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/kbuzzard/IISc-experiments.git | f2f2d7d14b3ec1957fcdd38cc0e3657df6850047 | IIScExperiments/Solutions/NatAddMulExperiments.lean | MyNat.add_assoc | [46, 1] | [49, 42] | rw [add_succ, add_succ, add_succ, hd] | case succ
a b d : MyNat
hd : a + b + d = a + (b + d)
⊢ a + b + succ d = a + (b + succ d) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
a b d : MyNat
hd : a + b + d = a + (b + d)
⊢ a + b + succ d = a + (b + succ d)
TACTIC:
|
https://github.com/kbuzzard/IISc-experiments.git | f2f2d7d14b3ec1957fcdd38cc0e3657df6850047 | IIScExperiments/Solutions/NatAddMulExperiments.lean | MyNat.add_add_add | [51, 1] | [52, 72] | rw [add_assoc, ← add_assoc b, add_comm b, add_assoc c, ← add_assoc a] | a b c d : MyNat
⊢ a + b + (c + d) = a + c + (b + d) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b c d : MyNat
⊢ a + b + (c + d) = a + c + (b + d)
TACTIC:
|
https://github.com/kbuzzard/IISc-experiments.git | f2f2d7d14b3ec1957fcdd38cc0e3657df6850047 | IIScExperiments/Solutions/NatAddMulExperiments.lean | MyNat.zero_mul | [75, 1] | [78, 32] | induction' a using MyNat.induction with d hd | a : MyNat
⊢ 0 * a = 0 | case zero
⊢ 0 * 0 = 0
case succ
d : MyNat
hd : 0 * d = 0
⊢ 0 * succ d = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
a : MyNat
⊢ 0 * a = 0
TACTIC:
|
https://github.com/kbuzzard/IISc-experiments.git | f2f2d7d14b3ec1957fcdd38cc0e3657df6850047 | IIScExperiments/Solutions/NatAddMulExperiments.lean | MyNat.zero_mul | [75, 1] | [78, 32] | rw [mul_zero] | case zero
⊢ 0 * 0 = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
⊢ 0 * 0 = 0
TACTIC:
|
https://github.com/kbuzzard/IISc-experiments.git | f2f2d7d14b3ec1957fcdd38cc0e3657df6850047 | IIScExperiments/Solutions/NatAddMulExperiments.lean | MyNat.zero_mul | [75, 1] | [78, 32] | rw [mul_succ, hd, add_zero] | case succ
d : MyNat
hd : 0 * d = 0
⊢ 0 * succ d = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
d : MyNat
hd : 0 * d = 0
⊢ 0 * succ d = 0
TACTIC:
|
https://github.com/kbuzzard/IISc-experiments.git | f2f2d7d14b3ec1957fcdd38cc0e3657df6850047 | IIScExperiments/Solutions/NatAddMulExperiments.lean | MyNat.succ_eq_add_one | [80, 1] | [81, 35] | rw [one_def, add_succ, add_zero] | a : MyNat
⊢ succ a = a + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a : MyNat
⊢ succ a = a + 1
TACTIC:
|
https://github.com/kbuzzard/IISc-experiments.git | f2f2d7d14b3ec1957fcdd38cc0e3657df6850047 | IIScExperiments/Solutions/NatAddMulExperiments.lean | MyNat.mul_one | [83, 1] | [84, 45] | rw [one_def, mul_succ, mul_zero, zero_add] | a : MyNat
⊢ a * 1 = a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a : MyNat
⊢ a * 1 = a
TACTIC:
|
https://github.com/kbuzzard/IISc-experiments.git | f2f2d7d14b3ec1957fcdd38cc0e3657df6850047 | IIScExperiments/Solutions/NatAddMulExperiments.lean | MyNat.one_mul | [86, 1] | [89, 47] | induction' a using MyNat.induction with d hd | a : MyNat
⊢ one * a = a | case zero
⊢ one * 0 = 0
case succ
d : MyNat
hd : one * d = d
⊢ one * succ d = succ d | Please generate a tactic in lean4 to solve the state.
STATE:
a : MyNat
⊢ one * a = a
TACTIC:
|
https://github.com/kbuzzard/IISc-experiments.git | f2f2d7d14b3ec1957fcdd38cc0e3657df6850047 | IIScExperiments/Solutions/NatAddMulExperiments.lean | MyNat.one_mul | [86, 1] | [89, 47] | rw [mul_zero] | case zero
⊢ one * 0 = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
⊢ one * 0 = 0
TACTIC:
|
https://github.com/kbuzzard/IISc-experiments.git | f2f2d7d14b3ec1957fcdd38cc0e3657df6850047 | IIScExperiments/Solutions/NatAddMulExperiments.lean | MyNat.one_mul | [86, 1] | [89, 47] | rw [mul_succ, hd, one, add_succ, add_zero] | case succ
d : MyNat
hd : one * d = d
⊢ one * succ d = succ d | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
d : MyNat
hd : one * d = d
⊢ one * succ d = succ d
TACTIC:
|
https://github.com/kbuzzard/IISc-experiments.git | f2f2d7d14b3ec1957fcdd38cc0e3657df6850047 | IIScExperiments/Solutions/NatAddMulExperiments.lean | MyNat.succ_mul | [91, 1] | [94, 79] | induction' b using MyNat.induction with d hd | a b : MyNat
⊢ succ a * b = a * b + b | case zero
a : MyNat
⊢ succ a * 0 = a * 0 + 0
case succ
a d : MyNat
hd : succ a * d = a * d + d
⊢ succ a * succ d = a * succ d + succ d | Please generate a tactic in lean4 to solve the state.
STATE:
a b : MyNat
⊢ succ a * b = a * b + b
TACTIC:
|
https://github.com/kbuzzard/IISc-experiments.git | f2f2d7d14b3ec1957fcdd38cc0e3657df6850047 | IIScExperiments/Solutions/NatAddMulExperiments.lean | MyNat.succ_mul | [91, 1] | [94, 79] | rw [mul_zero, mul_zero, add_zero] | case zero
a : MyNat
⊢ succ a * 0 = a * 0 + 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
a : MyNat
⊢ succ a * 0 = a * 0 + 0
TACTIC:
|
https://github.com/kbuzzard/IISc-experiments.git | f2f2d7d14b3ec1957fcdd38cc0e3657df6850047 | IIScExperiments/Solutions/NatAddMulExperiments.lean | MyNat.succ_mul | [91, 1] | [94, 79] | rw [mul_succ, hd, mul_succ, succ_eq_add_one, succ_eq_add_one, add_add_add] | case succ
a d : MyNat
hd : succ a * d = a * d + d
⊢ succ a * succ d = a * succ d + succ d | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
a d : MyNat
hd : succ a * d = a * d + d
⊢ succ a * succ d = a * succ d + succ d
TACTIC:
|
https://github.com/kbuzzard/IISc-experiments.git | f2f2d7d14b3ec1957fcdd38cc0e3657df6850047 | IIScExperiments/Solutions/GroupTheorySolutions.lean | MyGroup.mul_left_cancel | [17, 1] | [25, 42] | calc
b = 1 * b := by rw [one_mul]
_ = a⁻¹ * a * b := by rw [inv_mul_self]
_ = a⁻¹ * (a * b) := by rw [mul_assoc]
_ = a⁻¹ * (a * c) := by rw [h]
_ = a⁻¹ * a * c := by rw [mul_assoc]
_ = 1 * c := by rw [inv_mul_self]
_ = c := by rw [one_mul] | G : Type
inst✝ : MyGroup G
a b c : G
h : a * b = a * c
⊢ b = c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type
inst✝ : MyGroup G
a b c : G
h : a * b = a * c
⊢ b = c
TACTIC:
|
https://github.com/kbuzzard/IISc-experiments.git | f2f2d7d14b3ec1957fcdd38cc0e3657df6850047 | IIScExperiments/Solutions/GroupTheorySolutions.lean | MyGroup.mul_left_cancel | [17, 1] | [25, 42] | rw [one_mul] | G : Type
inst✝ : MyGroup G
a b c : G
h : a * b = a * c
⊢ b = 1 * b | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type
inst✝ : MyGroup G
a b c : G
h : a * b = a * c
⊢ b = 1 * b
TACTIC:
|
https://github.com/kbuzzard/IISc-experiments.git | f2f2d7d14b3ec1957fcdd38cc0e3657df6850047 | IIScExperiments/Solutions/GroupTheorySolutions.lean | MyGroup.mul_left_cancel | [17, 1] | [25, 42] | rw [inv_mul_self] | G : Type
inst✝ : MyGroup G
a b c : G
h : a * b = a * c
⊢ 1 * b = a⁻¹ * a * b | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type
inst✝ : MyGroup G
a b c : G
h : a * b = a * c
⊢ 1 * b = a⁻¹ * a * b
TACTIC:
|
https://github.com/kbuzzard/IISc-experiments.git | f2f2d7d14b3ec1957fcdd38cc0e3657df6850047 | IIScExperiments/Solutions/GroupTheorySolutions.lean | MyGroup.mul_left_cancel | [17, 1] | [25, 42] | rw [mul_assoc] | G : Type
inst✝ : MyGroup G
a b c : G
h : a * b = a * c
⊢ a⁻¹ * a * b = a⁻¹ * (a * b) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type
inst✝ : MyGroup G
a b c : G
h : a * b = a * c
⊢ a⁻¹ * a * b = a⁻¹ * (a * b)
TACTIC:
|
https://github.com/kbuzzard/IISc-experiments.git | f2f2d7d14b3ec1957fcdd38cc0e3657df6850047 | IIScExperiments/Solutions/GroupTheorySolutions.lean | MyGroup.mul_left_cancel | [17, 1] | [25, 42] | rw [h] | G : Type
inst✝ : MyGroup G
a b c : G
h : a * b = a * c
⊢ a⁻¹ * (a * b) = a⁻¹ * (a * c) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type
inst✝ : MyGroup G
a b c : G
h : a * b = a * c
⊢ a⁻¹ * (a * b) = a⁻¹ * (a * c)
TACTIC:
|
https://github.com/kbuzzard/IISc-experiments.git | f2f2d7d14b3ec1957fcdd38cc0e3657df6850047 | IIScExperiments/Solutions/GroupTheorySolutions.lean | MyGroup.mul_left_cancel | [17, 1] | [25, 42] | rw [mul_assoc] | G : Type
inst✝ : MyGroup G
a b c : G
h : a * b = a * c
⊢ a⁻¹ * (a * c) = a⁻¹ * a * c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type
inst✝ : MyGroup G
a b c : G
h : a * b = a * c
⊢ a⁻¹ * (a * c) = a⁻¹ * a * c
TACTIC:
|
https://github.com/kbuzzard/IISc-experiments.git | f2f2d7d14b3ec1957fcdd38cc0e3657df6850047 | IIScExperiments/Solutions/GroupTheorySolutions.lean | MyGroup.mul_left_cancel | [17, 1] | [25, 42] | rw [inv_mul_self] | G : Type
