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/Int/Lemmas.lean | Int.mk_self | [28, 1] | [31, 45] | rw [succ_mk_succ] | case succ
m : Nat
ih : m ⊖ m = 0
⊢ m + 1 ⊖ m + 1 = 0 | case succ
m : Nat
ih : m ⊖ m = 0
⊢ m ⊖ m = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
m : Nat
ih : m ⊖ m = 0
⊢ m + 1 ⊖ m + 1 = 0
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Int/Lemmas.lean | Int.mk_self | [28, 1] | [31, 45] | exact ih | case succ
m : Nat
ih : m ⊖ m = 0
⊢ m ⊖ m = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
m : Nat
ih : m ⊖ m = 0
⊢ m ⊖ m = 0
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Int/Lemmas.lean | Int.add_mk_add_left | [33, 1] | [36, 73] | induction k with
| zero => rw [Nat.zero_add, Nat.zero_add]
| succ k ih => rw [Nat.succ_add, Nat.succ_add, succ_mk_succ]; exact ih | k m n : Nat
⊢ k + m ⊖ k + n = m ⊖ n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
k m n : Nat
⊢ k + m ⊖ k + n = m ⊖ n
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Int/Lemmas.lean | Int.add_mk_add_left | [33, 1] | [36, 73] | rw [Nat.zero_add, Nat.zero_add] | case zero
m n : Nat
⊢ 0 + m ⊖ 0 + n = m ⊖ n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
m n : Nat
⊢ 0 + m ⊖ 0 + n = m ⊖ n
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Int/Lemmas.lean | Int.add_mk_add_left | [33, 1] | [36, 73] | rw [Nat.succ_add, Nat.succ_add, succ_mk_succ] | case succ
m n k : Nat
ih : k + m ⊖ k + n = m ⊖ n
⊢ k + 1 + m ⊖ k + 1 + n = m ⊖ n | case succ
m n k : Nat
ih : k + m ⊖ k + n = m ⊖ n
⊢ k + m ⊖ k + n = m ⊖ n | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
m n k : Nat
ih : k + m ⊖ k + n = m ⊖ n
⊢ k + 1 + m ⊖ k + 1 + n = m ⊖ n
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Int/Lemmas.lean | Int.add_mk_add_left | [33, 1] | [36, 73] | exact ih | case succ
m n k : Nat
ih : k + m ⊖ k + n = m ⊖ n
⊢ k + m ⊖ k + n = m ⊖ n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
m n k : Nat
ih : k + m ⊖ k + n = m ⊖ n
⊢ k + m ⊖ k + n = m ⊖ n
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Int/Lemmas.lean | Int.add_mk_add_right | [38, 1] | [41, 81] | induction k with
| zero => rw [Nat.add_zero, Nat.add_zero]
| succ k ih => rw [Nat.add_succ m k, Nat.add_succ n k, succ_mk_succ]; exact ih | k m n : Nat
⊢ m + k ⊖ n + k = m ⊖ n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
k m n : Nat
⊢ m + k ⊖ n + k = m ⊖ n
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Int/Lemmas.lean | Int.add_mk_add_right | [38, 1] | [41, 81] | rw [Nat.add_zero, Nat.add_zero] | case zero
m n : Nat
⊢ m + 0 ⊖ n + 0 = m ⊖ n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
m n : Nat
⊢ m + 0 ⊖ n + 0 = m ⊖ n
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Int/Lemmas.lean | Int.add_mk_add_right | [38, 1] | [41, 81] | rw [Nat.add_succ m k, Nat.add_succ n k, succ_mk_succ] | case succ
m n k : Nat
ih : m + k ⊖ n + k = m ⊖ n
⊢ m + (k + 1) ⊖ n + (k + 1) = m ⊖ n | case succ
m n k : Nat
ih : m + k ⊖ n + k = m ⊖ n
⊢ m + k ⊖ n + k = m ⊖ n | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
m n k : Nat
ih : m + k ⊖ n + k = m ⊖ n
⊢ m + (k + 1) ⊖ n + (k + 1) = m ⊖ n
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Int/Lemmas.lean | Int.add_mk_add_right | [38, 1] | [41, 81] | exact ih | case succ
m n k : Nat
ih : m + k ⊖ n + k = m ⊖ n
⊢ m + k ⊖ n + k = m ⊖ n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
m n k : Nat
ih : m + k ⊖ n + k = m ⊖ n
⊢ m + k ⊖ n + k = m ⊖ n
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Int/Lemmas.lean | Int.mk_add_ofNat | [43, 1] | [48, 74] | induction m, n with
| zero_zero => rw [zero_mk_zero, Int.zero_add, Nat.zero_add, mk_zero]
| zero_succ n => rw [zero_mk_succ, Nat.zero_add]; rfl
| succ_zero m => rw [succ_mk_zero, mk_zero]; rfl
| succ_succ m n ih => rw [succ_mk_succ, Nat.succ_add, succ_mk_succ, ih] | m n k : Nat
⊢ (m ⊖ n) + ofNat k = m + k ⊖ n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
m n k : Nat
⊢ (m ⊖ n) + ofNat k = m + k ⊖ n
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Int/Lemmas.lean | Int.mk_add_ofNat | [43, 1] | [48, 74] | rw [zero_mk_zero, Int.zero_add, Nat.zero_add, mk_zero] | case zero_zero
k : Nat
⊢ (0 ⊖ 0) + ofNat k = 0 + k ⊖ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case zero_zero
k : Nat
⊢ (0 ⊖ 0) + ofNat k = 0 + k ⊖ 0
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Int/Lemmas.lean | Int.mk_add_ofNat | [43, 1] | [48, 74] | rw [zero_mk_succ, Nat.zero_add] | case zero_succ
k n : Nat
a✝ : (0 ⊖ n) + ofNat k = 0 + k ⊖ n
⊢ (0 ⊖ n + 1) + ofNat k = 0 + k ⊖ n + 1 | case zero_succ
k n : Nat
a✝ : (0 ⊖ n) + ofNat k = 0 + k ⊖ n
⊢ -[n+1] + ofNat k = k ⊖ n + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case zero_succ
k n : Nat
a✝ : (0 ⊖ n) + ofNat k = 0 + k ⊖ n
⊢ (0 ⊖ n + 1) + ofNat k = 0 + k ⊖ n + 1
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Int/Lemmas.lean | Int.mk_add_ofNat | [43, 1] | [48, 74] | rfl | case zero_succ
k n : Nat
a✝ : (0 ⊖ n) + ofNat k = 0 + k ⊖ n
⊢ -[n+1] + ofNat k = k ⊖ n + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case zero_succ
k n : Nat
a✝ : (0 ⊖ n) + ofNat k = 0 + k ⊖ n
⊢ -[n+1] + ofNat k = k ⊖ n + 1
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Int/Lemmas.lean | Int.mk_add_ofNat | [43, 1] | [48, 74] | rw [succ_mk_zero, mk_zero] | case succ_zero
k m : Nat
a✝ : (m ⊖ 0) + ofNat k = m + k ⊖ 0
⊢ (m + 1 ⊖ 0) + ofNat k = m + 1 + k ⊖ 0 | case succ_zero
k m : Nat
a✝ : (m ⊖ 0) + ofNat k = m + k ⊖ 0
⊢ ofNat (m + 1) + ofNat k = ofNat (m + 1 + k) | Please generate a tactic in lean4 to solve the state.
