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/Pos/Lemmas.lean | Pos.lt_succ_of_le | [273, 11] | [274, 36] | exact Nat.lt_succ_of_le | x y : Pos
⊢ y.toNat ≥ x.toNat → y.toNat + 1 > x.toNat | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Pos
⊢ y.toNat ≥ x.toNat → y.toNat + 1 > x.toNat
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.lt_of_succ_le | [276, 11] | [277, 36] | by_toNat | x y : Pos
⊢ x + 1 ≤ y → x < y | x y : Pos
⊢ y.toNat ≥ x.toNat + 1 → y.toNat > x.toNat | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Pos
⊢ x + 1 ≤ y → x < y
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.lt_of_succ_le | [276, 11] | [277, 36] | exact Nat.lt_of_succ_le | x y : Pos
⊢ y.toNat ≥ x.toNat + 1 → y.toNat > x.toNat | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Pos
⊢ y.toNat ≥ x.toNat + 1 → y.toNat > x.toNat
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.succ_le_of_lt | [279, 11] | [280, 36] | by_toNat | x y : Pos
⊢ x < y → x + 1 ≤ y | x y : Pos
⊢ y.toNat > x.toNat → y.toNat ≥ x.toNat + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Pos
⊢ x < y → x + 1 ≤ y
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.succ_le_of_lt | [279, 11] | [280, 36] | exact Nat.succ_le_of_lt | x y : Pos
⊢ y.toNat > x.toNat → y.toNat ≥ x.toNat + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Pos
⊢ y.toNat > x.toNat → y.toNat ≥ x.toNat + 1
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.le_succ_of_le | [282, 11] | [283, 36] | by_toNat | x y : Pos
⊢ x ≤ y → x ≤ y + 1 | x y : Pos
⊢ y.toNat ≥ x.toNat → y.toNat + 1 ≥ x.toNat | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Pos
⊢ x ≤ y → x ≤ y + 1
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.le_succ_of_le | [282, 11] | [283, 36] | exact Nat.le_succ_of_le | x y : Pos
⊢ y.toNat ≥ x.toNat → y.toNat + 1 ≥ x.toNat | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Pos
⊢ y.toNat ≥ x.toNat → y.toNat + 1 ≥ x.toNat
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.le_of_succ_le | [285, 11] | [286, 36] | by_toNat | x y : Pos
⊢ x + 1 ≤ y → x ≤ y | x y : Pos
⊢ y.toNat ≥ x.toNat + 1 → y.toNat ≥ x.toNat | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Pos
⊢ x + 1 ≤ y → x ≤ y
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.le_of_succ_le | [285, 11] | [286, 36] | exact Nat.le_of_succ_le | x y : Pos
⊢ y.toNat ≥ x.toNat + 1 → y.toNat ≥ x.toNat | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Pos
⊢ y.toNat ≥ x.toNat + 1 → y.toNat ≥ x.toNat
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.succ_le_succ | [288, 11] | [289, 35] | by_toNat | x y : Pos
⊢ x ≤ y → x + 1 ≤ y + 1 | x y : Pos
⊢ y.toNat ≥ x.toNat → y.toNat + 1 ≥ x.toNat + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Pos
⊢ x ≤ y → x + 1 ≤ y + 1
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.succ_le_succ | [288, 11] | [289, 35] | exact Nat.succ_le_succ | x y : Pos
⊢ y.toNat ≥ x.toNat → y.toNat + 1 ≥ x.toNat + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Pos
⊢ y.toNat ≥ x.toNat → y.toNat + 1 ≥ x.toNat + 1
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.le_of_succ_le_succ | [291, 11] | [292, 41] | by_toNat | x y : Pos
⊢ x + 1 ≤ y + 1 → x ≤ y | x y : Pos
⊢ y.toNat + 1 ≥ x.toNat + 1 → y.toNat ≥ x.toNat | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Pos
⊢ x + 1 ≤ y + 1 → x ≤ y
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.le_of_succ_le_succ | [291, 11] | [292, 41] | exact Nat.le_of_succ_le_succ | x y : Pos
⊢ y.toNat + 1 ≥ x.toNat + 1 → y.toNat ≥ x.toNat | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Pos
⊢ y.toNat + 1 ≥ x.toNat + 1 → y.toNat ≥ x.toNat
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.succ_lt_succ | [294, 11] | [295, 35] | by_toNat | x y : Pos
⊢ x < y → x + 1 < y + 1 | x y : Pos
⊢ y.toNat > x.toNat → y.toNat + 1 > x.toNat + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Pos
⊢ x < y → x + 1 < y + 1
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.succ_lt_succ | [294, 11] | [295, 35] | exact Nat.succ_lt_succ | x y : Pos
⊢ y.toNat > x.toNat → y.toNat + 1 > x.toNat + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Pos
⊢ y.toNat > x.toNat → y.toNat + 1 > x.toNat + 1
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.lt_of_succ_lt_succ | [297, 11] | [298, 41] | by_toNat | x y : Pos
⊢ x + 1 < y + 1 → x < y | x y : Pos
⊢ y.toNat + 1 > x.toNat + 1 → y.toNat > x.toNat | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Pos
⊢ x + 1 < y + 1 → x < y
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.lt_of_succ_lt_succ | [297, 11] | [298, 41] | exact Nat.lt_of_succ_lt_succ | x y : Pos
⊢ y.toNat + 1 > x.toNat + 1 → y.toNat > x.toNat | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Pos
⊢ y.toNat + 1 > x.toNat + 1 → y.toNat > x.toNat
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.lt_add_right | [300, 11] | [303, 88] | induction y with
| one => exact Pos.lt_succ_self x
| succ y H => apply Pos.lt_trans H; rw [Pos.add_succ]; exact Pos.lt_succ_self (x + y) | x y : Pos
⊢ x < x + y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Pos
⊢ x < x + y
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.lt_add_right | [300, 11] | [303, 88] | exact Pos.lt_succ_self x | case one
