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/digama0/lean4lean.git | c534f13d8d25f5e1891b6d18cc76b601ee87aa66 | Lean4Lean/Theory/VExpr.lean | Lean4Lean.VExpr.inst_liftN | [272, 1] | [277, 66] | rintro rfl | e2 : VExpr
i k : Nat
h : ¬i < k
⊢ i + 1 = k → i < k | e2 : VExpr
i : Nat
h : ¬i < i + 1
⊢ i < i + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
e2 : VExpr
i k : Nat
h : ¬i < k
⊢ i + 1 = k → i < k
TACTIC:
|
https://github.com/digama0/lean4lean.git | c534f13d8d25f5e1891b6d18cc76b601ee87aa66 | Lean4Lean/Theory/VExpr.lean | Lean4Lean.VExpr.inst_liftN | [272, 1] | [277, 66] | apply Nat.lt_succ_self | e2 : VExpr
i : Nat
h : ¬i < i + 1
⊢ i < i + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
e2 : VExpr
i : Nat
h : ¬i < i + 1
⊢ i < i + 1
TACTIC:
|
https://github.com/digama0/lean4lean.git | c534f13d8d25f5e1891b6d18cc76b601ee87aa66 | Lean4Lean/Theory/VExpr.lean | Lean4Lean.VExpr.inst_liftN' | [279, 1] | [280, 37] | rw [← liftN'_liftN_hi, inst_liftN] | n k : Nat
e1 e2 : VExpr
⊢ inst (liftN (n + 1) e1 k) e2 k = liftN n e1 k | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n k : Nat
e1 e2 : VExpr
⊢ inst (liftN (n + 1) e1 k) e2 k = liftN n e1 k
TACTIC:
|
https://github.com/digama0/lean4lean.git | c534f13d8d25f5e1891b6d18cc76b601ee87aa66 | Lean4Lean/Theory/VExpr.lean | Lean4Lean.VExpr.LevelWF.inst | [284, 11] | [287, 72] | induction e1 generalizing k <;> simp_all [inst, instVar, LevelWF] | e1 e2 : VExpr
k U : Nat
h1 : LevelWF U e1
h2 : LevelWF U e2
⊢ LevelWF U (inst e1 e2 k) | case bvar
e2 : VExpr
U : Nat
h2 : LevelWF U e2
deBruijnIndex✝ k : Nat
⊢ LevelWF U
(if deBruijnIndex✝ < k then bvar deBruijnIndex✝
else if deBruijnIndex✝ = k then VExpr.liftN k e2 else bvar (deBruijnIndex✝ - 1)) | Please generate a tactic in lean4 to solve the state.
STATE:
e1 e2 : VExpr
k U : Nat
h1 : LevelWF U e1
h2 : LevelWF U e2
⊢ LevelWF U (inst e1 e2 k)
TACTIC:
|
https://github.com/digama0/lean4lean.git | c534f13d8d25f5e1891b6d18cc76b601ee87aa66 | Lean4Lean/Theory/VExpr.lean | Lean4Lean.VExpr.LevelWF.inst | [284, 11] | [287, 72] | case bvar => split <;> [trivial; split <;> [exact h2.liftN; trivial]] | e2 : VExpr
U : Nat
h2 : LevelWF U e2
deBruijnIndex✝ k : Nat
⊢ LevelWF U
(if deBruijnIndex✝ < k then bvar deBruijnIndex✝
else if deBruijnIndex✝ = k then VExpr.liftN k e2 else bvar (deBruijnIndex✝ - 1)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
e2 : VExpr
U : Nat
h2 : LevelWF U e2
deBruijnIndex✝ k : Nat
⊢ LevelWF U
(if deBruijnIndex✝ < k then bvar deBruijnIndex✝
else if deBruijnIndex✝ = k then VExpr.liftN k e2 else bvar (deBruijnIndex✝ - 1))
TACTIC:
|
https://github.com/digama0/lean4lean.git | c534f13d8d25f5e1891b6d18cc76b601ee87aa66 | Lean4Lean/Theory/VExpr.lean | Lean4Lean.VExpr.LevelWF.inst | [284, 11] | [287, 72] | split <;> [trivial; split <;> [exact h2.liftN; trivial]] | e2 : VExpr
U : Nat
h2 : LevelWF U e2
deBruijnIndex✝ k : Nat
⊢ LevelWF U
(if deBruijnIndex✝ < k then bvar deBruijnIndex✝
else if deBruijnIndex✝ = k then VExpr.liftN k e2 else bvar (deBruijnIndex✝ - 1)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
e2 : VExpr
U : Nat
h2 : LevelWF U e2
deBruijnIndex✝ k : Nat
⊢ LevelWF U
(if deBruijnIndex✝ < k then bvar deBruijnIndex✝
else if deBruijnIndex✝ = k then VExpr.liftN k e2 else bvar (deBruijnIndex✝ - 1))
TACTIC:
|
https://github.com/digama0/lean4lean.git | c534f13d8d25f5e1891b6d18cc76b601ee87aa66 | Lean4Lean/Theory/VExpr.lean | Lean4Lean.VExpr.unliftN_liftN | [294, 9] | [295, 49] | induction n <;> simp [unliftN, inst_liftN', *] | n : Nat
e : VExpr
k : Nat
⊢ unliftN (liftN n e k) n k = e | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : Nat
e : VExpr
k : Nat
⊢ unliftN (liftN n e k) n k = e
TACTIC:
|
https://github.com/digama0/lean4lean.git | c534f13d8d25f5e1891b6d18cc76b601ee87aa66 | Lean4Lean/Theory/VExpr.lean | Lean4Lean.VExpr.unliftN_add | [297, 1] | [298, 66] | induction n1 generalizing e <;> simp [unliftN, Nat.succ_add, *] | e : VExpr
n1 n2 k : Nat
⊢ unliftN e (n1 + n2) k = unliftN (unliftN e n1 k) n2 k | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
e : VExpr
n1 n2 k : Nat
⊢ unliftN e (n1 + n2) k = unliftN (unliftN e n1 k) n2 k
TACTIC:
|
https://github.com/digama0/lean4lean.git | c534f13d8d25f5e1891b6d18cc76b601ee87aa66 | Lean4Lean/Theory/VExpr.lean | Lean4Lean.VExpr.unliftN_succ' | [300, 1] | [301, 24] | rw [unliftN_add] | e : VExpr
n k : Nat
⊢ unliftN e (n + 1) k = inst (unliftN e n k) default k | e : VExpr
n k : Nat
⊢ unliftN (unliftN e n k) 1 k = inst (unliftN e n k) default k | Please generate a tactic in lean4 to solve the state.
STATE:
e : VExpr
n k : Nat
⊢ unliftN e (n + 1) k = inst (unliftN e n k) default k
TACTIC:
|
https://github.com/digama0/lean4lean.git | c534f13d8d25f5e1891b6d18cc76b601ee87aa66 | Lean4Lean/Theory/VExpr.lean | Lean4Lean.VExpr.unliftN_succ' | [300, 1] | [301, 24] | rfl | e : VExpr
n k : Nat
⊢ unliftN (unliftN e n k) 1 k = inst (unliftN e n k) default k | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
e : VExpr
n k : Nat
⊢ unliftN (unliftN e n k) 1 k = inst (unliftN e n k) default k
TACTIC:
|
https://github.com/digama0/lean4lean.git | c534f13d8d25f5e1891b6d18cc76b601ee87aa66 | Lean4Lean/Theory/VExpr.lean | Lean4Lean.VExpr.liftN_unliftN_hi | [303, 1] | [309, 34] | obtain ⟨k1, rfl⟩ := Nat.le_iff_exists_add'.1 h | k2 k1 n1 : Nat
e : VExpr
n2 : Nat
h : k2 ≤ k1
⊢ liftN n1 (unliftN e n2 k2) k1 = unliftN (liftN n1 e (k1 + n2)) n2 k2 | case intro
k2 n1 : Nat
e : VExpr
n2 k1 : Nat
h : k2 ≤ k1 + k2
⊢ liftN n1 (unliftN e n2 k2) (k1 + k2) = unliftN (liftN n1 e (k1 + k2 + n2)) n2 k2 | Please generate a tactic in lean4 to solve the state.
STATE:
k2 k1 n1 : Nat
e : VExpr
n2 : Nat
h : k2 ≤ k1
⊢ liftN n1 (unliftN e n2 k2) k1 = unliftN (liftN n1 e (k1 + n2)) n2 k2
TACTIC:
|
https://github.com/digama0/lean4lean.git | c534f13d8d25f5e1891b6d18cc76b601ee87aa66 | Lean4Lean/Theory/VExpr.lean | Lean4Lean.VExpr.liftN_unliftN_hi | [303, 1] | [309, 34] | induction n2 generalizing e with simp [unliftN]
| succ n2 ih =>
rw [ih, Nat.add_right_comm, liftN_instN_hi e default n1 (k1+n2) k2,
Nat.add_right_comm k1]; rfl | case intro
k2 n1 : Nat
e : VExpr
n2 k1 : Nat
h : k2 ≤ k1 + k2
⊢ liftN n1 (unliftN e n2 k2) (k1 + k2) = unliftN (liftN n1 e (k1 + k2 + n2)) n2 k2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
k2 n1 : Nat
e : VExpr
n2 k1 : Nat
h : k2 ≤ k1 + k2
⊢ liftN n1 (unliftN e n2 k2) (k1 + k2) = unliftN (liftN n1 e (k1 + k2 + n2)) n2 k2
TACTIC:
|
https://github.com/digama0/lean4lean.git | c534f13d8d25f5e1891b6d18cc76b601ee87aa66 | Lean4Lean/Theory/VExpr.lean | Lean4Lean.VExpr.liftN_unliftN_hi | [303, 1] | [309, 34] | rw [ih, Nat.add_right_comm, liftN_instN_hi e default n1 (k1+n2) k2,
Nat.add_right_comm k1] | case intro.succ
k2 n1 k1 : Nat
h : k2 ≤ k1 + k2
n2 : Nat
ih : ∀ {e : VExpr}, liftN n1 (unliftN e n2 k2) (k1 + k2) = unliftN (liftN n1 e (k1 + k2 + n2)) n2 k2
e : VExpr
⊢ liftN n1 (unliftN (inst e default k2) n2 k2) (k1 + k2) =
unliftN (inst (liftN n1 e (k1 + k2 + Nat.succ n2)) default k2) n2 k2 | case intro.succ
k2 n1 k1 : Nat
h : k2 ≤ k1 + k2
n2 : Nat
ih : ∀ {e : VExpr}, liftN n1 (unliftN e n2 k2) (k1 + k2) = unliftN (liftN n1 e (k1 + k2 + n2)) n2 k2
e : VExpr
⊢ unliftN (inst (liftN n1 e (k1 + k2 + n2 + 1)) (liftN n1 default (k1 + n2)) k2) n2 k2 =
unliftN (inst (liftN n1 e (k1 + k2 + Nat.succ n2)) default k2) n2 k2 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.succ
k2 n1 k1 : Nat
h : k2 ≤ k1 + k2
n2 : Nat
ih : ∀ {e : VExpr}, liftN n1 (unliftN e n2 k2) (k1 + k2) = unliftN (liftN n1 e (k1 + k2 + n2)) n2 k2
e : VExpr
⊢ liftN n1 (unliftN (inst e default k2) n2 k2) (k1 + k2) =
unliftN (inst (liftN n1 e (k1 + k2 + Nat.succ n2)) default k2) n2 k2
TACTIC:
|
https://github.com/digama0/lean4lean.git | c534f13d8d25f5e1891b6d18cc76b601ee87aa66 | Lean4Lean/Theory/VExpr.lean | Lean4Lean.VExpr.liftN_unliftN_hi | [303, 1] | [309, 34] | rfl | case intro.succ
k2 n1 k1 : Nat
h : k2 ≤ k1 + k2
n2 : Nat
ih : ∀ {e : VExpr}, liftN n1 (unliftN e n2 k2) (k1 + k2) = unliftN (liftN n1 e (k1 + k2 + n2)) n2 k2
e : VExpr
⊢ unliftN (inst (liftN n1 e (k1 + k2 + n2 + 1)) (liftN n1 default (k1 + n2)) k2) n2 k2 =
unliftN (inst (liftN n1 e (k1 + k2 + Nat.succ n2)) default k2) n2 k2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.succ
k2 n1 k1 : Nat
h : k2 ≤ k1 + k2
n2 : Nat
ih : ∀ {e : VExpr}, liftN n1 (unliftN e n2 k2) (k1 + k2) = unliftN (liftN n1 e (k1 + k2 + n2)) n2 k2
e : VExpr
⊢ unliftN (inst (liftN n1 e (k1 + k2 + n2 + 1)) (liftN n1 default (k1 + n2)) k2) n2 k2 =
unliftN (inst (liftN n1 e (k1 + k2 + Nat.succ n2)) default k2) n2 k2
TACTIC:
|
https://github.com/digama0/lean4lean.git | c534f13d8d25f5e1891b6d18cc76b601ee87aa66 | Lean4Lean/Theory/VExpr.lean | Lean4Lean.VExpr.Skips.liftN | [313, 11] | [313, 75] | simp [Skips] | n : Nat
e : VExpr
k : Nat
⊢ Skips (liftN n e k) n k | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : Nat
e : VExpr
k : Nat
⊢ Skips (liftN n e k) n k
TACTIC:
|
https://github.com/digama0/lean4lean.git | c534f13d8d25f5e1891b6d18cc76b601ee87aa66 | Lean4Lean/Theory/VExpr.lean | Lean4Lean.VExpr.Skips.zero | [318, 1] | [318, 61] | simp [Skips, unliftN] | e : VExpr
k : Nat
⊢ Skips e 0 k | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
e : VExpr
k : Nat
⊢ Skips e 0 k
TACTIC:
|
https://github.com/digama0/lean4lean.git | c534f13d8d25f5e1891b6d18cc76b601ee87aa66 | Lean4Lean/Theory/VExpr.lean | Lean4Lean.VExpr.liftN_inj | [320, 1] | [321, 76] | rw [← unliftN_liftN (e := e1), H, unliftN_liftN] | n : Nat
e1 : VExpr
k : Nat
e2 : VExpr
H : liftN n e1 k = liftN n e2 k
⊢ e1 = e2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : Nat
e1 : VExpr
k : Nat
e2 : VExpr
H : liftN n e1 k = liftN n e2 k
⊢ e1 = e2
TACTIC:
|
https://github.com/digama0/lean4lean.git | c534f13d8d25f5e1891b6d18cc76b601ee87aa66 | Lean4Lean/Theory/VExpr.lean | Lean4Lean.VExpr.liftVar_inj | [323, 1] | [324, 58] | simpa [liftN] using @liftN_inj n (.bvar i) k (.bvar i') | n i k i' : Nat
⊢ liftVar n i k = liftVar n i' k ↔ i = i' | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n i k i' : Nat
⊢ liftVar n i k = liftVar n i' k ↔ i = i'
TACTIC:
|
https://github.com/digama0/lean4lean.git | c534f13d8d25f5e1891b6d18cc76b601ee87aa66 | Lean4Lean/Theory/VExpr.lean | Lean4Lean.VExpr.Skips.of_liftN_hi | [326, 1] | [330, 76] | obtain ⟨k1, rfl⟩ := Nat.le_iff_exists_add'.1 h | n1 : Nat
e : VExpr
k1 n2 k2 : Nat
self : Skips (liftN n1 e k1) n2 k2
h : n2 + k2 ≤ k1
⊢ Skips e n2 k2 | case intro
n1 : Nat
e : VExpr
n2 k2 k1 : Nat
self : Skips (liftN n1 e (k1 + (n2 + k2))) n2 k2
h : n2 + k2 ≤ k1 + (n2 + k2)
⊢ Skips e n2 k2 | Please generate a tactic in lean4 to solve the state.