inst✝ : MyGroup G
a b c : G
h : a * b = a * c
⊢ a⁻¹ * a * c = 1 * c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type
inst✝ : MyGroup G
a b c : G
h : a * b = a * c
⊢ a⁻¹ * a * c = 1 * c
TACTIC:
|
https://github.com/kbuzzard/IISc-experiments.git | f2f2d7d14b3ec1957fcdd38cc0e3657df6850047 | IIScExperiments/Solutions/GroupTheorySolutions.lean | MyGroup.mul_left_cancel | [17, 1] | [25, 42] | rw [one_mul] | G : Type
inst✝ : MyGroup G
a b c : G
h : a * b = a * c
⊢ 1 * c = c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type
inst✝ : MyGroup G
a b c : G
h : a * b = a * c
⊢ 1 * c = c
TACTIC:
|
https://github.com/kbuzzard/IISc-experiments.git | f2f2d7d14b3ec1957fcdd38cc0e3657df6850047 | IIScExperiments/Solutions/GroupTheorySolutions.lean | MyGroup.mul_left_cancel' | [27, 1] | [28, 87] | rw [← one_mul b, ← inv_mul_self a, mul_assoc, h, ← mul_assoc, inv_mul_self, one_mul] | G : Type
inst✝ : MyGroup G
a b c : G
h : a * b = a * c
⊢ b = c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type
inst✝ : MyGroup G
a b c : G
h : a * b = a * c
⊢ b = c
TACTIC:
|
https://github.com/kbuzzard/IISc-experiments.git | f2f2d7d14b3ec1957fcdd38cc0e3657df6850047 | IIScExperiments/Solutions/GroupTheorySolutions.lean | MyGroup.mul_eq_of_eq_inv_mul | [30, 1] | [32, 45] | apply mul_left_cancel a⁻¹ | G : Type
inst✝ : MyGroup G
a b c : G
h : b = a⁻¹ * c
⊢ a * b = c | case h
G : Type
inst✝ : MyGroup G
a b c : G
h : b = a⁻¹ * c
⊢ a⁻¹ * (a * b) = a⁻¹ * c | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type
inst✝ : MyGroup G
a b c : G
h : b = a⁻¹ * c
⊢ a * b = c
TACTIC:
|
https://github.com/kbuzzard/IISc-experiments.git | f2f2d7d14b3ec1957fcdd38cc0e3657df6850047 | IIScExperiments/Solutions/GroupTheorySolutions.lean | MyGroup.mul_eq_of_eq_inv_mul | [30, 1] | [32, 45] | rw [← mul_assoc, inv_mul_self, one_mul, h] | case h
G : Type
inst✝ : MyGroup G
a b c : G
h : b = a⁻¹ * c
⊢ a⁻¹ * (a * b) = a⁻¹ * c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
G : Type
inst✝ : MyGroup G
a b c : G
h : b = a⁻¹ * c
⊢ a⁻¹ * (a * b) = a⁻¹ * c
TACTIC:
|
https://github.com/kbuzzard/IISc-experiments.git | f2f2d7d14b3ec1957fcdd38cc0e3657df6850047 | IIScExperiments/Solutions/GroupTheorySolutions.lean | MyGroup.mul_one | [34, 1] | [36, 20] | apply mul_eq_of_eq_inv_mul | G : Type
inst✝ : MyGroup G
a✝ b c a : G
⊢ a * 1 = a | case h
G : Type
inst✝ : MyGroup G
a✝ b c a : G
⊢ 1 = a⁻¹ * a | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type
inst✝ : MyGroup G
a✝ b c a : G
⊢ a * 1 = a
TACTIC:
|
https://github.com/kbuzzard/IISc-experiments.git | f2f2d7d14b3ec1957fcdd38cc0e3657df6850047 | IIScExperiments/Solutions/GroupTheorySolutions.lean | MyGroup.mul_one | [34, 1] | [36, 20] | rw [inv_mul_self] | case h
G : Type
inst✝ : MyGroup G
a✝ b c a : G
⊢ 1 = a⁻¹ * a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
G : Type
inst✝ : MyGroup G
a✝ b c a : G
⊢ 1 = a⁻¹ * a
TACTIC:
|
https://github.com/kbuzzard/IISc-experiments.git | f2f2d7d14b3ec1957fcdd38cc0e3657df6850047 | IIScExperiments/Solutions/GroupTheorySolutions.lean | MyGroup.mul_inv_self | [38, 1] | [40, 15] | apply mul_eq_of_eq_inv_mul | G : Type
inst✝ : MyGroup G
a✝ b c a : G
⊢ a * a⁻¹ = 1 | case h
G : Type
inst✝ : MyGroup G
a✝ b c a : G
⊢ a⁻¹ = a⁻¹ * 1 | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type
inst✝ : MyGroup G
a✝ b c a : G
⊢ a * a⁻¹ = 1
TACTIC:
|
https://github.com/kbuzzard/IISc-experiments.git | f2f2d7d14b3ec1957fcdd38cc0e3657df6850047 | IIScExperiments/Solutions/GroupTheorySolutions.lean | MyGroup.mul_inv_self | [38, 1] | [40, 15] | rw [mul_one] | case h
G : Type
inst✝ : MyGroup G
a✝ b c a : G
⊢ a⁻¹ = a⁻¹ * 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
G : Type
inst✝ : MyGroup G
a✝ b c a : G
⊢ a⁻¹ = a⁻¹ * 1
TACTIC:
|
https://github.com/kbuzzard/IISc-experiments.git | f2f2d7d14b3ec1957fcdd38cc0e3657df6850047 | IIScExperiments/Solutions/GroupTheorySolutions.lean | MyGroup.inv_mul_cancel_left | [44, 1] | [45, 21] | simp [← mul_assoc] | G : Type
inst✝ : MyGroup G
a b c : G
⊢ a⁻¹ * (a * b) = b | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type
inst✝ : MyGroup G
a b c : G
⊢ a⁻¹ * (a * b) = b
TACTIC:
|
https://github.com/kbuzzard/IISc-experiments.git | f2f2d7d14b3ec1957fcdd38cc0e3657df6850047 | IIScExperiments/Solutions/GroupTheorySolutions.lean | MyGroup.mul_inv_cancel_left | [47, 1] | [48, 21] | simp [← mul_assoc] | G : Type
inst✝ : MyGroup G
a b c : G
⊢ a * (a⁻¹ * b) = b | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type
inst✝ : MyGroup G
a b c : G
⊢ a * (a⁻¹ * b) = b
TACTIC:
|
https://github.com/kbuzzard/IISc-experiments.git | f2f2d7d14b3ec1957fcdd38cc0e3657df6850047 | IIScExperiments/Solutions/GroupTheorySolutions.lean | MyGroup.left_inv_eq_right_inv | [50, 1] | [53, 25] | have h : b * (a * c) = (b * a) * c := (mul_assoc b a c).symm | G : Type
inst✝ : MyGroup G
a✝ b✝ c✝ a b c : G
h1 : b * a = 1
h2 : a * c = 1
⊢ b = c | G : Type
inst✝ : MyGroup G
a✝ b✝ c✝ a b c : G
h1 : b * a = 1
h2 : a * c = 1
h : b * (a * c) = b * a * c
⊢ b = c | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type
inst✝ : MyGroup G
a✝ b✝ c✝ a b c : G
h1 : b * a = 1
h2 : a * c = 1
⊢ b = c
TACTIC:
|
https://github.com/kbuzzard/IISc-experiments.git | f2f2d7d14b3ec1957fcdd38cc0e3657df6850047 | IIScExperiments/Solutions/GroupTheorySolutions.lean | MyGroup.left_inv_eq_right_inv | [50, 1] | [53, 25] | simpa [h2, h1] using h | G : Type
inst✝ : MyGroup G
a✝ b✝ c✝ a b c : G
h1 : b * a = 1
h2 : a * c = 1
h : b * (a * c) = b * a * c
⊢ b = c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type
inst✝ : MyGroup G
a✝ b✝ c✝ a b c : G
h1 : b * a = 1
h2 : a * c = 1
h : b * (a * c) = b * a * c
⊢ b = c
TACTIC:
|
https://github.com/kbuzzard/IISc-experiments.git | f2f2d7d14b3ec1957fcdd38cc0e3657df6850047 | IIScExperiments/Solutions/GroupTheorySolutions.lean | MyGroup.mul_eq_one_iff_eq_inv | [55, 1] | [58, 8] | rintro rfl | G : Type
inst✝ : MyGroup G
a b c : G
⊢ a⁻¹ = b → a * b = 1 | G : Type
inst✝ : MyGroup G
a c : G
⊢ a * a⁻¹ = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type
inst✝ : MyGroup G
a b c : G
⊢ a⁻¹ = b → a * b = 1
TACTIC:
|
https://github.com/kbuzzard/IISc-experiments.git | f2f2d7d14b3ec1957fcdd38cc0e3657df6850047 | IIScExperiments/Solutions/GroupTheorySolutions.lean | MyGroup.mul_eq_one_iff_eq_inv | [55, 1] | [58, 8] | simp | G : Type
inst✝ : MyGroup G
a c : G
⊢ a * a⁻¹ = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type
inst✝ : MyGroup G
a c : G
⊢ a * a⁻¹ = 1
TACTIC:
|
https://github.com/kbuzzard/IISc-experiments.git | f2f2d7d14b3ec1957fcdd38cc0e3657df6850047 | IIScExperiments/Solutions/GroupTheorySolutions.lean | MyGroup.one_inv | [60, 1] | [61, 33] | simp [← mul_eq_one_iff_eq_inv] | G : Type
inst✝ : MyGroup G
a b c : G
⊢ 1⁻¹ = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type
inst✝ : MyGroup G
a b c : G
⊢ 1⁻¹ = 1
TACTIC:
|
https://github.com/kbuzzard/IISc-experiments.git | f2f2d7d14b3ec1957fcdd38cc0e3657df6850047 | IIScExperiments/Solutions/GroupTheorySolutions.lean | MyGroup.inv_inv | [63, 1] | [64, 33] | simp [← mul_eq_one_iff_eq_inv] | G : Type
inst✝ : MyGroup G
a b c : G
⊢ a⁻¹⁻¹ = a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type
inst✝ : MyGroup G
a b c : G
⊢ a⁻¹⁻¹ = a
TACTIC:
|
https://github.com/kbuzzard/IISc-experiments.git | f2f2d7d14b3ec1957fcdd38cc0e3657df6850047 | IIScExperiments/Solutions/GroupTheorySolutions.lean | MyGroup.mul_inv_rev | [66, 1] | [67, 33] | simp [← mul_eq_one_iff_eq_inv] | G : Type
inst✝ : MyGroup G
a b c : G
⊢ (a * b)⁻¹ = b⁻¹ * a⁻¹ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type
inst✝ : MyGroup G
a b c : G
⊢ (a * b)⁻¹ = b⁻¹ * a⁻¹
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/IntegerInduction.lean | IntegerInduction | [5, 1] | [17, 51] | constructor | p : ℤ → Prop
⊢ (∀ (n : ℤ), p n) ↔ (∃ k, p k) ∧ ∀ (m : ℤ), p m ↔ p (m + 1) | case mp
p : ℤ → Prop
⊢ (∀ (n : ℤ), p n) → (∃ k, p k) ∧ ∀ (m : ℤ), p m ↔ p (m + 1)
case mpr
p : ℤ → Prop
⊢ ((∃ k, p k) ∧ ∀ (m : ℤ), p m ↔ p (m + 1)) → ∀ (n : ℤ), p n | Please generate a tactic in lean4 to solve the state.
