m ^ n = 0 ∨ m = 0
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Lemmas.lean | Nat.pow_eq_one | [99, 1] | [111, 18] | simp | case zero
m : Nat
⊢ m ^ 0 = 1 ↔ m = 1 ∨ 0 = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
m : Nat
⊢ m ^ 0 = 1 ↔ m = 1 ∨ 0 = 0
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Lemmas.lean | Nat.pow_eq_one | [99, 1] | [111, 18] | rw [Nat.pow_succ] | case succ
m n : Nat
ih : m ^ n = 1 ↔ m = 1 ∨ n = 0
⊢ m ^ (n + 1) = 1 ↔ m = 1 ∨ n + 1 = 0 | case succ
m n : Nat
ih : m ^ n = 1 ↔ m = 1 ∨ n = 0
⊢ m ^ n * m = 1 ↔ m = 1 ∨ n + 1 = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
m n : Nat
ih : m ^ n = 1 ↔ m = 1 ∨ n = 0
⊢ m ^ (n + 1) = 1 ↔ m = 1 ∨ n + 1 = 0
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Lemmas.lean | Nat.pow_eq_one | [99, 1] | [111, 18] | rw [Nat.mul_eq_one] | case succ
m n : Nat
ih : m ^ n = 1 ↔ m = 1 ∨ n = 0
⊢ m ^ n * m = 1 ↔ m = 1 ∨ n + 1 = 0 | case succ
m n : Nat
ih : m ^ n = 1 ↔ m = 1 ∨ n = 0
⊢ m ^ n = 1 ∧ m = 1 ↔ m = 1 ∨ n + 1 = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
m n : Nat
ih : m ^ n = 1 ↔ m = 1 ∨ n = 0
⊢ m ^ n * m = 1 ↔ m = 1 ∨ n + 1 = 0
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Lemmas.lean | Nat.pow_eq_one | [99, 1] | [111, 18] | constructor | case succ
m n : Nat
ih : m ^ n = 1 ↔ m = 1 ∨ n = 0
⊢ m ^ n = 1 ∧ m = 1 ↔ m = 1 ∨ n + 1 = 0 | case succ.mp
m n : Nat
ih : m ^ n = 1 ↔ m = 1 ∨ n = 0
⊢ m ^ n = 1 ∧ m = 1 → m = 1 ∨ n + 1 = 0
case succ.mpr
m n : Nat
ih : m ^ n = 1 ↔ m = 1 ∨ n = 0
⊢ m = 1 ∨ n + 1 = 0 → m ^ n = 1 ∧ m = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
m n : Nat
ih : m ^ n = 1 ↔ m = 1 ∨ n = 0
⊢ m ^ n = 1 ∧ m = 1 ↔ m = 1 ∨ n + 1 = 0
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Lemmas.lean | Nat.pow_eq_one | [99, 1] | [111, 18] | intro ⟨_, h⟩ | case succ.mp
m n : Nat
ih : m ^ n = 1 ↔ m = 1 ∨ n = 0
⊢ m ^ n = 1 ∧ m = 1 → m = 1 ∨ n + 1 = 0 | case succ.mp
m n : Nat
ih : m ^ n = 1 ↔ m = 1 ∨ n = 0
left✝ : m ^ n = 1
h : m = 1
⊢ m = 1 ∨ n + 1 = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case succ.mp
m n : Nat
ih : m ^ n = 1 ↔ m = 1 ∨ n = 0
⊢ m ^ n = 1 ∧ m = 1 → m = 1 ∨ n + 1 = 0
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Lemmas.lean | Nat.pow_eq_one | [99, 1] | [111, 18] | exact .inl h | case succ.mp
m n : Nat
ih : m ^ n = 1 ↔ m = 1 ∨ n = 0
left✝ : m ^ n = 1
h : m = 1
⊢ m = 1 ∨ n + 1 = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ.mp
m n : Nat
ih : m ^ n = 1 ↔ m = 1 ∨ n = 0
left✝ : m ^ n = 1
h : m = 1
⊢ m = 1 ∨ n + 1 = 0
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Lemmas.lean | Nat.pow_eq_one | [99, 1] | [111, 18] | constructor | m n : Nat
ih : m ^ n = 1 ↔ m = 1 ∨ n = 0
x✝ : m = 1 ∨ n + 1 = 0
h : m = 1
⊢ m ^ n = 1 ∧ m = 1 | case left
m n : Nat
ih : m ^ n = 1 ↔ m = 1 ∨ n = 0
x✝ : m = 1 ∨ n + 1 = 0
h : m = 1
⊢ m ^ n = 1
case right
m n : Nat