STATE:
case succ_zero
k m : Nat
a✝ : (m ⊖ 0) + ofNat k = m + k ⊖ 0
⊢ (m + 1 ⊖ 0) + ofNat k = m + 1 + k ⊖ 0
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Int/Lemmas.lean | Int.mk_add_ofNat | [43, 1] | [48, 74] | rfl | case succ_zero
k m : Nat
a✝ : (m ⊖ 0) + ofNat k = m + k ⊖ 0
⊢ ofNat (m + 1) + ofNat k = ofNat (m + 1 + k) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ_zero
k m : Nat
a✝ : (m ⊖ 0) + ofNat k = m + k ⊖ 0
⊢ ofNat (m + 1) + ofNat k = ofNat (m + 1 + k)
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Int/Lemmas.lean | Int.mk_add_ofNat | [43, 1] | [48, 74] | rw [succ_mk_succ, Nat.succ_add, succ_mk_succ, ih] | case succ_succ
k m n : Nat
ih : (m ⊖ n) + ofNat k = m + k ⊖ n
⊢ (m + 1 ⊖ n + 1) + ofNat k = m + 1 + k ⊖ n + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ_succ
k m n : Nat
ih : (m ⊖ n) + ofNat k = m + k ⊖ n
⊢ (m + 1 ⊖ n + 1) + ofNat k = m + 1 + k ⊖ n + 1
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Int/Lemmas.lean | Int.mk_add_negSucc | [50, 1] | [55, 80] | induction m, n with
| zero_zero => rw [zero_mk_zero, Int.zero_add, Nat.zero_add]; rfl
| zero_succ n => rw [zero_mk_succ, zero_mk_succ, Nat.succ_add]; rfl
| succ_zero m => rw [succ_mk_zero, Nat.zero_add]; rfl
| succ_succ m n ih => rw [succ_mk_succ, Nat.succ_add, succ_mk_succ]; exact ih | m n k : Nat
⊢ (m ⊖ n) + -[k+1] = m ⊖ n + k + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
m n k : Nat
⊢ (m ⊖ n) + -[k+1] = m ⊖ n + k + 1
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Int/Lemmas.lean | Int.mk_add_negSucc | [50, 1] | [55, 80] | rw [zero_mk_zero, Int.zero_add, Nat.zero_add] | case zero_zero
k : Nat
⊢ (0 ⊖ 0) + -[k+1] = 0 ⊖ 0 + k + 1 | case zero_zero
k : Nat
⊢ -[k+1] = 0 ⊖ k + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case zero_zero
k : Nat
⊢ (0 ⊖ 0) + -[k+1] = 0 ⊖ 0 + k + 1
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Int/Lemmas.lean | Int.mk_add_negSucc | [50, 1] | [55, 80] | rfl | case zero_zero
k : Nat
⊢ -[k+1] = 0 ⊖ k + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case zero_zero
k : Nat
⊢ -[k+1] = 0 ⊖ k + 1
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Int/Lemmas.lean | Int.mk_add_negSucc | [50, 1] | [55, 80] | rw [zero_mk_succ, zero_mk_succ, Nat.succ_add] | case zero_succ
k n : Nat
a✝ : (0 ⊖ n) + -[k+1] = 0 ⊖ n + k + 1
⊢ (0 ⊖ n + 1) + -[k+1] = 0 ⊖ n + 1 + k + 1 | case zero_succ
k n : Nat
a✝ : (0 ⊖ n) + -[k+1] = 0 ⊖ n + k + 1
⊢ -[n+1] + -[k+1] = -[(n + k).succ+1] | Please generate a tactic in lean4 to solve the state.
STATE:
case zero_succ
k n : Nat
a✝ : (0 ⊖ n) + -[k+1] = 0 ⊖ n + k + 1
⊢ (0 ⊖ n + 1) + -[k+1] = 0 ⊖ n + 1 + k + 1
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Int/Lemmas.lean | Int.mk_add_negSucc | [50, 1] | [55, 80] | rfl | case zero_succ
k n : Nat
a✝ : (0 ⊖ n) + -[k+1] = 0 ⊖ n + k + 1
⊢ -[n+1] + -[k+1] = -[(n + k).succ+1] | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case zero_succ
k n : Nat
a✝ : (0 ⊖ n) + -[k+1] = 0 ⊖ n + k + 1
⊢ -[n+1] + -[k+1] = -[(n + k).succ+1]
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Int/Lemmas.lean | Int.mk_add_negSucc | [50, 1] | [55, 80] | rw [succ_mk_zero, Nat.zero_add] | case succ_zero
k m : Nat
a✝ : (m ⊖ 0) + -[k+1] = m ⊖ 0 + k + 1
⊢ (m + 1 ⊖ 0) + -[k+1] = m + 1 ⊖ 0 + k + 1 | case succ_zero
k m : Nat
a✝ : (m ⊖ 0) + -[k+1] = m ⊖ 0 + k + 1
⊢ ofNat (m + 1) + -[k+1] = m + 1 ⊖ k + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case succ_zero
k m : Nat
a✝ : (m ⊖ 0) + -[k+1] = m ⊖ 0 + k + 1
⊢ (m + 1 ⊖ 0) + -[k+1] = m + 1 ⊖ 0 + k + 1
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Int/Lemmas.lean | Int.mk_add_negSucc | [50, 1] | [55, 80] | rfl | case succ_zero
k m : Nat
a✝ : (m ⊖ 0) + -[k+1] = m ⊖ 0 + k + 1
⊢ ofNat (m + 1) + -[k+1] = m + 1 ⊖ k + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ_zero
k m : Nat
a✝ : (m ⊖ 0) + -[k+1] = m ⊖ 0 + k + 1
⊢ ofNat (m + 1) + -[k+1] = m + 1 ⊖ k + 1
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Int/Lemmas.lean | Int.mk_add_negSucc | [50, 1] | [55, 80] | rw [succ_mk_succ, Nat.succ_add, succ_mk_succ] | case succ_succ
k m n : Nat
ih : (m ⊖ n) + -[k+1] = m ⊖ n + k + 1
⊢ (m + 1 ⊖ n + 1) + -[k+1] = m + 1 ⊖ n + 1 + k + 1 | case succ_succ
k m n : Nat
ih : (m ⊖ n) + -[k+1] = m ⊖ n + k + 1
⊢ (m ⊖ n) + -[k+1] = m ⊖ n + k + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case succ_succ
k m n : Nat
ih : (m ⊖ n) + -[k+1] = m ⊖ n + k + 1
⊢ (m + 1 ⊖ n + 1) + -[k+1] = m + 1 ⊖ n + 1 + k + 1
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Int/Lemmas.lean | Int.mk_add_negSucc | [50, 1] | [55, 80] | exact ih | case succ_succ
k m n : Nat
ih : (m ⊖ n) + -[k+1] = m ⊖ n + k + 1
⊢ (m ⊖ n) + -[k+1] = m ⊖ n + k + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ_succ
k m n : Nat
ih : (m ⊖ n) + -[k+1] = m ⊖ n + k + 1
⊢ (m ⊖ n) + -[k+1] = m ⊖ n + k + 1
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Int/Lemmas.lean | Int.mk_add_mk | [57, 1] | [62, 108] | induction m₂, n₂ with
| zero_zero => rw [zero_mk_zero, Int.add_zero, Nat.add_zero, Nat.add_zero]
| zero_succ n₂ => rw [zero_mk_succ, Nat.add_succ, mk_add_negSucc]; rfl
| succ_zero m₂ => rw [mk_zero, mk_add_ofNat]; rfl
| succ_succ m₂ n₂ ih => rw [succ_mk_succ, Nat.add_succ m₁ m₂, Nat.add_succ n₁ n₂, succ_mk_succ]; exact... | m₁ n₁ m₂ n₂ : Nat
⊢ (m₁ ⊖ n₁) + (m₂ ⊖ n₂) = m₁ + m₂ ⊖ n₁ + n₂ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
m₁ n₁ m₂ n₂ : Nat
⊢ (m₁ ⊖ n₁) + (m₂ ⊖ n₂) = m₁ + m₂ ⊖ n₁ + n₂
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Int/Lemmas.lean | Int.mk_add_mk | [57, 1] | [62, 108] | rw [zero_mk_zero, Int.add_zero, Nat.add_zero, Nat.add_zero] | case zero_zero
m₁ n₁ : Nat
⊢ (m₁ ⊖ n₁) + (0 ⊖ 0) = m₁ + 0 ⊖ n₁ + 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case zero_zero
m₁ n₁ : Nat
⊢ (m₁ ⊖ n₁) + (0 ⊖ 0) = m₁ + 0 ⊖ n₁ + 0
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Int/Lemmas.lean | Int.mk_add_mk | [57, 1] | [62, 108] | rw [zero_mk_succ, Nat.add_succ, mk_add_negSucc] | case zero_succ
m₁ n₁ n₂ : Nat
a✝ : (m₁ ⊖ n₁) + (0 ⊖ n₂) = m₁ + 0 ⊖ n₁ + n₂
⊢ (m₁ ⊖ n₁) + (0 ⊖ n₂ + 1) = m₁ + 0 ⊖ n₁ + (n₂ + 1) | case zero_succ
m₁ n₁ n₂ : Nat
a✝ : (m₁ ⊖ n₁) + (0 ⊖ n₂) = m₁ + 0 ⊖ n₁ + n₂
⊢ m₁ ⊖ n₁ + n₂ + 1 = m₁ + 0 ⊖ (n₁ + n₂).succ | Please generate a tactic in lean4 to solve the state.