STATE:
p : ℤ → Prop
⊢ (∀ (n : ℤ), p n) ↔ (∃ k, p k) ∧ ∀ (m : ℤ), p m ↔ p (m + 1)
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/IntegerInduction.lean | IntegerInduction | [5, 1] | [17, 51] | intros h₀ | case mp
p : ℤ → Prop
⊢ (∀ (n : ℤ), p n) → (∃ k, p k) ∧ ∀ (m : ℤ), p m ↔ p (m + 1) | case mp
p : ℤ → Prop
h₀ : ∀ (n : ℤ), p n
⊢ (∃ k, p k) ∧ ∀ (m : ℤ), p m ↔ p (m + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
p : ℤ → Prop
⊢ (∀ (n : ℤ), p n) → (∃ k, p k) ∧ ∀ (m : ℤ), p m ↔ p (m + 1)
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/IntegerInduction.lean | IntegerInduction | [5, 1] | [17, 51] | constructor | case mp
p : ℤ → Prop
h₀ : ∀ (n : ℤ), p n
⊢ (∃ k, p k) ∧ ∀ (m : ℤ), p m ↔ p (m + 1) | case mp.left
p : ℤ → Prop
h₀ : ∀ (n : ℤ), p n
⊢ ∃ k, p k
case mp.right
p : ℤ → Prop
h₀ : ∀ (n : ℤ), p n
⊢ ∀ (m : ℤ), p m ↔ p (m + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
p : ℤ → Prop
h₀ : ∀ (n : ℤ), p n
⊢ (∃ k, p k) ∧ ∀ (m : ℤ), p m ↔ p (m + 1)
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/IntegerInduction.lean | IntegerInduction | [5, 1] | [17, 51] | use 391547 | case mp.left
p : ℤ → Prop
h₀ : ∀ (n : ℤ), p n
⊢ ∃ k, p k | case h
p : ℤ → Prop
h₀ : ∀ (n : ℤ), p n
⊢ p 391547 | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.left
p : ℤ → Prop
h₀ : ∀ (n : ℤ), p n
⊢ ∃ k, p k
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/IntegerInduction.lean | IntegerInduction | [5, 1] | [17, 51] | exact h₀ 391547 | case h
p : ℤ → Prop
h₀ : ∀ (n : ℤ), p n
⊢ p 391547 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
p : ℤ → Prop
h₀ : ∀ (n : ℤ), p n
⊢ p 391547
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/IntegerInduction.lean | IntegerInduction | [5, 1] | [17, 51] | intros m | case mp.right
p : ℤ → Prop
h₀ : ∀ (n : ℤ), p n
⊢ ∀ (m : ℤ), p m ↔ p (m + 1) | case mp.right
p : ℤ → Prop
h₀ : ∀ (n : ℤ), p n
m : ℤ
⊢ p m ↔ p (m + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.right
p : ℤ → Prop
h₀ : ∀ (n : ℤ), p n
⊢ ∀ (m : ℤ), p m ↔ p (m + 1)
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/IntegerInduction.lean | IntegerInduction | [5, 1] | [17, 51] | simp only [h₀ m, h₀ (m + 1)] | case mp.right
p : ℤ → Prop
h₀ : ∀ (n : ℤ), p n
m : ℤ
⊢ p m ↔ p (m + 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.right
p : ℤ → Prop
h₀ : ∀ (n : ℤ), p n
m : ℤ
⊢ p m ↔ p (m + 1)
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/IntegerInduction.lean | IntegerInduction | [5, 1] | [17, 51] | intros h₀ | case mpr
p : ℤ → Prop
⊢ ((∃ k, p k) ∧ ∀ (m : ℤ), p m ↔ p (m + 1)) → ∀ (n : ℤ), p n | case mpr
p : ℤ → Prop
h₀ : (∃ k, p k) ∧ ∀ (m : ℤ), p m ↔ p (m + 1)
⊢ ∀ (n : ℤ), p n | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
p : ℤ → Prop
⊢ ((∃ k, p k) ∧ ∀ (m : ℤ), p m ↔ p (m + 1)) → ∀ (n : ℤ), p n
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/IntegerInduction.lean | IntegerInduction | [5, 1] | [17, 51] | obtain ⟨he, hi⟩ := h₀ | case mpr
p : ℤ → Prop
h₀ : (∃ k, p k) ∧ ∀ (m : ℤ), p m ↔ p (m + 1)
⊢ ∀ (n : ℤ), p n | case mpr.intro
p : ℤ → Prop
he : ∃ k, p k
hi : ∀ (m : ℤ), p m ↔ p (m + 1)
⊢ ∀ (n : ℤ), p n | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
p : ℤ → Prop
h₀ : (∃ k, p k) ∧ ∀ (m : ℤ), p m ↔ p (m + 1)
⊢ ∀ (n : ℤ), p n
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/IntegerInduction.lean | IntegerInduction | [5, 1] | [17, 51] | obtain ⟨k, hk⟩ := he | case mpr.intro
p : ℤ → Prop
he : ∃ k, p k
hi : ∀ (m : ℤ), p m ↔ p (m + 1)
⊢ ∀ (n : ℤ), p n | case mpr.intro.intro
p : ℤ → Prop
hi : ∀ (m : ℤ), p m ↔ p (m + 1)
k : ℤ
hk : p k
⊢ ∀ (n : ℤ), p n | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro
p : ℤ → Prop
he : ∃ k, p k
hi : ∀ (m : ℤ), p m ↔ p (m + 1)
⊢ ∀ (n : ℤ), p n
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/IntegerInduction.lean | IntegerInduction | [5, 1] | [17, 51] | exact fun n => (Succ.rec_linear hi n k).mpr hk | case mpr.intro.intro
p : ℤ → Prop
hi : ∀ (m : ℤ), p m ↔ p (m + 1)
k : ℤ
hk : p k
⊢ ∀ (n : ℤ), p n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro.intro
p : ℤ → Prop
hi : ∀ (m : ℤ), p m ↔ p (m + 1)
k : ℤ
hk : p k
⊢ ∀ (n : ℤ), p n
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/IntegerInduction.lean | WavelengthRestate | [19, 1] | [38, 13] | constructor | p : ℤ → Prop
k : ℤ
⊢ (∀ (m : ℤ), p m ↔ p (m + k)) ↔ ∀ (m k₀ : ℤ), p m ↔ p (m + k₀ * k) | case mp
p : ℤ → Prop
k : ℤ
⊢ (∀ (m : ℤ), p m ↔ p (m + k)) → ∀ (m k₀ : ℤ), p m ↔ p (m + k₀ * k)
case mpr
p : ℤ → Prop
k : ℤ
⊢ (∀ (m k₀ : ℤ), p m ↔ p (m + k₀ * k)) → ∀ (m : ℤ), p m ↔ p (m + k) | Please generate a tactic in lean4 to solve the state.