ih : m ^ n = 1 ↔ m = 1 ∨ n = 0
x✝ : m = 1 ∨ n + 1 = 0
h : m = 1
⊢ m = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
m n : Nat
ih : m ^ n = 1 ↔ m = 1 ∨ n = 0
x✝ : m = 1 ∨ n + 1 = 0
h : m = 1
⊢ m ^ n = 1 ∧ m = 1
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Lemmas.lean | Nat.pow_eq_one | [99, 1] | [111, 18] | rw [ih] | case left
m n : Nat
ih : m ^ n = 1 ↔ m = 1 ∨ n = 0
x✝ : m = 1 ∨ n + 1 = 0
h : m = 1
⊢ m ^ n = 1 | case left
m n : Nat
ih : m ^ n = 1 ↔ m = 1 ∨ n = 0
x✝ : m = 1 ∨ n + 1 = 0
h : m = 1
⊢ m = 1 ∨ n = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case left
m n : Nat
ih : m ^ n = 1 ↔ m = 1 ∨ n = 0
x✝ : m = 1 ∨ n + 1 = 0
h : m = 1
⊢ m ^ n = 1
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Lemmas.lean | Nat.pow_eq_one | [99, 1] | [111, 18] | exact .inl h | case left
m n : Nat
ih : m ^ n = 1 ↔ m = 1 ∨ n = 0
x✝ : m = 1 ∨ n + 1 = 0
h : m = 1
⊢ m = 1 ∨ n = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case left
m n : Nat
ih : m ^ n = 1 ↔ m = 1 ∨ n = 0
x✝ : m = 1 ∨ n + 1 = 0
h : m = 1
⊢ m = 1 ∨ n = 0
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Lemmas.lean | Nat.pow_eq_one | [99, 1] | [111, 18] | exact h | case right
m n : Nat
ih : m ^ n = 1 ↔ m = 1 ∨ n = 0
x✝ : m = 1 ∨ n + 1 = 0
h : m = 1
⊢ m = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right
m n : Nat
ih : m ^ n = 1 ↔ m = 1 ∨ n = 0
x✝ : m = 1 ∨ n + 1 = 0
h : m = 1
⊢ m = 1
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Lemmas.lean | Nat.div_le_of_lt_mul_succ | [113, 1] | [118, 28] | have hk : 0 < k := match k with
| 0 => by rw [Nat.zero_mul] at h; absurd h; exact Nat.not_lt_zero _
| _+1 => Nat.zero_lt_succ _ | n k m : Nat
h : n < k * (m + 1)
⊢ n / k ≤ m | n k m : Nat
h : n < k * (m + 1)
hk : 0 < k
⊢ n / k ≤ m | Please generate a tactic in lean4 to solve the state.
STATE:
n k m : Nat
h : n < k * (m + 1)
⊢ n / k ≤ m
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Lemmas.lean | Nat.div_le_of_lt_mul_succ | [113, 1] | [118, 28] | rw [Nat.mul_comm, ←Nat.div_lt_iff_lt_mul hk] at h | n k m : Nat
h : n < k * (m + 1)
hk : 0 < k
⊢ n / k ≤ m | n k m : Nat
h : n / k < m + 1
hk : 0 < k
⊢ n / k ≤ m | Please generate a tactic in lean4 to solve the state.
STATE:
n k m : Nat
h : n < k * (m + 1)
hk : 0 < k
⊢ n / k ≤ m
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Lemmas.lean | Nat.div_le_of_lt_mul_succ | [113, 1] | [118, 28] | exact Nat.le_of_lt_succ h | n k m : Nat
h : n / k < m + 1
hk : 0 < k
⊢ n / k ≤ m | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n k m : Nat
h : n / k < m + 1
hk : 0 < k
⊢ n / k ≤ m
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Lemmas.lean | Nat.div_le_of_lt_mul_succ | [113, 1] | [118, 28] | rw [Nat.zero_mul] at h | n k m : Nat
h : n < 0 * (m + 1)
⊢ 0 < 0 | n k m : Nat
h : n < 0
⊢ 0 < 0 | Please generate a tactic in lean4 to solve the state.