STATE:
n1 : Nat
e : VExpr
k1 n2 k2 : Nat
self : Skips (liftN n1 e k1) n2 k2
h : n2 + k2 ≤ k1
⊢ Skips e n2 k2
TACTIC:
|
https://github.com/digama0/lean4lean.git | c534f13d8d25f5e1891b6d18cc76b601ee87aa66 | Lean4Lean/Theory/VExpr.lean | Lean4Lean.VExpr.Skips.of_liftN_hi | [326, 1] | [330, 76] | rwa [Skips, Nat.add_comm n2, ← Nat.add_assoc, ← liftN_unliftN_hi (Nat.le_add_left ..),
liftN'_comm (h := Nat.le_add_left ..), Nat.add_comm, liftN_inj] at self | case intro
n1 : Nat
e : VExpr
n2 k2 k1 : Nat
self : Skips (liftN n1 e (k1 + (n2 + k2))) n2 k2
h : n2 + k2 ≤ k1 + (n2 + k2)
⊢ Skips e n2 k2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
n1 : Nat
e : VExpr
n2 k2 k1 : Nat
self : Skips (liftN n1 e (k1 + (n2 + k2))) n2 k2
h : n2 + k2 ≤ k1 + (n2 + k2)
⊢ Skips e n2 k2
TACTIC:
|
https://github.com/digama0/lean4lean.git | c534f13d8d25f5e1891b6d18cc76b601ee87aa66 | Lean4Lean/Theory/VExpr.lean | Lean4Lean.VExpr.skips_add | [332, 1] | [334, 73] | simp [skips_iff_exists, ← liftN'_liftN_hi] | e : VExpr
n1 n2 k : Nat
⊢ Skips e (n1 + n2) k ↔ ∃ e', Skips e' n1 k ∧ e = liftN n2 e' k | e : VExpr
n1 n2 k : Nat
⊢ (∃ e', e = liftN n2 (liftN n1 e' k) k) ↔ ∃ e', (∃ e'_1, e' = liftN n1 e'_1 k) ∧ e = liftN n2 e' k | Please generate a tactic in lean4 to solve the state.
STATE:
e : VExpr
n1 n2 k : Nat
⊢ Skips e (n1 + n2) k ↔ ∃ e', Skips e' n1 k ∧ e = liftN n2 e' k
TACTIC:
|
https://github.com/digama0/lean4lean.git | c534f13d8d25f5e1891b6d18cc76b601ee87aa66 | Lean4Lean/Theory/VExpr.lean | Lean4Lean.VExpr.skips_add | [332, 1] | [334, 73] | exact ⟨fun ⟨_, h⟩ => ⟨_, ⟨_, rfl⟩, h⟩, fun ⟨_, ⟨_, rfl⟩, h⟩ => ⟨_, h⟩⟩ | e : VExpr
n1 n2 k : Nat
⊢ (∃ e', e = liftN n2 (liftN n1 e' k) k) ↔ ∃ e', (∃ e'_1, e' = liftN n1 e'_1 k) ∧ e = liftN n2 e' k | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
e : VExpr
n1 n2 k : Nat
⊢ (∃ e', e = liftN n2 (liftN n1 e' k) k) ↔ ∃ e', (∃ e'_1, e' = liftN n1 e'_1 k) ∧ e = liftN n2 e' k
TACTIC:
|
https://github.com/digama0/lean4lean.git | c534f13d8d25f5e1891b6d18cc76b601ee87aa66 | Lean4Lean/Theory/VExpr.lean | Lean4Lean.VExpr.skips_iff | [343, 1] | [381, 78] | simp [Skips, unliftN] | case zero
k : Nat
e : VExpr
⊢ Skips e Nat.zero k ↔ Skips' Nat.zero e k | case zero
k : Nat
e : VExpr
⊢ Skips' 0 e k | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
k : Nat
e : VExpr
⊢ Skips e Nat.zero k ↔ Skips' Nat.zero e k
TACTIC:
|
https://github.com/digama0/lean4lean.git | c534f13d8d25f5e1891b6d18cc76b601ee87aa66 | Lean4Lean/Theory/VExpr.lean | Lean4Lean.VExpr.skips_iff | [343, 1] | [381, 78] | induction e generalizing k <;> simp [Skips', *] | case zero
k : Nat
e : VExpr
⊢ Skips' 0 e k | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
k : Nat
e : VExpr
⊢ Skips' 0 e k
TACTIC:
|
https://github.com/digama0/lean4lean.git | c534f13d8d25f5e1891b6d18cc76b601ee87aa66 | Lean4Lean/Theory/VExpr.lean | Lean4Lean.VExpr.skips_iff | [343, 1] | [381, 78] | simp [Nat.succ_eq_add_one, skips_add, ih] | case succ
k n : Nat
ih : ∀ {e : VExpr}, Skips e n k ↔ Skips' n e k
e : VExpr
⊢ Skips e (Nat.succ n) k ↔ Skips' (Nat.succ n) e k | case succ
k n : Nat
ih : ∀ {e : VExpr}, Skips e n k ↔ Skips' n e k
e : VExpr
⊢ (∃ e', Skips' n e' k ∧ e = liftN 1 e' k) ↔ Skips' (n + 1) e k | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
k n : Nat
ih : ∀ {e : VExpr}, Skips e n k ↔ Skips' n e k
e : VExpr
⊢ Skips e (Nat.succ n) k ↔ Skips' (Nat.succ n) e k
TACTIC:
|
https://github.com/digama0/lean4lean.git | c534f13d8d25f5e1891b6d18cc76b601ee87aa66 | Lean4Lean/Theory/VExpr.lean | Lean4Lean.VExpr.skips_iff | [343, 1] | [381, 78] | clear ih | case succ
k n : Nat
ih : ∀ {e : VExpr}, Skips e n k ↔ Skips' n e k
e : VExpr
⊢ (∃ e', Skips' n e' k ∧ e = liftN 1 e' k) ↔ Skips' (n + 1) e k | case succ
k n : Nat
e : VExpr
⊢ (∃ e', Skips' n e' k ∧ e = liftN 1 e' k) ↔ Skips' (n + 1) e k | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
k n : Nat
ih : ∀ {e : VExpr}, Skips e n k ↔ Skips' n e k
e : VExpr
⊢ (∃ e', Skips' n e' k ∧ e = liftN 1 e' k) ↔ Skips' (n + 1) e k
TACTIC:
|
https://github.com/digama0/lean4lean.git | c534f13d8d25f5e1891b6d18cc76b601ee87aa66 | Lean4Lean/Theory/VExpr.lean | Lean4Lean.VExpr.skips_iff | [343, 1] | [381, 78] | refine ⟨fun ⟨e', h1, h2⟩ => ?_, fun h => ?_⟩ | case succ.bvar
n i k : Nat
⊢ (∃ e', Skips' n e' k ∧ bvar i = liftN 1 e' k) ↔ Skips' (n + 1) (bvar i) k | case succ.bvar.refine_1
n i k : Nat
x✝ : ∃ e', Skips' n e' k ∧ bvar i = liftN 1 e' k
e' : VExpr
h1 : Skips' n e' k
h2 : bvar i = liftN 1 e' k
⊢ Skips' (n + 1) (bvar i) k
case succ.bvar.refine_2
n i k : Nat
h : Skips' (n + 1) (bvar i) k
⊢ ∃ e', Skips' n e' k ∧ bvar i = liftN 1 e' k | Please generate a tactic in lean4 to solve the state.
STATE:
case succ.bvar
n i k : Nat
⊢ (∃ e', Skips' n e' k ∧ bvar i = liftN 1 e' k) ↔ Skips' (n + 1) (bvar i) k
TACTIC:
|
https://github.com/digama0/lean4lean.git | c534f13d8d25f5e1891b6d18cc76b601ee87aa66 | Lean4Lean/Theory/VExpr.lean | Lean4Lean.VExpr.skips_iff | [343, 1] | [381, 78] | cases e' <;> cases h2 | case succ.bvar.refine_1
n i k : Nat
x✝ : ∃ e', Skips' n e' k ∧ bvar i = liftN 1 e' k
e' : VExpr
h1 : Skips' n e' k
h2 : bvar i = liftN 1 e' k
⊢ Skips' (n + 1) (bvar i) k | case succ.bvar.refine_1.bvar.refl
n k deBruijnIndex✝ : Nat
h1 : Skips' n (bvar deBruijnIndex✝) k
x✝ : ∃ e', Skips' n e' k ∧ bvar (liftVar 1 deBruijnIndex✝ k) = liftN 1 e' k
⊢ Skips' (n + 1) (bvar (liftVar 1 deBruijnIndex✝ k)) k | Please generate a tactic in lean4 to solve the state.
STATE:
case succ.bvar.refine_1
n i k : Nat
x✝ : ∃ e', Skips' n e' k ∧ bvar i = liftN 1 e' k
e' : VExpr
h1 : Skips' n e' k
h2 : bvar i = liftN 1 e' k
⊢ Skips' (n + 1) (bvar i) k
TACTIC:
|
https://github.com/digama0/lean4lean.git | c534f13d8d25f5e1891b6d18cc76b601ee87aa66 | Lean4Lean/Theory/VExpr.lean | Lean4Lean.VExpr.skips_iff | [343, 1] | [381, 78] | simp [Skips', liftVar] | case succ.bvar.refine_1.bvar.refl
n k deBruijnIndex✝ : Nat
h1 : Skips' n (bvar deBruijnIndex✝) k
x✝ : ∃ e', Skips' n e' k ∧ bvar (liftVar 1 deBruijnIndex✝ k) = liftN 1 e' k
⊢ Skips' (n + 1) (bvar (liftVar 1 deBruijnIndex✝ k)) k | case succ.bvar.refine_1.bvar.refl
n k deBruijnIndex✝ : Nat
h1 : Skips' n (bvar deBruijnIndex✝) k
x✝ : ∃ e', Skips' n e' k ∧ bvar (liftVar 1 deBruijnIndex✝ k) = liftN 1 e' k
⊢ (if deBruijnIndex✝ < k then deBruijnIndex✝ else 1 + deBruijnIndex✝) < k + (n + 1) →
(if deBruijnIndex✝ < k then deBruijnIndex✝ else 1 + deBruijnIndex✝) < k | Please generate a tactic in lean4 to solve the state.
STATE:
case succ.bvar.refine_1.bvar.refl
n k deBruijnIndex✝ : Nat
h1 : Skips' n (bvar deBruijnIndex✝) k
x✝ : ∃ e', Skips' n e' k ∧ bvar (liftVar 1 deBruijnIndex✝ k) = liftN 1 e' k
⊢ Skips' (n + 1) (bvar (liftVar 1 deBruijnIndex✝ k)) k
TACTIC:
|
https://github.com/digama0/lean4lean.git | c534f13d8d25f5e1891b6d18cc76b601ee87aa66 | Lean4Lean/Theory/VExpr.lean | Lean4Lean.VExpr.skips_iff | [343, 1] | [381, 78] | split | case succ.bvar.refine_1.bvar.refl
n k deBruijnIndex✝ : Nat
h1 : Skips' n (bvar deBruijnIndex✝) k
x✝ : ∃ e', Skips' n e' k ∧ bvar (liftVar 1 deBruijnIndex✝ k) = liftN 1 e' k
⊢ (if deBruijnIndex✝ < k then deBruijnIndex✝ else 1 + deBruijnIndex✝) < k + (n + 1) →
(if deBruijnIndex✝ < k then deBruijnIndex✝ else 1 + deBruijnIndex✝) < k | case succ.bvar.refine_1.bvar.refl.inl
n k deBruijnIndex✝ : Nat
h1 : Skips' n (bvar deBruijnIndex✝) k
x✝ : ∃ e', Skips' n e' k ∧ bvar (liftVar 1 deBruijnIndex✝ k) = liftN 1 e' k
h✝ : deBruijnIndex✝ < k
⊢ deBruijnIndex✝ < k + (n + 1) → deBruijnIndex✝ < k
case succ.bvar.refine_1.bvar.refl.inr
n k deBruijnIndex✝ : Nat
h1 : Skips' n (bvar deBruijnIndex✝) k
x✝ : ∃ e', Skips' n e' k ∧ bvar (liftVar 1 deBruijnIndex✝ k) = liftN 1 e' k
h✝ : ¬deBruijnIndex✝ < k
⊢ 1 + deBruijnIndex✝ < k + (n + 1) → 1 + deBruijnIndex✝ < k | Please generate a tactic in lean4 to solve the state.
STATE:
case succ.bvar.refine_1.bvar.refl
n k deBruijnIndex✝ : Nat
h1 : Skips' n (bvar deBruijnIndex✝) k
x✝ : ∃ e', Skips' n e' k ∧ bvar (liftVar 1 deBruijnIndex✝ k) = liftN 1 e' k
⊢ (if deBruijnIndex✝ < k then deBruijnIndex✝ else 1 + deBruijnIndex✝) < k + (n + 1) →
(if deBruijnIndex✝ < k then deBruijnIndex✝ else 1 + deBruijnIndex✝) < k
TACTIC:
|
https://github.com/digama0/lean4lean.git | c534f13d8d25f5e1891b6d18cc76b601ee87aa66 | Lean4Lean/Theory/VExpr.lean | Lean4Lean.VExpr.skips_iff | [343, 1] | [381, 78] | intro | case succ.bvar.refine_1.bvar.refl.inl
n k deBruijnIndex✝ : Nat
h1 : Skips' n (bvar deBruijnIndex✝) k
x✝ : ∃ e', Skips' n e' k ∧ bvar (liftVar 1 deBruijnIndex✝ k) = liftN 1 e' k
h✝ : deBruijnIndex✝ < k
⊢ deBruijnIndex✝ < k + (n + 1) → deBruijnIndex✝ < k | case succ.bvar.refine_1.bvar.refl.inl
n k deBruijnIndex✝ : Nat
h1 : Skips' n (bvar deBruijnIndex✝) k
x✝ : ∃ e', Skips' n e' k ∧ bvar (liftVar 1 deBruijnIndex✝ k) = liftN 1 e' k
h✝ : deBruijnIndex✝ < k
a✝ : deBruijnIndex✝ < k + (n + 1)
⊢ deBruijnIndex✝ < k | Please generate a tactic in lean4 to solve the state.
STATE:
case succ.bvar.refine_1.bvar.refl.inl
n k deBruijnIndex✝ : Nat
h1 : Skips' n (bvar deBruijnIndex✝) k
x✝ : ∃ e', Skips' n e' k ∧ bvar (liftVar 1 deBruijnIndex✝ k) = liftN 1 e' k
h✝ : deBruijnIndex✝ < k
⊢ deBruijnIndex✝ < k + (n + 1) → deBruijnIndex✝ < k
TACTIC:
|
https://github.com/digama0/lean4lean.git | c534f13d8d25f5e1891b6d18cc76b601ee87aa66 | Lean4Lean/Theory/VExpr.lean | Lean4Lean.VExpr.skips_iff | [343, 1] | [381, 78] | assumption | case succ.bvar.refine_1.bvar.refl.inl
n k deBruijnIndex✝ : Nat
h1 : Skips' n (bvar deBruijnIndex✝) k
x✝ : ∃ e', Skips' n e' k ∧ bvar (liftVar 1 deBruijnIndex✝ k) = liftN 1 e' k
h✝ : deBruijnIndex✝ < k
a✝ : deBruijnIndex✝ < k + (n + 1)
⊢ deBruijnIndex✝ < k | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ.bvar.refine_1.bvar.refl.inl
n k deBruijnIndex✝ : Nat
h1 : Skips' n (bvar deBruijnIndex✝) k
x✝ : ∃ e', Skips' n e' k ∧ bvar (liftVar 1 deBruijnIndex✝ k) = liftN 1 e' k
h✝ : deBruijnIndex✝ < k
a✝ : deBruijnIndex✝ < k + (n + 1)
⊢ deBruijnIndex✝ < k
TACTIC:
|
https://github.com/digama0/lean4lean.git | c534f13d8d25f5e1891b6d18cc76b601ee87aa66 | Lean4Lean/Theory/VExpr.lean | Lean4Lean.VExpr.skips_iff | [343, 1] | [381, 78] | next h2 =>
rw [Nat.add_comm, ← Nat.add_assoc, Nat.succ_lt_succ_iff]
exact fun h => h2.elim (h1 h) | case succ.bvar.refine_1.bvar.refl.inr
n k deBruijnIndex✝ : Nat
h1 : Skips' n (bvar deBruijnIndex✝) k
x✝ : ∃ e', Skips' n e' k ∧ bvar (liftVar 1 deBruijnIndex✝ k) = liftN 1 e' k
h✝ : ¬deBruijnIndex✝ < k
⊢ 1 + deBruijnIndex✝ < k + (n + 1) → 1 + deBruijnIndex✝ < k | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ.bvar.refine_1.bvar.refl.inr
n k deBruijnIndex✝ : Nat
h1 : Skips' n (bvar deBruijnIndex✝) k
x✝ : ∃ e', Skips' n e' k ∧ bvar (liftVar 1 deBruijnIndex✝ k) = liftN 1 e' k
h✝ : ¬deBruijnIndex✝ < k
⊢ 1 + deBruijnIndex✝ < k + (n + 1) → 1 + deBruijnIndex✝ < k
TACTIC:
|
https://github.com/digama0/lean4lean.git | c534f13d8d25f5e1891b6d18cc76b601ee87aa66 | Lean4Lean/Theory/VExpr.lean | Lean4Lean.VExpr.skips_iff | [343, 1] | [381, 78] | rw [Nat.add_comm, ← Nat.add_assoc, Nat.succ_lt_succ_iff] | n k deBruijnIndex✝ : Nat
h1 : Skips' n (bvar deBruijnIndex✝) k
x✝ : ∃ e', Skips' n e' k ∧ bvar (liftVar 1 deBruijnIndex✝ k) = liftN 1 e' k
h2 : ¬deBruijnIndex✝ < k
⊢ 1 + deBruijnIndex✝ < k + (n + 1) → 1 + deBruijnIndex✝ < k | n k deBruijnIndex✝ : Nat
h1 : Skips' n (bvar deBruijnIndex✝) k
x✝ : ∃ e', Skips' n e' k ∧ bvar (liftVar 1 deBruijnIndex✝ k) = liftN 1 e' k
h2 : ¬deBruijnIndex✝ < k
⊢ deBruijnIndex✝ < k + n → deBruijnIndex✝ + 1 < k | Please generate a tactic in lean4 to solve the state.