STATE:
case zero_succ
m₁ n₁ n₂ : Nat
a✝ : (m₁ ⊖ n₁) + (0 ⊖ n₂) = m₁ + 0 ⊖ n₁ + n₂
⊢ (m₁ ⊖ n₁) + (0 ⊖ n₂ + 1) = m₁ + 0 ⊖ n₁ + (n₂ + 1)
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Int/Lemmas.lean | Int.mk_add_mk | [57, 1] | [62, 108] | rfl | case zero_succ
m₁ n₁ n₂ : Nat
a✝ : (m₁ ⊖ n₁) + (0 ⊖ n₂) = m₁ + 0 ⊖ n₁ + n₂
⊢ m₁ ⊖ n₁ + n₂ + 1 = m₁ + 0 ⊖ (n₁ + n₂).succ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case zero_succ
m₁ n₁ n₂ : Nat
a✝ : (m₁ ⊖ n₁) + (0 ⊖ n₂) = m₁ + 0 ⊖ n₁ + n₂
⊢ m₁ ⊖ n₁ + n₂ + 1 = m₁ + 0 ⊖ (n₁ + n₂).succ
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Int/Lemmas.lean | Int.mk_add_mk | [57, 1] | [62, 108] | rw [mk_zero, mk_add_ofNat] | case succ_zero
m₁ n₁ m₂ : Nat
a✝ : (m₁ ⊖ n₁) + (m₂ ⊖ 0) = m₁ + m₂ ⊖ n₁ + 0
⊢ (m₁ ⊖ n₁) + (m₂ + 1 ⊖ 0) = m₁ + (m₂ + 1) ⊖ n₁ + 0 | case succ_zero
m₁ n₁ m₂ : Nat
a✝ : (m₁ ⊖ n₁) + (m₂ ⊖ 0) = m₁ + m₂ ⊖ n₁ + 0
⊢ m₁ + (m₂ + 1) ⊖ n₁ = m₁ + (m₂ + 1) ⊖ n₁ + 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case succ_zero
m₁ n₁ m₂ : Nat
a✝ : (m₁ ⊖ n₁) + (m₂ ⊖ 0) = m₁ + m₂ ⊖ n₁ + 0
⊢ (m₁ ⊖ n₁) + (m₂ + 1 ⊖ 0) = m₁ + (m₂ + 1) ⊖ n₁ + 0
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Int/Lemmas.lean | Int.mk_add_mk | [57, 1] | [62, 108] | rfl | case succ_zero
m₁ n₁ m₂ : Nat
a✝ : (m₁ ⊖ n₁) + (m₂ ⊖ 0) = m₁ + m₂ ⊖ n₁ + 0
⊢ m₁ + (m₂ + 1) ⊖ n₁ = m₁ + (m₂ + 1) ⊖ n₁ + 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ_zero
m₁ n₁ m₂ : Nat
a✝ : (m₁ ⊖ n₁) + (m₂ ⊖ 0) = m₁ + m₂ ⊖ n₁ + 0
⊢ m₁ + (m₂ + 1) ⊖ n₁ = m₁ + (m₂ + 1) ⊖ n₁ + 0
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Int/Lemmas.lean | Int.mk_add_mk | [57, 1] | [62, 108] | rw [succ_mk_succ, Nat.add_succ m₁ m₂, Nat.add_succ n₁ n₂, succ_mk_succ] | case succ_succ
m₁ n₁ m₂ n₂ : Nat
ih : (m₁ ⊖ n₁) + (m₂ ⊖ n₂) = m₁ + m₂ ⊖ n₁ + n₂
⊢ (m₁ ⊖ n₁) + (m₂ + 1 ⊖ n₂ + 1) = m₁ + (m₂ + 1) ⊖ n₁ + (n₂ + 1) | case succ_succ
m₁ n₁ m₂ n₂ : Nat
ih : (m₁ ⊖ n₁) + (m₂ ⊖ n₂) = m₁ + m₂ ⊖ n₁ + n₂
⊢ (m₁ ⊖ n₁) + (m₂ ⊖ n₂) = m₁ + m₂ ⊖ n₁ + n₂ | Please generate a tactic in lean4 to solve the state.
STATE:
case succ_succ
m₁ n₁ m₂ n₂ : Nat
ih : (m₁ ⊖ n₁) + (m₂ ⊖ n₂) = m₁ + m₂ ⊖ n₁ + n₂
⊢ (m₁ ⊖ n₁) + (m₂ + 1 ⊖ n₂ + 1) = m₁ + (m₂ + 1) ⊖ n₁ + (n₂ + 1)
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Int/Lemmas.lean | Int.mk_add_mk | [57, 1] | [62, 108] | exact ih | case succ_succ
m₁ n₁ m₂ n₂ : Nat
ih : (m₁ ⊖ n₁) + (m₂ ⊖ n₂) = m₁ + m₂ ⊖ n₁ + n₂
⊢ (m₁ ⊖ n₁) + (m₂ ⊖ n₂) = m₁ + m₂ ⊖ n₁ + n₂ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ_succ
m₁ n₁ m₂ n₂ : Nat
ih : (m₁ ⊖ n₁) + (m₂ ⊖ n₂) = m₁ + m₂ ⊖ n₁ + n₂
⊢ (m₁ ⊖ n₁) + (m₂ ⊖ n₂) = m₁ + m₂ ⊖ n₁ + n₂
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Int/Lemmas.lean | Int.neg_mk | [64, 1] | [69, 66] | induction m, n with
| zero_zero => rw [zero_mk_zero]; rfl
| zero_succ n => rw [zero_mk_succ, succ_mk_zero]; rfl
| succ_zero m => rw [succ_mk_zero, zero_mk_succ]; rfl
| succ_succ m n ih => rw [succ_mk_succ, succ_mk_succ]; exact ih | m n : Nat
⊢ -(m ⊖ n) = n ⊖ m | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
m n : Nat
⊢ -(m ⊖ n) = n ⊖ m
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Int/Lemmas.lean | Int.neg_mk | [64, 1] | [69, 66] | rw [zero_mk_zero] | case zero_zero
⊢ -(0 ⊖ 0) = 0 ⊖ 0 | case zero_zero
⊢ -0 = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case zero_zero
⊢ -(0 ⊖ 0) = 0 ⊖ 0
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Int/Lemmas.lean | Int.neg_mk | [64, 1] | [69, 66] | rfl | case zero_zero
⊢ -0 = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case zero_zero
⊢ -0 = 0
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Int/Lemmas.lean | Int.neg_mk | [64, 1] | [69, 66] | rw [zero_mk_succ, succ_mk_zero] | case zero_succ
n : Nat
a✝ : -(0 ⊖ n) = n ⊖ 0
⊢ -(0 ⊖ n + 1) = n + 1 ⊖ 0 | case zero_succ
n : Nat
a✝ : -(0 ⊖ n) = n ⊖ 0
⊢ - -[n+1] = ofNat (n + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case zero_succ
n : Nat
a✝ : -(0 ⊖ n) = n ⊖ 0
⊢ -(0 ⊖ n + 1) = n + 1 ⊖ 0
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Int/Lemmas.lean | Int.neg_mk | [64, 1] | [69, 66] | rfl | case zero_succ
n : Nat
a✝ : -(0 ⊖ n) = n ⊖ 0
⊢ - -[n+1] = ofNat (n + 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case zero_succ
n : Nat
a✝ : -(0 ⊖ n) = n ⊖ 0
⊢ - -[n+1] = ofNat (n + 1)
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Int/Lemmas.lean | Int.neg_mk | [64, 1] | [69, 66] | rw [succ_mk_zero, zero_mk_succ] | case succ_zero
m : Nat
a✝ : -(m ⊖ 0) = 0 ⊖ m
⊢ -(m + 1 ⊖ 0) = 0 ⊖ m + 1 | case succ_zero
m : Nat
a✝ : -(m ⊖ 0) = 0 ⊖ m
⊢ -ofNat (m + 1) = -[m+1] | Please generate a tactic in lean4 to solve the state.