STATE:
n k m : Nat
h : n < 0 * (m + 1)
⊢ 0 < 0
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Lemmas.lean | Nat.div_le_of_lt_mul_succ | [113, 1] | [118, 28] | absurd h | n k m : Nat
h : n < 0
⊢ 0 < 0 | n k m : Nat
h : n < 0
⊢ ¬n < 0 | Please generate a tactic in lean4 to solve the state.
STATE:
n k m : Nat
h : n < 0
⊢ 0 < 0
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Lemmas.lean | Nat.div_le_of_lt_mul_succ | [113, 1] | [118, 28] | exact Nat.not_lt_zero _ | n k m : Nat
h : n < 0
⊢ ¬n < 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n k m : Nat
h : n < 0
⊢ ¬n < 0
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Lemmas.lean | Nat.div_le_iff_lt_mul_succ | [120, 1] | [130, 30] | rw [←Nat.div_add_mod n k, Nat.mul_succ] | k n m : Nat
k0 : 0 < k
h : n / k ≤ m
⊢ n < k * (m + 1) | k n m : Nat
k0 : 0 < k
h : n / k ≤ m
⊢ k * (n / k) + n % k < k * m + k | Please generate a tactic in lean4 to solve the state.
STATE:
k n m : Nat
k0 : 0 < k
h : n / k ≤ m
⊢ n < k * (m + 1)
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Lemmas.lean | Nat.div_le_iff_lt_mul_succ | [120, 1] | [130, 30] | apply Nat.lt_of_succ_le | k n m : Nat
k0 : 0 < k
h : n / k ≤ m
⊢ k * (n / k) + n % k < k * m + k | case h
k n m : Nat
k0 : 0 < k
h : n / k ≤ m
⊢ (k * (n / k) + n % k).succ ≤ k * m + k | Please generate a tactic in lean4 to solve the state.
STATE:
k n m : Nat
k0 : 0 < k
h : n / k ≤ m
⊢ k * (n / k) + n % k < k * m + k
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Lemmas.lean | Nat.div_le_iff_lt_mul_succ | [120, 1] | [130, 30] | apply Nat.add_le_add _ (Nat.mod_lt _ k0) | case h
k n m : Nat
k0 : 0 < k
h : n / k ≤ m
⊢ (k * (n / k) + n % k).succ ≤ k * m + k | k n m : Nat
k0 : 0 < k
h : n / k ≤ m
⊢ k * (n / k) ≤ k * m | Please generate a tactic in lean4 to solve the state.
STATE:
case h
k n m : Nat
k0 : 0 < k
h : n / k ≤ m
⊢ (k * (n / k) + n % k).succ ≤ k * m + k
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Lemmas.lean | Nat.div_le_iff_lt_mul_succ | [120, 1] | [130, 30] | exact Nat.mul_le_mul_left _ h | k n m : Nat
k0 : 0 < k
h : n / k ≤ m
⊢ k * (n / k) ≤ k * m | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
k n m : Nat
k0 : 0 < k
h : n / k ≤ m
⊢ k * (n / k) ≤ k * m
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Lemmas.lean | Nat.div_le_iff_lt_mul_succ | [120, 1] | [130, 30] | rw [Nat.mul_comm, ←Nat.div_lt_iff_lt_mul k0] at h | k n m : Nat
k0 : 0 < k
h : n < k * (m + 1)
⊢ n / k ≤ m | k n m : Nat
k0 : 0 < k
h : n / k < m + 1
⊢ n / k ≤ m | Please generate a tactic in lean4 to solve the state.
STATE:
k n m : Nat
k0 : 0 < k
h : n < k * (m + 1)
⊢ n / k ≤ m
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Lemmas.lean | Nat.div_le_iff_lt_mul_succ | [120, 1] | [130, 30] | exact Nat.le_of_lt_succ h | k n m : Nat
k0 : 0 < k
h : n / k < m + 1
⊢ n / k ≤ m | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
k n m : Nat
k0 : 0 < k
h : n / k < m + 1
⊢ n / k ≤ m
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Lemmas.lean | Nat.div_le_div_right | [132, 1] | [140, 44] | match k with
| 0 => rw [Nat.div_zero]; exact Nat.zero_le _
| k+1 =>
apply Nat.div_le_of_lt_mul_succ
apply Nat.lt_of_le_of_lt h
conv => lhs; rw [←Nat.div_add_mod n (k+1)]
rw [Nat.mul_succ, Nat.add_lt_add_iff_left]
exact Nat.mod_lt _ (Nat.zero_lt_succ _) | m n k : Nat
h : m ≤ n
⊢ m / k ≤ n / k | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
m n k : Nat
h : m ≤ n
⊢ m / k ≤ n / k
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Lemmas.lean | Nat.div_le_div_right | [132, 1] | [140, 44] | rw [Nat.div_zero] | m n k : Nat
h : m ≤ n
⊢ m / 0 ≤ n / 0 | m n k : Nat
h : m ≤ n
⊢ 0 ≤ n / 0 | Please generate a tactic in lean4 to solve the state.
