declaration
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theorem quartic_eq_zero_iff_of_q_eq_zero (ha : a ≠ 0)
(hp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2))
(hqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0)
(hr : r =
(16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4))
(ht : t ^ 2 = p ^ 2 - 4 * r)
(hv : v ^ 2 = (-p + t) / 2)
(hw : w ^ 2 = (-p - t) / 2) (x : K) :
a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = 0 ↔
x = v - b / (4 * a) ∨ x = -v - b / (4 * a) ∨ x = w - b / (4 * a) ∨ x = -w - b / (4 * a) := by
let y := x + b / (4 * a)
have h4 : (4 : K) = 2 ^ 2 := by norm_num
have h8 : (8 : K) = 2 ^ 3 := by norm_num
have h16 : (16 : K) = 2 ^ 4 := by norm_num
have h256 : (256 : K) = 2 ^ 8 := by norm_num
have h₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r) := by
rw [hp]
rw [hr]
simp [field_simps, y, ha, h4, h8, h16, h256]
linear_combination (1048576 * a ^ 10 * x + 262144 * a ^ 9 * b) * hqz
rw [h₁]
rw [ha.isUnit.mul_right_eq_zero]
calc
_ ↔ 1 * (y ^ 2 * y ^ 2) + p * y ^ 2 + r = 0 := by
apply Eq.congr_left
ring
_ ↔ y ^ 2 = (-p + t) / 2 ∨ y ^ 2 = (-p - t) / 2 := by
have ht' : discrim 1 p r = t * t := by rw [discrim]; linear_combination -ht
rw [quadratic_eq_zero_iff one_ne_zero ht']
rw [mul_one]
_ ↔ _ := by
simp_rw [y, ← hv, ← hw, pow_two, mul_self_eq_mul_self_iff, eq_sub_iff_add_eq, or_assoc]
|
/root/DuelModelResearch/mathlib4/Archive/Wiedijk100Theorems/SolutionOfCubicQuartic.lean
|
{
"open": [
"Polynomial"
],
"variables": [
"{K : Type*} [Field K] (a b c d e : K) {ω p q r s t u v w x y : K}",
"[Invertible (2 : K)] [Invertible (3 : K)]",
"[Invertible (2 : K)]"
]
}
|
[
{
"line": "let y := x + b / (4 * a)",
"before_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\n⊢ a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = 0 ↔\n x = v - b / (4 * a) ∨ x = -v - b / (4 * a) ∨ x = w - b / (4 * a) ∨ x = -w - b / (4 * a)",
"after_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\n⊢ a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = 0 ↔\n x = v - b / (4 * a) ∨ x = -v - b / (4 * a) ∨ x = w - b / (4 * a) ∨ x = -w - b / (4 * a)"
},
{
"line": "refine_lift\n let y := x + b / (4 * a);\n ?_",
"before_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\n⊢ a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = 0 ↔\n x = v - b / (4 * a) ∨ x = -v - b / (4 * a) ∨ x = w - b / (4 * a) ∨ x = -w - b / (4 * a)",
"after_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\n⊢ a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = 0 ↔\n x = v - b / (4 * a) ∨ x = -v - b / (4 * a) ∨ x = w - b / (4 * a) ∨ x = -w - b / (4 * a)"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (let y := x + b / (4 * a);\n ?_);\n rotate_right)",
"before_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\n⊢ a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = 0 ↔\n x = v - b / (4 * a) ∨ x = -v - b / (4 * a) ∨ x = w - b / (4 * a) ∨ x = -w - b / (4 * a)",
"after_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\n⊢ a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = 0 ↔\n x = v - b / (4 * a) ∨ x = -v - b / (4 * a) ∨ x = w - b / (4 * a) ∨ x = -w - b / (4 * a)"
},
{
"line": "refine\n no_implicit_lambda%\n (let y := x + b / (4 * a);\n ?_)",
"before_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\n⊢ a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = 0 ↔\n x = v - b / (4 * a) ∨ x = -v - b / (4 * a) ∨ x = w - b / (4 * a) ∨ x = -w - b / (4 * a)",
"after_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\n⊢ a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = 0 ↔\n x = v - b / (4 * a) ∨ x = -v - b / (4 * a) ∨ x = w - b / (4 * a) ∨ x = -w - b / (4 * a)"
},
{
"line": "rotate_right",
"before_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\n⊢ a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = 0 ↔\n x = v - b / (4 * a) ∨ x = -v - b / (4 * a) ∨ x = w - b / (4 * a) ∨ x = -w - b / (4 * a)",
"after_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\n⊢ a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = 0 ↔\n x = v - b / (4 * a) ∨ x = -v - b / (4 * a) ∨ x = w - b / (4 * a) ∨ x = -w - b / (4 * a)"
},
{
"line": "have h4 : (4 : K) = 2 ^ 2 := by norm_num",
"before_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\n⊢ a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = 0 ↔\n x = v - b / (4 * a) ∨ x = -v - b / (4 * a) ∨ x = w - b / (4 * a) ∨ x = -w - b / (4 * a)",
"after_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\n⊢ a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = 0 ↔\n x = v - b / (4 * a) ∨ x = -v - b / (4 * a) ∨ x = w - b / (4 * a) ∨ x = -w - b / (4 * a)"
},
{
"line": "focus\n refine\n no_implicit_lambda%\n (have h4 : (4 : K) = 2 ^ 2 := ?body✝;\n ?_)\n case body✝ => with_annotate_state\"by\" (norm_num)",
"before_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\n⊢ a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = 0 ↔\n x = v - b / (4 * a) ∨ x = -v - b / (4 * a) ∨ x = w - b / (4 * a) ∨ x = -w - b / (4 * a)",
"after_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\n⊢ a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = 0 ↔\n x = v - b / (4 * a) ∨ x = -v - b / (4 * a) ∨ x = w - b / (4 * a) ∨ x = -w - b / (4 * a)"
},
{
"line": "refine\n no_implicit_lambda%\n (have h4 : (4 : K) = 2 ^ 2 := ?body✝;\n ?_)",
"before_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\n⊢ a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = 0 ↔\n x = v - b / (4 * a) ∨ x = -v - b / (4 * a) ∨ x = w - b / (4 * a) ∨ x = -w - b / (4 * a)",
"after_state": "case body\nK : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\n⊢ 4 = 2 ^ 2\n---\nK : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\n⊢ a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = 0 ↔\n x = v - b / (4 * a) ∨ x = -v - b / (4 * a) ∨ x = w - b / (4 * a) ∨ x = -w - b / (4 * a)"
},
{
"line": "case body✝ => with_annotate_state\"by\" (norm_num)",
"before_state": "case body\nK : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\n⊢ 4 = 2 ^ 2\n---\nK : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\n⊢ a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = 0 ↔\n x = v - b / (4 * a) ∨ x = -v - b / (4 * a) ∨ x = w - b / (4 * a) ∨ x = -w - b / (4 * a)",
"after_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\n⊢ a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = 0 ↔\n x = v - b / (4 * a) ∨ x = -v - b / (4 * a) ∨ x = w - b / (4 * a) ∨ x = -w - b / (4 * a)"
},
{
"line": "with_annotate_state\"by\" (norm_num)",
"before_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\n⊢ 4 = 2 ^ 2",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\n⊢ 4 = 2 ^ 2",
"after_state": "No Goals!"
},
{
"line": "have h8 : (8 : K) = 2 ^ 3 := by norm_num",
"before_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\n⊢ a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = 0 ↔\n x = v - b / (4 * a) ∨ x = -v - b / (4 * a) ∨ x = w - b / (4 * a) ∨ x = -w - b / (4 * a)",
"after_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\n⊢ a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = 0 ↔\n x = v - b / (4 * a) ∨ x = -v - b / (4 * a) ∨ x = w - b / (4 * a) ∨ x = -w - b / (4 * a)"
},
{
"line": "focus\n refine\n no_implicit_lambda%\n (have h8 : (8 : K) = 2 ^ 3 := ?body✝;\n ?_)\n case body✝ => with_annotate_state\"by\" (norm_num)",
"before_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\n⊢ a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = 0 ↔\n x = v - b / (4 * a) ∨ x = -v - b / (4 * a) ∨ x = w - b / (4 * a) ∨ x = -w - b / (4 * a)",
"after_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\n⊢ a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = 0 ↔\n x = v - b / (4 * a) ∨ x = -v - b / (4 * a) ∨ x = w - b / (4 * a) ∨ x = -w - b / (4 * a)"
},
{
"line": "refine\n no_implicit_lambda%\n (have h8 : (8 : K) = 2 ^ 3 := ?body✝;\n ?_)",
"before_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\n⊢ a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = 0 ↔\n x = v - b / (4 * a) ∨ x = -v - b / (4 * a) ∨ x = w - b / (4 * a) ∨ x = -w - b / (4 * a)",
"after_state": "case body\nK : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\n⊢ 8 = 2 ^ 3\n---\nK : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\n⊢ a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = 0 ↔\n x = v - b / (4 * a) ∨ x = -v - b / (4 * a) ∨ x = w - b / (4 * a) ∨ x = -w - b / (4 * a)"
},
{
"line": "case body✝ => with_annotate_state\"by\" (norm_num)",
"before_state": "case body\nK : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\n⊢ 8 = 2 ^ 3\n---\nK : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\n⊢ a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = 0 ↔\n x = v - b / (4 * a) ∨ x = -v - b / (4 * a) ∨ x = w - b / (4 * a) ∨ x = -w - b / (4 * a)",
"after_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\n⊢ a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = 0 ↔\n x = v - b / (4 * a) ∨ x = -v - b / (4 * a) ∨ x = w - b / (4 * a) ∨ x = -w - b / (4 * a)"
},
{
"line": "with_annotate_state\"by\" (norm_num)",
"before_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\n⊢ 8 = 2 ^ 3",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\n⊢ 8 = 2 ^ 3",
"after_state": "No Goals!"
},
{
"line": "have h16 : (16 : K) = 2 ^ 4 := by norm_num",
"before_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\n⊢ a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = 0 ↔\n x = v - b / (4 * a) ∨ x = -v - b / (4 * a) ∨ x = w - b / (4 * a) ∨ x = -w - b / (4 * a)",
"after_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\n⊢ a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = 0 ↔\n x = v - b / (4 * a) ∨ x = -v - b / (4 * a) ∨ x = w - b / (4 * a) ∨ x = -w - b / (4 * a)"
},
{
"line": "focus\n refine\n no_implicit_lambda%\n (have h16 : (16 : K) = 2 ^ 4 := ?body✝;\n ?_)\n case body✝ => with_annotate_state\"by\" (norm_num)",
"before_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\n⊢ a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = 0 ↔\n x = v - b / (4 * a) ∨ x = -v - b / (4 * a) ∨ x = w - b / (4 * a) ∨ x = -w - b / (4 * a)",
"after_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\n⊢ a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = 0 ↔\n x = v - b / (4 * a) ∨ x = -v - b / (4 * a) ∨ x = w - b / (4 * a) ∨ x = -w - b / (4 * a)"
},
{
"line": "refine\n no_implicit_lambda%\n (have h16 : (16 : K) = 2 ^ 4 := ?body✝;\n ?_)",
"before_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\n⊢ a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = 0 ↔\n x = v - b / (4 * a) ∨ x = -v - b / (4 * a) ∨ x = w - b / (4 * a) ∨ x = -w - b / (4 * a)",
"after_state": "case body\nK : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\n⊢ 16 = 2 ^ 4\n---\nK : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\n⊢ a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = 0 ↔\n x = v - b / (4 * a) ∨ x = -v - b / (4 * a) ∨ x = w - b / (4 * a) ∨ x = -w - b / (4 * a)"
},
{
"line": "case body✝ => with_annotate_state\"by\" (norm_num)",
"before_state": "case body\nK : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\n⊢ 16 = 2 ^ 4\n---\nK : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\n⊢ a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = 0 ↔\n x = v - b / (4 * a) ∨ x = -v - b / (4 * a) ∨ x = w - b / (4 * a) ∨ x = -w - b / (4 * a)",
"after_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\n⊢ a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = 0 ↔\n x = v - b / (4 * a) ∨ x = -v - b / (4 * a) ∨ x = w - b / (4 * a) ∨ x = -w - b / (4 * a)"
},
{
"line": "with_annotate_state\"by\" (norm_num)",
"before_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\n⊢ 16 = 2 ^ 4",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\n⊢ 16 = 2 ^ 4",
"after_state": "No Goals!"
},
{
"line": "have h256 : (256 : K) = 2 ^ 8 := by norm_num",
"before_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\n⊢ a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = 0 ↔\n x = v - b / (4 * a) ∨ x = -v - b / (4 * a) ∨ x = w - b / (4 * a) ∨ x = -w - b / (4 * a)",
"after_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\n⊢ a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = 0 ↔\n x = v - b / (4 * a) ∨ x = -v - b / (4 * a) ∨ x = w - b / (4 * a) ∨ x = -w - b / (4 * a)"
},
{
"line": "focus\n refine\n no_implicit_lambda%\n (have h256 : (256 : K) = 2 ^ 8 := ?body✝;\n ?_)\n case body✝ => with_annotate_state\"by\" (norm_num)",
"before_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\n⊢ a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = 0 ↔\n x = v - b / (4 * a) ∨ x = -v - b / (4 * a) ∨ x = w - b / (4 * a) ∨ x = -w - b / (4 * a)",
"after_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\n⊢ a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = 0 ↔\n x = v - b / (4 * a) ∨ x = -v - b / (4 * a) ∨ x = w - b / (4 * a) ∨ x = -w - b / (4 * a)"
},
{
"line": "refine\n no_implicit_lambda%\n (have h256 : (256 : K) = 2 ^ 8 := ?body✝;\n ?_)",
"before_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\n⊢ a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = 0 ↔\n x = v - b / (4 * a) ∨ x = -v - b / (4 * a) ∨ x = w - b / (4 * a) ∨ x = -w - b / (4 * a)",
"after_state": "case body\nK : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\n⊢ 256 = 2 ^ 8\n---\nK : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\n⊢ a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = 0 ↔\n x = v - b / (4 * a) ∨ x = -v - b / (4 * a) ∨ x = w - b / (4 * a) ∨ x = -w - b / (4 * a)"
},
{
"line": "case body✝ => with_annotate_state\"by\" (norm_num)",
"before_state": "case body\nK : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\n⊢ 256 = 2 ^ 8\n---\nK : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\n⊢ a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = 0 ↔\n x = v - b / (4 * a) ∨ x = -v - b / (4 * a) ∨ x = w - b / (4 * a) ∨ x = -w - b / (4 * a)",
"after_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\n⊢ a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = 0 ↔\n x = v - b / (4 * a) ∨ x = -v - b / (4 * a) ∨ x = w - b / (4 * a) ∨ x = -w - b / (4 * a)"
},
{
"line": "with_annotate_state\"by\" (norm_num)",
"before_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\n⊢ 256 = 2 ^ 8",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\n⊢ 256 = 2 ^ 8",
"after_state": "No Goals!"
},
{
"line": "have h₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r) :=\n by\n rw [hp]\n rw [hr]\n simp [field_simps, y, ha, h4, h8, h16, h256]\n linear_combination (1048576 * a ^ 10 * x + 262144 * a ^ 9 * b) * hqz",
"before_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\n⊢ a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = 0 ↔\n x = v - b / (4 * a) ∨ x = -v - b / (4 * a) ∨ x = w - b / (4 * a) ∨ x = -w - b / (4 * a)",
"after_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\n⊢ a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = 0 ↔\n x = v - b / (4 * a) ∨ x = -v - b / (4 * a) ∨ x = w - b / (4 * a) ∨ x = -w - b / (4 * a)"
},
{
"line": "focus\n refine\n no_implicit_lambda%\n (have h₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r) := ?body✝;\n ?_)\n case body✝ =>\n with_annotate_state\"by\"\n ( rw [hp]\n rw [hr]\n simp [field_simps, y, ha, h4, h8, h16, h256]\n linear_combination (1048576 * a ^ 10 * x + 262144 * a ^ 9 * b) * hqz)",
"before_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\n⊢ a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = 0 ↔\n x = v - b / (4 * a) ∨ x = -v - b / (4 * a) ∨ x = w - b / (4 * a) ∨ x = -w - b / (4 * a)",
"after_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\n⊢ a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = 0 ↔\n x = v - b / (4 * a) ∨ x = -v - b / (4 * a) ∨ x = w - b / (4 * a) ∨ x = -w - b / (4 * a)"
},
{
"line": "refine\n no_implicit_lambda%\n (have h₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r) := ?body✝;\n ?_)",
"before_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\n⊢ a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = 0 ↔\n x = v - b / (4 * a) ∨ x = -v - b / (4 * a) ∨ x = w - b / (4 * a) ∨ x = -w - b / (4 * a)",
"after_state": "case body\nK : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\n⊢ a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\n---\nK : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\n⊢ a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = 0 ↔\n x = v - b / (4 * a) ∨ x = -v - b / (4 * a) ∨ x = w - b / (4 * a) ∨ x = -w - b / (4 * a)"
},
{
"line": "case body✝ =>\n with_annotate_state\"by\"\n ( rw [hp]\n rw [hr]\n simp [field_simps, y, ha, h4, h8, h16, h256]\n linear_combination (1048576 * a ^ 10 * x + 262144 * a ^ 9 * b) * hqz)",
"before_state": "case body\nK : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\n⊢ a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\n---\nK : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\n⊢ a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = 0 ↔\n x = v - b / (4 * a) ∨ x = -v - b / (4 * a) ∨ x = w - b / (4 * a) ∨ x = -w - b / (4 * a)",
"after_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\n⊢ a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = 0 ↔\n x = v - b / (4 * a) ∨ x = -v - b / (4 * a) ∨ x = w - b / (4 * a) ∨ x = -w - b / (4 * a)"
},
{
"line": "with_annotate_state\"by\"\n ( rw [hp]\n rw [hr]\n simp [field_simps, y, ha, h4, h8, h16, h256]\n linear_combination (1048576 * a ^ 10 * x + 262144 * a ^ 9 * b) * hqz)",
"before_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\n⊢ a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)",
"after_state": "No Goals!"
},
{
"line": "rw [hp]",
"before_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\n⊢ a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)",
"after_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\n⊢ a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2) * y ^ 2 + r)"
},
{
"line": "rewrite [hp]",
"before_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\n⊢ a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)",
"after_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\n⊢ a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2) * y ^ 2 + r)"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\n⊢ a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2) * y ^ 2 + r)",
"after_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\n⊢ a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2) * y ^ 2 + r)"
},
{
"line": "try (with_reducible rfl)",
"before_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\n⊢ a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2) * y ^ 2 + r)",
"after_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\n⊢ a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2) * y ^ 2 + r)"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\n⊢ a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2) * y ^ 2 + r)",
"after_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\n⊢ a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2) * y ^ 2 + r)"
},
{
"line": "with_reducible rfl",
"before_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\n⊢ a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2) * y ^ 2 + r)",
"after_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\n⊢ a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2) * y ^ 2 + r)"
},
{
"line": "rfl",
"before_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\n⊢ a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2) * y ^ 2 + r)",
"after_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\n⊢ a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2) * y ^ 2 + r)"
},
{
"line": "apply_rfl",
"before_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\n⊢ a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2) * y ^ 2 + r)",
"after_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\n⊢ a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2) * y ^ 2 + r)"
},
{
"line": "skip",
"before_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\n⊢ a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2) * y ^ 2 + r)",
"after_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\n⊢ a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2) * y ^ 2 + r)"
},
{
"line": "rw [hr]",
"before_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\n⊢ a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2) * y ^ 2 + r)",
"after_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\n⊢ a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e =\n a *\n (y ^ 4 + (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2) * y ^ 2 +\n (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4))"
},
{
"line": "rewrite [hr]",
"before_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\n⊢ a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2) * y ^ 2 + r)",
"after_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\n⊢ a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e =\n a *\n (y ^ 4 + (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2) * y ^ 2 +\n (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4))"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\n⊢ a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e =\n a *\n (y ^ 4 + (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2) * y ^ 2 +\n (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4))",
"after_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\n⊢ a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e =\n a *\n (y ^ 4 + (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2) * y ^ 2 +\n (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4))"
},
{
"line": "try (with_reducible rfl)",
"before_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\n⊢ a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e =\n a *\n (y ^ 4 + (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2) * y ^ 2 +\n (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4))",
"after_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\n⊢ a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e =\n a *\n (y ^ 4 + (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2) * y ^ 2 +\n (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4))"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\n⊢ a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e =\n a *\n (y ^ 4 + (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2) * y ^ 2 +\n (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4))",
"after_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\n⊢ a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e =\n a *\n (y ^ 4 + (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2) * y ^ 2 +\n (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4))"
},
{
"line": "with_reducible rfl",
"before_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\n⊢ a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e =\n a *\n (y ^ 4 + (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2) * y ^ 2 +\n (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4))",
"after_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\n⊢ a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e =\n a *\n (y ^ 4 + (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2) * y ^ 2 +\n (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4))"
},
{
"line": "rfl",
"before_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\n⊢ a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e =\n a *\n (y ^ 4 + (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2) * y ^ 2 +\n (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4))",
"after_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\n⊢ a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e =\n a *\n (y ^ 4 + (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2) * y ^ 2 +\n (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4))"
},
{
"line": "apply_rfl",
"before_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\n⊢ a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e =\n a *\n (y ^ 4 + (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2) * y ^ 2 +\n (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4))",
"after_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\n⊢ a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e =\n a *\n (y ^ 4 + (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2) * y ^ 2 +\n (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4))"
},
{
"line": "skip",
"before_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\n⊢ a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e =\n a *\n (y ^ 4 + (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2) * y ^ 2 +\n (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4))",
"after_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\n⊢ a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e =\n a *\n (y ^ 4 + (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2) * y ^ 2 +\n (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4))"
},
{
"line": "simp [field_simps, y, ha, h4, h8, h16, h256]",
"before_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\n⊢ a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e =\n a *\n (y ^ 4 + (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2) * y ^ 2 +\n (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4))",
"after_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\n⊢ (a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e) *\n ((2 ^ 2 * a) ^ 4 * (2 ^ 3 * a ^ 2 * (2 ^ 2 * a) ^ 2) * (2 ^ 8 * a ^ 4)) =\n a *\n (((x * (2 ^ 2 * a) + b) ^ 4 * (2 ^ 3 * a ^ 2 * (2 ^ 2 * a) ^ 2) +\n (2 ^ 3 * a * c - 3 * b ^ 2) * (x * (2 ^ 2 * a) + b) ^ 2 * (2 ^ 2 * a) ^ 4) *\n (2 ^ 8 * a ^ 4) +\n (2 ^ 4 * a * b ^ 2 * c + 2 ^ 8 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) *\n ((2 ^ 2 * a) ^ 4 * (2 ^ 3 * a ^ 2 * (2 ^ 2 * a) ^ 2)))"
},
{
"line": "linear_combination (1048576 * a ^ 10 * x + 262144 * a ^ 9 * b) * hqz",
"before_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\n⊢ (a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e) *\n ((2 ^ 2 * a) ^ 4 * (2 ^ 3 * a ^ 2 * (2 ^ 2 * a) ^ 2) * (2 ^ 8 * a ^ 4)) =\n a *\n (((x * (2 ^ 2 * a) + b) ^ 4 * (2 ^ 3 * a ^ 2 * (2 ^ 2 * a) ^ 2) +\n (2 ^ 3 * a * c - 3 * b ^ 2) * (x * (2 ^ 2 * a) + b) ^ 2 * (2 ^ 2 * a) ^ 4) *\n (2 ^ 8 * a ^ 4) +\n (2 ^ 4 * a * b ^ 2 * c + 2 ^ 8 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) *\n ((2 ^ 2 * a) ^ 4 * (2 ^ 3 * a ^ 2 * (2 ^ 2 * a) ^ 2)))",
"after_state": "No Goals!"
},
{
"line": "rw [h₁]",
"before_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\n⊢ a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = 0 ↔\n x = v - b / (4 * a) ∨ x = -v - b / (4 * a) ∨ x = w - b / (4 * a) ∨ x = -w - b / (4 * a)",
"after_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\n⊢ a * (y ^ 4 + p * y ^ 2 + r) = 0 ↔\n x = v - b / (4 * a) ∨ x = -v - b / (4 * a) ∨ x = w - b / (4 * a) ∨ x = -w - b / (4 * a)"
},
{
"line": "rewrite [h₁]",
"before_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\n⊢ a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = 0 ↔\n x = v - b / (4 * a) ∨ x = -v - b / (4 * a) ∨ x = w - b / (4 * a) ∨ x = -w - b / (4 * a)",
"after_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\n⊢ a * (y ^ 4 + p * y ^ 2 + r) = 0 ↔\n x = v - b / (4 * a) ∨ x = -v - b / (4 * a) ∨ x = w - b / (4 * a) ∨ x = -w - b / (4 * a)"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\n⊢ a * (y ^ 4 + p * y ^ 2 + r) = 0 ↔\n x = v - b / (4 * a) ∨ x = -v - b / (4 * a) ∨ x = w - b / (4 * a) ∨ x = -w - b / (4 * a)",
"after_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\n⊢ a * (y ^ 4 + p * y ^ 2 + r) = 0 ↔\n x = v - b / (4 * a) ∨ x = -v - b / (4 * a) ∨ x = w - b / (4 * a) ∨ x = -w - b / (4 * a)"
},
{
"line": "try (with_reducible rfl)",
"before_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\n⊢ a * (y ^ 4 + p * y ^ 2 + r) = 0 ↔\n x = v - b / (4 * a) ∨ x = -v - b / (4 * a) ∨ x = w - b / (4 * a) ∨ x = -w - b / (4 * a)",
"after_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\n⊢ a * (y ^ 4 + p * y ^ 2 + r) = 0 ↔\n x = v - b / (4 * a) ∨ x = -v - b / (4 * a) ∨ x = w - b / (4 * a) ∨ x = -w - b / (4 * a)"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\n⊢ a * (y ^ 4 + p * y ^ 2 + r) = 0 ↔\n x = v - b / (4 * a) ∨ x = -v - b / (4 * a) ∨ x = w - b / (4 * a) ∨ x = -w - b / (4 * a)",
"after_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\n⊢ a * (y ^ 4 + p * y ^ 2 + r) = 0 ↔\n x = v - b / (4 * a) ∨ x = -v - b / (4 * a) ∨ x = w - b / (4 * a) ∨ x = -w - b / (4 * a)"
},
{
"line": "with_reducible rfl",
"before_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\n⊢ a * (y ^ 4 + p * y ^ 2 + r) = 0 ↔\n x = v - b / (4 * a) ∨ x = -v - b / (4 * a) ∨ x = w - b / (4 * a) ∨ x = -w - b / (4 * a)",
"after_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\n⊢ a * (y ^ 4 + p * y ^ 2 + r) = 0 ↔\n x = v - b / (4 * a) ∨ x = -v - b / (4 * a) ∨ x = w - b / (4 * a) ∨ x = -w - b / (4 * a)"
},
{
"line": "rfl",
"before_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\n⊢ a * (y ^ 4 + p * y ^ 2 + r) = 0 ↔\n x = v - b / (4 * a) ∨ x = -v - b / (4 * a) ∨ x = w - b / (4 * a) ∨ x = -w - b / (4 * a)",
"after_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\n⊢ a * (y ^ 4 + p * y ^ 2 + r) = 0 ↔\n x = v - b / (4 * a) ∨ x = -v - b / (4 * a) ∨ x = w - b / (4 * a) ∨ x = -w - b / (4 * a)"
},
{
"line": "apply_rfl",
"before_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\n⊢ a * (y ^ 4 + p * y ^ 2 + r) = 0 ↔\n x = v - b / (4 * a) ∨ x = -v - b / (4 * a) ∨ x = w - b / (4 * a) ∨ x = -w - b / (4 * a)",
"after_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\n⊢ a * (y ^ 4 + p * y ^ 2 + r) = 0 ↔\n x = v - b / (4 * a) ∨ x = -v - b / (4 * a) ∨ x = w - b / (4 * a) ∨ x = -w - b / (4 * a)"
},
{
"line": "skip",
"before_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\n⊢ a * (y ^ 4 + p * y ^ 2 + r) = 0 ↔\n x = v - b / (4 * a) ∨ x = -v - b / (4 * a) ∨ x = w - b / (4 * a) ∨ x = -w - b / (4 * a)",
"after_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\n⊢ a * (y ^ 4 + p * y ^ 2 + r) = 0 ↔\n x = v - b / (4 * a) ∨ x = -v - b / (4 * a) ∨ x = w - b / (4 * a) ∨ x = -w - b / (4 * a)"
},
{
"line": "rw [ha.isUnit.mul_right_eq_zero]",
"before_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\n⊢ a * (y ^ 4 + p * y ^ 2 + r) = 0 ↔\n x = v - b / (4 * a) ∨ x = -v - b / (4 * a) ∨ x = w - b / (4 * a) ∨ x = -w - b / (4 * a)",
"after_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\n⊢ y ^ 4 + p * y ^ 2 + r = 0 ↔ x = v - b / (4 * a) ∨ x = -v - b / (4 * a) ∨ x = w - b / (4 * a) ∨ x = -w - b / (4 * a)"
},
{
"line": "rewrite [ha.isUnit.mul_right_eq_zero]",
"before_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\n⊢ a * (y ^ 4 + p * y ^ 2 + r) = 0 ↔\n x = v - b / (4 * a) ∨ x = -v - b / (4 * a) ∨ x = w - b / (4 * a) ∨ x = -w - b / (4 * a)",
"after_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\n⊢ y ^ 4 + p * y ^ 2 + r = 0 ↔ x = v - b / (4 * a) ∨ x = -v - b / (4 * a) ∨ x = w - b / (4 * a) ∨ x = -w - b / (4 * a)"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\n⊢ y ^ 4 + p * y ^ 2 + r = 0 ↔ x = v - b / (4 * a) ∨ x = -v - b / (4 * a) ∨ x = w - b / (4 * a) ∨ x = -w - b / (4 * a)",
"after_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\n⊢ y ^ 4 + p * y ^ 2 + r = 0 ↔ x = v - b / (4 * a) ∨ x = -v - b / (4 * a) ∨ x = w - b / (4 * a) ∨ x = -w - b / (4 * a)"
},
{
"line": "try (with_reducible rfl)",
"before_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\n⊢ y ^ 4 + p * y ^ 2 + r = 0 ↔ x = v - b / (4 * a) ∨ x = -v - b / (4 * a) ∨ x = w - b / (4 * a) ∨ x = -w - b / (4 * a)",
"after_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\n⊢ y ^ 4 + p * y ^ 2 + r = 0 ↔ x = v - b / (4 * a) ∨ x = -v - b / (4 * a) ∨ x = w - b / (4 * a) ∨ x = -w - b / (4 * a)"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\n⊢ y ^ 4 + p * y ^ 2 + r = 0 ↔ x = v - b / (4 * a) ∨ x = -v - b / (4 * a) ∨ x = w - b / (4 * a) ∨ x = -w - b / (4 * a)",
"after_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\n⊢ y ^ 4 + p * y ^ 2 + r = 0 ↔ x = v - b / (4 * a) ∨ x = -v - b / (4 * a) ∨ x = w - b / (4 * a) ∨ x = -w - b / (4 * a)"
},
{
"line": "with_reducible rfl",
"before_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\n⊢ y ^ 4 + p * y ^ 2 + r = 0 ↔ x = v - b / (4 * a) ∨ x = -v - b / (4 * a) ∨ x = w - b / (4 * a) ∨ x = -w - b / (4 * a)",
"after_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\n⊢ y ^ 4 + p * y ^ 2 + r = 0 ↔ x = v - b / (4 * a) ∨ x = -v - b / (4 * a) ∨ x = w - b / (4 * a) ∨ x = -w - b / (4 * a)"
},
{
"line": "rfl",
"before_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\n⊢ y ^ 4 + p * y ^ 2 + r = 0 ↔ x = v - b / (4 * a) ∨ x = -v - b / (4 * a) ∨ x = w - b / (4 * a) ∨ x = -w - b / (4 * a)",
"after_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\n⊢ y ^ 4 + p * y ^ 2 + r = 0 ↔ x = v - b / (4 * a) ∨ x = -v - b / (4 * a) ∨ x = w - b / (4 * a) ∨ x = -w - b / (4 * a)"
},
{
"line": "apply_rfl",
"before_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\n⊢ y ^ 4 + p * y ^ 2 + r = 0 ↔ x = v - b / (4 * a) ∨ x = -v - b / (4 * a) ∨ x = w - b / (4 * a) ∨ x = -w - b / (4 * a)",
"after_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\n⊢ y ^ 4 + p * y ^ 2 + r = 0 ↔ x = v - b / (4 * a) ∨ x = -v - b / (4 * a) ∨ x = w - b / (4 * a) ∨ x = -w - b / (4 * a)"
},
{
"line": "skip",
"before_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\n⊢ y ^ 4 + p * y ^ 2 + r = 0 ↔ x = v - b / (4 * a) ∨ x = -v - b / (4 * a) ∨ x = w - b / (4 * a) ∨ x = -w - b / (4 * a)",
"after_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\n⊢ y ^ 4 + p * y ^ 2 + r = 0 ↔ x = v - b / (4 * a) ∨ x = -v - b / (4 * a) ∨ x = w - b / (4 * a) ∨ x = -w - b / (4 * a)"
},
{
"line": "calc\n _ ↔ 1 * (y ^ 2 * y ^ 2) + p * y ^ 2 + r = 0 := by\n apply Eq.congr_left\n ring\n _ ↔ y ^ 2 = (-p + t) / 2 ∨ y ^ 2 = (-p - t) / 2 :=\n by\n have ht' : discrim 1 p r = t * t := by rw [discrim]; linear_combination -ht\n rw [quadratic_eq_zero_iff one_ne_zero ht']\n rw [mul_one]\n _ ↔ _ := by simp_rw [y, ← hv, ← hw, pow_two, mul_self_eq_mul_self_iff, eq_sub_iff_add_eq, or_assoc]",
"before_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\n⊢ y ^ 4 + p * y ^ 2 + r = 0 ↔ x = v - b / (4 * a) ∨ x = -v - b / (4 * a) ∨ x = w - b / (4 * a) ∨ x = -w - b / (4 * a)",
"after_state": "No Goals!"
},
{
"line": "apply Eq.congr_left",
"before_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\n⊢ y ^ 4 + p * y ^ 2 + r = 0 ↔ 1 * (y ^ 2 * y ^ 2) + p * y ^ 2 + r = 0",
"after_state": "case h\nK : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\n⊢ y ^ 4 + p * y ^ 2 + r = 1 * (y ^ 2 * y ^ 2) + p * y ^ 2 + r"
},
{
"line": "ring",
"before_state": "case h\nK : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\n⊢ y ^ 4 + p * y ^ 2 + r = 1 * (y ^ 2 * y ^ 2) + p * y ^ 2 + r",
"after_state": "No Goals!"
},
{
"line": "first\n| ring1\n|\n try_this ring_nf\"\\n\\nThe `ring` tactic failed to close the goal. Use `ring_nf` to obtain a normal form.\n \\nNote that `ring` works primarily in *commutative* rings. \\\n If you have a noncommutative ring, abelian group or module, consider using \\\n `noncomm_ring`, `abel` or `module` instead.\"",
"before_state": "case h\nK : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\n⊢ y ^ 4 + p * y ^ 2 + r = 1 * (y ^ 2 * y ^ 2) + p * y ^ 2 + r",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "case h\nK : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\n⊢ y ^ 4 + p * y ^ 2 + r = 1 * (y ^ 2 * y ^ 2) + p * y ^ 2 + r",
"after_state": "No Goals!"
},
{
"line": "have ht' : discrim 1 p r = t * t := by rw [discrim]; linear_combination -ht",
"before_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\n⊢ 1 * (y ^ 2 * y ^ 2) + p * y ^ 2 + r = 0 ↔ y ^ 2 = (-p + t) / 2 ∨ y ^ 2 = (-p - t) / 2",
"after_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\nht' : discrim 1 p r = t * t\n⊢ 1 * (y ^ 2 * y ^ 2) + p * y ^ 2 + r = 0 ↔ y ^ 2 = (-p + t) / 2 ∨ y ^ 2 = (-p - t) / 2"
},
{
"line": "focus\n refine\n no_implicit_lambda%\n (have ht' : discrim 1 p r = t * t := ?body✝;\n ?_)\n case body✝ => with_annotate_state\"by\" (rw [discrim]; linear_combination -ht)",
"before_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\n⊢ 1 * (y ^ 2 * y ^ 2) + p * y ^ 2 + r = 0 ↔ y ^ 2 = (-p + t) / 2 ∨ y ^ 2 = (-p - t) / 2",
"after_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\nht' : discrim 1 p r = t * t\n⊢ 1 * (y ^ 2 * y ^ 2) + p * y ^ 2 + r = 0 ↔ y ^ 2 = (-p + t) / 2 ∨ y ^ 2 = (-p - t) / 2"
},
{
"line": "refine\n no_implicit_lambda%\n (have ht' : discrim 1 p r = t * t := ?body✝;\n ?_)",
"before_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\n⊢ 1 * (y ^ 2 * y ^ 2) + p * y ^ 2 + r = 0 ↔ y ^ 2 = (-p + t) / 2 ∨ y ^ 2 = (-p - t) / 2",
"after_state": "case body\nK : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\n⊢ discrim 1 p r = t * t\n---\nK : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\nht' : discrim 1 p r = t * t\n⊢ 1 * (y ^ 2 * y ^ 2) + p * y ^ 2 + r = 0 ↔ y ^ 2 = (-p + t) / 2 ∨ y ^ 2 = (-p - t) / 2"
},
{
"line": "case body✝ => with_annotate_state\"by\" (rw [discrim]; linear_combination -ht)",
"before_state": "case body\nK : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\n⊢ discrim 1 p r = t * t\n---\nK : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\nht' : discrim 1 p r = t * t\n⊢ 1 * (y ^ 2 * y ^ 2) + p * y ^ 2 + r = 0 ↔ y ^ 2 = (-p + t) / 2 ∨ y ^ 2 = (-p - t) / 2",
"after_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\nht' : discrim 1 p r = t * t\n⊢ 1 * (y ^ 2 * y ^ 2) + p * y ^ 2 + r = 0 ↔ y ^ 2 = (-p + t) / 2 ∨ y ^ 2 = (-p - t) / 2"
},
{
"line": "with_annotate_state\"by\" (rw [discrim]; linear_combination -ht)",
"before_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\n⊢ discrim 1 p r = t * t",
"after_state": "No Goals!"
},
{
"line": "rw [discrim]",
"before_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\n⊢ discrim 1 p r = t * t",
"after_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\n⊢ p ^ 2 - 4 * 1 * r = t * t"
},
{
"line": "rewrite [discrim]",
"before_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\n⊢ discrim 1 p r = t * t",
"after_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\n⊢ p ^ 2 - 4 * 1 * r = t * t"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\n⊢ p ^ 2 - 4 * 1 * r = t * t",
"after_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\n⊢ p ^ 2 - 4 * 1 * r = t * t"
},
{
"line": "try (with_reducible rfl)",
"before_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\n⊢ p ^ 2 - 4 * 1 * r = t * t",
"after_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\n⊢ p ^ 2 - 4 * 1 * r = t * t"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\n⊢ p ^ 2 - 4 * 1 * r = t * t",
"after_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\n⊢ p ^ 2 - 4 * 1 * r = t * t"
},
{
"line": "with_reducible rfl",
"before_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\n⊢ p ^ 2 - 4 * 1 * r = t * t",
"after_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\n⊢ p ^ 2 - 4 * 1 * r = t * t"
},
{
"line": "rfl",
"before_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\n⊢ p ^ 2 - 4 * 1 * r = t * t",
"after_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\n⊢ p ^ 2 - 4 * 1 * r = t * t"
},
{
"line": "apply_rfl",
"before_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\n⊢ p ^ 2 - 4 * 1 * r = t * t",
"after_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\n⊢ p ^ 2 - 4 * 1 * r = t * t"
},
{
"line": "skip",
"before_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\n⊢ p ^ 2 - 4 * 1 * r = t * t",
"after_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\n⊢ p ^ 2 - 4 * 1 * r = t * t"
},
{
"line": "linear_combination -ht",
"before_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\n⊢ p ^ 2 - 4 * 1 * r = t * t",
"after_state": "No Goals!"
},
{
"line": "rw [quadratic_eq_zero_iff one_ne_zero ht']",
"before_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\nht' : discrim 1 p r = t * t\n⊢ 1 * (y ^ 2 * y ^ 2) + p * y ^ 2 + r = 0 ↔ y ^ 2 = (-p + t) / 2 ∨ y ^ 2 = (-p - t) / 2",
"after_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\nht' : discrim 1 p r = t * t\n⊢ y ^ 2 = (-p + t) / (2 * 1) ∨ y ^ 2 = (-p - t) / (2 * 1) ↔ y ^ 2 = (-p + t) / 2 ∨ y ^ 2 = (-p - t) / 2"
},
{
"line": "rewrite [quadratic_eq_zero_iff one_ne_zero ht']",
"before_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\nht' : discrim 1 p r = t * t\n⊢ 1 * (y ^ 2 * y ^ 2) + p * y ^ 2 + r = 0 ↔ y ^ 2 = (-p + t) / 2 ∨ y ^ 2 = (-p - t) / 2",
"after_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\nht' : discrim 1 p r = t * t\n⊢ y ^ 2 = (-p + t) / (2 * 1) ∨ y ^ 2 = (-p - t) / (2 * 1) ↔ y ^ 2 = (-p + t) / 2 ∨ y ^ 2 = (-p - t) / 2"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\nht' : discrim 1 p r = t * t\n⊢ y ^ 2 = (-p + t) / (2 * 1) ∨ y ^ 2 = (-p - t) / (2 * 1) ↔ y ^ 2 = (-p + t) / 2 ∨ y ^ 2 = (-p - t) / 2",
"after_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\nht' : discrim 1 p r = t * t\n⊢ y ^ 2 = (-p + t) / (2 * 1) ∨ y ^ 2 = (-p - t) / (2 * 1) ↔ y ^ 2 = (-p + t) / 2 ∨ y ^ 2 = (-p - t) / 2"
},
{
"line": "try (with_reducible rfl)",
"before_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\nht' : discrim 1 p r = t * t\n⊢ y ^ 2 = (-p + t) / (2 * 1) ∨ y ^ 2 = (-p - t) / (2 * 1) ↔ y ^ 2 = (-p + t) / 2 ∨ y ^ 2 = (-p - t) / 2",
"after_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\nht' : discrim 1 p r = t * t\n⊢ y ^ 2 = (-p + t) / (2 * 1) ∨ y ^ 2 = (-p - t) / (2 * 1) ↔ y ^ 2 = (-p + t) / 2 ∨ y ^ 2 = (-p - t) / 2"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\nht' : discrim 1 p r = t * t\n⊢ y ^ 2 = (-p + t) / (2 * 1) ∨ y ^ 2 = (-p - t) / (2 * 1) ↔ y ^ 2 = (-p + t) / 2 ∨ y ^ 2 = (-p - t) / 2",
"after_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\nht' : discrim 1 p r = t * t\n⊢ y ^ 2 = (-p + t) / (2 * 1) ∨ y ^ 2 = (-p - t) / (2 * 1) ↔ y ^ 2 = (-p + t) / 2 ∨ y ^ 2 = (-p - t) / 2"
},
{
"line": "with_reducible rfl",
"before_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\nht' : discrim 1 p r = t * t\n⊢ y ^ 2 = (-p + t) / (2 * 1) ∨ y ^ 2 = (-p - t) / (2 * 1) ↔ y ^ 2 = (-p + t) / 2 ∨ y ^ 2 = (-p - t) / 2",
"after_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\nht' : discrim 1 p r = t * t\n⊢ y ^ 2 = (-p + t) / (2 * 1) ∨ y ^ 2 = (-p - t) / (2 * 1) ↔ y ^ 2 = (-p + t) / 2 ∨ y ^ 2 = (-p - t) / 2"
},
{
"line": "rfl",
"before_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\nht' : discrim 1 p r = t * t\n⊢ y ^ 2 = (-p + t) / (2 * 1) ∨ y ^ 2 = (-p - t) / (2 * 1) ↔ y ^ 2 = (-p + t) / 2 ∨ y ^ 2 = (-p - t) / 2",
"after_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\nht' : discrim 1 p r = t * t\n⊢ y ^ 2 = (-p + t) / (2 * 1) ∨ y ^ 2 = (-p - t) / (2 * 1) ↔ y ^ 2 = (-p + t) / 2 ∨ y ^ 2 = (-p - t) / 2"
},
{
"line": "apply_rfl",
"before_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\nht' : discrim 1 p r = t * t\n⊢ y ^ 2 = (-p + t) / (2 * 1) ∨ y ^ 2 = (-p - t) / (2 * 1) ↔ y ^ 2 = (-p + t) / 2 ∨ y ^ 2 = (-p - t) / 2",
"after_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\nht' : discrim 1 p r = t * t\n⊢ y ^ 2 = (-p + t) / (2 * 1) ∨ y ^ 2 = (-p - t) / (2 * 1) ↔ y ^ 2 = (-p + t) / 2 ∨ y ^ 2 = (-p - t) / 2"
},
{
"line": "skip",
"before_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\nht' : discrim 1 p r = t * t\n⊢ y ^ 2 = (-p + t) / (2 * 1) ∨ y ^ 2 = (-p - t) / (2 * 1) ↔ y ^ 2 = (-p + t) / 2 ∨ y ^ 2 = (-p - t) / 2",
"after_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\nht' : discrim 1 p r = t * t\n⊢ y ^ 2 = (-p + t) / (2 * 1) ∨ y ^ 2 = (-p - t) / (2 * 1) ↔ y ^ 2 = (-p + t) / 2 ∨ y ^ 2 = (-p - t) / 2"
},
{
"line": "rw [mul_one]",
"before_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\nht' : discrim 1 p r = t * t\n⊢ y ^ 2 = (-p + t) / (2 * 1) ∨ y ^ 2 = (-p - t) / (2 * 1) ↔ y ^ 2 = (-p + t) / 2 ∨ y ^ 2 = (-p - t) / 2",
"after_state": "No Goals!"
},
{
"line": "rewrite [mul_one]",
"before_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\nht' : discrim 1 p r = t * t\n⊢ y ^ 2 = (-p + t) / (2 * 1) ∨ y ^ 2 = (-p - t) / (2 * 1) ↔ y ^ 2 = (-p + t) / 2 ∨ y ^ 2 = (-p - t) / 2",
"after_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\nht' : discrim 1 p r = t * t\n⊢ y ^ 2 = (-p + t) / 2 ∨ y ^ 2 = (-p - t) / 2 ↔ y ^ 2 = (-p + t) / 2 ∨ y ^ 2 = (-p - t) / 2"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\nht' : discrim 1 p r = t * t\n⊢ y ^ 2 = (-p + t) / 2 ∨ y ^ 2 = (-p - t) / 2 ↔ y ^ 2 = (-p + t) / 2 ∨ y ^ 2 = (-p - t) / 2",
"after_state": "No Goals!"
},
{
"line": "try (with_reducible rfl)",
"before_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\nht' : discrim 1 p r = t * t\n⊢ y ^ 2 = (-p + t) / 2 ∨ y ^ 2 = (-p - t) / 2 ↔ y ^ 2 = (-p + t) / 2 ∨ y ^ 2 = (-p - t) / 2",
"after_state": "No Goals!"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\nht' : discrim 1 p r = t * t\n⊢ y ^ 2 = (-p + t) / 2 ∨ y ^ 2 = (-p - t) / 2 ↔ y ^ 2 = (-p + t) / 2 ∨ y ^ 2 = (-p - t) / 2",
"after_state": "No Goals!"
},
{
"line": "with_reducible rfl",
"before_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\nht' : discrim 1 p r = t * t\n⊢ y ^ 2 = (-p + t) / 2 ∨ y ^ 2 = (-p - t) / 2 ↔ y ^ 2 = (-p + t) / 2 ∨ y ^ 2 = (-p - t) / 2",
"after_state": "No Goals!"
},
{
"line": "rfl",
"before_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\nht' : discrim 1 p r = t * t\n⊢ y ^ 2 = (-p + t) / 2 ∨ y ^ 2 = (-p - t) / 2 ↔ y ^ 2 = (-p + t) / 2 ∨ y ^ 2 = (-p - t) / 2",
"after_state": "No Goals!"
},
{
"line": "exact Iff.rfl✝",
"before_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\nht' : discrim 1 p r = t * t\n⊢ y ^ 2 = (-p + t) / 2 ∨ y ^ 2 = (-p - t) / 2 ↔ y ^ 2 = (-p + t) / 2 ∨ y ^ 2 = (-p - t) / 2",
"after_state": "No Goals!"
},
{
"line": "simp_rw [y, ← hv, ← hw, pow_two, mul_self_eq_mul_self_iff, eq_sub_iff_add_eq, or_assoc]",
"before_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\n⊢ y ^ 2 = (-p + t) / 2 ∨ y ^ 2 = (-p - t) / 2 ↔\n x = v - b / (4 * a) ∨ x = -v - b / (4 * a) ∨ x = w - b / (4 * a) ∨ x = -w - b / (4 * a)",
"after_state": "No Goals!"
},
{
"line": "simp (failIfUnchanged✝ := false✝) only",
"before_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\n⊢ y ^ 2 = (-p + t) / 2 ∨ y ^ 2 = (-p - t) / 2 ↔\n x = v - b / (4 * a) ∨ x = -v - b / (4 * a) ∨ x = w - b / (4 * a) ∨ x = -w - b / (4 * a)",
"after_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\n⊢ y ^ 2 = (-p + t) / 2 ∨ y ^ 2 = (-p - t) / 2 ↔\n x = v - b / (4 * a) ∨ x = -v - b / (4 * a) ∨ x = w - b / (4 * a) ∨ x = -w - b / (4 * a)"
},
{
"line": "simp only [y]",
"before_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\n⊢ y ^ 2 = (-p + t) / 2 ∨ y ^ 2 = (-p - t) / 2 ↔\n x = v - b / (4 * a) ∨ x = -v - b / (4 * a) ∨ x = w - b / (4 * a) ∨ x = -w - b / (4 * a)",
"after_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\n⊢ (x + b / (4 * a)) ^ 2 = (-p + t) / 2 ∨ (x + b / (4 * a)) ^ 2 = (-p - t) / 2 ↔\n x = v - b / (4 * a) ∨ x = -v - b / (4 * a) ∨ x = w - b / (4 * a) ∨ x = -w - b / (4 * a)"
},
{
"line": "simp only [← hv]",
"before_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\n⊢ (x + b / (4 * a)) ^ 2 = (-p + t) / 2 ∨ (x + b / (4 * a)) ^ 2 = (-p - t) / 2 ↔\n x = v - b / (4 * a) ∨ x = -v - b / (4 * a) ∨ x = w - b / (4 * a) ∨ x = -w - b / (4 * a)",
"after_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\n⊢ (x + b / (4 * a)) ^ 2 = v ^ 2 ∨ (x + b / (4 * a)) ^ 2 = (-p - t) / 2 ↔\n x = v - b / (4 * a) ∨ x = -v - b / (4 * a) ∨ x = w - b / (4 * a) ∨ x = -w - b / (4 * a)"
},
{
"line": "simp only [← hw]",
"before_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\n⊢ (x + b / (4 * a)) ^ 2 = v ^ 2 ∨ (x + b / (4 * a)) ^ 2 = (-p - t) / 2 ↔\n x = v - b / (4 * a) ∨ x = -v - b / (4 * a) ∨ x = w - b / (4 * a) ∨ x = -w - b / (4 * a)",
"after_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\n⊢ (x + b / (4 * a)) ^ 2 = v ^ 2 ∨ (x + b / (4 * a)) ^ 2 = w ^ 2 ↔\n x = v - b / (4 * a) ∨ x = -v - b / (4 * a) ∨ x = w - b / (4 * a) ∨ x = -w - b / (4 * a)"
},
{
"line": "simp only [pow_two]",
"before_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\n⊢ (x + b / (4 * a)) ^ 2 = v ^ 2 ∨ (x + b / (4 * a)) ^ 2 = w ^ 2 ↔\n x = v - b / (4 * a) ∨ x = -v - b / (4 * a) ∨ x = w - b / (4 * a) ∨ x = -w - b / (4 * a)",
"after_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\n⊢ (x + b / (4 * a)) * (x + b / (4 * a)) = v * v ∨ (x + b / (4 * a)) * (x + b / (4 * a)) = w * w ↔\n x = v - b / (4 * a) ∨ x = -v - b / (4 * a) ∨ x = w - b / (4 * a) ∨ x = -w - b / (4 * a)"
},
{
"line": "simp only [mul_self_eq_mul_self_iff]",
"before_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\n⊢ (x + b / (4 * a)) * (x + b / (4 * a)) = v * v ∨ (x + b / (4 * a)) * (x + b / (4 * a)) = w * w ↔\n x = v - b / (4 * a) ∨ x = -v - b / (4 * a) ∨ x = w - b / (4 * a) ∨ x = -w - b / (4 * a)",
"after_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\n⊢ (x + b / (4 * a) = v ∨ x + b / (4 * a) = -v) ∨ x + b / (4 * a) = w ∨ x + b / (4 * a) = -w ↔\n x = v - b / (4 * a) ∨ x = -v - b / (4 * a) ∨ x = w - b / (4 * a) ∨ x = -w - b / (4 * a)"
},
{
"line": "simp only [eq_sub_iff_add_eq]",
"before_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\n⊢ (x + b / (4 * a) = v ∨ x + b / (4 * a) = -v) ∨ x + b / (4 * a) = w ∨ x + b / (4 * a) = -w ↔\n x = v - b / (4 * a) ∨ x = -v - b / (4 * a) ∨ x = w - b / (4 * a) ∨ x = -w - b / (4 * a)",
"after_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\n⊢ (x + b / (4 * a) = v ∨ x + b / (4 * a) = -v) ∨ x + b / (4 * a) = w ∨ x + b / (4 * a) = -w ↔\n x + b / (4 * a) = v ∨ x + b / (4 * a) = -v ∨ x + b / (4 * a) = w ∨ x + b / (4 * a) = -w"
},
{
"line": "simp only [or_assoc]",
"before_state": "K : Type u_1\ninst✝³ : Field K\na b c d e p r t v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nha : a ≠ 0\nhp : p = (8 * a * c - 3 * b ^ 2) / (8 * a ^ 2)\nhqz : b ^ 3 - 4 * a * b * c + 8 * a ^ 2 * d = 0\nhr : r = (16 * a * b ^ 2 * c + 256 * a ^ 3 * e - 3 * b ^ 4 - 64 * a ^ 2 * b * d) / (256 * a ^ 4)\nht : t ^ 2 = p ^ 2 - 4 * r\nhv : v ^ 2 = (-p + t) / 2\nhw : w ^ 2 = (-p - t) / 2\nx : K\ny : K := x + b / (4 * a)\nh4 : 4 = 2 ^ 2\nh8 : 8 = 2 ^ 3\nh16 : 16 = 2 ^ 4\nh256 : 256 = 2 ^ 8\nh₁ : a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = a * (y ^ 4 + p * y ^ 2 + r)\n⊢ (x + b / (4 * a) = v ∨ x + b / (4 * a) = -v) ∨ x + b / (4 * a) = w ∨ x + b / (4 * a) = -w ↔\n x + b / (4 * a) = v ∨ x + b / (4 * a) = -v ∨ x + b / (4 * a) = w ∨ x + b / (4 * a) = -w",
"after_state": "No Goals!"
}
] |
lemma intervalIntegrable_min_const_sin_mul (a b : ℝ) :
IntervalIntegrable (fun (θ : ℝ) => min d (θ.sin * l)) ℙ a b := by
apply Continuous.intervalIntegrable
exact Continuous.min continuous_const (Continuous.mul Real.continuous_sin continuous_const)
|
/root/DuelModelResearch/mathlib4/Archive/Wiedijk100Theorems/BuffonsNeedle.lean
|
{
"open": [
"MeasureTheory (MeasureSpace IsProbabilityMeasure Measure pdf.IsUniform)",
"ProbabilityTheory Real"
],
"variables": [
""
]
}
|
[
{
"line": "apply Continuous.intervalIntegrable",
"before_state": "d l a b : ℝ\n⊢ IntervalIntegrable (fun θ => min d (sin θ * l)) ℙ a b",
"after_state": "case hu\nd l a b : ℝ\n⊢ Continuous fun θ => min d (sin θ * l)"
},
{
"line": "exact Continuous.min continuous_const (Continuous.mul Real.continuous_sin continuous_const)",
"before_state": "case hu\nd l a b : ℝ\n⊢ Continuous fun θ => min d (sin θ * l)",
"after_state": "No Goals!"