STATE:
p : ℤ → Prop
k : ℤ
⊢ (∀ (m : ℤ), p m ↔ p (m + k)) ↔ ∀ (m k₀ : ℤ), p m ↔ p (m + k₀ * k)
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/IntegerInduction.lean | WavelengthRestate | [19, 1] | [38, 13] | intros h | case mp
p : ℤ → Prop
k : ℤ
⊢ (∀ (m : ℤ), p m ↔ p (m + k)) → ∀ (m k₀ : ℤ), p m ↔ p (m + k₀ * k) | case mp
p : ℤ → Prop
k : ℤ
h : ∀ (m : ℤ), p m ↔ p (m + k)
⊢ ∀ (m k₀ : ℤ), p m ↔ p (m + k₀ * k) | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
p : ℤ → Prop
k : ℤ
⊢ (∀ (m : ℤ), p m ↔ p (m + k)) → ∀ (m k₀ : ℤ), p m ↔ p (m + k₀ * k)
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/IntegerInduction.lean | WavelengthRestate | [19, 1] | [38, 13] | rw [forall_swap, IntegerInduction] | case mp
p : ℤ → Prop
k : ℤ
h : ∀ (m : ℤ), p m ↔ p (m + k)
⊢ ∀ (m k₀ : ℤ), p m ↔ p (m + k₀ * k) | case mp
p : ℤ → Prop
k : ℤ
h : ∀ (m : ℤ), p m ↔ p (m + k)
⊢ (∃ k_1, ∀ (x : ℤ), p x ↔ p (x + k_1 * k)) ∧
∀ (m : ℤ), (∀ (x : ℤ), p x ↔ p (x + m * k)) ↔ ∀ (x : ℤ), p x ↔ p (x + (m + 1) * k) | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
p : ℤ → Prop
k : ℤ
h : ∀ (m : ℤ), p m ↔ p (m + k)
⊢ ∀ (m k₀ : ℤ), p m ↔ p (m + k₀ * k)
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/IntegerInduction.lean | WavelengthRestate | [19, 1] | [38, 13] | constructor | case mp
p : ℤ → Prop
k : ℤ
h : ∀ (m : ℤ), p m ↔ p (m + k)
⊢ (∃ k_1, ∀ (x : ℤ), p x ↔ p (x + k_1 * k)) ∧
∀ (m : ℤ), (∀ (x : ℤ), p x ↔ p (x + m * k)) ↔ ∀ (x : ℤ), p x ↔ p (x + (m + 1) * k) | case mp.left
p : ℤ → Prop
k : ℤ
h : ∀ (m : ℤ), p m ↔ p (m + k)
⊢ ∃ k_1, ∀ (x : ℤ), p x ↔ p (x + k_1 * k)
case mp.right
p : ℤ → Prop
k : ℤ
h : ∀ (m : ℤ), p m ↔ p (m + k)
⊢ ∀ (m : ℤ), (∀ (x : ℤ), p x ↔ p (x + m * k)) ↔ ∀ (x : ℤ), p x ↔ p (x + (m + 1) * k) | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
p : ℤ → Prop
k : ℤ
h : ∀ (m : ℤ), p m ↔ p (m + k)
⊢ (∃ k_1, ∀ (x : ℤ), p x ↔ p (x + k_1 * k)) ∧
∀ (m : ℤ), (∀ (x : ℤ), p x ↔ p (x + m * k)) ↔ ∀ (x : ℤ), p x ↔ p (x + (m + 1) * k)
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/IntegerInduction.lean | WavelengthRestate | [19, 1] | [38, 13] | use 0 | case mp.left
p : ℤ → Prop
k : ℤ
h : ∀ (m : ℤ), p m ↔ p (m + k)
⊢ ∃ k_1, ∀ (x : ℤ), p x ↔ p (x + k_1 * k) | case h
p : ℤ → Prop
k : ℤ
h : ∀ (m : ℤ), p m ↔ p (m + k)
⊢ ∀ (x : ℤ), p x ↔ p (x + 0 * k) | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.left
p : ℤ → Prop
k : ℤ
h : ∀ (m : ℤ), p m ↔ p (m + k)
⊢ ∃ k_1, ∀ (x : ℤ), p x ↔ p (x + k_1 * k)
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/IntegerInduction.lean | WavelengthRestate | [19, 1] | [38, 13] | simp only [zero_mul, add_zero, forall_const] | case h
p : ℤ → Prop
k : ℤ
h : ∀ (m : ℤ), p m ↔ p (m + k)
⊢ ∀ (x : ℤ), p x ↔ p (x + 0 * k) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
p : ℤ → Prop
k : ℤ
h : ∀ (m : ℤ), p m ↔ p (m + k)
⊢ ∀ (x : ℤ), p x ↔ p (x + 0 * k)
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/IntegerInduction.lean | WavelengthRestate | [19, 1] | [38, 13] | intros m | case mp.right
p : ℤ → Prop
k : ℤ
h : ∀ (m : ℤ), p m ↔ p (m + k)
⊢ ∀ (m : ℤ), (∀ (x : ℤ), p x ↔ p (x + m * k)) ↔ ∀ (x : ℤ), p x ↔ p (x + (m + 1) * k) | case mp.right
p : ℤ → Prop
k : ℤ
h : ∀ (m : ℤ), p m ↔ p (m + k)
m : ℤ
⊢ (∀ (x : ℤ), p x ↔ p (x + m * k)) ↔ ∀ (x : ℤ), p x ↔ p (x + (m + 1) * k) | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.right
p : ℤ → Prop
k : ℤ
h : ∀ (m : ℤ), p m ↔ p (m + k)
⊢ ∀ (m : ℤ), (∀ (x : ℤ), p x ↔ p (x + m * k)) ↔ ∀ (x : ℤ), p x ↔ p (x + (m + 1) * k)
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/IntegerInduction.lean | WavelengthRestate | [19, 1] | [38, 13] | constructor | case mp.right
p : ℤ → Prop
k : ℤ
h : ∀ (m : ℤ), p m ↔ p (m + k)
m : ℤ
⊢ (∀ (x : ℤ), p x ↔ p (x + m * k)) ↔ ∀ (x : ℤ), p x ↔ p (x + (m + 1) * k) | case mp.right.mp
p : ℤ → Prop
k : ℤ
h : ∀ (m : ℤ), p m ↔ p (m + k)
m : ℤ
⊢ (∀ (x : ℤ), p x ↔ p (x + m * k)) → ∀ (x : ℤ), p x ↔ p (x + (m + 1) * k)
case mp.right.mpr
p : ℤ → Prop
k : ℤ
h : ∀ (m : ℤ), p m ↔ p (m + k)
m : ℤ
⊢ (∀ (x : ℤ), p x ↔ p (x + (m + 1) * k)) → ∀ (x : ℤ), p x ↔ p (x + m * k) | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.right
p : ℤ → Prop
k : ℤ
h : ∀ (m : ℤ), p m ↔ p (m + k)
m : ℤ
⊢ (∀ (x : ℤ), p x ↔ p (x + m * k)) ↔ ∀ (x : ℤ), p x ↔ p (x + (m + 1) * k)
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/IntegerInduction.lean | WavelengthRestate | [19, 1] | [38, 13] | intros h₀ k₀ | case mp.right.mp
p : ℤ → Prop
k : ℤ
h : ∀ (m : ℤ), p m ↔ p (m + k)
m : ℤ
⊢ (∀ (x : ℤ), p x ↔ p (x + m * k)) → ∀ (x : ℤ), p x ↔ p (x + (m + 1) * k) | case mp.right.mp
p : ℤ → Prop
k : ℤ
h : ∀ (m : ℤ), p m ↔ p (m + k)
m : ℤ
h₀ : ∀ (x : ℤ), p x ↔ p (x + m * k)
k₀ : ℤ
⊢ p k₀ ↔ p (k₀ + (m + 1) * k) | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.right.mp
p : ℤ → Prop
k : ℤ
h : ∀ (m : ℤ), p m ↔ p (m + k)
m : ℤ
⊢ (∀ (x : ℤ), p x ↔ p (x + m * k)) → ∀ (x : ℤ), p x ↔ p (x + (m + 1) * k)
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/IntegerInduction.lean | WavelengthRestate | [19, 1] | [38, 13] | rw [h₀ k₀, h] | case mp.right.mp
p : ℤ → Prop
k : ℤ
h : ∀ (m : ℤ), p m ↔ p (m + k)
m : ℤ
h₀ : ∀ (x : ℤ), p x ↔ p (x + m * k)
k₀ : ℤ
⊢ p k₀ ↔ p (k₀ + (m + 1) * k) | case mp.right.mp
p : ℤ → Prop
k : ℤ
h : ∀ (m : ℤ), p m ↔ p (m + k)
m : ℤ
h₀ : ∀ (x : ℤ), p x ↔ p (x + m * k)
k₀ : ℤ
⊢ p (k₀ + m * k + k) ↔ p (k₀ + (m + 1) * k) | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.right.mp
p : ℤ → Prop
k : ℤ
h : ∀ (m : ℤ), p m ↔ p (m + k)
m : ℤ
h₀ : ∀ (x : ℤ), p x ↔ p (x + m * k)
k₀ : ℤ
⊢ p k₀ ↔ p (k₀ + (m + 1) * k)
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/IntegerInduction.lean | WavelengthRestate | [19, 1] | [38, 13] | ring_nf | case mp.right.mp
p : ℤ → Prop
k : ℤ
h : ∀ (m : ℤ), p m ↔ p (m + k)
m : ℤ
h₀ : ∀ (x : ℤ), p x ↔ p (x + m * k)
k₀ : ℤ
⊢ p (k₀ + m * k + k) ↔ p (k₀ + (m + 1) * k) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.right.mp
p : ℤ → Prop
k : ℤ
h : ∀ (m : ℤ), p m ↔ p (m + k)
m : ℤ
h₀ : ∀ (x : ℤ), p x ↔ p (x + m * k)
k₀ : ℤ
⊢ p (k₀ + m * k + k) ↔ p (k₀ + (m + 1) * k)
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/IntegerInduction.lean | WavelengthRestate | [19, 1] | [38, 13] | intros h₀ k₀ | case mp.right.mpr
p : ℤ → Prop
k : ℤ
h : ∀ (m : ℤ), p m ↔ p (m + k)
m : ℤ
⊢ (∀ (x : ℤ), p x ↔ p (x + (m + 1) * k)) → ∀ (x : ℤ), p x ↔ p (x + m * k) | case mp.right.mpr
p : ℤ → Prop
k : ℤ
h : ∀ (m : ℤ), p m ↔ p (m + k)
m : ℤ
h₀ : ∀ (x : ℤ), p x ↔ p (x + (m + 1) * k)
k₀ : ℤ
⊢ p k₀ ↔ p (k₀ + m * k) | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.right.mpr
p : ℤ → Prop
k : ℤ
h : ∀ (m : ℤ), p m ↔ p (m + k)
m : ℤ
⊢ (∀ (x : ℤ), p x ↔ p (x + (m + 1) * k)) → ∀ (x : ℤ), p x ↔ p (x + m * k)
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/IntegerInduction.lean | WavelengthRestate | [19, 1] | [38, 13] | rw [h₀ k₀, h (k₀ + m * k)] | case mp.right.mpr
p : ℤ → Prop
k : ℤ
h : ∀ (m : ℤ), p m ↔ p (m + k)
m : ℤ
h₀ : ∀ (x : ℤ), p x ↔ p (x + (m + 1) * k)
k₀ : ℤ
⊢ p k₀ ↔ p (k₀ + m * k) | case mp.right.mpr
p : ℤ → Prop
k : ℤ
h : ∀ (m : ℤ), p m ↔ p (m + k)
m : ℤ
h₀ : ∀ (x : ℤ), p x ↔ p (x + (m + 1) * k)
k₀ : ℤ
⊢ p (k₀ + (m + 1) * k) ↔ p (k₀ + m * k + k) | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.right.mpr
p : ℤ → Prop
k : ℤ
h : ∀ (m : ℤ), p m ↔ p (m + k)
m : ℤ
h₀ : ∀ (x : ℤ), p x ↔ p (x + (m + 1) * k)
k₀ : ℤ
⊢ p k₀ ↔ p (k₀ + m * k)
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/IntegerInduction.lean | WavelengthRestate | [19, 1] | [38, 13] | ring_nf | case mp.right.mpr