STATE:
n k deBruijnIndex✝ : Nat
h1 : Skips' n (bvar deBruijnIndex✝) k
x✝ : ∃ e', Skips' n e' k ∧ bvar (liftVar 1 deBruijnIndex✝ k) = liftN 1 e' k
h2 : ¬deBruijnIndex✝ < k
⊢ 1 + deBruijnIndex✝ < k + (n + 1) → 1 + deBruijnIndex✝ < k
TACTIC:
|
https://github.com/digama0/lean4lean.git | c534f13d8d25f5e1891b6d18cc76b601ee87aa66 | Lean4Lean/Theory/VExpr.lean | Lean4Lean.VExpr.skips_iff | [343, 1] | [381, 78] | exact fun h => h2.elim (h1 h) | n k deBruijnIndex✝ : Nat
h1 : Skips' n (bvar deBruijnIndex✝) k
x✝ : ∃ e', Skips' n e' k ∧ bvar (liftVar 1 deBruijnIndex✝ k) = liftN 1 e' k
h2 : ¬deBruijnIndex✝ < k
⊢ deBruijnIndex✝ < k + n → deBruijnIndex✝ + 1 < k | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n k deBruijnIndex✝ : Nat
h1 : Skips' n (bvar deBruijnIndex✝) k
x✝ : ∃ e', Skips' n e' k ∧ bvar (liftVar 1 deBruijnIndex✝ k) = liftN 1 e' k
h2 : ¬deBruijnIndex✝ < k
⊢ deBruijnIndex✝ < k + n → deBruijnIndex✝ + 1 < k
TACTIC:
|
https://github.com/digama0/lean4lean.git | c534f13d8d25f5e1891b6d18cc76b601ee87aa66 | Lean4Lean/Theory/VExpr.lean | Lean4Lean.VExpr.skips_iff | [343, 1] | [381, 78] | simp [Skips'] at h | case succ.bvar.refine_2
n i k : Nat
h : Skips' (n + 1) (bvar i) k
⊢ ∃ e', Skips' n e' k ∧ bvar i = liftN 1 e' k | case succ.bvar.refine_2
n i k : Nat
h : i < k + (n + 1) → i < k
⊢ ∃ e', Skips' n e' k ∧ bvar i = liftN 1 e' k | Please generate a tactic in lean4 to solve the state.
STATE:
case succ.bvar.refine_2
n i k : Nat
h : Skips' (n + 1) (bvar i) k
⊢ ∃ e', Skips' n e' k ∧ bvar i = liftN 1 e' k
TACTIC:
|
https://github.com/digama0/lean4lean.git | c534f13d8d25f5e1891b6d18cc76b601ee87aa66 | Lean4Lean/Theory/VExpr.lean | Lean4Lean.VExpr.skips_iff | [343, 1] | [381, 78] | exact ⟨.bvar i, fun _ => h h', by simp [liftN, liftVar, h h']⟩ | n i k : Nat
h : i < k + (n + 1) → i < k
h' : i < k + n + 1
⊢ ∃ e', Skips' n e' k ∧ bvar i = liftN 1 e' k | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n i k : Nat
h : i < k + (n + 1) → i < k
h' : i < k + n + 1
⊢ ∃ e', Skips' n e' k ∧ bvar i = liftN 1 e' k
TACTIC:
|
https://github.com/digama0/lean4lean.git | c534f13d8d25f5e1891b6d18cc76b601ee87aa66 | Lean4Lean/Theory/VExpr.lean | Lean4Lean.VExpr.skips_iff | [343, 1] | [381, 78] | simp [liftN, liftVar, h h'] | n i k : Nat
h : i < k + (n + 1) → i < k
h' : i < k + n + 1
⊢ bvar i = liftN 1 (bvar i) k | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n i k : Nat
h : i < k + (n + 1) → i < k
h' : i < k + n + 1
⊢ bvar i = liftN 1 (bvar i) k
TACTIC:
|
https://github.com/digama0/lean4lean.git | c534f13d8d25f5e1891b6d18cc76b601ee87aa66 | Lean4Lean/Theory/VExpr.lean | Lean4Lean.VExpr.skips_iff | [343, 1] | [381, 78] | have := Nat.not_lt.1 h' | n i k : Nat
h : i < k + (n + 1) → i < k
h' : ¬i < k + n + 1
⊢ ∃ e', Skips' n e' k ∧ bvar i = liftN 1 e' k | n i k : Nat
h : i < k + (n + 1) → i < k
h' : ¬i < k + n + 1
this : k + n + 1 ≤ i
⊢ ∃ e', Skips' n e' k ∧ bvar i = liftN 1 e' k | Please generate a tactic in lean4 to solve the state.
STATE:
n i k : Nat
h : i < k + (n + 1) → i < k
h' : ¬i < k + n + 1
⊢ ∃ e', Skips' n e' k ∧ bvar i = liftN 1 e' k
TACTIC:
|
https://github.com/digama0/lean4lean.git | c534f13d8d25f5e1891b6d18cc76b601ee87aa66 | Lean4Lean/Theory/VExpr.lean | Lean4Lean.VExpr.skips_iff | [343, 1] | [381, 78] | let i+1 := i | n i k : Nat
h : i < k + (n + 1) → i < k
h' : ¬i < k + n + 1
this : k + n + 1 ≤ i
⊢ ∃ e', Skips' n e' k ∧ bvar i = liftN 1 e' k | n i✝ k i : Nat
h : i + 1 < k + (n + 1) → i + 1 < k
h' : ¬i + 1 < k + n + 1
this : k + n + 1 ≤ i + 1
⊢ ∃ e', Skips' n e' k ∧ bvar (i + 1) = liftN 1 e' k | Please generate a tactic in lean4 to solve the state.
STATE:
n i k : Nat
h : i < k + (n + 1) → i < k
h' : ¬i < k + n + 1
this : k + n + 1 ≤ i
⊢ ∃ e', Skips' n e' k ∧ bvar i = liftN 1 e' k
TACTIC:
|
https://github.com/digama0/lean4lean.git | c534f13d8d25f5e1891b6d18cc76b601ee87aa66 | Lean4Lean/Theory/VExpr.lean | Lean4Lean.VExpr.skips_iff | [343, 1] | [381, 78] | rw [Nat.add_lt_add_iff_right] at h' | n i✝ k i : Nat
h : i + 1 < k + (n + 1) → i + 1 < k
h' : ¬i + 1 < k + n + 1
this : k + n + 1 ≤ i + 1
⊢ ∃ e', Skips' n e' k ∧ bvar (i + 1) = liftN 1 e' k | n i✝ k i : Nat
h : i + 1 < k + (n + 1) → i + 1 < k
h' : ¬i < k + n
this : k + n + 1 ≤ i + 1
⊢ ∃ e', Skips' n e' k ∧ bvar (i + 1) = liftN 1 e' k | Please generate a tactic in lean4 to solve the state.
STATE:
n i✝ k i : Nat
h : i + 1 < k + (n + 1) → i + 1 < k
h' : ¬i + 1 < k + n + 1
this : k + n + 1 ≤ i + 1
⊢ ∃ e', Skips' n e' k ∧ bvar (i + 1) = liftN 1 e' k
TACTIC:
|
https://github.com/digama0/lean4lean.git | c534f13d8d25f5e1891b6d18cc76b601ee87aa66 | Lean4Lean/Theory/VExpr.lean | Lean4Lean.VExpr.skips_iff | [343, 1] | [381, 78] | exact ⟨.bvar i, h'.elim, by simp [liftN, liftVar]; rw [if_neg this, Nat.add_comm]⟩ | n i✝ k i : Nat
h : i + 1 < k + (n + 1) → i + 1 < k
h' : ¬i < k + n
this✝ : k + n + 1 ≤ i + 1
this : ¬i < k
⊢ ∃ e', Skips' n e' k ∧ bvar (i + 1) = liftN 1 e' k | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n i✝ k i : Nat
h : i + 1 < k + (n + 1) → i + 1 < k
h' : ¬i < k + n
this✝ : k + n + 1 ≤ i + 1
this : ¬i < k
⊢ ∃ e', Skips' n e' k ∧ bvar (i + 1) = liftN 1 e' k
TACTIC:
|
https://github.com/digama0/lean4lean.git | c534f13d8d25f5e1891b6d18cc76b601ee87aa66 | Lean4Lean/Theory/VExpr.lean | Lean4Lean.VExpr.skips_iff | [343, 1] | [381, 78] | simp [liftN, liftVar] | n i✝ k i : Nat
h : i + 1 < k + (n + 1) → i + 1 < k
h' : ¬i < k + n
this✝ : k + n + 1 ≤ i + 1
this : ¬i < k
⊢ bvar (i + 1) = liftN 1 (bvar i) k | n i✝ k i : Nat
h : i + 1 < k + (n + 1) → i + 1 < k
h' : ¬i < k + n
this✝ : k + n + 1 ≤ i + 1
this : ¬i < k
⊢ i + 1 = if i < k then i else 1 + i | Please generate a tactic in lean4 to solve the state.
STATE:
n i✝ k i : Nat
h : i + 1 < k + (n + 1) → i + 1 < k
h' : ¬i < k + n
this✝ : k + n + 1 ≤ i + 1
this : ¬i < k
⊢ bvar (i + 1) = liftN 1 (bvar i) k
TACTIC:
|
https://github.com/digama0/lean4lean.git | c534f13d8d25f5e1891b6d18cc76b601ee87aa66 | Lean4Lean/Theory/VExpr.lean | Lean4Lean.VExpr.skips_iff | [343, 1] | [381, 78] | rw [if_neg this, Nat.add_comm] | n i✝ k i : Nat
h : i + 1 < k + (n + 1) → i + 1 < k
h' : ¬i < k + n
this✝ : k + n + 1 ≤ i + 1
this : ¬i < k
⊢ i + 1 = if i < k then i else 1 + i | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n i✝ k i : Nat
h : i + 1 < k + (n + 1) → i + 1 < k
h' : ¬i < k + n
this✝ : k + n + 1 ≤ i + 1
this : ¬i < k
⊢ i + 1 = if i < k then i else 1 + i
TACTIC:
|
https://github.com/digama0/lean4lean.git | c534f13d8d25f5e1891b6d18cc76b601ee87aa66 | Lean4Lean/Theory/VExpr.lean | Lean4Lean.VExpr.skips_iff | [343, 1] | [381, 78] | refine ⟨fun ⟨e', h1, h2⟩ => ?_, fun _ => ⟨.sort u, by simp [Skips', liftN]⟩⟩ | case succ.sort
n : Nat
u : VLevel
k : Nat
⊢ (∃ e', Skips' n e' k ∧ sort u = liftN 1 e' k) ↔ Skips' (n + 1) (sort u) k | case succ.sort
n : Nat
u : VLevel
k : Nat
x✝ : ∃ e', Skips' n e' k ∧ sort u = liftN 1 e' k
e' : VExpr
h1 : Skips' n e' k
h2 : sort u = liftN 1 e' k
⊢ Skips' (n + 1) (sort u) k | Please generate a tactic in lean4 to solve the state.
STATE:
case succ.sort
n : Nat
u : VLevel
k : Nat
⊢ (∃ e', Skips' n e' k ∧ sort u = liftN 1 e' k) ↔ Skips' (n + 1) (sort u) k
TACTIC:
|
https://github.com/digama0/lean4lean.git | c534f13d8d25f5e1891b6d18cc76b601ee87aa66 | Lean4Lean/Theory/VExpr.lean | Lean4Lean.VExpr.skips_iff | [343, 1] | [381, 78] | cases e' <;> cases h2 | case succ.sort
n : Nat
u : VLevel
k : Nat
x✝ : ∃ e', Skips' n e' k ∧ sort u = liftN 1 e' k
e' : VExpr
h1 : Skips' n e' k
h2 : sort u = liftN 1 e' k
⊢ Skips' (n + 1) (sort u) k | case succ.sort.sort.refl
n : Nat
u : VLevel
k : Nat
x✝ : ∃ e', Skips' n e' k ∧ sort u = liftN 1 e' k
h1 : Skips' n (sort u) k
⊢ Skips' (n + 1) (sort u) k | Please generate a tactic in lean4 to solve the state.
STATE:
case succ.sort
n : Nat
u : VLevel
k : Nat
x✝ : ∃ e', Skips' n e' k ∧ sort u = liftN 1 e' k
e' : VExpr
h1 : Skips' n e' k
h2 : sort u = liftN 1 e' k
⊢ Skips' (n + 1) (sort u) k
TACTIC:
|
https://github.com/digama0/lean4lean.git | c534f13d8d25f5e1891b6d18cc76b601ee87aa66 | Lean4Lean/Theory/VExpr.lean | Lean4Lean.VExpr.skips_iff | [343, 1] | [381, 78] | simp [Skips'] | case succ.sort.sort.refl
n : Nat
u : VLevel
k : Nat
x✝ : ∃ e', Skips' n e' k ∧ sort u = liftN 1 e' k
h1 : Skips' n (sort u) k
⊢ Skips' (n + 1) (sort u) k | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ.sort.sort.refl
n : Nat
u : VLevel
k : Nat
x✝ : ∃ e', Skips' n e' k ∧ sort u = liftN 1 e' k
h1 : Skips' n (sort u) k
⊢ Skips' (n + 1) (sort u) k
TACTIC:
|
https://github.com/digama0/lean4lean.git | c534f13d8d25f5e1891b6d18cc76b601ee87aa66 | Lean4Lean/Theory/VExpr.lean | Lean4Lean.VExpr.skips_iff | [343, 1] | [381, 78] | simp [Skips', liftN] | n : Nat
u : VLevel
k : Nat
x✝ : Skips' (n + 1) (sort u) k
⊢ Skips' n (sort u) k ∧ sort u = liftN 1 (sort u) k | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : Nat
u : VLevel
k : Nat
x✝ : Skips' (n + 1) (sort u) k
⊢ Skips' n (sort u) k ∧ sort u = liftN 1 (sort u) k
TACTIC:
|
https://github.com/digama0/lean4lean.git | c534f13d8d25f5e1891b6d18cc76b601ee87aa66 | Lean4Lean/Theory/VExpr.lean | Lean4Lean.VExpr.skips_iff | [343, 1] | [381, 78] | refine ⟨fun ⟨e', h1, h2⟩ => ?_, fun _ => ⟨.const c ls, by simp [Skips', liftN]⟩⟩ | case succ.const
n : Nat
c : Name
ls : List VLevel
k : Nat
⊢ (∃ e', Skips' n e' k ∧ const c ls = liftN 1 e' k) ↔ Skips' (n + 1) (const c ls) k | case succ.const
n : Nat
c : Name
ls : List VLevel
k : Nat
x✝ : ∃ e', Skips' n e' k ∧ const c ls = liftN 1 e' k
e' : VExpr
h1 : Skips' n e' k
h2 : const c ls = liftN 1 e' k
⊢ Skips' (n + 1) (const c ls) k | Please generate a tactic in lean4 to solve the state.
STATE:
case succ.const
n : Nat
c : Name
ls : List VLevel
k : Nat
⊢ (∃ e', Skips' n e' k ∧ const c ls = liftN 1 e' k) ↔ Skips' (n + 1) (const c ls) k
TACTIC:
|
https://github.com/digama0/lean4lean.git | c534f13d8d25f5e1891b6d18cc76b601ee87aa66 | Lean4Lean/Theory/VExpr.lean | Lean4Lean.VExpr.skips_iff | [343, 1] | [381, 78] | cases e' <;> cases h2 | case succ.const
n : Nat
c : Name
ls : List VLevel
k : Nat
x✝ : ∃ e', Skips' n e' k ∧ const c ls = liftN 1 e' k
e' : VExpr
h1 : Skips' n e' k
h2 : const c ls = liftN 1 e' k
⊢ Skips' (n + 1) (const c ls) k | case succ.const.const.refl
n : Nat
c : Name
ls : List VLevel
k : Nat
x✝ : ∃ e', Skips' n e' k ∧ const c ls = liftN 1 e' k
h1 : Skips' n (const c ls) k
⊢ Skips' (n + 1) (const c ls) k | Please generate a tactic in lean4 to solve the state.