x : Pos
⊢ x < x + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case one
x : Pos
⊢ x < x + 1
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.lt_add_right | [300, 11] | [303, 88] | apply Pos.lt_trans H | case succ
x y : Pos
H : x < x + y
⊢ x < x + (y + 1) | case succ
x y : Pos
H : x < x + y
⊢ x + y < x + (y + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
x y : Pos
H : x < x + y
⊢ x < x + (y + 1)
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.lt_add_right | [300, 11] | [303, 88] | rw [Pos.add_succ] | case succ
x y : Pos
H : x < x + y
⊢ x + y < x + (y + 1) | case succ
x y : Pos
H : x < x + y
⊢ x + y < x + y + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
x y : Pos
H : x < x + y
⊢ x + y < x + (y + 1)
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.lt_add_right | [300, 11] | [303, 88] | exact Pos.lt_succ_self (x + y) | case succ
x y : Pos
H : x < x + y
⊢ x + y < x + y + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
x y : Pos
H : x < x + y
⊢ x + y < x + y + 1
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.lt_add_left | [305, 11] | [308, 88] | induction y with
| one => rw [Pos.one_add_eq_add_one]; exact Pos.lt_succ_self x
| succ y H => apply Pos.lt_trans H; rw [Pos.succ_add]; exact Pos.lt_succ_self (y + x) | x y : Pos
⊢ x < y + x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Pos
⊢ x < y + x
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.lt_add_left | [305, 11] | [308, 88] | rw [Pos.one_add_eq_add_one] | case one
x : Pos
⊢ x < 1 + x | case one
x : Pos
⊢ x < x + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case one
x : Pos
⊢ x < 1 + x
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.lt_add_left | [305, 11] | [308, 88] | exact Pos.lt_succ_self x | case one
x : Pos
⊢ x < x + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case one
x : Pos
⊢ x < x + 1
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.lt_add_left | [305, 11] | [308, 88] | apply Pos.lt_trans H | case succ
x y : Pos
H : x < y + x
⊢ x < y + 1 + x | case succ
x y : Pos
H : x < y + x
⊢ y + x < y + 1 + x | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
x y : Pos
H : x < y + x
⊢ x < y + 1 + x
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.lt_add_left | [305, 11] | [308, 88] | rw [Pos.succ_add] | case succ
x y : Pos
H : x < y + x
⊢ y + x < y + 1 + x | case succ
x y : Pos
H : x < y + x
⊢ y + x < y + x + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
x y : Pos
H : x < y + x
⊢ y + x < y + 1 + x
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.lt_add_left | [305, 11] | [308, 88] | exact Pos.lt_succ_self (y + x) | case succ
x y : Pos
H : x < y + x
⊢ y + x < y + x + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
x y : Pos
H : x < y + x
⊢ y + x < y + x + 1
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.add_le_add_left | [310, 11] | [311, 51] | by_toNat | x y : Pos
⊢ x ≤ y → ∀ (z : Pos), z + x ≤ z + y | x y : Pos
⊢ y.toNat ≥ x.toNat → ∀ (z : Pos), z.toNat + y.toNat ≥ z.toNat + x.toNat | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Pos
⊢ x ≤ y → ∀ (z : Pos), z + x ≤ z + y
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.add_le_add_left | [310, 11] | [311, 51] | intro h z | x y : Pos
⊢ y.toNat ≥ x.toNat → ∀ (z : Pos), z.toNat + y.toNat ≥ z.toNat + x.toNat | x y : Pos
h : y.toNat ≥ x.toNat
z : Pos
⊢ z.toNat + y.toNat ≥ z.toNat + x.toNat | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Pos
⊢ y.toNat ≥ x.toNat → ∀ (z : Pos), z.toNat + y.toNat ≥ z.toNat + x.toNat
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.add_le_add_left | [310, 11] | [311, 51] | apply Nat.add_le_add_left h | x y : Pos
h : y.toNat ≥ x.toNat
z : Pos
⊢ z.toNat + y.toNat ≥ z.toNat + x.toNat | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Pos
h : y.toNat ≥ x.toNat
z : Pos
⊢ z.toNat + y.toNat ≥ z.toNat + x.toNat
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.add_le_add_right | [313, 11] | [314, 52] | by_toNat | x y : Pos
⊢ x ≤ y → ∀ (z : Pos), x + z ≤ y + z | x y : Pos
⊢ y.toNat ≥ x.toNat → ∀ (z : Pos), y.toNat + z.toNat ≥ x.toNat + z.toNat | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Pos
⊢ x ≤ y → ∀ (z : Pos), x + z ≤ y + z
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.add_le_add_right | [313, 11] | [314, 52] | intro h z | x y : Pos
⊢ y.toNat ≥ x.toNat → ∀ (z : Pos), y.toNat + z.toNat ≥ x.toNat + z.toNat | x y : Pos
h : y.toNat ≥ x.toNat
z : Pos
⊢ y.toNat + z.toNat ≥ x.toNat + z.toNat | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Pos
⊢ y.toNat ≥ x.toNat → ∀ (z : Pos), y.toNat + z.toNat ≥ x.toNat + z.toNat
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.add_le_add_right | [313, 11] | [314, 52] | apply Nat.add_le_add_right h | x y : Pos
h : y.toNat ≥ x.toNat
z : Pos
⊢ y.toNat + z.toNat ≥ x.toNat + z.toNat | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Pos
h : y.toNat ≥ x.toNat
z : Pos
⊢ y.toNat + z.toNat ≥ x.toNat + z.toNat
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.le_of_add_le_add_left | [316, 11] | [317, 44] | by_toNat | x y z : Pos
⊢ z + x ≤ z + y → x ≤ y | x y z : Pos
⊢ z.toNat + y.toNat ≥ z.toNat + x.toNat → y.toNat ≥ x.toNat | Please generate a tactic in lean4 to solve the state.