STATE:
m n k : Nat
h : m ≤ n
⊢ m / 0 ≤ n / 0
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Lemmas.lean | Nat.div_le_div_right | [132, 1] | [140, 44] | exact Nat.zero_le _ | m n k : Nat
h : m ≤ n
⊢ 0 ≤ n / 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
m n k : Nat
h : m ≤ n
⊢ 0 ≤ n / 0
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Lemmas.lean | Nat.div_le_div_right | [132, 1] | [140, 44] | apply Nat.div_le_of_lt_mul_succ | m n k✝ : Nat
h : m ≤ n
k : Nat
⊢ m / (k + 1) ≤ n / (k + 1) | case h
m n k✝ : Nat
h : m ≤ n
k : Nat
⊢ m < (k + 1) * (n / (k + 1) + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
m n k✝ : Nat
h : m ≤ n
k : Nat
⊢ m / (k + 1) ≤ n / (k + 1)
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Lemmas.lean | Nat.div_le_div_right | [132, 1] | [140, 44] | apply Nat.lt_of_le_of_lt h | case h
m n k✝ : Nat
h : m ≤ n
k : Nat
⊢ m < (k + 1) * (n / (k + 1) + 1) | case h
m n k✝ : Nat
h : m ≤ n
k : Nat
⊢ n < (k + 1) * (n / (k + 1) + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case h
m n k✝ : Nat
h : m ≤ n
k : Nat
⊢ m < (k + 1) * (n / (k + 1) + 1)
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Lemmas.lean | Nat.div_le_div_right | [132, 1] | [140, 44] | conv => lhs; rw [←Nat.div_add_mod n (k+1)] | case h
m n k✝ : Nat
h : m ≤ n
k : Nat
⊢ n < (k + 1) * (n / (k + 1) + 1) | case h
m n k✝ : Nat
h : m ≤ n
k : Nat
⊢ (k + 1) * (n / (k + 1)) + n % (k + 1) < (k + 1) * (n / (k + 1) + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case h
m n k✝ : Nat
h : m ≤ n
k : Nat
⊢ n < (k + 1) * (n / (k + 1) + 1)
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Lemmas.lean | Nat.div_le_div_right | [132, 1] | [140, 44] | rw [Nat.mul_succ, Nat.add_lt_add_iff_left] | case h
m n k✝ : Nat
h : m ≤ n
k : Nat
⊢ (k + 1) * (n / (k + 1)) + n % (k + 1) < (k + 1) * (n / (k + 1) + 1) | case h
m n k✝ : Nat
h : m ≤ n
k : Nat
⊢ n % (k + 1) < k + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case h
m n k✝ : Nat
h : m ≤ n
k : Nat
⊢ (k + 1) * (n / (k + 1)) + n % (k + 1) < (k + 1) * (n / (k + 1) + 1)
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Lemmas.lean | Nat.div_le_div_right | [132, 1] | [140, 44] | exact Nat.mod_lt _ (Nat.zero_lt_succ _) | case h
m n k✝ : Nat
h : m ≤ n
k : Nat
⊢ n % (k + 1) < k + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
m n k✝ : Nat
h : m ≤ n
k : Nat
⊢ n % (k + 1) < k + 1
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Fin.lean | Fin.all_zero | [10, 9] | [10, 84] | simp [Fin.all] | p : Fin 0 → Bool
⊢ Fin.all p = true | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
p : Fin 0 → Bool
⊢ Fin.all p = true
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Fin.lean | Fin.all_succ | [12, 1] | [13, 33] | simp [Fin.all, Fin.foldr_succ] | n : Nat
p : Fin (n + 1) → Bool
⊢ Fin.all p = (p 0 && Fin.all (p ∘ succ)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : Nat
p : Fin (n + 1) → Bool
⊢ Fin.all p = (p 0 && Fin.all (p ∘ succ))
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Fin.lean | Fin.forall_eq_true_of_all_eq_true | [15, 1] | [21, 77] | rw [Fin.all, Fin.foldr_succ, Bool.and_eq_true] at h | n : Nat
p : Fin (n + 1) → Bool
h : Fin.all p = true
isLt✝ : 0 < n + 1
⊢ p ⟨0, isLt✝⟩ = true | n : Nat
p : Fin (n + 1) → Bool
h : p 0 = true ∧ foldr n (fun i v => p i.succ && v) true = true
isLt✝ : 0 < n + 1
⊢ p ⟨0, isLt✝⟩ = true | Please generate a tactic in lean4 to solve the state.