}
] |
lemma integral_min_eq_two_mul :
∫ θ in (0)..π, min d (θ.sin * l) = 2 * ∫ θ in (0)..π / 2, min d (θ.sin * l) := by
rw [← intervalIntegral.integral_add_adjacent_intervals (b := π / 2) (c := π)]
conv => lhs; arg 2; arg 1; intro θ; rw [← neg_neg θ, Real.sin_neg]
· simp_rw [intervalIntegral.integral_comp_neg fun θ => min d (-θ.sin * l), ← Real.sin_add_pi,
intervalIntegral.integral_comp_add_right (fun θ => min d (θ.sin * l)), neg_add_cancel,
(by ring : -(π / 2) + π = π / 2), two_mul]
all_goals exact intervalIntegrable_min_const_sin_mul d l _ _
|
/root/DuelModelResearch/mathlib4/Archive/Wiedijk100Theorems/BuffonsNeedle.lean
|
{
"open": [
"MeasureTheory (MeasureSpace IsProbabilityMeasure Measure pdf.IsUniform)",
"ProbabilityTheory Real"
],
"variables": [
""
]
}
|
[
{
"line": "rw [← intervalIntegral.integral_add_adjacent_intervals (b := π / 2) (c := π)]",
"before_state": "d l : ℝ\n⊢ ∫ (θ : ℝ) in 0 ..π, min d (sin θ * l) = 2 * ∫ (θ : ℝ) in 0 ..π / 2, min d (sin θ * l)",
"after_state": "d l : ℝ\n⊢ (∫ (x : ℝ) in 0 ..π / 2, min d (sin x * l)) + ∫ (x : ℝ) in π / 2 ..π, min d (sin x * l) =\n 2 * ∫ (θ : ℝ) in 0 ..π / 2, min d (sin θ * l)\n---\ncase hab\nd l : ℝ\n⊢ IntervalIntegrable (fun θ => min d (sin θ * l)) ℙ 0 (π / 2)\n---\ncase hbc\nd l : ℝ\n⊢ IntervalIntegrable (fun θ => min d (sin θ * l)) ℙ (π / 2) π"
},
{
"line": "rewrite [← intervalIntegral.integral_add_adjacent_intervals (b := π / 2) (c := π)]",
"before_state": "d l : ℝ\n⊢ ∫ (θ : ℝ) in 0 ..π, min d (sin θ * l) = 2 * ∫ (θ : ℝ) in 0 ..π / 2, min d (sin θ * l)",
"after_state": "d l : ℝ\n⊢ (∫ (x : ℝ) in 0 ..π / 2, min d (sin x * l)) + ∫ (x : ℝ) in π / 2 ..π, min d (sin x * l) =\n 2 * ∫ (θ : ℝ) in 0 ..π / 2, min d (sin θ * l)\n---\ncase hab\nd l : ℝ\n⊢ IntervalIntegrable (fun θ => min d (sin θ * l)) ℙ 0 (π / 2)\n---\ncase hbc\nd l : ℝ\n⊢ IntervalIntegrable (fun θ => min d (sin θ * l)) ℙ (π / 2) π"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "d l : ℝ\n⊢ (∫ (x : ℝ) in 0 ..π / 2, min d (sin x * l)) + ∫ (x : ℝ) in π / 2 ..π, min d (sin x * l) =\n 2 * ∫ (θ : ℝ) in 0 ..π / 2, min d (sin θ * l)\n---\ncase hab\nd l : ℝ\n⊢ IntervalIntegrable (fun θ => min d (sin θ * l)) ℙ 0 (π / 2)\n---\ncase hbc\nd l : ℝ\n⊢ IntervalIntegrable (fun θ => min d (sin θ * l)) ℙ (π / 2) π",
"after_state": "d l : ℝ\n⊢ (∫ (x : ℝ) in 0 ..π / 2, min d (sin x * l)) + ∫ (x : ℝ) in π / 2 ..π, min d (sin x * l) =\n 2 * ∫ (θ : ℝ) in 0 ..π / 2, min d (sin θ * l)\n---\ncase hab\nd l : ℝ\n⊢ IntervalIntegrable (fun θ => min d (sin θ * l)) ℙ 0 (π / 2)\n---\ncase hbc\nd l : ℝ\n⊢ IntervalIntegrable (fun θ => min d (sin θ * l)) ℙ (π / 2) π"
},
{
"line": "try (with_reducible rfl)",
"before_state": "d l : ℝ\n⊢ (∫ (x : ℝ) in 0 ..π / 2, min d (sin x * l)) + ∫ (x : ℝ) in π / 2 ..π, min d (sin x * l) =\n 2 * ∫ (θ : ℝ) in 0 ..π / 2, min d (sin θ * l)\n---\ncase hab\nd l : ℝ\n⊢ IntervalIntegrable (fun θ => min d (sin θ * l)) ℙ 0 (π / 2)\n---\ncase hbc\nd l : ℝ\n⊢ IntervalIntegrable (fun θ => min d (sin θ * l)) ℙ (π / 2) π",
"after_state": "d l : ℝ\n⊢ (∫ (x : ℝ) in 0 ..π / 2, min d (sin x * l)) + ∫ (x : ℝ) in π / 2 ..π, min d (sin x * l) =\n 2 * ∫ (θ : ℝ) in 0 ..π / 2, min d (sin θ * l)\n---\ncase hab\nd l : ℝ\n⊢ IntervalIntegrable (fun θ => min d (sin θ * l)) ℙ 0 (π / 2)\n---\ncase hbc\nd l : ℝ\n⊢ IntervalIntegrable (fun θ => min d (sin θ * l)) ℙ (π / 2) π"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "d l : ℝ\n⊢ (∫ (x : ℝ) in 0 ..π / 2, min d (sin x * l)) + ∫ (x : ℝ) in π / 2 ..π, min d (sin x * l) =\n 2 * ∫ (θ : ℝ) in 0 ..π / 2, min d (sin θ * l)\n---\ncase hab\nd l : ℝ\n⊢ IntervalIntegrable (fun θ => min d (sin θ * l)) ℙ 0 (π / 2)\n---\ncase hbc\nd l : ℝ\n⊢ IntervalIntegrable (fun θ => min d (sin θ * l)) ℙ (π / 2) π",
"after_state": "d l : ℝ\n⊢ (∫ (x : ℝ) in 0 ..π / 2, min d (sin x * l)) + ∫ (x : ℝ) in π / 2 ..π, min d (sin x * l) =\n 2 * ∫ (θ : ℝ) in 0 ..π / 2, min d (sin θ * l)\n---\ncase hab\nd l : ℝ\n⊢ IntervalIntegrable (fun θ => min d (sin θ * l)) ℙ 0 (π / 2)\n---\ncase hbc\nd l : ℝ\n⊢ IntervalIntegrable (fun θ => min d (sin θ * l)) ℙ (π / 2) π"
},
{
"line": "with_reducible rfl",
"before_state": "d l : ℝ\n⊢ (∫ (x : ℝ) in 0 ..π / 2, min d (sin x * l)) + ∫ (x : ℝ) in π / 2 ..π, min d (sin x * l) =\n 2 * ∫ (θ : ℝ) in 0 ..π / 2, min d (sin θ * l)\n---\ncase hab\nd l : ℝ\n⊢ IntervalIntegrable (fun θ => min d (sin θ * l)) ℙ 0 (π / 2)\n---\ncase hbc\nd l : ℝ\n⊢ IntervalIntegrable (fun θ => min d (sin θ * l)) ℙ (π / 2) π",
"after_state": "d l : ℝ\n⊢ (∫ (x : ℝ) in 0 ..π / 2, min d (sin x * l)) + ∫ (x : ℝ) in π / 2 ..π, min d (sin x * l) =\n 2 * ∫ (θ : ℝ) in 0 ..π / 2, min d (sin θ * l)\n---\ncase hab\nd l : ℝ\n⊢ IntervalIntegrable (fun θ => min d (sin θ * l)) ℙ 0 (π / 2)\n---\ncase hbc\nd l : ℝ\n⊢ IntervalIntegrable (fun θ => min d (sin θ * l)) ℙ (π / 2) π"
},
{
"line": "rfl",
"before_state": "d l : ℝ\n⊢ (∫ (x : ℝ) in 0 ..π / 2, min d (sin x * l)) + ∫ (x : ℝ) in π / 2 ..π, min d (sin x * l) =\n 2 * ∫ (θ : ℝ) in 0 ..π / 2, min d (sin θ * l)\n---\ncase hab\nd l : ℝ\n⊢ IntervalIntegrable (fun θ => min d (sin θ * l)) ℙ 0 (π / 2)\n---\ncase hbc\nd l : ℝ\n⊢ IntervalIntegrable (fun θ => min d (sin θ * l)) ℙ (π / 2) π",
"after_state": "d l : ℝ\n⊢ (∫ (x : ℝ) in 0 ..π / 2, min d (sin x * l)) + ∫ (x : ℝ) in π / 2 ..π, min d (sin x * l) =\n 2 * ∫ (θ : ℝ) in 0 ..π / 2, min d (sin θ * l)\n---\ncase hab\nd l : ℝ\n⊢ IntervalIntegrable (fun θ => min d (sin θ * l)) ℙ 0 (π / 2)\n---\ncase hbc\nd l : ℝ\n⊢ IntervalIntegrable (fun θ => min d (sin θ * l)) ℙ (π / 2) π"
},
{
"line": "apply_rfl",
"before_state": "d l : ℝ\n⊢ (∫ (x : ℝ) in 0 ..π / 2, min d (sin x * l)) + ∫ (x : ℝ) in π / 2 ..π, min d (sin x * l) =\n 2 * ∫ (θ : ℝ) in 0 ..π / 2, min d (sin θ * l)\n---\ncase hab\nd l : ℝ\n⊢ IntervalIntegrable (fun θ => min d (sin θ * l)) ℙ 0 (π / 2)\n---\ncase hbc\nd l : ℝ\n⊢ IntervalIntegrable (fun θ => min d (sin θ * l)) ℙ (π / 2) π",
"after_state": "d l : ℝ\n⊢ (∫ (x : ℝ) in 0 ..π / 2, min d (sin x * l)) + ∫ (x : ℝ) in π / 2 ..π, min d (sin x * l) =\n 2 * ∫ (θ : ℝ) in 0 ..π / 2, min d (sin θ * l)\n---\ncase hab\nd l : ℝ\n⊢ IntervalIntegrable (fun θ => min d (sin θ * l)) ℙ 0 (π / 2)\n---\ncase hbc\nd l : ℝ\n⊢ IntervalIntegrable (fun θ => min d (sin θ * l)) ℙ (π / 2) π"
},
{
"line": "skip",
"before_state": "d l : ℝ\n⊢ (∫ (x : ℝ) in 0 ..π / 2, min d (sin x * l)) + ∫ (x : ℝ) in π / 2 ..π, min d (sin x * l) =\n 2 * ∫ (θ : ℝ) in 0 ..π / 2, min d (sin θ * l)\n---\ncase hab\nd l : ℝ\n⊢ IntervalIntegrable (fun θ => min d (sin θ * l)) ℙ 0 (π / 2)\n---\ncase hbc\nd l : ℝ\n⊢ IntervalIntegrable (fun θ => min d (sin θ * l)) ℙ (π / 2) π",
"after_state": "d l : ℝ\n⊢ (∫ (x : ℝ) in 0 ..π / 2, min d (sin x * l)) + ∫ (x : ℝ) in π / 2 ..π, min d (sin x * l) =\n 2 * ∫ (θ : ℝ) in 0 ..π / 2, min d (sin θ * l)\n---\ncase hab\nd l : ℝ\n⊢ IntervalIntegrable (fun θ => min d (sin θ * l)) ℙ 0 (π / 2)\n---\ncase hbc\nd l : ℝ\n⊢ IntervalIntegrable (fun θ => min d (sin θ * l)) ℙ (π / 2) π"
},
{
"line": "conv => lhs; arg 2; arg 1; intro θ; rw [← neg_neg θ, Real.sin_neg]",
"before_state": "d l : ℝ\n⊢ (∫ (x : ℝ) in 0 ..π / 2, min d (sin x * l)) + ∫ (x : ℝ) in π / 2 ..π, min d (sin x * l) =\n 2 * ∫ (θ : ℝ) in 0 ..π / 2, min d (sin θ * l)\n---\ncase hab\nd l : ℝ\n⊢ IntervalIntegrable (fun θ => min d (sin θ * l)) ℙ 0 (π / 2)\n---\ncase hbc\nd l : ℝ\n⊢ IntervalIntegrable (fun θ => min d (sin θ * l)) ℙ (π / 2) π",
"after_state": "d l : ℝ\n⊢ (∫ (x : ℝ) in 0 ..π / 2, min d (sin x * l)) + ∫ (θ : ℝ) in π / 2 ..π, min d (-sin (-θ) * l) =\n 2 * ∫ (θ : ℝ) in 0 ..π / 2, min d (sin θ * l)\n---\ncase hab\nd l : ℝ\n⊢ IntervalIntegrable (fun θ => min d (sin θ * l)) ℙ 0 (π / 2)\n---\ncase hbc\nd l : ℝ\n⊢ IntervalIntegrable (fun θ => min d (sin θ * l)) ℙ (π / 2) π"
},
{
"line": "lhs; arg 2; arg 1; intro θ; rw [← neg_neg θ, Real.sin_neg]",
"before_state": "d l : ℝ\n| (∫ (x : ℝ) in 0 ..π / 2, min d (sin x * l)) + ∫ (x : ℝ) in π / 2 ..π, min d (sin x * l) =\n 2 * ∫ (θ : ℝ) in 0 ..π / 2, min d (sin θ * l)",
"after_state": "case h\nd l θ : ℝ\n| min d (-sin (-θ) * l)"
},
{
"line": "lhs; arg 2; arg 1; intro θ; rw [← neg_neg θ, Real.sin_neg]",
"before_state": "d l : ℝ\n| (∫ (x : ℝ) in 0 ..π / 2, min d (sin x * l)) + ∫ (x : ℝ) in π / 2 ..π, min d (sin x * l) =\n 2 * ∫ (θ : ℝ) in 0 ..π / 2, min d (sin θ * l)",
"after_state": "case h\nd l θ : ℝ\n| min d (-sin (-θ) * l)"
},
{
"line": "lhs",
"before_state": "d l : ℝ\n| (∫ (x : ℝ) in 0 ..π / 2, min d (sin x * l)) + ∫ (x : ℝ) in π / 2 ..π, min d (sin x * l) =\n 2 * ∫ (θ : ℝ) in 0 ..π / 2, min d (sin θ * l)",
"after_state": "d l : ℝ\n| (∫ (x : ℝ) in 0 ..π / 2, min d (sin x * l)) + ∫ (x : ℝ) in π / 2 ..π, min d (sin x * l)"
},
{
"line": "arg 2",
"before_state": "d l : ℝ\n| (∫ (x : ℝ) in 0 ..π / 2, min d (sin x * l)) + ∫ (x : ℝ) in π / 2 ..π, min d (sin x * l)",
"after_state": "d l : ℝ\n| ∫ (x : ℝ) in π / 2 ..π, min d (sin x * l)"
},
{
"line": "arg 1",
"before_state": "d l : ℝ\n| ∫ (x : ℝ) in π / 2 ..π, min d (sin x * l)",
"after_state": "d l : ℝ\n| fun x => min d (sin x * l)"
},
{
"line": "intro θ",
"before_state": "d l : ℝ\n| fun x => min d (sin x * l)",
"after_state": "case h\nd l θ : ℝ\n| min d (sin θ * l)"
},
{
"line": "ext θ",
"before_state": "d l : ℝ\n| fun x => min d (sin x * l)",
"after_state": "case h\nd l θ : ℝ\n| min d (sin θ * l)"
},
{
"line": "rw [← neg_neg θ, Real.sin_neg]",
"before_state": "case h\nd l θ : ℝ\n| min d (sin θ * l)",
"after_state": "case h\nd l θ : ℝ\n| min d (-sin (-θ) * l)"
},
{
"line": "rewrite [← neg_neg θ, Real.sin_neg]",
"before_state": "case h\nd l θ : ℝ\n| min d (sin θ * l)",
"after_state": "case h\nd l θ : ℝ\n| min d (-sin (-θ) * l)"
},
{
"line": "try with_reducible rfl",
"before_state": "d l : ℝ\n⊢ (∫ (x : ℝ) in 0 ..π / 2, min d (sin x * l)) + ∫ (θ : ℝ) in π / 2 ..π, min d (-sin (-θ) * l) =\n 2 * ∫ (θ : ℝ) in 0 ..π / 2, min d (sin θ * l)\n---\ncase hab\nd l : ℝ\n⊢ IntervalIntegrable (fun θ => min d (sin θ * l)) ℙ 0 (π / 2)\n---\ncase hbc\nd l : ℝ\n⊢ IntervalIntegrable (fun θ => min d (sin θ * l)) ℙ (π / 2) π",
"after_state": "d l : ℝ\n⊢ (∫ (x : ℝ) in 0 ..π / 2, min d (sin x * l)) + ∫ (θ : ℝ) in π / 2 ..π, min d (-sin (-θ) * l) =\n 2 * ∫ (θ : ℝ) in 0 ..π / 2, min d (sin θ * l)\n---\ncase hab\nd l : ℝ\n⊢ IntervalIntegrable (fun θ => min d (sin θ * l)) ℙ 0 (π / 2)\n---\ncase hbc\nd l : ℝ\n⊢ IntervalIntegrable (fun θ => min d (sin θ * l)) ℙ (π / 2) π"
},
{
"line": "first\n| with_reducible rfl\n| skip",
"before_state": "d l : ℝ\n⊢ (∫ (x : ℝ) in 0 ..π / 2, min d (sin x * l)) + ∫ (θ : ℝ) in π / 2 ..π, min d (-sin (-θ) * l) =\n 2 * ∫ (θ : ℝ) in 0 ..π / 2, min d (sin θ * l)\n---\ncase hab\nd l : ℝ\n⊢ IntervalIntegrable (fun θ => min d (sin θ * l)) ℙ 0 (π / 2)\n---\ncase hbc\nd l : ℝ\n⊢ IntervalIntegrable (fun θ => min d (sin θ * l)) ℙ (π / 2) π",
"after_state": "d l : ℝ\n⊢ (∫ (x : ℝ) in 0 ..π / 2, min d (sin x * l)) + ∫ (θ : ℝ) in π / 2 ..π, min d (-sin (-θ) * l) =\n 2 * ∫ (θ : ℝ) in 0 ..π / 2, min d (sin θ * l)\n---\ncase hab\nd l : ℝ\n⊢ IntervalIntegrable (fun θ => min d (sin θ * l)) ℙ 0 (π / 2)\n---\ncase hbc\nd l : ℝ\n⊢ IntervalIntegrable (fun θ => min d (sin θ * l)) ℙ (π / 2) π"
},
{
"line": "with_reducible rfl",
"before_state": "d l : ℝ\n⊢ (∫ (x : ℝ) in 0 ..π / 2, min d (sin x * l)) + ∫ (θ : ℝ) in π / 2 ..π, min d (-sin (-θ) * l) =\n 2 * ∫ (θ : ℝ) in 0 ..π / 2, min d (sin θ * l)\n---\ncase hab\nd l : ℝ\n⊢ IntervalIntegrable (fun θ => min d (sin θ * l)) ℙ 0 (π / 2)\n---\ncase hbc\nd l : ℝ\n⊢ IntervalIntegrable (fun θ => min d (sin θ * l)) ℙ (π / 2) π",
"after_state": "d l : ℝ\n⊢ (∫ (x : ℝ) in 0 ..π / 2, min d (sin x * l)) + ∫ (θ : ℝ) in π / 2 ..π, min d (-sin (-θ) * l) =\n 2 * ∫ (θ : ℝ) in 0 ..π / 2, min d (sin θ * l)\n---\ncase hab\nd l : ℝ\n⊢ IntervalIntegrable (fun θ => min d (sin θ * l)) ℙ 0 (π / 2)\n---\ncase hbc\nd l : ℝ\n⊢ IntervalIntegrable (fun θ => min d (sin θ * l)) ℙ (π / 2) π"
},
{
"line": "rfl",
"before_state": "d l : ℝ\n⊢ (∫ (x : ℝ) in 0 ..π / 2, min d (sin x * l)) + ∫ (θ : ℝ) in π / 2 ..π, min d (-sin (-θ) * l) =\n 2 * ∫ (θ : ℝ) in 0 ..π / 2, min d (sin θ * l)\n---\ncase hab\nd l : ℝ\n⊢ IntervalIntegrable (fun θ => min d (sin θ * l)) ℙ 0 (π / 2)\n---\ncase hbc\nd l : ℝ\n⊢ IntervalIntegrable (fun θ => min d (sin θ * l)) ℙ (π / 2) π",
"after_state": "d l : ℝ\n⊢ (∫ (x : ℝ) in 0 ..π / 2, min d (sin x * l)) + ∫ (θ : ℝ) in π / 2 ..π, min d (-sin (-θ) * l) =\n 2 * ∫ (θ : ℝ) in 0 ..π / 2, min d (sin θ * l)\n---\ncase hab\nd l : ℝ\n⊢ IntervalIntegrable (fun θ => min d (sin θ * l)) ℙ 0 (π / 2)\n---\ncase hbc\nd l : ℝ\n⊢ IntervalIntegrable (fun θ => min d (sin θ * l)) ℙ (π / 2) π"
},
{
"line": "apply_rfl",
"before_state": "d l : ℝ\n⊢ (∫ (x : ℝ) in 0 ..π / 2, min d (sin x * l)) + ∫ (θ : ℝ) in π / 2 ..π, min d (-sin (-θ) * l) =\n 2 * ∫ (θ : ℝ) in 0 ..π / 2, min d (sin θ * l)\n---\ncase hab\nd l : ℝ\n⊢ IntervalIntegrable (fun θ => min d (sin θ * l)) ℙ 0 (π / 2)\n---\ncase hbc\nd l : ℝ\n⊢ IntervalIntegrable (fun θ => min d (sin θ * l)) ℙ (π / 2) π",
"after_state": "d l : ℝ\n⊢ (∫ (x : ℝ) in 0 ..π / 2, min d (sin x * l)) + ∫ (θ : ℝ) in π / 2 ..π, min d (-sin (-θ) * l) =\n 2 * ∫ (θ : ℝ) in 0 ..π / 2, min d (sin θ * l)\n---\ncase hab\nd l : ℝ\n⊢ IntervalIntegrable (fun θ => min d (sin θ * l)) ℙ 0 (π / 2)\n---\ncase hbc\nd l : ℝ\n⊢ IntervalIntegrable (fun θ => min d (sin θ * l)) ℙ (π / 2) π"
},
{
"line": "skip",
"before_state": "d l : ℝ\n⊢ (∫ (x : ℝ) in 0 ..π / 2, min d (sin x * l)) + ∫ (θ : ℝ) in π / 2 ..π, min d (-sin (-θ) * l) =\n 2 * ∫ (θ : ℝ) in 0 ..π / 2, min d (sin θ * l)\n---\ncase hab\nd l : ℝ\n⊢ IntervalIntegrable (fun θ => min d (sin θ * l)) ℙ 0 (π / 2)\n---\ncase hbc\nd l : ℝ\n⊢ IntervalIntegrable (fun θ => min d (sin θ * l)) ℙ (π / 2) π",
"after_state": "d l : ℝ\n⊢ (∫ (x : ℝ) in 0 ..π / 2, min d (sin x * l)) + ∫ (θ : ℝ) in π / 2 ..π, min d (-sin (-θ) * l) =\n 2 * ∫ (θ : ℝ) in 0 ..π / 2, min d (sin θ * l)\n---\ncase hab\nd l : ℝ\n⊢ IntervalIntegrable (fun θ => min d (sin θ * l)) ℙ 0 (π / 2)\n---\ncase hbc\nd l : ℝ\n⊢ IntervalIntegrable (fun θ => min d (sin θ * l)) ℙ (π / 2) π"
},
{
"line": "simp_rw [intervalIntegral.integral_comp_neg fun θ => min d (-θ.sin * l), ← Real.sin_add_pi,\n intervalIntegral.integral_comp_add_right (fun θ => min d (θ.sin * l)), neg_add_cancel,\n (by ring : -(π / 2) + π = π / 2), two_mul]",
"before_state": "d l : ℝ\n⊢ (∫ (x : ℝ) in 0 ..π / 2, min d (sin x * l)) + ∫ (θ : ℝ) in π / 2 ..π, min d (-sin (-θ) * l) =\n 2 * ∫ (θ : ℝ) in 0 ..π / 2, min d (sin θ * l)",
"after_state": "No Goals!"
},
{
"line": "simp (failIfUnchanged✝ := false✝) only",
"before_state": "d l : ℝ\n⊢ (∫ (x : ℝ) in 0 ..π / 2, min d (sin x * l)) + ∫ (θ : ℝ) in π / 2 ..π, min d (-sin (-θ) * l) =\n 2 * ∫ (θ : ℝ) in 0 ..π / 2, min d (sin θ * l)",
"after_state": "d l : ℝ\n⊢ (∫ (x : ℝ) in 0 ..π / 2, min d (sin x * l)) + ∫ (θ : ℝ) in π / 2 ..π, min d (-sin (-θ) * l) =\n 2 * ∫ (θ : ℝ) in 0 ..π / 2, min d (sin θ * l)"
},
{
"line": "simp only [intervalIntegral.integral_comp_neg fun θ => min d (-θ.sin * l)]",
"before_state": "d l : ℝ\n⊢ (∫ (x : ℝ) in 0 ..π / 2, min d (sin x * l)) + ∫ (θ : ℝ) in π / 2 ..π, min d (-sin (-θ) * l) =\n 2 * ∫ (θ : ℝ) in 0 ..π / 2, min d (sin θ * l)",
"after_state": "d l : ℝ\n⊢ (∫ (x : ℝ) in 0 ..π / 2, min d (sin x * l)) + ∫ (x : ℝ) in -π..-(π / 2), min d (-sin x * l) =\n 2 * ∫ (x : ℝ) in 0 ..π / 2, min d (sin x * l)"
},
{
"line": "simp only [← Real.sin_add_pi]",
"before_state": "d l : ℝ\n⊢ (∫ (x : ℝ) in 0 ..π / 2, min d (sin x * l)) + ∫ (x : ℝ) in -π..-(π / 2), min d (-sin x * l) =\n 2 * ∫ (x : ℝ) in 0 ..π / 2, min d (sin x * l)",
"after_state": "d l : ℝ\n⊢ (∫ (x : ℝ) in 0 ..π / 2, min d (sin x * l)) + ∫ (x : ℝ) in -π..-(π / 2), min d (sin (x + π) * l) =\n 2 * ∫ (x : ℝ) in 0 ..π / 2, min d (sin x * l)"
},
{
"line": "simp only [intervalIntegral.integral_comp_add_right (fun θ => min d (θ.sin * l))]",
"before_state": "d l : ℝ\n⊢ (∫ (x : ℝ) in 0 ..π / 2, min d (sin x * l)) + ∫ (x : ℝ) in -π..-(π / 2), min d (sin (x + π) * l) =\n 2 * ∫ (x : ℝ) in 0 ..π / 2, min d (sin x * l)",
"after_state": "d l : ℝ\n⊢ (∫ (x : ℝ) in 0 ..π / 2, min d (sin x * l)) + ∫ (x : ℝ) in -π + π..-(π / 2) + π, min d (sin x * l) =\n 2 * ∫ (x : ℝ) in 0 ..π / 2, min d (sin x * l)"
},
{
"line": "simp only [neg_add_cancel]",
"before_state": "d l : ℝ\n⊢ (∫ (x : ℝ) in 0 ..π / 2, min d (sin x * l)) + ∫ (x : ℝ) in -π + π..-(π / 2) + π, min d (sin x * l) =\n 2 * ∫ (x : ℝ) in 0 ..π / 2, min d (sin x * l)",
"after_state": "d l : ℝ\n⊢ (∫ (x : ℝ) in 0 ..π / 2, min d (sin x * l)) + ∫ (x : ℝ) in 0 ..-(π / 2) + π, min d (sin x * l) =\n 2 * ∫ (x : ℝ) in 0 ..π / 2, min d (sin x * l)"
},
{
"line": "simp only [(by ring : -(π / 2) + π = π / 2)]",
"before_state": "d l : ℝ\n⊢ (∫ (x : ℝ) in 0 ..π / 2, min d (sin x * l)) + ∫ (x : ℝ) in 0 ..-(π / 2) + π, min d (sin x * l) =\n 2 * ∫ (x : ℝ) in 0 ..π / 2, min d (sin x * l)",
"after_state": "d l : ℝ\n⊢ (∫ (x : ℝ) in 0 ..π / 2, min d (sin x * l)) + ∫ (x : ℝ) in 0 ..π / 2, min d (sin x * l) =\n 2 * ∫ (x : ℝ) in 0 ..π / 2, min d (sin x * l)"
},
{
"line": "ring",
"before_state": "d l : ℝ\n⊢ -(π / 2) + π = π / 2",
"after_state": "No Goals!"
},
{
"line": "first\n| ring1\n|\n try_this ring_nf\"\\n\\nThe `ring` tactic failed to close the goal. Use `ring_nf` to obtain a normal form.\n \\nNote that `ring` works primarily in *commutative* rings. \\\n If you have a noncommutative ring, abelian group or module, consider using \\\n `noncomm_ring`, `abel` or `module` instead.\"",
"before_state": "d l : ℝ\n⊢ -(π / 2) + π = π / 2",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "d l : ℝ\n⊢ -(π / 2) + π = π / 2",
"after_state": "No Goals!"
},
{
"line": "simp only [two_mul]",
"before_state": "d l : ℝ\n⊢ (∫ (x : ℝ) in 0 ..π / 2, min d (sin x * l)) + ∫ (x : ℝ) in 0 ..π / 2, min d (sin x * l) =\n 2 * ∫ (x : ℝ) in 0 ..π / 2, min d (sin x * l)",
"after_state": "No Goals!"
},
{
"line": "all_goals exact intervalIntegrable_min_const_sin_mul d l _ _",
"before_state": "case hab\nd l : ℝ\n⊢ IntervalIntegrable (fun θ => min d (sin θ * l)) ℙ 0 (π / 2)\n---\ncase hbc\nd l : ℝ\n⊢ IntervalIntegrable (fun θ => min d (sin θ * l)) ℙ (π / 2) π",
"after_state": "No Goals!"
},
{
"line": "exact intervalIntegrable_min_const_sin_mul d l _ _",
"before_state": "case hab\nd l : ℝ\n⊢ IntervalIntegrable (fun θ => min d (sin θ * l)) ℙ 0 (π / 2)",
"after_state": "No Goals!"
},
{
"line": "exact intervalIntegrable_min_const_sin_mul d l _ _",
"before_state": "case hbc\nd l : ℝ\n⊢ IntervalIntegrable (fun θ => min d (sin θ * l)) ℙ (π / 2) π",
"after_state": "No Goals!"
}
] |
lemma integral_zero_to_arcsin_min :
∫ θ in (0)..(d / l).arcsin, min d (θ.sin * l) = (1 - √(1 - (d / l) ^ 2)) * l := by
have : Set.EqOn (fun θ => min d (θ.sin * l)) (Real.sin · * l) (Set.uIcc 0 (d / l).arcsin) := by
intro θ ⟨hθ₁, hθ₂⟩
have : 0 ≤ (d / l).arcsin := Real.arcsin_nonneg.mpr (div_nonneg hd.le hl.le)
simp only [min_eq_left this] at hθ₁ hθ₂
simp only [max_eq_right this] at hθ₁ hθ₂
have hθ_mem : θ ∈ Set.Ioc (-(π / 2)) (π / 2) := by
exact ⟨lt_of_lt_of_le (neg_lt_zero.mpr (div_pos Real.pi_pos two_pos)) hθ₁,
le_trans hθ₂ (d / l).arcsin_mem_Icc.right⟩
simp_rw [min_eq_right ((le_div_iff₀ hl).mp ((Real.le_arcsin_iff_sin_le' hθ_mem).mp hθ₂))]
rw [intervalIntegral.integral_congr this]
rw [intervalIntegral.integral_mul_const]
rw [integral_sin]
rw [Real.cos_zero]
rw [Real.cos_arcsin]
|
/root/DuelModelResearch/mathlib4/Archive/Wiedijk100Theorems/BuffonsNeedle.lean
|
{
"open": [
"MeasureTheory (MeasureSpace IsProbabilityMeasure Measure pdf.IsUniform)",
"ProbabilityTheory Real"
],
"variables": [
""
]
}
|
[
{
"line": "have : Set.EqOn (fun θ => min d (θ.sin * l)) (Real.sin · * l) (Set.uIcc 0 (d / l).arcsin) :=\n by\n intro θ ⟨hθ₁, hθ₂⟩\n have : 0 ≤ (d / l).arcsin := Real.arcsin_nonneg.mpr (div_nonneg hd.le hl.le)\n simp only [min_eq_left this] at hθ₁ hθ₂\n simp only [max_eq_right this] at hθ₁ hθ₂\n have hθ_mem : θ ∈ Set.Ioc (-(π / 2)) (π / 2) := by\n exact\n ⟨lt_of_lt_of_le (neg_lt_zero.mpr (div_pos Real.pi_pos two_pos)) hθ₁, le_trans hθ₂ (d / l).arcsin_mem_Icc.right⟩\n simp_rw [min_eq_right ((le_div_iff₀ hl).mp ((Real.le_arcsin_iff_sin_le' hθ_mem).mp hθ₂))]",
"before_state": "d l : ℝ\n⊢ ∫ (θ : ℝ) in 0 ..arcsin (d / l), min d (sin θ * l) = (1 - √(1 - (d / l) ^ 2)) * l",
"after_state": "d l : ℝ\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => sin x * l) (Set.uIcc 0 (arcsin (d / l)))\n⊢ ∫ (θ : ℝ) in 0 ..arcsin (d / l), min d (sin θ * l) = (1 - √(1 - (d / l) ^ 2)) * l"
},
{
"line": "focus\n refine\n no_implicit_lambda%\n (have : Set.EqOn (fun θ => min d (θ.sin * l)) (Real.sin · * l) (Set.uIcc 0 (d / l).arcsin) := ?body✝;\n ?_)\n case body✝ =>\n with_annotate_state\"by\"\n ( intro θ ⟨hθ₁, hθ₂⟩\n have : 0 ≤ (d / l).arcsin := Real.arcsin_nonneg.mpr (div_nonneg hd.le hl.le)\n simp only [min_eq_left this] at hθ₁ hθ₂\n simp only [max_eq_right this] at hθ₁ hθ₂\n have hθ_mem : θ ∈ Set.Ioc (-(π / 2)) (π / 2) := by\n exact\n ⟨lt_of_lt_of_le (neg_lt_zero.mpr (div_pos Real.pi_pos two_pos)) hθ₁,\n le_trans hθ₂ (d / l).arcsin_mem_Icc.right⟩\n simp_rw [min_eq_right ((le_div_iff₀ hl).mp ((Real.le_arcsin_iff_sin_le' hθ_mem).mp hθ₂))])",
"before_state": "d l : ℝ\n⊢ ∫ (θ : ℝ) in 0 ..arcsin (d / l), min d (sin θ * l) = (1 - √(1 - (d / l) ^ 2)) * l",
"after_state": "d l : ℝ\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => sin x * l) (Set.uIcc 0 (arcsin (d / l)))\n⊢ ∫ (θ : ℝ) in 0 ..arcsin (d / l), min d (sin θ * l) = (1 - √(1 - (d / l) ^ 2)) * l"
},
{
"line": "refine\n no_implicit_lambda%\n (have : Set.EqOn (fun θ => min d (θ.sin * l)) (Real.sin · * l) (Set.uIcc 0 (d / l).arcsin) := ?body✝;\n ?_)",
"before_state": "d l : ℝ\n⊢ ∫ (θ : ℝ) in 0 ..arcsin (d / l), min d (sin θ * l) = (1 - √(1 - (d / l) ^ 2)) * l",
"after_state": "case body\nd l : ℝ\n⊢ Set.EqOn (fun θ => min d (sin θ * l)) (fun x => sin x * l) (Set.uIcc 0 (arcsin (d / l)))\n---\nd l : ℝ\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => sin x * l) (Set.uIcc 0 (arcsin (d / l)))\n⊢ ∫ (θ : ℝ) in 0 ..arcsin (d / l), min d (sin θ * l) = (1 - √(1 - (d / l) ^ 2)) * l"
},
{
"line": "case body✝ =>\n with_annotate_state\"by\"\n ( intro θ ⟨hθ₁, hθ₂⟩\n have : 0 ≤ (d / l).arcsin := Real.arcsin_nonneg.mpr (div_nonneg hd.le hl.le)\n simp only [min_eq_left this] at hθ₁ hθ₂\n simp only [max_eq_right this] at hθ₁ hθ₂\n have hθ_mem : θ ∈ Set.Ioc (-(π / 2)) (π / 2) := by\n exact\n ⟨lt_of_lt_of_le (neg_lt_zero.mpr (div_pos Real.pi_pos two_pos)) hθ₁,\n le_trans hθ₂ (d / l).arcsin_mem_Icc.right⟩\n simp_rw [min_eq_right ((le_div_iff₀ hl).mp ((Real.le_arcsin_iff_sin_le' hθ_mem).mp hθ₂))])",
"before_state": "case body\nd l : ℝ\n⊢ Set.EqOn (fun θ => min d (sin θ * l)) (fun x => sin x * l) (Set.uIcc 0 (arcsin (d / l)))\n---\nd l : ℝ\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => sin x * l) (Set.uIcc 0 (arcsin (d / l)))\n⊢ ∫ (θ : ℝ) in 0 ..arcsin (d / l), min d (sin θ * l) = (1 - √(1 - (d / l) ^ 2)) * l",
"after_state": "d l : ℝ\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => sin x * l) (Set.uIcc 0 (arcsin (d / l)))\n⊢ ∫ (θ : ℝ) in 0 ..arcsin (d / l), min d (sin θ * l) = (1 - √(1 - (d / l) ^ 2)) * l"
},
{
"line": "with_annotate_state\"by\"\n ( intro θ ⟨hθ₁, hθ₂⟩\n have : 0 ≤ (d / l).arcsin := Real.arcsin_nonneg.mpr (div_nonneg hd.le hl.le)\n simp only [min_eq_left this] at hθ₁ hθ₂\n simp only [max_eq_right this] at hθ₁ hθ₂\n have hθ_mem : θ ∈ Set.Ioc (-(π / 2)) (π / 2) := by\n exact\n ⟨lt_of_lt_of_le (neg_lt_zero.mpr (div_pos Real.pi_pos two_pos)) hθ₁, le_trans hθ₂ (d / l).arcsin_mem_Icc.right⟩\n simp_rw [min_eq_right ((le_div_iff₀ hl).mp ((Real.le_arcsin_iff_sin_le' hθ_mem).mp hθ₂))])",
"before_state": "d l : ℝ\n⊢ Set.EqOn (fun θ => min d (sin θ * l)) (fun x => sin x * l) (Set.uIcc 0 (arcsin (d / l)))",
"after_state": "d l θ : ℝ\nthis : 0 ≤ arcsin (d / l)\nhθ₁ : 0 ≤ θ\nhθ₂ : θ ≤ arcsin (d / l)\nhθ_mem : θ ∈ Set.Ioc (-(π / 2)) (π / 2)\n⊢ min d (sin θ * l) = sin θ * l"
},
{
"line": "intro θ ⟨hθ₁, hθ₂⟩",
"before_state": "d l : ℝ\n⊢ Set.EqOn (fun θ => min d (sin θ * l)) (fun x => sin x * l) (Set.uIcc 0 (arcsin (d / l)))",
"after_state": "d l θ : ℝ\nhθ₁ : min 0 (arcsin (d / l)) ≤ θ\nhθ₂ : θ ≤ max 0 (arcsin (d / l))\n⊢ (fun θ => min d (sin θ * l)) θ = (fun x => sin x * l) θ"
},
{
"line": "intro θ;\n intro ⟨hθ₁, hθ₂⟩",
"before_state": "d l : ℝ\n⊢ Set.EqOn (fun θ => min d (sin θ * l)) (fun x => sin x * l) (Set.uIcc 0 (arcsin (d / l)))",
"after_state": "d l θ : ℝ\nhθ₁ : min 0 (arcsin (d / l)) ≤ θ\nhθ₂ : θ ≤ max 0 (arcsin (d / l))\n⊢ (fun θ => min d (sin θ * l)) θ = (fun x => sin x * l) θ"
},
{
"line": "intro θ",
"before_state": "d l : ℝ\n⊢ Set.EqOn (fun θ => min d (sin θ * l)) (fun x => sin x * l) (Set.uIcc 0 (arcsin (d / l)))",
"after_state": "d l θ : ℝ\n⊢ θ ∈ Set.uIcc 0 (arcsin (d / l)) → (fun θ => min d (sin θ * l)) θ = (fun x => sin x * l) θ"
},
{
"line": "intro ⟨hθ₁, hθ₂⟩",
"before_state": "d l θ : ℝ\n⊢ θ ∈ Set.uIcc 0 (arcsin (d / l)) → (fun θ => min d (sin θ * l)) θ = (fun x => sin x * l) θ",
"after_state": "d l θ : ℝ\nhθ₁ : min 0 (arcsin (d / l)) ≤ θ\nhθ₂ : θ ≤ max 0 (arcsin (d / l))\n⊢ (fun θ => min d (sin θ * l)) θ = (fun x => sin x * l) θ"
},
{
"line": "intro h✝;\n match @h✝ with\n | ⟨hθ₁, hθ₂⟩ => ?_;\n try clear h✝",
"before_state": "d l θ : ℝ\n⊢ θ ∈ Set.uIcc 0 (arcsin (d / l)) → (fun θ => min d (sin θ * l)) θ = (fun x => sin x * l) θ",
"after_state": "d l θ : ℝ\nhθ₁ : min 0 (arcsin (d / l)) ≤ θ\nhθ₂ : θ ≤ max 0 (arcsin (d / l))\n⊢ (fun θ => min d (sin θ * l)) θ = (fun x => sin x * l) θ"
},
{
"line": "intro h✝",
"before_state": "d l θ : ℝ\n⊢ θ ∈ Set.uIcc 0 (arcsin (d / l)) → (fun θ => min d (sin θ * l)) θ = (fun x => sin x * l) θ",
"after_state": "d l θ : ℝ\nh✝ : θ ∈ Set.uIcc 0 (arcsin (d / l))\n⊢ (fun θ => min d (sin θ * l)) θ = (fun x => sin x * l) θ"
},
{
"line": "match @h✝ with\n| ⟨hθ₁, hθ₂⟩ => ?_",
"before_state": "d l θ : ℝ\nh✝ : θ ∈ Set.uIcc 0 (arcsin (d / l))\n⊢ (fun θ => min d (sin θ * l)) θ = (fun x => sin x * l) θ",
"after_state": "d l θ : ℝ\nh✝ : θ ∈ Set.uIcc 0 (arcsin (d / l))\nhθ₁ : min 0 (arcsin (d / l)) ≤ θ\nhθ₂ : θ ≤ max 0 (arcsin (d / l))\n⊢ (fun θ => min d (sin θ * l)) θ = (fun x => sin x * l) θ"
},
{
"line": "refine\n no_implicit_lambda%\n (match @h✝ with\n | ⟨hθ₁, hθ₂⟩ => ?_)",
"before_state": "d l θ : ℝ\nh✝ : θ ∈ Set.uIcc 0 (arcsin (d / l))\n⊢ (fun θ => min d (sin θ * l)) θ = (fun x => sin x * l) θ",
"after_state": "d l θ : ℝ\nh✝ : θ ∈ Set.uIcc 0 (arcsin (d / l))\nhθ₁ : min 0 (arcsin (d / l)) ≤ θ\nhθ₂ : θ ≤ max 0 (arcsin (d / l))\n⊢ (fun θ => min d (sin θ * l)) θ = (fun x => sin x * l) θ"
},
{
"line": "try clear h✝",
"before_state": "d l θ : ℝ\nh✝ : θ ∈ Set.uIcc 0 (arcsin (d / l))\nhθ₁ : min 0 (arcsin (d / l)) ≤ θ\nhθ₂ : θ ≤ max 0 (arcsin (d / l))\n⊢ (fun θ => min d (sin θ * l)) θ = (fun x => sin x * l) θ",
"after_state": "d l θ : ℝ\nhθ₁ : min 0 (arcsin (d / l)) ≤ θ\nhθ₂ : θ ≤ max 0 (arcsin (d / l))\n⊢ (fun θ => min d (sin θ * l)) θ = (fun x => sin x * l) θ"
},
{
"line": "first\n| clear h✝\n| skip",
"before_state": "d l θ : ℝ\nh✝ : θ ∈ Set.uIcc 0 (arcsin (d / l))\nhθ₁ : min 0 (arcsin (d / l)) ≤ θ\nhθ₂ : θ ≤ max 0 (arcsin (d / l))\n⊢ (fun θ => min d (sin θ * l)) θ = (fun x => sin x * l) θ",
"after_state": "d l θ : ℝ\nhθ₁ : min 0 (arcsin (d / l)) ≤ θ\nhθ₂ : θ ≤ max 0 (arcsin (d / l))\n⊢ (fun θ => min d (sin θ * l)) θ = (fun x => sin x * l) θ"
},
{
"line": "clear h✝",
"before_state": "d l θ : ℝ\nh✝ : θ ∈ Set.uIcc 0 (arcsin (d / l))\nhθ₁ : min 0 (arcsin (d / l)) ≤ θ\nhθ₂ : θ ≤ max 0 (arcsin (d / l))\n⊢ (fun θ => min d (sin θ * l)) θ = (fun x => sin x * l) θ",
"after_state": "d l θ : ℝ\nhθ₁ : min 0 (arcsin (d / l)) ≤ θ\nhθ₂ : θ ≤ max 0 (arcsin (d / l))\n⊢ (fun θ => min d (sin θ * l)) θ = (fun x => sin x * l) θ"
},
{
"line": "have : 0 ≤ (d / l).arcsin := Real.arcsin_nonneg.mpr (div_nonneg hd.le hl.le)",
"before_state": "d l θ : ℝ\nhθ₁ : min 0 (arcsin (d / l)) ≤ θ\nhθ₂ : θ ≤ max 0 (arcsin (d / l))\n⊢ (fun θ => min d (sin θ * l)) θ = (fun x => sin x * l) θ",
"after_state": "d l θ : ℝ\nhθ₁ : min 0 (arcsin (d / l)) ≤ θ\nhθ₂ : θ ≤ max 0 (arcsin (d / l))\nthis : 0 ≤ arcsin (d / l)\n⊢ (fun θ => min d (sin θ * l)) θ = (fun x => sin x * l) θ"
},
{
"line": "refine_lift\n have : 0 ≤ (d / l).arcsin := Real.arcsin_nonneg.mpr (div_nonneg hd.le hl.le);\n ?_",
"before_state": "d l θ : ℝ\nhθ₁ : min 0 (arcsin (d / l)) ≤ θ\nhθ₂ : θ ≤ max 0 (arcsin (d / l))\n⊢ (fun θ => min d (sin θ * l)) θ = (fun x => sin x * l) θ",
"after_state": "d l θ : ℝ\nhθ₁ : min 0 (arcsin (d / l)) ≤ θ\nhθ₂ : θ ≤ max 0 (arcsin (d / l))\nthis : 0 ≤ arcsin (d / l)\n⊢ (fun θ => min d (sin θ * l)) θ = (fun x => sin x * l) θ"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (have : 0 ≤ (d / l).arcsin := Real.arcsin_nonneg.mpr (div_nonneg hd.le hl.le);\n ?_);\n rotate_right)",
"before_state": "d l θ : ℝ\nhθ₁ : min 0 (arcsin (d / l)) ≤ θ\nhθ₂ : θ ≤ max 0 (arcsin (d / l))\n⊢ (fun θ => min d (sin θ * l)) θ = (fun x => sin x * l) θ",
"after_state": "d l θ : ℝ\nhθ₁ : min 0 (arcsin (d / l)) ≤ θ\nhθ₂ : θ ≤ max 0 (arcsin (d / l))\nthis : 0 ≤ arcsin (d / l)\n⊢ (fun θ => min d (sin θ * l)) θ = (fun x => sin x * l) θ"
},
{
"line": "refine\n no_implicit_lambda%\n (have : 0 ≤ (d / l).arcsin := Real.arcsin_nonneg.mpr (div_nonneg hd.le hl.le);\n ?_)",
"before_state": "d l θ : ℝ\nhθ₁ : min 0 (arcsin (d / l)) ≤ θ\nhθ₂ : θ ≤ max 0 (arcsin (d / l))\n⊢ (fun θ => min d (sin θ * l)) θ = (fun x => sin x * l) θ",
"after_state": "d l θ : ℝ\nhθ₁ : min 0 (arcsin (d / l)) ≤ θ\nhθ₂ : θ ≤ max 0 (arcsin (d / l))\nthis : 0 ≤ arcsin (d / l)\n⊢ (fun θ => min d (sin θ * l)) θ = (fun x => sin x * l) θ"
},
{
"line": "rotate_right",
"before_state": "d l θ : ℝ\nhθ₁ : min 0 (arcsin (d / l)) ≤ θ\nhθ₂ : θ ≤ max 0 (arcsin (d / l))\nthis : 0 ≤ arcsin (d / l)\n⊢ (fun θ => min d (sin θ * l)) θ = (fun x => sin x * l) θ",
"after_state": "d l θ : ℝ\nhθ₁ : min 0 (arcsin (d / l)) ≤ θ\nhθ₂ : θ ≤ max 0 (arcsin (d / l))\nthis : 0 ≤ arcsin (d / l)\n⊢ (fun θ => min d (sin θ * l)) θ = (fun x => sin x * l) θ"
},
{
"line": "simp only [min_eq_left this] at hθ₁ hθ₂",
"before_state": "d l θ : ℝ\nhθ₁ : min 0 (arcsin (d / l)) ≤ θ\nhθ₂ : θ ≤ max 0 (arcsin (d / l))\nthis : 0 ≤ arcsin (d / l)\n⊢ (fun θ => min d (sin θ * l)) θ = (fun x => sin x * l) θ",
"after_state": "d l θ : ℝ\nhθ₂ : θ ≤ max 0 (arcsin (d / l))\nthis : 0 ≤ arcsin (d / l)\nhθ₁ : 0 ≤ θ\n⊢ (fun θ => min d (sin θ * l)) θ = (fun x => sin x * l) θ"
},
{
"line": "simp only [max_eq_right this] at hθ₁ hθ₂",
"before_state": "d l θ : ℝ\nhθ₂ : θ ≤ max 0 (arcsin (d / l))\nthis : 0 ≤ arcsin (d / l)\nhθ₁ : 0 ≤ θ\n⊢ (fun θ => min d (sin θ * l)) θ = (fun x => sin x * l) θ",
"after_state": "d l θ : ℝ\nthis : 0 ≤ arcsin (d / l)\nhθ₁ : 0 ≤ θ\nhθ₂ : θ ≤ arcsin (d / l)\n⊢ (fun θ => min d (sin θ * l)) θ = (fun x => sin x * l) θ"
},
{
"line": "have hθ_mem : θ ∈ Set.Ioc (-(π / 2)) (π / 2) := by\n exact ⟨lt_of_lt_of_le (neg_lt_zero.mpr (div_pos Real.pi_pos two_pos)) hθ₁, le_trans hθ₂ (d / l).arcsin_mem_Icc.right⟩",
"before_state": "d l θ : ℝ\nthis : 0 ≤ arcsin (d / l)\nhθ₁ : 0 ≤ θ\nhθ₂ : θ ≤ arcsin (d / l)\n⊢ (fun θ => min d (sin θ * l)) θ = (fun x => sin x * l) θ",
"after_state": "d l θ : ℝ\nthis : 0 ≤ arcsin (d / l)\nhθ₁ : 0 ≤ θ\nhθ₂ : θ ≤ arcsin (d / l)\nhθ_mem : θ ∈ Set.Ioc (-(π / 2)) (π / 2)\n⊢ (fun θ => min d (sin θ * l)) θ = (fun x => sin x * l) θ"
},
{
"line": "focus\n refine\n no_implicit_lambda%\n (have hθ_mem : θ ∈ Set.Ioc (-(π / 2)) (π / 2) := ?body✝;\n ?_)\n case body✝ =>\n with_annotate_state\"by\"\n (exact\n ⟨lt_of_lt_of_le (neg_lt_zero.mpr (div_pos Real.pi_pos two_pos)) hθ₁,\n le_trans hθ₂ (d / l).arcsin_mem_Icc.right⟩)",
"before_state": "d l θ : ℝ\nthis : 0 ≤ arcsin (d / l)\nhθ₁ : 0 ≤ θ\nhθ₂ : θ ≤ arcsin (d / l)\n⊢ (fun θ => min d (sin θ * l)) θ = (fun x => sin x * l) θ",
"after_state": "d l θ : ℝ\nthis : 0 ≤ arcsin (d / l)\nhθ₁ : 0 ≤ θ\nhθ₂ : θ ≤ arcsin (d / l)\nhθ_mem : θ ∈ Set.Ioc (-(π / 2)) (π / 2)\n⊢ (fun θ => min d (sin θ * l)) θ = (fun x => sin x * l) θ"
},
{
"line": "refine\n no_implicit_lambda%\n (have hθ_mem : θ ∈ Set.Ioc (-(π / 2)) (π / 2) := ?body✝;\n ?_)",
"before_state": "d l θ : ℝ\nthis : 0 ≤ arcsin (d / l)\nhθ₁ : 0 ≤ θ\nhθ₂ : θ ≤ arcsin (d / l)\n⊢ (fun θ => min d (sin θ * l)) θ = (fun x => sin x * l) θ",
"after_state": "case body\nd l θ : ℝ\nthis : 0 ≤ arcsin (d / l)\nhθ₁ : 0 ≤ θ\nhθ₂ : θ ≤ arcsin (d / l)\n⊢ θ ∈ Set.Ioc (-(π / 2)) (π / 2)\n---\nd l θ : ℝ\nthis : 0 ≤ arcsin (d / l)\nhθ₁ : 0 ≤ θ\nhθ₂ : θ ≤ arcsin (d / l)\nhθ_mem : θ ∈ Set.Ioc (-(π / 2)) (π / 2)\n⊢ (fun θ => min d (sin θ * l)) θ = (fun x => sin x * l) θ"
},
{
"line": "case body✝ =>\n with_annotate_state\"by\"\n (exact\n ⟨lt_of_lt_of_le (neg_lt_zero.mpr (div_pos Real.pi_pos two_pos)) hθ₁, le_trans hθ₂ (d / l).arcsin_mem_Icc.right⟩)",
"before_state": "case body\nd l θ : ℝ\nthis : 0 ≤ arcsin (d / l)\nhθ₁ : 0 ≤ θ\nhθ₂ : θ ≤ arcsin (d / l)\n⊢ θ ∈ Set.Ioc (-(π / 2)) (π / 2)\n---\nd l θ : ℝ\nthis : 0 ≤ arcsin (d / l)\nhθ₁ : 0 ≤ θ\nhθ₂ : θ ≤ arcsin (d / l)\nhθ_mem : θ ∈ Set.Ioc (-(π / 2)) (π / 2)\n⊢ (fun θ => min d (sin θ * l)) θ = (fun x => sin x * l) θ",
"after_state": "d l θ : ℝ\nthis : 0 ≤ arcsin (d / l)\nhθ₁ : 0 ≤ θ\nhθ₂ : θ ≤ arcsin (d / l)\nhθ_mem : θ ∈ Set.Ioc (-(π / 2)) (π / 2)\n⊢ (fun θ => min d (sin θ * l)) θ = (fun x => sin x * l) θ"
},
{
"line": "with_annotate_state\"by\"\n (exact\n ⟨lt_of_lt_of_le (neg_lt_zero.mpr (div_pos Real.pi_pos two_pos)) hθ₁, le_trans hθ₂ (d / l).arcsin_mem_Icc.right⟩)",
"before_state": "d l θ : ℝ\nthis : 0 ≤ arcsin (d / l)\nhθ₁ : 0 ≤ θ\nhθ₂ : θ ≤ arcsin (d / l)\n⊢ θ ∈ Set.Ioc (-(π / 2)) (π / 2)",
"after_state": "No Goals!"
},
{
"line": "exact ⟨lt_of_lt_of_le (neg_lt_zero.mpr (div_pos Real.pi_pos two_pos)) hθ₁, le_trans hθ₂ (d / l).arcsin_mem_Icc.right⟩",
"before_state": "d l θ : ℝ\nthis : 0 ≤ arcsin (d / l)\nhθ₁ : 0 ≤ θ\nhθ₂ : θ ≤ arcsin (d / l)\n⊢ θ ∈ Set.Ioc (-(π / 2)) (π / 2)",
"after_state": "No Goals!"
},
{
"line": "simp_rw [min_eq_right ((le_div_iff₀ hl).mp ((Real.le_arcsin_iff_sin_le' hθ_mem).mp hθ₂))]",
"before_state": "d l θ : ℝ\nthis : 0 ≤ arcsin (d / l)\nhθ₁ : 0 ≤ θ\nhθ₂ : θ ≤ arcsin (d / l)\nhθ_mem : θ ∈ Set.Ioc (-(π / 2)) (π / 2)\n⊢ (fun θ => min d (sin θ * l)) θ = (fun x => sin x * l) θ",
"after_state": "d l θ : ℝ\nthis : 0 ≤ arcsin (d / l)\nhθ₁ : 0 ≤ θ\nhθ₂ : θ ≤ arcsin (d / l)\nhθ_mem : θ ∈ Set.Ioc (-(π / 2)) (π / 2)\n⊢ min d (sin θ * l) = sin θ * l"
},
{
"line": "simp (failIfUnchanged✝ := false✝) only",
"before_state": "d l θ : ℝ\nthis : 0 ≤ arcsin (d / l)\nhθ₁ : 0 ≤ θ\nhθ₂ : θ ≤ arcsin (d / l)\nhθ_mem : θ ∈ Set.Ioc (-(π / 2)) (π / 2)\n⊢ (fun θ => min d (sin θ * l)) θ = (fun x => sin x * l) θ",
"after_state": "d l θ : ℝ\nthis : 0 ≤ arcsin (d / l)\nhθ₁ : 0 ≤ θ\nhθ₂ : θ ≤ arcsin (d / l)\nhθ_mem : θ ∈ Set.Ioc (-(π / 2)) (π / 2)\n⊢ min d (sin θ * l) = sin θ * l"
},
{
"line": "simp only [min_eq_right ((le_div_iff₀ hl).mp ((Real.le_arcsin_iff_sin_le' hθ_mem).mp hθ₂))]",
"before_state": "d l θ : ℝ\nthis : 0 ≤ arcsin (d / l)\nhθ₁ : 0 ≤ θ\nhθ₂ : θ ≤ arcsin (d / l)\nhθ_mem : θ ∈ Set.Ioc (-(π / 2)) (π / 2)\n⊢ min d (sin θ * l) = sin θ * l",
"after_state": "d l θ : ℝ\nthis : 0 ≤ arcsin (d / l)\nhθ₁ : 0 ≤ θ\nhθ₂ : θ ≤ arcsin (d / l)\nhθ_mem : θ ∈ Set.Ioc (-(π / 2)) (π / 2)\n⊢ min d (sin θ * l) = sin θ * l"
},
{
"line": "rw [intervalIntegral.integral_congr this]",
"before_state": "d l : ℝ\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => sin x * l) (Set.uIcc 0 (arcsin (d / l)))\n⊢ ∫ (θ : ℝ) in 0 ..arcsin (d / l), min d (sin θ * l) = (1 - √(1 - (d / l) ^ 2)) * l",
"after_state": "d l : ℝ\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => sin x * l) (Set.uIcc 0 (arcsin (d / l)))\n⊢ ∫ (x : ℝ) in 0 ..arcsin (d / l), sin x * l = (1 - √(1 - (d / l) ^ 2)) * l"
},
{
"line": "rewrite [intervalIntegral.integral_congr this]",
"before_state": "d l : ℝ\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => sin x * l) (Set.uIcc 0 (arcsin (d / l)))\n⊢ ∫ (θ : ℝ) in 0 ..arcsin (d / l), min d (sin θ * l) = (1 - √(1 - (d / l) ^ 2)) * l",
"after_state": "d l : ℝ\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => sin x * l) (Set.uIcc 0 (arcsin (d / l)))\n⊢ ∫ (x : ℝ) in 0 ..arcsin (d / l), sin x * l = (1 - √(1 - (d / l) ^ 2)) * l"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "d l : ℝ\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => sin x * l) (Set.uIcc 0 (arcsin (d / l)))\n⊢ ∫ (x : ℝ) in 0 ..arcsin (d / l), sin x * l = (1 - √(1 - (d / l) ^ 2)) * l",
"after_state": "d l : ℝ\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => sin x * l) (Set.uIcc 0 (arcsin (d / l)))\n⊢ ∫ (x : ℝ) in 0 ..arcsin (d / l), sin x * l = (1 - √(1 - (d / l) ^ 2)) * l"
},
{
"line": "try (with_reducible rfl)",
"before_state": "d l : ℝ\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => sin x * l) (Set.uIcc 0 (arcsin (d / l)))\n⊢ ∫ (x : ℝ) in 0 ..arcsin (d / l), sin x * l = (1 - √(1 - (d / l) ^ 2)) * l",
"after_state": "d l : ℝ\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => sin x * l) (Set.uIcc 0 (arcsin (d / l)))\n⊢ ∫ (x : ℝ) in 0 ..arcsin (d / l), sin x * l = (1 - √(1 - (d / l) ^ 2)) * l"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "d l : ℝ\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => sin x * l) (Set.uIcc 0 (arcsin (d / l)))\n⊢ ∫ (x : ℝ) in 0 ..arcsin (d / l), sin x * l = (1 - √(1 - (d / l) ^ 2)) * l",
"after_state": "d l : ℝ\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => sin x * l) (Set.uIcc 0 (arcsin (d / l)))\n⊢ ∫ (x : ℝ) in 0 ..arcsin (d / l), sin x * l = (1 - √(1 - (d / l) ^ 2)) * l"
},
{
"line": "with_reducible rfl",
"before_state": "d l : ℝ\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => sin x * l) (Set.uIcc 0 (arcsin (d / l)))\n⊢ ∫ (x : ℝ) in 0 ..arcsin (d / l), sin x * l = (1 - √(1 - (d / l) ^ 2)) * l",
"after_state": "d l : ℝ\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => sin x * l) (Set.uIcc 0 (arcsin (d / l)))\n⊢ ∫ (x : ℝ) in 0 ..arcsin (d / l), sin x * l = (1 - √(1 - (d / l) ^ 2)) * l"
},
{
"line": "rfl",
"before_state": "d l : ℝ\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => sin x * l) (Set.uIcc 0 (arcsin (d / l)))\n⊢ ∫ (x : ℝ) in 0 ..arcsin (d / l), sin x * l = (1 - √(1 - (d / l) ^ 2)) * l",
"after_state": "d l : ℝ\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => sin x * l) (Set.uIcc 0 (arcsin (d / l)))\n⊢ ∫ (x : ℝ) in 0 ..arcsin (d / l), sin x * l = (1 - √(1 - (d / l) ^ 2)) * l"
},
{
"line": "apply_rfl",
"before_state": "d l : ℝ\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => sin x * l) (Set.uIcc 0 (arcsin (d / l)))\n⊢ ∫ (x : ℝ) in 0 ..arcsin (d / l), sin x * l = (1 - √(1 - (d / l) ^ 2)) * l",
"after_state": "d l : ℝ\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => sin x * l) (Set.uIcc 0 (arcsin (d / l)))\n⊢ ∫ (x : ℝ) in 0 ..arcsin (d / l), sin x * l = (1 - √(1 - (d / l) ^ 2)) * l"
},
{
"line": "skip",
"before_state": "d l : ℝ\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => sin x * l) (Set.uIcc 0 (arcsin (d / l)))\n⊢ ∫ (x : ℝ) in 0 ..arcsin (d / l), sin x * l = (1 - √(1 - (d / l) ^ 2)) * l",
"after_state": "d l : ℝ\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => sin x * l) (Set.uIcc 0 (arcsin (d / l)))\n⊢ ∫ (x : ℝ) in 0 ..arcsin (d / l), sin x * l = (1 - √(1 - (d / l) ^ 2)) * l"
},
{
"line": "rw [intervalIntegral.integral_mul_const]",
"before_state": "d l : ℝ\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => sin x * l) (Set.uIcc 0 (arcsin (d / l)))\n⊢ ∫ (x : ℝ) in 0 ..arcsin (d / l), sin x * l = (1 - √(1 - (d / l) ^ 2)) * l",
"after_state": "d l : ℝ\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => sin x * l) (Set.uIcc 0 (arcsin (d / l)))\n⊢ (∫ (x : ℝ) in 0 ..arcsin (d / l), sin x) * l = (1 - √(1 - (d / l) ^ 2)) * l"
},
{
"line": "rewrite [intervalIntegral.integral_mul_const]",
"before_state": "d l : ℝ\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => sin x * l) (Set.uIcc 0 (arcsin (d / l)))\n⊢ ∫ (x : ℝ) in 0 ..arcsin (d / l), sin x * l = (1 - √(1 - (d / l) ^ 2)) * l",
"after_state": "d l : ℝ\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => sin x * l) (Set.uIcc 0 (arcsin (d / l)))\n⊢ (∫ (x : ℝ) in 0 ..arcsin (d / l), sin x) * l = (1 - √(1 - (d / l) ^ 2)) * l"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "d l : ℝ\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => sin x * l) (Set.uIcc 0 (arcsin (d / l)))\n⊢ (∫ (x : ℝ) in 0 ..arcsin (d / l), sin x) * l = (1 - √(1 - (d / l) ^ 2)) * l",
"after_state": "d l : ℝ\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => sin x * l) (Set.uIcc 0 (arcsin (d / l)))\n⊢ (∫ (x : ℝ) in 0 ..arcsin (d / l), sin x) * l = (1 - √(1 - (d / l) ^ 2)) * l"
},
{
"line": "try (with_reducible rfl)",
"before_state": "d l : ℝ\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => sin x * l) (Set.uIcc 0 (arcsin (d / l)))\n⊢ (∫ (x : ℝ) in 0 ..arcsin (d / l), sin x) * l = (1 - √(1 - (d / l) ^ 2)) * l",
"after_state": "d l : ℝ\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => sin x * l) (Set.uIcc 0 (arcsin (d / l)))\n⊢ (∫ (x : ℝ) in 0 ..arcsin (d / l), sin x) * l = (1 - √(1 - (d / l) ^ 2)) * l"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "d l : ℝ\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => sin x * l) (Set.uIcc 0 (arcsin (d / l)))\n⊢ (∫ (x : ℝ) in 0 ..arcsin (d / l), sin x) * l = (1 - √(1 - (d / l) ^ 2)) * l",
"after_state": "d l : ℝ\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => sin x * l) (Set.uIcc 0 (arcsin (d / l)))\n⊢ (∫ (x : ℝ) in 0 ..arcsin (d / l), sin x) * l = (1 - √(1 - (d / l) ^ 2)) * l"
},
{
"line": "with_reducible rfl",
"before_state": "d l : ℝ\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => sin x * l) (Set.uIcc 0 (arcsin (d / l)))\n⊢ (∫ (x : ℝ) in 0 ..arcsin (d / l), sin x) * l = (1 - √(1 - (d / l) ^ 2)) * l",
"after_state": "d l : ℝ\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => sin x * l) (Set.uIcc 0 (arcsin (d / l)))\n⊢ (∫ (x : ℝ) in 0 ..arcsin (d / l), sin x) * l = (1 - √(1 - (d / l) ^ 2)) * l"
},
{
"line": "rfl",
"before_state": "d l : ℝ\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => sin x * l) (Set.uIcc 0 (arcsin (d / l)))\n⊢ (∫ (x : ℝ) in 0 ..arcsin (d / l), sin x) * l = (1 - √(1 - (d / l) ^ 2)) * l",
"after_state": "d l : ℝ\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => sin x * l) (Set.uIcc 0 (arcsin (d / l)))\n⊢ (∫ (x : ℝ) in 0 ..arcsin (d / l), sin x) * l = (1 - √(1 - (d / l) ^ 2)) * l"
},
{
"line": "apply_rfl",
"before_state": "d l : ℝ\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => sin x * l) (Set.uIcc 0 (arcsin (d / l)))\n⊢ (∫ (x : ℝ) in 0 ..arcsin (d / l), sin x) * l = (1 - √(1 - (d / l) ^ 2)) * l",
"after_state": "d l : ℝ\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => sin x * l) (Set.uIcc 0 (arcsin (d / l)))\n⊢ (∫ (x : ℝ) in 0 ..arcsin (d / l), sin x) * l = (1 - √(1 - (d / l) ^ 2)) * l"
},
{
"line": "skip",
"before_state": "d l : ℝ\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => sin x * l) (Set.uIcc 0 (arcsin (d / l)))\n⊢ (∫ (x : ℝ) in 0 ..arcsin (d / l), sin x) * l = (1 - √(1 - (d / l) ^ 2)) * l",
"after_state": "d l : ℝ\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => sin x * l) (Set.uIcc 0 (arcsin (d / l)))\n⊢ (∫ (x : ℝ) in 0 ..arcsin (d / l), sin x) * l = (1 - √(1 - (d / l) ^ 2)) * l"
},
{
"line": "rw [integral_sin]",
"before_state": "d l : ℝ\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => sin x * l) (Set.uIcc 0 (arcsin (d / l)))\n⊢ (∫ (x : ℝ) in 0 ..arcsin (d / l), sin x) * l = (1 - √(1 - (d / l) ^ 2)) * l",
"after_state": "d l : ℝ\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => sin x * l) (Set.uIcc 0 (arcsin (d / l)))\n⊢ (cos 0 - cos (arcsin (d / l))) * l = (1 - √(1 - (d / l) ^ 2)) * l"
},
{
"line": "rewrite [integral_sin]",
"before_state": "d l : ℝ\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => sin x * l) (Set.uIcc 0 (arcsin (d / l)))\n⊢ (∫ (x : ℝ) in 0 ..arcsin (d / l), sin x) * l = (1 - √(1 - (d / l) ^ 2)) * l",
"after_state": "d l : ℝ\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => sin x * l) (Set.uIcc 0 (arcsin (d / l)))\n⊢ (cos 0 - cos (arcsin (d / l))) * l = (1 - √(1 - (d / l) ^ 2)) * l"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "d l : ℝ\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => sin x * l) (Set.uIcc 0 (arcsin (d / l)))\n⊢ (cos 0 - cos (arcsin (d / l))) * l = (1 - √(1 - (d / l) ^ 2)) * l",
"after_state": "d l : ℝ\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => sin x * l) (Set.uIcc 0 (arcsin (d / l)))\n⊢ (cos 0 - cos (arcsin (d / l))) * l = (1 - √(1 - (d / l) ^ 2)) * l"
},
{
"line": "try (with_reducible rfl)",
"before_state": "d l : ℝ\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => sin x * l) (Set.uIcc 0 (arcsin (d / l)))\n⊢ (cos 0 - cos (arcsin (d / l))) * l = (1 - √(1 - (d / l) ^ 2)) * l",
"after_state": "d l : ℝ\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => sin x * l) (Set.uIcc 0 (arcsin (d / l)))\n⊢ (cos 0 - cos (arcsin (d / l))) * l = (1 - √(1 - (d / l) ^ 2)) * l"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "d l : ℝ\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => sin x * l) (Set.uIcc 0 (arcsin (d / l)))\n⊢ (cos 0 - cos (arcsin (d / l))) * l = (1 - √(1 - (d / l) ^ 2)) * l",
"after_state": "d l : ℝ\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => sin x * l) (Set.uIcc 0 (arcsin (d / l)))\n⊢ (cos 0 - cos (arcsin (d / l))) * l = (1 - √(1 - (d / l) ^ 2)) * l"
},
{
"line": "with_reducible rfl",
"before_state": "d l : ℝ\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => sin x * l) (Set.uIcc 0 (arcsin (d / l)))\n⊢ (cos 0 - cos (arcsin (d / l))) * l = (1 - √(1 - (d / l) ^ 2)) * l",
"after_state": "d l : ℝ\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => sin x * l) (Set.uIcc 0 (arcsin (d / l)))\n⊢ (cos 0 - cos (arcsin (d / l))) * l = (1 - √(1 - (d / l) ^ 2)) * l"
},
{
"line": "rfl",
"before_state": "d l : ℝ\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => sin x * l) (Set.uIcc 0 (arcsin (d / l)))\n⊢ (cos 0 - cos (arcsin (d / l))) * l = (1 - √(1 - (d / l) ^ 2)) * l",
"after_state": "d l : ℝ\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => sin x * l) (Set.uIcc 0 (arcsin (d / l)))\n⊢ (cos 0 - cos (arcsin (d / l))) * l = (1 - √(1 - (d / l) ^ 2)) * l"
},
{
"line": "apply_rfl",
"before_state": "d l : ℝ\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => sin x * l) (Set.uIcc 0 (arcsin (d / l)))\n⊢ (cos 0 - cos (arcsin (d / l))) * l = (1 - √(1 - (d / l) ^ 2)) * l",
"after_state": "d l : ℝ\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => sin x * l) (Set.uIcc 0 (arcsin (d / l)))\n⊢ (cos 0 - cos (arcsin (d / l))) * l = (1 - √(1 - (d / l) ^ 2)) * l"
},
{
"line": "skip",
"before_state": "d l : ℝ\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => sin x * l) (Set.uIcc 0 (arcsin (d / l)))\n⊢ (cos 0 - cos (arcsin (d / l))) * l = (1 - √(1 - (d / l) ^ 2)) * l",
"after_state": "d l : ℝ\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => sin x * l) (Set.uIcc 0 (arcsin (d / l)))\n⊢ (cos 0 - cos (arcsin (d / l))) * l = (1 - √(1 - (d / l) ^ 2)) * l"
},
{
"line": "rw [Real.cos_zero]",
"before_state": "d l : ℝ\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => sin x * l) (Set.uIcc 0 (arcsin (d / l)))\n⊢ (cos 0 - cos (arcsin (d / l))) * l = (1 - √(1 - (d / l) ^ 2)) * l",
"after_state": "d l : ℝ\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => sin x * l) (Set.uIcc 0 (arcsin (d / l)))\n⊢ (1 - cos (arcsin (d / l))) * l = (1 - √(1 - (d / l) ^ 2)) * l"
},
{
"line": "rewrite [Real.cos_zero]",
"before_state": "d l : ℝ\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => sin x * l) (Set.uIcc 0 (arcsin (d / l)))\n⊢ (cos 0 - cos (arcsin (d / l))) * l = (1 - √(1 - (d / l) ^ 2)) * l",
"after_state": "d l : ℝ\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => sin x * l) (Set.uIcc 0 (arcsin (d / l)))\n⊢ (1 - cos (arcsin (d / l))) * l = (1 - √(1 - (d / l) ^ 2)) * l"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "d l : ℝ\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => sin x * l) (Set.uIcc 0 (arcsin (d / l)))\n⊢ (1 - cos (arcsin (d / l))) * l = (1 - √(1 - (d / l) ^ 2)) * l",
"after_state": "d l : ℝ\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => sin x * l) (Set.uIcc 0 (arcsin (d / l)))\n⊢ (1 - cos (arcsin (d / l))) * l = (1 - √(1 - (d / l) ^ 2)) * l"
},
{
"line": "try (with_reducible rfl)",
"before_state": "d l : ℝ\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => sin x * l) (Set.uIcc 0 (arcsin (d / l)))\n⊢ (1 - cos (arcsin (d / l))) * l = (1 - √(1 - (d / l) ^ 2)) * l",
"after_state": "d l : ℝ\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => sin x * l) (Set.uIcc 0 (arcsin (d / l)))\n⊢ (1 - cos (arcsin (d / l))) * l = (1 - √(1 - (d / l) ^ 2)) * l"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "d l : ℝ\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => sin x * l) (Set.uIcc 0 (arcsin (d / l)))\n⊢ (1 - cos (arcsin (d / l))) * l = (1 - √(1 - (d / l) ^ 2)) * l",
"after_state": "d l : ℝ\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => sin x * l) (Set.uIcc 0 (arcsin (d / l)))\n⊢ (1 - cos (arcsin (d / l))) * l = (1 - √(1 - (d / l) ^ 2)) * l"
},
{
"line": "with_reducible rfl",
"before_state": "d l : ℝ\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => sin x * l) (Set.uIcc 0 (arcsin (d / l)))\n⊢ (1 - cos (arcsin (d / l))) * l = (1 - √(1 - (d / l) ^ 2)) * l",
"after_state": "d l : ℝ\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => sin x * l) (Set.uIcc 0 (arcsin (d / l)))\n⊢ (1 - cos (arcsin (d / l))) * l = (1 - √(1 - (d / l) ^ 2)) * l"
},
{
"line": "rfl",
"before_state": "d l : ℝ\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => sin x * l) (Set.uIcc 0 (arcsin (d / l)))\n⊢ (1 - cos (arcsin (d / l))) * l = (1 - √(1 - (d / l) ^ 2)) * l",
"after_state": "d l : ℝ\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => sin x * l) (Set.uIcc 0 (arcsin (d / l)))\n⊢ (1 - cos (arcsin (d / l))) * l = (1 - √(1 - (d / l) ^ 2)) * l"
},
{
"line": "apply_rfl",
"before_state": "d l : ℝ\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => sin x * l) (Set.uIcc 0 (arcsin (d / l)))\n⊢ (1 - cos (arcsin (d / l))) * l = (1 - √(1 - (d / l) ^ 2)) * l",
"after_state": "d l : ℝ\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => sin x * l) (Set.uIcc 0 (arcsin (d / l)))\n⊢ (1 - cos (arcsin (d / l))) * l = (1 - √(1 - (d / l) ^ 2)) * l"
},
{
"line": "skip",
"before_state": "d l : ℝ\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => sin x * l) (Set.uIcc 0 (arcsin (d / l)))\n⊢ (1 - cos (arcsin (d / l))) * l = (1 - √(1 - (d / l) ^ 2)) * l",
"after_state": "d l : ℝ\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => sin x * l) (Set.uIcc 0 (arcsin (d / l)))\n⊢ (1 - cos (arcsin (d / l))) * l = (1 - √(1 - (d / l) ^ 2)) * l"
},
{
"line": "rw [Real.cos_arcsin]",
"before_state": "d l : ℝ\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => sin x * l) (Set.uIcc 0 (arcsin (d / l)))\n⊢ (1 - cos (arcsin (d / l))) * l = (1 - √(1 - (d / l) ^ 2)) * l",
"after_state": "No Goals!"