STATE:
case succ.const
n : Nat
c : Name
ls : List VLevel
k : Nat
x✝ : ∃ e', Skips' n e' k ∧ const c ls = liftN 1 e' k
e' : VExpr
h1 : Skips' n e' k
h2 : const c ls = liftN 1 e' k
⊢ Skips' (n + 1) (const c ls) k
TACTIC:
|
https://github.com/digama0/lean4lean.git | c534f13d8d25f5e1891b6d18cc76b601ee87aa66 | Lean4Lean/Theory/VExpr.lean | Lean4Lean.VExpr.skips_iff | [343, 1] | [381, 78] | simp [Skips'] | case succ.const.const.refl
n : Nat
c : Name
ls : List VLevel
k : Nat
x✝ : ∃ e', Skips' n e' k ∧ const c ls = liftN 1 e' k
h1 : Skips' n (const c ls) k
⊢ Skips' (n + 1) (const c ls) k | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ.const.const.refl
n : Nat
c : Name
ls : List VLevel
k : Nat
x✝ : ∃ e', Skips' n e' k ∧ const c ls = liftN 1 e' k
h1 : Skips' n (const c ls) k
⊢ Skips' (n + 1) (const c ls) k
TACTIC:
|
https://github.com/digama0/lean4lean.git | c534f13d8d25f5e1891b6d18cc76b601ee87aa66 | Lean4Lean/Theory/VExpr.lean | Lean4Lean.VExpr.skips_iff | [343, 1] | [381, 78] | simp [Skips', liftN] | n : Nat
c : Name
ls : List VLevel
k : Nat
x✝ : Skips' (n + 1) (const c ls) k
⊢ Skips' n (const c ls) k ∧ const c ls = liftN 1 (const c ls) k | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : Nat
c : Name
ls : List VLevel
k : Nat
x✝ : Skips' (n + 1) (const c ls) k
⊢ Skips' n (const c ls) k ∧ const c ls = liftN 1 (const c ls) k
TACTIC:
|
https://github.com/digama0/lean4lean.git | c534f13d8d25f5e1891b6d18cc76b601ee87aa66 | Lean4Lean/Theory/VExpr.lean | Lean4Lean.VExpr.skips_iff | [343, 1] | [381, 78] | simp [Skips', ← fIH, ← aIH] | case succ.app
n : Nat
f a : VExpr
fIH : ∀ {k : Nat}, (∃ e', Skips' n e' k ∧ f = liftN 1 e' k) ↔ Skips' (n + 1) f k
aIH : ∀ {k : Nat}, (∃ e', Skips' n e' k ∧ a = liftN 1 e' k) ↔ Skips' (n + 1) a k
k : Nat
⊢ (∃ e', Skips' n e' k ∧ app f a = liftN 1 e' k) ↔ Skips' (n + 1) (app f a) k | case succ.app
n : Nat
f a : VExpr
fIH : ∀ {k : Nat}, (∃ e', Skips' n e' k ∧ f = liftN 1 e' k) ↔ Skips' (n + 1) f k
aIH : ∀ {k : Nat}, (∃ e', Skips' n e' k ∧ a = liftN 1 e' k) ↔ Skips' (n + 1) a k
k : Nat
⊢ (∃ e', Skips' n e' k ∧ app f a = liftN 1 e' k) ↔
(∃ e', Skips' n e' k ∧ f = liftN 1 e' k) ∧ ∃ e', Skips' n e' k ∧ a = liftN 1 e' k | Please generate a tactic in lean4 to solve the state.
STATE:
case succ.app
n : Nat
f a : VExpr
fIH : ∀ {k : Nat}, (∃ e', Skips' n e' k ∧ f = liftN 1 e' k) ↔ Skips' (n + 1) f k
aIH : ∀ {k : Nat}, (∃ e', Skips' n e' k ∧ a = liftN 1 e' k) ↔ Skips' (n + 1) a k
k : Nat
⊢ (∃ e', Skips' n e' k ∧ app f a = liftN 1 e' k) ↔ Skips' (n + 1) (app f a) k
TACTIC:
|
https://github.com/digama0/lean4lean.git | c534f13d8d25f5e1891b6d18cc76b601ee87aa66 | Lean4Lean/Theory/VExpr.lean | Lean4Lean.VExpr.skips_iff | [343, 1] | [381, 78] | refine ⟨fun ⟨e', h1, h2⟩ => ?_, ?_⟩ | case succ.app
n : Nat
f a : VExpr
fIH : ∀ {k : Nat}, (∃ e', Skips' n e' k ∧ f = liftN 1 e' k) ↔ Skips' (n + 1) f k
aIH : ∀ {k : Nat}, (∃ e', Skips' n e' k ∧ a = liftN 1 e' k) ↔ Skips' (n + 1) a k
k : Nat
⊢ (∃ e', Skips' n e' k ∧ app f a = liftN 1 e' k) ↔
(∃ e', Skips' n e' k ∧ f = liftN 1 e' k) ∧ ∃ e', Skips' n e' k ∧ a = liftN 1 e' k | case succ.app.refine_1
n : Nat
f a : VExpr
fIH : ∀ {k : Nat}, (∃ e', Skips' n e' k ∧ f = liftN 1 e' k) ↔ Skips' (n + 1) f k
aIH : ∀ {k : Nat}, (∃ e', Skips' n e' k ∧ a = liftN 1 e' k) ↔ Skips' (n + 1) a k
k : Nat
x✝ : ∃ e', Skips' n e' k ∧ app f a = liftN 1 e' k
e' : VExpr
h1 : Skips' n e' k
h2 : app f a = liftN 1 e' k
⊢ (∃ e', Skips' n e' k ∧ f = liftN 1 e' k) ∧ ∃ e', Skips' n e' k ∧ a = liftN 1 e' k
case succ.app.refine_2
n : Nat
f a : VExpr
fIH : ∀ {k : Nat}, (∃ e', Skips' n e' k ∧ f = liftN 1 e' k) ↔ Skips' (n + 1) f k
aIH : ∀ {k : Nat}, (∃ e', Skips' n e' k ∧ a = liftN 1 e' k) ↔ Skips' (n + 1) a k
k : Nat
⊢ ((∃ e', Skips' n e' k ∧ f = liftN 1 e' k) ∧ ∃ e', Skips' n e' k ∧ a = liftN 1 e' k) →
∃ e', Skips' n e' k ∧ app f a = liftN 1 e' k | Please generate a tactic in lean4 to solve the state.
STATE:
case succ.app
n : Nat
f a : VExpr
fIH : ∀ {k : Nat}, (∃ e', Skips' n e' k ∧ f = liftN 1 e' k) ↔ Skips' (n + 1) f k
aIH : ∀ {k : Nat}, (∃ e', Skips' n e' k ∧ a = liftN 1 e' k) ↔ Skips' (n + 1) a k
k : Nat
⊢ (∃ e', Skips' n e' k ∧ app f a = liftN 1 e' k) ↔
(∃ e', Skips' n e' k ∧ f = liftN 1 e' k) ∧ ∃ e', Skips' n e' k ∧ a = liftN 1 e' k
TACTIC:
|
https://github.com/digama0/lean4lean.git | c534f13d8d25f5e1891b6d18cc76b601ee87aa66 | Lean4Lean/Theory/VExpr.lean | Lean4Lean.VExpr.skips_iff | [343, 1] | [381, 78] | cases e' <;> cases h2 | case succ.app.refine_1
n : Nat
f a : VExpr
fIH : ∀ {k : Nat}, (∃ e', Skips' n e' k ∧ f = liftN 1 e' k) ↔ Skips' (n + 1) f k
aIH : ∀ {k : Nat}, (∃ e', Skips' n e' k ∧ a = liftN 1 e' k) ↔ Skips' (n + 1) a k
k : Nat
x✝ : ∃ e', Skips' n e' k ∧ app f a = liftN 1 e' k
e' : VExpr
h1 : Skips' n e' k
h2 : app f a = liftN 1 e' k
⊢ (∃ e', Skips' n e' k ∧ f = liftN 1 e' k) ∧ ∃ e', Skips' n e' k ∧ a = liftN 1 e' k | case succ.app.refine_1.app.refl
n k : Nat
fn✝ arg✝ : VExpr
h1 : Skips' n (app fn✝ arg✝) k
fIH : ∀ {k_1 : Nat}, (∃ e', Skips' n e' k_1 ∧ liftN 1 fn✝ k = liftN 1 e' k_1) ↔ Skips' (n + 1) (liftN 1 fn✝ k) k_1
aIH : ∀ {k_1 : Nat}, (∃ e', Skips' n e' k_1 ∧ liftN 1 arg✝ k = liftN 1 e' k_1) ↔ Skips' (n + 1) (liftN 1 arg✝ k) k_1
x✝ : ∃ e', Skips' n e' k ∧ app (liftN 1 fn✝ k) (liftN 1 arg✝ k) = liftN 1 e' k
⊢ (∃ e', Skips' n e' k ∧ liftN 1 fn✝ k = liftN 1 e' k) ∧ ∃ e', Skips' n e' k ∧ liftN 1 arg✝ k = liftN 1 e' k | Please generate a tactic in lean4 to solve the state.
STATE:
case succ.app.refine_1
n : Nat
f a : VExpr
fIH : ∀ {k : Nat}, (∃ e', Skips' n e' k ∧ f = liftN 1 e' k) ↔ Skips' (n + 1) f k
aIH : ∀ {k : Nat}, (∃ e', Skips' n e' k ∧ a = liftN 1 e' k) ↔ Skips' (n + 1) a k
k : Nat
x✝ : ∃ e', Skips' n e' k ∧ app f a = liftN 1 e' k
e' : VExpr
h1 : Skips' n e' k
h2 : app f a = liftN 1 e' k
⊢ (∃ e', Skips' n e' k ∧ f = liftN 1 e' k) ∧ ∃ e', Skips' n e' k ∧ a = liftN 1 e' k
TACTIC:
|
https://github.com/digama0/lean4lean.git | c534f13d8d25f5e1891b6d18cc76b601ee87aa66 | Lean4Lean/Theory/VExpr.lean | Lean4Lean.VExpr.skips_iff | [343, 1] | [381, 78] | exact ⟨⟨_, h1.1, rfl⟩, ⟨_, h1.2, rfl⟩⟩ | case succ.app.refine_1.app.refl
n k : Nat
fn✝ arg✝ : VExpr
h1 : Skips' n (app fn✝ arg✝) k
fIH : ∀ {k_1 : Nat}, (∃ e', Skips' n e' k_1 ∧ liftN 1 fn✝ k = liftN 1 e' k_1) ↔ Skips' (n + 1) (liftN 1 fn✝ k) k_1
aIH : ∀ {k_1 : Nat}, (∃ e', Skips' n e' k_1 ∧ liftN 1 arg✝ k = liftN 1 e' k_1) ↔ Skips' (n + 1) (liftN 1 arg✝ k) k_1
x✝ : ∃ e', Skips' n e' k ∧ app (liftN 1 fn✝ k) (liftN 1 arg✝ k) = liftN 1 e' k
⊢ (∃ e', Skips' n e' k ∧ liftN 1 fn✝ k = liftN 1 e' k) ∧ ∃ e', Skips' n e' k ∧ liftN 1 arg✝ k = liftN 1 e' k | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ.app.refine_1.app.refl
n k : Nat
fn✝ arg✝ : VExpr
h1 : Skips' n (app fn✝ arg✝) k
fIH : ∀ {k_1 : Nat}, (∃ e', Skips' n e' k_1 ∧ liftN 1 fn✝ k = liftN 1 e' k_1) ↔ Skips' (n + 1) (liftN 1 fn✝ k) k_1
aIH : ∀ {k_1 : Nat}, (∃ e', Skips' n e' k_1 ∧ liftN 1 arg✝ k = liftN 1 e' k_1) ↔ Skips' (n + 1) (liftN 1 arg✝ k) k_1
x✝ : ∃ e', Skips' n e' k ∧ app (liftN 1 fn✝ k) (liftN 1 arg✝ k) = liftN 1 e' k
⊢ (∃ e', Skips' n e' k ∧ liftN 1 fn✝ k = liftN 1 e' k) ∧ ∃ e', Skips' n e' k ∧ liftN 1 arg✝ k = liftN 1 e' k
TACTIC:
|
https://github.com/digama0/lean4lean.git | c534f13d8d25f5e1891b6d18cc76b601ee87aa66 | Lean4Lean/Theory/VExpr.lean | Lean4Lean.VExpr.skips_iff | [343, 1] | [381, 78] | rintro ⟨⟨e1, h1, rfl⟩, ⟨e2, h2, rfl⟩⟩ | case succ.app.refine_2
n : Nat
f a : VExpr
fIH : ∀ {k : Nat}, (∃ e', Skips' n e' k ∧ f = liftN 1 e' k) ↔ Skips' (n + 1) f k
aIH : ∀ {k : Nat}, (∃ e', Skips' n e' k ∧ a = liftN 1 e' k) ↔ Skips' (n + 1) a k
k : Nat
⊢ ((∃ e', Skips' n e' k ∧ f = liftN 1 e' k) ∧ ∃ e', Skips' n e' k ∧ a = liftN 1 e' k) →
∃ e', Skips' n e' k ∧ app f a = liftN 1 e' k | case succ.app.refine_2.intro.intro.intro.intro.intro
n k : Nat
e1 : VExpr
h1 : Skips' n e1 k
fIH : ∀ {k_1 : Nat}, (∃ e', Skips' n e' k_1 ∧ liftN 1 e1 k = liftN 1 e' k_1) ↔ Skips' (n + 1) (liftN 1 e1 k) k_1
e2 : VExpr
h2 : Skips' n e2 k
aIH : ∀ {k_1 : Nat}, (∃ e', Skips' n e' k_1 ∧ liftN 1 e2 k = liftN 1 e' k_1) ↔ Skips' (n + 1) (liftN 1 e2 k) k_1
⊢ ∃ e', Skips' n e' k ∧ app (liftN 1 e1 k) (liftN 1 e2 k) = liftN 1 e' k | Please generate a tactic in lean4 to solve the state.