STATE:
n : Nat
p : Fin (n + 1) → Bool
h : Fin.all p = true
isLt✝ : 0 < n + 1
⊢ p ⟨0, isLt✝⟩ = true
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Fin.lean | Fin.forall_eq_true_of_all_eq_true | [15, 1] | [21, 77] | exact h.left | n : Nat
p : Fin (n + 1) → Bool
h : p 0 = true ∧ foldr n (fun i v => p i.succ && v) true = true
isLt✝ : 0 < n + 1
⊢ p ⟨0, isLt✝⟩ = true | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : Nat
p : Fin (n + 1) → Bool
h : p 0 = true ∧ foldr n (fun i v => p i.succ && v) true = true
isLt✝ : 0 < n + 1
⊢ p ⟨0, isLt✝⟩ = true
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Fin.lean | Fin.forall_eq_true_of_all_eq_true | [15, 1] | [21, 77] | rw [Fin.all, Fin.foldr_succ, Bool.and_eq_true] at h | n : Nat
p : Fin (n + 1) → Bool
h : Fin.all p = true
i : Nat
hi : i + 1 < n + 1
⊢ p ⟨i + 1, hi⟩ = true | n : Nat
p : Fin (n + 1) → Bool
h : p 0 = true ∧ foldr n (fun i v => p i.succ && v) true = true
i : Nat
hi : i + 1 < n + 1
⊢ p ⟨i + 1, hi⟩ = true | Please generate a tactic in lean4 to solve the state.
STATE:
n : Nat
p : Fin (n + 1) → Bool
h : Fin.all p = true
i : Nat
hi : i + 1 < n + 1
⊢ p ⟨i + 1, hi⟩ = true
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Fin.lean | Fin.forall_eq_true_of_all_eq_true | [15, 1] | [21, 77] | exact forall_eq_true_of_all_eq_true h.right ⟨i, Nat.lt_of_succ_lt_succ hi⟩ | n : Nat
p : Fin (n + 1) → Bool
h : p 0 = true ∧ foldr n (fun i v => p i.succ && v) true = true
i : Nat
hi : i + 1 < n + 1
⊢ p ⟨i + 1, hi⟩ = true | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : Nat
p : Fin (n + 1) → Bool
h : p 0 = true ∧ foldr n (fun i v => p i.succ && v) true = true
i : Nat
hi : i + 1 < n + 1
⊢ p ⟨i + 1, hi⟩ = true
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Fin.lean | Fin.exists_eq_false_of_all_eq_false | [23, 1] | [30, 55] | simp at h | p : Fin 0 → Bool
h : Fin.all p = false
⊢ ∃ i, p i = false | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
p : Fin 0 → Bool
h : Fin.all p = false
⊢ ∃ i, p i = false
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Fin.lean | Fin.exists_eq_false_of_all_eq_false | [23, 1] | [30, 55] | rw [Fin.all, Fin.foldr_succ, Bool.and_eq_false_iff] at h | n : Nat
p : Fin (n + 1) → Bool
h : Fin.all p = false
⊢ ∃ i, p i = false | n : Nat
p : Fin (n + 1) → Bool
h : p 0 = false ∨ foldr n (fun i v => p i.succ && v) true = false
⊢ ∃ i, p i = false | Please generate a tactic in lean4 to solve the state.