STATE:
case succ_zero
m : Nat
a✝ : -(m ⊖ 0) = 0 ⊖ m
⊢ -(m + 1 ⊖ 0) = 0 ⊖ m + 1
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Int/Lemmas.lean | Int.neg_mk | [64, 1] | [69, 66] | rfl | case succ_zero
m : Nat
a✝ : -(m ⊖ 0) = 0 ⊖ m
⊢ -ofNat (m + 1) = -[m+1] | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ_zero
m : Nat
a✝ : -(m ⊖ 0) = 0 ⊖ m
⊢ -ofNat (m + 1) = -[m+1]
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Int/Lemmas.lean | Int.neg_mk | [64, 1] | [69, 66] | rw [succ_mk_succ, succ_mk_succ] | case succ_succ
m n : Nat
ih : -(m ⊖ n) = n ⊖ m
⊢ -(m + 1 ⊖ n + 1) = n + 1 ⊖ m + 1 | case succ_succ
m n : Nat
ih : -(m ⊖ n) = n ⊖ m
⊢ -(m ⊖ n) = n ⊖ m | Please generate a tactic in lean4 to solve the state.
STATE:
case succ_succ
m n : Nat
ih : -(m ⊖ n) = n ⊖ m
⊢ -(m + 1 ⊖ n + 1) = n + 1 ⊖ m + 1
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Int/Lemmas.lean | Int.neg_mk | [64, 1] | [69, 66] | exact ih | case succ_succ
m n : Nat
ih : -(m ⊖ n) = n ⊖ m
⊢ -(m ⊖ n) = n ⊖ m | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ_succ
m n : Nat
ih : -(m ⊖ n) = n ⊖ m
⊢ -(m ⊖ n) = n ⊖ m
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Int/Lemmas.lean | Int.mk_sub_mk | [71, 1] | [72, 78] | rw [neg_mk, mk_add_mk] | m₁ n₁ m₂ n₂ : Nat
⊢ (m₁ ⊖ n₁) + -(m₂ ⊖ n₂) = m₁ + n₂ ⊖ n₁ + m₂ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
m₁ n₁ m₂ n₂ : Nat
⊢ (m₁ ⊖ n₁) + -(m₂ ⊖ n₂) = m₁ + n₂ ⊖ n₁ + m₂
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Int/Lemmas.lean | Int.nonNeg_mk | [74, 1] | [94, 13] | rw [zero_mk_zero] | case zero_zero
⊢ (0 ⊖ 0).NonNeg ↔ 0 ≤ 0 | case zero_zero
⊢ NonNeg 0 ↔ 0 ≤ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case zero_zero
⊢ (0 ⊖ 0).NonNeg ↔ 0 ≤ 0
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Int/Lemmas.lean | Int.nonNeg_mk | [74, 1] | [94, 13] | constructor | case zero_zero
⊢ NonNeg 0 ↔ 0 ≤ 0 | case zero_zero.mp
⊢ NonNeg 0 → 0 ≤ 0
case zero_zero.mpr
⊢ 0 ≤ 0 → NonNeg 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case zero_zero
⊢ NonNeg 0 ↔ 0 ≤ 0
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Int/Lemmas.lean | Int.nonNeg_mk | [74, 1] | [94, 13] | intro | case zero_zero.mp
⊢ NonNeg 0 → 0 ≤ 0 | case zero_zero.mp
a✝ : NonNeg 0
⊢ 0 ≤ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case zero_zero.mp
⊢ NonNeg 0 → 0 ≤ 0
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Int/Lemmas.lean | Int.nonNeg_mk | [74, 1] | [94, 13] | exact Nat.le_refl .. | case zero_zero.mp
a✝ : NonNeg 0
⊢ 0 ≤ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case zero_zero.mp
a✝ : NonNeg 0
⊢ 0 ≤ 0
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Int/Lemmas.lean | Int.nonNeg_mk | [74, 1] | [94, 13] | intro | case zero_zero.mpr
⊢ 0 ≤ 0 → NonNeg 0 | case zero_zero.mpr
a✝ : 0 ≤ 0
⊢ NonNeg 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case zero_zero.mpr
⊢ 0 ≤ 0 → NonNeg 0
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Int/Lemmas.lean | Int.nonNeg_mk | [74, 1] | [94, 13] | apply NonNeg.mk | case zero_zero.mpr
a✝ : 0 ≤ 0
⊢ NonNeg 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case zero_zero.mpr
a✝ : 0 ≤ 0
⊢ NonNeg 0
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Int/Lemmas.lean | Int.nonNeg_mk | [74, 1] | [94, 13] | rw [zero_mk_succ] | case zero_succ
n : Nat
a✝ : (0 ⊖ n).NonNeg ↔ n ≤ 0
⊢ (0 ⊖ n + 1).NonNeg ↔ n + 1 ≤ 0 | case zero_succ
n : Nat
a✝ : (0 ⊖ n).NonNeg ↔ n ≤ 0
⊢ -[n+1].NonNeg ↔ n + 1 ≤ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case zero_succ
n : Nat
a✝ : (0 ⊖ n).NonNeg ↔ n ≤ 0
⊢ (0 ⊖ n + 1).NonNeg ↔ n + 1 ≤ 0
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Int/Lemmas.lean | Int.nonNeg_mk | [74, 1] | [94, 13] | constructor | case zero_succ
n : Nat
a✝ : (0 ⊖ n).NonNeg ↔ n ≤ 0
⊢ -[n+1].NonNeg ↔ n + 1 ≤ 0 | case zero_succ.mp
n : Nat
a✝ : (0 ⊖ n).NonNeg ↔ n ≤ 0
⊢ -[n+1].NonNeg → n + 1 ≤ 0
case zero_succ.mpr
n : Nat
a✝ : (0 ⊖ n).NonNeg ↔ n ≤ 0
⊢ n + 1 ≤ 0 → -[n+1].NonNeg | Please generate a tactic in lean4 to solve the state.