},
{
"line": "rewrite [Real.cos_arcsin]",
"before_state": "d l : ℝ\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => sin x * l) (Set.uIcc 0 (arcsin (d / l)))\n⊢ (1 - cos (arcsin (d / l))) * l = (1 - √(1 - (d / l) ^ 2)) * l",
"after_state": "d l : ℝ\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => sin x * l) (Set.uIcc 0 (arcsin (d / l)))\n⊢ (1 - √(1 - (d / l) ^ 2)) * l = (1 - √(1 - (d / l) ^ 2)) * l"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "d l : ℝ\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => sin x * l) (Set.uIcc 0 (arcsin (d / l)))\n⊢ (1 - √(1 - (d / l) ^ 2)) * l = (1 - √(1 - (d / l) ^ 2)) * l",
"after_state": "No Goals!"
},
{
"line": "try (with_reducible rfl)",
"before_state": "d l : ℝ\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => sin x * l) (Set.uIcc 0 (arcsin (d / l)))\n⊢ (1 - √(1 - (d / l) ^ 2)) * l = (1 - √(1 - (d / l) ^ 2)) * l",
"after_state": "No Goals!"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "d l : ℝ\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => sin x * l) (Set.uIcc 0 (arcsin (d / l)))\n⊢ (1 - √(1 - (d / l) ^ 2)) * l = (1 - √(1 - (d / l) ^ 2)) * l",
"after_state": "No Goals!"
},
{
"line": "with_reducible rfl",
"before_state": "d l : ℝ\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => sin x * l) (Set.uIcc 0 (arcsin (d / l)))\n⊢ (1 - √(1 - (d / l) ^ 2)) * l = (1 - √(1 - (d / l) ^ 2)) * l",
"after_state": "No Goals!"
},
{
"line": "rfl",
"before_state": "d l : ℝ\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => sin x * l) (Set.uIcc 0 (arcsin (d / l)))\n⊢ (1 - √(1 - (d / l) ^ 2)) * l = (1 - √(1 - (d / l) ^ 2)) * l",
"after_state": "No Goals!"
},
{
"line": "eq_refl",
"before_state": "d l : ℝ\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => sin x * l) (Set.uIcc 0 (arcsin (d / l)))\n⊢ (1 - √(1 - (d / l) ^ 2)) * l = (1 - √(1 - (d / l) ^ 2)) * l",
"after_state": "No Goals!"
}
] |
lemma integral_arcsin_to_pi_div_two_min (h : d ≤ l) :
∫ θ in (d / l).arcsin..(π / 2), min d (θ.sin * l) = (π / 2 - (d / l).arcsin) * d := by
have : Set.EqOn (fun θ => min d (θ.sin * l)) (fun _ => d) (Set.uIcc (d / l).arcsin (π / 2)) := by
intro θ ⟨hθ₁, hθ₂⟩
wlog hθ_ne_pi_div_two : θ ≠ π / 2
· simp only [ne_eq, not_not] at hθ_ne_pi_div_two
simp only [hθ_ne_pi_div_two]
simp only [Real.sin_pi_div_two]
simp only [one_mul]
simp only [min_eq_left h]
simp only [min_eq_left (d / l).arcsin_le_pi_div_two] at hθ₁ hθ₂
simp only [max_eq_right (d / l).arcsin_le_pi_div_two] at hθ₁ hθ₂
have hθ_mem : θ ∈ Set.Ico (-(π / 2)) (π / 2) := by
exact ⟨le_trans (Real.arcsin_mem_Icc (d / l)).left hθ₁, lt_of_le_of_ne hθ₂ hθ_ne_pi_div_two⟩
simp_rw [min_eq_left ((div_le_iff₀ hl).mp ((Real.arcsin_le_iff_le_sin' hθ_mem).mp hθ₁))]
rw [intervalIntegral.integral_congr this]
rw [intervalIntegral.integral_const]
rw [smul_eq_mul]
|
/root/DuelModelResearch/mathlib4/Archive/Wiedijk100Theorems/BuffonsNeedle.lean
|
{
"open": [
"MeasureTheory (MeasureSpace IsProbabilityMeasure Measure pdf.IsUniform)",
"ProbabilityTheory Real"
],
"variables": [
""
]
}
|
[
{
"line": "have : Set.EqOn (fun θ => min d (θ.sin * l)) (fun _ => d) (Set.uIcc (d / l).arcsin (π / 2)) :=\n by\n intro θ ⟨hθ₁, hθ₂⟩\n wlog hθ_ne_pi_div_two : θ ≠ π / 2\n · simp only [ne_eq, not_not] at hθ_ne_pi_div_two\n simp only [hθ_ne_pi_div_two]\n simp only [Real.sin_pi_div_two]\n simp only [one_mul]\n simp only [min_eq_left h]\n simp only [min_eq_left (d / l).arcsin_le_pi_div_two] at hθ₁ hθ₂\n simp only [max_eq_right (d / l).arcsin_le_pi_div_two] at hθ₁ hθ₂\n have hθ_mem : θ ∈ Set.Ico (-(π / 2)) (π / 2) := by\n exact ⟨le_trans (Real.arcsin_mem_Icc (d / l)).left hθ₁, lt_of_le_of_ne hθ₂ hθ_ne_pi_div_two⟩\n simp_rw [min_eq_left ((div_le_iff₀ hl).mp ((Real.arcsin_le_iff_le_sin' hθ_mem).mp hθ₁))]",
"before_state": "d l : ℝ\nh : d ≤ l\n⊢ ∫ (θ : ℝ) in arcsin (d / l)..π / 2, min d (sin θ * l) = (π / 2 - arcsin (d / l)) * d",
"after_state": "d l : ℝ\nh : d ≤ l\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => d) (Set.uIcc (arcsin (d / l)) (π / 2))\n⊢ ∫ (θ : ℝ) in arcsin (d / l)..π / 2, min d (sin θ * l) = (π / 2 - arcsin (d / l)) * d"
},
{
"line": "focus\n refine\n no_implicit_lambda%\n (have : Set.EqOn (fun θ => min d (θ.sin * l)) (fun _ => d) (Set.uIcc (d / l).arcsin (π / 2)) := ?body✝;\n ?_)\n case body✝ =>\n with_annotate_state\"by\"\n ( intro θ ⟨hθ₁, hθ₂⟩\n wlog hθ_ne_pi_div_two : θ ≠ π / 2\n · simp only [ne_eq, not_not] at hθ_ne_pi_div_two\n simp only [hθ_ne_pi_div_two]\n simp only [Real.sin_pi_div_two]\n simp only [one_mul]\n simp only [min_eq_left h]\n simp only [min_eq_left (d / l).arcsin_le_pi_div_two] at hθ₁ hθ₂\n simp only [max_eq_right (d / l).arcsin_le_pi_div_two] at hθ₁ hθ₂\n have hθ_mem : θ ∈ Set.Ico (-(π / 2)) (π / 2) := by\n exact ⟨le_trans (Real.arcsin_mem_Icc (d / l)).left hθ₁, lt_of_le_of_ne hθ₂ hθ_ne_pi_div_two⟩\n simp_rw [min_eq_left ((div_le_iff₀ hl).mp ((Real.arcsin_le_iff_le_sin' hθ_mem).mp hθ₁))])",
"before_state": "d l : ℝ\nh : d ≤ l\n⊢ ∫ (θ : ℝ) in arcsin (d / l)..π / 2, min d (sin θ * l) = (π / 2 - arcsin (d / l)) * d",
"after_state": "d l : ℝ\nh : d ≤ l\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => d) (Set.uIcc (arcsin (d / l)) (π / 2))\n⊢ ∫ (θ : ℝ) in arcsin (d / l)..π / 2, min d (sin θ * l) = (π / 2 - arcsin (d / l)) * d"
},
{
"line": "refine\n no_implicit_lambda%\n (have : Set.EqOn (fun θ => min d (θ.sin * l)) (fun _ => d) (Set.uIcc (d / l).arcsin (π / 2)) := ?body✝;\n ?_)",
"before_state": "d l : ℝ\nh : d ≤ l\n⊢ ∫ (θ : ℝ) in arcsin (d / l)..π / 2, min d (sin θ * l) = (π / 2 - arcsin (d / l)) * d",
"after_state": "case body\nd l : ℝ\nh : d ≤ l\n⊢ Set.EqOn (fun θ => min d (sin θ * l)) (fun x => d) (Set.uIcc (arcsin (d / l)) (π / 2))\n---\nd l : ℝ\nh : d ≤ l\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => d) (Set.uIcc (arcsin (d / l)) (π / 2))\n⊢ ∫ (θ : ℝ) in arcsin (d / l)..π / 2, min d (sin θ * l) = (π / 2 - arcsin (d / l)) * d"
},
{
"line": "case body✝ =>\n with_annotate_state\"by\"\n ( intro θ ⟨hθ₁, hθ₂⟩\n wlog hθ_ne_pi_div_two : θ ≠ π / 2\n · simp only [ne_eq, not_not] at hθ_ne_pi_div_two\n simp only [hθ_ne_pi_div_two]\n simp only [Real.sin_pi_div_two]\n simp only [one_mul]\n simp only [min_eq_left h]\n simp only [min_eq_left (d / l).arcsin_le_pi_div_two] at hθ₁ hθ₂\n simp only [max_eq_right (d / l).arcsin_le_pi_div_two] at hθ₁ hθ₂\n have hθ_mem : θ ∈ Set.Ico (-(π / 2)) (π / 2) := by\n exact ⟨le_trans (Real.arcsin_mem_Icc (d / l)).left hθ₁, lt_of_le_of_ne hθ₂ hθ_ne_pi_div_two⟩\n simp_rw [min_eq_left ((div_le_iff₀ hl).mp ((Real.arcsin_le_iff_le_sin' hθ_mem).mp hθ₁))])",
"before_state": "case body\nd l : ℝ\nh : d ≤ l\n⊢ Set.EqOn (fun θ => min d (sin θ * l)) (fun x => d) (Set.uIcc (arcsin (d / l)) (π / 2))\n---\nd l : ℝ\nh : d ≤ l\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => d) (Set.uIcc (arcsin (d / l)) (π / 2))\n⊢ ∫ (θ : ℝ) in arcsin (d / l)..π / 2, min d (sin θ * l) = (π / 2 - arcsin (d / l)) * d",
"after_state": "d l : ℝ\nh : d ≤ l\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => d) (Set.uIcc (arcsin (d / l)) (π / 2))\n⊢ ∫ (θ : ℝ) in arcsin (d / l)..π / 2, min d (sin θ * l) = (π / 2 - arcsin (d / l)) * d"
},
{
"line": "with_annotate_state\"by\"\n ( intro θ ⟨hθ₁, hθ₂⟩\n wlog hθ_ne_pi_div_two : θ ≠ π / 2\n · simp only [ne_eq, not_not] at hθ_ne_pi_div_two\n simp only [hθ_ne_pi_div_two]\n simp only [Real.sin_pi_div_two]\n simp only [one_mul]\n simp only [min_eq_left h]\n simp only [min_eq_left (d / l).arcsin_le_pi_div_two] at hθ₁ hθ₂\n simp only [max_eq_right (d / l).arcsin_le_pi_div_two] at hθ₁ hθ₂\n have hθ_mem : θ ∈ Set.Ico (-(π / 2)) (π / 2) := by\n exact ⟨le_trans (Real.arcsin_mem_Icc (d / l)).left hθ₁, lt_of_le_of_ne hθ₂ hθ_ne_pi_div_two⟩\n simp_rw [min_eq_left ((div_le_iff₀ hl).mp ((Real.arcsin_le_iff_le_sin' hθ_mem).mp hθ₁))])",
"before_state": "d l : ℝ\nh : d ≤ l\n⊢ Set.EqOn (fun θ => min d (sin θ * l)) (fun x => d) (Set.uIcc (arcsin (d / l)) (π / 2))",
"after_state": "d l : ℝ\nh : d ≤ l\nθ : ℝ\nhθ₁✝ : min (arcsin (d / l)) (π / 2) ≤ θ\nhθ₂✝ : θ ≤ max (arcsin (d / l)) (π / 2)\nhθ_ne_pi_div_two : θ ≠ π / 2\nhθ₁ : arcsin (d / l) ≤ θ\nhθ₂ : θ ≤ π / 2\nhθ_mem : θ ∈ Set.Ico (-(π / 2)) (π / 2)\n⊢ min d (sin θ * l) = d"
},
{
"line": "intro θ ⟨hθ₁, hθ₂⟩",
"before_state": "d l : ℝ\nh : d ≤ l\n⊢ Set.EqOn (fun θ => min d (sin θ * l)) (fun x => d) (Set.uIcc (arcsin (d / l)) (π / 2))",
"after_state": "d l : ℝ\nh : d ≤ l\nθ : ℝ\nhθ₁ : min (arcsin (d / l)) (π / 2) ≤ θ\nhθ₂ : θ ≤ max (arcsin (d / l)) (π / 2)\n⊢ (fun θ => min d (sin θ * l)) θ = (fun x => d) θ"
},
{
"line": "intro θ;\n intro ⟨hθ₁, hθ₂⟩",
"before_state": "d l : ℝ\nh : d ≤ l\n⊢ Set.EqOn (fun θ => min d (sin θ * l)) (fun x => d) (Set.uIcc (arcsin (d / l)) (π / 2))",
"after_state": "d l : ℝ\nh : d ≤ l\nθ : ℝ\nhθ₁ : min (arcsin (d / l)) (π / 2) ≤ θ\nhθ₂ : θ ≤ max (arcsin (d / l)) (π / 2)\n⊢ (fun θ => min d (sin θ * l)) θ = (fun x => d) θ"
},
{
"line": "intro θ",
"before_state": "d l : ℝ\nh : d ≤ l\n⊢ Set.EqOn (fun θ => min d (sin θ * l)) (fun x => d) (Set.uIcc (arcsin (d / l)) (π / 2))",
"after_state": "d l : ℝ\nh : d ≤ l\nθ : ℝ\n⊢ θ ∈ Set.uIcc (arcsin (d / l)) (π / 2) → (fun θ => min d (sin θ * l)) θ = (fun x => d) θ"
},
{
"line": "intro ⟨hθ₁, hθ₂⟩",
"before_state": "d l : ℝ\nh : d ≤ l\nθ : ℝ\n⊢ θ ∈ Set.uIcc (arcsin (d / l)) (π / 2) → (fun θ => min d (sin θ * l)) θ = (fun x => d) θ",
"after_state": "d l : ℝ\nh : d ≤ l\nθ : ℝ\nhθ₁ : min (arcsin (d / l)) (π / 2) ≤ θ\nhθ₂ : θ ≤ max (arcsin (d / l)) (π / 2)\n⊢ (fun θ => min d (sin θ * l)) θ = (fun x => d) θ"
},
{
"line": "intro h✝;\n match @h✝ with\n | ⟨hθ₁, hθ₂⟩ => ?_;\n try clear h✝",
"before_state": "d l : ℝ\nh : d ≤ l\nθ : ℝ\n⊢ θ ∈ Set.uIcc (arcsin (d / l)) (π / 2) → (fun θ => min d (sin θ * l)) θ = (fun x => d) θ",
"after_state": "d l : ℝ\nh : d ≤ l\nθ : ℝ\nhθ₁ : min (arcsin (d / l)) (π / 2) ≤ θ\nhθ₂ : θ ≤ max (arcsin (d / l)) (π / 2)\n⊢ (fun θ => min d (sin θ * l)) θ = (fun x => d) θ"
},
{
"line": "intro h✝",
"before_state": "d l : ℝ\nh : d ≤ l\nθ : ℝ\n⊢ θ ∈ Set.uIcc (arcsin (d / l)) (π / 2) → (fun θ => min d (sin θ * l)) θ = (fun x => d) θ",
"after_state": "d l : ℝ\nh : d ≤ l\nθ : ℝ\nh✝ : θ ∈ Set.uIcc (arcsin (d / l)) (π / 2)\n⊢ (fun θ => min d (sin θ * l)) θ = (fun x => d) θ"
},
{
"line": "match @h✝ with\n| ⟨hθ₁, hθ₂⟩ => ?_",
"before_state": "d l : ℝ\nh : d ≤ l\nθ : ℝ\nh✝ : θ ∈ Set.uIcc (arcsin (d / l)) (π / 2)\n⊢ (fun θ => min d (sin θ * l)) θ = (fun x => d) θ",
"after_state": "d l : ℝ\nh : d ≤ l\nθ : ℝ\nh✝ : θ ∈ Set.uIcc (arcsin (d / l)) (π / 2)\nhθ₁ : min (arcsin (d / l)) (π / 2) ≤ θ\nhθ₂ : θ ≤ max (arcsin (d / l)) (π / 2)\n⊢ (fun θ => min d (sin θ * l)) θ = (fun x => d) θ"
},
{
"line": "refine\n no_implicit_lambda%\n (match @h✝ with\n | ⟨hθ₁, hθ₂⟩ => ?_)",
"before_state": "d l : ℝ\nh : d ≤ l\nθ : ℝ\nh✝ : θ ∈ Set.uIcc (arcsin (d / l)) (π / 2)\n⊢ (fun θ => min d (sin θ * l)) θ = (fun x => d) θ",
"after_state": "d l : ℝ\nh : d ≤ l\nθ : ℝ\nh✝ : θ ∈ Set.uIcc (arcsin (d / l)) (π / 2)\nhθ₁ : min (arcsin (d / l)) (π / 2) ≤ θ\nhθ₂ : θ ≤ max (arcsin (d / l)) (π / 2)\n⊢ (fun θ => min d (sin θ * l)) θ = (fun x => d) θ"
},
{
"line": "try clear h✝",
"before_state": "d l : ℝ\nh : d ≤ l\nθ : ℝ\nh✝ : θ ∈ Set.uIcc (arcsin (d / l)) (π / 2)\nhθ₁ : min (arcsin (d / l)) (π / 2) ≤ θ\nhθ₂ : θ ≤ max (arcsin (d / l)) (π / 2)\n⊢ (fun θ => min d (sin θ * l)) θ = (fun x => d) θ",
"after_state": "d l : ℝ\nh : d ≤ l\nθ : ℝ\nhθ₁ : min (arcsin (d / l)) (π / 2) ≤ θ\nhθ₂ : θ ≤ max (arcsin (d / l)) (π / 2)\n⊢ (fun θ => min d (sin θ * l)) θ = (fun x => d) θ"
},
{
"line": "first\n| clear h✝\n| skip",
"before_state": "d l : ℝ\nh : d ≤ l\nθ : ℝ\nh✝ : θ ∈ Set.uIcc (arcsin (d / l)) (π / 2)\nhθ₁ : min (arcsin (d / l)) (π / 2) ≤ θ\nhθ₂ : θ ≤ max (arcsin (d / l)) (π / 2)\n⊢ (fun θ => min d (sin θ * l)) θ = (fun x => d) θ",
"after_state": "d l : ℝ\nh : d ≤ l\nθ : ℝ\nhθ₁ : min (arcsin (d / l)) (π / 2) ≤ θ\nhθ₂ : θ ≤ max (arcsin (d / l)) (π / 2)\n⊢ (fun θ => min d (sin θ * l)) θ = (fun x => d) θ"
},
{
"line": "clear h✝",
"before_state": "d l : ℝ\nh : d ≤ l\nθ : ℝ\nh✝ : θ ∈ Set.uIcc (arcsin (d / l)) (π / 2)\nhθ₁ : min (arcsin (d / l)) (π / 2) ≤ θ\nhθ₂ : θ ≤ max (arcsin (d / l)) (π / 2)\n⊢ (fun θ => min d (sin θ * l)) θ = (fun x => d) θ",
"after_state": "d l : ℝ\nh : d ≤ l\nθ : ℝ\nhθ₁ : min (arcsin (d / l)) (π / 2) ≤ θ\nhθ₂ : θ ≤ max (arcsin (d / l)) (π / 2)\n⊢ (fun θ => min d (sin θ * l)) θ = (fun x => d) θ"
},
{
"line": "wlog hθ_ne_pi_div_two : θ ≠ π / 2",
"before_state": "d l : ℝ\nh : d ≤ l\nθ : ℝ\nhθ₁ : min (arcsin (d / l)) (π / 2) ≤ θ\nhθ₂ : θ ≤ max (arcsin (d / l)) (π / 2)\n⊢ (fun θ => min d (sin θ * l)) θ = (fun x => d) θ",
"after_state": "case inr\nd l : ℝ\nh : d ≤ l\nθ : ℝ\nhθ₁ : min (arcsin (d / l)) (π / 2) ≤ θ\nhθ₂ : θ ≤ max (arcsin (d / l)) (π / 2)\nthis :\n ∀ {d l : ℝ} (h : d ≤ l) ⦃θ : ℝ⦄ (hθ₁ : min (arcsin (d / l)) (π / 2) ≤ θ) (hθ₂ : θ ≤ max (arcsin (d / l)) (π / 2)),\n θ ≠ π / 2 → (fun θ => min d (sin θ * l)) θ = (fun x => d) θ\nhθ_ne_pi_div_two : ¬θ ≠ π / 2\n⊢ (fun θ => min d (sin θ * l)) θ = (fun x => d) θ\n---\nd l : ℝ\nh : d ≤ l\nθ : ℝ\nhθ₁ : min (arcsin (d / l)) (π / 2) ≤ θ\nhθ₂ : θ ≤ max (arcsin (d / l)) (π / 2)\nhθ_ne_pi_div_two : θ ≠ π / 2\n⊢ (fun θ => min d (sin θ * l)) θ = (fun x => d) θ"
},
{
"line": "simp only [ne_eq, not_not] at hθ_ne_pi_div_two",
"before_state": "case inr\nd l : ℝ\nh : d ≤ l\nθ : ℝ\nhθ₁ : min (arcsin (d / l)) (π / 2) ≤ θ\nhθ₂ : θ ≤ max (arcsin (d / l)) (π / 2)\nthis :\n ∀ {d l : ℝ} (h : d ≤ l) ⦃θ : ℝ⦄ (hθ₁ : min (arcsin (d / l)) (π / 2) ≤ θ) (hθ₂ : θ ≤ max (arcsin (d / l)) (π / 2)),\n θ ≠ π / 2 → (fun θ => min d (sin θ * l)) θ = (fun x => d) θ\nhθ_ne_pi_div_two : ¬θ ≠ π / 2\n⊢ (fun θ => min d (sin θ * l)) θ = (fun x => d) θ",
"after_state": "case inr\nd l : ℝ\nh : d ≤ l\nθ : ℝ\nhθ₁ : min (arcsin (d / l)) (π / 2) ≤ θ\nhθ₂ : θ ≤ max (arcsin (d / l)) (π / 2)\nthis :\n ∀ {d l : ℝ} (h : d ≤ l) ⦃θ : ℝ⦄ (hθ₁ : min (arcsin (d / l)) (π / 2) ≤ θ) (hθ₂ : θ ≤ max (arcsin (d / l)) (π / 2)),\n θ ≠ π / 2 → (fun θ => min d (sin θ * l)) θ = (fun x => d) θ\nhθ_ne_pi_div_two : θ = π / 2\n⊢ (fun θ => min d (sin θ * l)) θ = (fun x => d) θ"
},
{
"line": "simp only [hθ_ne_pi_div_two]",
"before_state": "case inr\nd l : ℝ\nh : d ≤ l\nθ : ℝ\nhθ₁ : min (arcsin (d / l)) (π / 2) ≤ θ\nhθ₂ : θ ≤ max (arcsin (d / l)) (π / 2)\nthis :\n ∀ {d l : ℝ} (h : d ≤ l) ⦃θ : ℝ⦄ (hθ₁ : min (arcsin (d / l)) (π / 2) ≤ θ) (hθ₂ : θ ≤ max (arcsin (d / l)) (π / 2)),\n θ ≠ π / 2 → (fun θ => min d (sin θ * l)) θ = (fun x => d) θ\nhθ_ne_pi_div_two : θ = π / 2\n⊢ (fun θ => min d (sin θ * l)) θ = (fun x => d) θ",
"after_state": "case inr\nd l : ℝ\nh : d ≤ l\nθ : ℝ\nhθ₁ : min (arcsin (d / l)) (π / 2) ≤ θ\nhθ₂ : θ ≤ max (arcsin (d / l)) (π / 2)\nthis :\n ∀ {d l : ℝ} (h : d ≤ l) ⦃θ : ℝ⦄ (hθ₁ : min (arcsin (d / l)) (π / 2) ≤ θ) (hθ₂ : θ ≤ max (arcsin (d / l)) (π / 2)),\n θ ≠ π / 2 → (fun θ => min d (sin θ * l)) θ = (fun x => d) θ\nhθ_ne_pi_div_two : θ = π / 2\n⊢ min d (sin (π / 2) * l) = d"
},
{
"line": "simp only [Real.sin_pi_div_two]",
"before_state": "case inr\nd l : ℝ\nh : d ≤ l\nθ : ℝ\nhθ₁ : min (arcsin (d / l)) (π / 2) ≤ θ\nhθ₂ : θ ≤ max (arcsin (d / l)) (π / 2)\nthis :\n ∀ {d l : ℝ} (h : d ≤ l) ⦃θ : ℝ⦄ (hθ₁ : min (arcsin (d / l)) (π / 2) ≤ θ) (hθ₂ : θ ≤ max (arcsin (d / l)) (π / 2)),\n θ ≠ π / 2 → (fun θ => min d (sin θ * l)) θ = (fun x => d) θ\nhθ_ne_pi_div_two : θ = π / 2\n⊢ min d (sin (π / 2) * l) = d",
"after_state": "case inr\nd l : ℝ\nh : d ≤ l\nθ : ℝ\nhθ₁ : min (arcsin (d / l)) (π / 2) ≤ θ\nhθ₂ : θ ≤ max (arcsin (d / l)) (π / 2)\nthis :\n ∀ {d l : ℝ} (h : d ≤ l) ⦃θ : ℝ⦄ (hθ₁ : min (arcsin (d / l)) (π / 2) ≤ θ) (hθ₂ : θ ≤ max (arcsin (d / l)) (π / 2)),\n θ ≠ π / 2 → (fun θ => min d (sin θ * l)) θ = (fun x => d) θ\nhθ_ne_pi_div_two : θ = π / 2\n⊢ min d (1 * l) = d"
},
{
"line": "simp only [one_mul]",
"before_state": "case inr\nd l : ℝ\nh : d ≤ l\nθ : ℝ\nhθ₁ : min (arcsin (d / l)) (π / 2) ≤ θ\nhθ₂ : θ ≤ max (arcsin (d / l)) (π / 2)\nthis :\n ∀ {d l : ℝ} (h : d ≤ l) ⦃θ : ℝ⦄ (hθ₁ : min (arcsin (d / l)) (π / 2) ≤ θ) (hθ₂ : θ ≤ max (arcsin (d / l)) (π / 2)),\n θ ≠ π / 2 → (fun θ => min d (sin θ * l)) θ = (fun x => d) θ\nhθ_ne_pi_div_two : θ = π / 2\n⊢ min d (1 * l) = d",
"after_state": "case inr\nd l : ℝ\nh : d ≤ l\nθ : ℝ\nhθ₁ : min (arcsin (d / l)) (π / 2) ≤ θ\nhθ₂ : θ ≤ max (arcsin (d / l)) (π / 2)\nthis :\n ∀ {d l : ℝ} (h : d ≤ l) ⦃θ : ℝ⦄ (hθ₁ : min (arcsin (d / l)) (π / 2) ≤ θ) (hθ₂ : θ ≤ max (arcsin (d / l)) (π / 2)),\n θ ≠ π / 2 → (fun θ => min d (sin θ * l)) θ = (fun x => d) θ\nhθ_ne_pi_div_two : θ = π / 2\n⊢ min d l = d"
},
{
"line": "simp only [min_eq_left h]",
"before_state": "case inr\nd l : ℝ\nh : d ≤ l\nθ : ℝ\nhθ₁ : min (arcsin (d / l)) (π / 2) ≤ θ\nhθ₂ : θ ≤ max (arcsin (d / l)) (π / 2)\nthis :\n ∀ {d l : ℝ} (h : d ≤ l) ⦃θ : ℝ⦄ (hθ₁ : min (arcsin (d / l)) (π / 2) ≤ θ) (hθ₂ : θ ≤ max (arcsin (d / l)) (π / 2)),\n θ ≠ π / 2 → (fun θ => min d (sin θ * l)) θ = (fun x => d) θ\nhθ_ne_pi_div_two : θ = π / 2\n⊢ min d l = d",
"after_state": "No Goals!"
},
{
"line": "simp only [min_eq_left (d / l).arcsin_le_pi_div_two] at hθ₁ hθ₂",
"before_state": "d l : ℝ\nh : d ≤ l\nθ : ℝ\nhθ₁ : min (arcsin (d / l)) (π / 2) ≤ θ\nhθ₂ : θ ≤ max (arcsin (d / l)) (π / 2)\nhθ_ne_pi_div_two : θ ≠ π / 2\n⊢ (fun θ => min d (sin θ * l)) θ = (fun x => d) θ",
"after_state": "d l : ℝ\nh : d ≤ l\nθ : ℝ\nhθ₁✝ : min (arcsin (d / l)) (π / 2) ≤ θ\nhθ₂ : θ ≤ max (arcsin (d / l)) (π / 2)\nhθ_ne_pi_div_two : θ ≠ π / 2\nhθ₁ : arcsin (d / l) ≤ θ\n⊢ (fun θ => min d (sin θ * l)) θ = (fun x => d) θ"
},
{
"line": "simp only [max_eq_right (d / l).arcsin_le_pi_div_two] at hθ₁ hθ₂",
"before_state": "d l : ℝ\nh : d ≤ l\nθ : ℝ\nhθ₁✝ : min (arcsin (d / l)) (π / 2) ≤ θ\nhθ₂ : θ ≤ max (arcsin (d / l)) (π / 2)\nhθ_ne_pi_div_two : θ ≠ π / 2\nhθ₁ : arcsin (d / l) ≤ θ\n⊢ (fun θ => min d (sin θ * l)) θ = (fun x => d) θ",
"after_state": "d l : ℝ\nh : d ≤ l\nθ : ℝ\nhθ₁✝ : min (arcsin (d / l)) (π / 2) ≤ θ\nhθ₂✝ : θ ≤ max (arcsin (d / l)) (π / 2)\nhθ_ne_pi_div_two : θ ≠ π / 2\nhθ₁ : arcsin (d / l) ≤ θ\nhθ₂ : θ ≤ π / 2\n⊢ (fun θ => min d (sin θ * l)) θ = (fun x => d) θ"
},
{
"line": "have hθ_mem : θ ∈ Set.Ico (-(π / 2)) (π / 2) := by\n exact ⟨le_trans (Real.arcsin_mem_Icc (d / l)).left hθ₁, lt_of_le_of_ne hθ₂ hθ_ne_pi_div_two⟩",
"before_state": "d l : ℝ\nh : d ≤ l\nθ : ℝ\nhθ₁✝ : min (arcsin (d / l)) (π / 2) ≤ θ\nhθ₂✝ : θ ≤ max (arcsin (d / l)) (π / 2)\nhθ_ne_pi_div_two : θ ≠ π / 2\nhθ₁ : arcsin (d / l) ≤ θ\nhθ₂ : θ ≤ π / 2\n⊢ (fun θ => min d (sin θ * l)) θ = (fun x => d) θ",
"after_state": "d l : ℝ\nh : d ≤ l\nθ : ℝ\nhθ₁✝ : min (arcsin (d / l)) (π / 2) ≤ θ\nhθ₂✝ : θ ≤ max (arcsin (d / l)) (π / 2)\nhθ_ne_pi_div_two : θ ≠ π / 2\nhθ₁ : arcsin (d / l) ≤ θ\nhθ₂ : θ ≤ π / 2\nhθ_mem : θ ∈ Set.Ico (-(π / 2)) (π / 2)\n⊢ (fun θ => min d (sin θ * l)) θ = (fun x => d) θ"
},
{
"line": "focus\n refine\n no_implicit_lambda%\n (have hθ_mem : θ ∈ Set.Ico (-(π / 2)) (π / 2) := ?body✝;\n ?_)\n case body✝ =>\n with_annotate_state\"by\"\n (exact ⟨le_trans (Real.arcsin_mem_Icc (d / l)).left hθ₁, lt_of_le_of_ne hθ₂ hθ_ne_pi_div_two⟩)",
"before_state": "d l : ℝ\nh : d ≤ l\nθ : ℝ\nhθ₁✝ : min (arcsin (d / l)) (π / 2) ≤ θ\nhθ₂✝ : θ ≤ max (arcsin (d / l)) (π / 2)\nhθ_ne_pi_div_two : θ ≠ π / 2\nhθ₁ : arcsin (d / l) ≤ θ\nhθ₂ : θ ≤ π / 2\n⊢ (fun θ => min d (sin θ * l)) θ = (fun x => d) θ",
"after_state": "d l : ℝ\nh : d ≤ l\nθ : ℝ\nhθ₁✝ : min (arcsin (d / l)) (π / 2) ≤ θ\nhθ₂✝ : θ ≤ max (arcsin (d / l)) (π / 2)\nhθ_ne_pi_div_two : θ ≠ π / 2\nhθ₁ : arcsin (d / l) ≤ θ\nhθ₂ : θ ≤ π / 2\nhθ_mem : θ ∈ Set.Ico (-(π / 2)) (π / 2)\n⊢ (fun θ => min d (sin θ * l)) θ = (fun x => d) θ"
},
{
"line": "refine\n no_implicit_lambda%\n (have hθ_mem : θ ∈ Set.Ico (-(π / 2)) (π / 2) := ?body✝;\n ?_)",
"before_state": "d l : ℝ\nh : d ≤ l\nθ : ℝ\nhθ₁✝ : min (arcsin (d / l)) (π / 2) ≤ θ\nhθ₂✝ : θ ≤ max (arcsin (d / l)) (π / 2)\nhθ_ne_pi_div_two : θ ≠ π / 2\nhθ₁ : arcsin (d / l) ≤ θ\nhθ₂ : θ ≤ π / 2\n⊢ (fun θ => min d (sin θ * l)) θ = (fun x => d) θ",
"after_state": "case body\nd l : ℝ\nh : d ≤ l\nθ : ℝ\nhθ₁✝ : min (arcsin (d / l)) (π / 2) ≤ θ\nhθ₂✝ : θ ≤ max (arcsin (d / l)) (π / 2)\nhθ_ne_pi_div_two : θ ≠ π / 2\nhθ₁ : arcsin (d / l) ≤ θ\nhθ₂ : θ ≤ π / 2\n⊢ θ ∈ Set.Ico (-(π / 2)) (π / 2)\n---\nd l : ℝ\nh : d ≤ l\nθ : ℝ\nhθ₁✝ : min (arcsin (d / l)) (π / 2) ≤ θ\nhθ₂✝ : θ ≤ max (arcsin (d / l)) (π / 2)\nhθ_ne_pi_div_two : θ ≠ π / 2\nhθ₁ : arcsin (d / l) ≤ θ\nhθ₂ : θ ≤ π / 2\nhθ_mem : θ ∈ Set.Ico (-(π / 2)) (π / 2)\n⊢ (fun θ => min d (sin θ * l)) θ = (fun x => d) θ"
},
{
"line": "case body✝ =>\n with_annotate_state\"by\" (exact ⟨le_trans (Real.arcsin_mem_Icc (d / l)).left hθ₁, lt_of_le_of_ne hθ₂ hθ_ne_pi_div_two⟩)",
"before_state": "case body\nd l : ℝ\nh : d ≤ l\nθ : ℝ\nhθ₁✝ : min (arcsin (d / l)) (π / 2) ≤ θ\nhθ₂✝ : θ ≤ max (arcsin (d / l)) (π / 2)\nhθ_ne_pi_div_two : θ ≠ π / 2\nhθ₁ : arcsin (d / l) ≤ θ\nhθ₂ : θ ≤ π / 2\n⊢ θ ∈ Set.Ico (-(π / 2)) (π / 2)\n---\nd l : ℝ\nh : d ≤ l\nθ : ℝ\nhθ₁✝ : min (arcsin (d / l)) (π / 2) ≤ θ\nhθ₂✝ : θ ≤ max (arcsin (d / l)) (π / 2)\nhθ_ne_pi_div_two : θ ≠ π / 2\nhθ₁ : arcsin (d / l) ≤ θ\nhθ₂ : θ ≤ π / 2\nhθ_mem : θ ∈ Set.Ico (-(π / 2)) (π / 2)\n⊢ (fun θ => min d (sin θ * l)) θ = (fun x => d) θ",
"after_state": "d l : ℝ\nh : d ≤ l\nθ : ℝ\nhθ₁✝ : min (arcsin (d / l)) (π / 2) ≤ θ\nhθ₂✝ : θ ≤ max (arcsin (d / l)) (π / 2)\nhθ_ne_pi_div_two : θ ≠ π / 2\nhθ₁ : arcsin (d / l) ≤ θ\nhθ₂ : θ ≤ π / 2\nhθ_mem : θ ∈ Set.Ico (-(π / 2)) (π / 2)\n⊢ (fun θ => min d (sin θ * l)) θ = (fun x => d) θ"
},
{
"line": "with_annotate_state\"by\" (exact ⟨le_trans (Real.arcsin_mem_Icc (d / l)).left hθ₁, lt_of_le_of_ne hθ₂ hθ_ne_pi_div_two⟩)",
"before_state": "d l : ℝ\nh : d ≤ l\nθ : ℝ\nhθ₁✝ : min (arcsin (d / l)) (π / 2) ≤ θ\nhθ₂✝ : θ ≤ max (arcsin (d / l)) (π / 2)\nhθ_ne_pi_div_two : θ ≠ π / 2\nhθ₁ : arcsin (d / l) ≤ θ\nhθ₂ : θ ≤ π / 2\n⊢ θ ∈ Set.Ico (-(π / 2)) (π / 2)",
"after_state": "No Goals!"
},
{
"line": "exact ⟨le_trans (Real.arcsin_mem_Icc (d / l)).left hθ₁, lt_of_le_of_ne hθ₂ hθ_ne_pi_div_two⟩",
"before_state": "d l : ℝ\nh : d ≤ l\nθ : ℝ\nhθ₁✝ : min (arcsin (d / l)) (π / 2) ≤ θ\nhθ₂✝ : θ ≤ max (arcsin (d / l)) (π / 2)\nhθ_ne_pi_div_two : θ ≠ π / 2\nhθ₁ : arcsin (d / l) ≤ θ\nhθ₂ : θ ≤ π / 2\n⊢ θ ∈ Set.Ico (-(π / 2)) (π / 2)",
"after_state": "No Goals!"
},
{
"line": "simp_rw [min_eq_left ((div_le_iff₀ hl).mp ((Real.arcsin_le_iff_le_sin' hθ_mem).mp hθ₁))]",
"before_state": "d l : ℝ\nh : d ≤ l\nθ : ℝ\nhθ₁✝ : min (arcsin (d / l)) (π / 2) ≤ θ\nhθ₂✝ : θ ≤ max (arcsin (d / l)) (π / 2)\nhθ_ne_pi_div_two : θ ≠ π / 2\nhθ₁ : arcsin (d / l) ≤ θ\nhθ₂ : θ ≤ π / 2\nhθ_mem : θ ∈ Set.Ico (-(π / 2)) (π / 2)\n⊢ (fun θ => min d (sin θ * l)) θ = (fun x => d) θ",
"after_state": "d l : ℝ\nh : d ≤ l\nθ : ℝ\nhθ₁✝ : min (arcsin (d / l)) (π / 2) ≤ θ\nhθ₂✝ : θ ≤ max (arcsin (d / l)) (π / 2)\nhθ_ne_pi_div_two : θ ≠ π / 2\nhθ₁ : arcsin (d / l) ≤ θ\nhθ₂ : θ ≤ π / 2\nhθ_mem : θ ∈ Set.Ico (-(π / 2)) (π / 2)\n⊢ min d (sin θ * l) = d"
},
{
"line": "simp (failIfUnchanged✝ := false✝) only",
"before_state": "d l : ℝ\nh : d ≤ l\nθ : ℝ\nhθ₁✝ : min (arcsin (d / l)) (π / 2) ≤ θ\nhθ₂✝ : θ ≤ max (arcsin (d / l)) (π / 2)\nhθ_ne_pi_div_two : θ ≠ π / 2\nhθ₁ : arcsin (d / l) ≤ θ\nhθ₂ : θ ≤ π / 2\nhθ_mem : θ ∈ Set.Ico (-(π / 2)) (π / 2)\n⊢ (fun θ => min d (sin θ * l)) θ = (fun x => d) θ",
"after_state": "d l : ℝ\nh : d ≤ l\nθ : ℝ\nhθ₁✝ : min (arcsin (d / l)) (π / 2) ≤ θ\nhθ₂✝ : θ ≤ max (arcsin (d / l)) (π / 2)\nhθ_ne_pi_div_two : θ ≠ π / 2\nhθ₁ : arcsin (d / l) ≤ θ\nhθ₂ : θ ≤ π / 2\nhθ_mem : θ ∈ Set.Ico (-(π / 2)) (π / 2)\n⊢ min d (sin θ * l) = d"
},
{
"line": "simp only [min_eq_left ((div_le_iff₀ hl).mp ((Real.arcsin_le_iff_le_sin' hθ_mem).mp hθ₁))]",
"before_state": "d l : ℝ\nh : d ≤ l\nθ : ℝ\nhθ₁✝ : min (arcsin (d / l)) (π / 2) ≤ θ\nhθ₂✝ : θ ≤ max (arcsin (d / l)) (π / 2)\nhθ_ne_pi_div_two : θ ≠ π / 2\nhθ₁ : arcsin (d / l) ≤ θ\nhθ₂ : θ ≤ π / 2\nhθ_mem : θ ∈ Set.Ico (-(π / 2)) (π / 2)\n⊢ min d (sin θ * l) = d",
"after_state": "d l : ℝ\nh : d ≤ l\nθ : ℝ\nhθ₁✝ : min (arcsin (d / l)) (π / 2) ≤ θ\nhθ₂✝ : θ ≤ max (arcsin (d / l)) (π / 2)\nhθ_ne_pi_div_two : θ ≠ π / 2\nhθ₁ : arcsin (d / l) ≤ θ\nhθ₂ : θ ≤ π / 2\nhθ_mem : θ ∈ Set.Ico (-(π / 2)) (π / 2)\n⊢ min d (sin θ * l) = d"
},
{
"line": "rw [intervalIntegral.integral_congr this]",
"before_state": "d l : ℝ\nh : d ≤ l\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => d) (Set.uIcc (arcsin (d / l)) (π / 2))\n⊢ ∫ (θ : ℝ) in arcsin (d / l)..π / 2, min d (sin θ * l) = (π / 2 - arcsin (d / l)) * d",
"after_state": "d l : ℝ\nh : d ≤ l\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => d) (Set.uIcc (arcsin (d / l)) (π / 2))\n⊢ ∫ (x : ℝ) in arcsin (d / l)..π / 2, d = (π / 2 - arcsin (d / l)) * d"
},
{
"line": "rewrite [intervalIntegral.integral_congr this]",
"before_state": "d l : ℝ\nh : d ≤ l\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => d) (Set.uIcc (arcsin (d / l)) (π / 2))\n⊢ ∫ (θ : ℝ) in arcsin (d / l)..π / 2, min d (sin θ * l) = (π / 2 - arcsin (d / l)) * d",
"after_state": "d l : ℝ\nh : d ≤ l\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => d) (Set.uIcc (arcsin (d / l)) (π / 2))\n⊢ ∫ (x : ℝ) in arcsin (d / l)..π / 2, d = (π / 2 - arcsin (d / l)) * d"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "d l : ℝ\nh : d ≤ l\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => d) (Set.uIcc (arcsin (d / l)) (π / 2))\n⊢ ∫ (x : ℝ) in arcsin (d / l)..π / 2, d = (π / 2 - arcsin (d / l)) * d",
"after_state": "d l : ℝ\nh : d ≤ l\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => d) (Set.uIcc (arcsin (d / l)) (π / 2))\n⊢ ∫ (x : ℝ) in arcsin (d / l)..π / 2, d = (π / 2 - arcsin (d / l)) * d"
},
{
"line": "try (with_reducible rfl)",
"before_state": "d l : ℝ\nh : d ≤ l\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => d) (Set.uIcc (arcsin (d / l)) (π / 2))\n⊢ ∫ (x : ℝ) in arcsin (d / l)..π / 2, d = (π / 2 - arcsin (d / l)) * d",
"after_state": "d l : ℝ\nh : d ≤ l\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => d) (Set.uIcc (arcsin (d / l)) (π / 2))\n⊢ ∫ (x : ℝ) in arcsin (d / l)..π / 2, d = (π / 2 - arcsin (d / l)) * d"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "d l : ℝ\nh : d ≤ l\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => d) (Set.uIcc (arcsin (d / l)) (π / 2))\n⊢ ∫ (x : ℝ) in arcsin (d / l)..π / 2, d = (π / 2 - arcsin (d / l)) * d",
"after_state": "d l : ℝ\nh : d ≤ l\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => d) (Set.uIcc (arcsin (d / l)) (π / 2))\n⊢ ∫ (x : ℝ) in arcsin (d / l)..π / 2, d = (π / 2 - arcsin (d / l)) * d"
},
{
"line": "with_reducible rfl",
"before_state": "d l : ℝ\nh : d ≤ l\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => d) (Set.uIcc (arcsin (d / l)) (π / 2))\n⊢ ∫ (x : ℝ) in arcsin (d / l)..π / 2, d = (π / 2 - arcsin (d / l)) * d",
"after_state": "d l : ℝ\nh : d ≤ l\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => d) (Set.uIcc (arcsin (d / l)) (π / 2))\n⊢ ∫ (x : ℝ) in arcsin (d / l)..π / 2, d = (π / 2 - arcsin (d / l)) * d"
},
{
"line": "rfl",
"before_state": "d l : ℝ\nh : d ≤ l\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => d) (Set.uIcc (arcsin (d / l)) (π / 2))\n⊢ ∫ (x : ℝ) in arcsin (d / l)..π / 2, d = (π / 2 - arcsin (d / l)) * d",
"after_state": "d l : ℝ\nh : d ≤ l\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => d) (Set.uIcc (arcsin (d / l)) (π / 2))\n⊢ ∫ (x : ℝ) in arcsin (d / l)..π / 2, d = (π / 2 - arcsin (d / l)) * d"
},
{
"line": "apply_rfl",
"before_state": "d l : ℝ\nh : d ≤ l\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => d) (Set.uIcc (arcsin (d / l)) (π / 2))\n⊢ ∫ (x : ℝ) in arcsin (d / l)..π / 2, d = (π / 2 - arcsin (d / l)) * d",
"after_state": "d l : ℝ\nh : d ≤ l\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => d) (Set.uIcc (arcsin (d / l)) (π / 2))\n⊢ ∫ (x : ℝ) in arcsin (d / l)..π / 2, d = (π / 2 - arcsin (d / l)) * d"
},
{
"line": "skip",
"before_state": "d l : ℝ\nh : d ≤ l\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => d) (Set.uIcc (arcsin (d / l)) (π / 2))\n⊢ ∫ (x : ℝ) in arcsin (d / l)..π / 2, d = (π / 2 - arcsin (d / l)) * d",
"after_state": "d l : ℝ\nh : d ≤ l\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => d) (Set.uIcc (arcsin (d / l)) (π / 2))\n⊢ ∫ (x : ℝ) in arcsin (d / l)..π / 2, d = (π / 2 - arcsin (d / l)) * d"
},
{
"line": "rw [intervalIntegral.integral_const]",
"before_state": "d l : ℝ\nh : d ≤ l\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => d) (Set.uIcc (arcsin (d / l)) (π / 2))\n⊢ ∫ (x : ℝ) in arcsin (d / l)..π / 2, d = (π / 2 - arcsin (d / l)) * d",
"after_state": "d l : ℝ\nh : d ≤ l\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => d) (Set.uIcc (arcsin (d / l)) (π / 2))\n⊢ (π / 2 - arcsin (d / l)) • d = (π / 2 - arcsin (d / l)) * d"
},
{
"line": "rewrite [intervalIntegral.integral_const]",
"before_state": "d l : ℝ\nh : d ≤ l\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => d) (Set.uIcc (arcsin (d / l)) (π / 2))\n⊢ ∫ (x : ℝ) in arcsin (d / l)..π / 2, d = (π / 2 - arcsin (d / l)) * d",
"after_state": "d l : ℝ\nh : d ≤ l\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => d) (Set.uIcc (arcsin (d / l)) (π / 2))\n⊢ (π / 2 - arcsin (d / l)) • d = (π / 2 - arcsin (d / l)) * d"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "d l : ℝ\nh : d ≤ l\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => d) (Set.uIcc (arcsin (d / l)) (π / 2))\n⊢ (π / 2 - arcsin (d / l)) • d = (π / 2 - arcsin (d / l)) * d",
"after_state": "d l : ℝ\nh : d ≤ l\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => d) (Set.uIcc (arcsin (d / l)) (π / 2))\n⊢ (π / 2 - arcsin (d / l)) • d = (π / 2 - arcsin (d / l)) * d"
},
{
"line": "try (with_reducible rfl)",
"before_state": "d l : ℝ\nh : d ≤ l\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => d) (Set.uIcc (arcsin (d / l)) (π / 2))\n⊢ (π / 2 - arcsin (d / l)) • d = (π / 2 - arcsin (d / l)) * d",
"after_state": "d l : ℝ\nh : d ≤ l\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => d) (Set.uIcc (arcsin (d / l)) (π / 2))\n⊢ (π / 2 - arcsin (d / l)) • d = (π / 2 - arcsin (d / l)) * d"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "d l : ℝ\nh : d ≤ l\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => d) (Set.uIcc (arcsin (d / l)) (π / 2))\n⊢ (π / 2 - arcsin (d / l)) • d = (π / 2 - arcsin (d / l)) * d",
"after_state": "d l : ℝ\nh : d ≤ l\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => d) (Set.uIcc (arcsin (d / l)) (π / 2))\n⊢ (π / 2 - arcsin (d / l)) • d = (π / 2 - arcsin (d / l)) * d"
},
{
"line": "with_reducible rfl",
"before_state": "d l : ℝ\nh : d ≤ l\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => d) (Set.uIcc (arcsin (d / l)) (π / 2))\n⊢ (π / 2 - arcsin (d / l)) • d = (π / 2 - arcsin (d / l)) * d",
"after_state": "d l : ℝ\nh : d ≤ l\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => d) (Set.uIcc (arcsin (d / l)) (π / 2))\n⊢ (π / 2 - arcsin (d / l)) • d = (π / 2 - arcsin (d / l)) * d"
},
{
"line": "rfl",
"before_state": "d l : ℝ\nh : d ≤ l\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => d) (Set.uIcc (arcsin (d / l)) (π / 2))\n⊢ (π / 2 - arcsin (d / l)) • d = (π / 2 - arcsin (d / l)) * d",
"after_state": "d l : ℝ\nh : d ≤ l\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => d) (Set.uIcc (arcsin (d / l)) (π / 2))\n⊢ (π / 2 - arcsin (d / l)) • d = (π / 2 - arcsin (d / l)) * d"
},
{
"line": "apply_rfl",
"before_state": "d l : ℝ\nh : d ≤ l\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => d) (Set.uIcc (arcsin (d / l)) (π / 2))\n⊢ (π / 2 - arcsin (d / l)) • d = (π / 2 - arcsin (d / l)) * d",
"after_state": "d l : ℝ\nh : d ≤ l\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => d) (Set.uIcc (arcsin (d / l)) (π / 2))\n⊢ (π / 2 - arcsin (d / l)) • d = (π / 2 - arcsin (d / l)) * d"
},
{
"line": "skip",
"before_state": "d l : ℝ\nh : d ≤ l\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => d) (Set.uIcc (arcsin (d / l)) (π / 2))\n⊢ (π / 2 - arcsin (d / l)) • d = (π / 2 - arcsin (d / l)) * d",
"after_state": "d l : ℝ\nh : d ≤ l\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => d) (Set.uIcc (arcsin (d / l)) (π / 2))\n⊢ (π / 2 - arcsin (d / l)) • d = (π / 2 - arcsin (d / l)) * d"
},
{
"line": "rw [smul_eq_mul]",
"before_state": "d l : ℝ\nh : d ≤ l\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => d) (Set.uIcc (arcsin (d / l)) (π / 2))\n⊢ (π / 2 - arcsin (d / l)) • d = (π / 2 - arcsin (d / l)) * d",
"after_state": "No Goals!"
},
{
"line": "rewrite [smul_eq_mul]",
"before_state": "d l : ℝ\nh : d ≤ l\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => d) (Set.uIcc (arcsin (d / l)) (π / 2))\n⊢ (π / 2 - arcsin (d / l)) • d = (π / 2 - arcsin (d / l)) * d",
"after_state": "d l : ℝ\nh : d ≤ l\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => d) (Set.uIcc (arcsin (d / l)) (π / 2))\n⊢ (π / 2 - arcsin (d / l)) * d = (π / 2 - arcsin (d / l)) * d"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "d l : ℝ\nh : d ≤ l\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => d) (Set.uIcc (arcsin (d / l)) (π / 2))\n⊢ (π / 2 - arcsin (d / l)) * d = (π / 2 - arcsin (d / l)) * d",
"after_state": "No Goals!"
},
{
"line": "try (with_reducible rfl)",
"before_state": "d l : ℝ\nh : d ≤ l\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => d) (Set.uIcc (arcsin (d / l)) (π / 2))\n⊢ (π / 2 - arcsin (d / l)) * d = (π / 2 - arcsin (d / l)) * d",
"after_state": "No Goals!"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "d l : ℝ\nh : d ≤ l\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => d) (Set.uIcc (arcsin (d / l)) (π / 2))\n⊢ (π / 2 - arcsin (d / l)) * d = (π / 2 - arcsin (d / l)) * d",
"after_state": "No Goals!"
},
{
"line": "with_reducible rfl",
"before_state": "d l : ℝ\nh : d ≤ l\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => d) (Set.uIcc (arcsin (d / l)) (π / 2))\n⊢ (π / 2 - arcsin (d / l)) * d = (π / 2 - arcsin (d / l)) * d",
"after_state": "No Goals!"
},
{
"line": "rfl",
"before_state": "d l : ℝ\nh : d ≤ l\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => d) (Set.uIcc (arcsin (d / l)) (π / 2))\n⊢ (π / 2 - arcsin (d / l)) * d = (π / 2 - arcsin (d / l)) * d",
"after_state": "No Goals!"
},
{
"line": "eq_refl",
"before_state": "d l : ℝ\nh : d ≤ l\nthis : Set.EqOn (fun θ => min d (sin θ * l)) (fun x => d) (Set.uIcc (arcsin (d / l)) (π / 2))\n⊢ (π / 2 - arcsin (d / l)) * d = (π / 2 - arcsin (d / l)) * d",
"after_state": "No Goals!"
}
] |
theorem Theorems100.inverse_triangle_sum :
∀ n, ∑ k ∈ range n, (2 : ℚ) / (k * (k + 1)) = if n = 0 then 0 else 2 - (2 : ℚ) / n := by
refine sum_range_induction _ _ rfl ?_
rintro (_ | _)
· norm_num
field_simp
ring
|
/root/DuelModelResearch/mathlib4/Archive/Wiedijk100Theorems/InverseTriangleSum.lean
|
{
"open": [
"Finset"
],
"variables": []
}
|
[
{
"line": "refine sum_range_induction _ _ rfl ?_",
"before_state": "⊢ ∀ (n : ℕ), ∑ k ∈ range n, 2 / (↑k * (↑k + 1)) = if n = 0 then 0 else 2 - 2 / ↑n",
"after_state": "⊢ ∀ (n : ℕ), (if n + 1 = 0 then 0 else 2 - 2 / ↑(n + 1)) = (if n = 0 then 0 else 2 - 2 / ↑n) + 2 / (↑n * (↑n + 1))"
},
{
"line": "rintro (_ | _)",
"before_state": "⊢ ∀ (n : ℕ), (if n + 1 = 0 then 0 else 2 - 2 / ↑(n + 1)) = (if n = 0 then 0 else 2 - 2 / ↑n) + 2 / (↑n * (↑n + 1))",
"after_state": "case zero\n⊢ (if 0 + 1 = 0 then 0 else 2 - 2 / ↑(0 + 1)) = (if 0 = 0 then 0 else 2 - 2 / ↑0) + 2 / (↑0 * (↑0 + 1))\n---\ncase succ\nn✝ : ℕ\n⊢ (if n✝ + 1 + 1 = 0 then 0 else 2 - 2 / ↑(n✝ + 1 + 1)) =\n (if n✝ + 1 = 0 then 0 else 2 - 2 / ↑(n✝ + 1)) + 2 / (↑(n✝ + 1) * (↑(n✝ + 1) + 1))"
},
{
"line": "norm_num",
"before_state": "case zero\n⊢ (if 0 + 1 = 0 then 0 else 2 - 2 / ↑(0 + 1)) = (if 0 = 0 then 0 else 2 - 2 / ↑0) + 2 / (↑0 * (↑0 + 1))",
"after_state": "No Goals!"