p : ℤ → Prop
k : ℤ
h : ∀ (m : ℤ), p m ↔ p (m + k)
m : ℤ
h₀ : ∀ (x : ℤ), p x ↔ p (x + (m + 1) * k)
k₀ : ℤ
⊢ p (k₀ + (m + 1) * k) ↔ p (k₀ + m * k + k) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.right.mpr
p : ℤ → Prop
k : ℤ
h : ∀ (m : ℤ), p m ↔ p (m + k)
m : ℤ
h₀ : ∀ (x : ℤ), p x ↔ p (x + (m + 1) * k)
k₀ : ℤ
⊢ p (k₀ + (m + 1) * k) ↔ p (k₀ + m * k + k)
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/IntegerInduction.lean | WavelengthRestate | [19, 1] | [38, 13] | intros h m | case mpr
p : ℤ → Prop
k : ℤ
⊢ (∀ (m k₀ : ℤ), p m ↔ p (m + k₀ * k)) → ∀ (m : ℤ), p m ↔ p (m + k) | case mpr
p : ℤ → Prop
k : ℤ
h : ∀ (m k₀ : ℤ), p m ↔ p (m + k₀ * k)
m : ℤ
⊢ p m ↔ p (m + k) | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
p : ℤ → Prop
k : ℤ
⊢ (∀ (m k₀ : ℤ), p m ↔ p (m + k₀ * k)) → ∀ (m : ℤ), p m ↔ p (m + k)
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/IntegerInduction.lean | WavelengthRestate | [19, 1] | [38, 13] | have h₀ := h m 1 | case mpr
p : ℤ → Prop
k : ℤ
h : ∀ (m k₀ : ℤ), p m ↔ p (m + k₀ * k)
m : ℤ
⊢ p m ↔ p (m + k) | case mpr
p : ℤ → Prop
k : ℤ
h : ∀ (m k₀ : ℤ), p m ↔ p (m + k₀ * k)
m : ℤ
h₀ : p m ↔ p (m + 1 * k)
⊢ p m ↔ p (m + k) | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
p : ℤ → Prop
k : ℤ
h : ∀ (m k₀ : ℤ), p m ↔ p (m + k₀ * k)
m : ℤ
⊢ p m ↔ p (m + k)
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/IntegerInduction.lean | WavelengthRestate | [19, 1] | [38, 13] | simp only [one_mul] at h₀ | case mpr
p : ℤ → Prop
k : ℤ
h : ∀ (m k₀ : ℤ), p m ↔ p (m + k₀ * k)
m : ℤ
h₀ : p m ↔ p (m + 1 * k)
⊢ p m ↔ p (m + k) | case mpr
p : ℤ → Prop
k : ℤ
h : ∀ (m k₀ : ℤ), p m ↔ p (m + k₀ * k)
m : ℤ
h₀ : p m ↔ p (m + k)
⊢ p m ↔ p (m + k) | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
p : ℤ → Prop
k : ℤ
h : ∀ (m k₀ : ℤ), p m ↔ p (m + k₀ * k)
m : ℤ
h₀ : p m ↔ p (m + 1 * k)
⊢ p m ↔ p (m + k)
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/IntegerInduction.lean | WavelengthRestate | [19, 1] | [38, 13] | exact h₀ | case mpr
p : ℤ → Prop
k : ℤ
h : ∀ (m k₀ : ℤ), p m ↔ p (m + k₀ * k)
m : ℤ
h₀ : p m ↔ p (m + k)
⊢ p m ↔ p (m + k) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
p : ℤ → Prop
k : ℤ
h : ∀ (m k₀ : ℤ), p m ↔ p (m + k₀ * k)
m : ℤ
h₀ : p m ↔ p (m + k)
⊢ p m ↔ p (m + k)
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/IntegerInduction.lean | associated_gcd_gcd | [40, 1] | [41, 38] | exact IsBezout.associated_gcd_gcd ℤ | a b : ℤ
⊢ Associated (IsBezout.gcd a b) (GCDMonoid.gcd a b) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b : ℤ
⊢ Associated (IsBezout.gcd a b) (GCDMonoid.gcd a b)
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/IntegerInduction.lean | GcdLinearCombination | [43, 1] | [54, 13] | obtain ⟨m, n, h⟩ := IsBezout.gcd_eq_sum k₀ k₁ | k₀ k₁ : ℤ
⊢ ∃ m₀ m₁, ↑(k₀.gcd k₁) = m₀ * k₀ + m₁ * k₁ | case intro.intro
k₀ k₁ m n : ℤ
h : m * k₀ + n * k₁ = IsBezout.gcd k₀ k₁
⊢ ∃ m₀ m₁, ↑(k₀.gcd k₁) = m₀ * k₀ + m₁ * k₁ | Please generate a tactic in lean4 to solve the state.
STATE:
k₀ k₁ : ℤ
⊢ ∃ m₀ m₁, ↑(k₀.gcd k₁) = m₀ * k₀ + m₁ * k₁
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/IntegerInduction.lean | GcdLinearCombination | [43, 1] | [54, 13] | have := associated_gcd_gcd k₀ k₁ | case intro.intro
k₀ k₁ m n : ℤ
h : m * k₀ + n * k₁ = IsBezout.gcd k₀ k₁
⊢ ∃ m₀ m₁, ↑(k₀.gcd k₁) = m₀ * k₀ + m₁ * k₁ | case intro.intro
k₀ k₁ m n : ℤ
h : m * k₀ + n * k₁ = IsBezout.gcd k₀ k₁
this : Associated (IsBezout.gcd k₀ k₁) (GCDMonoid.gcd k₀ k₁)
⊢ ∃ m₀ m₁, ↑(k₀.gcd k₁) = m₀ * k₀ + m₁ * k₁ | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
k₀ k₁ m n : ℤ
h : m * k₀ + n * k₁ = IsBezout.gcd k₀ k₁
⊢ ∃ m₀ m₁, ↑(k₀.gcd k₁) = m₀ * k₀ + m₁ * k₁
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/IntegerInduction.lean | GcdLinearCombination | [43, 1] | [54, 13] | rw [Int.associated_iff] at this | case intro.intro
k₀ k₁ m n : ℤ
h : m * k₀ + n * k₁ = IsBezout.gcd k₀ k₁
this : Associated (IsBezout.gcd k₀ k₁) (GCDMonoid.gcd k₀ k₁)
⊢ ∃ m₀ m₁, ↑(k₀.gcd k₁) = m₀ * k₀ + m₁ * k₁ | case intro.intro
k₀ k₁ m n : ℤ
h : m * k₀ + n * k₁ = IsBezout.gcd k₀ k₁
this : IsBezout.gcd k₀ k₁ = GCDMonoid.gcd k₀ k₁ ∨ IsBezout.gcd k₀ k₁ = -GCDMonoid.gcd k₀ k₁
⊢ ∃ m₀ m₁, ↑(k₀.gcd k₁) = m₀ * k₀ + m₁ * k₁ | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
k₀ k₁ m n : ℤ
h : m * k₀ + n * k₁ = IsBezout.gcd k₀ k₁
this : Associated (IsBezout.gcd k₀ k₁) (GCDMonoid.gcd k₀ k₁)
⊢ ∃ m₀ m₁, ↑(k₀.gcd k₁) = m₀ * k₀ + m₁ * k₁
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/IntegerInduction.lean | GcdLinearCombination | [43, 1] | [54, 13] | cases this with
| inl h' =>
use m, n
rw [h, Int.coe_gcd, h']
| inr h' =>
use -m, -n
rw [Int.coe_gcd]
linarith | case intro.intro
k₀ k₁ m n : ℤ
h : m * k₀ + n * k₁ = IsBezout.gcd k₀ k₁
this : IsBezout.gcd k₀ k₁ = GCDMonoid.gcd k₀ k₁ ∨ IsBezout.gcd k₀ k₁ = -GCDMonoid.gcd k₀ k₁
⊢ ∃ m₀ m₁, ↑(k₀.gcd k₁) = m₀ * k₀ + m₁ * k₁ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
k₀ k₁ m n : ℤ
h : m * k₀ + n * k₁ = IsBezout.gcd k₀ k₁
this : IsBezout.gcd k₀ k₁ = GCDMonoid.gcd k₀ k₁ ∨ IsBezout.gcd k₀ k₁ = -GCDMonoid.gcd k₀ k₁
⊢ ∃ m₀ m₁, ↑(k₀.gcd k₁) = m₀ * k₀ + m₁ * k₁
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/IntegerInduction.lean | GcdLinearCombination | [43, 1] | [54, 13] | use m, n | case intro.intro.inl
k₀ k₁ m n : ℤ
h : m * k₀ + n * k₁ = IsBezout.gcd k₀ k₁
h' : IsBezout.gcd k₀ k₁ = GCDMonoid.gcd k₀ k₁
⊢ ∃ m₀ m₁, ↑(k₀.gcd k₁) = m₀ * k₀ + m₁ * k₁ | case h
k₀ k₁ m n : ℤ
h : m * k₀ + n * k₁ = IsBezout.gcd k₀ k₁
h' : IsBezout.gcd k₀ k₁ = GCDMonoid.gcd k₀ k₁
⊢ ↑(k₀.gcd k₁) = m * k₀ + n * k₁ | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.inl
k₀ k₁ m n : ℤ
h : m * k₀ + n * k₁ = IsBezout.gcd k₀ k₁
h' : IsBezout.gcd k₀ k₁ = GCDMonoid.gcd k₀ k₁
⊢ ∃ m₀ m₁, ↑(k₀.gcd k₁) = m₀ * k₀ + m₁ * k₁
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/IntegerInduction.lean | GcdLinearCombination | [43, 1] | [54, 13] | rw [h, Int.coe_gcd, h'] | case h
k₀ k₁ m n : ℤ
h : m * k₀ + n * k₁ = IsBezout.gcd k₀ k₁
h' : IsBezout.gcd k₀ k₁ = GCDMonoid.gcd k₀ k₁
⊢ ↑(k₀.gcd k₁) = m * k₀ + n * k₁ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
k₀ k₁ m n : ℤ
h : m * k₀ + n * k₁ = IsBezout.gcd k₀ k₁
h' : IsBezout.gcd k₀ k₁ = GCDMonoid.gcd k₀ k₁
⊢ ↑(k₀.gcd k₁) = m * k₀ + n * k₁
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/IntegerInduction.lean | GcdLinearCombination | [43, 1] | [54, 13] | use -m, -n | case intro.intro.inr
k₀ k₁ m n : ℤ
h : m * k₀ + n * k₁ = IsBezout.gcd k₀ k₁
h' : IsBezout.gcd k₀ k₁ = -GCDMonoid.gcd k₀ k₁
⊢ ∃ m₀ m₁, ↑(k₀.gcd k₁) = m₀ * k₀ + m₁ * k₁ | case h
k₀ k₁ m n : ℤ
h : m * k₀ + n * k₁ = IsBezout.gcd k₀ k₁
h' : IsBezout.gcd k₀ k₁ = -GCDMonoid.gcd k₀ k₁
⊢ ↑(k₀.gcd k₁) = -m * k₀ + -n * k₁ | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.inr
k₀ k₁ m n : ℤ
h : m * k₀ + n * k₁ = IsBezout.gcd k₀ k₁
h' : IsBezout.gcd k₀ k₁ = -GCDMonoid.gcd k₀ k₁
⊢ ∃ m₀ m₁, ↑(k₀.gcd k₁) = m₀ * k₀ + m₁ * k₁
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/IntegerInduction.lean | GcdLinearCombination | [43, 1] | [54, 13] | rw [Int.coe_gcd] | case h
k₀ k₁ m n : ℤ
h : m * k₀ + n * k₁ = IsBezout.gcd k₀ k₁
h' : IsBezout.gcd k₀ k₁ = -GCDMonoid.gcd k₀ k₁
⊢ ↑(k₀.gcd k₁) = -m * k₀ + -n * k₁ | case h
k₀ k₁ m n : ℤ
h : m * k₀ + n * k₁ = IsBezout.gcd k₀ k₁
h' : IsBezout.gcd k₀ k₁ = -GCDMonoid.gcd k₀ k₁
⊢ GCDMonoid.gcd k₀ k₁ = -m * k₀ + -n * k₁ | Please generate a tactic in lean4 to solve the state.