STATE:
x y z : Pos
⊢ z + x ≤ z + y → x ≤ y
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.le_of_add_le_add_left | [316, 11] | [317, 44] | exact Nat.le_of_add_le_add_left | x y z : Pos
⊢ z.toNat + y.toNat ≥ z.toNat + x.toNat → y.toNat ≥ x.toNat | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y z : Pos
⊢ z.toNat + y.toNat ≥ z.toNat + x.toNat → y.toNat ≥ x.toNat
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.le_of_add_le_add_right | [319, 11] | [320, 45] | by_toNat | x y z : Pos
⊢ x + z ≤ y + z → x ≤ y | x y z : Pos
⊢ y.toNat + z.toNat ≥ x.toNat + z.toNat → y.toNat ≥ x.toNat | Please generate a tactic in lean4 to solve the state.
STATE:
x y z : Pos
⊢ x + z ≤ y + z → x ≤ y
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.le_of_add_le_add_right | [319, 11] | [320, 45] | exact Nat.le_of_add_le_add_right | x y z : Pos
⊢ y.toNat + z.toNat ≥ x.toNat + z.toNat → y.toNat ≥ x.toNat | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y z : Pos
⊢ y.toNat + z.toNat ≥ x.toNat + z.toNat → y.toNat ≥ x.toNat
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.add_le_add | [322, 11] | [323, 33] | by_toNat | x₁ y₁ x₂ y₂ : Pos
⊢ x₁ ≤ y₁ → x₂ ≤ y₂ → x₁ + x₂ ≤ y₁ + y₂ | x₁ y₁ x₂ y₂ : Pos
⊢ y₁.toNat ≥ x₁.toNat → y₂.toNat ≥ x₂.toNat → y₁.toNat + y₂.toNat ≥ x₁.toNat + x₂.toNat | Please generate a tactic in lean4 to solve the state.
STATE:
x₁ y₁ x₂ y₂ : Pos
⊢ x₁ ≤ y₁ → x₂ ≤ y₂ → x₁ + x₂ ≤ y₁ + y₂
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.add_le_add | [322, 11] | [323, 33] | exact Nat.add_le_add | x₁ y₁ x₂ y₂ : Pos
⊢ y₁.toNat ≥ x₁.toNat → y₂.toNat ≥ x₂.toNat → y₁.toNat + y₂.toNat ≥ x₁.toNat + x₂.toNat | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x₁ y₁ x₂ y₂ : Pos
⊢ y₁.toNat ≥ x₁.toNat → y₂.toNat ≥ x₂.toNat → y₁.toNat + y₂.toNat ≥ x₁.toNat + x₂.toNat
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.add_lt_add_left | [325, 11] | [326, 51] | by_toNat | x y : Pos
⊢ x < y → ∀ (z : Pos), z + x < z + y | x y : Pos
⊢ y.toNat > x.toNat → ∀ (z : Pos), z.toNat + y.toNat > z.toNat + x.toNat | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Pos
⊢ x < y → ∀ (z : Pos), z + x < z + y
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.add_lt_add_left | [325, 11] | [326, 51] | intro h z | x y : Pos
⊢ y.toNat > x.toNat → ∀ (z : Pos), z.toNat + y.toNat > z.toNat + x.toNat | x y : Pos
h : y.toNat > x.toNat
z : Pos
⊢ z.toNat + y.toNat > z.toNat + x.toNat | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Pos
⊢ y.toNat > x.toNat → ∀ (z : Pos), z.toNat + y.toNat > z.toNat + x.toNat
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.add_lt_add_left | [325, 11] | [326, 51] | apply Nat.add_lt_add_left h | x y : Pos
h : y.toNat > x.toNat
z : Pos
⊢ z.toNat + y.toNat > z.toNat + x.toNat | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Pos
h : y.toNat > x.toNat
z : Pos
⊢ z.toNat + y.toNat > z.toNat + x.toNat
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.add_lt_add_right | [328, 11] | [329, 52] | by_toNat | x y : Pos
⊢ x < y → ∀ (z : Pos), x + z < y + z | x y : Pos
⊢ y.toNat > x.toNat → ∀ (z : Pos), y.toNat + z.toNat > x.toNat + z.toNat | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Pos
⊢ x < y → ∀ (z : Pos), x + z < y + z
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.add_lt_add_right | [328, 11] | [329, 52] | intro h z | x y : Pos
⊢ y.toNat > x.toNat → ∀ (z : Pos), y.toNat + z.toNat > x.toNat + z.toNat | x y : Pos
h : y.toNat > x.toNat
z : Pos
⊢ y.toNat + z.toNat > x.toNat + z.toNat | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Pos