},
{
"line": "field_simp",
"before_state": "case succ\nn✝ : ℕ\n⊢ (if n✝ + 1 + 1 = 0 then 0 else 2 - 2 / ↑(n✝ + 1 + 1)) =\n (if n✝ + 1 = 0 then 0 else 2 - 2 / ↑(n✝ + 1)) + 2 / (↑(n✝ + 1) * (↑(n✝ + 1) + 1))",
"after_state": "case succ\nn✝ : ℕ\n⊢ (2 * (↑n✝ + 1 + 1) - 2) * ((↑n✝ + 1) * ((↑n✝ + 1) * (↑n✝ + 1 + 1))) =\n ((2 * (↑n✝ + 1) - 2) * ((↑n✝ + 1) * (↑n✝ + 1 + 1)) + 2 * (↑n✝ + 1)) * (↑n✝ + 1 + 1)"
},
{
"line": "ring",
"before_state": "case succ\nn✝ : ℕ\n⊢ (2 * (↑n✝ + 1 + 1) - 2) * ((↑n✝ + 1) * ((↑n✝ + 1) * (↑n✝ + 1 + 1))) =\n ((2 * (↑n✝ + 1) - 2) * ((↑n✝ + 1) * (↑n✝ + 1 + 1)) + 2 * (↑n✝ + 1)) * (↑n✝ + 1 + 1)",
"after_state": "No Goals!"
},
{
"line": "first\n| ring1\n|\n try_this ring_nf\"\\n\\nThe `ring` tactic failed to close the goal. Use `ring_nf` to obtain a normal form.\n \\nNote that `ring` works primarily in *commutative* rings. \\\n If you have a noncommutative ring, abelian group or module, consider using \\\n `noncomm_ring`, `abel` or `module` instead.\"",
"before_state": "case succ\nn✝ : ℕ\n⊢ (2 * (↑n✝ + 1 + 1) - 2) * ((↑n✝ + 1) * ((↑n✝ + 1) * (↑n✝ + 1 + 1))) =\n ((2 * (↑n✝ + 1) - 2) * ((↑n✝ + 1) * (↑n✝ + 1 + 1)) + 2 * (↑n✝ + 1)) * (↑n✝ + 1 + 1)",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "case succ\nn✝ : ℕ\n⊢ (2 * (↑n✝ + 1 + 1) - 2) * ((↑n✝ + 1) * ((↑n✝ + 1) * (↑n✝ + 1 + 1))) =\n ((2 * (↑n✝ + 1) - 2) * ((↑n✝ + 1) * (↑n✝ + 1 + 1)) + 2 * (↑n✝ + 1)) * (↑n✝ + 1 + 1)",
"after_state": "No Goals!"
}
] |
theorem sigma_two_pow_eq_mersenne_succ (k : ℕ) : σ 1 (2 ^ k) = mersenne (k + 1) := by
simp_rw [sigma_one_apply, mersenne, show 2 = 1 + 1 from rfl, ← geom_sum_mul_add 1 (k + 1)]
norm_num
|
/root/DuelModelResearch/mathlib4/Archive/Wiedijk100Theorems/PerfectNumbers.lean
|
{
"open": [
"ArithmeticFunction Finset"
],
"variables": []
}
|
[
{
"line": "simp_rw [sigma_one_apply, mersenne, show 2 = 1 + 1 from rfl, ← geom_sum_mul_add 1 (k + 1)]",
"before_state": "k : ℕ\n⊢ (σ 1) (2 ^ k) = mersenne (k + 1)",
"after_state": "k : ℕ\n⊢ ∑ x ∈ ((1 + 1) ^ k).divisors, x = (∑ i ∈ range (k + 1), (1 + 1) ^ i) * 1 + 1 - 1"
},
{
"line": "simp (failIfUnchanged✝ := false✝) only",
"before_state": "k : ℕ\n⊢ (σ 1) (2 ^ k) = mersenne (k + 1)",
"after_state": "k : ℕ\n⊢ (σ 1) (2 ^ k) = mersenne (k + 1)"
},
{
"line": "simp only [sigma_one_apply]",
"before_state": "k : ℕ\n⊢ (σ 1) (2 ^ k) = mersenne (k + 1)",
"after_state": "k : ℕ\n⊢ ∑ d ∈ (2 ^ k).divisors, d = mersenne (k + 1)"
},
{
"line": "simp only [mersenne]",
"before_state": "k : ℕ\n⊢ ∑ d ∈ (2 ^ k).divisors, d = mersenne (k + 1)",
"after_state": "k : ℕ\n⊢ ∑ d ∈ (2 ^ k).divisors, d = 2 ^ (k + 1) - 1"
},
{
"line": "simp only [show 2 = 1 + 1 from rfl]",
"before_state": "k : ℕ\n⊢ ∑ d ∈ (2 ^ k).divisors, d = 2 ^ (k + 1) - 1",
"after_state": "k : ℕ\n⊢ ∑ x ∈ ((1 + 1) ^ k).divisors, x = (1 + 1) ^ (k + 1) - 1"
},
{
"line": "simp only [← geom_sum_mul_add 1 (k + 1)]",
"before_state": "k : ℕ\n⊢ ∑ x ∈ ((1 + 1) ^ k).divisors, x = (1 + 1) ^ (k + 1) - 1",
"after_state": "k : ℕ\n⊢ ∑ x ∈ ((1 + 1) ^ k).divisors, x = (∑ i ∈ range (k + 1), (1 + 1) ^ i) * 1 + 1 - 1"
},
{
"line": "norm_num",
"before_state": "k : ℕ\n⊢ ∑ x ∈ ((1 + 1) ^ k).divisors, x = (∑ i ∈ range (k + 1), (1 + 1) ^ i) * 1 + 1 - 1",
"after_state": "No Goals!"
}
] |
theorem ne_zero_of_prime_mersenne (k : ℕ) (pr : (mersenne (k + 1)).Prime) : k ≠ 0 := by
intro H
simp [H, mersenne, Nat.not_prime_one] at pr
|
/root/DuelModelResearch/mathlib4/Archive/Wiedijk100Theorems/PerfectNumbers.lean
|
{
"open": [
"ArithmeticFunction Finset"
],
"variables": []
}
|
[
{
"line": "intro H",
"before_state": "k : ℕ\npr : Nat.Prime (mersenne (k + 1))\n⊢ k ≠ 0",
"after_state": "k : ℕ\npr : Nat.Prime (mersenne (k + 1))\nH : k = 0\n⊢ False"
},
{
"line": "simp [H, mersenne, Nat.not_prime_one] at pr",
"before_state": "k : ℕ\npr : Nat.Prime (mersenne (k + 1))\nH : k = 0\n⊢ False",
"after_state": "No Goals!"
}
] |
theorem eq_two_pow_mul_odd {n : ℕ} (hpos : 0 < n) : ∃ k m : ℕ, n = 2 ^ k * m ∧ ¬Even m := by
have h := Nat.finiteMultiplicity_iff.2 ⟨Nat.prime_two.ne_one, hpos⟩
obtain ⟨m, hm⟩ := pow_multiplicity_dvd 2 n
use multiplicity 2 n, m
refine ⟨hm, ?_⟩
rw [even_iff_two_dvd]
have hg := h.not_pow_dvd_of_multiplicity_lt (Nat.lt_succ_self _)
contrapose! hg
rcases hg with ⟨k, rfl⟩
apply Dvd.intro k
rw [pow_succ]
rw [mul_assoc]
rw [← hm]
|
/root/DuelModelResearch/mathlib4/Archive/Wiedijk100Theorems/PerfectNumbers.lean
|
{
"open": [
"ArithmeticFunction Finset"
],
"variables": []
}
|
[
{
"line": "have h := Nat.finiteMultiplicity_iff.2 ⟨Nat.prime_two.ne_one, hpos⟩",
"before_state": "n : ℕ\nhpos : 0 < n\n⊢ ∃ k m, n = 2 ^ k * m ∧ ¬Even m",
"after_state": "n : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\n⊢ ∃ k m, n = 2 ^ k * m ∧ ¬Even m"
},
{
"line": "refine_lift\n have h := Nat.finiteMultiplicity_iff.2 ⟨Nat.prime_two.ne_one, hpos⟩;\n ?_",
"before_state": "n : ℕ\nhpos : 0 < n\n⊢ ∃ k m, n = 2 ^ k * m ∧ ¬Even m",
"after_state": "n : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\n⊢ ∃ k m, n = 2 ^ k * m ∧ ¬Even m"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (have h := Nat.finiteMultiplicity_iff.2 ⟨Nat.prime_two.ne_one, hpos⟩;\n ?_);\n rotate_right)",
"before_state": "n : ℕ\nhpos : 0 < n\n⊢ ∃ k m, n = 2 ^ k * m ∧ ¬Even m",
"after_state": "n : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\n⊢ ∃ k m, n = 2 ^ k * m ∧ ¬Even m"
},
{
"line": "refine\n no_implicit_lambda%\n (have h := Nat.finiteMultiplicity_iff.2 ⟨Nat.prime_two.ne_one, hpos⟩;\n ?_)",
"before_state": "n : ℕ\nhpos : 0 < n\n⊢ ∃ k m, n = 2 ^ k * m ∧ ¬Even m",
"after_state": "n : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\n⊢ ∃ k m, n = 2 ^ k * m ∧ ¬Even m"
},
{
"line": "rotate_right",
"before_state": "n : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\n⊢ ∃ k m, n = 2 ^ k * m ∧ ¬Even m",
"after_state": "n : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\n⊢ ∃ k m, n = 2 ^ k * m ∧ ¬Even m"
},
{
"line": "obtain ⟨m, hm⟩ := pow_multiplicity_dvd 2 n",
"before_state": "n : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\n⊢ ∃ k m, n = 2 ^ k * m ∧ ¬Even m",
"after_state": "case intro\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nm : ℕ\nhm : n = 2 ^ multiplicity 2 n * m\n⊢ ∃ k m, n = 2 ^ k * m ∧ ¬Even m"
},
{
"line": "use multiplicity 2 n, m",
"before_state": "case intro\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nm : ℕ\nhm : n = 2 ^ multiplicity 2 n * m\n⊢ ∃ k m, n = 2 ^ k * m ∧ ¬Even m",
"after_state": "case h\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nm : ℕ\nhm : n = 2 ^ multiplicity 2 n * m\n⊢ n = 2 ^ multiplicity 2 n * m ∧ ¬Even m"
},
{
"line": "refine without_cdot(multiplicity 2 n : ?m✝)",
"before_state": "case w\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nm : ℕ\nhm : n = 2 ^ multiplicity 2 n * m\n⊢ ℕ",
"after_state": "No Goals!"
},
{
"line": "refine without_cdot(m : ?m✝)",
"before_state": "case w\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nm : ℕ\nhm : n = 2 ^ multiplicity 2 n * m\n⊢ ℕ",
"after_state": "No Goals!"
},
{
"line": "try with_reducible use_discharger",
"before_state": "case h\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nm : ℕ\nhm : n = 2 ^ multiplicity 2 n * m\n⊢ n = 2 ^ multiplicity 2 n * m ∧ ¬Even m",
"after_state": "case h\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nm : ℕ\nhm : n = 2 ^ multiplicity 2 n * m\n⊢ n = 2 ^ multiplicity 2 n * m ∧ ¬Even m"
},
{
"line": "first\n| with_reducible use_discharger\n| skip",
"before_state": "case h\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nm : ℕ\nhm : n = 2 ^ multiplicity 2 n * m\n⊢ n = 2 ^ multiplicity 2 n * m ∧ ¬Even m",
"after_state": "case h\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nm : ℕ\nhm : n = 2 ^ multiplicity 2 n * m\n⊢ n = 2 ^ multiplicity 2 n * m ∧ ¬Even m"
},
{
"line": "with_reducible use_discharger",
"before_state": "case h\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nm : ℕ\nhm : n = 2 ^ multiplicity 2 n * m\n⊢ n = 2 ^ multiplicity 2 n * m ∧ ¬Even m",
"after_state": "case h\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nm : ℕ\nhm : n = 2 ^ multiplicity 2 n * m\n⊢ n = 2 ^ multiplicity 2 n * m ∧ ¬Even m"
},
{
"line": "use_discharger",
"before_state": "case h\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nm : ℕ\nhm : n = 2 ^ multiplicity 2 n * m\n⊢ n = 2 ^ multiplicity 2 n * m ∧ ¬Even m",
"after_state": "case h\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nm : ℕ\nhm : n = 2 ^ multiplicity 2 n * m\n⊢ n = 2 ^ multiplicity 2 n * m ∧ ¬Even m"
},
{
"line": "focus\n apply exists_prop.mpr✝\n with_annotate_state\"<;>\" skip\n all_goals use_discharger",
"before_state": "case h\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nm : ℕ\nhm : n = 2 ^ multiplicity 2 n * m\n⊢ n = 2 ^ multiplicity 2 n * m ∧ ¬Even m",
"after_state": "case h\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nm : ℕ\nhm : n = 2 ^ multiplicity 2 n * m\n⊢ n = 2 ^ multiplicity 2 n * m ∧ ¬Even m"
},
{
"line": "apply exists_prop.mpr✝",
"before_state": "case h\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nm : ℕ\nhm : n = 2 ^ multiplicity 2 n * m\n⊢ n = 2 ^ multiplicity 2 n * m ∧ ¬Even m",
"after_state": "case h\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nm : ℕ\nhm : n = 2 ^ multiplicity 2 n * m\n⊢ n = 2 ^ multiplicity 2 n * m ∧ ¬Even m"
},
{
"line": "skip",
"before_state": "case h\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nm : ℕ\nhm : n = 2 ^ multiplicity 2 n * m\n⊢ n = 2 ^ multiplicity 2 n * m ∧ ¬Even m",
"after_state": "case h\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nm : ℕ\nhm : n = 2 ^ multiplicity 2 n * m\n⊢ n = 2 ^ multiplicity 2 n * m ∧ ¬Even m"
},
{
"line": "refine ⟨hm, ?_⟩",
"before_state": "case h\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nm : ℕ\nhm : n = 2 ^ multiplicity 2 n * m\n⊢ n = 2 ^ multiplicity 2 n * m ∧ ¬Even m",
"after_state": "case h\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nm : ℕ\nhm : n = 2 ^ multiplicity 2 n * m\n⊢ ¬Even m"
},
{
"line": "rw [even_iff_two_dvd]",
"before_state": "case h\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nm : ℕ\nhm : n = 2 ^ multiplicity 2 n * m\n⊢ ¬Even m",
"after_state": "case h\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nm : ℕ\nhm : n = 2 ^ multiplicity 2 n * m\n⊢ ¬2 ∣ m"
},
{
"line": "rewrite [even_iff_two_dvd]",
"before_state": "case h\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nm : ℕ\nhm : n = 2 ^ multiplicity 2 n * m\n⊢ ¬Even m",
"after_state": "case h\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nm : ℕ\nhm : n = 2 ^ multiplicity 2 n * m\n⊢ ¬2 ∣ m"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case h\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nm : ℕ\nhm : n = 2 ^ multiplicity 2 n * m\n⊢ ¬2 ∣ m",
"after_state": "case h\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nm : ℕ\nhm : n = 2 ^ multiplicity 2 n * m\n⊢ ¬2 ∣ m"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case h\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nm : ℕ\nhm : n = 2 ^ multiplicity 2 n * m\n⊢ ¬2 ∣ m",
"after_state": "case h\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nm : ℕ\nhm : n = 2 ^ multiplicity 2 n * m\n⊢ ¬2 ∣ m"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case h\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nm : ℕ\nhm : n = 2 ^ multiplicity 2 n * m\n⊢ ¬2 ∣ m",
"after_state": "case h\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nm : ℕ\nhm : n = 2 ^ multiplicity 2 n * m\n⊢ ¬2 ∣ m"
},
{
"line": "with_reducible rfl",
"before_state": "case h\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nm : ℕ\nhm : n = 2 ^ multiplicity 2 n * m\n⊢ ¬2 ∣ m",
"after_state": "case h\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nm : ℕ\nhm : n = 2 ^ multiplicity 2 n * m\n⊢ ¬2 ∣ m"
},
{
"line": "rfl",
"before_state": "case h\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nm : ℕ\nhm : n = 2 ^ multiplicity 2 n * m\n⊢ ¬2 ∣ m",
"after_state": "case h\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nm : ℕ\nhm : n = 2 ^ multiplicity 2 n * m\n⊢ ¬2 ∣ m"
},
{
"line": "apply_rfl",
"before_state": "case h\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nm : ℕ\nhm : n = 2 ^ multiplicity 2 n * m\n⊢ ¬2 ∣ m",
"after_state": "case h\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nm : ℕ\nhm : n = 2 ^ multiplicity 2 n * m\n⊢ ¬2 ∣ m"
},
{
"line": "skip",
"before_state": "case h\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nm : ℕ\nhm : n = 2 ^ multiplicity 2 n * m\n⊢ ¬2 ∣ m",
"after_state": "case h\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nm : ℕ\nhm : n = 2 ^ multiplicity 2 n * m\n⊢ ¬2 ∣ m"
},
{
"line": "have hg := h.not_pow_dvd_of_multiplicity_lt (Nat.lt_succ_self _)",
"before_state": "case h\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nm : ℕ\nhm : n = 2 ^ multiplicity 2 n * m\n⊢ ¬2 ∣ m",
"after_state": "case h\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nm : ℕ\nhm : n = 2 ^ multiplicity 2 n * m\nhg : ¬2 ^ (multiplicity 2 n).succ ∣ n\n⊢ ¬2 ∣ m"
},
{
"line": "refine_lift\n have hg := h.not_pow_dvd_of_multiplicity_lt (Nat.lt_succ_self _);\n ?_",
"before_state": "case h\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nm : ℕ\nhm : n = 2 ^ multiplicity 2 n * m\n⊢ ¬2 ∣ m",
"after_state": "case h\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nm : ℕ\nhm : n = 2 ^ multiplicity 2 n * m\nhg : ¬2 ^ (multiplicity 2 n).succ ∣ n\n⊢ ¬2 ∣ m"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (have hg := h.not_pow_dvd_of_multiplicity_lt (Nat.lt_succ_self _);\n ?_);\n rotate_right)",
"before_state": "case h\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nm : ℕ\nhm : n = 2 ^ multiplicity 2 n * m\n⊢ ¬2 ∣ m",
"after_state": "case h\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nm : ℕ\nhm : n = 2 ^ multiplicity 2 n * m\nhg : ¬2 ^ (multiplicity 2 n).succ ∣ n\n⊢ ¬2 ∣ m"
},
{
"line": "refine\n no_implicit_lambda%\n (have hg := h.not_pow_dvd_of_multiplicity_lt (Nat.lt_succ_self _);\n ?_)",
"before_state": "case h\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nm : ℕ\nhm : n = 2 ^ multiplicity 2 n * m\n⊢ ¬2 ∣ m",
"after_state": "case h\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nm : ℕ\nhm : n = 2 ^ multiplicity 2 n * m\nhg : ¬2 ^ (multiplicity 2 n).succ ∣ n\n⊢ ¬2 ∣ m"
},
{
"line": "rotate_right",
"before_state": "case h\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nm : ℕ\nhm : n = 2 ^ multiplicity 2 n * m\nhg : ¬2 ^ (multiplicity 2 n).succ ∣ n\n⊢ ¬2 ∣ m",
"after_state": "case h\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nm : ℕ\nhm : n = 2 ^ multiplicity 2 n * m\nhg : ¬2 ^ (multiplicity 2 n).succ ∣ n\n⊢ ¬2 ∣ m"
},
{
"line": "contrapose! hg",
"before_state": "case h\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nm : ℕ\nhm : n = 2 ^ multiplicity 2 n * m\nhg : ¬2 ^ (multiplicity 2 n).succ ∣ n\n⊢ ¬2 ∣ m",
"after_state": "case h\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nm : ℕ\nhm : n = 2 ^ multiplicity 2 n * m\nhg : 2 ∣ m\n⊢ 2 ^ (multiplicity 2 n).succ ∣ n"
},
{
"line": "revert hg",
"before_state": "case h\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nm : ℕ\nhm : n = 2 ^ multiplicity 2 n * m\nhg : ¬2 ^ (multiplicity 2 n).succ ∣ n\n⊢ ¬2 ∣ m",
"after_state": "case h\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nm : ℕ\nhm : n = 2 ^ multiplicity 2 n * m\n⊢ ¬2 ^ (multiplicity 2 n).succ ∣ n → ¬2 ∣ m"
},
{
"line": "contrapose!",
"before_state": "case h\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nm : ℕ\nhm : n = 2 ^ multiplicity 2 n * m\n⊢ ¬2 ^ (multiplicity 2 n).succ ∣ n → ¬2 ∣ m",
"after_state": "case h\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nm : ℕ\nhm : n = 2 ^ multiplicity 2 n * m\n⊢ 2 ∣ m → 2 ^ (multiplicity 2 n).succ ∣ n"
},
{
"line": "contrapose",
"before_state": "case h\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nm : ℕ\nhm : n = 2 ^ multiplicity 2 n * m\n⊢ ¬2 ^ (multiplicity 2 n).succ ∣ n → ¬2 ∣ m",
"after_state": "case h\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nm : ℕ\nhm : n = 2 ^ multiplicity 2 n * m\n⊢ ¬¬2 ∣ m → ¬¬2 ^ (multiplicity 2 n).succ ∣ n"
},
{
"line": "refine mtr✝ ?_",
"before_state": "case h\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nm : ℕ\nhm : n = 2 ^ multiplicity 2 n * m\n⊢ ¬2 ^ (multiplicity 2 n).succ ∣ n → ¬2 ∣ m",
"after_state": "case h\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nm : ℕ\nhm : n = 2 ^ multiplicity 2 n * m\n⊢ ¬¬2 ∣ m → ¬¬2 ^ (multiplicity 2 n).succ ∣ n"
},
{
"line": "try push_neg",
"before_state": "case h\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nm : ℕ\nhm : n = 2 ^ multiplicity 2 n * m\n⊢ ¬¬2 ∣ m → ¬¬2 ^ (multiplicity 2 n).succ ∣ n",
"after_state": "case h\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nm : ℕ\nhm : n = 2 ^ multiplicity 2 n * m\n⊢ 2 ∣ m → 2 ^ (multiplicity 2 n).succ ∣ n"
},
{
"line": "first\n| push_neg\n| skip",
"before_state": "case h\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nm : ℕ\nhm : n = 2 ^ multiplicity 2 n * m\n⊢ ¬¬2 ∣ m → ¬¬2 ^ (multiplicity 2 n).succ ∣ n",
"after_state": "case h\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nm : ℕ\nhm : n = 2 ^ multiplicity 2 n * m\n⊢ 2 ∣ m → 2 ^ (multiplicity 2 n).succ ∣ n"
},
{
"line": "push_neg",
"before_state": "case h\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nm : ℕ\nhm : n = 2 ^ multiplicity 2 n * m\n⊢ ¬¬2 ∣ m → ¬¬2 ^ (multiplicity 2 n).succ ∣ n",
"after_state": "case h\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nm : ℕ\nhm : n = 2 ^ multiplicity 2 n * m\n⊢ 2 ∣ m → 2 ^ (multiplicity 2 n).succ ∣ n"
},
{
"line": "intro hg",
"before_state": "case h\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nm : ℕ\nhm : n = 2 ^ multiplicity 2 n * m\n⊢ 2 ∣ m → 2 ^ (multiplicity 2 n).succ ∣ n",
"after_state": "case h\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nm : ℕ\nhm : n = 2 ^ multiplicity 2 n * m\nhg : 2 ∣ m\n⊢ 2 ^ (multiplicity 2 n).succ ∣ n"
},
{
"line": "rcases hg with ⟨k, rfl⟩",
"before_state": "case h\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nm : ℕ\nhm : n = 2 ^ multiplicity 2 n * m\nhg : 2 ∣ m\n⊢ 2 ^ (multiplicity 2 n).succ ∣ n",
"after_state": "case h.intro\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nk : ℕ\nhm : n = 2 ^ multiplicity 2 n * (2 * k)\n⊢ 2 ^ (multiplicity 2 n).succ ∣ n"
},
{
"line": "apply Dvd.intro k",
"before_state": "case h.intro\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nk : ℕ\nhm : n = 2 ^ multiplicity 2 n * (2 * k)\n⊢ 2 ^ (multiplicity 2 n).succ ∣ n",
"after_state": "case h.intro\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nk : ℕ\nhm : n = 2 ^ multiplicity 2 n * (2 * k)\n⊢ 2 ^ (multiplicity 2 n).succ * k = n"
},
{
"line": "rw [pow_succ]",
"before_state": "case h.intro\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nk : ℕ\nhm : n = 2 ^ multiplicity 2 n * (2 * k)\n⊢ 2 ^ (multiplicity 2 n).succ * k = n",
"after_state": "case h.intro\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nk : ℕ\nhm : n = 2 ^ multiplicity 2 n * (2 * k)\n⊢ 2 ^ multiplicity 2 n * 2 * k = n"
},
{
"line": "rewrite [pow_succ]",
"before_state": "case h.intro\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nk : ℕ\nhm : n = 2 ^ multiplicity 2 n * (2 * k)\n⊢ 2 ^ (multiplicity 2 n).succ * k = n",
"after_state": "case h.intro\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nk : ℕ\nhm : n = 2 ^ multiplicity 2 n * (2 * k)\n⊢ 2 ^ multiplicity 2 n * 2 * k = n"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case h.intro\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nk : ℕ\nhm : n = 2 ^ multiplicity 2 n * (2 * k)\n⊢ 2 ^ multiplicity 2 n * 2 * k = n",
"after_state": "case h.intro\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nk : ℕ\nhm : n = 2 ^ multiplicity 2 n * (2 * k)\n⊢ 2 ^ multiplicity 2 n * 2 * k = n"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case h.intro\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nk : ℕ\nhm : n = 2 ^ multiplicity 2 n * (2 * k)\n⊢ 2 ^ multiplicity 2 n * 2 * k = n",
"after_state": "case h.intro\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nk : ℕ\nhm : n = 2 ^ multiplicity 2 n * (2 * k)\n⊢ 2 ^ multiplicity 2 n * 2 * k = n"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case h.intro\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nk : ℕ\nhm : n = 2 ^ multiplicity 2 n * (2 * k)\n⊢ 2 ^ multiplicity 2 n * 2 * k = n",
"after_state": "case h.intro\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nk : ℕ\nhm : n = 2 ^ multiplicity 2 n * (2 * k)\n⊢ 2 ^ multiplicity 2 n * 2 * k = n"
},
{
"line": "with_reducible rfl",
"before_state": "case h.intro\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nk : ℕ\nhm : n = 2 ^ multiplicity 2 n * (2 * k)\n⊢ 2 ^ multiplicity 2 n * 2 * k = n",
"after_state": "case h.intro\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nk : ℕ\nhm : n = 2 ^ multiplicity 2 n * (2 * k)\n⊢ 2 ^ multiplicity 2 n * 2 * k = n"
},
{
"line": "rfl",
"before_state": "case h.intro\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nk : ℕ\nhm : n = 2 ^ multiplicity 2 n * (2 * k)\n⊢ 2 ^ multiplicity 2 n * 2 * k = n",
"after_state": "case h.intro\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nk : ℕ\nhm : n = 2 ^ multiplicity 2 n * (2 * k)\n⊢ 2 ^ multiplicity 2 n * 2 * k = n"
},
{
"line": "apply_rfl",
"before_state": "case h.intro\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nk : ℕ\nhm : n = 2 ^ multiplicity 2 n * (2 * k)\n⊢ 2 ^ multiplicity 2 n * 2 * k = n",
"after_state": "case h.intro\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nk : ℕ\nhm : n = 2 ^ multiplicity 2 n * (2 * k)\n⊢ 2 ^ multiplicity 2 n * 2 * k = n"
},
{
"line": "skip",
"before_state": "case h.intro\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nk : ℕ\nhm : n = 2 ^ multiplicity 2 n * (2 * k)\n⊢ 2 ^ multiplicity 2 n * 2 * k = n",
"after_state": "case h.intro\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nk : ℕ\nhm : n = 2 ^ multiplicity 2 n * (2 * k)\n⊢ 2 ^ multiplicity 2 n * 2 * k = n"
},
{
"line": "rw [mul_assoc]",
"before_state": "case h.intro\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nk : ℕ\nhm : n = 2 ^ multiplicity 2 n * (2 * k)\n⊢ 2 ^ multiplicity 2 n * 2 * k = n",
"after_state": "case h.intro\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nk : ℕ\nhm : n = 2 ^ multiplicity 2 n * (2 * k)\n⊢ 2 ^ multiplicity 2 n * (2 * k) = n"
},
{
"line": "rewrite [mul_assoc]",
"before_state": "case h.intro\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nk : ℕ\nhm : n = 2 ^ multiplicity 2 n * (2 * k)\n⊢ 2 ^ multiplicity 2 n * 2 * k = n",
"after_state": "case h.intro\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nk : ℕ\nhm : n = 2 ^ multiplicity 2 n * (2 * k)\n⊢ 2 ^ multiplicity 2 n * (2 * k) = n"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case h.intro\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nk : ℕ\nhm : n = 2 ^ multiplicity 2 n * (2 * k)\n⊢ 2 ^ multiplicity 2 n * (2 * k) = n",
"after_state": "case h.intro\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nk : ℕ\nhm : n = 2 ^ multiplicity 2 n * (2 * k)\n⊢ 2 ^ multiplicity 2 n * (2 * k) = n"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case h.intro\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nk : ℕ\nhm : n = 2 ^ multiplicity 2 n * (2 * k)\n⊢ 2 ^ multiplicity 2 n * (2 * k) = n",
"after_state": "case h.intro\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nk : ℕ\nhm : n = 2 ^ multiplicity 2 n * (2 * k)\n⊢ 2 ^ multiplicity 2 n * (2 * k) = n"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case h.intro\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nk : ℕ\nhm : n = 2 ^ multiplicity 2 n * (2 * k)\n⊢ 2 ^ multiplicity 2 n * (2 * k) = n",
"after_state": "case h.intro\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nk : ℕ\nhm : n = 2 ^ multiplicity 2 n * (2 * k)\n⊢ 2 ^ multiplicity 2 n * (2 * k) = n"
},
{
"line": "with_reducible rfl",
"before_state": "case h.intro\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nk : ℕ\nhm : n = 2 ^ multiplicity 2 n * (2 * k)\n⊢ 2 ^ multiplicity 2 n * (2 * k) = n",
"after_state": "case h.intro\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nk : ℕ\nhm : n = 2 ^ multiplicity 2 n * (2 * k)\n⊢ 2 ^ multiplicity 2 n * (2 * k) = n"
},
{
"line": "rfl",
"before_state": "case h.intro\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nk : ℕ\nhm : n = 2 ^ multiplicity 2 n * (2 * k)\n⊢ 2 ^ multiplicity 2 n * (2 * k) = n",
"after_state": "case h.intro\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nk : ℕ\nhm : n = 2 ^ multiplicity 2 n * (2 * k)\n⊢ 2 ^ multiplicity 2 n * (2 * k) = n"
},
{
"line": "apply_rfl",
"before_state": "case h.intro\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nk : ℕ\nhm : n = 2 ^ multiplicity 2 n * (2 * k)\n⊢ 2 ^ multiplicity 2 n * (2 * k) = n",
"after_state": "case h.intro\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nk : ℕ\nhm : n = 2 ^ multiplicity 2 n * (2 * k)\n⊢ 2 ^ multiplicity 2 n * (2 * k) = n"
},
{
"line": "skip",
"before_state": "case h.intro\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nk : ℕ\nhm : n = 2 ^ multiplicity 2 n * (2 * k)\n⊢ 2 ^ multiplicity 2 n * (2 * k) = n",
"after_state": "case h.intro\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nk : ℕ\nhm : n = 2 ^ multiplicity 2 n * (2 * k)\n⊢ 2 ^ multiplicity 2 n * (2 * k) = n"
},
{
"line": "rw [← hm]",
"before_state": "case h.intro\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nk : ℕ\nhm : n = 2 ^ multiplicity 2 n * (2 * k)\n⊢ 2 ^ multiplicity 2 n * (2 * k) = n",
"after_state": "No Goals!"
},
{
"line": "rewrite [← hm]",
"before_state": "case h.intro\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nk : ℕ\nhm : n = 2 ^ multiplicity 2 n * (2 * k)\n⊢ 2 ^ multiplicity 2 n * (2 * k) = n",
"after_state": "case h.intro\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nk : ℕ\nhm : n = 2 ^ multiplicity 2 n * (2 * k)\n⊢ n = n"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case h.intro\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nk : ℕ\nhm : n = 2 ^ multiplicity 2 n * (2 * k)\n⊢ n = n",
"after_state": "No Goals!"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case h.intro\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nk : ℕ\nhm : n = 2 ^ multiplicity 2 n * (2 * k)\n⊢ n = n",
"after_state": "No Goals!"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case h.intro\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nk : ℕ\nhm : n = 2 ^ multiplicity 2 n * (2 * k)\n⊢ n = n",
"after_state": "No Goals!"
},
{
"line": "with_reducible rfl",
"before_state": "case h.intro\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nk : ℕ\nhm : n = 2 ^ multiplicity 2 n * (2 * k)\n⊢ n = n",
"after_state": "No Goals!"
},
{
"line": "rfl",
"before_state": "case h.intro\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nk : ℕ\nhm : n = 2 ^ multiplicity 2 n * (2 * k)\n⊢ n = n",
"after_state": "No Goals!"
},
{
"line": "eq_refl",
"before_state": "case h.intro\nn : ℕ\nhpos : 0 < n\nh : FiniteMultiplicity 2 n\nk : ℕ\nhm : n = 2 ^ multiplicity 2 n * (2 * k)\n⊢ n = n",
"after_state": "No Goals!"
}
] |
theorem eq_two_pow_mul_prime_mersenne_of_even_perfect {n : ℕ} (ev : Even n) (perf : Nat.Perfect n) :
∃ k : ℕ, Nat.Prime (mersenne (k + 1)) ∧ n = 2 ^ k * mersenne (k + 1) := by
have hpos := perf.2
rcases eq_two_pow_mul_odd hpos with ⟨k, m, rfl, hm⟩
use k
rw [even_iff_two_dvd] at hm
rw [Nat.perfect_iff_sum_divisors_eq_two_mul hpos] at perf
rw [← sigma_one_apply] at perf
rw [isMultiplicative_sigma.map_mul_of_coprime (Nat.prime_two.coprime_pow_of_not_dvd hm).symm] at perf
rw [sigma_two_pow_eq_mersenne_succ] at perf
rw [← mul_assoc] at perf
rw [← pow_succ'] at perf
obtain ⟨j, rfl⟩ := ((Odd.coprime_two_right (by simp)).pow_right _).dvd_of_dvd_mul_left
(Dvd.intro _ perf)
rw [← mul_assoc] at perf
rw [mul_comm _ (mersenne _)] at perf
rw [mul_assoc] at perf
have h := mul_left_cancel₀ (by positivity) perf
rw [sigma_one_apply] at h
rw [Nat.sum_divisors_eq_sum_properDivisors_add_self] at h
rw [← succ_mersenne] at h
rw [add_mul] at h
rw [one_mul] at h
rw [add_comm] at h
have hj := add_left_cancel h
cases Nat.sum_properDivisors_dvd (by rw [hj]; apply Dvd.intro_left (mersenne (k + 1)) rfl) with
| inl h_1 =>
have j1 : j = 1 := Eq.trans hj.symm h_1
rw [j1] at h_1
rw [mul_one] at h_1
rw [Nat.sum_properDivisors_eq_one_iff_prime] at h_1
simp [h_1, j1]
| inr h_1 =>
have jcon := Eq.trans hj.symm h_1
rw [← one_mul j] at jcon
rw [← mul_assoc] at jcon
rw [mul_one] at jcon
have jcon2 := mul_right_cancel₀ ?_ jcon
· exfalso
match k with
| 0 =>
apply hm
rw [← jcon2] at ev
rw [pow_zero] at ev
rw [one_mul] at ev
rw [one_mul] at ev
rw [← jcon2]
rw [one_mul]
exact even_iff_two_dvd.mp ev
| .succ k =>
apply ne_of_lt _ jcon2
rw [mersenne]
rw [← Nat.pred_eq_sub_one]
rw [Nat.lt_pred_iff]
rw [← pow_one (Nat.succ 1)]
apply pow_lt_pow_right₀ (Nat.lt_succ_self 1) (Nat.succ_lt_succ k.succ_pos)
contrapose! hm
simp [hm]
|
/root/DuelModelResearch/mathlib4/Archive/Wiedijk100Theorems/PerfectNumbers.lean
|
{
"open": [
"ArithmeticFunction Finset"
],
"variables": []
}
|
[
{
"line": "have hpos := perf.2",
"before_state": "n : ℕ\nev : Even n\nperf : n.Perfect\n⊢ ∃ k, Nat.Prime (mersenne (k + 1)) ∧ n = 2 ^ k * mersenne (k + 1)",
"after_state": "n : ℕ\nev : Even n\nperf : n.Perfect\nhpos : 0 < n\n⊢ ∃ k, Nat.Prime (mersenne (k + 1)) ∧ n = 2 ^ k * mersenne (k + 1)"
},
{
"line": "refine_lift\n have hpos := perf.2;\n ?_",
"before_state": "n : ℕ\nev : Even n\nperf : n.Perfect\n⊢ ∃ k, Nat.Prime (mersenne (k + 1)) ∧ n = 2 ^ k * mersenne (k + 1)",
"after_state": "n : ℕ\nev : Even n\nperf : n.Perfect\nhpos : 0 < n\n⊢ ∃ k, Nat.Prime (mersenne (k + 1)) ∧ n = 2 ^ k * mersenne (k + 1)"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (have hpos := perf.2;\n ?_);\n rotate_right)",
"before_state": "n : ℕ\nev : Even n\nperf : n.Perfect\n⊢ ∃ k, Nat.Prime (mersenne (k + 1)) ∧ n = 2 ^ k * mersenne (k + 1)",
"after_state": "n : ℕ\nev : Even n\nperf : n.Perfect\nhpos : 0 < n\n⊢ ∃ k, Nat.Prime (mersenne (k + 1)) ∧ n = 2 ^ k * mersenne (k + 1)"
},
{
"line": "refine\n no_implicit_lambda%\n (have hpos := perf.2;\n ?_)",
"before_state": "n : ℕ\nev : Even n\nperf : n.Perfect\n⊢ ∃ k, Nat.Prime (mersenne (k + 1)) ∧ n = 2 ^ k * mersenne (k + 1)",
"after_state": "n : ℕ\nev : Even n\nperf : n.Perfect\nhpos : 0 < n\n⊢ ∃ k, Nat.Prime (mersenne (k + 1)) ∧ n = 2 ^ k * mersenne (k + 1)"
},
{
"line": "rotate_right",
"before_state": "n : ℕ\nev : Even n\nperf : n.Perfect\nhpos : 0 < n\n⊢ ∃ k, Nat.Prime (mersenne (k + 1)) ∧ n = 2 ^ k * mersenne (k + 1)",
"after_state": "n : ℕ\nev : Even n\nperf : n.Perfect\nhpos : 0 < n\n⊢ ∃ k, Nat.Prime (mersenne (k + 1)) ∧ n = 2 ^ k * mersenne (k + 1)"
},
{
"line": "rcases eq_two_pow_mul_odd hpos with ⟨k, m, rfl, hm⟩",
"before_state": "n : ℕ\nev : Even n\nperf : n.Perfect\nhpos : 0 < n\n⊢ ∃ k, Nat.Prime (mersenne (k + 1)) ∧ n = 2 ^ k * mersenne (k + 1)",
"after_state": "No Goals!"
}
] |
theorem even_and_perfect_iff {n : ℕ} :
Even n ∧ Nat.Perfect n ↔ ∃ k : ℕ, Nat.Prime (mersenne (k + 1)) ∧
n = 2 ^ k * mersenne (k + 1) := by
constructor
· rintro ⟨ev, perf⟩
exact Nat.eq_two_pow_mul_prime_mersenne_of_even_perfect ev perf
· rintro ⟨k, pr, rfl⟩
exact ⟨even_two_pow_mul_mersenne_of_prime k pr, perfect_two_pow_mul_mersenne_of_prime k pr⟩
|
/root/DuelModelResearch/mathlib4/Archive/Wiedijk100Theorems/PerfectNumbers.lean
|
{
"open": [
"ArithmeticFunction Finset"
],
"variables": []
}
|
[
{
"line": "constructor",
"before_state": "n : ℕ\n⊢ Even n ∧ n.Perfect ↔ ∃ k, Nat.Prime (mersenne (k + 1)) ∧ n = 2 ^ k * mersenne (k + 1)",
"after_state": "case mp\nn : ℕ\n⊢ Even n ∧ n.Perfect → ∃ k, Nat.Prime (mersenne (k + 1)) ∧ n = 2 ^ k * mersenne (k + 1)\n---\ncase mpr\nn : ℕ\n⊢ (∃ k, Nat.Prime (mersenne (k + 1)) ∧ n = 2 ^ k * mersenne (k + 1)) → Even n ∧ n.Perfect"
},
{
"line": "rintro ⟨ev, perf⟩",
"before_state": "case mp\nn : ℕ\n⊢ Even n ∧ n.Perfect → ∃ k, Nat.Prime (mersenne (k + 1)) ∧ n = 2 ^ k * mersenne (k + 1)",
"after_state": "case mp.intro\nn : ℕ\nev : Even n\nperf : n.Perfect\n⊢ ∃ k, Nat.Prime (mersenne (k + 1)) ∧ n = 2 ^ k * mersenne (k + 1)"
},
{
"line": "exact Nat.eq_two_pow_mul_prime_mersenne_of_even_perfect ev perf",
"before_state": "case mp.intro\nn : ℕ\nev : Even n\nperf : n.Perfect\n⊢ ∃ k, Nat.Prime (mersenne (k + 1)) ∧ n = 2 ^ k * mersenne (k + 1)",
"after_state": "No Goals!"
},
{
"line": "rintro ⟨k, pr, rfl⟩",
"before_state": "case mpr\nn : ℕ\n⊢ (∃ k, Nat.Prime (mersenne (k + 1)) ∧ n = 2 ^ k * mersenne (k + 1)) → Even n ∧ n.Perfect",
"after_state": "case mpr.intro.intro\nk : ℕ\npr : Nat.Prime (mersenne (k + 1))\n⊢ Even (2 ^ k * mersenne (k + 1)) ∧ (2 ^ k * mersenne (k + 1)).Perfect"
},
{
"line": "exact ⟨even_two_pow_mul_mersenne_of_prime k pr, perfect_two_pow_mul_mersenne_of_prime k pr⟩",
"before_state": "case mpr.intro.intro\nk : ℕ\npr : Nat.Prime (mersenne (k + 1))\n⊢ Even (2 ^ k * mersenne (k + 1)) ∧ (2 ^ k * mersenne (k + 1)).Perfect",
"after_state": "No Goals!"
}
] |
theorem Real.tendsto_sum_one_div_prime_atTop :
Tendsto (fun n => ∑ p ∈ range n with p.Prime, 1 / (p : ℝ))
atTop atTop := by
-- Assume that the sum of the reciprocals of the primes converges.
by_contra h
-- Then there is a natural number `k` such that for all `x`, the sum of the reciprocals of primes
-- between `k` and `x` is less than 1/2.
obtain ⟨k, h1⟩ := sum_lt_half_of_not_tendsto h
-- Choose `x` sufficiently large for the argument below to work, and use a perfect square so we
-- can easily take the square root.
let x := 2 ^ (k + 1) * 2 ^ (k + 1)
-- We will partition `range x` into two subsets:
-- * `M`, the subset of those `e` for which `e + 1` is a product of powers of primes smaller
-- than or equal to `k`;
set M' := M x k with hM'
-- * `U`, the subset of those `e` for which there is a prime `p > k` that divides `e + 1`.
let P := {p ∈ range (x + 1) | k < p ∧ p.Prime}
set U' := U x k with hU'
-- This is indeed a partition, so `|U| + |M| = |range x| = x`.
have h2 : x = #U' + #M' := by
rw [← card_range x]
rw [hU']
rw [hM']
rw [← range_sdiff_eq_biUnion]
classical
exact (card_sdiff_add_card_eq_card (Finset.filter_subset _ _)).symm
-- But for the `x` we have chosen above, both `|U|` and `|M|` are less than or equal to `x / 2`,
-- and for U, the inequality is strict.
have h3 :=
calc
(#U' : ℝ) ≤ x * ∑ p ∈ P, 1 / (p : ℝ) := card_le_mul_sum
_ < x * (1 / 2) := mul_lt_mul_of_pos_left (h1 x) (by norm_num [x])
_ = x / 2 := mul_one_div (x : ℝ) 2
have h4 :=
calc
(#M' : ℝ) ≤ 2 ^ k * x.sqrt := by exact mod_cast card_le_two_pow_mul_sqrt
_ = 2 ^ k * (2 ^ (k + 1) : ℕ) := by rw [Nat.sqrt_eq]
_ = x / 2 := by field_simp [x, mul_right_comm, ← pow_succ]
refine lt_irrefl (x : ℝ) ?_
calc
(x : ℝ) = (#U' : ℝ) + (#M' : ℝ) := by assumption_mod_cast
_ < x / 2 + x / 2 := add_lt_add_of_lt_of_le h3 h4
_ = x := add_halves (x : ℝ)
|
/root/DuelModelResearch/mathlib4/Archive/Wiedijk100Theorems/SumOfPrimeReciprocalsDiverges.lean
|
{
"open": [
"Filter Finset",
"Classical in"
],
"variables": []
}
|
[
{
"line": "by_contra h",
"before_state": "⊢ Tendsto (fun n => ∑ p ∈ {p ∈ range n | Nat.Prime p}, 1 / ↑p) atTop atTop",
"after_state": "h : ¬Tendsto (fun n => ∑ p ∈ {p ∈ range n | Nat.Prime p}, 1 / ↑p) atTop atTop\n⊢ False"
},
{
"line": "first\n| guard_target = Not✝ _; intro h\n| refine (Decidable.byContradiction✝ fun h => ?_ :)\n| refine (Classical.byContradiction✝ fun h => ?_ :)",
"before_state": "⊢ Tendsto (fun n => ∑ p ∈ {p ∈ range n | Nat.Prime p}, 1 / ↑p) atTop atTop",
"after_state": "h : ¬Tendsto (fun n => ∑ p ∈ {p ∈ range n | Nat.Prime p}, 1 / ↑p) atTop atTop\n⊢ False"
},
{
"line": "guard_target = Not✝ _",
"before_state": "⊢ Tendsto (fun n => ∑ p ∈ {p ∈ range n | Nat.Prime p}, 1 / ↑p) atTop atTop",
"after_state": "⊢ Tendsto (fun n => ∑ p ∈ {p ∈ range n | Nat.Prime p}, 1 / ↑p) atTop atTop"
},
{
"line": "refine (Decidable.byContradiction✝ fun h => ?_ :)",
"before_state": "⊢ Tendsto (fun n => ∑ p ∈ {p ∈ range n | Nat.Prime p}, 1 / ↑p) atTop atTop",
"after_state": "⊢ Tendsto (fun n => ∑ p ∈ {p ∈ range n | Nat.Prime p}, 1 / ↑p) atTop atTop"
},
{
"line": "refine (Classical.byContradiction✝ fun h => ?_ :)",
"before_state": "⊢ Tendsto (fun n => ∑ p ∈ {p ∈ range n | Nat.Prime p}, 1 / ↑p) atTop atTop",
"after_state": "h : ¬Tendsto (fun n => ∑ p ∈ {p ∈ range n | Nat.Prime p}, 1 / ↑p) atTop atTop\n⊢ False"
},
{
"line": "obtain ⟨k, h1⟩ := sum_lt_half_of_not_tendsto h",
"before_state": "h : ¬Tendsto (fun n => ∑ p ∈ {p ∈ range n | Nat.Prime p}, 1 / ↑p) atTop atTop\n⊢ False",
"after_state": "No Goals!"
}
] |
theorem add_one_eq_one (x : WithZero Unit) : x + 1 = 1 :=
WithZero.cases_on x (by rfl) fun h => by rfl
|
/root/DuelModelResearch/mathlib4/Counterexamples/CharPZeroNeCharZero.lean
|
{
"open": [],
"variables": []
}
|
[
{
"line": "rfl",
"before_state": "x : WithZero Unit\n⊢ 0 + 1 = 1",
"after_state": "No Goals!"
},
{
"line": "eq_refl",
"before_state": "x : WithZero Unit\n⊢ 0 + 1 = 1",
"after_state": "No Goals!"
},
{
"line": "rfl",
"before_state": "x : WithZero Unit\nh : Unit\n⊢ ↑h + 1 = 1",
"after_state": "No Goals!"
},
{
"line": "eq_refl",
"before_state": "x : WithZero Unit\nh : Unit\n⊢ ↑h + 1 = 1",
"after_state": "No Goals!"
}
] |
theorem withZero_unit_charP_zero : CharP (WithZero Unit) 0 :=
⟨fun x => by cases x <;> simp⟩
|
/root/DuelModelResearch/mathlib4/Counterexamples/CharPZeroNeCharZero.lean
|
{
"open": [],
"variables": []
}
|
[
{
"line": "focus\n cases x\n with_annotate_state\"<;>\" skip\n all_goals simp",
"before_state": "x : ℕ\n⊢ ↑x = 0 ↔ 0 ∣ x",
"after_state": "No Goals!"
},
{
"line": "cases x",
"before_state": "x : ℕ\n⊢ ↑x = 0 ↔ 0 ∣ x",
"after_state": "case zero\n⊢ ↑0 = 0 ↔ 0 ∣ 0\n---\ncase succ\nn✝ : ℕ\n⊢ ↑(n✝ + 1) = 0 ↔ 0 ∣ n✝ + 1"
},
{
"line": "with_annotate_state\"<;>\" skip",
"before_state": "case zero\n⊢ ↑0 = 0 ↔ 0 ∣ 0\n---\ncase succ\nn✝ : ℕ\n⊢ ↑(n✝ + 1) = 0 ↔ 0 ∣ n✝ + 1",
"after_state": "case zero\n⊢ ↑0 = 0 ↔ 0 ∣ 0\n---\ncase succ\nn✝ : ℕ\n⊢ ↑(n✝ + 1) = 0 ↔ 0 ∣ n✝ + 1"
},
{
"line": "skip",
"before_state": "case zero\n⊢ ↑0 = 0 ↔ 0 ∣ 0\n---\ncase succ\nn✝ : ℕ\n⊢ ↑(n✝ + 1) = 0 ↔ 0 ∣ n✝ + 1",
"after_state": "case zero\n⊢ ↑0 = 0 ↔ 0 ∣ 0\n---\ncase succ\nn✝ : ℕ\n⊢ ↑(n✝ + 1) = 0 ↔ 0 ∣ n✝ + 1"
},
{
"line": "all_goals simp",
"before_state": "case zero\n⊢ ↑0 = 0 ↔ 0 ∣ 0\n---\ncase succ\nn✝ : ℕ\n⊢ ↑(n✝ + 1) = 0 ↔ 0 ∣ n✝ + 1",
"after_state": "No Goals!"
},
{
"line": "simp",
"before_state": "case zero\n⊢ ↑0 = 0 ↔ 0 ∣ 0",
"after_state": "No Goals!"
},
{
"line": "simp",
"before_state": "case succ\nn✝ : ℕ\n⊢ ↑(n✝ + 1) = 0 ↔ 0 ∣ n✝ + 1",
"after_state": "No Goals!"
}
] |
theorem star_sq : star * star ≈ star := by
have le : star * star ≤ star := by
rw [le_iff_forall_lf]
constructor <;>
intro i
· apply leftMoves_mul_cases i <;>
intro _ _
case' hl => rw [mul_moveLeft_inl]
case' hr => rw [mul_moveLeft_inr]
all_goals rw [lf_iff_game_lf]; simpa using zero_lf_star
· refine lf_zero.2 ⟨toRightMovesMul (Sum.inl default), ?_⟩
rintro (j | j) <;> -- Instance can't be inferred otherwise.
exact isEmptyElim j
constructor
case' right =>
rw [← neg_le_neg_iff];
apply (neg_mul _ _).symm.equiv.1.trans;
rw [neg_star]
assumption'
|
/root/DuelModelResearch/mathlib4/Counterexamples/GameMultiplication.lean
|
{
"open": [
"SetTheory PGame"
],
"variables": []
}
|
[
{
"line": "have le : star * star ≤ star := by\n rw [le_iff_forall_lf]\n constructor <;> intro i\n · apply leftMoves_mul_cases i <;> intro _ _\n case' hl => rw [mul_moveLeft_inl]\n case' hr => rw [mul_moveLeft_inr]\n all_goals rw [lf_iff_game_lf]; simpa using zero_lf_star\n · refine lf_zero.2 ⟨toRightMovesMul (Sum.inl default), ?_⟩\n rintro (j | j) <;>\n -- Instance can't be inferred otherwise.exact isEmptyElim j",
"before_state": "⊢ PGame.star * PGame.star ≈ PGame.star",
"after_state": "le : PGame.star * PGame.star ≤ PGame.star\n⊢ PGame.star * PGame.star ≈ PGame.star"
},
{
"line": "focus\n refine\n no_implicit_lambda%\n (have le : star * star ≤ star := ?body✝;\n ?_)\n case body✝ =>\n with_annotate_state\"by\"\n ( rw [le_iff_forall_lf]\n constructor <;> intro i\n · apply leftMoves_mul_cases i <;> intro _ _\n case' hl => rw [mul_moveLeft_inl]\n case' hr => rw [mul_moveLeft_inr]\n all_goals rw [lf_iff_game_lf]; simpa using zero_lf_star\n · refine lf_zero.2 ⟨toRightMovesMul (Sum.inl default), ?_⟩\n rintro (j | j) <;>\n -- Instance can't be inferred otherwise.exact isEmptyElim j)",
"before_state": "⊢ PGame.star * PGame.star ≈ PGame.star",
"after_state": "le : PGame.star * PGame.star ≤ PGame.star\n⊢ PGame.star * PGame.star ≈ PGame.star"
},
{
"line": "refine\n no_implicit_lambda%\n (have le : star * star ≤ star := ?body✝;\n ?_)",
"before_state": "⊢ PGame.star * PGame.star ≈ PGame.star",
"after_state": "case body\n⊢ PGame.star * PGame.star ≤ PGame.star\n---\nle : PGame.star * PGame.star ≤ PGame.star\n⊢ PGame.star * PGame.star ≈ PGame.star"
},
{
"line": "case body✝ =>\n with_annotate_state\"by\"\n ( rw [le_iff_forall_lf]\n constructor <;> intro i\n · apply leftMoves_mul_cases i <;> intro _ _\n case' hl => rw [mul_moveLeft_inl]\n case' hr => rw [mul_moveLeft_inr]\n all_goals rw [lf_iff_game_lf]; simpa using zero_lf_star\n · refine lf_zero.2 ⟨toRightMovesMul (Sum.inl default), ?_⟩\n rintro (j | j) <;>\n -- Instance can't be inferred otherwise.exact isEmptyElim j)",
"before_state": "case body\n⊢ PGame.star * PGame.star ≤ PGame.star\n---\nle : PGame.star * PGame.star ≤ PGame.star\n⊢ PGame.star * PGame.star ≈ PGame.star",
"after_state": "le : PGame.star * PGame.star ≤ PGame.star\n⊢ PGame.star * PGame.star ≈ PGame.star"
},
{
"line": "with_annotate_state\"by\"\n ( rw [le_iff_forall_lf]\n constructor <;> intro i\n · apply leftMoves_mul_cases i <;> intro _ _\n case' hl => rw [mul_moveLeft_inl]\n case' hr => rw [mul_moveLeft_inr]\n all_goals rw [lf_iff_game_lf]; simpa using zero_lf_star\n · refine lf_zero.2 ⟨toRightMovesMul (Sum.inl default), ?_⟩\n rintro (j | j) <;>\n -- Instance can't be inferred otherwise.exact isEmptyElim j)",
"before_state": "⊢ PGame.star * PGame.star ≤ PGame.star",
"after_state": "No Goals!"