STATE:
case h
k₀ k₁ m n : ℤ
h : m * k₀ + n * k₁ = IsBezout.gcd k₀ k₁
h' : IsBezout.gcd k₀ k₁ = -GCDMonoid.gcd k₀ k₁
⊢ ↑(k₀.gcd k₁) = -m * k₀ + -n * k₁
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/IntegerInduction.lean | GcdLinearCombination | [43, 1] | [54, 13] | linarith | case h
k₀ k₁ m n : ℤ
h : m * k₀ + n * k₁ = IsBezout.gcd k₀ k₁
h' : IsBezout.gcd k₀ k₁ = -GCDMonoid.gcd k₀ k₁
⊢ GCDMonoid.gcd k₀ k₁ = -m * k₀ + -n * k₁ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
k₀ k₁ m n : ℤ
h : m * k₀ + n * k₁ = IsBezout.gcd k₀ k₁
h' : IsBezout.gcd k₀ k₁ = -GCDMonoid.gcd k₀ k₁
⊢ GCDMonoid.gcd k₀ k₁ = -m * k₀ + -n * k₁
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/IntegerInduction.lean | WavelengthGcd | [56, 1] | [82, 12] | constructor | p : ℤ → Prop
k₀ k₁ : ℤ
⊢ (∀ (m : ℤ), p m ↔ p (m + ↑(k₀.gcd k₁))) ↔ (∀ (m : ℤ), p m ↔ p (m + k₀)) ∧ ∀ (m : ℤ), p m ↔ p (m + k₁) | case mp
p : ℤ → Prop
k₀ k₁ : ℤ
⊢ (∀ (m : ℤ), p m ↔ p (m + ↑(k₀.gcd k₁))) → (∀ (m : ℤ), p m ↔ p (m + k₀)) ∧ ∀ (m : ℤ), p m ↔ p (m + k₁)
case mpr
p : ℤ → Prop
k₀ k₁ : ℤ
⊢ ((∀ (m : ℤ), p m ↔ p (m + k₀)) ∧ ∀ (m : ℤ), p m ↔ p (m + k₁)) → ∀ (m : ℤ), p m ↔ p (m + ↑(k₀.gcd k₁)) | Please generate a tactic in lean4 to solve the state.
STATE:
p : ℤ → Prop
k₀ k₁ : ℤ
⊢ (∀ (m : ℤ), p m ↔ p (m + ↑(k₀.gcd k₁))) ↔ (∀ (m : ℤ), p m ↔ p (m + k₀)) ∧ ∀ (m : ℤ), p m ↔ p (m + k₁)
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/IntegerInduction.lean | WavelengthGcd | [56, 1] | [82, 12] | intros h₀ | case mp
p : ℤ → Prop
k₀ k₁ : ℤ
⊢ (∀ (m : ℤ), p m ↔ p (m + ↑(k₀.gcd k₁))) → (∀ (m : ℤ), p m ↔ p (m + k₀)) ∧ ∀ (m : ℤ), p m ↔ p (m + k₁) | case mp
p : ℤ → Prop
k₀ k₁ : ℤ
h₀ : ∀ (m : ℤ), p m ↔ p (m + ↑(k₀.gcd k₁))
⊢ (∀ (m : ℤ), p m ↔ p (m + k₀)) ∧ ∀ (m : ℤ), p m ↔ p (m + k₁) | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
p : ℤ → Prop
k₀ k₁ : ℤ
⊢ (∀ (m : ℤ), p m ↔ p (m + ↑(k₀.gcd k₁))) → (∀ (m : ℤ), p m ↔ p (m + k₀)) ∧ ∀ (m : ℤ), p m ↔ p (m + k₁)
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/IntegerInduction.lean | WavelengthGcd | [56, 1] | [82, 12] | rw [WavelengthRestate] at h₀ | case mp
p : ℤ → Prop
k₀ k₁ : ℤ
h₀ : ∀ (m : ℤ), p m ↔ p (m + ↑(k₀.gcd k₁))
⊢ (∀ (m : ℤ), p m ↔ p (m + k₀)) ∧ ∀ (m : ℤ), p m ↔ p (m + k₁) | case mp
p : ℤ → Prop
k₀ k₁ : ℤ
h₀ : ∀ (m k₀_1 : ℤ), p m ↔ p (m + k₀_1 * ↑(k₀.gcd k₁))
⊢ (∀ (m : ℤ), p m ↔ p (m + k₀)) ∧ ∀ (m : ℤ), p m ↔ p (m + k₁) | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
p : ℤ → Prop
k₀ k₁ : ℤ
h₀ : ∀ (m : ℤ), p m ↔ p (m + ↑(k₀.gcd k₁))
⊢ (∀ (m : ℤ), p m ↔ p (m + k₀)) ∧ ∀ (m : ℤ), p m ↔ p (m + k₁)
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/IntegerInduction.lean | WavelengthGcd | [56, 1] | [82, 12] | rw [WavelengthRestate] | case mp
p : ℤ → Prop
k₀ k₁ : ℤ
h₀ : ∀ (m k₀_1 : ℤ), p m ↔ p (m + k₀_1 * ↑(k₀.gcd k₁))
⊢ (∀ (m : ℤ), p m ↔ p (m + k₀)) ∧ ∀ (m : ℤ), p m ↔ p (m + k₁) | case mp
p : ℤ → Prop
k₀ k₁ : ℤ
h₀ : ∀ (m k₀_1 : ℤ), p m ↔ p (m + k₀_1 * ↑(k₀.gcd k₁))
⊢ (∀ (m k₀_1 : ℤ), p m ↔ p (m + k₀_1 * k₀)) ∧ ∀ (m : ℤ), p m ↔ p (m + k₁) | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
p : ℤ → Prop
k₀ k₁ : ℤ
h₀ : ∀ (m k₀_1 : ℤ), p m ↔ p (m + k₀_1 * ↑(k₀.gcd k₁))
⊢ (∀ (m : ℤ), p m ↔ p (m + k₀)) ∧ ∀ (m : ℤ), p m ↔ p (m + k₁)
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/IntegerInduction.lean | WavelengthGcd | [56, 1] | [82, 12] | constructor | case mp
p : ℤ → Prop
k₀ k₁ : ℤ
h₀ : ∀ (m k₀_1 : ℤ), p m ↔ p (m + k₀_1 * ↑(k₀.gcd k₁))
⊢ (∀ (m k₀_1 : ℤ), p m ↔ p (m + k₀_1 * k₀)) ∧ ∀ (m : ℤ), p m ↔ p (m + k₁) | case mp.left
p : ℤ → Prop
k₀ k₁ : ℤ
h₀ : ∀ (m k₀_1 : ℤ), p m ↔ p (m + k₀_1 * ↑(k₀.gcd k₁))
⊢ ∀ (m k₀_1 : ℤ), p m ↔ p (m + k₀_1 * k₀)
case mp.right
p : ℤ → Prop
k₀ k₁ : ℤ
h₀ : ∀ (m k₀_1 : ℤ), p m ↔ p (m + k₀_1 * ↑(k₀.gcd k₁))
⊢ ∀ (m : ℤ), p m ↔ p (m + k₁) | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
p : ℤ → Prop
k₀ k₁ : ℤ
h₀ : ∀ (m k₀_1 : ℤ), p m ↔ p (m + k₀_1 * ↑(k₀.gcd k₁))
⊢ (∀ (m k₀_1 : ℤ), p m ↔ p (m + k₀_1 * k₀)) ∧ ∀ (m : ℤ), p m ↔ p (m + k₁)
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/IntegerInduction.lean | WavelengthGcd | [56, 1] | [82, 12] | have h₁ : ↑(k₀.gcd k₁) ∣ k₀ := Int.gcd_dvd_left | case mp.left
p : ℤ → Prop
k₀ k₁ : ℤ
h₀ : ∀ (m k₀_1 : ℤ), p m ↔ p (m + k₀_1 * ↑(k₀.gcd k₁))
⊢ ∀ (m k₀_1 : ℤ), p m ↔ p (m + k₀_1 * k₀) | case mp.left
p : ℤ → Prop
k₀ k₁ : ℤ
h₀ : ∀ (m k₀_1 : ℤ), p m ↔ p (m + k₀_1 * ↑(k₀.gcd k₁))
h₁ : ↑(k₀.gcd k₁) ∣ k₀
⊢ ∀ (m k₀_1 : ℤ), p m ↔ p (m + k₀_1 * k₀) | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.left
p : ℤ → Prop
k₀ k₁ : ℤ
h₀ : ∀ (m k₀_1 : ℤ), p m ↔ p (m + k₀_1 * ↑(k₀.gcd k₁))
⊢ ∀ (m k₀_1 : ℤ), p m ↔ p (m + k₀_1 * k₀)
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/IntegerInduction.lean | WavelengthGcd | [56, 1] | [82, 12] | obtain ⟨w, hw⟩ := h₁ | case mp.left
p : ℤ → Prop
k₀ k₁ : ℤ
h₀ : ∀ (m k₀_1 : ℤ), p m ↔ p (m + k₀_1 * ↑(k₀.gcd k₁))
h₁ : ↑(k₀.gcd k₁) ∣ k₀
⊢ ∀ (m k₀_1 : ℤ), p m ↔ p (m + k₀_1 * k₀) | case mp.left.intro
p : ℤ → Prop
k₀ k₁ : ℤ
h₀ : ∀ (m k₀_1 : ℤ), p m ↔ p (m + k₀_1 * ↑(k₀.gcd k₁))
w : ℤ
hw : k₀ = ↑(k₀.gcd k₁) * w
⊢ ∀ (m k₀_1 : ℤ), p m ↔ p (m + k₀_1 * k₀) | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.left
p : ℤ → Prop
k₀ k₁ : ℤ
h₀ : ∀ (m k₀_1 : ℤ), p m ↔ p (m + k₀_1 * ↑(k₀.gcd k₁))
h₁ : ↑(k₀.gcd k₁) ∣ k₀
⊢ ∀ (m k₀_1 : ℤ), p m ↔ p (m + k₀_1 * k₀)
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/IntegerInduction.lean | WavelengthGcd | [56, 1] | [82, 12] | intros m k | case mp.left.intro
p : ℤ → Prop
k₀ k₁ : ℤ
h₀ : ∀ (m k₀_1 : ℤ), p m ↔ p (m + k₀_1 * ↑(k₀.gcd k₁))
w : ℤ
hw : k₀ = ↑(k₀.gcd k₁) * w
⊢ ∀ (m k₀_1 : ℤ), p m ↔ p (m + k₀_1 * k₀) | case mp.left.intro
p : ℤ → Prop
k₀ k₁ : ℤ