⊢ y.toNat > x.toNat → ∀ (z : Pos), y.toNat + z.toNat > x.toNat + z.toNat
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.add_lt_add_right | [328, 11] | [329, 52] | apply Nat.add_lt_add_right h | x y : Pos
h : y.toNat > x.toNat
z : Pos
⊢ y.toNat + z.toNat > x.toNat + z.toNat | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Pos
h : y.toNat > x.toNat
z : Pos
⊢ y.toNat + z.toNat > x.toNat + z.toNat
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.lt_of_add_lt_add_left | [331, 11] | [332, 44] | by_toNat | x y z : Pos
⊢ z + x < z + y → x < y | x y z : Pos
⊢ z.toNat + y.toNat > z.toNat + x.toNat → y.toNat > x.toNat | Please generate a tactic in lean4 to solve the state.
STATE:
x y z : Pos
⊢ z + x < z + y → x < y
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.lt_of_add_lt_add_left | [331, 11] | [332, 44] | exact Nat.lt_of_add_lt_add_left | x y z : Pos
⊢ z.toNat + y.toNat > z.toNat + x.toNat → y.toNat > x.toNat | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y z : Pos
⊢ z.toNat + y.toNat > z.toNat + x.toNat → y.toNat > x.toNat
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.lt_of_add_lt_add_right | [334, 11] | [335, 45] | by_toNat | x y z : Pos
⊢ x + z < y + z → x < y | x y z : Pos
⊢ y.toNat + z.toNat > x.toNat + z.toNat → y.toNat > x.toNat | Please generate a tactic in lean4 to solve the state.
STATE:
x y z : Pos
⊢ x + z < y + z → x < y
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.lt_of_add_lt_add_right | [334, 11] | [335, 45] | exact Nat.lt_of_add_lt_add_right | x y z : Pos
⊢ y.toNat + z.toNat > x.toNat + z.toNat → y.toNat > x.toNat | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y z : Pos
⊢ y.toNat + z.toNat > x.toNat + z.toNat → y.toNat > x.toNat
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.add_lt_add | [337, 11] | [338, 33] | by_toNat | x₁ y₁ x₂ y₂ : Pos
⊢ x₁ < y₁ → x₂ < y₂ → x₁ + x₂ < y₁ + y₂ | x₁ y₁ x₂ y₂ : Pos
⊢ y₁.toNat > x₁.toNat → y₂.toNat > x₂.toNat → y₁.toNat + y₂.toNat > x₁.toNat + x₂.toNat | Please generate a tactic in lean4 to solve the state.
STATE:
x₁ y₁ x₂ y₂ : Pos
⊢ x₁ < y₁ → x₂ < y₂ → x₁ + x₂ < y₁ + y₂
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.add_lt_add | [337, 11] | [338, 33] | exact Nat.add_lt_add | x₁ y₁ x₂ y₂ : Pos
⊢ y₁.toNat > x₁.toNat → y₂.toNat > x₂.toNat → y₁.toNat + y₂.toNat > x₁.toNat + x₂.toNat | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x₁ y₁ x₂ y₂ : Pos
⊢ y₁.toNat > x₁.toNat → y₂.toNat > x₂.toNat → y₁.toNat + y₂.toNat > x₁.toNat + x₂.toNat
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.mul_le_mul_left | [340, 11] | [341, 53] | by_toNat | x y : Pos
⊢ x ≤ y → ∀ (z : Pos), z * x ≤ z * y | x y : Pos
⊢ y.toNat ≥ x.toNat → ∀ (z : Pos), z.toNat * y.toNat ≥ z.toNat * x.toNat | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Pos
⊢ x ≤ y → ∀ (z : Pos), z * x ≤ z * y
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.mul_le_mul_left | [340, 11] | [341, 53] | intro h z | x y : Pos
⊢ y.toNat ≥ x.toNat → ∀ (z : Pos), z.toNat * y.toNat ≥ z.toNat * x.toNat | x y : Pos
h : y.toNat ≥ x.toNat
z : Pos
⊢ z.toNat * y.toNat ≥ z.toNat * x.toNat | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Pos
⊢ y.toNat ≥ x.toNat → ∀ (z : Pos), z.toNat * y.toNat ≥ z.toNat * x.toNat
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.mul_le_mul_left | [340, 11] | [341, 53] | apply Nat.mul_le_mul_left _ h | x y : Pos
h : y.toNat ≥ x.toNat
z : Pos
⊢ z.toNat * y.toNat ≥ z.toNat * x.toNat | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Pos
h : y.toNat ≥ x.toNat
z : Pos
⊢ z.toNat * y.toNat ≥ z.toNat * x.toNat