STATE:
n : Nat
p : Fin (n + 1) → Bool
h : Fin.all p = false
⊢ ∃ i, p i = false
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Fin.lean | Fin.exists_eq_false_of_all_eq_false | [23, 1] | [30, 55] | match h with
| .inl h => exists ⟨0, Nat.zero_lt_succ n⟩
| .inr h => match exists_eq_false_of_all_eq_false h with
| ⟨⟨i,hi⟩,hp⟩ => exists ⟨i+1, Nat.succ_lt_succ hi⟩ | n : Nat
p : Fin (n + 1) → Bool
h : p 0 = false ∨ foldr n (fun i v => p i.succ && v) true = false
⊢ ∃ i, p i = false | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : Nat
p : Fin (n + 1) → Bool
h : p 0 = false ∨ foldr n (fun i v => p i.succ && v) true = false
⊢ ∃ i, p i = false
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Fin.lean | Fin.exists_eq_false_of_all_eq_false | [23, 1] | [30, 55] | exists ⟨0, Nat.zero_lt_succ n⟩ | n : Nat
p : Fin (n + 1) → Bool
h✝ : p 0 = false ∨ foldr n (fun i v => p i.succ && v) true = false
h : p 0 = false
⊢ ∃ i, p i = false | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : Nat
p : Fin (n + 1) → Bool
h✝ : p 0 = false ∨ foldr n (fun i v => p i.succ && v) true = false
h : p 0 = false
⊢ ∃ i, p i = false
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Fin.lean | Fin.exists_eq_false_of_all_eq_false | [23, 1] | [30, 55] | match exists_eq_false_of_all_eq_false h with
| ⟨⟨i,hi⟩,hp⟩ => exists ⟨i+1, Nat.succ_lt_succ hi⟩ | n : Nat
p : Fin (n + 1) → Bool
h✝ : p 0 = false ∨ foldr n (fun i v => p i.succ && v) true = false
h : foldr n (fun i v => p i.succ && v) true = false
⊢ ∃ i, p i = false | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : Nat
p : Fin (n + 1) → Bool
h✝ : p 0 = false ∨ foldr n (fun i v => p i.succ && v) true = false
h : foldr n (fun i v => p i.succ && v) true = false
⊢ ∃ i, p i = false
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Fin.lean | Fin.exists_eq_false_of_all_eq_false | [23, 1] | [30, 55] | exists ⟨i+1, Nat.succ_lt_succ hi⟩ | n : Nat
p : Fin (n + 1) → Bool
h✝ : p 0 = false ∨ foldr n (fun i v => p i.succ && v) true = false
h : foldr n (fun i v => p i.succ && v) true = false
i : Nat
hi : i < n
hp : p ⟨i, hi⟩.succ = false
⊢ ∃ i, p i = false | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : Nat
p : Fin (n + 1) → Bool
h✝ : p 0 = false ∨ foldr n (fun i v => p i.succ && v) true = false
h : foldr n (fun i v => p i.succ && v) true = false
i : Nat
hi : i < n
hp : p ⟨i, hi⟩.succ = false
⊢ ∃ i, p i = false
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Fin.lean | Fin.decide_forall_eq_all | [43, 1] | [58, 16] | match h : Fin.all fun i => decide (p i) with
| true =>
have h := forall_eq_true_of_all_eq_true h
apply decide_eq_true
intro i
apply of_decide_eq_true
exact h i
| false =>
match exists_eq_false_of_all_eq_false h with
| ⟨i, hi⟩ =>
have hi := of_decide_eq_false hi
apply decide_eq_false
intro h
... | n : Nat
p : Fin n → Prop
inst✝¹ : DecidablePred p
inst✝ : Decidable (∀ (i : Fin n), p i)
⊢ decide (∀ (i : Fin n), p i) = Fin.all fun i => decide (p i) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : Nat
p : Fin n → Prop
inst✝¹ : DecidablePred p
inst✝ : Decidable (∀ (i : Fin n), p i)
⊢ decide (∀ (i : Fin n), p i) = Fin.all fun i => decide (p i)
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Fin.lean | Fin.decide_forall_eq_all | [43, 1] | [58, 16] | have h := forall_eq_true_of_all_eq_true h | n : Nat
p : Fin n → Prop
inst✝¹ : DecidablePred p
inst✝ : Decidable (∀ (i : Fin n), p i)
h : (Fin.all fun i => decide (p i)) = true
⊢ decide (∀ (i : Fin n), p i) = true | n : Nat
p : Fin n → Prop
inst✝¹ : DecidablePred p
inst✝ : Decidable (∀ (i : Fin n), p i)
h✝ : (Fin.all fun i => decide (p i)) = true
h : ∀ (i : Fin n), decide (p i) = true
⊢ decide (∀ (i : Fin n), p i) = true | Please generate a tactic in lean4 to solve the state.
STATE:
n : Nat
p : Fin n → Prop
inst✝¹ : DecidablePred p
inst✝ : Decidable (∀ (i : Fin n), p i)
h : (Fin.all fun i => decide (p i)) = true
⊢ decide (∀ (i : Fin n), p i) = true
TACTIC:
|
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