STATE:
case zero_succ
n : Nat
a✝ : (0 ⊖ n).NonNeg ↔ n ≤ 0
⊢ -[n+1].NonNeg ↔ n + 1 ≤ 0
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Int/Lemmas.lean | Int.nonNeg_mk | [74, 1] | [94, 13] | intro | case zero_succ.mp
n : Nat
a✝ : (0 ⊖ n).NonNeg ↔ n ≤ 0
⊢ -[n+1].NonNeg → n + 1 ≤ 0 | case zero_succ.mp
n : Nat
a✝¹ : (0 ⊖ n).NonNeg ↔ n ≤ 0
a✝ : -[n+1].NonNeg
⊢ n + 1 ≤ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case zero_succ.mp
n : Nat
a✝ : (0 ⊖ n).NonNeg ↔ n ≤ 0
⊢ -[n+1].NonNeg → n + 1 ≤ 0
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Int/Lemmas.lean | Int.nonNeg_mk | [74, 1] | [94, 13] | contradiction | case zero_succ.mp
n : Nat
a✝¹ : (0 ⊖ n).NonNeg ↔ n ≤ 0
a✝ : -[n+1].NonNeg
⊢ n + 1 ≤ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case zero_succ.mp
n : Nat
a✝¹ : (0 ⊖ n).NonNeg ↔ n ≤ 0
a✝ : -[n+1].NonNeg
⊢ n + 1 ≤ 0
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Int/Lemmas.lean | Int.nonNeg_mk | [74, 1] | [94, 13] | intro | case zero_succ.mpr
n : Nat
a✝ : (0 ⊖ n).NonNeg ↔ n ≤ 0
⊢ n + 1 ≤ 0 → -[n+1].NonNeg | case zero_succ.mpr
n : Nat
a✝¹ : (0 ⊖ n).NonNeg ↔ n ≤ 0
a✝ : n + 1 ≤ 0
⊢ -[n+1].NonNeg | Please generate a tactic in lean4 to solve the state.
STATE:
case zero_succ.mpr
n : Nat
a✝ : (0 ⊖ n).NonNeg ↔ n ≤ 0
⊢ n + 1 ≤ 0 → -[n+1].NonNeg
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Int/Lemmas.lean | Int.nonNeg_mk | [74, 1] | [94, 13] | contradiction | case zero_succ.mpr
n : Nat
a✝¹ : (0 ⊖ n).NonNeg ↔ n ≤ 0
a✝ : n + 1 ≤ 0
⊢ -[n+1].NonNeg | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case zero_succ.mpr
n : Nat
a✝¹ : (0 ⊖ n).NonNeg ↔ n ≤ 0
a✝ : n + 1 ≤ 0
⊢ -[n+1].NonNeg
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Int/Lemmas.lean | Int.nonNeg_mk | [74, 1] | [94, 13] | rw [succ_mk_zero] | case succ_zero
m : Nat
a✝ : (m ⊖ 0).NonNeg ↔ 0 ≤ m
⊢ (m + 1 ⊖ 0).NonNeg ↔ 0 ≤ m + 1 | case succ_zero
m : Nat
a✝ : (m ⊖ 0).NonNeg ↔ 0 ≤ m
⊢ (ofNat (m + 1)).NonNeg ↔ 0 ≤ m + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case succ_zero
m : Nat
a✝ : (m ⊖ 0).NonNeg ↔ 0 ≤ m
⊢ (m + 1 ⊖ 0).NonNeg ↔ 0 ≤ m + 1
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Int/Lemmas.lean | Int.nonNeg_mk | [74, 1] | [94, 13] | constructor | case succ_zero
m : Nat
a✝ : (m ⊖ 0).NonNeg ↔ 0 ≤ m
⊢ (ofNat (m + 1)).NonNeg ↔ 0 ≤ m + 1 | case succ_zero.mp
m : Nat
a✝ : (m ⊖ 0).NonNeg ↔ 0 ≤ m
⊢ (ofNat (m + 1)).NonNeg → 0 ≤ m + 1
case succ_zero.mpr
m : Nat
a✝ : (m ⊖ 0).NonNeg ↔ 0 ≤ m
⊢ 0 ≤ m + 1 → (ofNat (m + 1)).NonNeg | Please generate a tactic in lean4 to solve the state.
STATE:
case succ_zero
m : Nat
a✝ : (m ⊖ 0).NonNeg ↔ 0 ≤ m
⊢ (ofNat (m + 1)).NonNeg ↔ 0 ≤ m + 1
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Int/Lemmas.lean | Int.nonNeg_mk | [74, 1] | [94, 13] | intro | case succ_zero.mp
m : Nat
a✝ : (m ⊖ 0).NonNeg ↔ 0 ≤ m
⊢ (ofNat (m + 1)).NonNeg → 0 ≤ m + 1 | case succ_zero.mp
m : Nat
a✝¹ : (m ⊖ 0).NonNeg ↔ 0 ≤ m
a✝ : (ofNat (m + 1)).NonNeg
⊢ 0 ≤ m + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case succ_zero.mp
m : Nat
a✝ : (m ⊖ 0).NonNeg ↔ 0 ≤ m
⊢ (ofNat (m + 1)).NonNeg → 0 ≤ m + 1
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Int/Lemmas.lean | Int.nonNeg_mk | [74, 1] | [94, 13] | apply Nat.zero_le | case succ_zero.mp
m : Nat
a✝¹ : (m ⊖ 0).NonNeg ↔ 0 ≤ m
a✝ : (ofNat (m + 1)).NonNeg
⊢ 0 ≤ m + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ_zero.mp
m : Nat
a✝¹ : (m ⊖ 0).NonNeg ↔ 0 ≤ m
a✝ : (ofNat (m + 1)).NonNeg
⊢ 0 ≤ m + 1
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Int/Lemmas.lean | Int.nonNeg_mk | [74, 1] | [94, 13] | intro | case succ_zero.mpr
m : Nat
a✝ : (m ⊖ 0).NonNeg ↔ 0 ≤ m
⊢ 0 ≤ m + 1 → (ofNat (m + 1)).NonNeg | case succ_zero.mpr
m : Nat
a✝¹ : (m ⊖ 0).NonNeg ↔ 0 ≤ m
a✝ : 0 ≤ m + 1
⊢ (ofNat (m + 1)).NonNeg | Please generate a tactic in lean4 to solve the state.
STATE:
case succ_zero.mpr
m : Nat
a✝ : (m ⊖ 0).NonNeg ↔ 0 ≤ m
⊢ 0 ≤ m + 1 → (ofNat (m + 1)).NonNeg
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Int/Lemmas.lean | Int.nonNeg_mk | [74, 1] | [94, 13] | apply NonNeg.mk | case succ_zero.mpr
m : Nat
a✝¹ : (m ⊖ 0).NonNeg ↔ 0 ≤ m
a✝ : 0 ≤ m + 1
⊢ (ofNat (m + 1)).NonNeg | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ_zero.mpr
m : Nat
a✝¹ : (m ⊖ 0).NonNeg ↔ 0 ≤ m
a✝ : 0 ≤ m + 1
⊢ (ofNat (m + 1)).NonNeg
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Int/Lemmas.lean | Int.nonNeg_mk | [74, 1] | [94, 13] | rw [succ_mk_succ] | case succ_succ
m n : Nat
ih : (m ⊖ n).NonNeg ↔ n ≤ m
⊢ (m + 1 ⊖ n + 1).NonNeg ↔ n + 1 ≤ m + 1 | case succ_succ
m n : Nat
ih : (m ⊖ n).NonNeg ↔ n ≤ m
⊢ (m ⊖ n).NonNeg ↔ n + 1 ≤ m + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case succ_succ
m n : Nat
ih : (m ⊖ n).NonNeg ↔ n ≤ m
⊢ (m + 1 ⊖ n + 1).NonNeg ↔ n + 1 ≤ m + 1
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Int/Lemmas.lean | Int.nonNeg_mk | [74, 1] | [94, 13] | rw [Nat.succ_le_succ_iff] | case succ_succ
m n : Nat
ih : (m ⊖ n).NonNeg ↔ n ≤ m
⊢ (m ⊖ n).NonNeg ↔ n + 1 ≤ m + 1 | case succ_succ
m n : Nat
ih : (m ⊖ n).NonNeg ↔ n ≤ m
⊢ (m ⊖ n).NonNeg ↔ n ≤ m | Please generate a tactic in lean4 to solve the state.