},
{
"line": "rw [le_iff_forall_lf]",
"before_state": "⊢ PGame.star * PGame.star ≤ PGame.star",
"after_state": "⊢ (∀ (i : (PGame.star * PGame.star).LeftMoves), (PGame.star * PGame.star).moveLeft i ⧏ PGame.star) ∧\n ∀ (j : PGame.star.RightMoves), PGame.star * PGame.star ⧏ PGame.star.moveRight j"
},
{
"line": "rewrite [le_iff_forall_lf]",
"before_state": "⊢ PGame.star * PGame.star ≤ PGame.star",
"after_state": "⊢ (∀ (i : (PGame.star * PGame.star).LeftMoves), (PGame.star * PGame.star).moveLeft i ⧏ PGame.star) ∧\n ∀ (j : PGame.star.RightMoves), PGame.star * PGame.star ⧏ PGame.star.moveRight j"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "⊢ (∀ (i : (PGame.star * PGame.star).LeftMoves), (PGame.star * PGame.star).moveLeft i ⧏ PGame.star) ∧\n ∀ (j : PGame.star.RightMoves), PGame.star * PGame.star ⧏ PGame.star.moveRight j",
"after_state": "⊢ (∀ (i : (PGame.star * PGame.star).LeftMoves), (PGame.star * PGame.star).moveLeft i ⧏ PGame.star) ∧\n ∀ (j : PGame.star.RightMoves), PGame.star * PGame.star ⧏ PGame.star.moveRight j"
},
{
"line": "try (with_reducible rfl)",
"before_state": "⊢ (∀ (i : (PGame.star * PGame.star).LeftMoves), (PGame.star * PGame.star).moveLeft i ⧏ PGame.star) ∧\n ∀ (j : PGame.star.RightMoves), PGame.star * PGame.star ⧏ PGame.star.moveRight j",
"after_state": "⊢ (∀ (i : (PGame.star * PGame.star).LeftMoves), (PGame.star * PGame.star).moveLeft i ⧏ PGame.star) ∧\n ∀ (j : PGame.star.RightMoves), PGame.star * PGame.star ⧏ PGame.star.moveRight j"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "⊢ (∀ (i : (PGame.star * PGame.star).LeftMoves), (PGame.star * PGame.star).moveLeft i ⧏ PGame.star) ∧\n ∀ (j : PGame.star.RightMoves), PGame.star * PGame.star ⧏ PGame.star.moveRight j",
"after_state": "⊢ (∀ (i : (PGame.star * PGame.star).LeftMoves), (PGame.star * PGame.star).moveLeft i ⧏ PGame.star) ∧\n ∀ (j : PGame.star.RightMoves), PGame.star * PGame.star ⧏ PGame.star.moveRight j"
},
{
"line": "with_reducible rfl",
"before_state": "⊢ (∀ (i : (PGame.star * PGame.star).LeftMoves), (PGame.star * PGame.star).moveLeft i ⧏ PGame.star) ∧\n ∀ (j : PGame.star.RightMoves), PGame.star * PGame.star ⧏ PGame.star.moveRight j",
"after_state": "⊢ (∀ (i : (PGame.star * PGame.star).LeftMoves), (PGame.star * PGame.star).moveLeft i ⧏ PGame.star) ∧\n ∀ (j : PGame.star.RightMoves), PGame.star * PGame.star ⧏ PGame.star.moveRight j"
},
{
"line": "rfl",
"before_state": "⊢ (∀ (i : (PGame.star * PGame.star).LeftMoves), (PGame.star * PGame.star).moveLeft i ⧏ PGame.star) ∧\n ∀ (j : PGame.star.RightMoves), PGame.star * PGame.star ⧏ PGame.star.moveRight j",
"after_state": "⊢ (∀ (i : (PGame.star * PGame.star).LeftMoves), (PGame.star * PGame.star).moveLeft i ⧏ PGame.star) ∧\n ∀ (j : PGame.star.RightMoves), PGame.star * PGame.star ⧏ PGame.star.moveRight j"
},
{
"line": "apply_rfl",
"before_state": "⊢ (∀ (i : (PGame.star * PGame.star).LeftMoves), (PGame.star * PGame.star).moveLeft i ⧏ PGame.star) ∧\n ∀ (j : PGame.star.RightMoves), PGame.star * PGame.star ⧏ PGame.star.moveRight j",
"after_state": "⊢ (∀ (i : (PGame.star * PGame.star).LeftMoves), (PGame.star * PGame.star).moveLeft i ⧏ PGame.star) ∧\n ∀ (j : PGame.star.RightMoves), PGame.star * PGame.star ⧏ PGame.star.moveRight j"
},
{
"line": "skip",
"before_state": "⊢ (∀ (i : (PGame.star * PGame.star).LeftMoves), (PGame.star * PGame.star).moveLeft i ⧏ PGame.star) ∧\n ∀ (j : PGame.star.RightMoves), PGame.star * PGame.star ⧏ PGame.star.moveRight j",
"after_state": "⊢ (∀ (i : (PGame.star * PGame.star).LeftMoves), (PGame.star * PGame.star).moveLeft i ⧏ PGame.star) ∧\n ∀ (j : PGame.star.RightMoves), PGame.star * PGame.star ⧏ PGame.star.moveRight j"
},
{
"line": "focus\n constructor\n with_annotate_state\"<;>\" skip\n all_goals intro i",
"before_state": "⊢ (∀ (i : (PGame.star * PGame.star).LeftMoves), (PGame.star * PGame.star).moveLeft i ⧏ PGame.star) ∧\n ∀ (j : PGame.star.RightMoves), PGame.star * PGame.star ⧏ PGame.star.moveRight j",
"after_state": "case left\ni : (PGame.star * PGame.star).LeftMoves\n⊢ (PGame.star * PGame.star).moveLeft i ⧏ PGame.star\n---\ncase right\ni : PGame.star.RightMoves\n⊢ PGame.star * PGame.star ⧏ PGame.star.moveRight i"
},
{
"line": "constructor",
"before_state": "⊢ (∀ (i : (PGame.star * PGame.star).LeftMoves), (PGame.star * PGame.star).moveLeft i ⧏ PGame.star) ∧\n ∀ (j : PGame.star.RightMoves), PGame.star * PGame.star ⧏ PGame.star.moveRight j",
"after_state": "case left\n⊢ ∀ (i : (PGame.star * PGame.star).LeftMoves), (PGame.star * PGame.star).moveLeft i ⧏ PGame.star\n---\ncase right\n⊢ ∀ (j : PGame.star.RightMoves), PGame.star * PGame.star ⧏ PGame.star.moveRight j"
},
{
"line": "with_annotate_state\"<;>\" skip",
"before_state": "case left\n⊢ ∀ (i : (PGame.star * PGame.star).LeftMoves), (PGame.star * PGame.star).moveLeft i ⧏ PGame.star\n---\ncase right\n⊢ ∀ (j : PGame.star.RightMoves), PGame.star * PGame.star ⧏ PGame.star.moveRight j",
"after_state": "case left\n⊢ ∀ (i : (PGame.star * PGame.star).LeftMoves), (PGame.star * PGame.star).moveLeft i ⧏ PGame.star\n---\ncase right\n⊢ ∀ (j : PGame.star.RightMoves), PGame.star * PGame.star ⧏ PGame.star.moveRight j"
},
{
"line": "skip",
"before_state": "case left\n⊢ ∀ (i : (PGame.star * PGame.star).LeftMoves), (PGame.star * PGame.star).moveLeft i ⧏ PGame.star\n---\ncase right\n⊢ ∀ (j : PGame.star.RightMoves), PGame.star * PGame.star ⧏ PGame.star.moveRight j",
"after_state": "case left\n⊢ ∀ (i : (PGame.star * PGame.star).LeftMoves), (PGame.star * PGame.star).moveLeft i ⧏ PGame.star\n---\ncase right\n⊢ ∀ (j : PGame.star.RightMoves), PGame.star * PGame.star ⧏ PGame.star.moveRight j"
},
{
"line": "all_goals intro i",
"before_state": "case left\n⊢ ∀ (i : (PGame.star * PGame.star).LeftMoves), (PGame.star * PGame.star).moveLeft i ⧏ PGame.star\n---\ncase right\n⊢ ∀ (j : PGame.star.RightMoves), PGame.star * PGame.star ⧏ PGame.star.moveRight j",
"after_state": "case left\ni : (PGame.star * PGame.star).LeftMoves\n⊢ (PGame.star * PGame.star).moveLeft i ⧏ PGame.star\n---\ncase right\ni : PGame.star.RightMoves\n⊢ PGame.star * PGame.star ⧏ PGame.star.moveRight i"
},
{
"line": "intro i",
"before_state": "case left\n⊢ ∀ (i : (PGame.star * PGame.star).LeftMoves), (PGame.star * PGame.star).moveLeft i ⧏ PGame.star",
"after_state": "case left\ni : (PGame.star * PGame.star).LeftMoves\n⊢ (PGame.star * PGame.star).moveLeft i ⧏ PGame.star"
},
{
"line": "intro i",
"before_state": "case right\n⊢ ∀ (j : PGame.star.RightMoves), PGame.star * PGame.star ⧏ PGame.star.moveRight j",
"after_state": "case right\ni : PGame.star.RightMoves\n⊢ PGame.star * PGame.star ⧏ PGame.star.moveRight i"
},
{
"line": "focus\n apply leftMoves_mul_cases i\n with_annotate_state\"<;>\" skip\n all_goals intro _ _",
"before_state": "case left\ni : (PGame.star * PGame.star).LeftMoves\n⊢ (PGame.star * PGame.star).moveLeft i ⧏ PGame.star",
"after_state": "case left.hl\ni : (PGame.star * PGame.star).LeftMoves\nix✝ iy✝ : PGame.star.LeftMoves\n⊢ (PGame.star * PGame.star).moveLeft (toLeftMovesMul (Sum.inl (ix✝, iy✝))) ⧏ PGame.star\n---\ncase left.hr\ni : (PGame.star * PGame.star).LeftMoves\njx✝ jy✝ : PGame.star.RightMoves\n⊢ (PGame.star * PGame.star).moveLeft (toLeftMovesMul (Sum.inr (jx✝, jy✝))) ⧏ PGame.star"
},
{
"line": "apply leftMoves_mul_cases i",
"before_state": "case left\ni : (PGame.star * PGame.star).LeftMoves\n⊢ (PGame.star * PGame.star).moveLeft i ⧏ PGame.star",
"after_state": "case left.hl\ni : (PGame.star * PGame.star).LeftMoves\n⊢ ∀ (ix iy : PGame.star.LeftMoves), (PGame.star * PGame.star).moveLeft (toLeftMovesMul (Sum.inl (ix, iy))) ⧏ PGame.star\n---\ncase left.hr\ni : (PGame.star * PGame.star).LeftMoves\n⊢ ∀ (jx jy : PGame.star.RightMoves), (PGame.star * PGame.star).moveLeft (toLeftMovesMul (Sum.inr (jx, jy))) ⧏ PGame.star"
},
{
"line": "with_annotate_state\"<;>\" skip",
"before_state": "case left.hl\ni : (PGame.star * PGame.star).LeftMoves\n⊢ ∀ (ix iy : PGame.star.LeftMoves), (PGame.star * PGame.star).moveLeft (toLeftMovesMul (Sum.inl (ix, iy))) ⧏ PGame.star\n---\ncase left.hr\ni : (PGame.star * PGame.star).LeftMoves\n⊢ ∀ (jx jy : PGame.star.RightMoves), (PGame.star * PGame.star).moveLeft (toLeftMovesMul (Sum.inr (jx, jy))) ⧏ PGame.star",
"after_state": "case left.hl\ni : (PGame.star * PGame.star).LeftMoves\n⊢ ∀ (ix iy : PGame.star.LeftMoves), (PGame.star * PGame.star).moveLeft (toLeftMovesMul (Sum.inl (ix, iy))) ⧏ PGame.star\n---\ncase left.hr\ni : (PGame.star * PGame.star).LeftMoves\n⊢ ∀ (jx jy : PGame.star.RightMoves), (PGame.star * PGame.star).moveLeft (toLeftMovesMul (Sum.inr (jx, jy))) ⧏ PGame.star"
},
{
"line": "skip",
"before_state": "case left.hl\ni : (PGame.star * PGame.star).LeftMoves\n⊢ ∀ (ix iy : PGame.star.LeftMoves), (PGame.star * PGame.star).moveLeft (toLeftMovesMul (Sum.inl (ix, iy))) ⧏ PGame.star\n---\ncase left.hr\ni : (PGame.star * PGame.star).LeftMoves\n⊢ ∀ (jx jy : PGame.star.RightMoves), (PGame.star * PGame.star).moveLeft (toLeftMovesMul (Sum.inr (jx, jy))) ⧏ PGame.star",
"after_state": "case left.hl\ni : (PGame.star * PGame.star).LeftMoves\n⊢ ∀ (ix iy : PGame.star.LeftMoves), (PGame.star * PGame.star).moveLeft (toLeftMovesMul (Sum.inl (ix, iy))) ⧏ PGame.star\n---\ncase left.hr\ni : (PGame.star * PGame.star).LeftMoves\n⊢ ∀ (jx jy : PGame.star.RightMoves), (PGame.star * PGame.star).moveLeft (toLeftMovesMul (Sum.inr (jx, jy))) ⧏ PGame.star"
},
{
"line": "all_goals intro _ _",
"before_state": "case left.hl\ni : (PGame.star * PGame.star).LeftMoves\n⊢ ∀ (ix iy : PGame.star.LeftMoves), (PGame.star * PGame.star).moveLeft (toLeftMovesMul (Sum.inl (ix, iy))) ⧏ PGame.star\n---\ncase left.hr\ni : (PGame.star * PGame.star).LeftMoves\n⊢ ∀ (jx jy : PGame.star.RightMoves), (PGame.star * PGame.star).moveLeft (toLeftMovesMul (Sum.inr (jx, jy))) ⧏ PGame.star",
"after_state": "case left.hl\ni : (PGame.star * PGame.star).LeftMoves\nix✝ iy✝ : PGame.star.LeftMoves\n⊢ (PGame.star * PGame.star).moveLeft (toLeftMovesMul (Sum.inl (ix✝, iy✝))) ⧏ PGame.star\n---\ncase left.hr\ni : (PGame.star * PGame.star).LeftMoves\njx✝ jy✝ : PGame.star.RightMoves\n⊢ (PGame.star * PGame.star).moveLeft (toLeftMovesMul (Sum.inr (jx✝, jy✝))) ⧏ PGame.star"
},
{
"line": "intro _ _",
"before_state": "case left.hl\ni : (PGame.star * PGame.star).LeftMoves\n⊢ ∀ (ix iy : PGame.star.LeftMoves), (PGame.star * PGame.star).moveLeft (toLeftMovesMul (Sum.inl (ix, iy))) ⧏ PGame.star",
"after_state": "case left.hl\ni : (PGame.star * PGame.star).LeftMoves\nix✝ iy✝ : PGame.star.LeftMoves\n⊢ (PGame.star * PGame.star).moveLeft (toLeftMovesMul (Sum.inl (ix✝, iy✝))) ⧏ PGame.star"
},
{
"line": "intro _;\n intro _",
"before_state": "case left.hl\ni : (PGame.star * PGame.star).LeftMoves\n⊢ ∀ (ix iy : PGame.star.LeftMoves), (PGame.star * PGame.star).moveLeft (toLeftMovesMul (Sum.inl (ix, iy))) ⧏ PGame.star",
"after_state": "case left.hl\ni : (PGame.star * PGame.star).LeftMoves\nix✝ iy✝ : PGame.star.LeftMoves\n⊢ (PGame.star * PGame.star).moveLeft (toLeftMovesMul (Sum.inl (ix✝, iy✝))) ⧏ PGame.star"
},
{
"line": "intro _",
"before_state": "case left.hl\ni : (PGame.star * PGame.star).LeftMoves\n⊢ ∀ (ix iy : PGame.star.LeftMoves), (PGame.star * PGame.star).moveLeft (toLeftMovesMul (Sum.inl (ix, iy))) ⧏ PGame.star",
"after_state": "case left.hl\ni : (PGame.star * PGame.star).LeftMoves\nix✝ : PGame.star.LeftMoves\n⊢ ∀ (iy : PGame.star.LeftMoves), (PGame.star * PGame.star).moveLeft (toLeftMovesMul (Sum.inl (ix✝, iy))) ⧏ PGame.star"
},
{
"line": "intro _",
"before_state": "case left.hl\ni : (PGame.star * PGame.star).LeftMoves\nix✝ : PGame.star.LeftMoves\n⊢ ∀ (iy : PGame.star.LeftMoves), (PGame.star * PGame.star).moveLeft (toLeftMovesMul (Sum.inl (ix✝, iy))) ⧏ PGame.star",
"after_state": "case left.hl\ni : (PGame.star * PGame.star).LeftMoves\nix✝ iy✝ : PGame.star.LeftMoves\n⊢ (PGame.star * PGame.star).moveLeft (toLeftMovesMul (Sum.inl (ix✝, iy✝))) ⧏ PGame.star"
},
{
"line": "intro _ _",
"before_state": "case left.hr\ni : (PGame.star * PGame.star).LeftMoves\n⊢ ∀ (jx jy : PGame.star.RightMoves), (PGame.star * PGame.star).moveLeft (toLeftMovesMul (Sum.inr (jx, jy))) ⧏ PGame.star",
"after_state": "case left.hr\ni : (PGame.star * PGame.star).LeftMoves\njx✝ jy✝ : PGame.star.RightMoves\n⊢ (PGame.star * PGame.star).moveLeft (toLeftMovesMul (Sum.inr (jx✝, jy✝))) ⧏ PGame.star"
},
{
"line": "intro _;\n intro _",
"before_state": "case left.hr\ni : (PGame.star * PGame.star).LeftMoves\n⊢ ∀ (jx jy : PGame.star.RightMoves), (PGame.star * PGame.star).moveLeft (toLeftMovesMul (Sum.inr (jx, jy))) ⧏ PGame.star",
"after_state": "case left.hr\ni : (PGame.star * PGame.star).LeftMoves\njx✝ jy✝ : PGame.star.RightMoves\n⊢ (PGame.star * PGame.star).moveLeft (toLeftMovesMul (Sum.inr (jx✝, jy✝))) ⧏ PGame.star"
},
{
"line": "intro _",
"before_state": "case left.hr\ni : (PGame.star * PGame.star).LeftMoves\n⊢ ∀ (jx jy : PGame.star.RightMoves), (PGame.star * PGame.star).moveLeft (toLeftMovesMul (Sum.inr (jx, jy))) ⧏ PGame.star",
"after_state": "case left.hr\ni : (PGame.star * PGame.star).LeftMoves\njx✝ : PGame.star.RightMoves\n⊢ ∀ (jy : PGame.star.RightMoves), (PGame.star * PGame.star).moveLeft (toLeftMovesMul (Sum.inr (jx✝, jy))) ⧏ PGame.star"
},
{
"line": "intro _",
"before_state": "case left.hr\ni : (PGame.star * PGame.star).LeftMoves\njx✝ : PGame.star.RightMoves\n⊢ ∀ (jy : PGame.star.RightMoves), (PGame.star * PGame.star).moveLeft (toLeftMovesMul (Sum.inr (jx✝, jy))) ⧏ PGame.star",
"after_state": "case left.hr\ni : (PGame.star * PGame.star).LeftMoves\njx✝ jy✝ : PGame.star.RightMoves\n⊢ (PGame.star * PGame.star).moveLeft (toLeftMovesMul (Sum.inr (jx✝, jy✝))) ⧏ PGame.star"
},
{
"line": "case' hl => rw [mul_moveLeft_inl]",
"before_state": "case left.hl\ni : (PGame.star * PGame.star).LeftMoves\nix✝ iy✝ : PGame.star.LeftMoves\n⊢ (PGame.star * PGame.star).moveLeft (toLeftMovesMul (Sum.inl (ix✝, iy✝))) ⧏ PGame.star\n---\ncase left.hr\ni : (PGame.star * PGame.star).LeftMoves\njx✝ jy✝ : PGame.star.RightMoves\n⊢ (PGame.star * PGame.star).moveLeft (toLeftMovesMul (Sum.inr (jx✝, jy✝))) ⧏ PGame.star",
"after_state": "case left.hl\ni : (PGame.star * PGame.star).LeftMoves\nix✝ iy✝ : PGame.star.LeftMoves\n⊢ PGame.star.moveLeft ix✝ * PGame.star + PGame.star * PGame.star.moveLeft iy✝ -\n PGame.star.moveLeft ix✝ * PGame.star.moveLeft iy✝ ⧏\n PGame.star\n---\ncase left.hr\ni : (PGame.star * PGame.star).LeftMoves\njx✝ jy✝ : PGame.star.RightMoves\n⊢ (PGame.star * PGame.star).moveLeft (toLeftMovesMul (Sum.inr (jx✝, jy✝))) ⧏ PGame.star"
},
{
"line": "rw [mul_moveLeft_inl]",
"before_state": "case left.hl\ni : (PGame.star * PGame.star).LeftMoves\nix✝ iy✝ : PGame.star.LeftMoves\n⊢ (PGame.star * PGame.star).moveLeft (toLeftMovesMul (Sum.inl (ix✝, iy✝))) ⧏ PGame.star",
"after_state": "case left.hl\ni : (PGame.star * PGame.star).LeftMoves\nix✝ iy✝ : PGame.star.LeftMoves\n⊢ PGame.star.moveLeft ix✝ * PGame.star + PGame.star * PGame.star.moveLeft iy✝ -\n PGame.star.moveLeft ix✝ * PGame.star.moveLeft iy✝ ⧏\n PGame.star"
},
{
"line": "rewrite [mul_moveLeft_inl]",
"before_state": "case left.hl\ni : (PGame.star * PGame.star).LeftMoves\nix✝ iy✝ : PGame.star.LeftMoves\n⊢ (PGame.star * PGame.star).moveLeft (toLeftMovesMul (Sum.inl (ix✝, iy✝))) ⧏ PGame.star",
"after_state": "case left.hl\ni : (PGame.star * PGame.star).LeftMoves\nix✝ iy✝ : PGame.star.LeftMoves\n⊢ PGame.star.moveLeft ix✝ * PGame.star + PGame.star * PGame.star.moveLeft iy✝ -\n PGame.star.moveLeft ix✝ * PGame.star.moveLeft iy✝ ⧏\n PGame.star"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case left.hl\ni : (PGame.star * PGame.star).LeftMoves\nix✝ iy✝ : PGame.star.LeftMoves\n⊢ PGame.star.moveLeft ix✝ * PGame.star + PGame.star * PGame.star.moveLeft iy✝ -\n PGame.star.moveLeft ix✝ * PGame.star.moveLeft iy✝ ⧏\n PGame.star",
"after_state": "case left.hl\ni : (PGame.star * PGame.star).LeftMoves\nix✝ iy✝ : PGame.star.LeftMoves\n⊢ PGame.star.moveLeft ix✝ * PGame.star + PGame.star * PGame.star.moveLeft iy✝ -\n PGame.star.moveLeft ix✝ * PGame.star.moveLeft iy✝ ⧏\n PGame.star"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case left.hl\ni : (PGame.star * PGame.star).LeftMoves\nix✝ iy✝ : PGame.star.LeftMoves\n⊢ PGame.star.moveLeft ix✝ * PGame.star + PGame.star * PGame.star.moveLeft iy✝ -\n PGame.star.moveLeft ix✝ * PGame.star.moveLeft iy✝ ⧏\n PGame.star",
"after_state": "case left.hl\ni : (PGame.star * PGame.star).LeftMoves\nix✝ iy✝ : PGame.star.LeftMoves\n⊢ PGame.star.moveLeft ix✝ * PGame.star + PGame.star * PGame.star.moveLeft iy✝ -\n PGame.star.moveLeft ix✝ * PGame.star.moveLeft iy✝ ⧏\n PGame.star"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case left.hl\ni : (PGame.star * PGame.star).LeftMoves\nix✝ iy✝ : PGame.star.LeftMoves\n⊢ PGame.star.moveLeft ix✝ * PGame.star + PGame.star * PGame.star.moveLeft iy✝ -\n PGame.star.moveLeft ix✝ * PGame.star.moveLeft iy✝ ⧏\n PGame.star",
"after_state": "case left.hl\ni : (PGame.star * PGame.star).LeftMoves\nix✝ iy✝ : PGame.star.LeftMoves\n⊢ PGame.star.moveLeft ix✝ * PGame.star + PGame.star * PGame.star.moveLeft iy✝ -\n PGame.star.moveLeft ix✝ * PGame.star.moveLeft iy✝ ⧏\n PGame.star"
},
{
"line": "with_reducible rfl",
"before_state": "case left.hl\ni : (PGame.star * PGame.star).LeftMoves\nix✝ iy✝ : PGame.star.LeftMoves\n⊢ PGame.star.moveLeft ix✝ * PGame.star + PGame.star * PGame.star.moveLeft iy✝ -\n PGame.star.moveLeft ix✝ * PGame.star.moveLeft iy✝ ⧏\n PGame.star",
"after_state": "case left.hl\ni : (PGame.star * PGame.star).LeftMoves\nix✝ iy✝ : PGame.star.LeftMoves\n⊢ PGame.star.moveLeft ix✝ * PGame.star + PGame.star * PGame.star.moveLeft iy✝ -\n PGame.star.moveLeft ix✝ * PGame.star.moveLeft iy✝ ⧏\n PGame.star"
},
{
"line": "rfl",
"before_state": "case left.hl\ni : (PGame.star * PGame.star).LeftMoves\nix✝ iy✝ : PGame.star.LeftMoves\n⊢ PGame.star.moveLeft ix✝ * PGame.star + PGame.star * PGame.star.moveLeft iy✝ -\n PGame.star.moveLeft ix✝ * PGame.star.moveLeft iy✝ ⧏\n PGame.star",
"after_state": "case left.hl\ni : (PGame.star * PGame.star).LeftMoves\nix✝ iy✝ : PGame.star.LeftMoves\n⊢ PGame.star.moveLeft ix✝ * PGame.star + PGame.star * PGame.star.moveLeft iy✝ -\n PGame.star.moveLeft ix✝ * PGame.star.moveLeft iy✝ ⧏\n PGame.star"
},
{
"line": "apply_rfl",
"before_state": "case left.hl\ni : (PGame.star * PGame.star).LeftMoves\nix✝ iy✝ : PGame.star.LeftMoves\n⊢ PGame.star.moveLeft ix✝ * PGame.star + PGame.star * PGame.star.moveLeft iy✝ -\n PGame.star.moveLeft ix✝ * PGame.star.moveLeft iy✝ ⧏\n PGame.star",
"after_state": "case left.hl\ni : (PGame.star * PGame.star).LeftMoves\nix✝ iy✝ : PGame.star.LeftMoves\n⊢ PGame.star.moveLeft ix✝ * PGame.star + PGame.star * PGame.star.moveLeft iy✝ -\n PGame.star.moveLeft ix✝ * PGame.star.moveLeft iy✝ ⧏\n PGame.star"
},
{
"line": "skip",
"before_state": "case left.hl\ni : (PGame.star * PGame.star).LeftMoves\nix✝ iy✝ : PGame.star.LeftMoves\n⊢ PGame.star.moveLeft ix✝ * PGame.star + PGame.star * PGame.star.moveLeft iy✝ -\n PGame.star.moveLeft ix✝ * PGame.star.moveLeft iy✝ ⧏\n PGame.star",
"after_state": "case left.hl\ni : (PGame.star * PGame.star).LeftMoves\nix✝ iy✝ : PGame.star.LeftMoves\n⊢ PGame.star.moveLeft ix✝ * PGame.star + PGame.star * PGame.star.moveLeft iy✝ -\n PGame.star.moveLeft ix✝ * PGame.star.moveLeft iy✝ ⧏\n PGame.star"
},
{
"line": "case' hr => rw [mul_moveLeft_inr]",
"before_state": "case left.hl\ni : (PGame.star * PGame.star).LeftMoves\nix✝ iy✝ : PGame.star.LeftMoves\n⊢ PGame.star.moveLeft ix✝ * PGame.star + PGame.star * PGame.star.moveLeft iy✝ -\n PGame.star.moveLeft ix✝ * PGame.star.moveLeft iy✝ ⧏\n PGame.star\n---\ncase left.hr\ni : (PGame.star * PGame.star).LeftMoves\njx✝ jy✝ : PGame.star.RightMoves\n⊢ (PGame.star * PGame.star).moveLeft (toLeftMovesMul (Sum.inr (jx✝, jy✝))) ⧏ PGame.star",
"after_state": "case left.hr\ni : (PGame.star * PGame.star).LeftMoves\njx✝ jy✝ : PGame.star.RightMoves\n⊢ PGame.star.moveRight jx✝ * PGame.star + PGame.star * PGame.star.moveRight jy✝ -\n PGame.star.moveRight jx✝ * PGame.star.moveRight jy✝ ⧏\n PGame.star\n---\ncase left.hl\ni : (PGame.star * PGame.star).LeftMoves\nix✝ iy✝ : PGame.star.LeftMoves\n⊢ PGame.star.moveLeft ix✝ * PGame.star + PGame.star * PGame.star.moveLeft iy✝ -\n PGame.star.moveLeft ix✝ * PGame.star.moveLeft iy✝ ⧏\n PGame.star"
},
{
"line": "rw [mul_moveLeft_inr]",
"before_state": "case left.hr\ni : (PGame.star * PGame.star).LeftMoves\njx✝ jy✝ : PGame.star.RightMoves\n⊢ (PGame.star * PGame.star).moveLeft (toLeftMovesMul (Sum.inr (jx✝, jy✝))) ⧏ PGame.star",
"after_state": "case left.hr\ni : (PGame.star * PGame.star).LeftMoves\njx✝ jy✝ : PGame.star.RightMoves\n⊢ PGame.star.moveRight jx✝ * PGame.star + PGame.star * PGame.star.moveRight jy✝ -\n PGame.star.moveRight jx✝ * PGame.star.moveRight jy✝ ⧏\n PGame.star"
},
{
"line": "rewrite [mul_moveLeft_inr]",
"before_state": "case left.hr\ni : (PGame.star * PGame.star).LeftMoves\njx✝ jy✝ : PGame.star.RightMoves\n⊢ (PGame.star * PGame.star).moveLeft (toLeftMovesMul (Sum.inr (jx✝, jy✝))) ⧏ PGame.star",
"after_state": "case left.hr\ni : (PGame.star * PGame.star).LeftMoves\njx✝ jy✝ : PGame.star.RightMoves\n⊢ PGame.star.moveRight jx✝ * PGame.star + PGame.star * PGame.star.moveRight jy✝ -\n PGame.star.moveRight jx✝ * PGame.star.moveRight jy✝ ⧏\n PGame.star"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case left.hr\ni : (PGame.star * PGame.star).LeftMoves\njx✝ jy✝ : PGame.star.RightMoves\n⊢ PGame.star.moveRight jx✝ * PGame.star + PGame.star * PGame.star.moveRight jy✝ -\n PGame.star.moveRight jx✝ * PGame.star.moveRight jy✝ ⧏\n PGame.star",
"after_state": "case left.hr\ni : (PGame.star * PGame.star).LeftMoves\njx✝ jy✝ : PGame.star.RightMoves\n⊢ PGame.star.moveRight jx✝ * PGame.star + PGame.star * PGame.star.moveRight jy✝ -\n PGame.star.moveRight jx✝ * PGame.star.moveRight jy✝ ⧏\n PGame.star"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case left.hr\ni : (PGame.star * PGame.star).LeftMoves\njx✝ jy✝ : PGame.star.RightMoves\n⊢ PGame.star.moveRight jx✝ * PGame.star + PGame.star * PGame.star.moveRight jy✝ -\n PGame.star.moveRight jx✝ * PGame.star.moveRight jy✝ ⧏\n PGame.star",
"after_state": "case left.hr\ni : (PGame.star * PGame.star).LeftMoves\njx✝ jy✝ : PGame.star.RightMoves\n⊢ PGame.star.moveRight jx✝ * PGame.star + PGame.star * PGame.star.moveRight jy✝ -\n PGame.star.moveRight jx✝ * PGame.star.moveRight jy✝ ⧏\n PGame.star"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case left.hr\ni : (PGame.star * PGame.star).LeftMoves\njx✝ jy✝ : PGame.star.RightMoves\n⊢ PGame.star.moveRight jx✝ * PGame.star + PGame.star * PGame.star.moveRight jy✝ -\n PGame.star.moveRight jx✝ * PGame.star.moveRight jy✝ ⧏\n PGame.star",
"after_state": "case left.hr\ni : (PGame.star * PGame.star).LeftMoves\njx✝ jy✝ : PGame.star.RightMoves\n⊢ PGame.star.moveRight jx✝ * PGame.star + PGame.star * PGame.star.moveRight jy✝ -\n PGame.star.moveRight jx✝ * PGame.star.moveRight jy✝ ⧏\n PGame.star"
},
{
"line": "with_reducible rfl",
"before_state": "case left.hr\ni : (PGame.star * PGame.star).LeftMoves\njx✝ jy✝ : PGame.star.RightMoves\n⊢ PGame.star.moveRight jx✝ * PGame.star + PGame.star * PGame.star.moveRight jy✝ -\n PGame.star.moveRight jx✝ * PGame.star.moveRight jy✝ ⧏\n PGame.star",
"after_state": "case left.hr\ni : (PGame.star * PGame.star).LeftMoves\njx✝ jy✝ : PGame.star.RightMoves\n⊢ PGame.star.moveRight jx✝ * PGame.star + PGame.star * PGame.star.moveRight jy✝ -\n PGame.star.moveRight jx✝ * PGame.star.moveRight jy✝ ⧏\n PGame.star"
},
{
"line": "rfl",
"before_state": "case left.hr\ni : (PGame.star * PGame.star).LeftMoves\njx✝ jy✝ : PGame.star.RightMoves\n⊢ PGame.star.moveRight jx✝ * PGame.star + PGame.star * PGame.star.moveRight jy✝ -\n PGame.star.moveRight jx✝ * PGame.star.moveRight jy✝ ⧏\n PGame.star",
"after_state": "case left.hr\ni : (PGame.star * PGame.star).LeftMoves\njx✝ jy✝ : PGame.star.RightMoves\n⊢ PGame.star.moveRight jx✝ * PGame.star + PGame.star * PGame.star.moveRight jy✝ -\n PGame.star.moveRight jx✝ * PGame.star.moveRight jy✝ ⧏\n PGame.star"
},
{
"line": "apply_rfl",
"before_state": "case left.hr\ni : (PGame.star * PGame.star).LeftMoves\njx✝ jy✝ : PGame.star.RightMoves\n⊢ PGame.star.moveRight jx✝ * PGame.star + PGame.star * PGame.star.moveRight jy✝ -\n PGame.star.moveRight jx✝ * PGame.star.moveRight jy✝ ⧏\n PGame.star",
"after_state": "case left.hr\ni : (PGame.star * PGame.star).LeftMoves\njx✝ jy✝ : PGame.star.RightMoves\n⊢ PGame.star.moveRight jx✝ * PGame.star + PGame.star * PGame.star.moveRight jy✝ -\n PGame.star.moveRight jx✝ * PGame.star.moveRight jy✝ ⧏\n PGame.star"
},
{
"line": "skip",
"before_state": "case left.hr\ni : (PGame.star * PGame.star).LeftMoves\njx✝ jy✝ : PGame.star.RightMoves\n⊢ PGame.star.moveRight jx✝ * PGame.star + PGame.star * PGame.star.moveRight jy✝ -\n PGame.star.moveRight jx✝ * PGame.star.moveRight jy✝ ⧏\n PGame.star",
"after_state": "case left.hr\ni : (PGame.star * PGame.star).LeftMoves\njx✝ jy✝ : PGame.star.RightMoves\n⊢ PGame.star.moveRight jx✝ * PGame.star + PGame.star * PGame.star.moveRight jy✝ -\n PGame.star.moveRight jx✝ * PGame.star.moveRight jy✝ ⧏\n PGame.star"
},
{
"line": "all_goals rw [lf_iff_game_lf]; simpa using zero_lf_star",
"before_state": "case left.hr\ni : (PGame.star * PGame.star).LeftMoves\njx✝ jy✝ : PGame.star.RightMoves\n⊢ PGame.star.moveRight jx✝ * PGame.star + PGame.star * PGame.star.moveRight jy✝ -\n PGame.star.moveRight jx✝ * PGame.star.moveRight jy✝ ⧏\n PGame.star\n---\ncase left.hl\ni : (PGame.star * PGame.star).LeftMoves\nix✝ iy✝ : PGame.star.LeftMoves\n⊢ PGame.star.moveLeft ix✝ * PGame.star + PGame.star * PGame.star.moveLeft iy✝ -\n PGame.star.moveLeft ix✝ * PGame.star.moveLeft iy✝ ⧏\n PGame.star",
"after_state": "No Goals!"
},
{
"line": "rw [lf_iff_game_lf]",
"before_state": "case left.hr\ni : (PGame.star * PGame.star).LeftMoves\njx✝ jy✝ : PGame.star.RightMoves\n⊢ PGame.star.moveRight jx✝ * PGame.star + PGame.star * PGame.star.moveRight jy✝ -\n PGame.star.moveRight jx✝ * PGame.star.moveRight jy✝ ⧏\n PGame.star",
"after_state": "case left.hr\ni : (PGame.star * PGame.star).LeftMoves\njx✝ jy✝ : PGame.star.RightMoves\n⊢ Game.LF\n ⟦PGame.star.moveRight jx✝ * PGame.star + PGame.star * PGame.star.moveRight jy✝ -\n PGame.star.moveRight jx✝ * PGame.star.moveRight jy✝⟧\n ⟦PGame.star⟧"
},
{
"line": "rewrite [lf_iff_game_lf]",
"before_state": "case left.hr\ni : (PGame.star * PGame.star).LeftMoves\njx✝ jy✝ : PGame.star.RightMoves\n⊢ PGame.star.moveRight jx✝ * PGame.star + PGame.star * PGame.star.moveRight jy✝ -\n PGame.star.moveRight jx✝ * PGame.star.moveRight jy✝ ⧏\n PGame.star",
"after_state": "case left.hr\ni : (PGame.star * PGame.star).LeftMoves\njx✝ jy✝ : PGame.star.RightMoves\n⊢ Game.LF\n ⟦PGame.star.moveRight jx✝ * PGame.star + PGame.star * PGame.star.moveRight jy✝ -\n PGame.star.moveRight jx✝ * PGame.star.moveRight jy✝⟧\n ⟦PGame.star⟧"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case left.hr\ni : (PGame.star * PGame.star).LeftMoves\njx✝ jy✝ : PGame.star.RightMoves\n⊢ Game.LF\n ⟦PGame.star.moveRight jx✝ * PGame.star + PGame.star * PGame.star.moveRight jy✝ -\n PGame.star.moveRight jx✝ * PGame.star.moveRight jy✝⟧\n ⟦PGame.star⟧",
"after_state": "case left.hr\ni : (PGame.star * PGame.star).LeftMoves\njx✝ jy✝ : PGame.star.RightMoves\n⊢ Game.LF\n ⟦PGame.star.moveRight jx✝ * PGame.star + PGame.star * PGame.star.moveRight jy✝ -\n PGame.star.moveRight jx✝ * PGame.star.moveRight jy✝⟧\n ⟦PGame.star⟧"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case left.hr\ni : (PGame.star * PGame.star).LeftMoves\njx✝ jy✝ : PGame.star.RightMoves\n⊢ Game.LF\n ⟦PGame.star.moveRight jx✝ * PGame.star + PGame.star * PGame.star.moveRight jy✝ -\n PGame.star.moveRight jx✝ * PGame.star.moveRight jy✝⟧\n ⟦PGame.star⟧",
"after_state": "case left.hr\ni : (PGame.star * PGame.star).LeftMoves\njx✝ jy✝ : PGame.star.RightMoves\n⊢ Game.LF\n ⟦PGame.star.moveRight jx✝ * PGame.star + PGame.star * PGame.star.moveRight jy✝ -\n PGame.star.moveRight jx✝ * PGame.star.moveRight jy✝⟧\n ⟦PGame.star⟧"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case left.hr\ni : (PGame.star * PGame.star).LeftMoves\njx✝ jy✝ : PGame.star.RightMoves\n⊢ Game.LF\n ⟦PGame.star.moveRight jx✝ * PGame.star + PGame.star * PGame.star.moveRight jy✝ -\n PGame.star.moveRight jx✝ * PGame.star.moveRight jy✝⟧\n ⟦PGame.star⟧",
"after_state": "case left.hr\ni : (PGame.star * PGame.star).LeftMoves\njx✝ jy✝ : PGame.star.RightMoves\n⊢ Game.LF\n ⟦PGame.star.moveRight jx✝ * PGame.star + PGame.star * PGame.star.moveRight jy✝ -\n PGame.star.moveRight jx✝ * PGame.star.moveRight jy✝⟧\n ⟦PGame.star⟧"
},
{
"line": "with_reducible rfl",
"before_state": "case left.hr\ni : (PGame.star * PGame.star).LeftMoves\njx✝ jy✝ : PGame.star.RightMoves\n⊢ Game.LF\n ⟦PGame.star.moveRight jx✝ * PGame.star + PGame.star * PGame.star.moveRight jy✝ -\n PGame.star.moveRight jx✝ * PGame.star.moveRight jy✝⟧\n ⟦PGame.star⟧",
"after_state": "case left.hr\ni : (PGame.star * PGame.star).LeftMoves\njx✝ jy✝ : PGame.star.RightMoves\n⊢ Game.LF\n ⟦PGame.star.moveRight jx✝ * PGame.star + PGame.star * PGame.star.moveRight jy✝ -\n PGame.star.moveRight jx✝ * PGame.star.moveRight jy✝⟧\n ⟦PGame.star⟧"
},
{
"line": "rfl",
"before_state": "case left.hr\ni : (PGame.star * PGame.star).LeftMoves\njx✝ jy✝ : PGame.star.RightMoves\n⊢ Game.LF\n ⟦PGame.star.moveRight jx✝ * PGame.star + PGame.star * PGame.star.moveRight jy✝ -\n PGame.star.moveRight jx✝ * PGame.star.moveRight jy✝⟧\n ⟦PGame.star⟧",
"after_state": "case left.hr\ni : (PGame.star * PGame.star).LeftMoves\njx✝ jy✝ : PGame.star.RightMoves\n⊢ Game.LF\n ⟦PGame.star.moveRight jx✝ * PGame.star + PGame.star * PGame.star.moveRight jy✝ -\n PGame.star.moveRight jx✝ * PGame.star.moveRight jy✝⟧\n ⟦PGame.star⟧"
},
{
"line": "apply_rfl",
"before_state": "case left.hr\ni : (PGame.star * PGame.star).LeftMoves\njx✝ jy✝ : PGame.star.RightMoves\n⊢ Game.LF\n ⟦PGame.star.moveRight jx✝ * PGame.star + PGame.star * PGame.star.moveRight jy✝ -\n PGame.star.moveRight jx✝ * PGame.star.moveRight jy✝⟧\n ⟦PGame.star⟧",
"after_state": "case left.hr\ni : (PGame.star * PGame.star).LeftMoves\njx✝ jy✝ : PGame.star.RightMoves\n⊢ Game.LF\n ⟦PGame.star.moveRight jx✝ * PGame.star + PGame.star * PGame.star.moveRight jy✝ -\n PGame.star.moveRight jx✝ * PGame.star.moveRight jy✝⟧\n ⟦PGame.star⟧"
},
{
"line": "skip",
"before_state": "case left.hr\ni : (PGame.star * PGame.star).LeftMoves\njx✝ jy✝ : PGame.star.RightMoves\n⊢ Game.LF\n ⟦PGame.star.moveRight jx✝ * PGame.star + PGame.star * PGame.star.moveRight jy✝ -\n PGame.star.moveRight jx✝ * PGame.star.moveRight jy✝⟧\n ⟦PGame.star⟧",
"after_state": "case left.hr\ni : (PGame.star * PGame.star).LeftMoves\njx✝ jy✝ : PGame.star.RightMoves\n⊢ Game.LF\n ⟦PGame.star.moveRight jx✝ * PGame.star + PGame.star * PGame.star.moveRight jy✝ -\n PGame.star.moveRight jx✝ * PGame.star.moveRight jy✝⟧\n ⟦PGame.star⟧"
},
{
"line": "simpa using zero_lf_star",
"before_state": "case left.hr\ni : (PGame.star * PGame.star).LeftMoves\njx✝ jy✝ : PGame.star.RightMoves\n⊢ Game.LF\n ⟦PGame.star.moveRight jx✝ * PGame.star + PGame.star * PGame.star.moveRight jy✝ -\n PGame.star.moveRight jx✝ * PGame.star.moveRight jy✝⟧\n ⟦PGame.star⟧",
"after_state": "No Goals!"
},
{
"line": "rw [lf_iff_game_lf]",
"before_state": "case left.hl\ni : (PGame.star * PGame.star).LeftMoves\nix✝ iy✝ : PGame.star.LeftMoves\n⊢ PGame.star.moveLeft ix✝ * PGame.star + PGame.star * PGame.star.moveLeft iy✝ -\n PGame.star.moveLeft ix✝ * PGame.star.moveLeft iy✝ ⧏\n PGame.star",
"after_state": "case left.hl\ni : (PGame.star * PGame.star).LeftMoves\nix✝ iy✝ : PGame.star.LeftMoves\n⊢ Game.LF\n ⟦PGame.star.moveLeft ix✝ * PGame.star + PGame.star * PGame.star.moveLeft iy✝ -\n PGame.star.moveLeft ix✝ * PGame.star.moveLeft iy✝⟧\n ⟦PGame.star⟧"
},
{
"line": "rewrite [lf_iff_game_lf]",
"before_state": "case left.hl\ni : (PGame.star * PGame.star).LeftMoves\nix✝ iy✝ : PGame.star.LeftMoves\n⊢ PGame.star.moveLeft ix✝ * PGame.star + PGame.star * PGame.star.moveLeft iy✝ -\n PGame.star.moveLeft ix✝ * PGame.star.moveLeft iy✝ ⧏\n PGame.star",
"after_state": "case left.hl\ni : (PGame.star * PGame.star).LeftMoves\nix✝ iy✝ : PGame.star.LeftMoves\n⊢ Game.LF\n ⟦PGame.star.moveLeft ix✝ * PGame.star + PGame.star * PGame.star.moveLeft iy✝ -\n PGame.star.moveLeft ix✝ * PGame.star.moveLeft iy✝⟧\n ⟦PGame.star⟧"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case left.hl\ni : (PGame.star * PGame.star).LeftMoves\nix✝ iy✝ : PGame.star.LeftMoves\n⊢ Game.LF\n ⟦PGame.star.moveLeft ix✝ * PGame.star + PGame.star * PGame.star.moveLeft iy✝ -\n PGame.star.moveLeft ix✝ * PGame.star.moveLeft iy✝⟧\n ⟦PGame.star⟧",
"after_state": "case left.hl\ni : (PGame.star * PGame.star).LeftMoves\nix✝ iy✝ : PGame.star.LeftMoves\n⊢ Game.LF\n ⟦PGame.star.moveLeft ix✝ * PGame.star + PGame.star * PGame.star.moveLeft iy✝ -\n PGame.star.moveLeft ix✝ * PGame.star.moveLeft iy✝⟧\n ⟦PGame.star⟧"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case left.hl\ni : (PGame.star * PGame.star).LeftMoves\nix✝ iy✝ : PGame.star.LeftMoves\n⊢ Game.LF\n ⟦PGame.star.moveLeft ix✝ * PGame.star + PGame.star * PGame.star.moveLeft iy✝ -\n PGame.star.moveLeft ix✝ * PGame.star.moveLeft iy✝⟧\n ⟦PGame.star⟧",
"after_state": "case left.hl\ni : (PGame.star * PGame.star).LeftMoves\nix✝ iy✝ : PGame.star.LeftMoves\n⊢ Game.LF\n ⟦PGame.star.moveLeft ix✝ * PGame.star + PGame.star * PGame.star.moveLeft iy✝ -\n PGame.star.moveLeft ix✝ * PGame.star.moveLeft iy✝⟧\n ⟦PGame.star⟧"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case left.hl\ni : (PGame.star * PGame.star).LeftMoves\nix✝ iy✝ : PGame.star.LeftMoves\n⊢ Game.LF\n ⟦PGame.star.moveLeft ix✝ * PGame.star + PGame.star * PGame.star.moveLeft iy✝ -\n PGame.star.moveLeft ix✝ * PGame.star.moveLeft iy✝⟧\n ⟦PGame.star⟧",
"after_state": "case left.hl\ni : (PGame.star * PGame.star).LeftMoves\nix✝ iy✝ : PGame.star.LeftMoves\n⊢ Game.LF\n ⟦PGame.star.moveLeft ix✝ * PGame.star + PGame.star * PGame.star.moveLeft iy✝ -\n PGame.star.moveLeft ix✝ * PGame.star.moveLeft iy✝⟧\n ⟦PGame.star⟧"
},
{
"line": "with_reducible rfl",
"before_state": "case left.hl\ni : (PGame.star * PGame.star).LeftMoves\nix✝ iy✝ : PGame.star.LeftMoves\n⊢ Game.LF\n ⟦PGame.star.moveLeft ix✝ * PGame.star + PGame.star * PGame.star.moveLeft iy✝ -\n PGame.star.moveLeft ix✝ * PGame.star.moveLeft iy✝⟧\n ⟦PGame.star⟧",
"after_state": "case left.hl\ni : (PGame.star * PGame.star).LeftMoves\nix✝ iy✝ : PGame.star.LeftMoves\n⊢ Game.LF\n ⟦PGame.star.moveLeft ix✝ * PGame.star + PGame.star * PGame.star.moveLeft iy✝ -\n PGame.star.moveLeft ix✝ * PGame.star.moveLeft iy✝⟧\n ⟦PGame.star⟧"
},
{
"line": "rfl",
"before_state": "case left.hl\ni : (PGame.star * PGame.star).LeftMoves\nix✝ iy✝ : PGame.star.LeftMoves\n⊢ Game.LF\n ⟦PGame.star.moveLeft ix✝ * PGame.star + PGame.star * PGame.star.moveLeft iy✝ -\n PGame.star.moveLeft ix✝ * PGame.star.moveLeft iy✝⟧\n ⟦PGame.star⟧",
"after_state": "case left.hl\ni : (PGame.star * PGame.star).LeftMoves\nix✝ iy✝ : PGame.star.LeftMoves\n⊢ Game.LF\n ⟦PGame.star.moveLeft ix✝ * PGame.star + PGame.star * PGame.star.moveLeft iy✝ -\n PGame.star.moveLeft ix✝ * PGame.star.moveLeft iy✝⟧\n ⟦PGame.star⟧"
},
{
"line": "apply_rfl",
"before_state": "case left.hl\ni : (PGame.star * PGame.star).LeftMoves\nix✝ iy✝ : PGame.star.LeftMoves\n⊢ Game.LF\n ⟦PGame.star.moveLeft ix✝ * PGame.star + PGame.star * PGame.star.moveLeft iy✝ -\n PGame.star.moveLeft ix✝ * PGame.star.moveLeft iy✝⟧\n ⟦PGame.star⟧",
"after_state": "case left.hl\ni : (PGame.star * PGame.star).LeftMoves\nix✝ iy✝ : PGame.star.LeftMoves\n⊢ Game.LF\n ⟦PGame.star.moveLeft ix✝ * PGame.star + PGame.star * PGame.star.moveLeft iy✝ -\n PGame.star.moveLeft ix✝ * PGame.star.moveLeft iy✝⟧\n ⟦PGame.star⟧"
},
{
"line": "skip",
"before_state": "case left.hl\ni : (PGame.star * PGame.star).LeftMoves\nix✝ iy✝ : PGame.star.LeftMoves\n⊢ Game.LF\n ⟦PGame.star.moveLeft ix✝ * PGame.star + PGame.star * PGame.star.moveLeft iy✝ -\n PGame.star.moveLeft ix✝ * PGame.star.moveLeft iy✝⟧\n ⟦PGame.star⟧",
"after_state": "case left.hl\ni : (PGame.star * PGame.star).LeftMoves\nix✝ iy✝ : PGame.star.LeftMoves\n⊢ Game.LF\n ⟦PGame.star.moveLeft ix✝ * PGame.star + PGame.star * PGame.star.moveLeft iy✝ -\n PGame.star.moveLeft ix✝ * PGame.star.moveLeft iy✝⟧\n ⟦PGame.star⟧"
},
{
"line": "simpa using zero_lf_star",
"before_state": "case left.hl\ni : (PGame.star * PGame.star).LeftMoves\nix✝ iy✝ : PGame.star.LeftMoves\n⊢ Game.LF\n ⟦PGame.star.moveLeft ix✝ * PGame.star + PGame.star * PGame.star.moveLeft iy✝ -\n PGame.star.moveLeft ix✝ * PGame.star.moveLeft iy✝⟧\n ⟦PGame.star⟧",
"after_state": "No Goals!"
},
{
"line": "refine lf_zero.2 ⟨toRightMovesMul (Sum.inl default), ?_⟩",
"before_state": "case right\ni : PGame.star.RightMoves\n⊢ PGame.star * PGame.star ⧏ PGame.star.moveRight i",
"after_state": "case right\ni : PGame.star.RightMoves\n⊢ ∀ (i : ((PGame.star * PGame.star).moveRight (toRightMovesMul (Sum.inl default))).LeftMoves),\n ((PGame.star * PGame.star).moveRight (toRightMovesMul (Sum.inl default))).moveLeft i ⧏ 0"
},
{
"line": "focus\n rintro (j | j)\n with_annotate_state\"<;>\" skip\n all_goals exact isEmptyElim j",
"before_state": "case right\ni : PGame.star.RightMoves\n⊢ ∀ (i : ((PGame.star * PGame.star).moveRight (toRightMovesMul (Sum.inl default))).LeftMoves),\n ((PGame.star * PGame.star).moveRight (toRightMovesMul (Sum.inl default))).moveLeft i ⧏ 0",
"after_state": "No Goals!"
},
{
"line": "rintro (j | j)",
"before_state": "case right\ni : PGame.star.RightMoves\n⊢ ∀ (i : ((PGame.star * PGame.star).moveRight (toRightMovesMul (Sum.inl default))).LeftMoves),\n ((PGame.star * PGame.star).moveRight (toRightMovesMul (Sum.inl default))).moveLeft i ⧏ 0",
"after_state": "case right.inl\ni : PGame.star.RightMoves\nj :\n (PEmpty.{?u.11384 + 1} × PUnit.{?u.11384 + 1} ⊕ PEmpty.{?u.11384 + 1} × PUnit.{?u.11384 + 1}) ⊕\n PUnit.{?u.11384 + 1} × PEmpty.{?u.11384 + 1} ⊕ PUnit.{?u.11384 + 1} × PEmpty.{?u.11384 + 1}\n⊢ ((PGame.star * PGame.star).moveRight (toRightMovesMul (Sum.inl default))).moveLeft (Sum.inl j) ⧏ 0\n---\ncase right.inr\ni : PGame.star.RightMoves\nj : PEmpty.{?u.11384 + 1} × PEmpty.{?u.11384 + 1} ⊕ PEmpty.{?u.11384 + 1} × PEmpty.{?u.11384 + 1}\n⊢ ((PGame.star * PGame.star).moveRight (toRightMovesMul (Sum.inl default))).moveLeft (Sum.inr j) ⧏ 0"
},
{
"line": "with_annotate_state\"<;>\" skip",
"before_state": "case right.inl\ni : PGame.star.RightMoves\nj :\n (PEmpty.{?u.11384 + 1} × PUnit.{?u.11384 + 1} ⊕ PEmpty.{?u.11384 + 1} × PUnit.{?u.11384 + 1}) ⊕\n PUnit.{?u.11384 + 1} × PEmpty.{?u.11384 + 1} ⊕ PUnit.{?u.11384 + 1} × PEmpty.{?u.11384 + 1}\n⊢ ((PGame.star * PGame.star).moveRight (toRightMovesMul (Sum.inl default))).moveLeft (Sum.inl j) ⧏ 0\n---\ncase right.inr\ni : PGame.star.RightMoves\nj : PEmpty.{?u.11384 + 1} × PEmpty.{?u.11384 + 1} ⊕ PEmpty.{?u.11384 + 1} × PEmpty.{?u.11384 + 1}\n⊢ ((PGame.star * PGame.star).moveRight (toRightMovesMul (Sum.inl default))).moveLeft (Sum.inr j) ⧏ 0",
"after_state": "case right.inl\ni : PGame.star.RightMoves\nj :\n (PEmpty.{?u.11384 + 1} × PUnit.{?u.11384 + 1} ⊕ PEmpty.{?u.11384 + 1} × PUnit.{?u.11384 + 1}) ⊕\n PUnit.{?u.11384 + 1} × PEmpty.{?u.11384 + 1} ⊕ PUnit.{?u.11384 + 1} × PEmpty.{?u.11384 + 1}\n⊢ ((PGame.star * PGame.star).moveRight (toRightMovesMul (Sum.inl default))).moveLeft (Sum.inl j) ⧏ 0\n---\ncase right.inr\ni : PGame.star.RightMoves\nj : PEmpty.{?u.11384 + 1} × PEmpty.{?u.11384 + 1} ⊕ PEmpty.{?u.11384 + 1} × PEmpty.{?u.11384 + 1}\n⊢ ((PGame.star * PGame.star).moveRight (toRightMovesMul (Sum.inl default))).moveLeft (Sum.inr j) ⧏ 0"
},
{
"line": "skip",
"before_state": "case right.inl\ni : PGame.star.RightMoves\nj :\n (PEmpty.{?u.11384 + 1} × PUnit.{?u.11384 + 1} ⊕ PEmpty.{?u.11384 + 1} × PUnit.{?u.11384 + 1}) ⊕\n PUnit.{?u.11384 + 1} × PEmpty.{?u.11384 + 1} ⊕ PUnit.{?u.11384 + 1} × PEmpty.{?u.11384 + 1}\n⊢ ((PGame.star * PGame.star).moveRight (toRightMovesMul (Sum.inl default))).moveLeft (Sum.inl j) ⧏ 0\n---\ncase right.inr\ni : PGame.star.RightMoves\nj : PEmpty.{?u.11384 + 1} × PEmpty.{?u.11384 + 1} ⊕ PEmpty.{?u.11384 + 1} × PEmpty.{?u.11384 + 1}\n⊢ ((PGame.star * PGame.star).moveRight (toRightMovesMul (Sum.inl default))).moveLeft (Sum.inr j) ⧏ 0",
"after_state": "case right.inl\ni : PGame.star.RightMoves\nj :\n (PEmpty.{?u.11384 + 1} × PUnit.{?u.11384 + 1} ⊕ PEmpty.{?u.11384 + 1} × PUnit.{?u.11384 + 1}) ⊕\n PUnit.{?u.11384 + 1} × PEmpty.{?u.11384 + 1} ⊕ PUnit.{?u.11384 + 1} × PEmpty.{?u.11384 + 1}\n⊢ ((PGame.star * PGame.star).moveRight (toRightMovesMul (Sum.inl default))).moveLeft (Sum.inl j) ⧏ 0\n---\ncase right.inr\ni : PGame.star.RightMoves\nj : PEmpty.{?u.11384 + 1} × PEmpty.{?u.11384 + 1} ⊕ PEmpty.{?u.11384 + 1} × PEmpty.{?u.11384 + 1}\n⊢ ((PGame.star * PGame.star).moveRight (toRightMovesMul (Sum.inl default))).moveLeft (Sum.inr j) ⧏ 0"
},
{
"line": "all_goals exact isEmptyElim j",
"before_state": "case right.inl\ni : PGame.star.RightMoves\nj :\n (PEmpty.{?u.11384 + 1} × PUnit.{?u.11384 + 1} ⊕ PEmpty.{?u.11384 + 1} × PUnit.{?u.11384 + 1}) ⊕\n PUnit.{?u.11384 + 1} × PEmpty.{?u.11384 + 1} ⊕ PUnit.{?u.11384 + 1} × PEmpty.{?u.11384 + 1}\n⊢ ((PGame.star * PGame.star).moveRight (toRightMovesMul (Sum.inl default))).moveLeft (Sum.inl j) ⧏ 0\n---\ncase right.inr\ni : PGame.star.RightMoves\nj : PEmpty.{?u.11384 + 1} × PEmpty.{?u.11384 + 1} ⊕ PEmpty.{?u.11384 + 1} × PEmpty.{?u.11384 + 1}\n⊢ ((PGame.star * PGame.star).moveRight (toRightMovesMul (Sum.inl default))).moveLeft (Sum.inr j) ⧏ 0",
"after_state": "No Goals!"
},
{
"line": "exact isEmptyElim j",
"before_state": "case right.inl\ni : PGame.star.RightMoves\nj :\n (PEmpty.{?u.11384 + 1} × PUnit.{?u.11384 + 1} ⊕ PEmpty.{?u.11384 + 1} × PUnit.{?u.11384 + 1}) ⊕\n PUnit.{?u.11384 + 1} × PEmpty.{?u.11384 + 1} ⊕ PUnit.{?u.11384 + 1} × PEmpty.{?u.11384 + 1}\n⊢ ((PGame.star * PGame.star).moveRight (toRightMovesMul (Sum.inl default))).moveLeft (Sum.inl j) ⧏ 0",
"after_state": "No Goals!"
},
{
"line": "exact isEmptyElim j",
"before_state": "case right.inr\ni : PGame.star.RightMoves\nj : PEmpty.{?u.11384 + 1} × PEmpty.{?u.11384 + 1} ⊕ PEmpty.{?u.11384 + 1} × PEmpty.{?u.11384 + 1}\n⊢ ((PGame.star * PGame.star).moveRight (toRightMovesMul (Sum.inl default))).moveLeft (Sum.inr j) ⧏ 0",
"after_state": "No Goals!"
},
{
"line": "constructor",
"before_state": "le : PGame.star * PGame.star ≤ PGame.star\n⊢ PGame.star * PGame.star ≈ PGame.star",
"after_state": "case left\nle : PGame.star * PGame.star ≤ PGame.star\n⊢ PGame.star * PGame.star ≤ PGame.star\n---\ncase right\nle : PGame.star * PGame.star ≤ PGame.star\n⊢ PGame.star ≤ PGame.star * PGame.star"
},
{
"line": "case' right => rw [← neg_le_neg_iff]; apply (neg_mul _ _).symm.equiv.1.trans; rw [neg_star]",
"before_state": "case left\nle : PGame.star * PGame.star ≤ PGame.star\n⊢ PGame.star * PGame.star ≤ PGame.star\n---\ncase right\nle : PGame.star * PGame.star ≤ PGame.star\n⊢ PGame.star ≤ PGame.star * PGame.star",
"after_state": "case right\nle : PGame.star * PGame.star ≤ PGame.star\n⊢ PGame.star * PGame.star ≤ PGame.star\n---\ncase left\nle : PGame.star * PGame.star ≤ PGame.star\n⊢ PGame.star * PGame.star ≤ PGame.star"
},
{
"line": "rw [← neg_le_neg_iff]",
"before_state": "case right\nle : PGame.star * PGame.star ≤ PGame.star\n⊢ PGame.star ≤ PGame.star * PGame.star",
"after_state": "case right\nle : PGame.star * PGame.star ≤ PGame.star\n⊢ -(PGame.star * PGame.star) ≤ -PGame.star"
},
{
"line": "rewrite [← neg_le_neg_iff]",
"before_state": "case right\nle : PGame.star * PGame.star ≤ PGame.star\n⊢ PGame.star ≤ PGame.star * PGame.star",
"after_state": "case right\nle : PGame.star * PGame.star ≤ PGame.star\n⊢ -(PGame.star * PGame.star) ≤ -PGame.star"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case right\nle : PGame.star * PGame.star ≤ PGame.star\n⊢ -(PGame.star * PGame.star) ≤ -PGame.star",
"after_state": "case right\nle : PGame.star * PGame.star ≤ PGame.star\n⊢ -(PGame.star * PGame.star) ≤ -PGame.star"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case right\nle : PGame.star * PGame.star ≤ PGame.star\n⊢ -(PGame.star * PGame.star) ≤ -PGame.star",
"after_state": "case right\nle : PGame.star * PGame.star ≤ PGame.star\n⊢ -(PGame.star * PGame.star) ≤ -PGame.star"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case right\nle : PGame.star * PGame.star ≤ PGame.star\n⊢ -(PGame.star * PGame.star) ≤ -PGame.star",
"after_state": "case right\nle : PGame.star * PGame.star ≤ PGame.star\n⊢ -(PGame.star * PGame.star) ≤ -PGame.star"
},
{
"line": "with_reducible rfl",
"before_state": "case right\nle : PGame.star * PGame.star ≤ PGame.star\n⊢ -(PGame.star * PGame.star) ≤ -PGame.star",
"after_state": "case right\nle : PGame.star * PGame.star ≤ PGame.star\n⊢ -(PGame.star * PGame.star) ≤ -PGame.star"
},
{
"line": "rfl",
"before_state": "case right\nle : PGame.star * PGame.star ≤ PGame.star\n⊢ -(PGame.star * PGame.star) ≤ -PGame.star",
"after_state": "case right\nle : PGame.star * PGame.star ≤ PGame.star\n⊢ -(PGame.star * PGame.star) ≤ -PGame.star"
},
{
"line": "apply_rfl",
"before_state": "case right\nle : PGame.star * PGame.star ≤ PGame.star\n⊢ -(PGame.star * PGame.star) ≤ -PGame.star",
"after_state": "case right\nle : PGame.star * PGame.star ≤ PGame.star\n⊢ -(PGame.star * PGame.star) ≤ -PGame.star"
},
{
"line": "skip",
"before_state": "case right\nle : PGame.star * PGame.star ≤ PGame.star\n⊢ -(PGame.star * PGame.star) ≤ -PGame.star",
"after_state": "case right\nle : PGame.star * PGame.star ≤ PGame.star\n⊢ -(PGame.star * PGame.star) ≤ -PGame.star"
},
{
"line": "apply (neg_mul _ _).symm.equiv.1.trans",
"before_state": "case right\nle : PGame.star * PGame.star ≤ PGame.star\n⊢ -(PGame.star * PGame.star) ≤ -PGame.star",
"after_state": "case right\nle : PGame.star * PGame.star ≤ PGame.star\n⊢ -PGame.star * PGame.star ≤ -PGame.star"
},
{
"line": "rw [neg_star]",
"before_state": "case right\nle : PGame.star * PGame.star ≤ PGame.star\n⊢ -PGame.star * PGame.star ≤ -PGame.star",
"after_state": "case right\nle : PGame.star * PGame.star ≤ PGame.star\n⊢ PGame.star * PGame.star ≤ PGame.star"
},
{
"line": "rewrite [neg_star]",
"before_state": "case right\nle : PGame.star * PGame.star ≤ PGame.star\n⊢ -PGame.star * PGame.star ≤ -PGame.star",
"after_state": "case right\nle : PGame.star * PGame.star ≤ PGame.star\n⊢ PGame.star * PGame.star ≤ PGame.star"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case right\nle : PGame.star * PGame.star ≤ PGame.star\n⊢ PGame.star * PGame.star ≤ PGame.star",
"after_state": "case right\nle : PGame.star * PGame.star ≤ PGame.star\n⊢ PGame.star * PGame.star ≤ PGame.star"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case right\nle : PGame.star * PGame.star ≤ PGame.star\n⊢ PGame.star * PGame.star ≤ PGame.star",
"after_state": "case right\nle : PGame.star * PGame.star ≤ PGame.star\n⊢ PGame.star * PGame.star ≤ PGame.star"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case right\nle : PGame.star * PGame.star ≤ PGame.star\n⊢ PGame.star * PGame.star ≤ PGame.star",
"after_state": "case right\nle : PGame.star * PGame.star ≤ PGame.star\n⊢ PGame.star * PGame.star ≤ PGame.star"
},
{
"line": "with_reducible rfl",
"before_state": "case right\nle : PGame.star * PGame.star ≤ PGame.star\n⊢ PGame.star * PGame.star ≤ PGame.star",
"after_state": "case right\nle : PGame.star * PGame.star ≤ PGame.star\n⊢ PGame.star * PGame.star ≤ PGame.star"
},
{
"line": "rfl",
"before_state": "case right\nle : PGame.star * PGame.star ≤ PGame.star\n⊢ PGame.star * PGame.star ≤ PGame.star",
"after_state": "case right\nle : PGame.star * PGame.star ≤ PGame.star\n⊢ PGame.star * PGame.star ≤ PGame.star"
},
{
"line": "apply_rfl",
"before_state": "case right\nle : PGame.star * PGame.star ≤ PGame.star\n⊢ PGame.star * PGame.star ≤ PGame.star",
"after_state": "case right\nle : PGame.star * PGame.star ≤ PGame.star\n⊢ PGame.star * PGame.star ≤ PGame.star"
},
{
"line": "skip",
"before_state": "case right\nle : PGame.star * PGame.star ≤ PGame.star\n⊢ PGame.star * PGame.star ≤ PGame.star",
"after_state": "case right\nle : PGame.star * PGame.star ≤ PGame.star\n⊢ PGame.star * PGame.star ≤ PGame.star"
},
{
"line": "assumption'",
"before_state": "case right\nle : PGame.star * PGame.star ≤ PGame.star\n⊢ PGame.star * PGame.star ≤ PGame.star\n---\ncase left\nle : PGame.star * PGame.star ≤ PGame.star\n⊢ PGame.star * PGame.star ≤ PGame.star",
"after_state": "No Goals!"
},
{
"line": "any_goals assumption",
"before_state": "case right\nle : PGame.star * PGame.star ≤ PGame.star\n⊢ PGame.star * PGame.star ≤ PGame.star\n---\ncase left\nle : PGame.star * PGame.star ≤ PGame.star\n⊢ PGame.star * PGame.star ≤ PGame.star",
"after_state": "No Goals!"
},
{
"line": "assumption",
"before_state": "case right\nle : PGame.star * PGame.star ≤ PGame.star\n⊢ PGame.star * PGame.star ≤ PGame.star",
"after_state": "No Goals!"
},
{
"line": "assumption",
"before_state": "case left\nle : PGame.star * PGame.star ≤ PGame.star\n⊢ PGame.star * PGame.star ≤ PGame.star",
"after_state": "No Goals!"
}
] |
theorem not_irrational_rpow :
¬ ∀ a b : ℝ, Irrational a → Irrational b → 0 < a → Irrational (a ^ b) := by
push_neg
by_cases hc : Irrational (√2 ^ √2)
· use (√2 ^ √2), √2, hc, irrational_sqrt_two, by positivity
rw [← rpow_mul] <;> norm_num
rw [mul_self_sqrt] <;> norm_num
rw [rpow_two] <;> norm_num
rw [sq_sqrt] <;> norm_num
· use √2, √2, irrational_sqrt_two, irrational_sqrt_two, by positivity, hc
|
/root/DuelModelResearch/mathlib4/Counterexamples/IrrationalPowerOfIrrational.lean
|
{
"open": [
"Real"
],
"variables": []
}
|
[
{
"line": "push_neg",
"before_state": "⊢ ¬∀ (a b : ℝ), Irrational a → Irrational b → 0 < a → Irrational (a ^ b)",
"after_state": "⊢ ∃ a b, Irrational a ∧ Irrational b ∧ 0 < a ∧ ¬Irrational (a ^ b)"
},
{
"line": "by_cases hc : Irrational (√2 ^ √2)",
"before_state": "⊢ ∃ a b, Irrational a ∧ Irrational b ∧ 0 < a ∧ ¬Irrational (a ^ b)",
"after_state": "case pos\nhc : Irrational (√2 ^ √2)\n⊢ ∃ a b, Irrational a ∧ Irrational b ∧ 0 < a ∧ ¬Irrational (a ^ b)\n---\ncase neg\nhc : ¬Irrational (√2 ^ √2)\n⊢ ∃ a b, Irrational a ∧ Irrational b ∧ 0 < a ∧ ¬Irrational (a ^ b)"
},
{
"line": "open Classical✝ in refine if hc : Irrational (√2 ^ √2) then ?pos✝ else ?neg✝",
"before_state": "⊢ ∃ a b, Irrational a ∧ Irrational b ∧ 0 < a ∧ ¬Irrational (a ^ b)",
"after_state": "case pos\nhc : Irrational (√2 ^ √2)\n⊢ ∃ a b, Irrational a ∧ Irrational b ∧ 0 < a ∧ ¬Irrational (a ^ b)\n---\ncase neg\nhc : ¬Irrational (√2 ^ √2)\n⊢ ∃ a b, Irrational a ∧ Irrational b ∧ 0 < a ∧ ¬Irrational (a ^ b)"
},
{
"line": "refine if hc : Irrational (√2 ^ √2) then ?pos✝ else ?neg✝",
"before_state": "⊢ ∃ a b, Irrational a ∧ Irrational b ∧ 0 < a ∧ ¬Irrational (a ^ b)",
"after_state": "case pos\nhc : Irrational (√2 ^ √2)\n⊢ ∃ a b, Irrational a ∧ Irrational b ∧ 0 < a ∧ ¬Irrational (a ^ b)\n---\ncase neg\nhc : ¬Irrational (√2 ^ √2)\n⊢ ∃ a b, Irrational a ∧ Irrational b ∧ 0 < a ∧ ¬Irrational (a ^ b)"
},
{
"line": "use (√2 ^ √2), √2, hc, irrational_sqrt_two, by positivity",
"before_state": "case pos\nhc : Irrational (√2 ^ √2)\n⊢ ∃ a b, Irrational a ∧ Irrational b ∧ 0 < a ∧ ¬Irrational (a ^ b)",
"after_state": "case right\nhc : Irrational (√2 ^ √2)\n⊢ ¬Irrational ((√2 ^ √2) ^ √2)"
},
{
"line": "refine without_cdot((√2 ^ √2) : ?m✝)",
"before_state": "case w\nhc : Irrational (√2 ^ √2)\n⊢ ℝ",
"after_state": "No Goals!"