STATE:
case succ.app.refine_2
n : Nat
f a : VExpr
fIH : ∀ {k : Nat}, (∃ e', Skips' n e' k ∧ f = liftN 1 e' k) ↔ Skips' (n + 1) f k
aIH : ∀ {k : Nat}, (∃ e', Skips' n e' k ∧ a = liftN 1 e' k) ↔ Skips' (n + 1) a k
k : Nat
⊢ ((∃ e', Skips' n e' k ∧ f = liftN 1 e' k) ∧ ∃ e', Skips' n e' k ∧ a = liftN 1 e' k) →
∃ e', Skips' n e' k ∧ app f a = liftN 1 e' k
TACTIC:
|
https://github.com/digama0/lean4lean.git | c534f13d8d25f5e1891b6d18cc76b601ee87aa66 | Lean4Lean/Theory/VExpr.lean | Lean4Lean.VExpr.skips_iff | [343, 1] | [381, 78] | exact ⟨.app .., ⟨h1, h2⟩, rfl⟩ | case succ.app.refine_2.intro.intro.intro.intro.intro
n k : Nat
e1 : VExpr
h1 : Skips' n e1 k
fIH : ∀ {k_1 : Nat}, (∃ e', Skips' n e' k_1 ∧ liftN 1 e1 k = liftN 1 e' k_1) ↔ Skips' (n + 1) (liftN 1 e1 k) k_1
e2 : VExpr
h2 : Skips' n e2 k
aIH : ∀ {k_1 : Nat}, (∃ e', Skips' n e' k_1 ∧ liftN 1 e2 k = liftN 1 e' k_1) ↔ Skips' (n + 1) (liftN 1 e2 k) k_1
⊢ ∃ e', Skips' n e' k ∧ app (liftN 1 e1 k) (liftN 1 e2 k) = liftN 1 e' k | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ.app.refine_2.intro.intro.intro.intro.intro
n k : Nat
e1 : VExpr
h1 : Skips' n e1 k
fIH : ∀ {k_1 : Nat}, (∃ e', Skips' n e' k_1 ∧ liftN 1 e1 k = liftN 1 e' k_1) ↔ Skips' (n + 1) (liftN 1 e1 k) k_1
e2 : VExpr
h2 : Skips' n e2 k
aIH : ∀ {k_1 : Nat}, (∃ e', Skips' n e' k_1 ∧ liftN 1 e2 k = liftN 1 e' k_1) ↔ Skips' (n + 1) (liftN 1 e2 k) k_1
⊢ ∃ e', Skips' n e' k ∧ app (liftN 1 e1 k) (liftN 1 e2 k) = liftN 1 e' k
TACTIC:
|
https://github.com/digama0/lean4lean.git | c534f13d8d25f5e1891b6d18cc76b601ee87aa66 | Lean4Lean/Theory/VExpr.lean | Lean4Lean.VExpr.skips_iff | [343, 1] | [381, 78] | simp [Skips', ← fIH, ← aIH] | case succ.forallE
n : Nat
f a : VExpr
fIH : ∀ {k : Nat}, (∃ e', Skips' n e' k ∧ f = liftN 1 e' k) ↔ Skips' (n + 1) f k
aIH : ∀ {k : Nat}, (∃ e', Skips' n e' k ∧ a = liftN 1 e' k) ↔ Skips' (n + 1) a k
k : Nat
⊢ (∃ e', Skips' n e' k ∧ forallE f a = liftN 1 e' k) ↔ Skips' (n + 1) (forallE f a) k | case succ.forallE
n : Nat
f a : VExpr
fIH : ∀ {k : Nat}, (∃ e', Skips' n e' k ∧ f = liftN 1 e' k) ↔ Skips' (n + 1) f k
aIH : ∀ {k : Nat}, (∃ e', Skips' n e' k ∧ a = liftN 1 e' k) ↔ Skips' (n + 1) a k
k : Nat
⊢ (∃ e', Skips' n e' k ∧ forallE f a = liftN 1 e' k) ↔
(∃ e', Skips' n e' k ∧ f = liftN 1 e' k) ∧ ∃ e', Skips' n e' (k + 1) ∧ a = liftN 1 e' (k + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case succ.forallE
n : Nat
f a : VExpr
fIH : ∀ {k : Nat}, (∃ e', Skips' n e' k ∧ f = liftN 1 e' k) ↔ Skips' (n + 1) f k
aIH : ∀ {k : Nat}, (∃ e', Skips' n e' k ∧ a = liftN 1 e' k) ↔ Skips' (n + 1) a k
k : Nat
⊢ (∃ e', Skips' n e' k ∧ forallE f a = liftN 1 e' k) ↔ Skips' (n + 1) (forallE f a) k
TACTIC:
|
https://github.com/digama0/lean4lean.git | c534f13d8d25f5e1891b6d18cc76b601ee87aa66 | Lean4Lean/Theory/VExpr.lean | Lean4Lean.VExpr.skips_iff | [343, 1] | [381, 78] | refine ⟨fun ⟨e', h1, h2⟩ => ?_, ?_⟩ | case succ.forallE
n : Nat
f a : VExpr
fIH : ∀ {k : Nat}, (∃ e', Skips' n e' k ∧ f = liftN 1 e' k) ↔ Skips' (n + 1) f k
aIH : ∀ {k : Nat}, (∃ e', Skips' n e' k ∧ a = liftN 1 e' k) ↔ Skips' (n + 1) a k
k : Nat
⊢ (∃ e', Skips' n e' k ∧ forallE f a = liftN 1 e' k) ↔
(∃ e', Skips' n e' k ∧ f = liftN 1 e' k) ∧ ∃ e', Skips' n e' (k + 1) ∧ a = liftN 1 e' (k + 1) | case succ.forallE.refine_1
n : Nat
f a : VExpr
fIH : ∀ {k : Nat}, (∃ e', Skips' n e' k ∧ f = liftN 1 e' k) ↔ Skips' (n + 1) f k
aIH : ∀ {k : Nat}, (∃ e', Skips' n e' k ∧ a = liftN 1 e' k) ↔ Skips' (n + 1) a k
k : Nat
x✝ : ∃ e', Skips' n e' k ∧ forallE f a = liftN 1 e' k
e' : VExpr
h1 : Skips' n e' k
h2 : forallE f a = liftN 1 e' k
⊢ (∃ e', Skips' n e' k ∧ f = liftN 1 e' k) ∧ ∃ e', Skips' n e' (k + 1) ∧ a = liftN 1 e' (k + 1)
case succ.forallE.refine_2
n : Nat
f a : VExpr
fIH : ∀ {k : Nat}, (∃ e', Skips' n e' k ∧ f = liftN 1 e' k) ↔ Skips' (n + 1) f k
aIH : ∀ {k : Nat}, (∃ e', Skips' n e' k ∧ a = liftN 1 e' k) ↔ Skips' (n + 1) a k
k : Nat
⊢ ((∃ e', Skips' n e' k ∧ f = liftN 1 e' k) ∧ ∃ e', Skips' n e' (k + 1) ∧ a = liftN 1 e' (k + 1)) →
∃ e', Skips' n e' k ∧ forallE f a = liftN 1 e' k | Please generate a tactic in lean4 to solve the state.
STATE:
case succ.forallE
n : Nat
f a : VExpr
fIH : ∀ {k : Nat}, (∃ e', Skips' n e' k ∧ f = liftN 1 e' k) ↔ Skips' (n + 1) f k
aIH : ∀ {k : Nat}, (∃ e', Skips' n e' k ∧ a = liftN 1 e' k) ↔ Skips' (n + 1) a k
k : Nat
⊢ (∃ e', Skips' n e' k ∧ forallE f a = liftN 1 e' k) ↔
(∃ e', Skips' n e' k ∧ f = liftN 1 e' k) ∧ ∃ e', Skips' n e' (k + 1) ∧ a = liftN 1 e' (k + 1)
TACTIC:
|
https://github.com/digama0/lean4lean.git | c534f13d8d25f5e1891b6d18cc76b601ee87aa66 | Lean4Lean/Theory/VExpr.lean | Lean4Lean.VExpr.skips_iff | [343, 1] | [381, 78] | cases e' <;> cases h2 | case succ.forallE.refine_1
n : Nat
f a : VExpr
fIH : ∀ {k : Nat}, (∃ e', Skips' n e' k ∧ f = liftN 1 e' k) ↔ Skips' (n + 1) f k
aIH : ∀ {k : Nat}, (∃ e', Skips' n e' k ∧ a = liftN 1 e' k) ↔ Skips' (n + 1) a k
k : Nat
x✝ : ∃ e', Skips' n e' k ∧ forallE f a = liftN 1 e' k
e' : VExpr
h1 : Skips' n e' k
h2 : forallE f a = liftN 1 e' k
⊢ (∃ e', Skips' n e' k ∧ f = liftN 1 e' k) ∧ ∃ e', Skips' n e' (k + 1) ∧ a = liftN 1 e' (k + 1) | case succ.forallE.refine_1.forallE.refl
n k : Nat
binderType✝ body✝ : VExpr
h1 : Skips' n (forallE binderType✝ body✝) k
fIH :
∀ {k_1 : Nat},
(∃ e', Skips' n e' k_1 ∧ liftN 1 binderType✝ k = liftN 1 e' k_1) ↔ Skips' (n + 1) (liftN 1 binderType✝ k) k_1
aIH :
∀ {k_1 : Nat},
(∃ e', Skips' n e' k_1 ∧ liftN 1 body✝ (k + 1) = liftN 1 e' k_1) ↔ Skips' (n + 1) (liftN 1 body✝ (k + 1)) k_1
x✝ : ∃ e', Skips' n e' k ∧ forallE (liftN 1 binderType✝ k) (liftN 1 body✝ (k + 1)) = liftN 1 e' k
⊢ (∃ e', Skips' n e' k ∧ liftN 1 binderType✝ k = liftN 1 e' k) ∧
∃ e', Skips' n e' (k + 1) ∧ liftN 1 body✝ (k + 1) = liftN 1 e' (k + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case succ.forallE.refine_1
n : Nat
f a : VExpr
fIH : ∀ {k : Nat}, (∃ e', Skips' n e' k ∧ f = liftN 1 e' k) ↔ Skips' (n + 1) f k
aIH : ∀ {k : Nat}, (∃ e', Skips' n e' k ∧ a = liftN 1 e' k) ↔ Skips' (n + 1) a k
k : Nat
x✝ : ∃ e', Skips' n e' k ∧ forallE f a = liftN 1 e' k
e' : VExpr
h1 : Skips' n e' k
h2 : forallE f a = liftN 1 e' k
⊢ (∃ e', Skips' n e' k ∧ f = liftN 1 e' k) ∧ ∃ e', Skips' n e' (k + 1) ∧ a = liftN 1 e' (k + 1)
TACTIC:
|
https://github.com/digama0/lean4lean.git | c534f13d8d25f5e1891b6d18cc76b601ee87aa66 | Lean4Lean/Theory/VExpr.lean | Lean4Lean.VExpr.skips_iff | [343, 1] | [381, 78] | exact ⟨⟨_, h1.1, rfl⟩, ⟨_, h1.2, rfl⟩⟩ | case succ.forallE.refine_1.forallE.refl
n k : Nat
binderType✝ body✝ : VExpr
h1 : Skips' n (forallE binderType✝ body✝) k
fIH :
∀ {k_1 : Nat},
(∃ e', Skips' n e' k_1 ∧ liftN 1 binderType✝ k = liftN 1 e' k_1) ↔ Skips' (n + 1) (liftN 1 binderType✝ k) k_1
aIH :
∀ {k_1 : Nat},
(∃ e', Skips' n e' k_1 ∧ liftN 1 body✝ (k + 1) = liftN 1 e' k_1) ↔ Skips' (n + 1) (liftN 1 body✝ (k + 1)) k_1
x✝ : ∃ e', Skips' n e' k ∧ forallE (liftN 1 binderType✝ k) (liftN 1 body✝ (k + 1)) = liftN 1 e' k
⊢ (∃ e', Skips' n e' k ∧ liftN 1 binderType✝ k = liftN 1 e' k) ∧
∃ e', Skips' n e' (k + 1) ∧ liftN 1 body✝ (k + 1) = liftN 1 e' (k + 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ.forallE.refine_1.forallE.refl
n k : Nat
binderType✝ body✝ : VExpr
h1 : Skips' n (forallE binderType✝ body✝) k
fIH :
∀ {k_1 : Nat},
(∃ e', Skips' n e' k_1 ∧ liftN 1 binderType✝ k = liftN 1 e' k_1) ↔ Skips' (n + 1) (liftN 1 binderType✝ k) k_1
aIH :
∀ {k_1 : Nat},
(∃ e', Skips' n e' k_1 ∧ liftN 1 body✝ (k + 1) = liftN 1 e' k_1) ↔ Skips' (n + 1) (liftN 1 body✝ (k + 1)) k_1
x✝ : ∃ e', Skips' n e' k ∧ forallE (liftN 1 binderType✝ k) (liftN 1 body✝ (k + 1)) = liftN 1 e' k
⊢ (∃ e', Skips' n e' k ∧ liftN 1 binderType✝ k = liftN 1 e' k) ∧
∃ e', Skips' n e' (k + 1) ∧ liftN 1 body✝ (k + 1) = liftN 1 e' (k + 1)
TACTIC:
|
https://github.com/digama0/lean4lean.git | c534f13d8d25f5e1891b6d18cc76b601ee87aa66 | Lean4Lean/Theory/VExpr.lean | Lean4Lean.VExpr.skips_iff | [343, 1] | [381, 78] | rintro ⟨⟨e1, h1, rfl⟩, ⟨e2, h2, rfl⟩⟩ | case succ.forallE.refine_2
n : Nat
f a : VExpr
fIH : ∀ {k : Nat}, (∃ e', Skips' n e' k ∧ f = liftN 1 e' k) ↔ Skips' (n + 1) f k
aIH : ∀ {k : Nat}, (∃ e', Skips' n e' k ∧ a = liftN 1 e' k) ↔ Skips' (n + 1) a k
k : Nat
⊢ ((∃ e', Skips' n e' k ∧ f = liftN 1 e' k) ∧ ∃ e', Skips' n e' (k + 1) ∧ a = liftN 1 e' (k + 1)) →
∃ e', Skips' n e' k ∧ forallE f a = liftN 1 e' k | case succ.forallE.refine_2.intro.intro.intro.intro.intro
n k : Nat
e1 : VExpr
h1 : Skips' n e1 k
fIH : ∀ {k_1 : Nat}, (∃ e', Skips' n e' k_1 ∧ liftN 1 e1 k = liftN 1 e' k_1) ↔ Skips' (n + 1) (liftN 1 e1 k) k_1
e2 : VExpr
h2 : Skips' n e2 (k + 1)
aIH :
∀ {k_1 : Nat}, (∃ e', Skips' n e' k_1 ∧ liftN 1 e2 (k + 1) = liftN 1 e' k_1) ↔ Skips' (n + 1) (liftN 1 e2 (k + 1)) k_1
⊢ ∃ e', Skips' n e' k ∧ forallE (liftN 1 e1 k) (liftN 1 e2 (k + 1)) = liftN 1 e' k | Please generate a tactic in lean4 to solve the state.