STATE:
case succ_succ
m n : Nat
ih : (m ⊖ n).NonNeg ↔ n ≤ m
⊢ (m ⊖ n).NonNeg ↔ n + 1 ≤ m + 1
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Int/Lemmas.lean | Int.nonNeg_mk | [74, 1] | [94, 13] | exact ih | case succ_succ
m n : Nat
ih : (m ⊖ n).NonNeg ↔ n ≤ m
⊢ (m ⊖ n).NonNeg ↔ n ≤ m | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ_succ
m n : Nat
ih : (m ⊖ n).NonNeg ↔ n ≤ m
⊢ (m ⊖ n).NonNeg ↔ n ≤ m
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Int/Lemmas.lean | Int.mk_le_mk | [96, 1] | [99, 6] | simp only [LE.le, Int.le] | m₁ n₁ m₂ n₂ : Nat
⊢ m₁ ⊖ n₁ ≤ m₂ ⊖ n₂ ↔ n₂ + m₁ ≤ m₂ + n₁ | m₁ n₁ m₂ n₂ : Nat
⊢ ((m₂ ⊖ n₂) - (m₁ ⊖ n₁)).NonNeg ↔ (n₂ + m₁).le (m₂ + n₁) | Please generate a tactic in lean4 to solve the state.
STATE:
m₁ n₁ m₂ n₂ : Nat
⊢ m₁ ⊖ n₁ ≤ m₂ ⊖ n₂ ↔ n₂ + m₁ ≤ m₂ + n₁
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Int/Lemmas.lean | Int.mk_le_mk | [96, 1] | [99, 6] | rw [mk_sub_mk, nonNeg_mk] | m₁ n₁ m₂ n₂ : Nat
⊢ ((m₂ ⊖ n₂) - (m₁ ⊖ n₁)).NonNeg ↔ (n₂ + m₁).le (m₂ + n₁) | m₁ n₁ m₂ n₂ : Nat
⊢ n₂ + m₁ ≤ m₂ + n₁ ↔ (n₂ + m₁).le (m₂ + n₁) | Please generate a tactic in lean4 to solve the state.
STATE:
m₁ n₁ m₂ n₂ : Nat
⊢ ((m₂ ⊖ n₂) - (m₁ ⊖ n₁)).NonNeg ↔ (n₂ + m₁).le (m₂ + n₁)
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Int/Lemmas.lean | Int.mk_le_mk | [96, 1] | [99, 6] | rfl | m₁ n₁ m₂ n₂ : Nat
⊢ n₂ + m₁ ≤ m₂ + n₁ ↔ (n₂ + m₁).le (m₂ + n₁) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
m₁ n₁ m₂ n₂ : Nat
⊢ n₂ + m₁ ≤ m₂ + n₁ ↔ (n₂ + m₁).le (m₂ + n₁)
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Int/Lemmas.lean | Int.mk_lt_mk | [101, 1] | [104, 6] | simp only [LT.lt, Int.lt, Nat.lt] | m₁ n₁ m₂ n₂ : Nat
⊢ m₁ ⊖ n₁ < m₂ ⊖ n₂ ↔ n₂ + m₁ < m₂ + n₁ | m₁ n₁ m₂ n₂ : Nat
⊢ (m₁ ⊖ n₁) + 1 ≤ m₂ ⊖ n₂ ↔ (n₂ + m₁).succ.le (m₂ + n₁) | Please generate a tactic in lean4 to solve the state.
STATE:
m₁ n₁ m₂ n₂ : Nat
⊢ m₁ ⊖ n₁ < m₂ ⊖ n₂ ↔ n₂ + m₁ < m₂ + n₁
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Int/Lemmas.lean | Int.mk_lt_mk | [101, 1] | [104, 6] | rw [←one_mk_zero, mk_add_mk, mk_le_mk, Nat.add_succ, Nat.add_zero] | m₁ n₁ m₂ n₂ : Nat
⊢ (m₁ ⊖ n₁) + 1 ≤ m₂ ⊖ n₂ ↔ (n₂ + m₁).succ.le (m₂ + n₁) | m₁ n₁ m₂ n₂ : Nat
⊢ (n₂ + m₁).succ ≤ m₂ + n₁ ↔ (n₂ + m₁).succ.le (m₂ + n₁) | Please generate a tactic in lean4 to solve the state.
STATE:
m₁ n₁ m₂ n₂ : Nat
⊢ (m₁ ⊖ n₁) + 1 ≤ m₂ ⊖ n₂ ↔ (n₂ + m₁).succ.le (m₂ + n₁)
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Int/Lemmas.lean | Int.mk_lt_mk | [101, 1] | [104, 6] | rfl | m₁ n₁ m₂ n₂ : Nat
⊢ (n₂ + m₁).succ ≤ m₂ + n₁ ↔ (n₂ + m₁).succ.le (m₂ + n₁) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
m₁ n₁ m₂ n₂ : Nat
⊢ (n₂ + m₁).succ ≤ m₂ + n₁ ↔ (n₂ + m₁).succ.le (m₂ + n₁)
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Int/Lemmas.lean | Int.add_neg_self_left | [106, 11] | [108, 68] | cases i using Int.casesMkOn with
| mk mi ni => rw [neg_mk, mk_add_mk, Nat.add_comm mi ni, mk_self] | i : Int
⊢ -i + i = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
i : Int
⊢ -i + i = 0
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Int/Lemmas.lean | Int.add_neg_self_left | [106, 11] | [108, 68] | rw [neg_mk, mk_add_mk, Nat.add_comm mi ni, mk_self] | case mk
mi ni : Nat
⊢ -(mi ⊖ ni) + (mi ⊖ ni) = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mk
mi ni : Nat
⊢ -(mi ⊖ ni) + (mi ⊖ ni) = 0
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Int/Lemmas.lean | Int.add_neg_self_right | [110, 11] | [112, 68] | cases i using Int.casesMkOn with
| mk mi ni => rw [neg_mk, mk_add_mk, Nat.add_comm mi ni, mk_self] | i : Int
⊢ i + -i = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
i : Int
⊢ i + -i = 0
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Int/Lemmas.lean | Int.add_neg_self_right | [110, 11] | [112, 68] | rw [neg_mk, mk_add_mk, Nat.add_comm mi ni, mk_self] | case mk
mi ni : Nat
⊢ (mi ⊖ ni) + -(mi ⊖ ni) = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mk
mi ni : Nat
⊢ (mi ⊖ ni) + -(mi ⊖ ni) = 0
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Int/Lemmas.lean | Int.sub_add_assoc | [114, 11] | [120, 52] | rw [Int.sub_eq] | i j k : Int
⊢ i - j + k = i + -j + k | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
i j k : Int
⊢ i - j + k = i + -j + k
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Int/Lemmas.lean | Int.sub_add_assoc | [114, 11] | [120, 52] | rw [Int.add_assoc] | i j k : Int
⊢ i + -j + k = i + (-j + k) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
i j k : Int
⊢ i + -j + k = i + (-j + k)
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Int/Lemmas.lean | Int.sub_add_assoc | [114, 11] | [120, 52] | rw [Int.neg_neg] | i j k : Int
⊢ i + (-j + k) = i + (-j + - -k) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
i j k : Int
⊢ i + (-j + k) = i + (-j + - -k)
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Int/Lemmas.lean | Int.sub_add_assoc | [114, 11] | [120, 52] | rw [Int.neg_add] | i j k : Int
⊢ i + (-j + - -k) = i + -(j + -k) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
i j k : Int
⊢ i + (-j + - -k) = i + -(j + -k)
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Int/Lemmas.lean | Int.sub_add_assoc | [114, 11] | [120, 52] | rw [Int.sub_eq, Int.sub_eq] | i j k : Int
⊢ i + -(j + -k) = i - (j - k) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
i j k : Int
⊢ i + -(j + -k) = i - (j - k)
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Int/Lemmas.lean | Int.mk_mul_ofNat | [122, 1] | [127, 98] | induction m, n with
| zero_zero => rw [Nat.zero_mul, zero_mk_zero, Int.zero_mul]
| zero_succ n => rw [Nat.zero_mul, zero_mk, zero_mk]; rfl
| succ_zero m => rw [Nat.zero_mul, mk_zero, mk_zero]; rfl
| succ_succ m n ih => rw [Nat.succ_mul, Nat.succ_mul, succ_mk_succ, add_mk_add_right]; exact ih | m n k : Nat
⊢ (m ⊖ n) * ofNat k = m * k ⊖ n * k | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
m n k : Nat
⊢ (m ⊖ n) * ofNat k = m * k ⊖ n * k
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Int/Lemmas.lean | Int.mk_mul_ofNat | [122, 1] | [127, 98] | rw [Nat.zero_mul, zero_mk_zero, Int.zero_mul] | case zero_zero
k : Nat
⊢ (0 ⊖ 0) * ofNat k = 0 * k ⊖ 0 * k | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case zero_zero
k : Nat
⊢ (0 ⊖ 0) * ofNat k = 0 * k ⊖ 0 * k
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Int/Lemmas.lean | Int.mk_mul_ofNat | [122, 1] | [127, 98] | rw [Nat.zero_mul, zero_mk, zero_mk] | case zero_succ
k n : Nat
a✝ : (0 ⊖ n) * ofNat k = 0 * k ⊖ n * k
⊢ (0 ⊖ n + 1) * ofNat k = 0 * k ⊖ (n + 1) * k | case zero_succ
k n : Nat
a✝ : (0 ⊖ n) * ofNat k = 0 * k ⊖ n * k
⊢ negOfNat (n + 1) * ofNat k = negOfNat ((n + 1) * k) | Please generate a tactic in lean4 to solve the state.