},
{
"line": "refine without_cdot(√2 : ?m✝)",
"before_state": "case w\nhc : Irrational (√2 ^ √2)\n⊢ ℝ",
"after_state": "No Goals!"
},
{
"line": "refine without_cdot(hc : ?m✝)",
"before_state": "case left\nhc : Irrational (√2 ^ √2)\n⊢ Irrational (√2 ^ √2)",
"after_state": "No Goals!"
},
{
"line": "refine without_cdot(irrational_sqrt_two : ?m✝)",
"before_state": "case left\nhc : Irrational (√2 ^ √2)\n⊢ Irrational √2",
"after_state": "No Goals!"
},
{
"line": "refine without_cdot(by positivity : ?m✝)",
"before_state": "case left\nhc : Irrational (√2 ^ √2)\n⊢ 0 < √2 ^ √2",
"after_state": "No Goals!"
},
{
"line": "positivity",
"before_state": "hc : Irrational (√2 ^ √2)\n⊢ 0 < √2 ^ √2",
"after_state": "No Goals!"
},
{
"line": "try with_reducible use_discharger",
"before_state": "case right\nhc : Irrational (√2 ^ √2)\n⊢ ¬Irrational ((√2 ^ √2) ^ √2)",
"after_state": "case right\nhc : Irrational (√2 ^ √2)\n⊢ ¬Irrational ((√2 ^ √2) ^ √2)"
},
{
"line": "first\n| with_reducible use_discharger\n| skip",
"before_state": "case right\nhc : Irrational (√2 ^ √2)\n⊢ ¬Irrational ((√2 ^ √2) ^ √2)",
"after_state": "case right\nhc : Irrational (√2 ^ √2)\n⊢ ¬Irrational ((√2 ^ √2) ^ √2)"
},
{
"line": "with_reducible use_discharger",
"before_state": "case right\nhc : Irrational (√2 ^ √2)\n⊢ ¬Irrational ((√2 ^ √2) ^ √2)",
"after_state": "case right\nhc : Irrational (√2 ^ √2)\n⊢ ¬Irrational ((√2 ^ √2) ^ √2)"
},
{
"line": "use_discharger",
"before_state": "case right\nhc : Irrational (√2 ^ √2)\n⊢ ¬Irrational ((√2 ^ √2) ^ √2)",
"after_state": "case right\nhc : Irrational (√2 ^ √2)\n⊢ ¬Irrational ((√2 ^ √2) ^ √2)"
},
{
"line": "focus\n apply exists_prop.mpr✝\n with_annotate_state\"<;>\" skip\n all_goals use_discharger",
"before_state": "case right\nhc : Irrational (√2 ^ √2)\n⊢ ¬Irrational ((√2 ^ √2) ^ √2)",
"after_state": "case right\nhc : Irrational (√2 ^ √2)\n⊢ ¬Irrational ((√2 ^ √2) ^ √2)"
},
{
"line": "apply exists_prop.mpr✝",
"before_state": "case right\nhc : Irrational (√2 ^ √2)\n⊢ ¬Irrational ((√2 ^ √2) ^ √2)",
"after_state": "case right\nhc : Irrational (√2 ^ √2)\n⊢ ¬Irrational ((√2 ^ √2) ^ √2)"
},
{
"line": "skip",
"before_state": "case right\nhc : Irrational (√2 ^ √2)\n⊢ ¬Irrational ((√2 ^ √2) ^ √2)",
"after_state": "case right\nhc : Irrational (√2 ^ √2)\n⊢ ¬Irrational ((√2 ^ √2) ^ √2)"
},
{
"line": "focus\n rw [← rpow_mul]\n with_annotate_state\"<;>\" skip\n all_goals norm_num",
"before_state": "case right\nhc : Irrational (√2 ^ √2)\n⊢ ¬Irrational ((√2 ^ √2) ^ √2)",
"after_state": "No Goals!"
},
{
"line": "rw [← rpow_mul]",
"before_state": "case right\nhc : Irrational (√2 ^ √2)\n⊢ ¬Irrational ((√2 ^ √2) ^ √2)",
"after_state": "case right\nhc : Irrational (√2 ^ √2)\n⊢ ¬Irrational (√2 ^ (√2 * √2))\n---\ncase right.hx\nhc : Irrational (√2 ^ √2)\n⊢ 0 ≤ √2"
},
{
"line": "rewrite [← rpow_mul]",
"before_state": "case right\nhc : Irrational (√2 ^ √2)\n⊢ ¬Irrational ((√2 ^ √2) ^ √2)",
"after_state": "case right\nhc : Irrational (√2 ^ √2)\n⊢ ¬Irrational (√2 ^ (√2 * √2))\n---\ncase right.hx\nhc : Irrational (√2 ^ √2)\n⊢ 0 ≤ √2"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case right\nhc : Irrational (√2 ^ √2)\n⊢ ¬Irrational (√2 ^ (√2 * √2))\n---\ncase right.hx\nhc : Irrational (√2 ^ √2)\n⊢ 0 ≤ √2",
"after_state": "case right\nhc : Irrational (√2 ^ √2)\n⊢ ¬Irrational (√2 ^ (√2 * √2))\n---\ncase right.hx\nhc : Irrational (√2 ^ √2)\n⊢ 0 ≤ √2"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case right\nhc : Irrational (√2 ^ √2)\n⊢ ¬Irrational (√2 ^ (√2 * √2))\n---\ncase right.hx\nhc : Irrational (√2 ^ √2)\n⊢ 0 ≤ √2",
"after_state": "case right\nhc : Irrational (√2 ^ √2)\n⊢ ¬Irrational (√2 ^ (√2 * √2))\n---\ncase right.hx\nhc : Irrational (√2 ^ √2)\n⊢ 0 ≤ √2"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case right\nhc : Irrational (√2 ^ √2)\n⊢ ¬Irrational (√2 ^ (√2 * √2))\n---\ncase right.hx\nhc : Irrational (√2 ^ √2)\n⊢ 0 ≤ √2",
"after_state": "case right\nhc : Irrational (√2 ^ √2)\n⊢ ¬Irrational (√2 ^ (√2 * √2))\n---\ncase right.hx\nhc : Irrational (√2 ^ √2)\n⊢ 0 ≤ √2"
},
{
"line": "with_reducible rfl",
"before_state": "case right\nhc : Irrational (√2 ^ √2)\n⊢ ¬Irrational (√2 ^ (√2 * √2))\n---\ncase right.hx\nhc : Irrational (√2 ^ √2)\n⊢ 0 ≤ √2",
"after_state": "case right\nhc : Irrational (√2 ^ √2)\n⊢ ¬Irrational (√2 ^ (√2 * √2))\n---\ncase right.hx\nhc : Irrational (√2 ^ √2)\n⊢ 0 ≤ √2"
},
{
"line": "rfl",
"before_state": "case right\nhc : Irrational (√2 ^ √2)\n⊢ ¬Irrational (√2 ^ (√2 * √2))\n---\ncase right.hx\nhc : Irrational (√2 ^ √2)\n⊢ 0 ≤ √2",
"after_state": "case right\nhc : Irrational (√2 ^ √2)\n⊢ ¬Irrational (√2 ^ (√2 * √2))\n---\ncase right.hx\nhc : Irrational (√2 ^ √2)\n⊢ 0 ≤ √2"
},
{
"line": "apply_rfl",
"before_state": "case right\nhc : Irrational (√2 ^ √2)\n⊢ ¬Irrational (√2 ^ (√2 * √2))\n---\ncase right.hx\nhc : Irrational (√2 ^ √2)\n⊢ 0 ≤ √2",
"after_state": "case right\nhc : Irrational (√2 ^ √2)\n⊢ ¬Irrational (√2 ^ (√2 * √2))\n---\ncase right.hx\nhc : Irrational (√2 ^ √2)\n⊢ 0 ≤ √2"
},
{
"line": "skip",
"before_state": "case right\nhc : Irrational (√2 ^ √2)\n⊢ ¬Irrational (√2 ^ (√2 * √2))\n---\ncase right.hx\nhc : Irrational (√2 ^ √2)\n⊢ 0 ≤ √2",
"after_state": "case right\nhc : Irrational (√2 ^ √2)\n⊢ ¬Irrational (√2 ^ (√2 * √2))\n---\ncase right.hx\nhc : Irrational (√2 ^ √2)\n⊢ 0 ≤ √2"
},
{
"line": "with_annotate_state\"<;>\" skip",
"before_state": "case right\nhc : Irrational (√2 ^ √2)\n⊢ ¬Irrational (√2 ^ (√2 * √2))\n---\ncase right.hx\nhc : Irrational (√2 ^ √2)\n⊢ 0 ≤ √2",
"after_state": "case right\nhc : Irrational (√2 ^ √2)\n⊢ ¬Irrational (√2 ^ (√2 * √2))\n---\ncase right.hx\nhc : Irrational (√2 ^ √2)\n⊢ 0 ≤ √2"
},
{
"line": "skip",
"before_state": "case right\nhc : Irrational (√2 ^ √2)\n⊢ ¬Irrational (√2 ^ (√2 * √2))\n---\ncase right.hx\nhc : Irrational (√2 ^ √2)\n⊢ 0 ≤ √2",
"after_state": "case right\nhc : Irrational (√2 ^ √2)\n⊢ ¬Irrational (√2 ^ (√2 * √2))\n---\ncase right.hx\nhc : Irrational (√2 ^ √2)\n⊢ 0 ≤ √2"
},
{
"line": "all_goals norm_num",
"before_state": "case right\nhc : Irrational (√2 ^ √2)\n⊢ ¬Irrational (√2 ^ (√2 * √2))\n---\ncase right.hx\nhc : Irrational (√2 ^ √2)\n⊢ 0 ≤ √2",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "case right\nhc : Irrational (√2 ^ √2)\n⊢ ¬Irrational (√2 ^ (√2 * √2))",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "case right.hx\nhc : Irrational (√2 ^ √2)\n⊢ 0 ≤ √2",
"after_state": "No Goals!"
},
{
"line": "use √2, √2, irrational_sqrt_two, irrational_sqrt_two, by positivity, hc",
"before_state": "case neg\nhc : ¬Irrational (√2 ^ √2)\n⊢ ∃ a b, Irrational a ∧ Irrational b ∧ 0 < a ∧ ¬Irrational (a ^ b)",
"after_state": "No Goals!"
},
{
"line": "refine without_cdot(√2 : ?m✝)",
"before_state": "case w\nhc : ¬Irrational (√2 ^ √2)\n⊢ ℝ",
"after_state": "No Goals!"
},
{
"line": "refine without_cdot(√2 : ?m✝)",
"before_state": "case w\nhc : ¬Irrational (√2 ^ √2)\n⊢ ℝ",
"after_state": "No Goals!"
},
{
"line": "refine without_cdot(irrational_sqrt_two : ?m✝)",
"before_state": "case left\nhc : ¬Irrational (√2 ^ √2)\n⊢ Irrational √2",
"after_state": "No Goals!"
},
{
"line": "refine without_cdot(irrational_sqrt_two : ?m✝)",
"before_state": "case left\nhc : ¬Irrational (√2 ^ √2)\n⊢ Irrational √2",
"after_state": "No Goals!"
},
{
"line": "refine without_cdot(by positivity : ?m✝)",
"before_state": "case left\nhc : ¬Irrational (√2 ^ √2)\n⊢ 0 < √2",
"after_state": "No Goals!"
},
{
"line": "positivity",
"before_state": "hc : ¬Irrational (√2 ^ √2)\n⊢ 0 < √2",
"after_state": "No Goals!"
},
{
"line": "refine without_cdot(hc : ?m✝)",
"before_state": "case right\nhc : ¬Irrational (√2 ^ √2)\n⊢ ¬Irrational (√2 ^ √2)",
"after_state": "No Goals!"
}
] |
theorem LinearMap.BilinForm.not_injOn_toQuadraticForm_isSymm.{u} :
¬∀ {R M : Type u} [CommSemiring R] [AddCommMonoid M], ∀ [Module R M],
Set.InjOn (toQuadraticMap : BilinForm R M → QuadraticForm R M) {B | B.IsSymm} := by
intro h
let F := ULift.{u} (ZMod 2)
apply B_ne_zero F
apply h (isSymm_B F) isSymm_zero
rw [toQuadraticMap_zero]
rw [toQuadraticMap_eq_zero]
exact isAlt_B F
|
/root/DuelModelResearch/mathlib4/Counterexamples/QuadraticForm.lean
|
{
"open": [
"LinearMap",
"LinearMap.BilinForm",
"LinearMap (BilinForm)",
"LinearMap.BilinMap"
],
"variables": [
"(F : Type*) [CommRing F]"
]
}
|
[
{
"line": "intro h",
"before_state": "⊢ ¬∀ {R M : Type u} [inst : CommSemiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M],\n Set.InjOn toQuadraticMap {B | LinearMap.IsSymm B}",
"after_state": "h :\n ∀ {R M : Type u} [inst : CommSemiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M],\n Set.InjOn toQuadraticMap {B | LinearMap.IsSymm B}\n⊢ False"
},
{
"line": "let F := ULift.{u} (ZMod 2)",
"before_state": "h :\n ∀ {R M : Type u} [inst : CommSemiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M],\n Set.InjOn toQuadraticMap {B | LinearMap.IsSymm B}\n⊢ False",
"after_state": "h :\n ∀ {R M : Type u} [inst : CommSemiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M],\n Set.InjOn toQuadraticMap {B | LinearMap.IsSymm B}\nF : Type u := ULift.{u, 0} (ZMod 2)\n⊢ False"
},
{
"line": "refine_lift\n let F := ULift.{u} (ZMod 2);\n ?_",
"before_state": "h :\n ∀ {R M : Type u} [inst : CommSemiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M],\n Set.InjOn toQuadraticMap {B | LinearMap.IsSymm B}\n⊢ False",
"after_state": "h :\n ∀ {R M : Type u} [inst : CommSemiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M],\n Set.InjOn toQuadraticMap {B | LinearMap.IsSymm B}\nF : Type u := ULift.{u, 0} (ZMod 2)\n⊢ False"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (let F := ULift.{u} (ZMod 2);\n ?_);\n rotate_right)",
"before_state": "h :\n ∀ {R M : Type u} [inst : CommSemiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M],\n Set.InjOn toQuadraticMap {B | LinearMap.IsSymm B}\n⊢ False",
"after_state": "h :\n ∀ {R M : Type u} [inst : CommSemiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M],\n Set.InjOn toQuadraticMap {B | LinearMap.IsSymm B}\nF : Type u := ULift.{u, 0} (ZMod 2)\n⊢ False"
},
{
"line": "refine\n no_implicit_lambda%\n (let F := ULift.{u} (ZMod 2);\n ?_)",
"before_state": "h :\n ∀ {R M : Type u} [inst : CommSemiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M],\n Set.InjOn toQuadraticMap {B | LinearMap.IsSymm B}\n⊢ False",
"after_state": "h :\n ∀ {R M : Type u} [inst : CommSemiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M],\n Set.InjOn toQuadraticMap {B | LinearMap.IsSymm B}\nF : Type u := ULift.{u, 0} (ZMod 2)\n⊢ False"
},
{
"line": "rotate_right",
"before_state": "h :\n ∀ {R M : Type u} [inst : CommSemiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M],\n Set.InjOn toQuadraticMap {B | LinearMap.IsSymm B}\nF : Type u := ULift.{u, 0} (ZMod 2)\n⊢ False",
"after_state": "h :\n ∀ {R M : Type u} [inst : CommSemiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M],\n Set.InjOn toQuadraticMap {B | LinearMap.IsSymm B}\nF : Type u := ULift.{u, 0} (ZMod 2)\n⊢ False"
},
{
"line": "apply B_ne_zero F",
"before_state": "h :\n ∀ {R M : Type u} [inst : CommSemiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M],\n Set.InjOn toQuadraticMap {B | LinearMap.IsSymm B}\nF : Type u := ULift.{u, 0} (ZMod 2)\n⊢ False",
"after_state": "No Goals!"
}
] |
theorem map_toReal_nhds (a : ℝₗ) : map toReal (𝓝 a) = 𝓝[≥] toReal a := by
refine ((nhds_basis_Ico a).map _).eq_of_same_basis ?_
simpa only [toReal.image_eq_preimage] using nhdsGE_basis_Ico (toReal a)
|
/root/DuelModelResearch/mathlib4/Counterexamples/SorgenfreyLine.lean
|
{
"open": [
"Set Filter TopologicalSpace",
"scoped Topology Filter Cardinal",
"scoped SorgenfreyLine"
],
"variables": []
}
|
[
{
"line": "refine ((nhds_basis_Ico a).map _).eq_of_same_basis ?_",
"before_state": "ℝₗ : Type u_1\nα✝ : Type u_2\ntoReal : ℝₗ → α✝\na : ℝₗ\n⊢ map toReal (𝓝 a) = 𝓝[≥] toReal a",
"after_state": "No Goals!"
}
] |
private lemma no_strictly_decreasing {α : Type*} [Preorder α] [WellFoundedLT α] (f : ℕ → α) {n₀ : ℕ}
(hf : ∀ n ≥ n₀, f (n + 1) < f n) : False := by
let g (n : ℕ) : α := f (n₀ + n)
have : (· > ·) ↪r (· < ·) := RelEmbedding.natGT g (fun n ↦ hf _ (by simp))
exact this.not_wellFounded_of_decreasing_seq wellFounded_lt
|
/root/DuelModelResearch/mathlib4/Counterexamples/AharoniKorman.lean
|
{
"open": [
"Hollom"
],
"variables": []
}
|
[
{
"line": "let g (n : ℕ) : α := f (n₀ + n)",
"before_state": "α : Type u_1\ninst✝¹ : Preorder α\ninst✝ : WellFoundedLT α\nf : ℕ → α\nn₀ : ℕ\nhf : ∀ n ≥ n₀, f (n + 1) < f n\n⊢ False",
"after_state": "α : Type u_1\ninst✝¹ : Preorder α\ninst✝ : WellFoundedLT α\nf : ℕ → α\nn₀ : ℕ\nhf : ∀ n ≥ n₀, f (n + 1) < f n\ng : ℕ → α := fun n => f (n₀ + n)\n⊢ False"
},
{
"line": "refine_lift\n let g (n : ℕ) : α := f (n₀ + n);\n ?_",
"before_state": "α : Type u_1\ninst✝¹ : Preorder α\ninst✝ : WellFoundedLT α\nf : ℕ → α\nn₀ : ℕ\nhf : ∀ n ≥ n₀, f (n + 1) < f n\n⊢ False",
"after_state": "α : Type u_1\ninst✝¹ : Preorder α\ninst✝ : WellFoundedLT α\nf : ℕ → α\nn₀ : ℕ\nhf : ∀ n ≥ n₀, f (n + 1) < f n\ng : ℕ → α := fun n => f (n₀ + n)\n⊢ False"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (let g (n : ℕ) : α := f (n₀ + n);\n ?_);\n rotate_right)",
"before_state": "α : Type u_1\ninst✝¹ : Preorder α\ninst✝ : WellFoundedLT α\nf : ℕ → α\nn₀ : ℕ\nhf : ∀ n ≥ n₀, f (n + 1) < f n\n⊢ False",
"after_state": "α : Type u_1\ninst✝¹ : Preorder α\ninst✝ : WellFoundedLT α\nf : ℕ → α\nn₀ : ℕ\nhf : ∀ n ≥ n₀, f (n + 1) < f n\ng : ℕ → α := fun n => f (n₀ + n)\n⊢ False"
},
{
"line": "refine\n no_implicit_lambda%\n (let g (n : ℕ) : α := f (n₀ + n);\n ?_)",
"before_state": "α : Type u_1\ninst✝¹ : Preorder α\ninst✝ : WellFoundedLT α\nf : ℕ → α\nn₀ : ℕ\nhf : ∀ n ≥ n₀, f (n + 1) < f n\n⊢ False",
"after_state": "α : Type u_1\ninst✝¹ : Preorder α\ninst✝ : WellFoundedLT α\nf : ℕ → α\nn₀ : ℕ\nhf : ∀ n ≥ n₀, f (n + 1) < f n\ng : ℕ → α := fun n => f (n₀ + n)\n⊢ False"
},
{
"line": "rotate_right",
"before_state": "α : Type u_1\ninst✝¹ : Preorder α\ninst✝ : WellFoundedLT α\nf : ℕ → α\nn₀ : ℕ\nhf : ∀ n ≥ n₀, f (n + 1) < f n\ng : ℕ → α := fun n => f (n₀ + n)\n⊢ False",
"after_state": "α : Type u_1\ninst✝¹ : Preorder α\ninst✝ : WellFoundedLT α\nf : ℕ → α\nn₀ : ℕ\nhf : ∀ n ≥ n₀, f (n + 1) < f n\ng : ℕ → α := fun n => f (n₀ + n)\n⊢ False"
},
{
"line": "have : (· > ·) ↪r (· < ·) := RelEmbedding.natGT g (fun n ↦ hf _ (by simp))",
"before_state": "α : Type u_1\ninst✝¹ : Preorder α\ninst✝ : WellFoundedLT α\nf : ℕ → α\nn₀ : ℕ\nhf : ∀ n ≥ n₀, f (n + 1) < f n\ng : ℕ → α := fun n => f (n₀ + n)\n⊢ False",
"after_state": "α : Type u_1\ninst✝¹ : Preorder α\ninst✝ : WellFoundedLT α\nf : ℕ → α\nn₀ : ℕ\nhf : ∀ n ≥ n₀, f (n + 1) < f n\ng : ℕ → α := fun n => f (n₀ + n)\nthis : (fun x1 x2 => x1 > x2) ↪r fun x1 x2 => x1 < x2\n⊢ False"
},
{
"line": "refine_lift\n have : (· > ·) ↪r (· < ·) := RelEmbedding.natGT g (fun n ↦ hf _ (by simp));\n ?_",
"before_state": "α : Type u_1\ninst✝¹ : Preorder α\ninst✝ : WellFoundedLT α\nf : ℕ → α\nn₀ : ℕ\nhf : ∀ n ≥ n₀, f (n + 1) < f n\ng : ℕ → α := fun n => f (n₀ + n)\n⊢ False",
"after_state": "α : Type u_1\ninst✝¹ : Preorder α\ninst✝ : WellFoundedLT α\nf : ℕ → α\nn₀ : ℕ\nhf : ∀ n ≥ n₀, f (n + 1) < f n\ng : ℕ → α := fun n => f (n₀ + n)\nthis : (fun x1 x2 => x1 > x2) ↪r fun x1 x2 => x1 < x2\n⊢ False"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (have : (· > ·) ↪r (· < ·) := RelEmbedding.natGT g (fun n ↦ hf _ (by simp));\n ?_);\n rotate_right)",
"before_state": "α : Type u_1\ninst✝¹ : Preorder α\ninst✝ : WellFoundedLT α\nf : ℕ → α\nn₀ : ℕ\nhf : ∀ n ≥ n₀, f (n + 1) < f n\ng : ℕ → α := fun n => f (n₀ + n)\n⊢ False",
"after_state": "α : Type u_1\ninst✝¹ : Preorder α\ninst✝ : WellFoundedLT α\nf : ℕ → α\nn₀ : ℕ\nhf : ∀ n ≥ n₀, f (n + 1) < f n\ng : ℕ → α := fun n => f (n₀ + n)\nthis : (fun x1 x2 => x1 > x2) ↪r fun x1 x2 => x1 < x2\n⊢ False"
},
{
"line": "refine\n no_implicit_lambda%\n (have : (· > ·) ↪r (· < ·) := RelEmbedding.natGT g (fun n ↦ hf _ (by simp));\n ?_)",
"before_state": "α : Type u_1\ninst✝¹ : Preorder α\ninst✝ : WellFoundedLT α\nf : ℕ → α\nn₀ : ℕ\nhf : ∀ n ≥ n₀, f (n + 1) < f n\ng : ℕ → α := fun n => f (n₀ + n)\n⊢ False",
"after_state": "α : Type u_1\ninst✝¹ : Preorder α\ninst✝ : WellFoundedLT α\nf : ℕ → α\nn₀ : ℕ\nhf : ∀ n ≥ n₀, f (n + 1) < f n\ng : ℕ → α := fun n => f (n₀ + n)\nthis : (fun x1 x2 => x1 > x2) ↪r fun x1 x2 => x1 < x2\n⊢ False"
},
{
"line": "simp",
"before_state": "α : Type u_1\ninst✝¹ : Preorder α\ninst✝ : WellFoundedLT α\nf : ℕ → α\nn₀ : ℕ\nhf : ∀ n ≥ n₀, f (n + 1) < f n\ng : ℕ → α := fun n => f (n₀ + n)\nn : ℕ\n⊢ n₀.add n ≥ n₀",
"after_state": "No Goals!"
},
{
"line": "rotate_right",
"before_state": "α : Type u_1\ninst✝¹ : Preorder α\ninst✝ : WellFoundedLT α\nf : ℕ → α\nn₀ : ℕ\nhf : ∀ n ≥ n₀, f (n + 1) < f n\ng : ℕ → α := fun n => f (n₀ + n)\nthis : (fun x1 x2 => x1 > x2) ↪r fun x1 x2 => x1 < x2\n⊢ False",
"after_state": "α : Type u_1\ninst✝¹ : Preorder α\ninst✝ : WellFoundedLT α\nf : ℕ → α\nn₀ : ℕ\nhf : ∀ n ≥ n₀, f (n + 1) < f n\ng : ℕ → α := fun n => f (n₀ + n)\nthis : (fun x1 x2 => x1 > x2) ↪r fun x1 x2 => x1 < x2\n⊢ False"
},
{
"line": "exact this.not_wellFounded_of_decreasing_seq wellFounded_lt",
"before_state": "α : Type u_1\ninst✝¹ : Preorder α\ninst✝ : WellFoundedLT α\nf : ℕ → α\nn₀ : ℕ\nhf : ∀ n ≥ n₀, f (n + 1) < f n\ng : ℕ → α := fun n => f (n₀ + n)\nthis : (fun x1 x2 => x1 > x2) ↪r fun x1 x2 => x1 < x2\n⊢ False",
"after_state": "No Goals!"
}
] |
theorem mem_zmod_2 (a : ZMod 2) : a = 0 ∨ a = 1 := by
rcases a with ⟨_ | _, _ | _ | _ | _⟩
· exact Or.inl rfl
· exact Or.inr rfl
|
/root/DuelModelResearch/mathlib4/Counterexamples/CanonicallyOrderedCommSemiringTwoMul.lean
|
{
"open": [],
"variables": []
}
|
[
{
"line": "rcases a with ⟨_ | _, _ | _ | _ | _⟩",
"before_state": "a : ZMod 2\n⊢ a = 0 ∨ a = 1",
"after_state": "case mk.zero.step.refl\n⊢ ⟨0, ⋯⟩ = 0 ∨ ⟨0, ⋯⟩ = 1\n---\ncase mk.succ.refl\n⊢ ⟨0 + 1, ⋯⟩ = 0 ∨ ⟨0 + 1, ⋯⟩ = 1"
},
{
"line": "exact Or.inl rfl",
"before_state": "case mk.zero.step.refl\n⊢ ⟨0, ⋯⟩ = 0 ∨ ⟨0, ⋯⟩ = 1",
"after_state": "No Goals!"
},
{
"line": "exact Or.inr rfl",
"before_state": "case mk.succ.refl\n⊢ ⟨0 + 1, ⋯⟩ = 0 ∨ ⟨0 + 1, ⋯⟩ = 1",
"after_state": "No Goals!"
}
] |
theorem add_self_zmod_2 (a : ZMod 2) : a + a = 0 := by rcases mem_zmod_2 a with (rfl | rfl) <;> rfl
|
/root/DuelModelResearch/mathlib4/Counterexamples/CanonicallyOrderedCommSemiringTwoMul.lean
|
{
"open": [],
"variables": []
}
|
[
{
"line": "focus\n rcases mem_zmod_2 a with (rfl | rfl)\n with_annotate_state\"<;>\" skip\n all_goals rfl",
"before_state": "a : ZMod 2\n⊢ a + a = 0",
"after_state": "No Goals!"
},
{
"line": "rcases mem_zmod_2 a with (rfl | rfl)",
"before_state": "a : ZMod 2\n⊢ a + a = 0",
"after_state": "No Goals!"
}
] |
theorem add_L {a b : ℕ × ZMod 2} (ha : a ≠ (0, 1)) (hb : b ≠ (0, 1)) : a + b ≠ (0, 1) := by
rcases a with ⟨a, a2⟩
rcases b with ⟨b, b2⟩
match b with
| 0 =>
rcases mem_zmod_2 b2 with (rfl | rfl)
· simp [ha, -Prod.mk.injEq]
· cases hb rfl
| b + 1 =>
simp [(a + b).succ_ne_zero]
|
/root/DuelModelResearch/mathlib4/Counterexamples/CanonicallyOrderedCommSemiringTwoMul.lean
|
{
"open": [
"Nxzmod2 Subtype"
],
"variables": [
"{a b : ℕ × ZMod 2}"
]
}
|
[
{
"line": "rcases a with ⟨a, a2⟩",
"before_state": "a b : ℕ × ZMod 2\nha : a ≠ (0, 1)\nhb : b ≠ (0, 1)\n⊢ a + b ≠ (0, 1)",
"after_state": "case mk\nb : ℕ × ZMod 2\nhb : b ≠ (0, 1)\na : ℕ\na2 : ZMod 2\nha : (a, a2) ≠ (0, 1)\n⊢ (a, a2) + b ≠ (0, 1)"
},
{
"line": "rcases b with ⟨b, b2⟩",
"before_state": "case mk\nb : ℕ × ZMod 2\nhb : b ≠ (0, 1)\na : ℕ\na2 : ZMod 2\nha : (a, a2) ≠ (0, 1)\n⊢ (a, a2) + b ≠ (0, 1)",
"after_state": "case mk.mk\na : ℕ\na2 : ZMod 2\nha : (a, a2) ≠ (0, 1)\nb : ℕ\nb2 : ZMod 2\nhb : (b, b2) ≠ (0, 1)\n⊢ (a, a2) + (b, b2) ≠ (0, 1)"
},
{
"line": "match b with\n| 0 =>\n rcases mem_zmod_2 b2 with (rfl | rfl)\n · simp [ha, -Prod.mk.injEq]\n · cases hb rfl\n| b + 1 => simp [(a + b).succ_ne_zero]",
"before_state": "case mk.mk\na : ℕ\na2 : ZMod 2\nha : (a, a2) ≠ (0, 1)\nb : ℕ\nb2 : ZMod 2\nhb : (b, b2) ≠ (0, 1)\n⊢ (a, a2) + (b, b2) ≠ (0, 1)",
"after_state": "No Goals!"
},
{
"line": "refine\n no_implicit_lambda%\n (match b with\n | 0 => ?rhs✝\n | b + 1 => ?rhs✝¹)",
"before_state": "case mk.mk\na : ℕ\na2 : ZMod 2\nha : (a, a2) ≠ (0, 1)\nb : ℕ\nb2 : ZMod 2\nhb : (b, b2) ≠ (0, 1)\n⊢ (a, a2) + (b, b2) ≠ (0, 1)",
"after_state": "case rhs\na : ℕ\na2 : ZMod 2\nha : (a, a2) ≠ (0, 1)\nb : ℕ\nb2 : ZMod 2\nhb : (0, b2) ≠ (0, 1)\n⊢ (a, a2) + (0, b2) ≠ (0, 1)\n---\ncase rhs\na : ℕ\na2 : ZMod 2\nha : (a, a2) ≠ (0, 1)\nb✝ : ℕ\nb2 : ZMod 2\nb : ℕ\nhb : (b + 1, b2) ≠ (0, 1)\n⊢ (a, a2) + (b + 1, b2) ≠ (0, 1)"
},
{
"line": "case rhs✝ =>\n with_annotate_state[\"|\" \"=>\"] skip\n rcases mem_zmod_2 b2 with (rfl | rfl)\n · simp [ha, -Prod.mk.injEq]\n · cases hb rfl",
"before_state": "case rhs\na : ℕ\na2 : ZMod 2\nha : (a, a2) ≠ (0, 1)\nb : ℕ\nb2 : ZMod 2\nhb : (0, b2) ≠ (0, 1)\n⊢ (a, a2) + (0, b2) ≠ (0, 1)\n---\ncase rhs\na : ℕ\na2 : ZMod 2\nha : (a, a2) ≠ (0, 1)\nb✝ : ℕ\nb2 : ZMod 2\nb : ℕ\nhb : (b + 1, b2) ≠ (0, 1)\n⊢ (a, a2) + (b + 1, b2) ≠ (0, 1)",
"after_state": "case rhs\na : ℕ\na2 : ZMod 2\nha : (a, a2) ≠ (0, 1)\nb✝ : ℕ\nb2 : ZMod 2\nb : ℕ\nhb : (b + 1, b2) ≠ (0, 1)\n⊢ (a, a2) + (b + 1, b2) ≠ (0, 1)"
},
{
"line": "with_annotate_state[\"|\" \"=>\"] skip",
"before_state": "a : ℕ\na2 : ZMod 2\nha : (a, a2) ≠ (0, 1)\nb : ℕ\nb2 : ZMod 2\nhb : (0, b2) ≠ (0, 1)\n⊢ (a, a2) + (0, b2) ≠ (0, 1)",
"after_state": "a : ℕ\na2 : ZMod 2\nha : (a, a2) ≠ (0, 1)\nb : ℕ\nb2 : ZMod 2\nhb : (0, b2) ≠ (0, 1)\n⊢ (a, a2) + (0, b2) ≠ (0, 1)"
},
{
"line": "skip",
"before_state": "a : ℕ\na2 : ZMod 2\nha : (a, a2) ≠ (0, 1)\nb : ℕ\nb2 : ZMod 2\nhb : (0, b2) ≠ (0, 1)\n⊢ (a, a2) + (0, b2) ≠ (0, 1)",
"after_state": "a : ℕ\na2 : ZMod 2\nha : (a, a2) ≠ (0, 1)\nb : ℕ\nb2 : ZMod 2\nhb : (0, b2) ≠ (0, 1)\n⊢ (a, a2) + (0, b2) ≠ (0, 1)"
},
{
"line": "rcases mem_zmod_2 b2 with (rfl | rfl)",
"before_state": "a : ℕ\na2 : ZMod 2\nha : (a, a2) ≠ (0, 1)\nb : ℕ\nb2 : ZMod 2\nhb : (0, b2) ≠ (0, 1)\n⊢ (a, a2) + (0, b2) ≠ (0, 1)",
"after_state": "No Goals!"
},
{
"line": "case rhs✝ =>\n with_annotate_state[\"|\" \"=>\"] skip\n simp [(a + b).succ_ne_zero]",
"before_state": "case rhs\na : ℕ\na2 : ZMod 2\nha : (a, a2) ≠ (0, 1)\nb✝ : ℕ\nb2 : ZMod 2\nb : ℕ\nhb : (b + 1, b2) ≠ (0, 1)\n⊢ (a, a2) + (b + 1, b2) ≠ (0, 1)",
"after_state": "No Goals!"
},
{
"line": "with_annotate_state[\"|\" \"=>\"] skip",
"before_state": "a : ℕ\na2 : ZMod 2\nha : (a, a2) ≠ (0, 1)\nb✝ : ℕ\nb2 : ZMod 2\nb : ℕ\nhb : (b + 1, b2) ≠ (0, 1)\n⊢ (a, a2) + (b + 1, b2) ≠ (0, 1)",
"after_state": "a : ℕ\na2 : ZMod 2\nha : (a, a2) ≠ (0, 1)\nb✝ : ℕ\nb2 : ZMod 2\nb : ℕ\nhb : (b + 1, b2) ≠ (0, 1)\n⊢ (a, a2) + (b + 1, b2) ≠ (0, 1)"
},
{
"line": "skip",
"before_state": "a : ℕ\na2 : ZMod 2\nha : (a, a2) ≠ (0, 1)\nb✝ : ℕ\nb2 : ZMod 2\nb : ℕ\nhb : (b + 1, b2) ≠ (0, 1)\n⊢ (a, a2) + (b + 1, b2) ≠ (0, 1)",
"after_state": "a : ℕ\na2 : ZMod 2\nha : (a, a2) ≠ (0, 1)\nb✝ : ℕ\nb2 : ZMod 2\nb : ℕ\nhb : (b + 1, b2) ≠ (0, 1)\n⊢ (a, a2) + (b + 1, b2) ≠ (0, 1)"
},
{
"line": "simp [(a + b).succ_ne_zero]",
"before_state": "a : ℕ\na2 : ZMod 2\nha : (a, a2) ≠ (0, 1)\nb✝ : ℕ\nb2 : ZMod 2\nb : ℕ\nhb : (b + 1, b2) ≠ (0, 1)\n⊢ (a, a2) + (b + 1, b2) ≠ (0, 1)",
"after_state": "No Goals!"
}
] |
theorem mul_L {a b : ℕ × ZMod 2} (ha : a ≠ (0, 1)) (hb : b ≠ (0, 1)) : a * b ≠ (0, 1) := by
rcases a with ⟨a, a2⟩
rcases b with ⟨b, b2⟩
cases b
· rcases mem_zmod_2 b2 with (rfl | rfl) <;> rcases mem_zmod_2 a2 with (rfl | rfl) <;>
simp only [Prod.mk_mul_mk]
simp only [mul_zero]
simp only [mul_one]
simp only [ne_eq]
simp only [Prod.mk.injEq]
simp only [zero_ne_one]
simp only [and_false]
simp only [not_false_eq_true]
simp only [not_true_eq_false]
exact hb rfl
cases a
· rcases mem_zmod_2 b2 with (rfl | rfl) <;> rcases mem_zmod_2 a2 with (rfl | rfl) <;>
simp only [Prod.mk_mul_mk]
simp only [mul_zero]
simp only [zero_mul]
simp only [mul_one]
simp only [ne_eq]
simp only [Prod.mk.injEq]
simp only [zero_ne_one]
simp only [and_false]
simp only [not_false_eq_true]
simp only [not_true_eq_false]
exact ha rfl
· simp [mul_ne_zero _ _, Nat.succ_ne_zero _]
|
/root/DuelModelResearch/mathlib4/Counterexamples/CanonicallyOrderedCommSemiringTwoMul.lean
|
{
"open": [
"Nxzmod2 Subtype"
],
"variables": [
"{a b : ℕ × ZMod 2}"
]
}
|
[
{
"line": "rcases a with ⟨a, a2⟩",
"before_state": "a b : ℕ × ZMod 2\nha : a ≠ (0, 1)\nhb : b ≠ (0, 1)\n⊢ a * b ≠ (0, 1)",
"after_state": "case mk\nb : ℕ × ZMod 2\nhb : b ≠ (0, 1)\na : ℕ\na2 : ZMod 2\nha : (a, a2) ≠ (0, 1)\n⊢ (a, a2) * b ≠ (0, 1)"
},
{
"line": "rcases b with ⟨b, b2⟩",
"before_state": "case mk\nb : ℕ × ZMod 2\nhb : b ≠ (0, 1)\na : ℕ\na2 : ZMod 2\nha : (a, a2) ≠ (0, 1)\n⊢ (a, a2) * b ≠ (0, 1)",
"after_state": "case mk.mk\na : ℕ\na2 : ZMod 2\nha : (a, a2) ≠ (0, 1)\nb : ℕ\nb2 : ZMod 2\nhb : (b, b2) ≠ (0, 1)\n⊢ (a, a2) * (b, b2) ≠ (0, 1)"
},
{
"line": "cases b",
"before_state": "case mk.mk\na : ℕ\na2 : ZMod 2\nha : (a, a2) ≠ (0, 1)\nb : ℕ\nb2 : ZMod 2\nhb : (b, b2) ≠ (0, 1)\n⊢ (a, a2) * (b, b2) ≠ (0, 1)",
"after_state": "case mk.mk.zero\na : ℕ\na2 : ZMod 2\nha : (a, a2) ≠ (0, 1)\nb2 : ZMod 2\nhb : (0, b2) ≠ (0, 1)\n⊢ (a, a2) * (0, b2) ≠ (0, 1)\n---\ncase mk.mk.succ\na : ℕ\na2 : ZMod 2\nha : (a, a2) ≠ (0, 1)\nb2 : ZMod 2\nn✝ : ℕ\nhb : (n✝ + 1, b2) ≠ (0, 1)\n⊢ (a, a2) * (n✝ + 1, b2) ≠ (0, 1)"
},
{
"line": "focus\n rcases mem_zmod_2 b2 with (rfl | rfl) <;> rcases mem_zmod_2 a2 with (rfl | rfl)\n with_annotate_state\"<;>\" skip\n all_goals simp only [Prod.mk_mul_mk]",
"before_state": "case mk.mk.zero\na : ℕ\na2 : ZMod 2\nha : (a, a2) ≠ (0, 1)\nb2 : ZMod 2\nhb : (0, b2) ≠ (0, 1)\n⊢ (a, a2) * (0, b2) ≠ (0, 1)",
"after_state": "No Goals!"
},
{
"line": "focus\n rcases mem_zmod_2 b2 with (rfl | rfl)\n with_annotate_state\"<;>\" skip\n all_goals rcases mem_zmod_2 a2 with (rfl | rfl)",
"before_state": "case mk.mk.zero\na : ℕ\na2 : ZMod 2\nha : (a, a2) ≠ (0, 1)\nb2 : ZMod 2\nhb : (0, b2) ≠ (0, 1)\n⊢ (a, a2) * (0, b2) ≠ (0, 1)",
"after_state": "No Goals!"
},
{
"line": "rcases mem_zmod_2 b2 with (rfl | rfl)",
"before_state": "case mk.mk.zero\na : ℕ\na2 : ZMod 2\nha : (a, a2) ≠ (0, 1)\nb2 : ZMod 2\nhb : (0, b2) ≠ (0, 1)\n⊢ (a, a2) * (0, b2) ≠ (0, 1)",
"after_state": "No Goals!"
}
] |
theorem UnitsInt.one_ne_neg_one : (1 : ℤˣ) ≠ -1 := by decide
|
/root/DuelModelResearch/mathlib4/Counterexamples/DirectSumIsInternal.lean
|
{
"open": [],
"variables": []
}
|
[
{
"line": "decide",
"before_state": "⊢ 1 ≠ -1",
"after_state": "No Goals!"
}
] |
theorem mem_withSign_one {x : ℤ} : x ∈ ℤ≥0 ↔ 0 ≤ x :=
show _ ≤ (_ : ℤˣ) • x ↔ _ by rw [one_smul]
|
/root/DuelModelResearch/mathlib4/Counterexamples/DirectSumIsInternal.lean
|
{
"open": [],
"variables": []
}
|
[
{
"line": "rw [one_smul]",
"before_state": "x : ℤ\n⊢ ?m.1883 ≤ ?m.1900 • x ↔ ?m.1976",
"after_state": "No Goals!"
},
{
"line": "rewrite [one_smul]",
"before_state": "x : ℤ\n⊢ ?m.1883 ≤ ?m.1900 • x ↔ ?m.1976",
"after_state": "x : ℤ\n⊢ ?m.1883 ≤ x ↔ ?m.1976"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "x : ℤ\n⊢ ?m.1883 ≤ x ↔ ?m.1976",
"after_state": "No Goals!"
},
{
"line": "try (with_reducible rfl)",
"before_state": "x : ℤ\n⊢ ?m.1883 ≤ x ↔ ?m.1976",
"after_state": "No Goals!"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "x : ℤ\n⊢ ?m.1883 ≤ x ↔ ?m.1976",
"after_state": "No Goals!"
},
{
"line": "with_reducible rfl",
"before_state": "x : ℤ\n⊢ ?m.1883 ≤ x ↔ ?m.1976",
"after_state": "No Goals!"
},
{
"line": "rfl",
"before_state": "x : ℤ\n⊢ ?m.1883 ≤ x ↔ ?m.1976",
"after_state": "No Goals!"
},
{
"line": "exact Iff.rfl✝",
"before_state": "x : ℤ\n⊢ ?m.1883 ≤ x ↔ ?m.1976",
"after_state": "No Goals!"
}
] |
theorem mem_withSign_neg_one {x : ℤ} : x ∈ ℤ≤0 ↔ x ≤ 0 :=
show _ ≤ (_ : ℤˣ) • x ↔ _ by rw [Units.neg_smul, le_neg, one_smul, neg_zero]
|
/root/DuelModelResearch/mathlib4/Counterexamples/DirectSumIsInternal.lean
|
{
"open": [],
"variables": []
}
|
[
{
"line": "rw [Units.neg_smul, le_neg, one_smul, neg_zero]",
"before_state": "x : ℤ\n⊢ ?m.1875 ≤ ?m.1892 • x ↔ ?m.1968",
"after_state": "No Goals!"
},
{
"line": "rewrite [Units.neg_smul, le_neg, one_smul, neg_zero]",
"before_state": "x : ℤ\n⊢ ?m.1875 ≤ ?m.1892 • x ↔ ?m.1968",
"after_state": "x : ℤ\n⊢ x ≤ 0 ↔ ?m.1968"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "x : ℤ\n⊢ x ≤ 0 ↔ ?m.1968",
"after_state": "No Goals!"
},
{
"line": "try (with_reducible rfl)",
"before_state": "x : ℤ\n⊢ x ≤ 0 ↔ ?m.1968",
"after_state": "No Goals!"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "x : ℤ\n⊢ x ≤ 0 ↔ ?m.1968",
"after_state": "No Goals!"
},
{
"line": "with_reducible rfl",
"before_state": "x : ℤ\n⊢ x ≤ 0 ↔ ?m.1968",
"after_state": "No Goals!"
},
{
"line": "rfl",
"before_state": "x : ℤ\n⊢ x ≤ 0 ↔ ?m.1968",
"after_state": "No Goals!"
},
{
"line": "exact Iff.rfl✝",
"before_state": "x : ℤ\n⊢ x ≤ 0 ↔ ?m.1968",
"after_state": "No Goals!"
}
] |
lemma Int.eq_of_pow_sub_le {d : ℕ} {m n : ℤ} (hd1 : 1 < d)
(h : |(d : ℝ) ^ (-m) - d ^ (-n)| < d ^ (-n - 1)) : m = n := by
have hd0 : 0 < d := one_pos.trans hd1
replace h : |(1 : ℝ) - d ^ (n - m)| < (d : ℝ)⁻¹ := by
rw [← mul_lt_mul_iff_of_pos_left (a := (d : ℝ) ^ (-n)) (zpow_pos _ _)]
rw [← abs_of_nonneg (a := (d : ℝ) ^ (-n)) (le_of_lt <| zpow_pos _ _)]
rw [← abs_mul]
rw [mul_sub]
rw [mul_one]
rw [← zpow_add₀ <| Nat.cast_ne_zero.mpr (ne_of_gt hd0)]
rw [sub_eq_add_neg (b := m)]
rw [neg_add_cancel_left]
rw [← abs_neg]
rw [neg_sub]
rw [abs_of_nonneg (a := (d : ℝ) ^ (-n)) (le_of_lt <| zpow_pos _ _)]
rw [← zpow_neg_one]
rw [← zpow_add₀ <| Nat.cast_ne_zero.mpr (ne_of_gt hd0)]
rw [← sub_eq_add_neg]
exact h
all_goals exact Nat.cast_pos'.mpr hd0
by_cases H : (m : ℤ) ≤ n
· obtain ⟨a, ha⟩ := Int.eq_ofNat_of_zero_le (sub_nonneg.mpr H)
rw [ha] at h
rw [← mul_lt_mul_iff_of_pos_left (a := (d : ℝ)) <| Nat.cast_pos'.mpr hd0] at h
rw [mul_inv_cancel₀ <| Nat.cast_ne_zero.mpr (ne_of_gt hd0)] at h
rw [← abs_of_nonneg (a := (d : ℝ)) <| Nat.cast_nonneg' d] at h
rw [← abs_mul] at h
rw [show |(d : ℝ) * (1 - |(d : ℝ)| ^ (a : ℤ))| = |(d : ℤ) * (1 - |(d : ℤ)| ^ a)| by norm_cast] at h
rw [← Int.cast_one (R := ℝ)] at h
rw [Int.cast_lt] at h
rw [Int.abs_lt_one_iff] at h
rw [Int.mul_eq_zero] at h
rw [sub_eq_zero] at h
rw [eq_comm (a := 1)] at h
rw [pow_eq_one_iff_cases] at h
simp only [Nat.cast_eq_zero] at h
simp only [ne_of_gt hd0] at h
simp only [Nat.abs_cast] at h
simp only [Nat.cast_eq_one] at h
simp only [ne_of_gt hd1] at h
simp only [Int.reduceNeg] at h
simp only [reduceCtorEq] at h
simp only [false_and] at h
simp only [or_self] at h
simp only [or_false] at h
simp only [false_or] at h
rwa [h, Nat.cast_zero, sub_eq_zero, eq_comm] at ha
· have h1 : (d : ℝ) ^ (n - m) ≤ 1 - (d : ℝ)⁻¹ := calc
(d : ℝ) ^ (n - m) ≤ (d : ℝ)⁻¹ := by
rw [← zpow_neg_one]
apply zpow_right_mono₀ <| Nat.one_le_cast.mpr hd0
linarith
_ ≤ 1 - (d : ℝ)⁻¹ := by
rw [inv_eq_one_div]
rw [one_sub_div <| Nat.cast_ne_zero.mpr (ne_of_gt hd0)]
rw [div_le_div_iff_of_pos_right <| Nat.cast_pos'.mpr hd0]
rw [le_sub_iff_add_le]
norm_cast
linarith [sub_lt_of_abs_sub_lt_right (a := (1 : ℝ)) (b := d ^ (n - m)) (c := d⁻¹) h]
|
/root/DuelModelResearch/mathlib4/Counterexamples/DiscreteTopologyNonDiscreteUniformity.lean
|
{
"open": [
"Set Function Filter Metric"
],
"variables": []
}
|
[
{
"line": "have hd0 : 0 < d := one_pos.trans hd1",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\n⊢ m = n",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ m = n"
},
{
"line": "refine_lift\n have hd0 : 0 < d := one_pos.trans hd1;\n ?_",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\n⊢ m = n",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ m = n"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (have hd0 : 0 < d := one_pos.trans hd1;\n ?_);\n rotate_right)",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\n⊢ m = n",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ m = n"
},
{
"line": "refine\n no_implicit_lambda%\n (have hd0 : 0 < d := one_pos.trans hd1;\n ?_)",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\n⊢ m = n",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ m = n"
},
{
"line": "rotate_right",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ m = n",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ m = n"
},
{
"line": "replace h : |(1 : ℝ) - d ^ (n - m)| < (d : ℝ)⁻¹ :=\n by\n rw [← mul_lt_mul_iff_of_pos_left (a := (d : ℝ) ^ (-n)) (zpow_pos _ _)]\n rw [← abs_of_nonneg (a := (d : ℝ) ^ (-n)) (le_of_lt <| zpow_pos _ _)]\n rw [← abs_mul]\n rw [mul_sub]\n rw [mul_one]\n rw [← zpow_add₀ <| Nat.cast_ne_zero.mpr (ne_of_gt hd0)]\n rw [sub_eq_add_neg (b := m)]\n rw [neg_add_cancel_left]\n rw [← abs_neg]\n rw [neg_sub]\n rw [abs_of_nonneg (a := (d : ℝ) ^ (-n)) (le_of_lt <| zpow_pos _ _)]\n rw [← zpow_neg_one]\n rw [← zpow_add₀ <| Nat.cast_ne_zero.mpr (ne_of_gt hd0)]\n rw [← sub_eq_add_neg]\n exact h\n all_goals exact Nat.cast_pos'.mpr hd0",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ m = n",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\n⊢ m = n"
},
{
"line": "have h : |(1 : ℝ) - d ^ (n - m)| < (d : ℝ)⁻¹ :=\n by\n rw [← mul_lt_mul_iff_of_pos_left (a := (d : ℝ) ^ (-n)) (zpow_pos _ _)]\n rw [← abs_of_nonneg (a := (d : ℝ) ^ (-n)) (le_of_lt <| zpow_pos _ _)]\n rw [← abs_mul]\n rw [mul_sub]\n rw [mul_one]\n rw [← zpow_add₀ <| Nat.cast_ne_zero.mpr (ne_of_gt hd0)]\n rw [sub_eq_add_neg (b := m)]\n rw [neg_add_cancel_left]\n rw [← abs_neg]\n rw [neg_sub]\n rw [abs_of_nonneg (a := (d : ℝ) ^ (-n)) (le_of_lt <| zpow_pos _ _)]\n rw [← zpow_neg_one]\n rw [← zpow_add₀ <| Nat.cast_ne_zero.mpr (ne_of_gt hd0)]\n rw [← sub_eq_add_neg]\n exact h\n all_goals exact Nat.cast_pos'.mpr hd0",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ m = n",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh✝ : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\n⊢ m = n"
},
{
"line": "focus\n refine\n no_implicit_lambda%\n (have h : |(1 : ℝ) - d ^ (n - m)| < (d : ℝ)⁻¹ := ?body✝;\n ?_)\n case body✝ =>\n with_annotate_state\"by\"\n ( rw [← mul_lt_mul_iff_of_pos_left (a := (d : ℝ) ^ (-n)) (zpow_pos _ _)]\n rw [← abs_of_nonneg (a := (d : ℝ) ^ (-n)) (le_of_lt <| zpow_pos _ _)]\n rw [← abs_mul]\n rw [mul_sub]\n rw [mul_one]\n rw [← zpow_add₀ <| Nat.cast_ne_zero.mpr (ne_of_gt hd0)]\n rw [sub_eq_add_neg (b := m)]\n rw [neg_add_cancel_left]\n rw [← abs_neg]\n rw [neg_sub]\n rw [abs_of_nonneg (a := (d : ℝ) ^ (-n)) (le_of_lt <| zpow_pos _ _)]\n rw [← zpow_neg_one]\n rw [← zpow_add₀ <| Nat.cast_ne_zero.mpr (ne_of_gt hd0)]\n rw [← sub_eq_add_neg]\n exact h\n all_goals exact Nat.cast_pos'.mpr hd0)",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ m = n",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh✝ : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\n⊢ m = n"
},
{
"line": "refine\n no_implicit_lambda%\n (have h : |(1 : ℝ) - d ^ (n - m)| < (d : ℝ)⁻¹ := ?body✝;\n ?_)",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ m = n",
"after_state": "case body\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |1 - ↑d ^ (n - m)| < (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh✝ : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\n⊢ m = n"
},
{
"line": "case body✝ =>\n with_annotate_state\"by\"\n ( rw [← mul_lt_mul_iff_of_pos_left (a := (d : ℝ) ^ (-n)) (zpow_pos _ _)]\n rw [← abs_of_nonneg (a := (d : ℝ) ^ (-n)) (le_of_lt <| zpow_pos _ _)]\n rw [← abs_mul]\n rw [mul_sub]\n rw [mul_one]\n rw [← zpow_add₀ <| Nat.cast_ne_zero.mpr (ne_of_gt hd0)]\n rw [sub_eq_add_neg (b := m)]\n rw [neg_add_cancel_left]\n rw [← abs_neg]\n rw [neg_sub]\n rw [abs_of_nonneg (a := (d : ℝ) ^ (-n)) (le_of_lt <| zpow_pos _ _)]\n rw [← zpow_neg_one]\n rw [← zpow_add₀ <| Nat.cast_ne_zero.mpr (ne_of_gt hd0)]\n rw [← sub_eq_add_neg]\n exact h\n all_goals exact Nat.cast_pos'.mpr hd0)",
"before_state": "case body\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |1 - ↑d ^ (n - m)| < (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh✝ : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\n⊢ m = n",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh✝ : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\n⊢ m = n"
},
{
"line": "with_annotate_state\"by\"\n ( rw [← mul_lt_mul_iff_of_pos_left (a := (d : ℝ) ^ (-n)) (zpow_pos _ _)]\n rw [← abs_of_nonneg (a := (d : ℝ) ^ (-n)) (le_of_lt <| zpow_pos _ _)]\n rw [← abs_mul]\n rw [mul_sub]\n rw [mul_one]\n rw [← zpow_add₀ <| Nat.cast_ne_zero.mpr (ne_of_gt hd0)]\n rw [sub_eq_add_neg (b := m)]\n rw [neg_add_cancel_left]\n rw [← abs_neg]\n rw [neg_sub]\n rw [abs_of_nonneg (a := (d : ℝ) ^ (-n)) (le_of_lt <| zpow_pos _ _)]\n rw [← zpow_neg_one]\n rw [← zpow_add₀ <| Nat.cast_ne_zero.mpr (ne_of_gt hd0)]\n rw [← sub_eq_add_neg]\n exact h\n all_goals exact Nat.cast_pos'.mpr hd0)",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |1 - ↑d ^ (n - m)| < (↑d)⁻¹",
"after_state": "No Goals!"