STATE:
case succ.forallE.refine_2
n : Nat
f a : VExpr
fIH : ∀ {k : Nat}, (∃ e', Skips' n e' k ∧ f = liftN 1 e' k) ↔ Skips' (n + 1) f k
aIH : ∀ {k : Nat}, (∃ e', Skips' n e' k ∧ a = liftN 1 e' k) ↔ Skips' (n + 1) a k
k : Nat
⊢ ((∃ e', Skips' n e' k ∧ f = liftN 1 e' k) ∧ ∃ e', Skips' n e' (k + 1) ∧ a = liftN 1 e' (k + 1)) →
∃ e', Skips' n e' k ∧ forallE f a = liftN 1 e' k
TACTIC:
|
https://github.com/digama0/lean4lean.git | c534f13d8d25f5e1891b6d18cc76b601ee87aa66 | Lean4Lean/Theory/VExpr.lean | Lean4Lean.VExpr.skips_iff | [343, 1] | [381, 78] | exact ⟨.forallE .., ⟨h1, h2⟩, rfl⟩ | case succ.forallE.refine_2.intro.intro.intro.intro.intro
n k : Nat
e1 : VExpr
h1 : Skips' n e1 k
fIH : ∀ {k_1 : Nat}, (∃ e', Skips' n e' k_1 ∧ liftN 1 e1 k = liftN 1 e' k_1) ↔ Skips' (n + 1) (liftN 1 e1 k) k_1
e2 : VExpr
h2 : Skips' n e2 (k + 1)
aIH :
∀ {k_1 : Nat}, (∃ e', Skips' n e' k_1 ∧ liftN 1 e2 (k + 1) = liftN 1 e' k_1) ↔ Skips' (n + 1) (liftN 1 e2 (k + 1)) k_1
⊢ ∃ e', Skips' n e' k ∧ forallE (liftN 1 e1 k) (liftN 1 e2 (k + 1)) = liftN 1 e' k | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ.forallE.refine_2.intro.intro.intro.intro.intro
n k : Nat
e1 : VExpr
h1 : Skips' n e1 k
fIH : ∀ {k_1 : Nat}, (∃ e', Skips' n e' k_1 ∧ liftN 1 e1 k = liftN 1 e' k_1) ↔ Skips' (n + 1) (liftN 1 e1 k) k_1
e2 : VExpr
h2 : Skips' n e2 (k + 1)
aIH :
∀ {k_1 : Nat}, (∃ e', Skips' n e' k_1 ∧ liftN 1 e2 (k + 1) = liftN 1 e' k_1) ↔ Skips' (n + 1) (liftN 1 e2 (k + 1)) k_1
⊢ ∃ e', Skips' n e' k ∧ forallE (liftN 1 e1 k) (liftN 1 e2 (k + 1)) = liftN 1 e' k
TACTIC:
|
https://github.com/digama0/lean4lean.git | c534f13d8d25f5e1891b6d18cc76b601ee87aa66 | Lean4Lean/Theory/VExpr.lean | Lean4Lean.VExpr.skips_iff | [343, 1] | [381, 78] | simp [Skips', ← fIH, ← aIH] | case succ.lam
n : Nat
f a : VExpr
fIH : ∀ {k : Nat}, (∃ e', Skips' n e' k ∧ f = liftN 1 e' k) ↔ Skips' (n + 1) f k
aIH : ∀ {k : Nat}, (∃ e', Skips' n e' k ∧ a = liftN 1 e' k) ↔ Skips' (n + 1) a k
k : Nat
⊢ (∃ e', Skips' n e' k ∧ lam f a = liftN 1 e' k) ↔ Skips' (n + 1) (lam f a) k | case succ.lam
n : Nat
f a : VExpr
fIH : ∀ {k : Nat}, (∃ e', Skips' n e' k ∧ f = liftN 1 e' k) ↔ Skips' (n + 1) f k
aIH : ∀ {k : Nat}, (∃ e', Skips' n e' k ∧ a = liftN 1 e' k) ↔ Skips' (n + 1) a k
k : Nat
⊢ (∃ e', Skips' n e' k ∧ lam f a = liftN 1 e' k) ↔
(∃ e', Skips' n e' k ∧ f = liftN 1 e' k) ∧ ∃ e', Skips' n e' (k + 1) ∧ a = liftN 1 e' (k + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case succ.lam
n : Nat
f a : VExpr
fIH : ∀ {k : Nat}, (∃ e', Skips' n e' k ∧ f = liftN 1 e' k) ↔ Skips' (n + 1) f k
aIH : ∀ {k : Nat}, (∃ e', Skips' n e' k ∧ a = liftN 1 e' k) ↔ Skips' (n + 1) a k
k : Nat
⊢ (∃ e', Skips' n e' k ∧ lam f a = liftN 1 e' k) ↔ Skips' (n + 1) (lam f a) k
TACTIC:
|
https://github.com/digama0/lean4lean.git | c534f13d8d25f5e1891b6d18cc76b601ee87aa66 | Lean4Lean/Theory/VExpr.lean | Lean4Lean.VExpr.skips_iff | [343, 1] | [381, 78] | refine ⟨fun ⟨e', h1, h2⟩ => ?_, ?_⟩ | case succ.lam
n : Nat
f a : VExpr
fIH : ∀ {k : Nat}, (∃ e', Skips' n e' k ∧ f = liftN 1 e' k) ↔ Skips' (n + 1) f k
aIH : ∀ {k : Nat}, (∃ e', Skips' n e' k ∧ a = liftN 1 e' k) ↔ Skips' (n + 1) a k
k : Nat
⊢ (∃ e', Skips' n e' k ∧ lam f a = liftN 1 e' k) ↔
(∃ e', Skips' n e' k ∧ f = liftN 1 e' k) ∧ ∃ e', Skips' n e' (k + 1) ∧ a = liftN 1 e' (k + 1) | case succ.lam.refine_1
n : Nat
f a : VExpr
fIH : ∀ {k : Nat}, (∃ e', Skips' n e' k ∧ f = liftN 1 e' k) ↔ Skips' (n + 1) f k
aIH : ∀ {k : Nat}, (∃ e', Skips' n e' k ∧ a = liftN 1 e' k) ↔ Skips' (n + 1) a k
k : Nat
x✝ : ∃ e', Skips' n e' k ∧ lam f a = liftN 1 e' k
e' : VExpr
h1 : Skips' n e' k
h2 : lam f a = liftN 1 e' k
⊢ (∃ e', Skips' n e' k ∧ f = liftN 1 e' k) ∧ ∃ e', Skips' n e' (k + 1) ∧ a = liftN 1 e' (k + 1)
case succ.lam.refine_2
n : Nat
f a : VExpr
fIH : ∀ {k : Nat}, (∃ e', Skips' n e' k ∧ f = liftN 1 e' k) ↔ Skips' (n + 1) f k
aIH : ∀ {k : Nat}, (∃ e', Skips' n e' k ∧ a = liftN 1 e' k) ↔ Skips' (n + 1) a k
k : Nat
⊢ ((∃ e', Skips' n e' k ∧ f = liftN 1 e' k) ∧ ∃ e', Skips' n e' (k + 1) ∧ a = liftN 1 e' (k + 1)) →
∃ e', Skips' n e' k ∧ lam f a = liftN 1 e' k | Please generate a tactic in lean4 to solve the state.
STATE:
case succ.lam
n : Nat
f a : VExpr
fIH : ∀ {k : Nat}, (∃ e', Skips' n e' k ∧ f = liftN 1 e' k) ↔ Skips' (n + 1) f k
aIH : ∀ {k : Nat}, (∃ e', Skips' n e' k ∧ a = liftN 1 e' k) ↔ Skips' (n + 1) a k
k : Nat
⊢ (∃ e', Skips' n e' k ∧ lam f a = liftN 1 e' k) ↔
(∃ e', Skips' n e' k ∧ f = liftN 1 e' k) ∧ ∃ e', Skips' n e' (k + 1) ∧ a = liftN 1 e' (k + 1)
TACTIC:
|
https://github.com/digama0/lean4lean.git | c534f13d8d25f5e1891b6d18cc76b601ee87aa66 | Lean4Lean/Theory/VExpr.lean | Lean4Lean.VExpr.skips_iff | [343, 1] | [381, 78] | cases e' <;> cases h2 | case succ.lam.refine_1
n : Nat
f a : VExpr
fIH : ∀ {k : Nat}, (∃ e', Skips' n e' k ∧ f = liftN 1 e' k) ↔ Skips' (n + 1) f k
aIH : ∀ {k : Nat}, (∃ e', Skips' n e' k ∧ a = liftN 1 e' k) ↔ Skips' (n + 1) a k
k : Nat
x✝ : ∃ e', Skips' n e' k ∧ lam f a = liftN 1 e' k
e' : VExpr
h1 : Skips' n e' k
h2 : lam f a = liftN 1 e' k
⊢ (∃ e', Skips' n e' k ∧ f = liftN 1 e' k) ∧ ∃ e', Skips' n e' (k + 1) ∧ a = liftN 1 e' (k + 1) | case succ.lam.refine_1.lam.refl
n k : Nat
binderType✝ body✝ : VExpr
h1 : Skips' n (lam binderType✝ body✝) k
fIH :
∀ {k_1 : Nat},
(∃ e', Skips' n e' k_1 ∧ liftN 1 binderType✝ k = liftN 1 e' k_1) ↔ Skips' (n + 1) (liftN 1 binderType✝ k) k_1
aIH :
∀ {k_1 : Nat},
(∃ e', Skips' n e' k_1 ∧ liftN 1 body✝ (k + 1) = liftN 1 e' k_1) ↔ Skips' (n + 1) (liftN 1 body✝ (k + 1)) k_1
x✝ : ∃ e', Skips' n e' k ∧ lam (liftN 1 binderType✝ k) (liftN 1 body✝ (k + 1)) = liftN 1 e' k
⊢ (∃ e', Skips' n e' k ∧ liftN 1 binderType✝ k = liftN 1 e' k) ∧
∃ e', Skips' n e' (k + 1) ∧ liftN 1 body✝ (k + 1) = liftN 1 e' (k + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case succ.lam.refine_1
n : Nat
f a : VExpr
fIH : ∀ {k : Nat}, (∃ e', Skips' n e' k ∧ f = liftN 1 e' k) ↔ Skips' (n + 1) f k
aIH : ∀ {k : Nat}, (∃ e', Skips' n e' k ∧ a = liftN 1 e' k) ↔ Skips' (n + 1) a k
k : Nat
x✝ : ∃ e', Skips' n e' k ∧ lam f a = liftN 1 e' k
e' : VExpr
h1 : Skips' n e' k
h2 : lam f a = liftN 1 e' k
⊢ (∃ e', Skips' n e' k ∧ f = liftN 1 e' k) ∧ ∃ e', Skips' n e' (k + 1) ∧ a = liftN 1 e' (k + 1)
TACTIC:
|
https://github.com/digama0/lean4lean.git | c534f13d8d25f5e1891b6d18cc76b601ee87aa66 | Lean4Lean/Theory/VExpr.lean | Lean4Lean.VExpr.skips_iff | [343, 1] | [381, 78] | exact ⟨⟨_, h1.1, rfl⟩, ⟨_, h1.2, rfl⟩⟩ | case succ.lam.refine_1.lam.refl
n k : Nat
binderType✝ body✝ : VExpr
h1 : Skips' n (lam binderType✝ body✝) k
fIH :
∀ {k_1 : Nat},
(∃ e', Skips' n e' k_1 ∧ liftN 1 binderType✝ k = liftN 1 e' k_1) ↔ Skips' (n + 1) (liftN 1 binderType✝ k) k_1
aIH :
∀ {k_1 : Nat},
(∃ e', Skips' n e' k_1 ∧ liftN 1 body✝ (k + 1) = liftN 1 e' k_1) ↔ Skips' (n + 1) (liftN 1 body✝ (k + 1)) k_1
x✝ : ∃ e', Skips' n e' k ∧ lam (liftN 1 binderType✝ k) (liftN 1 body✝ (k + 1)) = liftN 1 e' k
⊢ (∃ e', Skips' n e' k ∧ liftN 1 binderType✝ k = liftN 1 e' k) ∧
∃ e', Skips' n e' (k + 1) ∧ liftN 1 body✝ (k + 1) = liftN 1 e' (k + 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ.lam.refine_1.lam.refl
n k : Nat
binderType✝ body✝ : VExpr
h1 : Skips' n (lam binderType✝ body✝) k
fIH :
∀ {k_1 : Nat},
(∃ e', Skips' n e' k_1 ∧ liftN 1 binderType✝ k = liftN 1 e' k_1) ↔ Skips' (n + 1) (liftN 1 binderType✝ k) k_1
aIH :
∀ {k_1 : Nat},
(∃ e', Skips' n e' k_1 ∧ liftN 1 body✝ (k + 1) = liftN 1 e' k_1) ↔ Skips' (n + 1) (liftN 1 body✝ (k + 1)) k_1
x✝ : ∃ e', Skips' n e' k ∧ lam (liftN 1 binderType✝ k) (liftN 1 body✝ (k + 1)) = liftN 1 e' k
⊢ (∃ e', Skips' n e' k ∧ liftN 1 binderType✝ k = liftN 1 e' k) ∧
∃ e', Skips' n e' (k + 1) ∧ liftN 1 body✝ (k + 1) = liftN 1 e' (k + 1)
TACTIC:
|
https://github.com/digama0/lean4lean.git | c534f13d8d25f5e1891b6d18cc76b601ee87aa66 | Lean4Lean/Theory/VExpr.lean | Lean4Lean.VExpr.skips_iff | [343, 1] | [381, 78] | rintro ⟨⟨e1, h1, rfl⟩, ⟨e2, h2, rfl⟩⟩ | case succ.lam.refine_2
n : Nat
f a : VExpr
fIH : ∀ {k : Nat}, (∃ e', Skips' n e' k ∧ f = liftN 1 e' k) ↔ Skips' (n + 1) f k
aIH : ∀ {k : Nat}, (∃ e', Skips' n e' k ∧ a = liftN 1 e' k) ↔ Skips' (n + 1) a k
k : Nat
⊢ ((∃ e', Skips' n e' k ∧ f = liftN 1 e' k) ∧ ∃ e', Skips' n e' (k + 1) ∧ a = liftN 1 e' (k + 1)) →
∃ e', Skips' n e' k ∧ lam f a = liftN 1 e' k | case succ.lam.refine_2.intro.intro.intro.intro.intro
n k : Nat
e1 : VExpr
h1 : Skips' n e1 k
fIH : ∀ {k_1 : Nat}, (∃ e', Skips' n e' k_1 ∧ liftN 1 e1 k = liftN 1 e' k_1) ↔ Skips' (n + 1) (liftN 1 e1 k) k_1
e2 : VExpr
h2 : Skips' n e2 (k + 1)
aIH :
∀ {k_1 : Nat}, (∃ e', Skips' n e' k_1 ∧ liftN 1 e2 (k + 1) = liftN 1 e' k_1) ↔ Skips' (n + 1) (liftN 1 e2 (k + 1)) k_1
⊢ ∃ e', Skips' n e' k ∧ lam (liftN 1 e1 k) (liftN 1 e2 (k + 1)) = liftN 1 e' k | Please generate a tactic in lean4 to solve the state.
STATE:
case succ.lam.refine_2
n : Nat
f a : VExpr
fIH : ∀ {k : Nat}, (∃ e', Skips' n e' k ∧ f = liftN 1 e' k) ↔ Skips' (n + 1) f k
aIH : ∀ {k : Nat}, (∃ e', Skips' n e' k ∧ a = liftN 1 e' k) ↔ Skips' (n + 1) a k
k : Nat
⊢ ((∃ e', Skips' n e' k ∧ f = liftN 1 e' k) ∧ ∃ e', Skips' n e' (k + 1) ∧ a = liftN 1 e' (k + 1)) →
∃ e', Skips' n e' k ∧ lam f a = liftN 1 e' k
TACTIC:
|
https://github.com/digama0/lean4lean.git | c534f13d8d25f5e1891b6d18cc76b601ee87aa66 | Lean4Lean/Theory/VExpr.lean | Lean4Lean.VExpr.skips_iff | [343, 1] | [381, 78] | exact ⟨.lam .., ⟨h1, h2⟩, rfl⟩ | case succ.lam.refine_2.intro.intro.intro.intro.intro
n k : Nat
e1 : VExpr
h1 : Skips' n e1 k
fIH : ∀ {k_1 : Nat}, (∃ e', Skips' n e' k_1 ∧ liftN 1 e1 k = liftN 1 e' k_1) ↔ Skips' (n + 1) (liftN 1 e1 k) k_1
e2 : VExpr
h2 : Skips' n e2 (k + 1)
aIH :
∀ {k_1 : Nat}, (∃ e', Skips' n e' k_1 ∧ liftN 1 e2 (k + 1) = liftN 1 e' k_1) ↔ Skips' (n + 1) (liftN 1 e2 (k + 1)) k_1
⊢ ∃ e', Skips' n e' k ∧ lam (liftN 1 e1 k) (liftN 1 e2 (k + 1)) = liftN 1 e' k | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ.lam.refine_2.intro.intro.intro.intro.intro
n k : Nat
e1 : VExpr
h1 : Skips' n e1 k
fIH : ∀ {k_1 : Nat}, (∃ e', Skips' n e' k_1 ∧ liftN 1 e1 k = liftN 1 e' k_1) ↔ Skips' (n + 1) (liftN 1 e1 k) k_1
e2 : VExpr
h2 : Skips' n e2 (k + 1)
aIH :
∀ {k_1 : Nat}, (∃ e', Skips' n e' k_1 ∧ liftN 1 e2 (k + 1) = liftN 1 e' k_1) ↔ Skips' (n + 1) (liftN 1 e2 (k + 1)) k_1
⊢ ∃ e', Skips' n e' k ∧ lam (liftN 1 e1 k) (liftN 1 e2 (k + 1)) = liftN 1 e' k
TACTIC:
|
https://github.com/digama0/lean4lean.git | c534f13d8d25f5e1891b6d18cc76b601ee87aa66 | Lean4Lean/Theory/VExpr.lean | Lean4Lean.VExpr.of_liftN_eq_liftN | [383, 1] | [390, 38] | have : (liftN n1 e1 (k1+n2+k2)).Skips n2 k2 := h ▸ .liftN | n1 : Nat
e1 : VExpr
k1 n2 k2 : Nat
e2 : VExpr
h : liftN n1 e1 (k1 + n2 + k2) = liftN n2 e2 k2
⊢ ∃ e', e1 = liftN n2 e' k2 ∧ e2 = liftN n1 e' (k1 + k2) | n1 : Nat
e1 : VExpr
k1 n2 k2 : Nat
e2 : VExpr
h : liftN n1 e1 (k1 + n2 + k2) = liftN n2 e2 k2
this : Skips (liftN n1 e1 (k1 + n2 + k2)) n2 k2
⊢ ∃ e', e1 = liftN n2 e' k2 ∧ e2 = liftN n1 e' (k1 + k2) | Please generate a tactic in lean4 to solve the state.
STATE:
n1 : Nat
e1 : VExpr
k1 n2 k2 : Nat
e2 : VExpr
h : liftN n1 e1 (k1 + n2 + k2) = liftN n2 e2 k2
⊢ ∃ e', e1 = liftN n2 e' k2 ∧ e2 = liftN n1 e' (k1 + k2)
TACTIC:
|
https://github.com/digama0/lean4lean.git | c534f13d8d25f5e1891b6d18cc76b601ee87aa66 | Lean4Lean/Theory/VExpr.lean | Lean4Lean.VExpr.of_liftN_eq_liftN | [383, 1] | [390, 38] | obtain ⟨e', rfl⟩ := skips_iff_exists.1 <|
this.of_liftN_hi (Nat.add_assoc .. ▸ Nat.le_add_left ..) | n1 : Nat
e1 : VExpr
k1 n2 k2 : Nat
e2 : VExpr
h : liftN n1 e1 (k1 + n2 + k2) = liftN n2 e2 k2
this : Skips (liftN n1 e1 (k1 + n2 + k2)) n2 k2
⊢ ∃ e', e1 = liftN n2 e' k2 ∧ e2 = liftN n1 e' (k1 + k2) | case intro
n1 k1 n2 k2 : Nat
e2 e' : VExpr
h : liftN n1 (liftN n2 e' k2) (k1 + n2 + k2) = liftN n2 e2 k2
this : Skips (liftN n1 (liftN n2 e' k2) (k1 + n2 + k2)) n2 k2
⊢ ∃ e'_1, liftN n2 e' k2 = liftN n2 e'_1 k2 ∧ e2 = liftN n1 e'_1 (k1 + k2) | Please generate a tactic in lean4 to solve the state.