STATE:
case zero_succ
k n : Nat
a✝ : (0 ⊖ n) * ofNat k = 0 * k ⊖ n * k
⊢ (0 ⊖ n + 1) * ofNat k = 0 * k ⊖ (n + 1) * k
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Int/Lemmas.lean | Int.mk_mul_ofNat | [122, 1] | [127, 98] | rfl | case zero_succ
k n : Nat
a✝ : (0 ⊖ n) * ofNat k = 0 * k ⊖ n * k
⊢ negOfNat (n + 1) * ofNat k = negOfNat ((n + 1) * k) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case zero_succ
k n : Nat
a✝ : (0 ⊖ n) * ofNat k = 0 * k ⊖ n * k
⊢ negOfNat (n + 1) * ofNat k = negOfNat ((n + 1) * k)
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Int/Lemmas.lean | Int.mk_mul_ofNat | [122, 1] | [127, 98] | rw [Nat.zero_mul, mk_zero, mk_zero] | case succ_zero
k m : Nat
a✝ : (m ⊖ 0) * ofNat k = m * k ⊖ 0 * k
⊢ (m + 1 ⊖ 0) * ofNat k = (m + 1) * k ⊖ 0 * k | case succ_zero
k m : Nat
a✝ : (m ⊖ 0) * ofNat k = m * k ⊖ 0 * k
⊢ ofNat (m + 1) * ofNat k = ofNat ((m + 1) * k) | Please generate a tactic in lean4 to solve the state.
STATE:
case succ_zero
k m : Nat
a✝ : (m ⊖ 0) * ofNat k = m * k ⊖ 0 * k
⊢ (m + 1 ⊖ 0) * ofNat k = (m + 1) * k ⊖ 0 * k
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Int/Lemmas.lean | Int.mk_mul_ofNat | [122, 1] | [127, 98] | rfl | case succ_zero
k m : Nat
a✝ : (m ⊖ 0) * ofNat k = m * k ⊖ 0 * k
⊢ ofNat (m + 1) * ofNat k = ofNat ((m + 1) * k) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ_zero
k m : Nat
a✝ : (m ⊖ 0) * ofNat k = m * k ⊖ 0 * k
⊢ ofNat (m + 1) * ofNat k = ofNat ((m + 1) * k)
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Int/Lemmas.lean | Int.mk_mul_ofNat | [122, 1] | [127, 98] | rw [Nat.succ_mul, Nat.succ_mul, succ_mk_succ, add_mk_add_right] | case succ_succ
k m n : Nat
ih : (m ⊖ n) * ofNat k = m * k ⊖ n * k
⊢ (m + 1 ⊖ n + 1) * ofNat k = (m + 1) * k ⊖ (n + 1) * k | case succ_succ
k m n : Nat
ih : (m ⊖ n) * ofNat k = m * k ⊖ n * k
⊢ (m ⊖ n) * ofNat k = m * k ⊖ n * k | Please generate a tactic in lean4 to solve the state.
STATE:
case succ_succ
k m n : Nat
ih : (m ⊖ n) * ofNat k = m * k ⊖ n * k
⊢ (m + 1 ⊖ n + 1) * ofNat k = (m + 1) * k ⊖ (n + 1) * k
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Int/Lemmas.lean | Int.mk_mul_ofNat | [122, 1] | [127, 98] | exact ih | case succ_succ
k m n : Nat
ih : (m ⊖ n) * ofNat k = m * k ⊖ n * k
⊢ (m ⊖ n) * ofNat k = m * k ⊖ n * k | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ_succ
k m n : Nat
ih : (m ⊖ n) * ofNat k = m * k ⊖ n * k
⊢ (m ⊖ n) * ofNat k = m * k ⊖ n * k
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Int/Lemmas.lean | Int.mk_mul_negSucc | [129, 1] | [134, 98] | induction m, n with
| zero_zero => rw [Nat.zero_mul, zero_mk_zero, Int.zero_mul]
| zero_succ n => rw [Nat.zero_mul, zero_mk, mk_zero]; rfl
| succ_zero m => rw [Nat.zero_mul, mk_zero, zero_mk]; rfl
| succ_succ m n ih => rw [Nat.succ_mul, Nat.succ_mul, succ_mk_succ, add_mk_add_right]; exact ih | m n k : Nat
⊢ (m ⊖ n) * -[k+1] = n * (k + 1) ⊖ m * (k + 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
m n k : Nat
⊢ (m ⊖ n) * -[k+1] = n * (k + 1) ⊖ m * (k + 1)
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Int/Lemmas.lean | Int.mk_mul_negSucc | [129, 1] | [134, 98] | rw [Nat.zero_mul, zero_mk_zero, Int.zero_mul] | case zero_zero
k : Nat
⊢ (0 ⊖ 0) * -[k+1] = 0 * (k + 1) ⊖ 0 * (k + 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case zero_zero
k : Nat
⊢ (0 ⊖ 0) * -[k+1] = 0 * (k + 1) ⊖ 0 * (k + 1)
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Int/Lemmas.lean | Int.mk_mul_negSucc | [129, 1] | [134, 98] | rw [Nat.zero_mul, zero_mk, mk_zero] | case zero_succ
k n : Nat
a✝ : (0 ⊖ n) * -[k+1] = n * (k + 1) ⊖ 0 * (k + 1)
⊢ (0 ⊖ n + 1) * -[k+1] = (n + 1) * (k + 1) ⊖ 0 * (k + 1) | case zero_succ
k n : Nat
a✝ : (0 ⊖ n) * -[k+1] = n * (k + 1) ⊖ 0 * (k + 1)
⊢ negOfNat (n + 1) * -[k+1] = ofNat ((n + 1) * (k + 1)) | Please generate a tactic in lean4 to solve the state.