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.mul_le_mul_right | [343, 11] | [344, 54] | by_toNat | x y : Pos
⊢ x ≤ y → ∀ (z : Pos), x * z ≤ y * z | x y : Pos
⊢ y.toNat ≥ x.toNat → ∀ (z : Pos), y.toNat * z.toNat ≥ x.toNat * z.toNat | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Pos
⊢ x ≤ y → ∀ (z : Pos), x * z ≤ y * z
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.mul_le_mul_right | [343, 11] | [344, 54] | intro h z | x y : Pos
⊢ y.toNat ≥ x.toNat → ∀ (z : Pos), y.toNat * z.toNat ≥ x.toNat * z.toNat | x y : Pos
h : y.toNat ≥ x.toNat
z : Pos
⊢ y.toNat * z.toNat ≥ x.toNat * z.toNat | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Pos
⊢ y.toNat ≥ x.toNat → ∀ (z : Pos), y.toNat * z.toNat ≥ x.toNat * z.toNat
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.mul_le_mul_right | [343, 11] | [344, 54] | apply Nat.mul_le_mul_right _ h | x y : Pos
h : y.toNat ≥ x.toNat
z : Pos
⊢ y.toNat * z.toNat ≥ x.toNat * z.toNat | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Pos
h : y.toNat ≥ x.toNat
z : Pos
⊢ y.toNat * z.toNat ≥ x.toNat * z.toNat
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.le_of_mul_le_mul_left | [346, 11] | [347, 78] | by_toNat | x y z : Pos
⊢ z * x ≤ z * y → x ≤ y | x y z : Pos
⊢ z.toNat * y.toNat ≥ z.toNat * x.toNat → y.toNat ≥ x.toNat | Please generate a tactic in lean4 to solve the state.
STATE:
x y z : Pos
⊢ z * x ≤ z * y → x ≤ y
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.le_of_mul_le_mul_left | [346, 11] | [347, 78] | intro h | x y z : Pos
⊢ z.toNat * y.toNat ≥ z.toNat * x.toNat → y.toNat ≥ x.toNat | x y z : Pos
h : z.toNat * y.toNat ≥ z.toNat * x.toNat
⊢ y.toNat ≥ x.toNat | Please generate a tactic in lean4 to solve the state.
STATE:
x y z : Pos
⊢ z.toNat * y.toNat ≥ z.toNat * x.toNat → y.toNat ≥ x.toNat
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.le_of_mul_le_mul_left | [346, 11] | [347, 78] | exact Nat.le_of_mul_le_mul_of_pos_left (Nat.zero_lt _) h | x y z : Pos
h : z.toNat * y.toNat ≥ z.toNat * x.toNat
⊢ y.toNat ≥ x.toNat | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y z : Pos
h : z.toNat * y.toNat ≥ z.toNat * x.toNat
⊢ y.toNat ≥ x.toNat
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.le_of_mul_le_mul_right | [349, 11] | [350, 79] | by_toNat | x y z : Pos
⊢ x * z ≤ y * z → x ≤ y | x y z : Pos
⊢ y.toNat * z.toNat ≥ x.toNat * z.toNat → y.toNat ≥ x.toNat | Please generate a tactic in lean4 to solve the state.
STATE:
x y z : Pos
⊢ x * z ≤ y * z → x ≤ y
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.le_of_mul_le_mul_right | [349, 11] | [350, 79] | intro h | x y z : Pos
⊢ y.toNat * z.toNat ≥ x.toNat * z.toNat → y.toNat ≥ x.toNat | x y z : Pos
h : y.toNat * z.toNat ≥ x.toNat * z.toNat
⊢ y.toNat ≥ x.toNat | Please generate a tactic in lean4 to solve the state.
STATE:
x y z : Pos
⊢ y.toNat * z.toNat ≥ x.toNat * z.toNat → y.toNat ≥ x.toNat
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.le_of_mul_le_mul_right | [349, 11] | [350, 79] | exact Nat.le_of_mul_le_mul_of_pos_right (Nat.zero_lt _) h | x y z : Pos
h : y.toNat * z.toNat ≥ x.toNat * z.toNat
⊢ y.toNat ≥ x.toNat | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y z : Pos
h : y.toNat * z.toNat ≥ x.toNat * z.toNat
⊢ y.toNat ≥ x.toNat
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.mul_le_mul | [352, 11] | [353, 33] | by_toNat | x₁ y₁ x₂ y₂ : Pos
⊢ x₁ ≤ y₁ → x₂ ≤ y₂ → x₁ * x₂ ≤ y₁ * y₂ | x₁ y₁ x₂ y₂ : Pos
⊢ y₁.toNat ≥ x₁.toNat → y₂.toNat ≥ x₂.toNat → y₁.toNat * y₂.toNat ≥ x₁.toNat * x₂.toNat | Please generate a tactic in lean4 to solve the state.