STATE:
n1 : Nat
e1 : VExpr
k1 n2 k2 : Nat
e2 : VExpr
h : liftN n1 e1 (k1 + n2 + k2) = liftN n2 e2 k2
this : Skips (liftN n1 e1 (k1 + n2 + k2)) n2 k2
⊢ ∃ e', e1 = liftN n2 e' k2 ∧ e2 = liftN n1 e' (k1 + k2)
TACTIC:
|
https://github.com/digama0/lean4lean.git | c534f13d8d25f5e1891b6d18cc76b601ee87aa66 | Lean4Lean/Theory/VExpr.lean | Lean4Lean.VExpr.of_liftN_eq_liftN | [383, 1] | [390, 38] | refine ⟨e', rfl, ?_⟩ | case intro
n1 k1 n2 k2 : Nat
e2 e' : VExpr
h : liftN n1 (liftN n2 e' k2) (k1 + n2 + k2) = liftN n2 e2 k2
this : Skips (liftN n1 (liftN n2 e' k2) (k1 + n2 + k2)) n2 k2
⊢ ∃ e'_1, liftN n2 e' k2 = liftN n2 e'_1 k2 ∧ e2 = liftN n1 e'_1 (k1 + k2) | case intro
n1 k1 n2 k2 : Nat
e2 e' : VExpr
h : liftN n1 (liftN n2 e' k2) (k1 + n2 + k2) = liftN n2 e2 k2
this : Skips (liftN n1 (liftN n2 e' k2) (k1 + n2 + k2)) n2 k2
⊢ e2 = liftN n1 e' (k1 + k2) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
n1 k1 n2 k2 : Nat
e2 e' : VExpr
h : liftN n1 (liftN n2 e' k2) (k1 + n2 + k2) = liftN n2 e2 k2
this : Skips (liftN n1 (liftN n2 e' k2) (k1 + n2 + k2)) n2 k2
⊢ ∃ e'_1, liftN n2 e' k2 = liftN n2 e'_1 k2 ∧ e2 = liftN n1 e'_1 (k1 + k2)
TACTIC:
|
https://github.com/digama0/lean4lean.git | c534f13d8d25f5e1891b6d18cc76b601ee87aa66 | Lean4Lean/Theory/VExpr.lean | Lean4Lean.VExpr.of_liftN_eq_liftN | [383, 1] | [390, 38] | rw [← liftN_inj, ← h, liftN'_comm (n1 := n1) (h := Nat.le_add_left ..),
Nat.add_left_comm, Nat.add_assoc] | case intro
n1 k1 n2 k2 : Nat
e2 e' : VExpr
h : liftN n1 (liftN n2 e' k2) (k1 + n2 + k2) = liftN n2 e2 k2
this : Skips (liftN n1 (liftN n2 e' k2) (k1 + n2 + k2)) n2 k2
⊢ e2 = liftN n1 e' (k1 + k2) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
n1 k1 n2 k2 : Nat
e2 e' : VExpr
h : liftN n1 (liftN n2 e' k2) (k1 + n2 + k2) = liftN n2 e2 k2
this : Skips (liftN n1 (liftN n2 e' k2) (k1 + n2 + k2)) n2 k2
⊢ e2 = liftN n1 e' (k1 + k2)
TACTIC:
|
https://github.com/digama0/lean4lean.git | c534f13d8d25f5e1891b6d18cc76b601ee87aa66 | Lean4Lean/Theory/VExpr.lean | Lean4Lean.VExpr.instL_instN | [392, 9] | [394, 56] | induction e1 generalizing k <;> simp [instL, inst, *] | k : Nat
ls : List VLevel
e1 e2 : VExpr
⊢ instL ls (inst e1 e2 k) = inst (instL ls e1) (instL ls e2) k | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
k : Nat
ls : List VLevel
e1 e2 : VExpr
⊢ instL ls (inst e1 e2 k) = inst (instL ls e1) (instL ls e2) k
TACTIC:
|
https://github.com/digama0/lean4lean.git | c534f13d8d25f5e1891b6d18cc76b601ee87aa66 | Lean4Lean/Theory/VExpr.lean | Lean4Lean.VExpr.instL_unliftN | [396, 1] | [398, 43] | induction n generalizing e with simp [unliftN]
| succ _ ih => rw [ih, instL_instN]; rfl | ls : List VLevel
e : VExpr
n k : Nat
⊢ instL ls (unliftN e n k) = unliftN (instL ls e) n k | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ls : List VLevel
e : VExpr
n k : Nat
⊢ instL ls (unliftN e n k) = unliftN (instL ls e) n k
TACTIC:
|
https://github.com/digama0/lean4lean.git | c534f13d8d25f5e1891b6d18cc76b601ee87aa66 | Lean4Lean/Theory/VExpr.lean | Lean4Lean.VExpr.instL_unliftN | [396, 1] | [398, 43] | rw [ih, instL_instN] | case succ
ls : List VLevel
k n✝ : Nat
ih : ∀ {e : VExpr}, instL ls (unliftN e n✝ k) = unliftN (instL ls e) n✝ k
e : VExpr
⊢ instL ls (unliftN (inst e default k) n✝ k) = unliftN (inst (instL ls e) default k) n✝ k | case succ
ls : List VLevel
k n✝ : Nat
ih : ∀ {e : VExpr}, instL ls (unliftN e n✝ k) = unliftN (instL ls e) n✝ k
e : VExpr
⊢ unliftN (inst (instL ls e) (instL ls default) k) n✝ k = unliftN (inst (instL ls e) default k) n✝ k | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
ls : List VLevel
k n✝ : Nat
ih : ∀ {e : VExpr}, instL ls (unliftN e n✝ k) = unliftN (instL ls e) n✝ k
e : VExpr
⊢ instL ls (unliftN (inst e default k) n✝ k) = unliftN (inst (instL ls e) default k) n✝ k
TACTIC:
|
https://github.com/digama0/lean4lean.git | c534f13d8d25f5e1891b6d18cc76b601ee87aa66 | Lean4Lean/Theory/VExpr.lean | Lean4Lean.VExpr.instL_unliftN | [396, 1] | [398, 43] | rfl | case succ
ls : List VLevel
k n✝ : Nat
ih : ∀ {e : VExpr}, instL ls (unliftN e n✝ k) = unliftN (instL ls e) n✝ k
e : VExpr
⊢ unliftN (inst (instL ls e) (instL ls default) k) n✝ k = unliftN (inst (instL ls e) default k) n✝ k | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
ls : List VLevel
k n✝ : Nat
ih : ∀ {e : VExpr}, instL ls (unliftN e n✝ k) = unliftN (instL ls e) n✝ k
e : VExpr
⊢ unliftN (inst (instL ls e) (instL ls default) k) n✝ k = unliftN (inst (instL ls e) default k) n✝ k
TACTIC:
|
https://github.com/digama0/lean4lean.git | c534f13d8d25f5e1891b6d18cc76b601ee87aa66 | Lean4Lean/Theory/VExpr.lean | Lean4Lean.VExpr.Skips.of_instL | [400, 1] | [402, 51] | rw [skips_iff] at self ⊢ | ls : List VLevel
e : VExpr
n k : Nat
self : Skips (instL ls e) n k
⊢ Skips e n k | ls : List VLevel
e : VExpr
n k : Nat
self : Skips' n (instL ls e) k
⊢ Skips' n e k | Please generate a tactic in lean4 to solve the state.
STATE:
ls : List VLevel
e : VExpr
n k : Nat
self : Skips (instL ls e) n k
⊢ Skips e n k
TACTIC:
|
https://github.com/digama0/lean4lean.git | c534f13d8d25f5e1891b6d18cc76b601ee87aa66 | Lean4Lean/Theory/VExpr.lean | Lean4Lean.VExpr.Skips.of_instL | [400, 1] | [402, 51] | induction e generalizing k <;> simp_all [Skips'] | ls : List VLevel
e : VExpr
n k : Nat
self : Skips' n (instL ls e) k
⊢ Skips' n e k | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ls : List VLevel
e : VExpr
n k : Nat
self : Skips' n (instL ls e) k
⊢ Skips' n e k
TACTIC:
|
https://github.com/digama0/lean4lean.git | c534f13d8d25f5e1891b6d18cc76b601ee87aa66 | Lean4Lean/Theory/VExpr.lean | Lean4Lean.VExpr.of_liftN_eq_instL | [404, 1] | [409, 35] | have : (instL ls e2).Skips n k := h ▸ .liftN | n : Nat
e1 : VExpr
k : Nat
ls : List VLevel
e2 : VExpr
h : liftN n e1 k = instL ls e2
⊢ ∃ e', e1 = instL ls e' ∧ e2 = liftN n e' k | n : Nat
e1 : VExpr
k : Nat
ls : List VLevel
e2 : VExpr
h : liftN n e1 k = instL ls e2
this : Skips (instL ls e2) n k
⊢ ∃ e', e1 = instL ls e' ∧ e2 = liftN n e' k | Please generate a tactic in lean4 to solve the state.
STATE:
n : Nat
e1 : VExpr
k : Nat
ls : List VLevel
e2 : VExpr
h : liftN n e1 k = instL ls e2
⊢ ∃ e', e1 = instL ls e' ∧ e2 = liftN n e' k
TACTIC:
|
https://github.com/digama0/lean4lean.git | c534f13d8d25f5e1891b6d18cc76b601ee87aa66 | Lean4Lean/Theory/VExpr.lean | Lean4Lean.VExpr.of_liftN_eq_instL | [404, 1] | [409, 35] | obtain ⟨e', rfl⟩ := skips_iff_exists.1 this.of_instL | n : Nat
e1 : VExpr
k : Nat
ls : List VLevel
e2 : VExpr
h : liftN n e1 k = instL ls e2
this : Skips (instL ls e2) n k
⊢ ∃ e', e1 = instL ls e' ∧ e2 = liftN n e' k | case intro
n : Nat
e1 : VExpr
k : Nat
ls : List VLevel
e' : VExpr
h : liftN n e1 k = instL ls (liftN n e' k)
this : Skips (instL ls (liftN n e' k)) n k
⊢ ∃ e'_1, e1 = instL ls e'_1 ∧ liftN n e' k = liftN n e'_1 k | Please generate a tactic in lean4 to solve the state.
STATE:
n : Nat
e1 : VExpr
k : Nat
ls : List VLevel
e2 : VExpr
h : liftN n e1 k = instL ls e2
this : Skips (instL ls e2) n k
⊢ ∃ e', e1 = instL ls e' ∧ e2 = liftN n e' k
TACTIC:
|
https://github.com/digama0/lean4lean.git | c534f13d8d25f5e1891b6d18cc76b601ee87aa66 | Lean4Lean/Theory/VExpr.lean | Lean4Lean.VExpr.of_liftN_eq_instL | [404, 1] | [409, 35] | refine ⟨e', ?_, rfl⟩ | case intro
n : Nat
e1 : VExpr
k : Nat
ls : List VLevel
e' : VExpr
h : liftN n e1 k = instL ls (liftN n e' k)
this : Skips (instL ls (liftN n e' k)) n k
⊢ ∃ e'_1, e1 = instL ls e'_1 ∧ liftN n e' k = liftN n e'_1 k | case intro
n : Nat
e1 : VExpr
k : Nat
ls : List VLevel
e' : VExpr
h : liftN n e1 k = instL ls (liftN n e' k)
this : Skips (instL ls (liftN n e' k)) n k
⊢ e1 = instL ls e' | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
n : Nat
e1 : VExpr
k : Nat
ls : List VLevel
e' : VExpr
h : liftN n e1 k = instL ls (liftN n e' k)
this : Skips (instL ls (liftN n e' k)) n k
⊢ ∃ e'_1, e1 = instL ls e'_1 ∧ liftN n e' k = liftN n e'_1 k
TACTIC:
|
https://github.com/digama0/lean4lean.git | c534f13d8d25f5e1891b6d18cc76b601ee87aa66 | Lean4Lean/Theory/VExpr.lean | Lean4Lean.VExpr.of_liftN_eq_instL | [404, 1] | [409, 35] | rw [← liftN_inj, h, instL_liftN] | case intro
n : Nat
e1 : VExpr
k : Nat
ls : List VLevel
e' : VExpr
h : liftN n e1 k = instL ls (liftN n e' k)
this : Skips (instL ls (liftN n e' k)) n k
⊢ e1 = instL ls e' | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
n : Nat
e1 : VExpr
k : Nat
ls : List VLevel
e' : VExpr
h : liftN n e1 k = instL ls (liftN n e' k)
this : Skips (instL ls (liftN n e' k)) n k
⊢ e1 = instL ls e'
TACTIC:
|
https://github.com/digama0/lean4lean.git | c534f13d8d25f5e1891b6d18cc76b601ee87aa66 | Lean4Lean/Theory/VExpr.lean | Lean4Lean.VExpr.ClosedN.instN_eq | [411, 1] | [413, 18] | conv => lhs; rw [← self.liftN_eq (n := 1) h] | e1 : VExpr
k j : Nat
e2 : VExpr
self : ClosedN e1 k
h : k ≤ j
⊢ inst e1 e2 j = e1 | e1 : VExpr
k j : Nat
e2 : VExpr
self : ClosedN e1 k
h : k ≤ j
⊢ inst (liftN 1 e1 j) e2 j = e1 | Please generate a tactic in lean4 to solve the state.
STATE:
e1 : VExpr
k j : Nat
e2 : VExpr
self : ClosedN e1 k
h : k ≤ j
⊢ inst e1 e2 j = e1
TACTIC:
|
https://github.com/digama0/lean4lean.git | c534f13d8d25f5e1891b6d18cc76b601ee87aa66 | Lean4Lean/Theory/VExpr.lean | Lean4Lean.VExpr.ClosedN.instN_eq | [411, 1] | [413, 18] | rw [inst_liftN] | e1 : VExpr
k j : Nat
e2 : VExpr
self : ClosedN e1 k
h : k ≤ j
⊢ inst (liftN 1 e1 j) e2 j = e1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
e1 : VExpr
k j : Nat
e2 : VExpr
self : ClosedN e1 k
h : k ≤ j
⊢ inst (liftN 1 e1 j) e2 j = e1
TACTIC:
|
https://github.com/digama0/lean4lean.git | c534f13d8d25f5e1891b6d18cc76b601ee87aa66 | Lean4Lean/Theory/VExpr.lean | Lean4Lean.VExpr.ClosedN.instN | [415, 1] | [427, 75] | simp [inst, instVar] | e : VExpr
k j : Nat
e2 : VExpr
h1 : ClosedN e (k + j + 1)
h2 : ClosedN e2 k
i : Nat
h : ClosedN (bvar i) (k + j + 1)
⊢ ClosedN (inst (bvar i) e2 j) (k + j) | e : VExpr
k j : Nat
e2 : VExpr
h1 : ClosedN e (k + j + 1)
h2 : ClosedN e2 k
i : Nat
h : ClosedN (bvar i) (k + j + 1)
⊢ ClosedN (if i < j then bvar i else if i = j then liftN j e2 else bvar (i - 1)) (k + j) | Please generate a tactic in lean4 to solve the state.
STATE:
e : VExpr
k j : Nat
e2 : VExpr
h1 : ClosedN e (k + j + 1)
h2 : ClosedN e2 k
i : Nat
h : ClosedN (bvar i) (k + j + 1)
⊢ ClosedN (inst (bvar i) e2 j) (k + j)
TACTIC:
|
https://github.com/digama0/lean4lean.git | c534f13d8d25f5e1891b6d18cc76b601ee87aa66 | Lean4Lean/Theory/VExpr.lean | Lean4Lean.VExpr.ClosedN.instN | [415, 1] | [427, 75] | split <;> rename_i h1 | e : VExpr
k j : Nat
e2 : VExpr
h1 : ClosedN e (k + j + 1)
h2 : ClosedN e2 k
i : Nat
h : ClosedN (bvar i) (k + j + 1)
⊢ ClosedN (if i < j then bvar i else if i = j then liftN j e2 else bvar (i - 1)) (k + j) | case inl
e : VExpr
k j : Nat
e2 : VExpr
h1✝ : ClosedN e (k + j + 1)
h2 : ClosedN e2 k
i : Nat
h : ClosedN (bvar i) (k + j + 1)
h1 : i < j
⊢ ClosedN (bvar i) (k + j)
case inr
e : VExpr
k j : Nat
e2 : VExpr
h1✝ : ClosedN e (k + j + 1)
h2 : ClosedN e2 k
i : Nat
h : ClosedN (bvar i) (k + j + 1)
h1 : ¬i < j
⊢ ClosedN (if i = j then liftN j e2 else bvar (i - 1)) (k + j) | Please generate a tactic in lean4 to solve the state.