h₀ : ∀ (m k₀_1 : ℤ), p m ↔ p (m + k₀_1 * ↑(k₀.gcd k₁))
w : ℤ
hw : k₀ = ↑(k₀.gcd k₁) * w
m k : ℤ
⊢ p m ↔ p (m + k * k₀) | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.left.intro
p : ℤ → Prop
k₀ k₁ : ℤ
h₀ : ∀ (m k₀_1 : ℤ), p m ↔ p (m + k₀_1 * ↑(k₀.gcd k₁))
w : ℤ
hw : k₀ = ↑(k₀.gcd k₁) * w
⊢ ∀ (m k₀_1 : ℤ), p m ↔ p (m + k₀_1 * k₀)
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/IntegerInduction.lean | WavelengthGcd | [56, 1] | [82, 12] | rw [h₀ m (w * k)] | case mp.left.intro
p : ℤ → Prop
k₀ k₁ : ℤ
h₀ : ∀ (m k₀_1 : ℤ), p m ↔ p (m + k₀_1 * ↑(k₀.gcd k₁))
w : ℤ
hw : k₀ = ↑(k₀.gcd k₁) * w
m k : ℤ
⊢ p m ↔ p (m + k * k₀) | case mp.left.intro
p : ℤ → Prop
k₀ k₁ : ℤ
h₀ : ∀ (m k₀_1 : ℤ), p m ↔ p (m + k₀_1 * ↑(k₀.gcd k₁))
w : ℤ
hw : k₀ = ↑(k₀.gcd k₁) * w
m k : ℤ
⊢ p (m + w * k * ↑(k₀.gcd k₁)) ↔ p (m + k * k₀) | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.left.intro
p : ℤ → Prop
k₀ k₁ : ℤ
h₀ : ∀ (m k₀_1 : ℤ), p m ↔ p (m + k₀_1 * ↑(k₀.gcd k₁))
w : ℤ
hw : k₀ = ↑(k₀.gcd k₁) * w
m k : ℤ
⊢ p m ↔ p (m + k * k₀)
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/IntegerInduction.lean | WavelengthGcd | [56, 1] | [82, 12] | nth_rw 2 [hw] | case mp.left.intro
p : ℤ → Prop
k₀ k₁ : ℤ
h₀ : ∀ (m k₀_1 : ℤ), p m ↔ p (m + k₀_1 * ↑(k₀.gcd k₁))
w : ℤ
hw : k₀ = ↑(k₀.gcd k₁) * w
m k : ℤ
⊢ p (m + w * k * ↑(k₀.gcd k₁)) ↔ p (m + k * k₀) | case mp.left.intro
p : ℤ → Prop
k₀ k₁ : ℤ
h₀ : ∀ (m k₀_1 : ℤ), p m ↔ p (m + k₀_1 * ↑(k₀.gcd k₁))
w : ℤ
hw : k₀ = ↑(k₀.gcd k₁) * w
m k : ℤ
⊢ p (m + w * k * ↑(k₀.gcd k₁)) ↔ p (m + k * (↑(k₀.gcd k₁) * w)) | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.left.intro
p : ℤ → Prop
k₀ k₁ : ℤ
h₀ : ∀ (m k₀_1 : ℤ), p m ↔ p (m + k₀_1 * ↑(k₀.gcd k₁))
w : ℤ
hw : k₀ = ↑(k₀.gcd k₁) * w
m k : ℤ
⊢ p (m + w * k * ↑(k₀.gcd k₁)) ↔ p (m + k * k₀)
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/IntegerInduction.lean | WavelengthGcd | [56, 1] | [82, 12] | ring_nf | case mp.left.intro
p : ℤ → Prop
k₀ k₁ : ℤ
h₀ : ∀ (m k₀_1 : ℤ), p m ↔ p (m + k₀_1 * ↑(k₀.gcd k₁))
w : ℤ
hw : k₀ = ↑(k₀.gcd k₁) * w
m k : ℤ
⊢ p (m + w * k * ↑(k₀.gcd k₁)) ↔ p (m + k * (↑(k₀.gcd k₁) * w)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.left.intro
p : ℤ → Prop
k₀ k₁ : ℤ
h₀ : ∀ (m k₀_1 : ℤ), p m ↔ p (m + k₀_1 * ↑(k₀.gcd k₁))
w : ℤ
hw : k₀ = ↑(k₀.gcd k₁) * w
m k : ℤ
⊢ p (m + w * k * ↑(k₀.gcd k₁)) ↔ p (m + k * (↑(k₀.gcd k₁) * w))
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/IntegerInduction.lean | WavelengthGcd | [56, 1] | [82, 12] | have h₁ : ↑(k₀.gcd k₁) ∣ k₁ := Int.gcd_dvd_right | case mp.right
p : ℤ → Prop
k₀ k₁ : ℤ
h₀ : ∀ (m k₀_1 : ℤ), p m ↔ p (m + k₀_1 * ↑(k₀.gcd k₁))
⊢ ∀ (m : ℤ), p m ↔ p (m + k₁) | case mp.right
p : ℤ → Prop
k₀ k₁ : ℤ
h₀ : ∀ (m k₀_1 : ℤ), p m ↔ p (m + k₀_1 * ↑(k₀.gcd k₁))
h₁ : ↑(k₀.gcd k₁) ∣ k₁
⊢ ∀ (m : ℤ), p m ↔ p (m + k₁) | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.right
p : ℤ → Prop
k₀ k₁ : ℤ
h₀ : ∀ (m k₀_1 : ℤ), p m ↔ p (m + k₀_1 * ↑(k₀.gcd k₁))
⊢ ∀ (m : ℤ), p m ↔ p (m + k₁)
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/IntegerInduction.lean | WavelengthGcd | [56, 1] | [82, 12] | obtain ⟨w, hw⟩ := h₁ | case mp.right
p : ℤ → Prop
k₀ k₁ : ℤ
h₀ : ∀ (m k₀_1 : ℤ), p m ↔ p (m + k₀_1 * ↑(k₀.gcd k₁))
h₁ : ↑(k₀.gcd k₁) ∣ k₁
⊢ ∀ (m : ℤ), p m ↔ p (m + k₁) | case mp.right.intro
p : ℤ → Prop
k₀ k₁ : ℤ
h₀ : ∀ (m k₀_1 : ℤ), p m ↔ p (m + k₀_1 * ↑(k₀.gcd k₁))
w : ℤ
hw : k₁ = ↑(k₀.gcd k₁) * w
⊢ ∀ (m : ℤ), p m ↔ p (m + k₁) | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.right
p : ℤ → Prop
k₀ k₁ : ℤ
h₀ : ∀ (m k₀_1 : ℤ), p m ↔ p (m + k₀_1 * ↑(k₀.gcd k₁))
h₁ : ↑(k₀.gcd k₁) ∣ k₁
⊢ ∀ (m : ℤ), p m ↔ p (m + k₁)
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/IntegerInduction.lean | WavelengthGcd | [56, 1] | [82, 12] | rw [WavelengthRestate] | case mp.right.intro
p : ℤ → Prop
k₀ k₁ : ℤ
h₀ : ∀ (m k₀_1 : ℤ), p m ↔ p (m + k₀_1 * ↑(k₀.gcd k₁))
w : ℤ
hw : k₁ = ↑(k₀.gcd k₁) * w
⊢ ∀ (m : ℤ), p m ↔ p (m + k₁) | case mp.right.intro
p : ℤ → Prop
k₀ k₁ : ℤ
h₀ : ∀ (m k₀_1 : ℤ), p m ↔ p (m + k₀_1 * ↑(k₀.gcd k₁))
w : ℤ
hw : k₁ = ↑(k₀.gcd k₁) * w
⊢ ∀ (m k₀ : ℤ), p m ↔ p (m + k₀ * k₁) | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.right.intro
p : ℤ → Prop
k₀ k₁ : ℤ
h₀ : ∀ (m k₀_1 : ℤ), p m ↔ p (m + k₀_1 * ↑(k₀.gcd k₁))
w : ℤ
hw : k₁ = ↑(k₀.gcd k₁) * w
⊢ ∀ (m : ℤ), p m ↔ p (m + k₁)
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/IntegerInduction.lean | WavelengthGcd | [56, 1] | [82, 12] | intros m k | case mp.right.intro
p : ℤ → Prop
k₀ k₁ : ℤ
h₀ : ∀ (m k₀_1 : ℤ), p m ↔ p (m + k₀_1 * ↑(k₀.gcd k₁))
w : ℤ
hw : k₁ = ↑(k₀.gcd k₁) * w
⊢ ∀ (m k₀ : ℤ), p m ↔ p (m + k₀ * k₁) | case mp.right.intro
p : ℤ → Prop
k₀ k₁ : ℤ
h₀ : ∀ (m k₀_1 : ℤ), p m ↔ p (m + k₀_1 * ↑(k₀.gcd k₁))
w : ℤ
hw : k₁ = ↑(k₀.gcd k₁) * w
m k : ℤ
⊢ p m ↔ p (m + k * k₁) | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.right.intro
p : ℤ → Prop
k₀ k₁ : ℤ
h₀ : ∀ (m k₀_1 : ℤ), p m ↔ p (m + k₀_1 * ↑(k₀.gcd k₁))
w : ℤ
hw : k₁ = ↑(k₀.gcd k₁) * w
⊢ ∀ (m k₀ : ℤ), p m ↔ p (m + k₀ * k₁)
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/IntegerInduction.lean | WavelengthGcd | [56, 1] | [82, 12] | rw [h₀ m (w * k)] | case mp.right.intro
p : ℤ → Prop
k₀ k₁ : ℤ
h₀ : ∀ (m k₀_1 : ℤ), p m ↔ p (m + k₀_1 * ↑(k₀.gcd k₁))
w : ℤ
hw : k₁ = ↑(k₀.gcd k₁) * w
m k : ℤ
⊢ p m ↔ p (m + k * k₁) | case mp.right.intro
p : ℤ → Prop
k₀ k₁ : ℤ
h₀ : ∀ (m k₀_1 : ℤ), p m ↔ p (m + k₀_1 * ↑(k₀.gcd k₁))
w : ℤ
hw : k₁ = ↑(k₀.gcd k₁) * w
m k : ℤ
⊢ p (m + w * k * ↑(k₀.gcd k₁)) ↔ p (m + k * k₁) | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.right.intro
p : ℤ → Prop
k₀ k₁ : ℤ
h₀ : ∀ (m k₀_1 : ℤ), p m ↔ p (m + k₀_1 * ↑(k₀.gcd k₁))
w : ℤ
hw : k₁ = ↑(k₀.gcd k₁) * w
m k : ℤ