STATE:
x₁ y₁ x₂ y₂ : Pos
⊢ x₁ ≤ y₁ → x₂ ≤ y₂ → x₁ * x₂ ≤ y₁ * y₂
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.mul_le_mul | [352, 11] | [353, 33] | exact Nat.mul_le_mul | x₁ y₁ x₂ y₂ : Pos
⊢ y₁.toNat ≥ x₁.toNat → y₂.toNat ≥ x₂.toNat → y₁.toNat * y₂.toNat ≥ x₁.toNat * x₂.toNat | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x₁ y₁ x₂ y₂ : Pos
⊢ y₁.toNat ≥ x₁.toNat → y₂.toNat ≥ x₂.toNat → y₁.toNat * y₂.toNat ≥ x₁.toNat * x₂.toNat
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.mul_lt_mul_left | [355, 11] | [356, 79] | by_toNat | x y : Pos
⊢ x < y → ∀ (z : Pos), z * x < z * y | x y : Pos
⊢ y.toNat > x.toNat → ∀ (z : Pos), z.toNat * y.toNat > z.toNat * x.toNat | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Pos
⊢ x < y → ∀ (z : Pos), z * x < z * y
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.mul_lt_mul_left | [355, 11] | [356, 79] | intro h z | x y : Pos
⊢ y.toNat > x.toNat → ∀ (z : Pos), z.toNat * y.toNat > z.toNat * x.toNat | x y : Pos
h : y.toNat > x.toNat
z : Pos
⊢ z.toNat * y.toNat > z.toNat * x.toNat | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Pos
⊢ y.toNat > x.toNat → ∀ (z : Pos), z.toNat * y.toNat > z.toNat * x.toNat
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.mul_lt_mul_left | [355, 11] | [356, 79] | apply Nat.mul_lt_mul_of_pos_left h | x y : Pos
h : y.toNat > x.toNat
z : Pos
⊢ z.toNat * y.toNat > z.toNat * x.toNat | x y : Pos
h : y.toNat > x.toNat
z : Pos
⊢ z.toNat > 0 | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Pos
h : y.toNat > x.toNat
z : Pos
⊢ z.toNat * y.toNat > z.toNat * x.toNat
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.mul_lt_mul_left | [355, 11] | [356, 79] | exact Nat.zero_lt _ | x y : Pos
h : y.toNat > x.toNat
z : Pos
⊢ z.toNat > 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Pos
h : y.toNat > x.toNat
z : Pos
⊢ z.toNat > 0
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.mul_lt_mul_right | [358, 11] | [359, 80] | by_toNat | x y : Pos
⊢ x < y → ∀ (z : Pos), x * z < y * z | x y : Pos
⊢ y.toNat > x.toNat → ∀ (z : Pos), y.toNat * z.toNat > x.toNat * z.toNat | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Pos
⊢ x < y → ∀ (z : Pos), x * z < y * z
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.mul_lt_mul_right | [358, 11] | [359, 80] | intro h z | x y : Pos
⊢ y.toNat > x.toNat → ∀ (z : Pos), y.toNat * z.toNat > x.toNat * z.toNat | x y : Pos
h : y.toNat > x.toNat
z : Pos
⊢ y.toNat * z.toNat > x.toNat * z.toNat | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Pos
⊢ y.toNat > x.toNat → ∀ (z : Pos), y.toNat * z.toNat > x.toNat * z.toNat
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.mul_lt_mul_right | [358, 11] | [359, 80] | apply Nat.mul_lt_mul_of_pos_right h | x y : Pos
h : y.toNat > x.toNat
z : Pos
⊢ y.toNat * z.toNat > x.toNat * z.toNat | x y : Pos
h : y.toNat > x.toNat
z : Pos
⊢ z.toNat > 0 | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Pos
h : y.toNat > x.toNat
z : Pos
⊢ y.toNat * z.toNat > x.toNat * z.toNat
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.mul_lt_mul_right | [358, 11] | [359, 80] | exact Nat.zero_lt _ | x y : Pos
h : y.toNat > x.toNat
z : Pos
⊢ z.toNat > 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Pos
h : y.toNat > x.toNat
z : Pos
⊢ z.toNat > 0
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.lt_of_mul_lt_mul_left | [361, 11] | [362, 44] | by_toNat | x y z : Pos
⊢ z * x < z * y → x < y | x y z : Pos
⊢ z.toNat * y.toNat > z.toNat * x.toNat → y.toNat > x.toNat | Please generate a tactic in lean4 to solve the state.
STATE:
x y z : Pos
⊢ z * x < z * y → x < y
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.lt_of_mul_lt_mul_left | [361, 11] | [362, 44] | exact Nat.lt_of_mul_lt_mul_left | x y z : Pos
⊢ z.toNat * y.toNat > z.toNat * x.toNat → y.toNat > x.toNat | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y z : Pos
⊢ z.toNat * y.toNat > z.toNat * x.toNat → y.toNat > x.toNat
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.lt_of_mul_lt_mul_right | [364, 11] | [365, 45] | by_toNat | x y z : Pos
⊢ x * z < y * z → x < y | x y z : Pos
⊢ y.toNat * z.toNat > x.toNat * z.toNat → y.toNat > x.toNat | Please generate a tactic in lean4 to solve the state.