STATE:
e : VExpr
k j : Nat
e2 : VExpr
h1 : ClosedN e (k + j + 1)
h2 : ClosedN e2 k
i : Nat
h : ClosedN (bvar i) (k + j + 1)
⊢ ClosedN (if i < j then bvar i else if i = j then liftN j e2 else bvar (i - 1)) (k + j)
TACTIC:
|
https://github.com/digama0/lean4lean.git | c534f13d8d25f5e1891b6d18cc76b601ee87aa66 | Lean4Lean/Theory/VExpr.lean | Lean4Lean.VExpr.ClosedN.instN | [415, 1] | [427, 75] | split <;> rename_i h1' | case inr
e : VExpr
k j : Nat
e2 : VExpr
h1✝ : ClosedN e (k + j + 1)
h2 : ClosedN e2 k
i : Nat
h : ClosedN (bvar i) (k + j + 1)
h1 : ¬i < j
⊢ ClosedN (if i = j then liftN j e2 else bvar (i - 1)) (k + j) | case inr.inl
e : VExpr
k j : Nat
e2 : VExpr
h1✝ : ClosedN e (k + j + 1)
h2 : ClosedN e2 k
i : Nat
h : ClosedN (bvar i) (k + j + 1)
h1 : ¬i < j
h1' : i = j
⊢ ClosedN (liftN j e2) (k + j)
case inr.inr
e : VExpr
k j : Nat
e2 : VExpr
h1✝ : ClosedN e (k + j + 1)
h2 : ClosedN e2 k
i : Nat
h : ClosedN (bvar i) (k + j + 1)
h1 : ¬i < j
h1' : ¬i = j
⊢ ClosedN (bvar (i - 1)) (k + j) | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
e : VExpr
k j : Nat
e2 : VExpr
h1✝ : ClosedN e (k + j + 1)
h2 : ClosedN e2 k
i : Nat
h : ClosedN (bvar i) (k + j + 1)
h1 : ¬i < j
⊢ ClosedN (if i = j then liftN j e2 else bvar (i - 1)) (k + j)
TACTIC:
|
https://github.com/digama0/lean4lean.git | c534f13d8d25f5e1891b6d18cc76b601ee87aa66 | Lean4Lean/Theory/VExpr.lean | Lean4Lean.VExpr.ClosedN.instN | [415, 1] | [427, 75] | exact Nat.lt_of_lt_of_le h1 (Nat.le_add_left ..) | case inl
e : VExpr
k j : Nat
e2 : VExpr
h1✝ : ClosedN e (k + j + 1)
h2 : ClosedN e2 k
i : Nat
h : ClosedN (bvar i) (k + j + 1)
h1 : i < j
⊢ ClosedN (bvar i) (k + j) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
e : VExpr
k j : Nat
e2 : VExpr
h1✝ : ClosedN e (k + j + 1)
h2 : ClosedN e2 k
i : Nat
h : ClosedN (bvar i) (k + j + 1)
h1 : i < j
⊢ ClosedN (bvar i) (k + j)
TACTIC:
|
https://github.com/digama0/lean4lean.git | c534f13d8d25f5e1891b6d18cc76b601ee87aa66 | Lean4Lean/Theory/VExpr.lean | Lean4Lean.VExpr.ClosedN.instN | [415, 1] | [427, 75] | exact h2.liftN | case inr.inl
e : VExpr
k j : Nat
e2 : VExpr
h1✝ : ClosedN e (k + j + 1)
h2 : ClosedN e2 k
i : Nat
h : ClosedN (bvar i) (k + j + 1)
h1 : ¬i < j
h1' : i = j
⊢ ClosedN (liftN j e2) (k + j) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inl
e : VExpr
k j : Nat
e2 : VExpr
h1✝ : ClosedN e (k + j + 1)
h2 : ClosedN e2 k
i : Nat
h : ClosedN (bvar i) (k + j + 1)
h1 : ¬i < j
h1' : i = j
⊢ ClosedN (liftN j e2) (k + j)
TACTIC:
|
https://github.com/digama0/lean4lean.git | c534f13d8d25f5e1891b6d18cc76b601ee87aa66 | Lean4Lean/Theory/VExpr.lean | Lean4Lean.VExpr.ClosedN.instN | [415, 1] | [427, 75] | have hk := Nat.lt_of_le_of_ne (Nat.not_lt.1 h1) (Ne.symm h1') | case inr.inr
e : VExpr
k j : Nat
e2 : VExpr
h1✝ : ClosedN e (k + j + 1)
h2 : ClosedN e2 k
i : Nat
h : ClosedN (bvar i) (k + j + 1)
h1 : ¬i < j
h1' : ¬i = j
⊢ ClosedN (bvar (i - 1)) (k + j) | case inr.inr
e : VExpr
k j : Nat
e2 : VExpr
h1✝ : ClosedN e (k + j + 1)
h2 : ClosedN e2 k
i : Nat
h : ClosedN (bvar i) (k + j + 1)
h1 : ¬i < j
h1' : ¬i = j
hk : j < i
⊢ ClosedN (bvar (i - 1)) (k + j) | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr
e : VExpr
k j : Nat
e2 : VExpr
h1✝ : ClosedN e (k + j + 1)
h2 : ClosedN e2 k
i : Nat
h : ClosedN (bvar i) (k + j + 1)
h1 : ¬i < j
h1' : ¬i = j
⊢ ClosedN (bvar (i - 1)) (k + j)
TACTIC:
|
https://github.com/digama0/lean4lean.git | c534f13d8d25f5e1891b6d18cc76b601ee87aa66 | Lean4Lean/Theory/VExpr.lean | Lean4Lean.VExpr.ClosedN.instN | [415, 1] | [427, 75] | let i+1 := i | case inr.inr
e : VExpr
k j : Nat
e2 : VExpr
h1✝ : ClosedN e (k + j + 1)
h2 : ClosedN e2 k
i : Nat
h : ClosedN (bvar i) (k + j + 1)
h1 : ¬i < j
h1' : ¬i = j
hk : j < i
⊢ ClosedN (bvar (i - 1)) (k + j) | case inr.inr
e : VExpr
k j : Nat
e2 : VExpr
h1✝ : ClosedN e (k + j + 1)
h2 : ClosedN e2 k
i✝ i : Nat
h : ClosedN (bvar (i + 1)) (k + j + 1)
h1 : ¬i + 1 < j
h1' : ¬i + 1 = j
hk : j < i + 1
⊢ ClosedN (bvar (i + 1 - 1)) (k + j) | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr
e : VExpr
k j : Nat
e2 : VExpr
h1✝ : ClosedN e (k + j + 1)
h2 : ClosedN e2 k
i : Nat
h : ClosedN (bvar i) (k + j + 1)
h1 : ¬i < j
h1' : ¬i = j
hk : j < i
⊢ ClosedN (bvar (i - 1)) (k + j)
TACTIC:
|
https://github.com/digama0/lean4lean.git | c534f13d8d25f5e1891b6d18cc76b601ee87aa66 | Lean4Lean/Theory/VExpr.lean | Lean4Lean.VExpr.ClosedN.instN | [415, 1] | [427, 75] | exact Nat.lt_of_succ_lt_succ h | case inr.inr
e : VExpr
k j : Nat
e2 : VExpr
h1✝ : ClosedN e (k + j + 1)
h2 : ClosedN e2 k
i✝ i : Nat
h : ClosedN (bvar (i + 1)) (k + j + 1)
h1 : ¬i + 1 < j
h1' : ¬i + 1 = j
hk : j < i + 1
⊢ ClosedN (bvar (i + 1 - 1)) (k + j) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr
e : VExpr
k j : Nat
e2 : VExpr
h1✝ : ClosedN e (k + j + 1)
h2 : ClosedN e2 k
i✝ i : Nat
h : ClosedN (bvar (i + 1)) (k + j + 1)
h1 : ¬i + 1 < j
h1' : ¬i + 1 = j
hk : j < i + 1
⊢ ClosedN (bvar (i + 1 - 1)) (k + j)
TACTIC:
|
https://github.com/digama0/lean4lean.git | c534f13d8d25f5e1891b6d18cc76b601ee87aa66 | Lean4Lean/Theory/VExpr.lean | Lean4Lean.VExpr.inst_instVar_hi | [432, 1] | [453, 66] | simp [instVar] | i : Nat
e2 e3 : VExpr
k j : Nat
⊢ inst (instVar i e2 k) e3 (j + k) = inst (instVar i e3 (j + k + 1)) (inst e2 e3 j) k | i : Nat
e2 e3 : VExpr
k j : Nat
⊢ inst (if i < k then bvar i else if i = k then liftN k e2 else bvar (i - 1)) e3 (j + k) =
inst (if i < j + k + 1 then bvar i else if i = j + k + 1 then liftN (j + k + 1) e3 else bvar (i - 1)) (inst e2 e3 j)
k | Please generate a tactic in lean4 to solve the state.
STATE:
i : Nat
e2 e3 : VExpr
k j : Nat
⊢ inst (instVar i e2 k) e3 (j + k) = inst (instVar i e3 (j + k + 1)) (inst e2 e3 j) k
TACTIC:
|
https://github.com/digama0/lean4lean.git | c534f13d8d25f5e1891b6d18cc76b601ee87aa66 | Lean4Lean/Theory/VExpr.lean | Lean4Lean.VExpr.inst_instVar_hi | [432, 1] | [453, 66] | split <;> rename_i h | i : Nat
e2 e3 : VExpr
k j : Nat
⊢ inst (if i < k then bvar i else if i = k then liftN k e2 else bvar (i - 1)) e3 (j + k) =
inst (if i < j + k + 1 then bvar i else if i = j + k + 1 then liftN (j + k + 1) e3 else bvar (i - 1)) (inst e2 e3 j)
k | case inl
i : Nat
e2 e3 : VExpr
k j : Nat
h : i < k
⊢ inst (bvar i) e3 (j + k) =
inst (if i < j + k + 1 then bvar i else if i = j + k + 1 then liftN (j + k + 1) e3 else bvar (i - 1)) (inst e2 e3 j)
k
case inr
i : Nat
e2 e3 : VExpr
k j : Nat
h : ¬i < k
⊢ inst (if i = k then liftN k e2 else bvar (i - 1)) e3 (j + k) =
inst (if i < j + k + 1 then bvar i else if i = j + k + 1 then liftN (j + k + 1) e3 else bvar (i - 1)) (inst e2 e3 j)
k | Please generate a tactic in lean4 to solve the state.
STATE:
i : Nat
e2 e3 : VExpr
k j : Nat
⊢ inst (if i < k then bvar i else if i = k then liftN k e2 else bvar (i - 1)) e3 (j + k) =
inst (if i < j + k + 1 then bvar i else if i = j + k + 1 then liftN (j + k + 1) e3 else bvar (i - 1)) (inst e2 e3 j)
k
TACTIC:
|
https://github.com/digama0/lean4lean.git | c534f13d8d25f5e1891b6d18cc76b601ee87aa66 | Lean4Lean/Theory/VExpr.lean | Lean4Lean.VExpr.inst_instVar_hi | [432, 1] | [453, 66] | split <;> rename_i h' | case inr
i : Nat
e2 e3 : VExpr
k j : Nat
h : ¬i < k
⊢ inst (if i = k then liftN k e2 else bvar (i - 1)) e3 (j + k) =
inst (if i < j + k + 1 then bvar i else if i = j + k + 1 then liftN (j + k + 1) e3 else bvar (i - 1)) (inst e2 e3 j)
k | case inr.inl
i : Nat
e2 e3 : VExpr
k j : Nat
h : ¬i < k
h' : i = k
⊢ inst (liftN k e2) e3 (j + k) =
inst (if i < j + k + 1 then bvar i else if i = j + k + 1 then liftN (j + k + 1) e3 else bvar (i - 1)) (inst e2 e3 j)
k
case inr.inr
i : Nat
e2 e3 : VExpr
k j : Nat
h : ¬i < k
h' : ¬i = k
⊢ inst (bvar (i - 1)) e3 (j + k) =
inst (if i < j + k + 1 then bvar i else if i = j + k + 1 then liftN (j + k + 1) e3 else bvar (i - 1)) (inst e2 e3 j)
k | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
i : Nat
e2 e3 : VExpr
k j : Nat
h : ¬i < k
⊢ inst (if i = k then liftN k e2 else bvar (i - 1)) e3 (j + k) =
inst (if i < j + k + 1 then bvar i else if i = j + k + 1 then liftN (j + k + 1) e3 else bvar (i - 1)) (inst e2 e3 j)
k
TACTIC:
|
https://github.com/digama0/lean4lean.git | c534f13d8d25f5e1891b6d18cc76b601ee87aa66 | Lean4Lean/Theory/VExpr.lean | Lean4Lean.VExpr.inst_instVar_hi | [432, 1] | [453, 66] | simp [Nat.lt_succ_of_lt, inst, instVar, h, Nat.lt_of_lt_of_le h (Nat.le_add_left k j)] | case inl
i : Nat
e2 e3 : VExpr
k j : Nat
h : i < k
⊢ inst (bvar i) e3 (j + k) =
inst (if i < j + k + 1 then bvar i else if i = j + k + 1 then liftN (j + k + 1) e3 else bvar (i - 1)) (inst e2 e3 j)
k | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
i : Nat
e2 e3 : VExpr
k j : Nat
h : i < k
⊢ inst (bvar i) e3 (j + k) =
inst (if i < j + k + 1 then bvar i else if i = j + k + 1 then liftN (j + k + 1) e3 else bvar (i - 1)) (inst e2 e3 j)
k
TACTIC:
|
https://github.com/digama0/lean4lean.git | c534f13d8d25f5e1891b6d18cc76b601ee87aa66 | Lean4Lean/Theory/VExpr.lean | Lean4Lean.VExpr.inst_instVar_hi | [432, 1] | [453, 66] | subst i | case inr.inl
i : Nat
e2 e3 : VExpr
k j : Nat
h : ¬i < k
h' : i = k
⊢ inst (liftN k e2) e3 (j + k) =
inst (if i < j + k + 1 then bvar i else if i = j + k + 1 then liftN (j + k + 1) e3 else bvar (i - 1)) (inst e2 e3 j)
k | case inr.inl
e2 e3 : VExpr
k j : Nat
h : ¬k < k
⊢ inst (liftN k e2) e3 (j + k) =
inst (if k < j + k + 1 then bvar k else if k = j + k + 1 then liftN (j + k + 1) e3 else bvar (k - 1)) (inst e2 e3 j)
k | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inl
i : Nat
e2 e3 : VExpr
k j : Nat
h : ¬i < k
h' : i = k
⊢ inst (liftN k e2) e3 (j + k) =
inst (if i < j + k + 1 then bvar i else if i = j + k + 1 then liftN (j + k + 1) e3 else bvar (i - 1)) (inst e2 e3 j)
k
TACTIC:
|
https://github.com/digama0/lean4lean.git | c534f13d8d25f5e1891b6d18cc76b601ee87aa66 | Lean4Lean/Theory/VExpr.lean | Lean4Lean.VExpr.inst_instVar_hi | [432, 1] | [453, 66] | simp [Nat.lt_succ_of_le, Nat.le_add_left, inst, instVar] | case inr.inl
e2 e3 : VExpr
k j : Nat
h : ¬k < k
⊢ inst (liftN k e2) e3 (j + k) =
inst (if k < j + k + 1 then bvar k else if k = j + k + 1 then liftN (j + k + 1) e3 else bvar (k - 1)) (inst e2 e3 j)
k | case inr.inl
e2 e3 : VExpr
k j : Nat
h : ¬k < k
⊢ inst (liftN k e2) e3 (j + k) = liftN k (inst e2 e3 j) | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inl
e2 e3 : VExpr
k j : Nat
h : ¬k < k
⊢ inst (liftN k e2) e3 (j + k) =
inst (if k < j + k + 1 then bvar k else if k = j + k + 1 then liftN (j + k + 1) e3 else bvar (k - 1)) (inst e2 e3 j)
k
TACTIC:
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.