⊢ p m ↔ p (m + k * k₁)
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/IntegerInduction.lean | WavelengthGcd | [56, 1] | [82, 12] | nth_rw 2 [hw] | case mp.right.intro
p : ℤ → Prop
k₀ k₁ : ℤ
h₀ : ∀ (m k₀_1 : ℤ), p m ↔ p (m + k₀_1 * ↑(k₀.gcd k₁))
w : ℤ
hw : k₁ = ↑(k₀.gcd k₁) * w
m k : ℤ
⊢ p (m + w * k * ↑(k₀.gcd k₁)) ↔ p (m + k * k₁) | case mp.right.intro
p : ℤ → Prop
k₀ k₁ : ℤ
h₀ : ∀ (m k₀_1 : ℤ), p m ↔ p (m + k₀_1 * ↑(k₀.gcd k₁))
w : ℤ
hw : k₁ = ↑(k₀.gcd k₁) * w
m k : ℤ
⊢ p (m + w * k * ↑(k₀.gcd k₁)) ↔ p (m + k * (↑(k₀.gcd k₁) * w)) | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.right.intro
p : ℤ → Prop
k₀ k₁ : ℤ
h₀ : ∀ (m k₀_1 : ℤ), p m ↔ p (m + k₀_1 * ↑(k₀.gcd k₁))
w : ℤ
hw : k₁ = ↑(k₀.gcd k₁) * w
m k : ℤ
⊢ p (m + w * k * ↑(k₀.gcd k₁)) ↔ p (m + k * k₁)
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/IntegerInduction.lean | WavelengthGcd | [56, 1] | [82, 12] | ring_nf | case mp.right.intro
p : ℤ → Prop
k₀ k₁ : ℤ
h₀ : ∀ (m k₀_1 : ℤ), p m ↔ p (m + k₀_1 * ↑(k₀.gcd k₁))
w : ℤ
hw : k₁ = ↑(k₀.gcd k₁) * w
m k : ℤ
⊢ p (m + w * k * ↑(k₀.gcd k₁)) ↔ p (m + k * (↑(k₀.gcd k₁) * w)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.right.intro
p : ℤ → Prop
k₀ k₁ : ℤ
h₀ : ∀ (m k₀_1 : ℤ), p m ↔ p (m + k₀_1 * ↑(k₀.gcd k₁))
w : ℤ
hw : k₁ = ↑(k₀.gcd k₁) * w
m k : ℤ
⊢ p (m + w * k * ↑(k₀.gcd k₁)) ↔ p (m + k * (↑(k₀.gcd k₁) * w))
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/IntegerInduction.lean | WavelengthGcd | [56, 1] | [82, 12] | intros h₀ | case mpr
p : ℤ → Prop
k₀ k₁ : ℤ
⊢ ((∀ (m : ℤ), p m ↔ p (m + k₀)) ∧ ∀ (m : ℤ), p m ↔ p (m + k₁)) → ∀ (m : ℤ), p m ↔ p (m + ↑(k₀.gcd k₁)) | case mpr
p : ℤ → Prop
k₀ k₁ : ℤ
h₀ : (∀ (m : ℤ), p m ↔ p (m + k₀)) ∧ ∀ (m : ℤ), p m ↔ p (m + k₁)
⊢ ∀ (m : ℤ), p m ↔ p (m + ↑(k₀.gcd k₁)) | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
p : ℤ → Prop
k₀ k₁ : ℤ
⊢ ((∀ (m : ℤ), p m ↔ p (m + k₀)) ∧ ∀ (m : ℤ), p m ↔ p (m + k₁)) → ∀ (m : ℤ), p m ↔ p (m + ↑(k₀.gcd k₁))
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/IntegerInduction.lean | WavelengthGcd | [56, 1] | [82, 12] | obtain ⟨h₁, h₂⟩ := h₀ | case mpr
p : ℤ → Prop
k₀ k₁ : ℤ
h₀ : (∀ (m : ℤ), p m ↔ p (m + k₀)) ∧ ∀ (m : ℤ), p m ↔ p (m + k₁)
⊢ ∀ (m : ℤ), p m ↔ p (m + ↑(k₀.gcd k₁)) | case mpr.intro
p : ℤ → Prop
k₀ k₁ : ℤ
h₁ : ∀ (m : ℤ), p m ↔ p (m + k₀)
h₂ : ∀ (m : ℤ), p m ↔ p (m + k₁)
⊢ ∀ (m : ℤ), p m ↔ p (m + ↑(k₀.gcd k₁)) | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
p : ℤ → Prop
k₀ k₁ : ℤ
h₀ : (∀ (m : ℤ), p m ↔ p (m + k₀)) ∧ ∀ (m : ℤ), p m ↔ p (m + k₁)
⊢ ∀ (m : ℤ), p m ↔ p (m + ↑(k₀.gcd k₁))
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/IntegerInduction.lean | WavelengthGcd | [56, 1] | [82, 12] | rw [WavelengthRestate] at * | case mpr.intro
p : ℤ → Prop
k₀ k₁ : ℤ
h₁ : ∀ (m : ℤ), p m ↔ p (m + k₀)
h₂ : ∀ (m : ℤ), p m ↔ p (m + k₁)
⊢ ∀ (m : ℤ), p m ↔ p (m + ↑(k₀.gcd k₁)) | case mpr.intro
p : ℤ → Prop
k₀ k₁ : ℤ
h₁ : ∀ (m k₀_1 : ℤ), p m ↔ p (m + k₀_1 * k₀)
h₂ : ∀ (m k₀ : ℤ), p m ↔ p (m + k₀ * k₁)
⊢ ∀ (m k₀_1 : ℤ), p m ↔ p (m + k₀_1 * ↑(k₀.gcd k₁)) | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro
p : ℤ → Prop
k₀ k₁ : ℤ
h₁ : ∀ (m : ℤ), p m ↔ p (m + k₀)
h₂ : ∀ (m : ℤ), p m ↔ p (m + k₁)
⊢ ∀ (m : ℤ), p m ↔ p (m + ↑(k₀.gcd k₁))
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/IntegerInduction.lean | WavelengthGcd | [56, 1] | [82, 12] | intros m j | case mpr.intro
p : ℤ → Prop
k₀ k₁ : ℤ
h₁ : ∀ (m k₀_1 : ℤ), p m ↔ p (m + k₀_1 * k₀)
h₂ : ∀ (m k₀ : ℤ), p m ↔ p (m + k₀ * k₁)
⊢ ∀ (m k₀_1 : ℤ), p m ↔ p (m + k₀_1 * ↑(k₀.gcd k₁)) | case mpr.intro
p : ℤ → Prop
k₀ k₁ : ℤ
h₁ : ∀ (m k₀_1 : ℤ), p m ↔ p (m + k₀_1 * k₀)
h₂ : ∀ (m k₀ : ℤ), p m ↔ p (m + k₀ * k₁)
m j : ℤ
⊢ p m ↔ p (m + j * ↑(k₀.gcd k₁)) | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro
p : ℤ → Prop
k₀ k₁ : ℤ
h₁ : ∀ (m k₀_1 : ℤ), p m ↔ p (m + k₀_1 * k₀)
h₂ : ∀ (m k₀ : ℤ), p m ↔ p (m + k₀ * k₁)
⊢ ∀ (m k₀_1 : ℤ), p m ↔ p (m + k₀_1 * ↑(k₀.gcd k₁))
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/IntegerInduction.lean | WavelengthGcd | [56, 1] | [82, 12] | obtain ⟨w₀, w₁, h₃⟩ := GcdLinearCombination k₀ k₁ | case mpr.intro
p : ℤ → Prop
k₀ k₁ : ℤ
h₁ : ∀ (m k₀_1 : ℤ), p m ↔ p (m + k₀_1 * k₀)
h₂ : ∀ (m k₀ : ℤ), p m ↔ p (m + k₀ * k₁)
m j : ℤ
⊢ p m ↔ p (m + j * ↑(k₀.gcd k₁)) | case mpr.intro.intro.intro
p : ℤ → Prop
k₀ k₁ : ℤ
h₁ : ∀ (m k₀_1 : ℤ), p m ↔ p (m + k₀_1 * k₀)
h₂ : ∀ (m k₀ : ℤ), p m ↔ p (m + k₀ * k₁)
m j w₀ w₁ : ℤ
h₃ : ↑(k₀.gcd k₁) = w₀ * k₀ + w₁ * k₁
⊢ p m ↔ p (m + j * ↑(k₀.gcd k₁)) | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro
p : ℤ → Prop
k₀ k₁ : ℤ
h₁ : ∀ (m k₀_1 : ℤ), p m ↔ p (m + k₀_1 * k₀)
h₂ : ∀ (m k₀ : ℤ), p m ↔ p (m + k₀ * k₁)
m j : ℤ
⊢ p m ↔ p (m + j * ↑(k₀.gcd k₁))
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/IntegerInduction.lean | WavelengthGcd | [56, 1] | [82, 12] | rw [h₃, h₁ m (j * w₀), h₂ (m + j * w₀ * k₀) (j * w₁)] | case mpr.intro.intro.intro
p : ℤ → Prop
k₀ k₁ : ℤ
h₁ : ∀ (m k₀_1 : ℤ), p m ↔ p (m + k₀_1 * k₀)
h₂ : ∀ (m k₀ : ℤ), p m ↔ p (m + k₀ * k₁)
m j w₀ w₁ : ℤ
h₃ : ↑(k₀.gcd k₁) = w₀ * k₀ + w₁ * k₁
⊢ p m ↔ p (m + j * ↑(k₀.gcd k₁)) | case mpr.intro.intro.intro
p : ℤ → Prop
k₀ k₁ : ℤ
h₁ : ∀ (m k₀_1 : ℤ), p m ↔ p (m + k₀_1 * k₀)
h₂ : ∀ (m k₀ : ℤ), p m ↔ p (m + k₀ * k₁)
m j w₀ w₁ : ℤ
h₃ : ↑(k₀.gcd k₁) = w₀ * k₀ + w₁ * k₁
⊢ p (m + j * w₀ * k₀ + j * w₁ * k₁) ↔ p (m + j * (w₀ * k₀ + w₁ * k₁)) | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro.intro.intro
p : ℤ → Prop
k₀ k₁ : ℤ
h₁ : ∀ (m k₀_1 : ℤ), p m ↔ p (m + k₀_1 * k₀)
h₂ : ∀ (m k₀ : ℤ), p m ↔ p (m + k₀ * k₁)
m j w₀ w₁ : ℤ
h₃ : ↑(k₀.gcd k₁) = w₀ * k₀ + w₁ * k₁
⊢ p m ↔ p (m + j * ↑(k₀.gcd k₁))
TACTIC:
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.