STATE:
x y z : Pos
⊢ x * z < y * z → x < y
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.lt_of_mul_lt_mul_right | [364, 11] | [365, 45] | exact Nat.lt_of_mul_lt_mul_right | x y z : Pos
⊢ y.toNat * z.toNat > x.toNat * z.toNat → y.toNat > x.toNat | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y z : Pos
⊢ y.toNat * z.toNat > x.toNat * z.toNat → y.toNat > x.toNat
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.mul_lt_mul_of_lt_of_lt | [367, 11] | [368, 45] | by_toNat | x₁ y₁ x₂ y₂ : Pos
⊢ x₁ < y₁ → x₂ < y₂ → x₁ * x₂ < y₁ * y₂ | x₁ y₁ x₂ y₂ : Pos
⊢ y₁.toNat > x₁.toNat → y₂.toNat > x₂.toNat → y₁.toNat * y₂.toNat > x₁.toNat * x₂.toNat | Please generate a tactic in lean4 to solve the state.
STATE:
x₁ y₁ x₂ y₂ : Pos
⊢ x₁ < y₁ → x₂ < y₂ → x₁ * x₂ < y₁ * y₂
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.mul_lt_mul_of_lt_of_lt | [367, 11] | [368, 45] | exact Nat.mul_lt_mul_of_lt_of_lt | x₁ y₁ x₂ y₂ : Pos
⊢ y₁.toNat > x₁.toNat → y₂.toNat > x₂.toNat → y₁.toNat * y₂.toNat > x₁.toNat * x₂.toNat | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x₁ y₁ x₂ y₂ : Pos
⊢ y₁.toNat > x₁.toNat → y₂.toNat > x₂.toNat → y₁.toNat * y₂.toNat > x₁.toNat * x₂.toNat
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.pow_le_pow_left | [370, 11] | [371, 80] | by_toNat | x y : Nat
⊢ x ≤ y → ∀ (z : Pos), z ^ x ≤ z ^ y | x y : Nat
⊢ x ≤ y → ∀ (z : Pos), z.toNat ^ y ≥ z.toNat ^ x | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Nat
⊢ x ≤ y → ∀ (z : Pos), z ^ x ≤ z ^ y
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.pow_le_pow_left | [370, 11] | [371, 80] | intro h z | x y : Nat
⊢ x ≤ y → ∀ (z : Pos), z.toNat ^ y ≥ z.toNat ^ x | x y : Nat
h : x ≤ y
z : Pos
⊢ z.toNat ^ y ≥ z.toNat ^ x | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Nat
⊢ x ≤ y → ∀ (z : Pos), z.toNat ^ y ≥ z.toNat ^ x
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.pow_le_pow_left | [370, 11] | [371, 80] | exact Nat.pow_le_pow_of_pos_left h (Nat.zero_lt z.toNat) | x y : Nat
h : x ≤ y
z : Pos
⊢ z.toNat ^ y ≥ z.toNat ^ x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Nat
h : x ≤ y
z : Pos
⊢ z.toNat ^ y ≥ z.toNat ^ x
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.pow_le_pow_right | [373, 11] | [374, 38] | by_toNat | x y : Pos
⊢ x ≤ y → ∀ (z : Nat), x ^ z ≤ y ^ z | x y : Pos
⊢ y.toNat ≥ x.toNat → ∀ (z : Nat), y.toNat ^ z ≥ x.toNat ^ z | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Pos
⊢ x ≤ y → ∀ (z : Nat), x ^ z ≤ y ^ z
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.pow_le_pow_right | [373, 11] | [374, 38] | exact Nat.pow_le_pow_left | x y : Pos
⊢ y.toNat ≥ x.toNat → ∀ (z : Nat), y.toNat ^ z ≥ x.toNat ^ z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Pos
⊢ y.toNat ≥ x.toNat → ∀ (z : Nat), y.toNat ^ z ≥ x.toNat ^ z
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.pow_lt_pow_of_pos_left | [376, 11] | [377, 46] | by_toNat | x y : Pos
⊢ x < y → ∀ {z : Nat}, z > 0 → x ^ z < y ^ z | x y : Pos
⊢ y.toNat > x.toNat → ∀ {z : Nat}, z > 0 → y.toNat ^ z > x.toNat ^ z | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Pos
⊢ x < y → ∀ {z : Nat}, z > 0 → x ^ z < y ^ z
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.pow_lt_pow_of_pos_left | [376, 11] | [377, 46] | exact Nat.pow_lt_pow_of_pos_right | x y : Pos
⊢ y.toNat > x.toNat → ∀ {z : Nat}, z > 0 → y.toNat ^ z > x.toNat ^ z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Pos
⊢ y.toNat > x.toNat → ∀ {z : Nat}, z > 0 → y.toNat ^ z > x.toNat ^ z
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.lt.dest | [379, 11] | [395, 28] | intro h | x y : Pos
⊢ x < y → ∃ z, x + z = y | x y : Pos
h : x < y
⊢ ∃ z, x + z = y | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Pos
⊢ x < y → ∃ z, x + z = y
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.lt.dest | [379, 11] | [395, 28] | induction x, y with
| one_one =>
absurd h
exact Pos.not_lt_one 1
| one_succ y =>
exists y
exact Pos.one_add_eq_add_one y
| succ_one x =>
absurd h
exact Pos.not_lt_one (x+1)