STATE:
case zero_succ
k n : Nat
a✝ : (0 ⊖ n) * -[k+1] = n * (k + 1) ⊖ 0 * (k + 1)
⊢ (0 ⊖ n + 1) * -[k+1] = (n + 1) * (k + 1) ⊖ 0 * (k + 1)
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Int/Lemmas.lean | Int.mk_mul_negSucc | [129, 1] | [134, 98] | rfl | case zero_succ
k n : Nat
a✝ : (0 ⊖ n) * -[k+1] = n * (k + 1) ⊖ 0 * (k + 1)
⊢ negOfNat (n + 1) * -[k+1] = ofNat ((n + 1) * (k + 1)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case zero_succ
k n : Nat
a✝ : (0 ⊖ n) * -[k+1] = n * (k + 1) ⊖ 0 * (k + 1)
⊢ negOfNat (n + 1) * -[k+1] = ofNat ((n + 1) * (k + 1))
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Int/Lemmas.lean | Int.mk_mul_negSucc | [129, 1] | [134, 98] | rw [Nat.zero_mul, mk_zero, zero_mk] | case succ_zero
k m : Nat
a✝ : (m ⊖ 0) * -[k+1] = 0 * (k + 1) ⊖ m * (k + 1)
⊢ (m + 1 ⊖ 0) * -[k+1] = 0 * (k + 1) ⊖ (m + 1) * (k + 1) | case succ_zero
k m : Nat
a✝ : (m ⊖ 0) * -[k+1] = 0 * (k + 1) ⊖ m * (k + 1)
⊢ ofNat (m + 1) * -[k+1] = negOfNat ((m + 1) * (k + 1)) | Please generate a tactic in lean4 to solve the state.
STATE:
case succ_zero
k m : Nat
a✝ : (m ⊖ 0) * -[k+1] = 0 * (k + 1) ⊖ m * (k + 1)
⊢ (m + 1 ⊖ 0) * -[k+1] = 0 * (k + 1) ⊖ (m + 1) * (k + 1)
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Int/Lemmas.lean | Int.mk_mul_negSucc | [129, 1] | [134, 98] | rfl | case succ_zero
k m : Nat
a✝ : (m ⊖ 0) * -[k+1] = 0 * (k + 1) ⊖ m * (k + 1)
⊢ ofNat (m + 1) * -[k+1] = negOfNat ((m + 1) * (k + 1)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ_zero
k m : Nat
a✝ : (m ⊖ 0) * -[k+1] = 0 * (k + 1) ⊖ m * (k + 1)
⊢ ofNat (m + 1) * -[k+1] = negOfNat ((m + 1) * (k + 1))
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Int/Lemmas.lean | Int.mk_mul_negSucc | [129, 1] | [134, 98] | rw [Nat.succ_mul, Nat.succ_mul, succ_mk_succ, add_mk_add_right] | case succ_succ
k m n : Nat
ih : (m ⊖ n) * -[k+1] = n * (k + 1) ⊖ m * (k + 1)
⊢ (m + 1 ⊖ n + 1) * -[k+1] = (n + 1) * (k + 1) ⊖ (m + 1) * (k + 1) | case succ_succ
k m n : Nat
ih : (m ⊖ n) * -[k+1] = n * (k + 1) ⊖ m * (k + 1)
⊢ (m ⊖ n) * -[k+1] = n * (k + 1) ⊖ m * (k + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case succ_succ
k m n : Nat
ih : (m ⊖ n) * -[k+1] = n * (k + 1) ⊖ m * (k + 1)
⊢ (m + 1 ⊖ n + 1) * -[k+1] = (n + 1) * (k + 1) ⊖ (m + 1) * (k + 1)
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Int/Lemmas.lean | Int.mk_mul_negSucc | [129, 1] | [134, 98] | exact ih | case succ_succ
k m n : Nat
ih : (m ⊖ n) * -[k+1] = n * (k + 1) ⊖ m * (k + 1)
⊢ (m ⊖ n) * -[k+1] = n * (k + 1) ⊖ m * (k + 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ_succ
k m n : Nat
ih : (m ⊖ n) * -[k+1] = n * (k + 1) ⊖ m * (k + 1)
⊢ (m ⊖ n) * -[k+1] = n * (k + 1) ⊖ m * (k + 1)
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Int/Lemmas.lean | Int.mk_mul_mk | [136, 1] | [141, 176] | induction m₂, n₂ with
| zero_zero => simp only [Nat.zero_mul, Nat.mul_zero, Nat.add_zero, Nat.zero_add, zero_mk_zero, Int.mul_zero]
| zero_succ n₂ => simp only [Nat.zero_mul, Nat.mul_zero, Nat.add_zero, Nat.zero_add, zero_mk_succ, mk_mul_negSucc]
| succ_zero m₂ => simp only [Nat.zero_mul, Nat.mul_zero, Nat.add_zero, Na... | m₁ n₁ m₂ n₂ : Nat
⊢ (m₁ ⊖ n₁) * (m₂ ⊖ n₂) = m₁ * m₂ + n₁ * n₂ ⊖ m₁ * n₂ + n₁ * m₂ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
m₁ n₁ m₂ n₂ : Nat
⊢ (m₁ ⊖ n₁) * (m₂ ⊖ n₂) = m₁ * m₂ + n₁ * n₂ ⊖ m₁ * n₂ + n₁ * m₂
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Int/Lemmas.lean | Int.mk_mul_mk | [136, 1] | [141, 176] | simp only [Nat.zero_mul, Nat.mul_zero, Nat.add_zero, Nat.zero_add, zero_mk_zero, Int.mul_zero] | case zero_zero
m₁ n₁ : Nat
⊢ (m₁ ⊖ n₁) * (0 ⊖ 0) = m₁ * 0 + n₁ * 0 ⊖ m₁ * 0 + n₁ * 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case zero_zero
m₁ n₁ : Nat
⊢ (m₁ ⊖ n₁) * (0 ⊖ 0) = m₁ * 0 + n₁ * 0 ⊖ m₁ * 0 + n₁ * 0
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Int/Lemmas.lean | Int.mk_mul_mk | [136, 1] | [141, 176] | simp only [Nat.zero_mul, Nat.mul_zero, Nat.add_zero, Nat.zero_add, zero_mk_succ, mk_mul_negSucc] | case zero_succ
m₁ n₁ n₂ : Nat
a✝ : (m₁ ⊖ n₁) * (0 ⊖ n₂) = m₁ * 0 + n₁ * n₂ ⊖ m₁ * n₂ + n₁ * 0
⊢ (m₁ ⊖ n₁) * (0 ⊖ n₂ + 1) = m₁ * 0 + n₁ * (n₂ + 1) ⊖ m₁ * (n₂ + 1) + n₁ * 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case zero_succ
m₁ n₁ n₂ : Nat
a✝ : (m₁ ⊖ n₁) * (0 ⊖ n₂) = m₁ * 0 + n₁ * n₂ ⊖ m₁ * n₂ + n₁ * 0
⊢ (m₁ ⊖ n₁) * (0 ⊖ n₂ + 1) = m₁ * 0 + n₁ * (n₂ + 1) ⊖ m₁ * (n₂ + 1) + n₁ * 0
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Int/Lemmas.lean | Int.mk_mul_mk | [136, 1] | [141, 176] | simp only [Nat.zero_mul, Nat.mul_zero, Nat.add_zero, Nat.zero_add, succ_mk_zero, mk_mul_ofNat, Nat.mul_comm] | case succ_zero
m₁ n₁ m₂ : Nat
a✝ : (m₁ ⊖ n₁) * (m₂ ⊖ 0) = m₁ * m₂ + n₁ * 0 ⊖ m₁ * 0 + n₁ * m₂
⊢ (m₁ ⊖ n₁) * (m₂ + 1 ⊖ 0) = m₁ * (m₂ + 1) + n₁ * 0 ⊖ m₁ * 0 + n₁ * (m₂ + 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ_zero
m₁ n₁ m₂ : Nat
a✝ : (m₁ ⊖ n₁) * (m₂ ⊖ 0) = m₁ * m₂ + n₁ * 0 ⊖ m₁ * 0 + n₁ * m₂
⊢ (m₁ ⊖ n₁) * (m₂ + 1 ⊖ 0) = m₁ * (m₂ + 1) + n₁ * 0 ⊖ m₁ * 0 + n₁ * (m₂ + 1)
TACTIC:
|
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