},
{
"line": "rw [← mul_lt_mul_iff_of_pos_left (a := (d : ℝ) ^ (-n)) (zpow_pos _ _)]",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |1 - ↑d ^ (n - m)| < (↑d)⁻¹",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ ↑d ^ (-n) * |1 - ↑d ^ (n - m)| < ↑d ^ (-n) * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "rewrite [← mul_lt_mul_iff_of_pos_left (a := (d : ℝ) ^ (-n)) (zpow_pos _ _)]",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |1 - ↑d ^ (n - m)| < (↑d)⁻¹",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ ↑d ^ (-n) * |1 - ↑d ^ (n - m)| < ↑d ^ (-n) * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ ↑d ^ (-n) * |1 - ↑d ^ (n - m)| < ↑d ^ (-n) * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ ↑d ^ (-n) * |1 - ↑d ^ (n - m)| < ↑d ^ (-n) * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "try (with_reducible rfl)",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ ↑d ^ (-n) * |1 - ↑d ^ (n - m)| < ↑d ^ (-n) * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ ↑d ^ (-n) * |1 - ↑d ^ (n - m)| < ↑d ^ (-n) * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ ↑d ^ (-n) * |1 - ↑d ^ (n - m)| < ↑d ^ (-n) * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ ↑d ^ (-n) * |1 - ↑d ^ (n - m)| < ↑d ^ (-n) * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "with_reducible rfl",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ ↑d ^ (-n) * |1 - ↑d ^ (n - m)| < ↑d ^ (-n) * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ ↑d ^ (-n) * |1 - ↑d ^ (n - m)| < ↑d ^ (-n) * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "rfl",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ ↑d ^ (-n) * |1 - ↑d ^ (n - m)| < ↑d ^ (-n) * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ ↑d ^ (-n) * |1 - ↑d ^ (n - m)| < ↑d ^ (-n) * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "apply_rfl",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ ↑d ^ (-n) * |1 - ↑d ^ (n - m)| < ↑d ^ (-n) * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ ↑d ^ (-n) * |1 - ↑d ^ (n - m)| < ↑d ^ (-n) * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "skip",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ ↑d ^ (-n) * |1 - ↑d ^ (n - m)| < ↑d ^ (-n) * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ ↑d ^ (-n) * |1 - ↑d ^ (n - m)| < ↑d ^ (-n) * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "rw [← abs_of_nonneg (a := (d : ℝ) ^ (-n)) (le_of_lt <| zpow_pos _ _)]",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ ↑d ^ (-n) * |1 - ↑d ^ (n - m)| < ↑d ^ (-n) * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n)| * |1 - ↑d ^ (n - m)| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "rewrite [← abs_of_nonneg (a := (d : ℝ) ^ (-n)) (le_of_lt <| zpow_pos _ _)]",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ ↑d ^ (-n) * |1 - ↑d ^ (n - m)| < ↑d ^ (-n) * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n)| * |1 - ↑d ^ (n - m)| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n)| * |1 - ↑d ^ (n - m)| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n)| * |1 - ↑d ^ (n - m)| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "try (with_reducible rfl)",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n)| * |1 - ↑d ^ (n - m)| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n)| * |1 - ↑d ^ (n - m)| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n)| * |1 - ↑d ^ (n - m)| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n)| * |1 - ↑d ^ (n - m)| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "with_reducible rfl",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n)| * |1 - ↑d ^ (n - m)| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n)| * |1 - ↑d ^ (n - m)| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "rfl",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n)| * |1 - ↑d ^ (n - m)| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n)| * |1 - ↑d ^ (n - m)| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "apply_rfl",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n)| * |1 - ↑d ^ (n - m)| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n)| * |1 - ↑d ^ (n - m)| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "skip",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n)| * |1 - ↑d ^ (n - m)| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n)| * |1 - ↑d ^ (n - m)| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "rw [← abs_mul]",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n)| * |1 - ↑d ^ (n - m)| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n) * (1 - ↑d ^ (n - m))| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "rewrite [← abs_mul]",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n)| * |1 - ↑d ^ (n - m)| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n) * (1 - ↑d ^ (n - m))| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n) * (1 - ↑d ^ (n - m))| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n) * (1 - ↑d ^ (n - m))| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "try (with_reducible rfl)",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n) * (1 - ↑d ^ (n - m))| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n) * (1 - ↑d ^ (n - m))| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n) * (1 - ↑d ^ (n - m))| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n) * (1 - ↑d ^ (n - m))| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "with_reducible rfl",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n) * (1 - ↑d ^ (n - m))| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n) * (1 - ↑d ^ (n - m))| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "rfl",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n) * (1 - ↑d ^ (n - m))| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n) * (1 - ↑d ^ (n - m))| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "apply_rfl",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n) * (1 - ↑d ^ (n - m))| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n) * (1 - ↑d ^ (n - m))| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "skip",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n) * (1 - ↑d ^ (n - m))| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n) * (1 - ↑d ^ (n - m))| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "rw [mul_sub]",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n) * (1 - ↑d ^ (n - m))| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n) * 1 - ↑d ^ (-n) * ↑d ^ (n - m)| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "rewrite [mul_sub]",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n) * (1 - ↑d ^ (n - m))| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n) * 1 - ↑d ^ (-n) * ↑d ^ (n - m)| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n) * 1 - ↑d ^ (-n) * ↑d ^ (n - m)| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n) * 1 - ↑d ^ (-n) * ↑d ^ (n - m)| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "try (with_reducible rfl)",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n) * 1 - ↑d ^ (-n) * ↑d ^ (n - m)| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n) * 1 - ↑d ^ (-n) * ↑d ^ (n - m)| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n) * 1 - ↑d ^ (-n) * ↑d ^ (n - m)| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n) * 1 - ↑d ^ (-n) * ↑d ^ (n - m)| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "with_reducible rfl",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n) * 1 - ↑d ^ (-n) * ↑d ^ (n - m)| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n) * 1 - ↑d ^ (-n) * ↑d ^ (n - m)| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "rfl",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n) * 1 - ↑d ^ (-n) * ↑d ^ (n - m)| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n) * 1 - ↑d ^ (-n) * ↑d ^ (n - m)| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "apply_rfl",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n) * 1 - ↑d ^ (-n) * ↑d ^ (n - m)| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n) * 1 - ↑d ^ (-n) * ↑d ^ (n - m)| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "skip",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n) * 1 - ↑d ^ (-n) * ↑d ^ (n - m)| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n) * 1 - ↑d ^ (-n) * ↑d ^ (n - m)| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "rw [mul_one]",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n) * 1 - ↑d ^ (-n) * ↑d ^ (n - m)| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n) - ↑d ^ (-n) * ↑d ^ (n - m)| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "rewrite [mul_one]",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n) * 1 - ↑d ^ (-n) * ↑d ^ (n - m)| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n) - ↑d ^ (-n) * ↑d ^ (n - m)| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n) - ↑d ^ (-n) * ↑d ^ (n - m)| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n) - ↑d ^ (-n) * ↑d ^ (n - m)| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "try (with_reducible rfl)",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n) - ↑d ^ (-n) * ↑d ^ (n - m)| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n) - ↑d ^ (-n) * ↑d ^ (n - m)| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n) - ↑d ^ (-n) * ↑d ^ (n - m)| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n) - ↑d ^ (-n) * ↑d ^ (n - m)| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "with_reducible rfl",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n) - ↑d ^ (-n) * ↑d ^ (n - m)| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n) - ↑d ^ (-n) * ↑d ^ (n - m)| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "rfl",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n) - ↑d ^ (-n) * ↑d ^ (n - m)| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n) - ↑d ^ (-n) * ↑d ^ (n - m)| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "apply_rfl",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n) - ↑d ^ (-n) * ↑d ^ (n - m)| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n) - ↑d ^ (-n) * ↑d ^ (n - m)| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "skip",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n) - ↑d ^ (-n) * ↑d ^ (n - m)| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n) - ↑d ^ (-n) * ↑d ^ (n - m)| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "rw [← zpow_add₀ <| Nat.cast_ne_zero.mpr (ne_of_gt hd0)]",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n) - ↑d ^ (-n) * ↑d ^ (n - m)| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n) - ↑d ^ (-n + (n - m))| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "rewrite [← zpow_add₀ <| Nat.cast_ne_zero.mpr (ne_of_gt hd0)]",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n) - ↑d ^ (-n) * ↑d ^ (n - m)| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n) - ↑d ^ (-n + (n - m))| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n) - ↑d ^ (-n + (n - m))| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n) - ↑d ^ (-n + (n - m))| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "try (with_reducible rfl)",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n) - ↑d ^ (-n + (n - m))| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n) - ↑d ^ (-n + (n - m))| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n) - ↑d ^ (-n + (n - m))| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n) - ↑d ^ (-n + (n - m))| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "with_reducible rfl",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n) - ↑d ^ (-n + (n - m))| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n) - ↑d ^ (-n + (n - m))| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "rfl",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n) - ↑d ^ (-n + (n - m))| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n) - ↑d ^ (-n + (n - m))| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "apply_rfl",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n) - ↑d ^ (-n + (n - m))| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n) - ↑d ^ (-n + (n - m))| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "skip",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n) - ↑d ^ (-n + (n - m))| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n) - ↑d ^ (-n + (n - m))| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "rw [sub_eq_add_neg (b := m)]",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n) - ↑d ^ (-n + (n - m))| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n) - ↑d ^ (-n + (n + -m))| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "rewrite [sub_eq_add_neg (b := m)]",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n) - ↑d ^ (-n + (n - m))| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n) - ↑d ^ (-n + (n + -m))| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n) - ↑d ^ (-n + (n + -m))| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n) - ↑d ^ (-n + (n + -m))| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "try (with_reducible rfl)",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n) - ↑d ^ (-n + (n + -m))| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n) - ↑d ^ (-n + (n + -m))| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n) - ↑d ^ (-n + (n + -m))| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n) - ↑d ^ (-n + (n + -m))| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "with_reducible rfl",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n) - ↑d ^ (-n + (n + -m))| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n) - ↑d ^ (-n + (n + -m))| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "rfl",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n) - ↑d ^ (-n + (n + -m))| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n) - ↑d ^ (-n + (n + -m))| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "apply_rfl",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n) - ↑d ^ (-n + (n + -m))| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n) - ↑d ^ (-n + (n + -m))| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "skip",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n) - ↑d ^ (-n + (n + -m))| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n) - ↑d ^ (-n + (n + -m))| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "rw [neg_add_cancel_left]",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n) - ↑d ^ (-n + (n + -m))| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n) - ↑d ^ (-m)| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "rewrite [neg_add_cancel_left]",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n) - ↑d ^ (-n + (n + -m))| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n) - ↑d ^ (-m)| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n) - ↑d ^ (-m)| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n) - ↑d ^ (-m)| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "try (with_reducible rfl)",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n) - ↑d ^ (-m)| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n) - ↑d ^ (-m)| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n) - ↑d ^ (-m)| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n) - ↑d ^ (-m)| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "with_reducible rfl",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n) - ↑d ^ (-m)| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n) - ↑d ^ (-m)| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "rfl",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n) - ↑d ^ (-m)| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n) - ↑d ^ (-m)| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "apply_rfl",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n) - ↑d ^ (-m)| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n) - ↑d ^ (-m)| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "skip",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n) - ↑d ^ (-m)| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n) - ↑d ^ (-m)| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "rw [← abs_neg]",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n) - ↑d ^ (-m)| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |(-(↑d ^ (-n) - ↑d ^ (-m)))| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "rewrite [← abs_neg]",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-n) - ↑d ^ (-m)| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |(-(↑d ^ (-n) - ↑d ^ (-m)))| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |(-(↑d ^ (-n) - ↑d ^ (-m)))| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |(-(↑d ^ (-n) - ↑d ^ (-m)))| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "try (with_reducible rfl)",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |(-(↑d ^ (-n) - ↑d ^ (-m)))| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |(-(↑d ^ (-n) - ↑d ^ (-m)))| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |(-(↑d ^ (-n) - ↑d ^ (-m)))| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |(-(↑d ^ (-n) - ↑d ^ (-m)))| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "with_reducible rfl",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |(-(↑d ^ (-n) - ↑d ^ (-m)))| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |(-(↑d ^ (-n) - ↑d ^ (-m)))| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "rfl",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |(-(↑d ^ (-n) - ↑d ^ (-m)))| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |(-(↑d ^ (-n) - ↑d ^ (-m)))| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "apply_rfl",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |(-(↑d ^ (-n) - ↑d ^ (-m)))| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |(-(↑d ^ (-n) - ↑d ^ (-m)))| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "skip",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |(-(↑d ^ (-n) - ↑d ^ (-m)))| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |(-(↑d ^ (-n) - ↑d ^ (-m)))| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "rw [neg_sub]",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |(-(↑d ^ (-n) - ↑d ^ (-m)))| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-m) - ↑d ^ (-n)| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "rewrite [neg_sub]",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |(-(↑d ^ (-n) - ↑d ^ (-m)))| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-m) - ↑d ^ (-n)| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-m) - ↑d ^ (-n)| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-m) - ↑d ^ (-n)| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "try (with_reducible rfl)",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-m) - ↑d ^ (-n)| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-m) - ↑d ^ (-n)| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-m) - ↑d ^ (-n)| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-m) - ↑d ^ (-n)| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "with_reducible rfl",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-m) - ↑d ^ (-n)| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-m) - ↑d ^ (-n)| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "rfl",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-m) - ↑d ^ (-n)| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-m) - ↑d ^ (-n)| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "apply_rfl",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-m) - ↑d ^ (-n)| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-m) - ↑d ^ (-n)| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "skip",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-m) - ↑d ^ (-n)| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-m) - ↑d ^ (-n)| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "rw [abs_of_nonneg (a := (d : ℝ) ^ (-n)) (le_of_lt <| zpow_pos _ _)]",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-m) - ↑d ^ (-n)| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n) * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "rewrite [abs_of_nonneg (a := (d : ℝ) ^ (-n)) (le_of_lt <| zpow_pos _ _)]",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-m) - ↑d ^ (-n)| < |↑d ^ (-n)| * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n) * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n) * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n) * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "try (with_reducible rfl)",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n) * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n) * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n) * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n) * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "with_reducible rfl",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n) * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n) * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "rfl",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n) * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n) * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "apply_rfl",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n) * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n) * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "skip",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n) * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n) * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "rw [← zpow_neg_one]",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n) * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n) * ↑d ^ (-1)\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "rewrite [← zpow_neg_one]",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n) * (↑d)⁻¹\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n) * ↑d ^ (-1)\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n) * ↑d ^ (-1)\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n) * ↑d ^ (-1)\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "try (with_reducible rfl)",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n) * ↑d ^ (-1)\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n) * ↑d ^ (-1)\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n) * ↑d ^ (-1)\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n) * ↑d ^ (-1)\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "with_reducible rfl",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n) * ↑d ^ (-1)\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n) * ↑d ^ (-1)\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "rfl",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n) * ↑d ^ (-1)\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n) * ↑d ^ (-1)\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "apply_rfl",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n) * ↑d ^ (-1)\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n) * ↑d ^ (-1)\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "skip",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n) * ↑d ^ (-1)\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n) * ↑d ^ (-1)\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "rw [← zpow_add₀ <| Nat.cast_ne_zero.mpr (ne_of_gt hd0)]",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n) * ↑d ^ (-1)\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n + -1)\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "rewrite [← zpow_add₀ <| Nat.cast_ne_zero.mpr (ne_of_gt hd0)]",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n) * ↑d ^ (-1)\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n + -1)\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n + -1)\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n + -1)\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "try (with_reducible rfl)",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n + -1)\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n + -1)\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n + -1)\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n + -1)\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "with_reducible rfl",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n + -1)\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n + -1)\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "rfl",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n + -1)\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n + -1)\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "apply_rfl",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n + -1)\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n + -1)\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "skip",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n + -1)\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n + -1)\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "rw [← sub_eq_add_neg]",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n + -1)\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "rewrite [← sub_eq_add_neg]",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n + -1)\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "try (with_reducible rfl)",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "with_reducible rfl",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "rfl",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "apply_rfl",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "skip",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "exact h",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d"
},
{
"line": "all_goals exact Nat.cast_pos'.mpr hd0",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d\n---\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "No Goals!"
},
{
"line": "exact Nat.cast_pos'.mpr hd0",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "No Goals!"
},
{
"line": "exact Nat.cast_pos'.mpr hd0",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "No Goals!"
},
{
"line": "exact Nat.cast_pos'.mpr hd0",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nh : |↑d ^ (-m) - ↑d ^ (-n)| < ↑d ^ (-n - 1)\nhd0 : 0 < d\n⊢ 0 < ↑d",
"after_state": "No Goals!"
},
{
"line": "by_cases H : (m : ℤ) ≤ n",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\n⊢ m = n",
"after_state": "case pos\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : m ≤ n\n⊢ m = n\n---\ncase neg\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ m = n"
},
{
"line": "open Classical✝ in refine if H : (m : ℤ) ≤ n then ?pos✝ else ?neg✝",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\n⊢ m = n",
"after_state": "case pos\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : m ≤ n\n⊢ m = n\n---\ncase neg\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ m = n"
},
{
"line": "refine if H : (m : ℤ) ≤ n then ?pos✝ else ?neg✝",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\n⊢ m = n",
"after_state": "case pos\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : m ≤ n\n⊢ m = n\n---\ncase neg\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ m = n"
},
{
"line": "obtain ⟨a, ha⟩ := Int.eq_ofNat_of_zero_le (sub_nonneg.mpr H)",
"before_state": "case pos\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : m ≤ n\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : m ≤ n\na : ℕ\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "rw [ha] at h",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : m ≤ n\na : ℕ\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : |1 - ↑d ^ ↑a| < (↑d)⁻¹\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "rewrite [ha] at h",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : m ≤ n\na : ℕ\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : |1 - ↑d ^ ↑a| < (↑d)⁻¹\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : |1 - ↑d ^ ↑a| < (↑d)⁻¹\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : |1 - ↑d ^ ↑a| < (↑d)⁻¹\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : |1 - ↑d ^ ↑a| < (↑d)⁻¹\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : |1 - ↑d ^ ↑a| < (↑d)⁻¹\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : |1 - ↑d ^ ↑a| < (↑d)⁻¹\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : |1 - ↑d ^ ↑a| < (↑d)⁻¹\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "with_reducible rfl",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : |1 - ↑d ^ ↑a| < (↑d)⁻¹\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : |1 - ↑d ^ ↑a| < (↑d)⁻¹\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "rfl",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : |1 - ↑d ^ ↑a| < (↑d)⁻¹\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : |1 - ↑d ^ ↑a| < (↑d)⁻¹\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "apply_rfl",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : |1 - ↑d ^ ↑a| < (↑d)⁻¹\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : |1 - ↑d ^ ↑a| < (↑d)⁻¹\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "skip",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : |1 - ↑d ^ ↑a| < (↑d)⁻¹\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : |1 - ↑d ^ ↑a| < (↑d)⁻¹\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "rw [← mul_lt_mul_iff_of_pos_left (a := (d : ℝ)) <| Nat.cast_pos'.mpr hd0] at h",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : |1 - ↑d ^ ↑a| < (↑d)⁻¹\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d * |1 - ↑d ^ ↑a| < ↑d * (↑d)⁻¹\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "rewrite [← mul_lt_mul_iff_of_pos_left (a := (d : ℝ)) <| Nat.cast_pos'.mpr hd0] at h",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : |1 - ↑d ^ ↑a| < (↑d)⁻¹\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d * |1 - ↑d ^ ↑a| < ↑d * (↑d)⁻¹\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d * |1 - ↑d ^ ↑a| < ↑d * (↑d)⁻¹\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d * |1 - ↑d ^ ↑a| < ↑d * (↑d)⁻¹\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d * |1 - ↑d ^ ↑a| < ↑d * (↑d)⁻¹\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d * |1 - ↑d ^ ↑a| < ↑d * (↑d)⁻¹\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d * |1 - ↑d ^ ↑a| < ↑d * (↑d)⁻¹\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d * |1 - ↑d ^ ↑a| < ↑d * (↑d)⁻¹\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "with_reducible rfl",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d * |1 - ↑d ^ ↑a| < ↑d * (↑d)⁻¹\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d * |1 - ↑d ^ ↑a| < ↑d * (↑d)⁻¹\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "rfl",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d * |1 - ↑d ^ ↑a| < ↑d * (↑d)⁻¹\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d * |1 - ↑d ^ ↑a| < ↑d * (↑d)⁻¹\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "apply_rfl",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d * |1 - ↑d ^ ↑a| < ↑d * (↑d)⁻¹\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d * |1 - ↑d ^ ↑a| < ↑d * (↑d)⁻¹\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "skip",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d * |1 - ↑d ^ ↑a| < ↑d * (↑d)⁻¹\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d * |1 - ↑d ^ ↑a| < ↑d * (↑d)⁻¹\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "rw [mul_inv_cancel₀ <| Nat.cast_ne_zero.mpr (ne_of_gt hd0)] at h",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d * |1 - ↑d ^ ↑a| < ↑d * (↑d)⁻¹\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d * |1 - ↑d ^ ↑a| < 1\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "rewrite [mul_inv_cancel₀ <| Nat.cast_ne_zero.mpr (ne_of_gt hd0)] at h",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d * |1 - ↑d ^ ↑a| < ↑d * (↑d)⁻¹\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d * |1 - ↑d ^ ↑a| < 1\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d * |1 - ↑d ^ ↑a| < 1\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d * |1 - ↑d ^ ↑a| < 1\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d * |1 - ↑d ^ ↑a| < 1\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d * |1 - ↑d ^ ↑a| < 1\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d * |1 - ↑d ^ ↑a| < 1\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d * |1 - ↑d ^ ↑a| < 1\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "with_reducible rfl",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d * |1 - ↑d ^ ↑a| < 1\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d * |1 - ↑d ^ ↑a| < 1\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "rfl",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d * |1 - ↑d ^ ↑a| < 1\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d * |1 - ↑d ^ ↑a| < 1\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "apply_rfl",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d * |1 - ↑d ^ ↑a| < 1\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d * |1 - ↑d ^ ↑a| < 1\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "skip",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d * |1 - ↑d ^ ↑a| < 1\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d * |1 - ↑d ^ ↑a| < 1\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "rw [← abs_of_nonneg (a := (d : ℝ)) <| Nat.cast_nonneg' d] at h",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d * |1 - ↑d ^ ↑a| < 1\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : |↑d| * |1 - |↑d| ^ ↑a| < 1\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "rewrite [← abs_of_nonneg (a := (d : ℝ)) <| Nat.cast_nonneg' d] at h",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d * |1 - ↑d ^ ↑a| < 1\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : |↑d| * |1 - |↑d| ^ ↑a| < 1\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : |↑d| * |1 - |↑d| ^ ↑a| < 1\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : |↑d| * |1 - |↑d| ^ ↑a| < 1\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : |↑d| * |1 - |↑d| ^ ↑a| < 1\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : |↑d| * |1 - |↑d| ^ ↑a| < 1\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : |↑d| * |1 - |↑d| ^ ↑a| < 1\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : |↑d| * |1 - |↑d| ^ ↑a| < 1\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "with_reducible rfl",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : |↑d| * |1 - |↑d| ^ ↑a| < 1\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : |↑d| * |1 - |↑d| ^ ↑a| < 1\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "rfl",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : |↑d| * |1 - |↑d| ^ ↑a| < 1\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : |↑d| * |1 - |↑d| ^ ↑a| < 1\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "apply_rfl",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : |↑d| * |1 - |↑d| ^ ↑a| < 1\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : |↑d| * |1 - |↑d| ^ ↑a| < 1\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "skip",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : |↑d| * |1 - |↑d| ^ ↑a| < 1\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : |↑d| * |1 - |↑d| ^ ↑a| < 1\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "rw [← abs_mul] at h",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : |↑d| * |1 - |↑d| ^ ↑a| < 1\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : |↑d * (1 - |↑d| ^ ↑a)| < 1\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "rewrite [← abs_mul] at h",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : |↑d| * |1 - |↑d| ^ ↑a| < 1\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : |↑d * (1 - |↑d| ^ ↑a)| < 1\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : |↑d * (1 - |↑d| ^ ↑a)| < 1\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : |↑d * (1 - |↑d| ^ ↑a)| < 1\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : |↑d * (1 - |↑d| ^ ↑a)| < 1\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : |↑d * (1 - |↑d| ^ ↑a)| < 1\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : |↑d * (1 - |↑d| ^ ↑a)| < 1\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : |↑d * (1 - |↑d| ^ ↑a)| < 1\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "with_reducible rfl",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : |↑d * (1 - |↑d| ^ ↑a)| < 1\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : |↑d * (1 - |↑d| ^ ↑a)| < 1\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "rfl",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : |↑d * (1 - |↑d| ^ ↑a)| < 1\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : |↑d * (1 - |↑d| ^ ↑a)| < 1\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "apply_rfl",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : |↑d * (1 - |↑d| ^ ↑a)| < 1\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : |↑d * (1 - |↑d| ^ ↑a)| < 1\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "skip",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : |↑d * (1 - |↑d| ^ ↑a)| < 1\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : |↑d * (1 - |↑d| ^ ↑a)| < 1\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "rw [show |(d : ℝ) * (1 - |(d : ℝ)| ^ (a : ℤ))| = |(d : ℤ) * (1 - |(d : ℤ)| ^ a)| by norm_cast] at h",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : |↑d * (1 - |↑d| ^ ↑a)| < 1\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑|↑d * (1 - |↑d| ^ a)| < 1\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "rewrite [show |(d : ℝ) * (1 - |(d : ℝ)| ^ (a : ℤ))| = |(d : ℤ) * (1 - |(d : ℤ)| ^ a)| by norm_cast] at h",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : |↑d * (1 - |↑d| ^ ↑a)| < 1\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑|↑d * (1 - |↑d| ^ a)| < 1\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "norm_cast",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : |↑d * (1 - |↑d| ^ ↑a)| < 1\nha : n - m = ↑a\n⊢ |↑d * (1 - |↑d| ^ ↑a)| = ↑|↑d * (1 - |↑d| ^ a)|",
"after_state": "No Goals!"
},
{
"line": "focus\n norm_cast0\n with_annotate_state\"<;>\" skip\n all_goals try trivial",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : |↑d * (1 - |↑d| ^ ↑a)| < 1\nha : n - m = ↑a\n⊢ |↑d * (1 - |↑d| ^ ↑a)| = ↑|↑d * (1 - |↑d| ^ a)|",
"after_state": "No Goals!"
},
{
"line": "norm_cast0",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : |↑d * (1 - |↑d| ^ ↑a)| < 1\nha : n - m = ↑a\n⊢ |↑d * (1 - |↑d| ^ ↑a)| = ↑|↑d * (1 - |↑d| ^ a)|",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : |↑d * (1 - |↑d| ^ ↑a)| < 1\nha : n - m = ↑a\n⊢ |↑d * subNatNat 1 (d ^ a)| = |↑d * subNatNat 1 (d ^ a)|"
},
{
"line": "with_annotate_state\"<;>\" skip",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : |↑d * (1 - |↑d| ^ ↑a)| < 1\nha : n - m = ↑a\n⊢ |↑d * subNatNat 1 (d ^ a)| = |↑d * subNatNat 1 (d ^ a)|",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : |↑d * (1 - |↑d| ^ ↑a)| < 1\nha : n - m = ↑a\n⊢ |↑d * subNatNat 1 (d ^ a)| = |↑d * subNatNat 1 (d ^ a)|"
},
{
"line": "skip",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : |↑d * (1 - |↑d| ^ ↑a)| < 1\nha : n - m = ↑a\n⊢ |↑d * subNatNat 1 (d ^ a)| = |↑d * subNatNat 1 (d ^ a)|",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : |↑d * (1 - |↑d| ^ ↑a)| < 1\nha : n - m = ↑a\n⊢ |↑d * subNatNat 1 (d ^ a)| = |↑d * subNatNat 1 (d ^ a)|"
},
{
"line": "all_goals try trivial",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : |↑d * (1 - |↑d| ^ ↑a)| < 1\nha : n - m = ↑a\n⊢ |↑d * subNatNat 1 (d ^ a)| = |↑d * subNatNat 1 (d ^ a)|",
"after_state": "No Goals!"
},
{
"line": "try trivial",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : |↑d * (1 - |↑d| ^ ↑a)| < 1\nha : n - m = ↑a\n⊢ |↑d * subNatNat 1 (d ^ a)| = |↑d * subNatNat 1 (d ^ a)|",
"after_state": "No Goals!"
},
{
"line": "first\n| trivial\n| skip",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : |↑d * (1 - |↑d| ^ ↑a)| < 1\nha : n - m = ↑a\n⊢ |↑d * subNatNat 1 (d ^ a)| = |↑d * subNatNat 1 (d ^ a)|",
"after_state": "No Goals!"
},
{
"line": "trivial",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : |↑d * (1 - |↑d| ^ ↑a)| < 1\nha : n - m = ↑a\n⊢ |↑d * subNatNat 1 (d ^ a)| = |↑d * subNatNat 1 (d ^ a)|",
"after_state": "No Goals!"
},
{
"line": "rfl",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : |↑d * (1 - |↑d| ^ ↑a)| < 1\nha : n - m = ↑a\n⊢ |↑d * subNatNat 1 (d ^ a)| = |↑d * subNatNat 1 (d ^ a)|",
"after_state": "No Goals!"
},
{
"line": "eq_refl",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : |↑d * (1 - |↑d| ^ ↑a)| < 1\nha : n - m = ↑a\n⊢ |↑d * subNatNat 1 (d ^ a)| = |↑d * subNatNat 1 (d ^ a)|",
"after_state": "No Goals!"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑|↑d * (1 - |↑d| ^ a)| < 1\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑|↑d * (1 - |↑d| ^ a)| < 1\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑|↑d * (1 - |↑d| ^ a)| < 1\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑|↑d * (1 - |↑d| ^ a)| < 1\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑|↑d * (1 - |↑d| ^ a)| < 1\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑|↑d * (1 - |↑d| ^ a)| < 1\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "with_reducible rfl",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑|↑d * (1 - |↑d| ^ a)| < 1\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑|↑d * (1 - |↑d| ^ a)| < 1\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "rfl",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑|↑d * (1 - |↑d| ^ a)| < 1\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑|↑d * (1 - |↑d| ^ a)| < 1\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "apply_rfl",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑|↑d * (1 - |↑d| ^ a)| < 1\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑|↑d * (1 - |↑d| ^ a)| < 1\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "skip",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑|↑d * (1 - |↑d| ^ a)| < 1\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑|↑d * (1 - |↑d| ^ a)| < 1\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "rw [← Int.cast_one (R := ℝ)] at h",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑|↑d * (1 - |↑d| ^ a)| < 1\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑|↑d * (1 - |↑d| ^ a)| < ↑1\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "rewrite [← Int.cast_one (R := ℝ)] at h",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑|↑d * (1 - |↑d| ^ a)| < 1\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑|↑d * (1 - |↑d| ^ a)| < ↑1\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑|↑d * (1 - |↑d| ^ a)| < ↑1\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑|↑d * (1 - |↑d| ^ a)| < ↑1\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑|↑d * (1 - |↑d| ^ a)| < ↑1\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑|↑d * (1 - |↑d| ^ a)| < ↑1\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑|↑d * (1 - |↑d| ^ a)| < ↑1\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑|↑d * (1 - |↑d| ^ a)| < ↑1\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "with_reducible rfl",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑|↑d * (1 - |↑d| ^ a)| < ↑1\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑|↑d * (1 - |↑d| ^ a)| < ↑1\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "rfl",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑|↑d * (1 - |↑d| ^ a)| < ↑1\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑|↑d * (1 - |↑d| ^ a)| < ↑1\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "apply_rfl",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑|↑d * (1 - |↑d| ^ a)| < ↑1\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑|↑d * (1 - |↑d| ^ a)| < ↑1\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "skip",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑|↑d * (1 - |↑d| ^ a)| < ↑1\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑|↑d * (1 - |↑d| ^ a)| < ↑1\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "rw [Int.cast_lt] at h",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑|↑d * (1 - |↑d| ^ a)| < ↑1\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : |↑d * (1 - |↑d| ^ a)| < 1\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "rewrite [Int.cast_lt] at h",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑|↑d * (1 - |↑d| ^ a)| < ↑1\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : |↑d * (1 - |↑d| ^ a)| < 1\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : |↑d * (1 - |↑d| ^ a)| < 1\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : |↑d * (1 - |↑d| ^ a)| < 1\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : |↑d * (1 - |↑d| ^ a)| < 1\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : |↑d * (1 - |↑d| ^ a)| < 1\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : |↑d * (1 - |↑d| ^ a)| < 1\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : |↑d * (1 - |↑d| ^ a)| < 1\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "with_reducible rfl",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : |↑d * (1 - |↑d| ^ a)| < 1\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : |↑d * (1 - |↑d| ^ a)| < 1\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "rfl",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : |↑d * (1 - |↑d| ^ a)| < 1\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : |↑d * (1 - |↑d| ^ a)| < 1\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "apply_rfl",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : |↑d * (1 - |↑d| ^ a)| < 1\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : |↑d * (1 - |↑d| ^ a)| < 1\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "skip",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : |↑d * (1 - |↑d| ^ a)| < 1\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : |↑d * (1 - |↑d| ^ a)| < 1\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "rw [Int.abs_lt_one_iff] at h",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : |↑d * (1 - |↑d| ^ a)| < 1\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d * (1 - |↑d| ^ a) = 0\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "rewrite [Int.abs_lt_one_iff] at h",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : |↑d * (1 - |↑d| ^ a)| < 1\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d * (1 - |↑d| ^ a) = 0\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d * (1 - |↑d| ^ a) = 0\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d * (1 - |↑d| ^ a) = 0\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d * (1 - |↑d| ^ a) = 0\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d * (1 - |↑d| ^ a) = 0\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d * (1 - |↑d| ^ a) = 0\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d * (1 - |↑d| ^ a) = 0\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "with_reducible rfl",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d * (1 - |↑d| ^ a) = 0\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d * (1 - |↑d| ^ a) = 0\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "rfl",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d * (1 - |↑d| ^ a) = 0\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d * (1 - |↑d| ^ a) = 0\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "apply_rfl",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d * (1 - |↑d| ^ a) = 0\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d * (1 - |↑d| ^ a) = 0\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "skip",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d * (1 - |↑d| ^ a) = 0\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d * (1 - |↑d| ^ a) = 0\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "rw [Int.mul_eq_zero] at h",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d * (1 - |↑d| ^ a) = 0\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d = 0 ∨ 1 - |↑d| ^ a = 0\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "rewrite [Int.mul_eq_zero] at h",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d * (1 - |↑d| ^ a) = 0\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d = 0 ∨ 1 - |↑d| ^ a = 0\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d = 0 ∨ 1 - |↑d| ^ a = 0\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d = 0 ∨ 1 - |↑d| ^ a = 0\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d = 0 ∨ 1 - |↑d| ^ a = 0\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d = 0 ∨ 1 - |↑d| ^ a = 0\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d = 0 ∨ 1 - |↑d| ^ a = 0\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d = 0 ∨ 1 - |↑d| ^ a = 0\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "with_reducible rfl",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d = 0 ∨ 1 - |↑d| ^ a = 0\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d = 0 ∨ 1 - |↑d| ^ a = 0\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "rfl",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d = 0 ∨ 1 - |↑d| ^ a = 0\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d = 0 ∨ 1 - |↑d| ^ a = 0\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "apply_rfl",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d = 0 ∨ 1 - |↑d| ^ a = 0\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d = 0 ∨ 1 - |↑d| ^ a = 0\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "skip",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d = 0 ∨ 1 - |↑d| ^ a = 0\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d = 0 ∨ 1 - |↑d| ^ a = 0\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "rw [sub_eq_zero] at h",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d = 0 ∨ 1 - |↑d| ^ a = 0\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d = 0 ∨ 1 = |↑d| ^ a\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "rewrite [sub_eq_zero] at h",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d = 0 ∨ 1 - |↑d| ^ a = 0\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d = 0 ∨ 1 = |↑d| ^ a\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d = 0 ∨ 1 = |↑d| ^ a\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d = 0 ∨ 1 = |↑d| ^ a\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d = 0 ∨ 1 = |↑d| ^ a\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d = 0 ∨ 1 = |↑d| ^ a\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d = 0 ∨ 1 = |↑d| ^ a\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d = 0 ∨ 1 = |↑d| ^ a\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "with_reducible rfl",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d = 0 ∨ 1 = |↑d| ^ a\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d = 0 ∨ 1 = |↑d| ^ a\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "rfl",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d = 0 ∨ 1 = |↑d| ^ a\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d = 0 ∨ 1 = |↑d| ^ a\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "apply_rfl",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d = 0 ∨ 1 = |↑d| ^ a\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d = 0 ∨ 1 = |↑d| ^ a\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "skip",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d = 0 ∨ 1 = |↑d| ^ a\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d = 0 ∨ 1 = |↑d| ^ a\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "rw [eq_comm (a := 1)] at h",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d = 0 ∨ 1 = |↑d| ^ a\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d = 0 ∨ |↑d| ^ a = 1\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "rewrite [eq_comm (a := 1)] at h",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d = 0 ∨ 1 = |↑d| ^ a\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d = 0 ∨ |↑d| ^ a = 1\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d = 0 ∨ |↑d| ^ a = 1\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d = 0 ∨ |↑d| ^ a = 1\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d = 0 ∨ |↑d| ^ a = 1\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d = 0 ∨ |↑d| ^ a = 1\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d = 0 ∨ |↑d| ^ a = 1\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d = 0 ∨ |↑d| ^ a = 1\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "with_reducible rfl",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d = 0 ∨ |↑d| ^ a = 1\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d = 0 ∨ |↑d| ^ a = 1\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "rfl",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d = 0 ∨ |↑d| ^ a = 1\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d = 0 ∨ |↑d| ^ a = 1\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "apply_rfl",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d = 0 ∨ |↑d| ^ a = 1\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d = 0 ∨ |↑d| ^ a = 1\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "skip",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d = 0 ∨ |↑d| ^ a = 1\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d = 0 ∨ |↑d| ^ a = 1\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "rw [pow_eq_one_iff_cases] at h",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d = 0 ∨ |↑d| ^ a = 1\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d = 0 ∨ a = 0 ∨ |↑d| = 1 ∨ |↑d| = -1 ∧ Even a\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "rewrite [pow_eq_one_iff_cases] at h",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d = 0 ∨ |↑d| ^ a = 1\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d = 0 ∨ a = 0 ∨ |↑d| = 1 ∨ |↑d| = -1 ∧ Even a\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d = 0 ∨ a = 0 ∨ |↑d| = 1 ∨ |↑d| = -1 ∧ Even a\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d = 0 ∨ a = 0 ∨ |↑d| = 1 ∨ |↑d| = -1 ∧ Even a\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d = 0 ∨ a = 0 ∨ |↑d| = 1 ∨ |↑d| = -1 ∧ Even a\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d = 0 ∨ a = 0 ∨ |↑d| = 1 ∨ |↑d| = -1 ∧ Even a\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d = 0 ∨ a = 0 ∨ |↑d| = 1 ∨ |↑d| = -1 ∧ Even a\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d = 0 ∨ a = 0 ∨ |↑d| = 1 ∨ |↑d| = -1 ∧ Even a\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "with_reducible rfl",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d = 0 ∨ a = 0 ∨ |↑d| = 1 ∨ |↑d| = -1 ∧ Even a\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d = 0 ∨ a = 0 ∨ |↑d| = 1 ∨ |↑d| = -1 ∧ Even a\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "rfl",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d = 0 ∨ a = 0 ∨ |↑d| = 1 ∨ |↑d| = -1 ∧ Even a\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d = 0 ∨ a = 0 ∨ |↑d| = 1 ∨ |↑d| = -1 ∧ Even a\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "apply_rfl",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d = 0 ∨ a = 0 ∨ |↑d| = 1 ∨ |↑d| = -1 ∧ Even a\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d = 0 ∨ a = 0 ∨ |↑d| = 1 ∨ |↑d| = -1 ∧ Even a\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "skip",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d = 0 ∨ a = 0 ∨ |↑d| = 1 ∨ |↑d| = -1 ∧ Even a\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d = 0 ∨ a = 0 ∨ |↑d| = 1 ∨ |↑d| = -1 ∧ Even a\nha : n - m = ↑a\n⊢ m = n"
},
{
"line": "simp only [Nat.cast_eq_zero] at h",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nh : ↑d = 0 ∨ a = 0 ∨ |↑d| = 1 ∨ |↑d| = -1 ∧ Even a\nha : n - m = ↑a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nha : n - m = ↑a\nh : d = 0 ∨ a = 0 ∨ |↑d| = 1 ∨ |↑d| = -1 ∧ Even a\n⊢ m = n"
},
{
"line": "simp only [ne_of_gt hd0] at h",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nha : n - m = ↑a\nh : d = 0 ∨ a = 0 ∨ |↑d| = 1 ∨ |↑d| = -1 ∧ Even a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nha : n - m = ↑a\nh : False ∨ a = 0 ∨ |↑d| = 1 ∨ |↑d| = -1 ∧ Even a\n⊢ m = n"
},
{
"line": "simp only [Nat.abs_cast] at h",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nha : n - m = ↑a\nh : False ∨ a = 0 ∨ |↑d| = 1 ∨ |↑d| = -1 ∧ Even a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nha : n - m = ↑a\nh : False ∨ a = 0 ∨ ↑d = 1 ∨ ↑d = -1 ∧ Even a\n⊢ m = n"
},
{
"line": "simp only [Nat.cast_eq_one] at h",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nha : n - m = ↑a\nh : False ∨ a = 0 ∨ ↑d = 1 ∨ ↑d = -1 ∧ Even a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nha : n - m = ↑a\nh : False ∨ a = 0 ∨ d = 1 ∨ ↑d = -1 ∧ Even a\n⊢ m = n"
},
{
"line": "simp only [ne_of_gt hd1] at h",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nha : n - m = ↑a\nh : False ∨ a = 0 ∨ d = 1 ∨ ↑d = -1 ∧ Even a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nha : n - m = ↑a\nh : False ∨ a = 0 ∨ False ∨ ↑d = -1 ∧ Even a\n⊢ m = n"
},
{
"line": "simp only [Int.reduceNeg] at h",
"before_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nha : n - m = ↑a\nh : False ∨ a = 0 ∨ False ∨ ↑d = -1 ∧ Even a\n⊢ m = n",
"after_state": "case pos.intro\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nH : m ≤ n\na : ℕ\nha : n - m = ↑a\nh : False ∨ a = 0 ∨ False ∨ ↑d = -1 ∧ Even a\n⊢ m = n"
},
{
"line": "have h1 : (d : ℝ) ^ (n - m) ≤ 1 - (d : ℝ)⁻¹ :=\n calc\n (d : ℝ) ^ (n - m) ≤ (d : ℝ)⁻¹ := by\n rw [← zpow_neg_one]\n apply zpow_right_mono₀ <| Nat.one_le_cast.mpr hd0\n linarith\n _ ≤ 1 - (d : ℝ)⁻¹ := by\n rw [inv_eq_one_div]\n rw [one_sub_div <| Nat.cast_ne_zero.mpr (ne_of_gt hd0)]\n rw [div_le_div_iff_of_pos_right <| Nat.cast_pos'.mpr hd0]\n rw [le_sub_iff_add_le]\n norm_cast",
"before_state": "case neg\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ m = n",
"after_state": "case neg\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\nh1 : ↑d ^ (n - m) ≤ 1 - (↑d)⁻¹\n⊢ m = n"
},
{
"line": "refine_lift\n have h1 : (d : ℝ) ^ (n - m) ≤ 1 - (d : ℝ)⁻¹ :=\n calc\n (d : ℝ) ^ (n - m) ≤ (d : ℝ)⁻¹ := by\n rw [← zpow_neg_one]\n apply zpow_right_mono₀ <| Nat.one_le_cast.mpr hd0\n linarith\n _ ≤ 1 - (d : ℝ)⁻¹ := by\n rw [inv_eq_one_div]\n rw [one_sub_div <| Nat.cast_ne_zero.mpr (ne_of_gt hd0)]\n rw [div_le_div_iff_of_pos_right <| Nat.cast_pos'.mpr hd0]\n rw [le_sub_iff_add_le]\n norm_cast;\n ?_",
"before_state": "case neg\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ m = n",
"after_state": "case neg\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\nh1 : ↑d ^ (n - m) ≤ 1 - (↑d)⁻¹\n⊢ m = n"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (have h1 : (d : ℝ) ^ (n - m) ≤ 1 - (d : ℝ)⁻¹ :=\n calc\n (d : ℝ) ^ (n - m) ≤ (d : ℝ)⁻¹ := by\n rw [← zpow_neg_one]\n apply zpow_right_mono₀ <| Nat.one_le_cast.mpr hd0\n linarith\n _ ≤ 1 - (d : ℝ)⁻¹ := by\n rw [inv_eq_one_div]\n rw [one_sub_div <| Nat.cast_ne_zero.mpr (ne_of_gt hd0)]\n rw [div_le_div_iff_of_pos_right <| Nat.cast_pos'.mpr hd0]\n rw [le_sub_iff_add_le]\n norm_cast;\n ?_);\n rotate_right)",
"before_state": "case neg\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ m = n",
"after_state": "case neg\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\nh1 : ↑d ^ (n - m) ≤ 1 - (↑d)⁻¹\n⊢ m = n"
},
{
"line": "refine\n no_implicit_lambda%\n (have h1 : (d : ℝ) ^ (n - m) ≤ 1 - (d : ℝ)⁻¹ :=\n calc\n (d : ℝ) ^ (n - m) ≤ (d : ℝ)⁻¹ := by\n rw [← zpow_neg_one]\n apply zpow_right_mono₀ <| Nat.one_le_cast.mpr hd0\n linarith\n _ ≤ 1 - (d : ℝ)⁻¹ := by\n rw [inv_eq_one_div]\n rw [one_sub_div <| Nat.cast_ne_zero.mpr (ne_of_gt hd0)]\n rw [div_le_div_iff_of_pos_right <| Nat.cast_pos'.mpr hd0]\n rw [le_sub_iff_add_le]\n norm_cast;\n ?_)",
"before_state": "case neg\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ m = n",
"after_state": "case neg\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\nh1 : ↑d ^ (n - m) ≤ 1 - (↑d)⁻¹\n⊢ m = n"
},
{
"line": "rw [← zpow_neg_one]",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ ↑d ^ (n - m) ≤ (↑d)⁻¹",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ ↑d ^ (n - m) ≤ ↑d ^ (-1)"
},
{
"line": "rewrite [← zpow_neg_one]",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ ↑d ^ (n - m) ≤ (↑d)⁻¹",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ ↑d ^ (n - m) ≤ ↑d ^ (-1)"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ ↑d ^ (n - m) ≤ ↑d ^ (-1)",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ ↑d ^ (n - m) ≤ ↑d ^ (-1)"
},
{
"line": "try (with_reducible rfl)",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ ↑d ^ (n - m) ≤ ↑d ^ (-1)",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ ↑d ^ (n - m) ≤ ↑d ^ (-1)"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ ↑d ^ (n - m) ≤ ↑d ^ (-1)",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ ↑d ^ (n - m) ≤ ↑d ^ (-1)"
},
{
"line": "with_reducible rfl",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ ↑d ^ (n - m) ≤ ↑d ^ (-1)",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ ↑d ^ (n - m) ≤ ↑d ^ (-1)"
},
{
"line": "rfl",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ ↑d ^ (n - m) ≤ ↑d ^ (-1)",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ ↑d ^ (n - m) ≤ ↑d ^ (-1)"
},
{
"line": "apply_rfl",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ ↑d ^ (n - m) ≤ ↑d ^ (-1)",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ ↑d ^ (n - m) ≤ ↑d ^ (-1)"
},
{
"line": "skip",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ ↑d ^ (n - m) ≤ ↑d ^ (-1)",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ ↑d ^ (n - m) ≤ ↑d ^ (-1)"
},
{
"line": "apply zpow_right_mono₀ <| Nat.one_le_cast.mpr hd0",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ ↑d ^ (n - m) ≤ ↑d ^ (-1)",
"after_state": "case a\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ n - m ≤ -1"
},
{
"line": "linarith",
"before_state": "case a\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ n - m ≤ -1",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\na✝ : n - m > -1\n⊢ -1 + (n + 1 - m) + (-1 + 1 - (n - m)) = 0",
"after_state": "No Goals!"
},
{
"line": "rw [inv_eq_one_div]",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ (↑d)⁻¹ ≤ 1 - (↑d)⁻¹",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ 1 / ↑d ≤ 1 - 1 / ↑d"
},
{
"line": "rewrite [inv_eq_one_div]",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ (↑d)⁻¹ ≤ 1 - (↑d)⁻¹",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ 1 / ↑d ≤ 1 - 1 / ↑d"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ 1 / ↑d ≤ 1 - 1 / ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ 1 / ↑d ≤ 1 - 1 / ↑d"
},
{
"line": "try (with_reducible rfl)",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ 1 / ↑d ≤ 1 - 1 / ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ 1 / ↑d ≤ 1 - 1 / ↑d"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ 1 / ↑d ≤ 1 - 1 / ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ 1 / ↑d ≤ 1 - 1 / ↑d"
},
{
"line": "with_reducible rfl",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ 1 / ↑d ≤ 1 - 1 / ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ 1 / ↑d ≤ 1 - 1 / ↑d"
},
{
"line": "rfl",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ 1 / ↑d ≤ 1 - 1 / ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ 1 / ↑d ≤ 1 - 1 / ↑d"
},
{
"line": "apply_rfl",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ 1 / ↑d ≤ 1 - 1 / ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ 1 / ↑d ≤ 1 - 1 / ↑d"
},
{
"line": "skip",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ 1 / ↑d ≤ 1 - 1 / ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ 1 / ↑d ≤ 1 - 1 / ↑d"
},
{
"line": "rw [one_sub_div <| Nat.cast_ne_zero.mpr (ne_of_gt hd0)]",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ 1 / ↑d ≤ 1 - 1 / ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ 1 / ↑d ≤ (↑d - 1) / ↑d"
},
{
"line": "rewrite [one_sub_div <| Nat.cast_ne_zero.mpr (ne_of_gt hd0)]",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ 1 / ↑d ≤ 1 - 1 / ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ 1 / ↑d ≤ (↑d - 1) / ↑d"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ 1 / ↑d ≤ (↑d - 1) / ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ 1 / ↑d ≤ (↑d - 1) / ↑d"
},
{
"line": "try (with_reducible rfl)",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ 1 / ↑d ≤ (↑d - 1) / ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ 1 / ↑d ≤ (↑d - 1) / ↑d"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ 1 / ↑d ≤ (↑d - 1) / ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ 1 / ↑d ≤ (↑d - 1) / ↑d"
},
{
"line": "with_reducible rfl",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ 1 / ↑d ≤ (↑d - 1) / ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ 1 / ↑d ≤ (↑d - 1) / ↑d"
},
{
"line": "rfl",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ 1 / ↑d ≤ (↑d - 1) / ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ 1 / ↑d ≤ (↑d - 1) / ↑d"
},
{
"line": "apply_rfl",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ 1 / ↑d ≤ (↑d - 1) / ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ 1 / ↑d ≤ (↑d - 1) / ↑d"
},
{
"line": "skip",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ 1 / ↑d ≤ (↑d - 1) / ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ 1 / ↑d ≤ (↑d - 1) / ↑d"
},
{
"line": "rw [div_le_div_iff_of_pos_right <| Nat.cast_pos'.mpr hd0]",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ 1 / ↑d ≤ (↑d - 1) / ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ 1 ≤ ↑d - 1"
},
{
"line": "rewrite [div_le_div_iff_of_pos_right <| Nat.cast_pos'.mpr hd0]",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ 1 / ↑d ≤ (↑d - 1) / ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ 1 ≤ ↑d - 1"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ 1 ≤ ↑d - 1",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ 1 ≤ ↑d - 1"
},
{
"line": "try (with_reducible rfl)",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ 1 ≤ ↑d - 1",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ 1 ≤ ↑d - 1"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ 1 ≤ ↑d - 1",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ 1 ≤ ↑d - 1"
},
{
"line": "with_reducible rfl",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ 1 ≤ ↑d - 1",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ 1 ≤ ↑d - 1"
},
{
"line": "rfl",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ 1 ≤ ↑d - 1",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ 1 ≤ ↑d - 1"
},
{
"line": "apply_rfl",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ 1 ≤ ↑d - 1",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ 1 ≤ ↑d - 1"
},
{
"line": "skip",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ 1 ≤ ↑d - 1",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ 1 ≤ ↑d - 1"
},
{
"line": "rw [le_sub_iff_add_le]",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ 1 ≤ ↑d - 1",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ 1 + 1 ≤ ↑d"
},
{
"line": "rewrite [le_sub_iff_add_le]",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ 1 ≤ ↑d - 1",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ 1 + 1 ≤ ↑d"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ 1 + 1 ≤ ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ 1 + 1 ≤ ↑d"
},
{
"line": "try (with_reducible rfl)",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ 1 + 1 ≤ ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ 1 + 1 ≤ ↑d"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ 1 + 1 ≤ ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ 1 + 1 ≤ ↑d"
},
{
"line": "with_reducible rfl",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ 1 + 1 ≤ ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ 1 + 1 ≤ ↑d"
},
{
"line": "rfl",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ 1 + 1 ≤ ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ 1 + 1 ≤ ↑d"
},
{
"line": "apply_rfl",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ 1 + 1 ≤ ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ 1 + 1 ≤ ↑d"
},
{
"line": "skip",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ 1 + 1 ≤ ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ 1 + 1 ≤ ↑d"
},
{
"line": "norm_cast",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ 1 + 1 ≤ ↑d",
"after_state": "No Goals!"
},
{
"line": "focus\n norm_cast0\n with_annotate_state\"<;>\" skip\n all_goals try trivial",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ 1 + 1 ≤ ↑d",
"after_state": "No Goals!"
},
{
"line": "norm_cast0",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ 1 + 1 ≤ ↑d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ 1 + 1 ≤ d"
},
{
"line": "with_annotate_state\"<;>\" skip",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ 1 + 1 ≤ d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ 1 + 1 ≤ d"
},
{
"line": "skip",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ 1 + 1 ≤ d",
"after_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ 1 + 1 ≤ d"
},
{
"line": "all_goals try trivial",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ 1 + 1 ≤ d",
"after_state": "No Goals!"
},
{
"line": "try trivial",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ 1 + 1 ≤ d",
"after_state": "No Goals!"
},
{
"line": "first\n| trivial\n| skip",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ 1 + 1 ≤ d",
"after_state": "No Goals!"
},
{
"line": "trivial",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ 1 + 1 ≤ d",
"after_state": "No Goals!"
},
{
"line": "assumption",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\n⊢ 1 + 1 ≤ d",
"after_state": "No Goals!"
},
{
"line": "rotate_right",
"before_state": "case neg\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\nh1 : ↑d ^ (n - m) ≤ 1 - (↑d)⁻¹\n⊢ m = n",
"after_state": "case neg\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\nh1 : ↑d ^ (n - m) ≤ 1 - (↑d)⁻¹\n⊢ m = n"
},
{
"line": "linarith [sub_lt_of_abs_sub_lt_right (a := (1 : ℝ)) (b := d ^ (n - m)) (c := d⁻¹) h]",
"before_state": "case neg\nd : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\nh1 : ↑d ^ (n - m) ≤ 1 - (↑d)⁻¹\n⊢ m = n",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\nh1 : ↑d ^ (n - m) ≤ 1 - (↑d)⁻¹\na✝ : m < n\n⊢ 2 * -1 + (n + 1 - m) + (m + 1 - n) = 0",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\nh1 : ↑d ^ (n - m) ≤ 1 - (↑d)⁻¹\na✝ : m < n\n⊢ 2 > 0",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "d : ℕ\nm n : ℤ\nhd1 : 1 < d\nhd0 : 0 < d\nh : |1 - ↑d ^ (n - m)| < (↑d)⁻¹\nH : ¬m ≤ n\nh1 : ↑d ^ (n - m) ≤ 1 - (↑d)⁻¹\na✝ : n < m\n⊢ 1 * ↑d ^ (n - m) - (1 * 1 - 1 * (↑d)⁻¹) + (1 * 1 - 1 * (↑d)⁻¹ - 1 * ↑d ^ (n - m)) = 0",
"after_state": "No Goals!"
}
] |
lemma ball_eq_singleton {n : ℕ} : Metric.ball n ((2 : ℝ) ^ (-n - 1 : ℤ)) = {n} := by
ext m
constructor
· zify [zpow_natCast, mem_ball, dist_def, mem_singleton_iff]
apply Int.eq_of_pow_sub_le one_lt_two
· intro H
rw [H]
rw [mem_ball]
rw [dist_self]
apply zpow_pos two_pos
|
/root/DuelModelResearch/mathlib4/Counterexamples/DiscreteTopologyNonDiscreteUniformity.lean
|
{
"open": [
"Set Function Filter Metric"
],
"variables": []
}
|
[
{
"line": "ext m",
"before_state": "n : ℕ\n⊢ ball n (2 ^ (-↑n - 1)) = {n}",
"after_state": "case h\nn m : ℕ\n⊢ m ∈ ball n (2 ^ (-↑n - 1)) ↔ m ∈ {n}"
},
{
"line": "constructor",
"before_state": "case h\nn m : ℕ\n⊢ m ∈ ball n (2 ^ (-↑n - 1)) ↔ m ∈ {n}",
"after_state": "case h.mp\nn m : ℕ\n⊢ m ∈ ball n (2 ^ (-↑n - 1)) → m ∈ {n}\n---\ncase h.mpr\nn m : ℕ\n⊢ m ∈ {n} → m ∈ ball n (2 ^ (-↑n - 1))"
},
{
"line": "zify [zpow_natCast, mem_ball, dist_def, mem_singleton_iff]",
"before_state": "case h.mp\nn m : ℕ\n⊢ m ∈ ball n (2 ^ (-↑n - 1)) → m ∈ {n}",
"after_state": "case h.mp\nn m : ℕ\n⊢ dist m n < 2 ^ (-↑n - 1) → ↑m = ↑n"
},
{
"line": "simp -decide✝ only [zify_simps✝, push_cast✝, zpow_natCast, mem_ball, dist_def, mem_singleton_iff]",
"before_state": "case h.mp\nn m : ℕ\n⊢ m ∈ ball n (2 ^ (-↑n - 1)) → m ∈ {n}",
"after_state": "case h.mp\nn m : ℕ\n⊢ dist m n < 2 ^ (-↑n - 1) → ↑m = ↑n"
},
{
"line": "apply Int.eq_of_pow_sub_le one_lt_two",
"before_state": "case h.mp\nn m : ℕ\n⊢ dist m n < 2 ^ (-↑n - 1) → ↑m = ↑n",
"after_state": "No Goals!"
},
{
"line": "intro H",
"before_state": "case h.mpr\nn m : ℕ\n⊢ m ∈ {n} → m ∈ ball n (2 ^ (-↑n - 1))",
"after_state": "case h.mpr\nn m : ℕ\nH : m ∈ {n}\n⊢ m ∈ ball n (2 ^ (-↑n - 1))"
},
{
"line": "rw [H]",
"before_state": "case h.mpr\nn m : ℕ\nH : m ∈ {n}\n⊢ m ∈ ball n (2 ^ (-↑n - 1))",
"after_state": "case h.mpr\nn m : ℕ\nH : m ∈ {n}\n⊢ n ∈ ball n (2 ^ (-↑n - 1))"
},
{
"line": "rewrite [H]",
"before_state": "case h.mpr\nn m : ℕ\nH : m ∈ {n}\n⊢ m ∈ ball n (2 ^ (-↑n - 1))",
"after_state": "case h.mpr\nn m : ℕ\nH : m ∈ {n}\n⊢ n ∈ ball n (2 ^ (-↑n - 1))"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case h.mpr\nn m : ℕ\nH : m ∈ {n}\n⊢ n ∈ ball n (2 ^ (-↑n - 1))",
"after_state": "case h.mpr\nn m : ℕ\nH : m ∈ {n}\n⊢ n ∈ ball n (2 ^ (-↑n - 1))"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case h.mpr\nn m : ℕ\nH : m ∈ {n}\n⊢ n ∈ ball n (2 ^ (-↑n - 1))",
"after_state": "case h.mpr\nn m : ℕ\nH : m ∈ {n}\n⊢ n ∈ ball n (2 ^ (-↑n - 1))"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case h.mpr\nn m : ℕ\nH : m ∈ {n}\n⊢ n ∈ ball n (2 ^ (-↑n - 1))",
"after_state": "case h.mpr\nn m : ℕ\nH : m ∈ {n}\n⊢ n ∈ ball n (2 ^ (-↑n - 1))"
},
{
"line": "with_reducible rfl",
"before_state": "case h.mpr\nn m : ℕ\nH : m ∈ {n}\n⊢ n ∈ ball n (2 ^ (-↑n - 1))",
"after_state": "case h.mpr\nn m : ℕ\nH : m ∈ {n}\n⊢ n ∈ ball n (2 ^ (-↑n - 1))"
},
{
"line": "rfl",
"before_state": "case h.mpr\nn m : ℕ\nH : m ∈ {n}\n⊢ n ∈ ball n (2 ^ (-↑n - 1))",
"after_state": "case h.mpr\nn m : ℕ\nH : m ∈ {n}\n⊢ n ∈ ball n (2 ^ (-↑n - 1))"
},
{
"line": "apply_rfl",
"before_state": "case h.mpr\nn m : ℕ\nH : m ∈ {n}\n⊢ n ∈ ball n (2 ^ (-↑n - 1))",
"after_state": "case h.mpr\nn m : ℕ\nH : m ∈ {n}\n⊢ n ∈ ball n (2 ^ (-↑n - 1))"
},
{
"line": "skip",
"before_state": "case h.mpr\nn m : ℕ\nH : m ∈ {n}\n⊢ n ∈ ball n (2 ^ (-↑n - 1))",
"after_state": "case h.mpr\nn m : ℕ\nH : m ∈ {n}\n⊢ n ∈ ball n (2 ^ (-↑n - 1))"
},
{
"line": "rw [mem_ball]",
"before_state": "case h.mpr\nn m : ℕ\nH : m ∈ {n}\n⊢ n ∈ ball n (2 ^ (-↑n - 1))",
"after_state": "case h.mpr\nn m : ℕ\nH : m ∈ {n}\n⊢ dist n n < 2 ^ (-↑n - 1)"
},
{
"line": "rewrite [mem_ball]",
"before_state": "case h.mpr\nn m : ℕ\nH : m ∈ {n}\n⊢ n ∈ ball n (2 ^ (-↑n - 1))",
"after_state": "case h.mpr\nn m : ℕ\nH : m ∈ {n}\n⊢ dist n n < 2 ^ (-↑n - 1)"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case h.mpr\nn m : ℕ\nH : m ∈ {n}\n⊢ dist n n < 2 ^ (-↑n - 1)",
"after_state": "case h.mpr\nn m : ℕ\nH : m ∈ {n}\n⊢ dist n n < 2 ^ (-↑n - 1)"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case h.mpr\nn m : ℕ\nH : m ∈ {n}\n⊢ dist n n < 2 ^ (-↑n - 1)",
"after_state": "case h.mpr\nn m : ℕ\nH : m ∈ {n}\n⊢ dist n n < 2 ^ (-↑n - 1)"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case h.mpr\nn m : ℕ\nH : m ∈ {n}\n⊢ dist n n < 2 ^ (-↑n - 1)",
"after_state": "case h.mpr\nn m : ℕ\nH : m ∈ {n}\n⊢ dist n n < 2 ^ (-↑n - 1)"
},
{
"line": "with_reducible rfl",
"before_state": "case h.mpr\nn m : ℕ\nH : m ∈ {n}\n⊢ dist n n < 2 ^ (-↑n - 1)",
"after_state": "case h.mpr\nn m : ℕ\nH : m ∈ {n}\n⊢ dist n n < 2 ^ (-↑n - 1)"
},
{
"line": "rfl",
"before_state": "case h.mpr\nn m : ℕ\nH : m ∈ {n}\n⊢ dist n n < 2 ^ (-↑n - 1)",
"after_state": "case h.mpr\nn m : ℕ\nH : m ∈ {n}\n⊢ dist n n < 2 ^ (-↑n - 1)"
},
{
"line": "apply_rfl",
"before_state": "case h.mpr\nn m : ℕ\nH : m ∈ {n}\n⊢ dist n n < 2 ^ (-↑n - 1)",
"after_state": "case h.mpr\nn m : ℕ\nH : m ∈ {n}\n⊢ dist n n < 2 ^ (-↑n - 1)"
},
{
"line": "skip",
"before_state": "case h.mpr\nn m : ℕ\nH : m ∈ {n}\n⊢ dist n n < 2 ^ (-↑n - 1)",
"after_state": "case h.mpr\nn m : ℕ\nH : m ∈ {n}\n⊢ dist n n < 2 ^ (-↑n - 1)"
},
{
"line": "rw [dist_self]",
"before_state": "case h.mpr\nn m : ℕ\nH : m ∈ {n}\n⊢ dist n n < 2 ^ (-↑n - 1)",
"after_state": "case h.mpr\nn m : ℕ\nH : m ∈ {n}\n⊢ 0 < 2 ^ (-↑n - 1)"
},
{
"line": "rewrite [dist_self]",
"before_state": "case h.mpr\nn m : ℕ\nH : m ∈ {n}\n⊢ dist n n < 2 ^ (-↑n - 1)",
"after_state": "case h.mpr\nn m : ℕ\nH : m ∈ {n}\n⊢ 0 < 2 ^ (-↑n - 1)"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case h.mpr\nn m : ℕ\nH : m ∈ {n}\n⊢ 0 < 2 ^ (-↑n - 1)",
"after_state": "case h.mpr\nn m : ℕ\nH : m ∈ {n}\n⊢ 0 < 2 ^ (-↑n - 1)"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case h.mpr\nn m : ℕ\nH : m ∈ {n}\n⊢ 0 < 2 ^ (-↑n - 1)",
"after_state": "case h.mpr\nn m : ℕ\nH : m ∈ {n}\n⊢ 0 < 2 ^ (-↑n - 1)"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case h.mpr\nn m : ℕ\nH : m ∈ {n}\n⊢ 0 < 2 ^ (-↑n - 1)",
"after_state": "case h.mpr\nn m : ℕ\nH : m ∈ {n}\n⊢ 0 < 2 ^ (-↑n - 1)"
},
{
"line": "with_reducible rfl",
"before_state": "case h.mpr\nn m : ℕ\nH : m ∈ {n}\n⊢ 0 < 2 ^ (-↑n - 1)",
"after_state": "case h.mpr\nn m : ℕ\nH : m ∈ {n}\n⊢ 0 < 2 ^ (-↑n - 1)"
},
{
"line": "rfl",
"before_state": "case h.mpr\nn m : ℕ\nH : m ∈ {n}\n⊢ 0 < 2 ^ (-↑n - 1)",
"after_state": "case h.mpr\nn m : ℕ\nH : m ∈ {n}\n⊢ 0 < 2 ^ (-↑n - 1)"
},
{
"line": "apply_rfl",
"before_state": "case h.mpr\nn m : ℕ\nH : m ∈ {n}\n⊢ 0 < 2 ^ (-↑n - 1)",
"after_state": "case h.mpr\nn m : ℕ\nH : m ∈ {n}\n⊢ 0 < 2 ^ (-↑n - 1)"
},
{
"line": "skip",
"before_state": "case h.mpr\nn m : ℕ\nH : m ∈ {n}\n⊢ 0 < 2 ^ (-↑n - 1)",
"after_state": "case h.mpr\nn m : ℕ\nH : m ∈ {n}\n⊢ 0 < 2 ^ (-↑n - 1)"
},
{
"line": "apply zpow_pos two_pos",
"before_state": "case h.mpr\nn m : ℕ\nH : m ∈ {n}\n⊢ 0 < 2 ^ (-↑n - 1)",
"after_state": "No Goals!"