| succ_succ x y ih =>
match ih (Pos.lt_of_succ_lt_succ h) with
| ⟨z, hz⟩ =>
exists z
rw [Pos.succ_add, hz] | x y : Pos
h : x < y
⊢ ∃ z, x + z = y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Pos
h : x < y
⊢ ∃ z, x + z = y
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.lt.dest | [379, 11] | [395, 28] | absurd h | case one_one
h : 1 < 1
⊢ ∃ z, 1 + z = 1 | case one_one
h : 1 < 1
⊢ ¬1 < 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case one_one
h : 1 < 1
⊢ ∃ z, 1 + z = 1
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.lt.dest | [379, 11] | [395, 28] | exact Pos.not_lt_one 1 | case one_one
h : 1 < 1
⊢ ¬1 < 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case one_one
h : 1 < 1
⊢ ¬1 < 1
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.lt.dest | [379, 11] | [395, 28] | exists y | case one_succ
y : Pos
a✝ : 1 < y → ∃ z, 1 + z = y
h : 1 < y + 1
⊢ ∃ z, 1 + z = y + 1 | case one_succ
y : Pos
a✝ : 1 < y → ∃ z, 1 + z = y
h : 1 < y + 1
⊢ 1 + y = y + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case one_succ
y : Pos
a✝ : 1 < y → ∃ z, 1 + z = y
h : 1 < y + 1
⊢ ∃ z, 1 + z = y + 1
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.lt.dest | [379, 11] | [395, 28] | exact Pos.one_add_eq_add_one y | case one_succ
y : Pos
a✝ : 1 < y → ∃ z, 1 + z = y
h : 1 < y + 1
⊢ 1 + y = y + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case one_succ
y : Pos
a✝ : 1 < y → ∃ z, 1 + z = y
h : 1 < y + 1
⊢ 1 + y = y + 1
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.lt.dest | [379, 11] | [395, 28] | absurd h | case succ_one
x : Pos
a✝ : x < 1 → ∃ z, x + z = 1
h : x + 1 < 1
⊢ ∃ z, x + 1 + z = 1 | case succ_one
x : Pos
a✝ : x < 1 → ∃ z, x + z = 1
h : x + 1 < 1
⊢ ¬x + 1 < 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case succ_one
x : Pos
a✝ : x < 1 → ∃ z, x + z = 1
h : x + 1 < 1
⊢ ∃ z, x + 1 + z = 1
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.lt.dest | [379, 11] | [395, 28] | exact Pos.not_lt_one (x+1) | case succ_one
x : Pos
a✝ : x < 1 → ∃ z, x + z = 1
h : x + 1 < 1
⊢ ¬x + 1 < 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ_one
x : Pos
a✝ : x < 1 → ∃ z, x + z = 1
h : x + 1 < 1
⊢ ¬x + 1 < 1
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.lt.dest | [379, 11] | [395, 28] | match ih (Pos.lt_of_succ_lt_succ h) with
| ⟨z, hz⟩ =>
exists z
rw [Pos.succ_add, hz] | case succ_succ
x y : Pos
ih : x < y → ∃ z, x + z = y
h : x + 1 < y + 1
⊢ ∃ z, x + 1 + z = y + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ_succ
x y : Pos
ih : x < y → ∃ z, x + z = y
h : x + 1 < y + 1
⊢ ∃ z, x + 1 + z = y + 1
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.lt.dest | [379, 11] | [395, 28] | exists z | x y : Pos
ih : x < y → ∃ z, x + z = y
h : x + 1 < y + 1
z : Pos
hz : x + z = y
⊢ ∃ z, x + 1 + z = y + 1 | x y : Pos
ih : x < y → ∃ z, x + z = y
h : x + 1 < y + 1
z : Pos
hz : x + z = y
⊢ x + 1 + z = y + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Pos
ih : x < y → ∃ z, x + z = y
h : x + 1 < y + 1
z : Pos
hz : x + z = y
⊢ ∃ z, x + 1 + z = y + 1
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.lt.dest | [379, 11] | [395, 28] | rw [Pos.succ_add, hz] | x y : Pos
ih : x < y → ∃ z, x + z = y
h : x + 1 < y + 1
z : Pos
hz : x + z = y
⊢ x + 1 + z = y + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Pos
ih : x < y → ∃ z, x + z = y
h : x + 1 < y + 1
z : Pos
hz : x + z = y
⊢ x + 1 + z = y + 1
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.lt.intro | [397, 11] | [398, 47] | intro h | x y z : Pos
⊢ x + z = y → x < y | x y z : Pos
h : x + z = y
⊢ x < y | Please generate a tactic in lean4 to solve the state.
STATE:
x y z : Pos
⊢ x + z = y → x < y
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.lt.intro | [397, 11] | [398, 47] | rw [←h] | x y z : Pos
h : x + z = y
⊢ x < y | x y z : Pos
h : x + z = y
⊢ x < x + z | Please generate a tactic in lean4 to solve the state.
STATE:
x y z : Pos
h : x + z = y
⊢ x < y
TACTIC:
|
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