}
] |
lemma idIsCauchy : CauchySeq (id : ℕ → ℕ) := by
rw [Metric.cauchySeq_iff]
refine fun ε ↦ Metric.cauchySeq_iff.mp
(@cauchySeq_of_le_geometric_two ℝ _ 1 (fun n ↦ 2 ^(-n : ℤ)) fun n ↦ le_of_eq ?_) ε
simp only [zpow_natCast]
simp only [Nat.cast_add]
simp only [Nat.cast_one]
simp only [neg_add_rev]
simp only [Int.reduceNeg]
simp only [one_div]
rw [Real.dist_eq]
rw [zpow_add' <| Or.intro_left _ two_ne_zero]
calc
|2 ^ (- n : ℤ) - 2 ^ (-1 : ℤ) * 2 ^ (- n : ℤ)|
_ = |(1 - (2 : ℝ)⁻¹) * 2 ^ (- n : ℤ)| := by rw [← one_sub_mul, zpow_neg_one]
_ = |2⁻¹ * 2 ^ (-(n : ℤ))| := by congr; rw [inv_eq_one_div 2, sub_half 1]
_ = 2⁻¹ / 2 ^ n := by rw [zpow_neg, abs_mul, abs_inv, abs_inv, inv_eq_one_div,
Nat.abs_ofNat, one_div, zpow_natCast, abs_pow, ← div_eq_mul_inv, Nat.abs_ofNat]
|
/root/DuelModelResearch/mathlib4/Counterexamples/DiscreteTopologyNonDiscreteUniformity.lean
|
{
"open": [
"Set Function Filter Metric"
],
"variables": []
}
|
[
{
"line": "rw [Metric.cauchySeq_iff]",
"before_state": "⊢ CauchySeq id",
"after_state": "⊢ ∀ ε > 0, ∃ N, ∀ m ≥ N, ∀ n ≥ N, dist (id m) (id n) < ε"
},
{
"line": "rewrite [Metric.cauchySeq_iff]",
"before_state": "⊢ CauchySeq id",
"after_state": "⊢ ∀ ε > 0, ∃ N, ∀ m ≥ N, ∀ n ≥ N, dist (id m) (id n) < ε"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "⊢ ∀ ε > 0, ∃ N, ∀ m ≥ N, ∀ n ≥ N, dist (id m) (id n) < ε",
"after_state": "⊢ ∀ ε > 0, ∃ N, ∀ m ≥ N, ∀ n ≥ N, dist (id m) (id n) < ε"
},
{
"line": "try (with_reducible rfl)",
"before_state": "⊢ ∀ ε > 0, ∃ N, ∀ m ≥ N, ∀ n ≥ N, dist (id m) (id n) < ε",
"after_state": "⊢ ∀ ε > 0, ∃ N, ∀ m ≥ N, ∀ n ≥ N, dist (id m) (id n) < ε"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "⊢ ∀ ε > 0, ∃ N, ∀ m ≥ N, ∀ n ≥ N, dist (id m) (id n) < ε",
"after_state": "⊢ ∀ ε > 0, ∃ N, ∀ m ≥ N, ∀ n ≥ N, dist (id m) (id n) < ε"
},
{
"line": "with_reducible rfl",
"before_state": "⊢ ∀ ε > 0, ∃ N, ∀ m ≥ N, ∀ n ≥ N, dist (id m) (id n) < ε",
"after_state": "⊢ ∀ ε > 0, ∃ N, ∀ m ≥ N, ∀ n ≥ N, dist (id m) (id n) < ε"
},
{
"line": "rfl",
"before_state": "⊢ ∀ ε > 0, ∃ N, ∀ m ≥ N, ∀ n ≥ N, dist (id m) (id n) < ε",
"after_state": "⊢ ∀ ε > 0, ∃ N, ∀ m ≥ N, ∀ n ≥ N, dist (id m) (id n) < ε"
},
{
"line": "apply_rfl",
"before_state": "⊢ ∀ ε > 0, ∃ N, ∀ m ≥ N, ∀ n ≥ N, dist (id m) (id n) < ε",
"after_state": "⊢ ∀ ε > 0, ∃ N, ∀ m ≥ N, ∀ n ≥ N, dist (id m) (id n) < ε"
},
{
"line": "skip",
"before_state": "⊢ ∀ ε > 0, ∃ N, ∀ m ≥ N, ∀ n ≥ N, dist (id m) (id n) < ε",
"after_state": "⊢ ∀ ε > 0, ∃ N, ∀ m ≥ N, ∀ n ≥ N, dist (id m) (id n) < ε"
},
{
"line": "refine fun ε ↦\n Metric.cauchySeq_iff.mp (@cauchySeq_of_le_geometric_two ℝ _ 1 (fun n ↦ 2 ^ (-n : ℤ)) fun n ↦ le_of_eq ?_) ε",
"before_state": "⊢ ∀ ε > 0, ∃ N, ∀ m ≥ N, ∀ n ≥ N, dist (id m) (id n) < ε",
"after_state": "No Goals!"
}
] |
lemma mem_range_fundamentalEntourage (S : Set (ℕ × ℕ)) :
S ∈ (range fundamentalEntourage) ↔ ∃ n, fundamentalEntourage n = S := by
simp only [Set.mem_range]
simp only [Eq.symm]
|
/root/DuelModelResearch/mathlib4/Counterexamples/DiscreteTopologyNonDiscreteUniformity.lean
|
{
"open": [
"Set Function Filter Metric"
],
"variables": []
}
|
[
{
"line": "simp only [Set.mem_range]",
"before_state": "ι✝ : Sort u_1\nfundamentalEntourage : ι✝ → Set (ℕ × ℕ)\nS : Set (ℕ × ℕ)\n⊢ S ∈ range fundamentalEntourage ↔ ∃ n, fundamentalEntourage n = S",
"after_state": "No Goals!"
}
] |
theorem forgetEpsilons_apply (p : ℤ[ε]) : forgetEpsilons p = coeff p 0 :=
rfl
|
/root/DuelModelResearch/mathlib4/Counterexamples/MapFloor.lean
|
{
"open": [
"Function Int Polynomial",
"scoped Polynomial"
],
"variables": []
}
|
[
{
"line": "get_elem_tactic",
"before_state": "ε : ?m.12\nforgetEpsilons : ?m.64\n⊢ ?m.74 ℤ ε",
"after_state": "No Goals!"
},
{
"line": "first\n| done\n| assumption\n| get_elem_tactic_trivial\n|\n fail \"failed to prove index is valid, possible solutions:\n - Use `have`-expressions to prove the index is valid\n - Use `a[i]!` notation instead, runtime check is performed, and 'Panic' error message is produced if index is not valid\n - Use `a[i]?` notation instead, result is an `Option` type\n - Use `a[i]'h` notation instead, where `h` is a proof that index is valid\"",
"before_state": "ε : ?m.12\nforgetEpsilons : ?m.64\n⊢ ?m.74 ℤ ε",
"after_state": "No Goals!"
},
{
"line": "done",
"before_state": "ε : ?m.12\nforgetEpsilons : ?m.64\n⊢ ?m.74 ℤ ε",
"after_state": "No Goals!"
},
{
"line": "assumption",
"before_state": "ε : ?m.12\nforgetEpsilons : ?m.64\n⊢ ?m.74 ℤ ε",
"after_state": "No Goals!"
}
] |
theorem norm_indicator_le_one (s : Set α) (x : α) : ‖(indicator s (1 : α → ℝ)) x‖ ≤ 1 := by
simp only [Set.indicator]; split_ifs <;> norm_num
simp only [Pi.one_apply]; split_ifs <;> norm_num
|
/root/DuelModelResearch/mathlib4/Counterexamples/Phillips.lean
|
{
"open": [
"Set BoundedContinuousFunction MeasureTheory",
"Cardinal (aleph)",
"scoped Cardinal BoundedContinuousFunction",
"BoundedAdditiveMeasure"
],
"variables": [
"{α : Type u}"
]
}
|
[
{
"line": "simp only [Set.indicator]",
"before_state": "α : Type u\ns : Set α\nx : α\n⊢ ‖s.indicator 1 x‖ ≤ 1",
"after_state": "α : Type u\ns : Set α\nx : α\n⊢ ‖if x ∈ s then 1 x else 0‖ ≤ 1"
},
{
"line": "focus\n split_ifs\n with_annotate_state\"<;>\" skip\n all_goals norm_num",
"before_state": "α : Type u\ns : Set α\nx : α\n⊢ ‖if x ∈ s then 1 x else 0‖ ≤ 1",
"after_state": "No Goals!"
},
{
"line": "split_ifs",
"before_state": "α : Type u\ns : Set α\nx : α\n⊢ ‖if x ∈ s then 1 x else 0‖ ≤ 1",
"after_state": "case pos\nα : Type u\ns : Set α\nx : α\nh✝ : x ∈ s\n⊢ ‖1 x‖ ≤ 1\n---\ncase neg\nα : Type u\ns : Set α\nx : α\nh✝ : x ∉ s\n⊢ ‖0‖ ≤ 1"
},
{
"line": "by_cases h✝ : ?m✝",
"before_state": "α : Type u\ns : Set α\nx : α\n⊢ ‖if x ∈ s then 1 x else 0‖ ≤ 1",
"after_state": "case pos\nα : Type u\ns : Set α\nx : α\nh✝ : x ∈ s\n⊢ ‖if x ∈ s then 1 x else 0‖ ≤ 1\n---\ncase neg\nα : Type u\ns : Set α\nx : α\nh✝ : x ∉ s\n⊢ ‖if x ∈ s then 1 x else 0‖ ≤ 1"
},
{
"line": "open Classical✝ in refine if h✝ : ?m✝ then ?pos✝ else ?neg✝",
"before_state": "α : Type u\ns : Set α\nx : α\n⊢ ‖if x ∈ s then 1 x else 0‖ ≤ 1",
"after_state": "case pos\nα : Type u\ns : Set α\nx : α\nh✝ : x ∈ s\n⊢ ‖if x ∈ s then 1 x else 0‖ ≤ 1\n---\ncase neg\nα : Type u\ns : Set α\nx : α\nh✝ : x ∉ s\n⊢ ‖if x ∈ s then 1 x else 0‖ ≤ 1"
},
{
"line": "refine if h✝ : ?m✝ then ?pos✝ else ?neg✝",
"before_state": "α : Type u\ns : Set α\nx : α\n⊢ ‖if x ∈ s then 1 x else 0‖ ≤ 1",
"after_state": "case pos\nα : Type u\ns : Set α\nx : α\nh✝ : x ∈ s\n⊢ ‖if x ∈ s then 1 x else 0‖ ≤ 1\n---\ncase neg\nα : Type u\ns : Set α\nx : α\nh✝ : x ∉ s\n⊢ ‖if x ∈ s then 1 x else 0‖ ≤ 1"
},
{
"line": "with_annotate_state\"<;>\" skip",
"before_state": "case pos\nα : Type u\ns : Set α\nx : α\nh✝ : x ∈ s\n⊢ ‖1 x‖ ≤ 1\n---\ncase neg\nα : Type u\ns : Set α\nx : α\nh✝ : x ∉ s\n⊢ ‖0‖ ≤ 1",
"after_state": "case pos\nα : Type u\ns : Set α\nx : α\nh✝ : x ∈ s\n⊢ ‖1 x‖ ≤ 1\n---\ncase neg\nα : Type u\ns : Set α\nx : α\nh✝ : x ∉ s\n⊢ ‖0‖ ≤ 1"
},
{
"line": "skip",
"before_state": "case pos\nα : Type u\ns : Set α\nx : α\nh✝ : x ∈ s\n⊢ ‖1 x‖ ≤ 1\n---\ncase neg\nα : Type u\ns : Set α\nx : α\nh✝ : x ∉ s\n⊢ ‖0‖ ≤ 1",
"after_state": "case pos\nα : Type u\ns : Set α\nx : α\nh✝ : x ∈ s\n⊢ ‖1 x‖ ≤ 1\n---\ncase neg\nα : Type u\ns : Set α\nx : α\nh✝ : x ∉ s\n⊢ ‖0‖ ≤ 1"
},
{
"line": "all_goals norm_num",
"before_state": "case pos\nα : Type u\ns : Set α\nx : α\nh✝ : x ∈ s\n⊢ ‖1 x‖ ≤ 1\n---\ncase neg\nα : Type u\ns : Set α\nx : α\nh✝ : x ∉ s\n⊢ ‖0‖ ≤ 1",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "case pos\nα : Type u\ns : Set α\nx : α\nh✝ : x ∈ s\n⊢ ‖1 x‖ ≤ 1",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "case neg\nα : Type u\ns : Set α\nx : α\nh✝ : x ∉ s\n⊢ ‖0‖ ≤ 1",
"after_state": "No Goals!"
}
] |
theorem sierpinski_pathological_family (Hcont : #ℝ = ℵ₁) :
∃ f : ℝ → Set ℝ, (∀ x, (univ \ f x).Countable) ∧ ∀ y, {x : ℝ | y ∈ f x}.Countable := by
rcases Cardinal.ord_eq ℝ with ⟨r, hr, H⟩
refine ⟨fun x => {y | r x y}, fun x => ?_, fun y => ?_⟩
· have : univ \ {y | r x y} = {y | r y x} ∪ {x} := by
ext y
simp only [true_and]
simp only [mem_univ]
simp only [mem_setOf_eq]
simp only [mem_insert_iff]
simp only [union_singleton]
simp only [mem_diff]
rcases trichotomous_of r x y with (h | rfl | h)
· simp only [h, not_or, false_iff, not_true]
constructor
· rintro rfl; exact irrefl_of r y h
· exact asymm h
· simp only [true_or, eq_self_iff_true, iff_true]; exact irrefl x
· simp only [h, iff_true, or_true]; exact asymm h
rw [this]
apply Countable.union _ (countable_singleton _)
rw [Cardinal.countable_iff_lt_aleph_one]
rw [← Hcont]
exact Cardinal.card_typein_lt r x H
· rw [Cardinal.countable_iff_lt_aleph_one, ← Hcont]
exact Cardinal.card_typein_lt r y H
|
/root/DuelModelResearch/mathlib4/Counterexamples/Phillips.lean
|
{
"open": [
"Set BoundedContinuousFunction MeasureTheory",
"Cardinal (aleph)",
"scoped Cardinal BoundedContinuousFunction",
"BoundedAdditiveMeasure"
],
"variables": [
"{α : Type u}"
]
}
|
[
{
"line": "rcases Cardinal.ord_eq ℝ with ⟨r, hr, H⟩",
"before_state": "Hcont : #ℝ = ℵ_ 1\n⊢ ∃ f, (∀ (x : ℝ), (univ \\ f x).Countable) ∧ ∀ (y : ℝ), {x | y ∈ f x}.Countable",
"after_state": "case intro.intro\nHcont : #ℝ = ℵ_ 1\nr : ℝ → ℝ → Prop\nhr : IsWellOrder ℝ r\nH : (#ℝ).ord = Ordinal.type r\n⊢ ∃ f, (∀ (x : ℝ), (univ \\ f x).Countable) ∧ ∀ (y : ℝ), {x | y ∈ f x}.Countable"
},
{
"line": "refine ⟨fun x => {y | r x y}, fun x => ?_, fun y => ?_⟩",
"before_state": "case intro.intro\nHcont : #ℝ = ℵ_ 1\nr : ℝ → ℝ → Prop\nhr : IsWellOrder ℝ r\nH : (#ℝ).ord = Ordinal.type r\n⊢ ∃ f, (∀ (x : ℝ), (univ \\ f x).Countable) ∧ ∀ (y : ℝ), {x | y ∈ f x}.Countable",
"after_state": "case intro.intro.refine_1\nHcont : #ℝ = ℵ_ 1\nr : ℝ → ℝ → Prop\nhr : IsWellOrder ℝ r\nH : (#ℝ).ord = Ordinal.type r\nx : ℝ\n⊢ (univ \\ (fun x => {y | r x y}) x).Countable\n---\ncase intro.intro.refine_2\nHcont : #ℝ = ℵ_ 1\nr : ℝ → ℝ → Prop\nhr : IsWellOrder ℝ r\nH : (#ℝ).ord = Ordinal.type r\ny : ℝ\n⊢ {x | y ∈ (fun x => {y | r x y}) x}.Countable"
},
{
"line": "have : univ \\ {y | r x y} = {y | r y x} ∪ { x } := by\n ext y\n simp only [true_and]\n simp only [mem_univ]\n simp only [mem_setOf_eq]\n simp only [mem_insert_iff]\n simp only [union_singleton]\n simp only [mem_diff]\n rcases trichotomous_of r x y with (h | rfl | h)\n · simp only [h, not_or, false_iff, not_true]\n constructor\n · rintro rfl; exact irrefl_of r y h\n · exact asymm h\n · simp only [true_or, eq_self_iff_true, iff_true]; exact irrefl x\n · simp only [h, iff_true, or_true]; exact asymm h",
"before_state": "case intro.intro.refine_1\nHcont : #ℝ = ℵ_ 1\nr : ℝ → ℝ → Prop\nhr : IsWellOrder ℝ r\nH : (#ℝ).ord = Ordinal.type r\nx : ℝ\n⊢ (univ \\ (fun x => {y | r x y}) x).Countable",
"after_state": "case intro.intro.refine_1\nHcont : #ℝ = ℵ_ 1\nr : ℝ → ℝ → Prop\nhr : IsWellOrder ℝ r\nH : (#ℝ).ord = Ordinal.type r\nx : ℝ\nthis : univ \\ {y | r x y} = {y | r y x} ∪ {x}\n⊢ (univ \\ (fun x => {y | r x y}) x).Countable"
},
{
"line": "focus\n refine\n no_implicit_lambda%\n (have : univ \\ {y | r x y} = {y | r y x} ∪ { x } := ?body✝;\n ?_)\n case body✝ =>\n with_annotate_state\"by\"\n ( ext y\n simp only [true_and]\n simp only [mem_univ]\n simp only [mem_setOf_eq]\n simp only [mem_insert_iff]\n simp only [union_singleton]\n simp only [mem_diff]\n rcases trichotomous_of r x y with (h | rfl | h)\n · simp only [h, not_or, false_iff, not_true]\n constructor\n · rintro rfl; exact irrefl_of r y h\n · exact asymm h\n · simp only [true_or, eq_self_iff_true, iff_true]; exact irrefl x\n · simp only [h, iff_true, or_true]; exact asymm h)",
"before_state": "case intro.intro.refine_1\nHcont : #ℝ = ℵ_ 1\nr : ℝ → ℝ → Prop\nhr : IsWellOrder ℝ r\nH : (#ℝ).ord = Ordinal.type r\nx : ℝ\n⊢ (univ \\ (fun x => {y | r x y}) x).Countable",
"after_state": "case intro.intro.refine_1\nHcont : #ℝ = ℵ_ 1\nr : ℝ → ℝ → Prop\nhr : IsWellOrder ℝ r\nH : (#ℝ).ord = Ordinal.type r\nx : ℝ\nthis : univ \\ {y | r x y} = {y | r y x} ∪ {x}\n⊢ (univ \\ (fun x => {y | r x y}) x).Countable"
},
{
"line": "refine\n no_implicit_lambda%\n (have : univ \\ {y | r x y} = {y | r y x} ∪ { x } := ?body✝;\n ?_)",
"before_state": "case intro.intro.refine_1\nHcont : #ℝ = ℵ_ 1\nr : ℝ → ℝ → Prop\nhr : IsWellOrder ℝ r\nH : (#ℝ).ord = Ordinal.type r\nx : ℝ\n⊢ (univ \\ (fun x => {y | r x y}) x).Countable",
"after_state": "case body\nHcont : #ℝ = ℵ_ 1\nr : ℝ → ℝ → Prop\nhr : IsWellOrder ℝ r\nH : (#ℝ).ord = Ordinal.type r\nx : ℝ\n⊢ univ \\ {y | r x y} = {y | r y x} ∪ {x}\n---\ncase intro.intro.refine_1\nHcont : #ℝ = ℵ_ 1\nr : ℝ → ℝ → Prop\nhr : IsWellOrder ℝ r\nH : (#ℝ).ord = Ordinal.type r\nx : ℝ\nthis : univ \\ {y | r x y} = {y | r y x} ∪ {x}\n⊢ (univ \\ (fun x => {y | r x y}) x).Countable"
},
{
"line": "case body✝ =>\n with_annotate_state\"by\"\n ( ext y\n simp only [true_and]\n simp only [mem_univ]\n simp only [mem_setOf_eq]\n simp only [mem_insert_iff]\n simp only [union_singleton]\n simp only [mem_diff]\n rcases trichotomous_of r x y with (h | rfl | h)\n · simp only [h, not_or, false_iff, not_true]\n constructor\n · rintro rfl; exact irrefl_of r y h\n · exact asymm h\n · simp only [true_or, eq_self_iff_true, iff_true]; exact irrefl x\n · simp only [h, iff_true, or_true]; exact asymm h)",
"before_state": "case body\nHcont : #ℝ = ℵ_ 1\nr : ℝ → ℝ → Prop\nhr : IsWellOrder ℝ r\nH : (#ℝ).ord = Ordinal.type r\nx : ℝ\n⊢ univ \\ {y | r x y} = {y | r y x} ∪ {x}\n---\ncase intro.intro.refine_1\nHcont : #ℝ = ℵ_ 1\nr : ℝ → ℝ → Prop\nhr : IsWellOrder ℝ r\nH : (#ℝ).ord = Ordinal.type r\nx : ℝ\nthis : univ \\ {y | r x y} = {y | r y x} ∪ {x}\n⊢ (univ \\ (fun x => {y | r x y}) x).Countable",
"after_state": "case intro.intro.refine_1\nHcont : #ℝ = ℵ_ 1\nr : ℝ → ℝ → Prop\nhr : IsWellOrder ℝ r\nH : (#ℝ).ord = Ordinal.type r\nx : ℝ\nthis : univ \\ {y | r x y} = {y | r y x} ∪ {x}\n⊢ (univ \\ (fun x => {y | r x y}) x).Countable"
},
{
"line": "with_annotate_state\"by\"\n ( ext y\n simp only [true_and]\n simp only [mem_univ]\n simp only [mem_setOf_eq]\n simp only [mem_insert_iff]\n simp only [union_singleton]\n simp only [mem_diff]\n rcases trichotomous_of r x y with (h | rfl | h)\n · simp only [h, not_or, false_iff, not_true]\n constructor\n · rintro rfl; exact irrefl_of r y h\n · exact asymm h\n · simp only [true_or, eq_self_iff_true, iff_true]; exact irrefl x\n · simp only [h, iff_true, or_true]; exact asymm h)",
"before_state": "Hcont : #ℝ = ℵ_ 1\nr : ℝ → ℝ → Prop\nhr : IsWellOrder ℝ r\nH : (#ℝ).ord = Ordinal.type r\nx : ℝ\n⊢ univ \\ {y | r x y} = {y | r y x} ∪ {x}",
"after_state": "case h\nHcont : #ℝ = ℵ_ 1\nr : ℝ → ℝ → Prop\nhr : IsWellOrder ℝ r\nH : (#ℝ).ord = Ordinal.type r\nx y : ℝ\n⊢ y ∈ univ \\ {y | r x y} ↔ y ∈ {y | r y x} ∪ {x}"
},
{
"line": "ext y",
"before_state": "Hcont : #ℝ = ℵ_ 1\nr : ℝ → ℝ → Prop\nhr : IsWellOrder ℝ r\nH : (#ℝ).ord = Ordinal.type r\nx : ℝ\n⊢ univ \\ {y | r x y} = {y | r y x} ∪ {x}",
"after_state": "case h\nHcont : #ℝ = ℵ_ 1\nr : ℝ → ℝ → Prop\nhr : IsWellOrder ℝ r\nH : (#ℝ).ord = Ordinal.type r\nx y : ℝ\n⊢ y ∈ univ \\ {y | r x y} ↔ y ∈ {y | r y x} ∪ {x}"
},
{
"line": "simp only [true_and]",
"before_state": "case h\nHcont : #ℝ = ℵ_ 1\nr : ℝ → ℝ → Prop\nhr : IsWellOrder ℝ r\nH : (#ℝ).ord = Ordinal.type r\nx y : ℝ\n⊢ y ∈ univ \\ {y | r x y} ↔ y ∈ {y | r y x} ∪ {x}",
"after_state": "case h\nHcont : #ℝ = ℵ_ 1\nr : ℝ → ℝ → Prop\nhr : IsWellOrder ℝ r\nH : (#ℝ).ord = Ordinal.type r\nx y : ℝ\n⊢ y ∈ univ \\ {y | r x y} ↔ y ∈ {y | r y x} ∪ {x}"
},
{
"line": "rw [this]",
"before_state": "case intro.intro.refine_1\nHcont : #ℝ = ℵ_ 1\nr : ℝ → ℝ → Prop\nhr : IsWellOrder ℝ r\nH : (#ℝ).ord = Ordinal.type r\nx : ℝ\nthis : univ \\ {y | r x y} = {y | r y x} ∪ {x}\n⊢ (univ \\ (fun x => {y | r x y}) x).Countable",
"after_state": "case intro.intro.refine_1\nHcont : #ℝ = ℵ_ 1\nr : ℝ → ℝ → Prop\nhr : IsWellOrder ℝ r\nH : (#ℝ).ord = Ordinal.type r\nx : ℝ\nthis : univ \\ {y | r x y} = {y | r y x} ∪ {x}\n⊢ ({y | r y x} ∪ {x}).Countable"
},
{
"line": "rewrite [this]",
"before_state": "case intro.intro.refine_1\nHcont : #ℝ = ℵ_ 1\nr : ℝ → ℝ → Prop\nhr : IsWellOrder ℝ r\nH : (#ℝ).ord = Ordinal.type r\nx : ℝ\nthis : univ \\ {y | r x y} = {y | r y x} ∪ {x}\n⊢ (univ \\ (fun x => {y | r x y}) x).Countable",
"after_state": "case intro.intro.refine_1\nHcont : #ℝ = ℵ_ 1\nr : ℝ → ℝ → Prop\nhr : IsWellOrder ℝ r\nH : (#ℝ).ord = Ordinal.type r\nx : ℝ\nthis : univ \\ {y | r x y} = {y | r y x} ∪ {x}\n⊢ ({y | r y x} ∪ {x}).Countable"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case intro.intro.refine_1\nHcont : #ℝ = ℵ_ 1\nr : ℝ → ℝ → Prop\nhr : IsWellOrder ℝ r\nH : (#ℝ).ord = Ordinal.type r\nx : ℝ\nthis : univ \\ {y | r x y} = {y | r y x} ∪ {x}\n⊢ ({y | r y x} ∪ {x}).Countable",
"after_state": "case intro.intro.refine_1\nHcont : #ℝ = ℵ_ 1\nr : ℝ → ℝ → Prop\nhr : IsWellOrder ℝ r\nH : (#ℝ).ord = Ordinal.type r\nx : ℝ\nthis : univ \\ {y | r x y} = {y | r y x} ∪ {x}\n⊢ ({y | r y x} ∪ {x}).Countable"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case intro.intro.refine_1\nHcont : #ℝ = ℵ_ 1\nr : ℝ → ℝ → Prop\nhr : IsWellOrder ℝ r\nH : (#ℝ).ord = Ordinal.type r\nx : ℝ\nthis : univ \\ {y | r x y} = {y | r y x} ∪ {x}\n⊢ ({y | r y x} ∪ {x}).Countable",
"after_state": "case intro.intro.refine_1\nHcont : #ℝ = ℵ_ 1\nr : ℝ → ℝ → Prop\nhr : IsWellOrder ℝ r\nH : (#ℝ).ord = Ordinal.type r\nx : ℝ\nthis : univ \\ {y | r x y} = {y | r y x} ∪ {x}\n⊢ ({y | r y x} ∪ {x}).Countable"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case intro.intro.refine_1\nHcont : #ℝ = ℵ_ 1\nr : ℝ → ℝ → Prop\nhr : IsWellOrder ℝ r\nH : (#ℝ).ord = Ordinal.type r\nx : ℝ\nthis : univ \\ {y | r x y} = {y | r y x} ∪ {x}\n⊢ ({y | r y x} ∪ {x}).Countable",
"after_state": "case intro.intro.refine_1\nHcont : #ℝ = ℵ_ 1\nr : ℝ → ℝ → Prop\nhr : IsWellOrder ℝ r\nH : (#ℝ).ord = Ordinal.type r\nx : ℝ\nthis : univ \\ {y | r x y} = {y | r y x} ∪ {x}\n⊢ ({y | r y x} ∪ {x}).Countable"
},
{
"line": "with_reducible rfl",
"before_state": "case intro.intro.refine_1\nHcont : #ℝ = ℵ_ 1\nr : ℝ → ℝ → Prop\nhr : IsWellOrder ℝ r\nH : (#ℝ).ord = Ordinal.type r\nx : ℝ\nthis : univ \\ {y | r x y} = {y | r y x} ∪ {x}\n⊢ ({y | r y x} ∪ {x}).Countable",
"after_state": "case intro.intro.refine_1\nHcont : #ℝ = ℵ_ 1\nr : ℝ → ℝ → Prop\nhr : IsWellOrder ℝ r\nH : (#ℝ).ord = Ordinal.type r\nx : ℝ\nthis : univ \\ {y | r x y} = {y | r y x} ∪ {x}\n⊢ ({y | r y x} ∪ {x}).Countable"
},
{
"line": "rfl",
"before_state": "case intro.intro.refine_1\nHcont : #ℝ = ℵ_ 1\nr : ℝ → ℝ → Prop\nhr : IsWellOrder ℝ r\nH : (#ℝ).ord = Ordinal.type r\nx : ℝ\nthis : univ \\ {y | r x y} = {y | r y x} ∪ {x}\n⊢ ({y | r y x} ∪ {x}).Countable",
"after_state": "case intro.intro.refine_1\nHcont : #ℝ = ℵ_ 1\nr : ℝ → ℝ → Prop\nhr : IsWellOrder ℝ r\nH : (#ℝ).ord = Ordinal.type r\nx : ℝ\nthis : univ \\ {y | r x y} = {y | r y x} ∪ {x}\n⊢ ({y | r y x} ∪ {x}).Countable"
},
{
"line": "apply_rfl",
"before_state": "case intro.intro.refine_1\nHcont : #ℝ = ℵ_ 1\nr : ℝ → ℝ → Prop\nhr : IsWellOrder ℝ r\nH : (#ℝ).ord = Ordinal.type r\nx : ℝ\nthis : univ \\ {y | r x y} = {y | r y x} ∪ {x}\n⊢ ({y | r y x} ∪ {x}).Countable",
"after_state": "case intro.intro.refine_1\nHcont : #ℝ = ℵ_ 1\nr : ℝ → ℝ → Prop\nhr : IsWellOrder ℝ r\nH : (#ℝ).ord = Ordinal.type r\nx : ℝ\nthis : univ \\ {y | r x y} = {y | r y x} ∪ {x}\n⊢ ({y | r y x} ∪ {x}).Countable"
},
{
"line": "skip",
"before_state": "case intro.intro.refine_1\nHcont : #ℝ = ℵ_ 1\nr : ℝ → ℝ → Prop\nhr : IsWellOrder ℝ r\nH : (#ℝ).ord = Ordinal.type r\nx : ℝ\nthis : univ \\ {y | r x y} = {y | r y x} ∪ {x}\n⊢ ({y | r y x} ∪ {x}).Countable",
"after_state": "case intro.intro.refine_1\nHcont : #ℝ = ℵ_ 1\nr : ℝ → ℝ → Prop\nhr : IsWellOrder ℝ r\nH : (#ℝ).ord = Ordinal.type r\nx : ℝ\nthis : univ \\ {y | r x y} = {y | r y x} ∪ {x}\n⊢ ({y | r y x} ∪ {x}).Countable"
},
{
"line": "apply Countable.union _ (countable_singleton _)",
"before_state": "case intro.intro.refine_1\nHcont : #ℝ = ℵ_ 1\nr : ℝ → ℝ → Prop\nhr : IsWellOrder ℝ r\nH : (#ℝ).ord = Ordinal.type r\nx : ℝ\nthis : univ \\ {y | r x y} = {y | r y x} ∪ {x}\n⊢ ({y | r y x} ∪ {x}).Countable",
"after_state": "Hcont : #ℝ = ℵ_ 1\nr : ℝ → ℝ → Prop\nhr : IsWellOrder ℝ r\nH : (#ℝ).ord = Ordinal.type r\nx : ℝ\nthis : univ \\ {y | r x y} = {y | r y x} ∪ {x}\n⊢ {y | r y x}.Countable"
},
{
"line": "rw [Cardinal.countable_iff_lt_aleph_one]",
"before_state": "Hcont : #ℝ = ℵ_ 1\nr : ℝ → ℝ → Prop\nhr : IsWellOrder ℝ r\nH : (#ℝ).ord = Ordinal.type r\nx : ℝ\nthis : univ \\ {y | r x y} = {y | r y x} ∪ {x}\n⊢ {y | r y x}.Countable",
"after_state": "Hcont : #ℝ = ℵ_ 1\nr : ℝ → ℝ → Prop\nhr : IsWellOrder ℝ r\nH : (#ℝ).ord = Ordinal.type r\nx : ℝ\nthis : univ \\ {y | r x y} = {y | r y x} ∪ {x}\n⊢ #↑{y | r y x} < ℵ_ 1"
},
{
"line": "rewrite [Cardinal.countable_iff_lt_aleph_one]",
"before_state": "Hcont : #ℝ = ℵ_ 1\nr : ℝ → ℝ → Prop\nhr : IsWellOrder ℝ r\nH : (#ℝ).ord = Ordinal.type r\nx : ℝ\nthis : univ \\ {y | r x y} = {y | r y x} ∪ {x}\n⊢ {y | r y x}.Countable",
"after_state": "Hcont : #ℝ = ℵ_ 1\nr : ℝ → ℝ → Prop\nhr : IsWellOrder ℝ r\nH : (#ℝ).ord = Ordinal.type r\nx : ℝ\nthis : univ \\ {y | r x y} = {y | r y x} ∪ {x}\n⊢ #↑{y | r y x} < ℵ_ 1"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "Hcont : #ℝ = ℵ_ 1\nr : ℝ → ℝ → Prop\nhr : IsWellOrder ℝ r\nH : (#ℝ).ord = Ordinal.type r\nx : ℝ\nthis : univ \\ {y | r x y} = {y | r y x} ∪ {x}\n⊢ #↑{y | r y x} < ℵ_ 1",
"after_state": "Hcont : #ℝ = ℵ_ 1\nr : ℝ → ℝ → Prop\nhr : IsWellOrder ℝ r\nH : (#ℝ).ord = Ordinal.type r\nx : ℝ\nthis : univ \\ {y | r x y} = {y | r y x} ∪ {x}\n⊢ #↑{y | r y x} < ℵ_ 1"
},
{
"line": "try (with_reducible rfl)",
"before_state": "Hcont : #ℝ = ℵ_ 1\nr : ℝ → ℝ → Prop\nhr : IsWellOrder ℝ r\nH : (#ℝ).ord = Ordinal.type r\nx : ℝ\nthis : univ \\ {y | r x y} = {y | r y x} ∪ {x}\n⊢ #↑{y | r y x} < ℵ_ 1",
"after_state": "Hcont : #ℝ = ℵ_ 1\nr : ℝ → ℝ → Prop\nhr : IsWellOrder ℝ r\nH : (#ℝ).ord = Ordinal.type r\nx : ℝ\nthis : univ \\ {y | r x y} = {y | r y x} ∪ {x}\n⊢ #↑{y | r y x} < ℵ_ 1"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "Hcont : #ℝ = ℵ_ 1\nr : ℝ → ℝ → Prop\nhr : IsWellOrder ℝ r\nH : (#ℝ).ord = Ordinal.type r\nx : ℝ\nthis : univ \\ {y | r x y} = {y | r y x} ∪ {x}\n⊢ #↑{y | r y x} < ℵ_ 1",
"after_state": "Hcont : #ℝ = ℵ_ 1\nr : ℝ → ℝ → Prop\nhr : IsWellOrder ℝ r\nH : (#ℝ).ord = Ordinal.type r\nx : ℝ\nthis : univ \\ {y | r x y} = {y | r y x} ∪ {x}\n⊢ #↑{y | r y x} < ℵ_ 1"
},
{
"line": "with_reducible rfl",
"before_state": "Hcont : #ℝ = ℵ_ 1\nr : ℝ → ℝ → Prop\nhr : IsWellOrder ℝ r\nH : (#ℝ).ord = Ordinal.type r\nx : ℝ\nthis : univ \\ {y | r x y} = {y | r y x} ∪ {x}\n⊢ #↑{y | r y x} < ℵ_ 1",
"after_state": "Hcont : #ℝ = ℵ_ 1\nr : ℝ → ℝ → Prop\nhr : IsWellOrder ℝ r\nH : (#ℝ).ord = Ordinal.type r\nx : ℝ\nthis : univ \\ {y | r x y} = {y | r y x} ∪ {x}\n⊢ #↑{y | r y x} < ℵ_ 1"
},
{
"line": "rfl",
"before_state": "Hcont : #ℝ = ℵ_ 1\nr : ℝ → ℝ → Prop\nhr : IsWellOrder ℝ r\nH : (#ℝ).ord = Ordinal.type r\nx : ℝ\nthis : univ \\ {y | r x y} = {y | r y x} ∪ {x}\n⊢ #↑{y | r y x} < ℵ_ 1",
"after_state": "Hcont : #ℝ = ℵ_ 1\nr : ℝ → ℝ → Prop\nhr : IsWellOrder ℝ r\nH : (#ℝ).ord = Ordinal.type r\nx : ℝ\nthis : univ \\ {y | r x y} = {y | r y x} ∪ {x}\n⊢ #↑{y | r y x} < ℵ_ 1"
},
{
"line": "apply_rfl",
"before_state": "Hcont : #ℝ = ℵ_ 1\nr : ℝ → ℝ → Prop\nhr : IsWellOrder ℝ r\nH : (#ℝ).ord = Ordinal.type r\nx : ℝ\nthis : univ \\ {y | r x y} = {y | r y x} ∪ {x}\n⊢ #↑{y | r y x} < ℵ_ 1",
"after_state": "Hcont : #ℝ = ℵ_ 1\nr : ℝ → ℝ → Prop\nhr : IsWellOrder ℝ r\nH : (#ℝ).ord = Ordinal.type r\nx : ℝ\nthis : univ \\ {y | r x y} = {y | r y x} ∪ {x}\n⊢ #↑{y | r y x} < ℵ_ 1"
},
{
"line": "skip",
"before_state": "Hcont : #ℝ = ℵ_ 1\nr : ℝ → ℝ → Prop\nhr : IsWellOrder ℝ r\nH : (#ℝ).ord = Ordinal.type r\nx : ℝ\nthis : univ \\ {y | r x y} = {y | r y x} ∪ {x}\n⊢ #↑{y | r y x} < ℵ_ 1",
"after_state": "Hcont : #ℝ = ℵ_ 1\nr : ℝ → ℝ → Prop\nhr : IsWellOrder ℝ r\nH : (#ℝ).ord = Ordinal.type r\nx : ℝ\nthis : univ \\ {y | r x y} = {y | r y x} ∪ {x}\n⊢ #↑{y | r y x} < ℵ_ 1"
},
{
"line": "rw [← Hcont]",
"before_state": "Hcont : #ℝ = ℵ_ 1\nr : ℝ → ℝ → Prop\nhr : IsWellOrder ℝ r\nH : (#ℝ).ord = Ordinal.type r\nx : ℝ\nthis : univ \\ {y | r x y} = {y | r y x} ∪ {x}\n⊢ #↑{y | r y x} < ℵ_ 1",
"after_state": "Hcont : #ℝ = ℵ_ 1\nr : ℝ → ℝ → Prop\nhr : IsWellOrder ℝ r\nH : (#ℝ).ord = Ordinal.type r\nx : ℝ\nthis : univ \\ {y | r x y} = {y | r y x} ∪ {x}\n⊢ #↑{y | r y x} < #ℝ"
},
{
"line": "rewrite [← Hcont]",
"before_state": "Hcont : #ℝ = ℵ_ 1\nr : ℝ → ℝ → Prop\nhr : IsWellOrder ℝ r\nH : (#ℝ).ord = Ordinal.type r\nx : ℝ\nthis : univ \\ {y | r x y} = {y | r y x} ∪ {x}\n⊢ #↑{y | r y x} < ℵ_ 1",
"after_state": "Hcont : #ℝ = ℵ_ 1\nr : ℝ → ℝ → Prop\nhr : IsWellOrder ℝ r\nH : (#ℝ).ord = Ordinal.type r\nx : ℝ\nthis : univ \\ {y | r x y} = {y | r y x} ∪ {x}\n⊢ #↑{y | r y x} < #ℝ"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "Hcont : #ℝ = ℵ_ 1\nr : ℝ → ℝ → Prop\nhr : IsWellOrder ℝ r\nH : (#ℝ).ord = Ordinal.type r\nx : ℝ\nthis : univ \\ {y | r x y} = {y | r y x} ∪ {x}\n⊢ #↑{y | r y x} < #ℝ",
"after_state": "Hcont : #ℝ = ℵ_ 1\nr : ℝ → ℝ → Prop\nhr : IsWellOrder ℝ r\nH : (#ℝ).ord = Ordinal.type r\nx : ℝ\nthis : univ \\ {y | r x y} = {y | r y x} ∪ {x}\n⊢ #↑{y | r y x} < #ℝ"
},
{
"line": "try (with_reducible rfl)",
"before_state": "Hcont : #ℝ = ℵ_ 1\nr : ℝ → ℝ → Prop\nhr : IsWellOrder ℝ r\nH : (#ℝ).ord = Ordinal.type r\nx : ℝ\nthis : univ \\ {y | r x y} = {y | r y x} ∪ {x}\n⊢ #↑{y | r y x} < #ℝ",
"after_state": "Hcont : #ℝ = ℵ_ 1\nr : ℝ → ℝ → Prop\nhr : IsWellOrder ℝ r\nH : (#ℝ).ord = Ordinal.type r\nx : ℝ\nthis : univ \\ {y | r x y} = {y | r y x} ∪ {x}\n⊢ #↑{y | r y x} < #ℝ"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "Hcont : #ℝ = ℵ_ 1\nr : ℝ → ℝ → Prop\nhr : IsWellOrder ℝ r\nH : (#ℝ).ord = Ordinal.type r\nx : ℝ\nthis : univ \\ {y | r x y} = {y | r y x} ∪ {x}\n⊢ #↑{y | r y x} < #ℝ",
"after_state": "Hcont : #ℝ = ℵ_ 1\nr : ℝ → ℝ → Prop\nhr : IsWellOrder ℝ r\nH : (#ℝ).ord = Ordinal.type r\nx : ℝ\nthis : univ \\ {y | r x y} = {y | r y x} ∪ {x}\n⊢ #↑{y | r y x} < #ℝ"
},
{
"line": "with_reducible rfl",
"before_state": "Hcont : #ℝ = ℵ_ 1\nr : ℝ → ℝ → Prop\nhr : IsWellOrder ℝ r\nH : (#ℝ).ord = Ordinal.type r\nx : ℝ\nthis : univ \\ {y | r x y} = {y | r y x} ∪ {x}\n⊢ #↑{y | r y x} < #ℝ",
"after_state": "Hcont : #ℝ = ℵ_ 1\nr : ℝ → ℝ → Prop\nhr : IsWellOrder ℝ r\nH : (#ℝ).ord = Ordinal.type r\nx : ℝ\nthis : univ \\ {y | r x y} = {y | r y x} ∪ {x}\n⊢ #↑{y | r y x} < #ℝ"
},
{
"line": "rfl",
"before_state": "Hcont : #ℝ = ℵ_ 1\nr : ℝ → ℝ → Prop\nhr : IsWellOrder ℝ r\nH : (#ℝ).ord = Ordinal.type r\nx : ℝ\nthis : univ \\ {y | r x y} = {y | r y x} ∪ {x}\n⊢ #↑{y | r y x} < #ℝ",
"after_state": "Hcont : #ℝ = ℵ_ 1\nr : ℝ → ℝ → Prop\nhr : IsWellOrder ℝ r\nH : (#ℝ).ord = Ordinal.type r\nx : ℝ\nthis : univ \\ {y | r x y} = {y | r y x} ∪ {x}\n⊢ #↑{y | r y x} < #ℝ"
},
{
"line": "apply_rfl",
"before_state": "Hcont : #ℝ = ℵ_ 1\nr : ℝ → ℝ → Prop\nhr : IsWellOrder ℝ r\nH : (#ℝ).ord = Ordinal.type r\nx : ℝ\nthis : univ \\ {y | r x y} = {y | r y x} ∪ {x}\n⊢ #↑{y | r y x} < #ℝ",
"after_state": "Hcont : #ℝ = ℵ_ 1\nr : ℝ → ℝ → Prop\nhr : IsWellOrder ℝ r\nH : (#ℝ).ord = Ordinal.type r\nx : ℝ\nthis : univ \\ {y | r x y} = {y | r y x} ∪ {x}\n⊢ #↑{y | r y x} < #ℝ"
},
{
"line": "skip",
"before_state": "Hcont : #ℝ = ℵ_ 1\nr : ℝ → ℝ → Prop\nhr : IsWellOrder ℝ r\nH : (#ℝ).ord = Ordinal.type r\nx : ℝ\nthis : univ \\ {y | r x y} = {y | r y x} ∪ {x}\n⊢ #↑{y | r y x} < #ℝ",
"after_state": "Hcont : #ℝ = ℵ_ 1\nr : ℝ → ℝ → Prop\nhr : IsWellOrder ℝ r\nH : (#ℝ).ord = Ordinal.type r\nx : ℝ\nthis : univ \\ {y | r x y} = {y | r y x} ∪ {x}\n⊢ #↑{y | r y x} < #ℝ"
},
{
"line": "exact Cardinal.card_typein_lt r x H",
"before_state": "Hcont : #ℝ = ℵ_ 1\nr : ℝ → ℝ → Prop\nhr : IsWellOrder ℝ r\nH : (#ℝ).ord = Ordinal.type r\nx : ℝ\nthis : univ \\ {y | r x y} = {y | r y x} ∪ {x}\n⊢ #↑{y | r y x} < #ℝ",
"after_state": "No Goals!"
},
{
"line": "rw [Cardinal.countable_iff_lt_aleph_one, ← Hcont]",
"before_state": "case intro.intro.refine_2\nHcont : #ℝ = ℵ_ 1\nr : ℝ → ℝ → Prop\nhr : IsWellOrder ℝ r\nH : (#ℝ).ord = Ordinal.type r\ny : ℝ\n⊢ {x | y ∈ (fun x => {y | r x y}) x}.Countable",
"after_state": "case intro.intro.refine_2\nHcont : #ℝ = ℵ_ 1\nr : ℝ → ℝ → Prop\nhr : IsWellOrder ℝ r\nH : (#ℝ).ord = Ordinal.type r\ny : ℝ\n⊢ #↑{x | y ∈ (fun x => {y | r x y}) x} < #ℝ"
},
{
"line": "rewrite [Cardinal.countable_iff_lt_aleph_one, ← Hcont]",
"before_state": "case intro.intro.refine_2\nHcont : #ℝ = ℵ_ 1\nr : ℝ → ℝ → Prop\nhr : IsWellOrder ℝ r\nH : (#ℝ).ord = Ordinal.type r\ny : ℝ\n⊢ {x | y ∈ (fun x => {y | r x y}) x}.Countable",
"after_state": "case intro.intro.refine_2\nHcont : #ℝ = ℵ_ 1\nr : ℝ → ℝ → Prop\nhr : IsWellOrder ℝ r\nH : (#ℝ).ord = Ordinal.type r\ny : ℝ\n⊢ #↑{x | y ∈ (fun x => {y | r x y}) x} < #ℝ"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case intro.intro.refine_2\nHcont : #ℝ = ℵ_ 1\nr : ℝ → ℝ → Prop\nhr : IsWellOrder ℝ r\nH : (#ℝ).ord = Ordinal.type r\ny : ℝ\n⊢ #↑{x | y ∈ (fun x => {y | r x y}) x} < #ℝ",
"after_state": "case intro.intro.refine_2\nHcont : #ℝ = ℵ_ 1\nr : ℝ → ℝ → Prop\nhr : IsWellOrder ℝ r\nH : (#ℝ).ord = Ordinal.type r\ny : ℝ\n⊢ #↑{x | y ∈ (fun x => {y | r x y}) x} < #ℝ"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case intro.intro.refine_2\nHcont : #ℝ = ℵ_ 1\nr : ℝ → ℝ → Prop\nhr : IsWellOrder ℝ r\nH : (#ℝ).ord = Ordinal.type r\ny : ℝ\n⊢ #↑{x | y ∈ (fun x => {y | r x y}) x} < #ℝ",
"after_state": "case intro.intro.refine_2\nHcont : #ℝ = ℵ_ 1\nr : ℝ → ℝ → Prop\nhr : IsWellOrder ℝ r\nH : (#ℝ).ord = Ordinal.type r\ny : ℝ\n⊢ #↑{x | y ∈ (fun x => {y | r x y}) x} < #ℝ"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case intro.intro.refine_2\nHcont : #ℝ = ℵ_ 1\nr : ℝ → ℝ → Prop\nhr : IsWellOrder ℝ r\nH : (#ℝ).ord = Ordinal.type r\ny : ℝ\n⊢ #↑{x | y ∈ (fun x => {y | r x y}) x} < #ℝ",
"after_state": "case intro.intro.refine_2\nHcont : #ℝ = ℵ_ 1\nr : ℝ → ℝ → Prop\nhr : IsWellOrder ℝ r\nH : (#ℝ).ord = Ordinal.type r\ny : ℝ\n⊢ #↑{x | y ∈ (fun x => {y | r x y}) x} < #ℝ"
},
{
"line": "with_reducible rfl",
"before_state": "case intro.intro.refine_2\nHcont : #ℝ = ℵ_ 1\nr : ℝ → ℝ → Prop\nhr : IsWellOrder ℝ r\nH : (#ℝ).ord = Ordinal.type r\ny : ℝ\n⊢ #↑{x | y ∈ (fun x => {y | r x y}) x} < #ℝ",
"after_state": "case intro.intro.refine_2\nHcont : #ℝ = ℵ_ 1\nr : ℝ → ℝ → Prop\nhr : IsWellOrder ℝ r\nH : (#ℝ).ord = Ordinal.type r\ny : ℝ\n⊢ #↑{x | y ∈ (fun x => {y | r x y}) x} < #ℝ"
},
{
"line": "rfl",
"before_state": "case intro.intro.refine_2\nHcont : #ℝ = ℵ_ 1\nr : ℝ → ℝ → Prop\nhr : IsWellOrder ℝ r\nH : (#ℝ).ord = Ordinal.type r\ny : ℝ\n⊢ #↑{x | y ∈ (fun x => {y | r x y}) x} < #ℝ",
"after_state": "case intro.intro.refine_2\nHcont : #ℝ = ℵ_ 1\nr : ℝ → ℝ → Prop\nhr : IsWellOrder ℝ r\nH : (#ℝ).ord = Ordinal.type r\ny : ℝ\n⊢ #↑{x | y ∈ (fun x => {y | r x y}) x} < #ℝ"
},
{
"line": "apply_rfl",
"before_state": "case intro.intro.refine_2\nHcont : #ℝ = ℵ_ 1\nr : ℝ → ℝ → Prop\nhr : IsWellOrder ℝ r\nH : (#ℝ).ord = Ordinal.type r\ny : ℝ\n⊢ #↑{x | y ∈ (fun x => {y | r x y}) x} < #ℝ",
"after_state": "case intro.intro.refine_2\nHcont : #ℝ = ℵ_ 1\nr : ℝ → ℝ → Prop\nhr : IsWellOrder ℝ r\nH : (#ℝ).ord = Ordinal.type r\ny : ℝ\n⊢ #↑{x | y ∈ (fun x => {y | r x y}) x} < #ℝ"
},
{
"line": "skip",
"before_state": "case intro.intro.refine_2\nHcont : #ℝ = ℵ_ 1\nr : ℝ → ℝ → Prop\nhr : IsWellOrder ℝ r\nH : (#ℝ).ord = Ordinal.type r\ny : ℝ\n⊢ #↑{x | y ∈ (fun x => {y | r x y}) x} < #ℝ",
"after_state": "case intro.intro.refine_2\nHcont : #ℝ = ℵ_ 1\nr : ℝ → ℝ → Prop\nhr : IsWellOrder ℝ r\nH : (#ℝ).ord = Ordinal.type r\ny : ℝ\n⊢ #↑{x | y ∈ (fun x => {y | r x y}) x} < #ℝ"
},
{
"line": "exact Cardinal.card_typein_lt r y H",
"before_state": "case intro.intro.refine_2\nHcont : #ℝ = ℵ_ 1\nr : ℝ → ℝ → Prop\nhr : IsWellOrder ℝ r\nH : (#ℝ).ord = Ordinal.type r\ny : ℝ\n⊢ #↑{x | y ∈ (fun x => {y | r x y}) x} < #ℝ",
"after_state": "No Goals!"
}
] |
theorem zero_divisors_of_periodic {R A} [Nontrivial R] [Ring R] [AddMonoid A] {n : ℕ} (a : A)
(n2 : 2 ≤ n) (na : n • a = a) (na1 : (n - 1) • a ≠ 0) :
∃ f g : R[A], f ≠ 0 ∧ g ≠ 0 ∧ f * g = 0 := by
refine ⟨single a 1, single ((n - 1) • a) 1 - single 0 1, by simp, ?_, ?_⟩
· exact sub_ne_zero.mpr (by simpa [single, AddMonoidAlgebra, single_eq_single_iff])
· rw [mul_sub, AddMonoidAlgebra.single_mul_single, AddMonoidAlgebra.single_mul_single,
sub_eq_zero, add_zero, ← succ_nsmul', Nat.sub_add_cancel (one_le_two.trans n2), na]
|
/root/DuelModelResearch/mathlib4/Counterexamples/ZeroDivisorsInAddMonoidAlgebras.lean
|
{
"open": [
"Finsupp hiding single",
"AddMonoidAlgebra"
],
"variables": []
}
|
[
{
"line": "refine ⟨single a 1, single ((n - 1) • a) 1 - single 0 1, by simp, ?_, ?_⟩",
"before_state": "R : Type u_1\nA : Type u_2\ninst✝² : Nontrivial R\ninst✝¹ : Ring R\ninst✝ : AddMonoid A\nn : ℕ\na : A\nn2 : 2 ≤ n\nna : n • a = a\nna1 : (n - 1) • a ≠ 0\n⊢ ∃ f g, f ≠ 0 ∧ g ≠ 0 ∧ f * g = 0",
"after_state": "case refine_1\nR : Type u_1\nA : Type u_2\ninst✝² : Nontrivial R\ninst✝¹ : Ring R\ninst✝ : AddMonoid A\nn : ℕ\na : A\nn2 : 2 ≤ n\nna : n • a = a\nna1 : (n - 1) • a ≠ 0\n⊢ single ((n - 1) • a) 1 - single 0 1 ≠ 0\n---\ncase refine_2\nR : Type u_1\nA : Type u_2\ninst✝² : Nontrivial R\ninst✝¹ : Ring R\ninst✝ : AddMonoid A\nn : ℕ\na : A\nn2 : 2 ≤ n\nna : n • a = a\nna1 : (n - 1) • a ≠ 0\n⊢ single a 1 * (single ((n - 1) • a) 1 - single 0 1) = 0"
},
{
"line": "simp",
"before_state": "R : Type u_1\nA : Type u_2\ninst✝² : Nontrivial R\ninst✝¹ : Ring R\ninst✝ : AddMonoid A\nn : ℕ\na : A\nn2 : 2 ≤ n\nna : n • a = a\nna1 : (n - 1) • a ≠ 0\n⊢ single a 1 ≠ 0",
"after_state": "No Goals!"
},
{
"line": "exact sub_ne_zero.mpr (by simpa [single, AddMonoidAlgebra, single_eq_single_iff])",
"before_state": "case refine_1\nR : Type u_1\nA : Type u_2\ninst✝² : Nontrivial R\ninst✝¹ : Ring R\ninst✝ : AddMonoid A\nn : ℕ\na : A\nn2 : 2 ≤ n\nna : n • a = a\nna1 : (n - 1) • a ≠ 0\n⊢ single ((n - 1) • a) 1 - single 0 1 ≠ 0",
"after_state": "No Goals!"
},
{
"line": "simpa [single, AddMonoidAlgebra, single_eq_single_iff]",
"before_state": "R : Type u_1\nA : Type u_2\ninst✝² : Nontrivial R\ninst✝¹ : Ring R\ninst✝ : AddMonoid A\nn : ℕ\na : A\nn2 : 2 ≤ n\nna : n • a = a\nna1 : (n - 1) • a ≠ 0\n⊢ single ((n - 1) • a) 1 ≠ single 0 1",
"after_state": "No Goals!"
},
{
"line": "rw [mul_sub, AddMonoidAlgebra.single_mul_single, AddMonoidAlgebra.single_mul_single, sub_eq_zero, add_zero, ←\n succ_nsmul', Nat.sub_add_cancel (one_le_two.trans n2), na]",
"before_state": "case refine_2\nR : Type u_1\nA : Type u_2\ninst✝² : Nontrivial R\ninst✝¹ : Ring R\ninst✝ : AddMonoid A\nn : ℕ\na : A\nn2 : 2 ≤ n\nna : n • a = a\nna1 : (n - 1) • a ≠ 0\n⊢ single a 1 * (single ((n - 1) • a) 1 - single 0 1) = 0",
"after_state": "No Goals!"
},
{
"line": "rewrite [mul_sub, AddMonoidAlgebra.single_mul_single, AddMonoidAlgebra.single_mul_single, sub_eq_zero, add_zero, ←\n succ_nsmul', Nat.sub_add_cancel (one_le_two.trans n2), na]",
"before_state": "case refine_2\nR : Type u_1\nA : Type u_2\ninst✝² : Nontrivial R\ninst✝¹ : Ring R\ninst✝ : AddMonoid A\nn : ℕ\na : A\nn2 : 2 ≤ n\nna : n • a = a\nna1 : (n - 1) • a ≠ 0\n⊢ single a 1 * (single ((n - 1) • a) 1 - single 0 1) = 0",
"after_state": "case refine_2\nR : Type u_1\nA : Type u_2\ninst✝² : Nontrivial R\ninst✝¹ : Ring R\ninst✝ : AddMonoid A\nn : ℕ\na : A\nn2 : 2 ≤ n\nna : n • a = a\nna1 : (n - 1) • a ≠ 0\n⊢ single a (1 * 1) = single a (1 * 1)"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case refine_2\nR : Type u_1\nA : Type u_2\ninst✝² : Nontrivial R\ninst✝¹ : Ring R\ninst✝ : AddMonoid A\nn : ℕ\na : A\nn2 : 2 ≤ n\nna : n • a = a\nna1 : (n - 1) • a ≠ 0\n⊢ single a (1 * 1) = single a (1 * 1)",
"after_state": "No Goals!"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case refine_2\nR : Type u_1\nA : Type u_2\ninst✝² : Nontrivial R\ninst✝¹ : Ring R\ninst✝ : AddMonoid A\nn : ℕ\na : A\nn2 : 2 ≤ n\nna : n • a = a\nna1 : (n - 1) • a ≠ 0\n⊢ single a (1 * 1) = single a (1 * 1)",
"after_state": "No Goals!"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case refine_2\nR : Type u_1\nA : Type u_2\ninst✝² : Nontrivial R\ninst✝¹ : Ring R\ninst✝ : AddMonoid A\nn : ℕ\na : A\nn2 : 2 ≤ n\nna : n • a = a\nna1 : (n - 1) • a ≠ 0\n⊢ single a (1 * 1) = single a (1 * 1)",
"after_state": "No Goals!"
},
{
"line": "with_reducible rfl",
"before_state": "case refine_2\nR : Type u_1\nA : Type u_2\ninst✝² : Nontrivial R\ninst✝¹ : Ring R\ninst✝ : AddMonoid A\nn : ℕ\na : A\nn2 : 2 ≤ n\nna : n • a = a\nna1 : (n - 1) • a ≠ 0\n⊢ single a (1 * 1) = single a (1 * 1)",
"after_state": "No Goals!"
},
{
"line": "rfl",
"before_state": "case refine_2\nR : Type u_1\nA : Type u_2\ninst✝² : Nontrivial R\ninst✝¹ : Ring R\ninst✝ : AddMonoid A\nn : ℕ\na : A\nn2 : 2 ≤ n\nna : n • a = a\nna1 : (n - 1) • a ≠ 0\n⊢ single a (1 * 1) = single a (1 * 1)",
"after_state": "No Goals!"
},
{
"line": "eq_refl",
"before_state": "case refine_2\nR : Type u_1\nA : Type u_2\ninst✝² : Nontrivial R\ninst✝¹ : Ring R\ninst✝ : AddMonoid A\nn : ℕ\na : A\nn2 : 2 ≤ n\nna : n • a = a\nna1 : (n - 1) • a ≠ 0\n⊢ single a (1 * 1) = single a (1 * 1)",
"after_state": "No Goals!"
}
] |
example : LinearOrder F := by infer_instance
|
/root/DuelModelResearch/mathlib4/Counterexamples/ZeroDivisorsInAddMonoidAlgebras.lean
|
{
"open": [
"Finsupp hiding single",
"AddMonoidAlgebra",
"Lean Elab Command in"
],
"variables": []
}
|
[
{
"line": "infer_instance",
"before_state": "F : Type ?u.7\n⊢ LinearOrder F",
"after_state": "No Goals!"
},
{
"line": "exact inferInstance✝",
"before_state": "F : Type ?u.7\n⊢ LinearOrder F",
"after_state": "No Goals!"
}
] |
example : AddMonoid F := by infer_instance
|
/root/DuelModelResearch/mathlib4/Counterexamples/ZeroDivisorsInAddMonoidAlgebras.lean
|
{
"open": [
"Finsupp hiding single",
"AddMonoidAlgebra",
"Lean Elab Command in"
],
"variables": []
}
|
[
{
"line": "infer_instance",
"before_state": "F : Type ?u.7\n⊢ AddMonoid F",
"after_state": "No Goals!"
},
{
"line": "exact inferInstance✝",
"before_state": "F : Type ?u.7\n⊢ AddMonoid F",
"after_state": "No Goals!"
}
] |
example : ¬UniqueProds ℕ := by
rintro ⟨h⟩
refine not_not.mpr (h (Finset.singleton_nonempty 0) (Finset.insert_nonempty 0 {1})) ?_
simp [UniqueMul, not_or]
|
/root/DuelModelResearch/mathlib4/Counterexamples/ZeroDivisorsInAddMonoidAlgebras.lean
|
{
"open": [
"Finsupp hiding single",
"AddMonoidAlgebra",
"Lean Elab Command in"
],
"variables": []
}
|
[
{
"line": "rintro ⟨h⟩",
"before_state": "⊢ ¬UniqueProds ℕ",
"after_state": "case mk\nh : ∀ {A B : Finset ℕ}, A.Nonempty → B.Nonempty → ∃ a0 ∈ A, ∃ b0 ∈ B, UniqueMul A B a0 b0\n⊢ False"
},
{
"line": "refine not_not.mpr (h (Finset.singleton_nonempty 0) (Finset.insert_nonempty 0 { 1 })) ?_",
"before_state": "case mk\nh : ∀ {A B : Finset ℕ}, A.Nonempty → B.Nonempty → ∃ a0 ∈ A, ∃ b0 ∈ B, UniqueMul A B a0 b0\n⊢ False",
"after_state": "case mk\nh : ∀ {A B : Finset ℕ}, A.Nonempty → B.Nonempty → ∃ a0 ∈ A, ∃ b0 ∈ B, UniqueMul A B a0 b0\n⊢ ¬∃ a0 ∈ {0}, ∃ b0 ∈ {0, 1}, UniqueMul {0} {0, 1} a0 b0"
},
{
"line": "simp [UniqueMul, not_or]",
"before_state": "case mk\nh : ∀ {A B : Finset ℕ}, A.Nonempty → B.Nonempty → ∃ a0 ∈ A, ∃ b0 ∈ B, UniqueMul A B a0 b0\n⊢ ¬∃ a0 ∈ {0}, ∃ b0 ∈ {0, 1}, UniqueMul {0} {0, 1} a0 b0",
"after_state": "No Goals!"
}
] |
example (n : ℕ) (n2 : 2 ≤ n) : ¬UniqueSums (ZMod n) := by
haveI : Fintype (ZMod n) := @ZMod.fintype n ⟨(zero_lt_two.trans_le n2).ne'⟩
haveI : Nontrivial (ZMod n) := CharP.nontrivial_of_char_ne_one (one_lt_two.trans_le n2).ne'
rintro ⟨h⟩
refine not_not.mpr (h Finset.univ_nonempty Finset.univ_nonempty) ?_
suffices ∀ x y : ZMod n, ∃ x' y' : ZMod n, x' + y' = x + y ∧ (x' = x → ¬y' = y) by
simpa [UniqueAdd]
exact fun x y => ⟨x - 1, y + 1, sub_add_add_cancel _ _ _, by simp⟩
|
/root/DuelModelResearch/mathlib4/Counterexamples/ZeroDivisorsInAddMonoidAlgebras.lean
|
{
"open": [
"Finsupp hiding single",
"AddMonoidAlgebra",
"Lean Elab Command in"
],
"variables": []
}
|
[
{
"line": "haveI : Fintype (ZMod n) := @ZMod.fintype n ⟨(zero_lt_two.trans_le n2).ne'⟩",
"before_state": "n : ℕ\nn2 : 2 ≤ n\n⊢ ¬UniqueSums (ZMod n)",
"after_state": "n : ℕ\nn2 : 2 ≤ n\nthis : Fintype (ZMod n)\n⊢ ¬UniqueSums (ZMod n)"
},
{
"line": "refine_lift\n haveI : Fintype (ZMod n) := @ZMod.fintype n ⟨(zero_lt_two.trans_le n2).ne'⟩;\n ?_",
"before_state": "n : ℕ\nn2 : 2 ≤ n\n⊢ ¬UniqueSums (ZMod n)",
"after_state": "n : ℕ\nn2 : 2 ≤ n\nthis : Fintype (ZMod n)\n⊢ ¬UniqueSums (ZMod n)"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n haveI : Fintype (ZMod n) := @ZMod.fintype n ⟨(zero_lt_two.trans_le n2).ne'⟩;\n ?_;\n rotate_right)",
"before_state": "n : ℕ\nn2 : 2 ≤ n\n⊢ ¬UniqueSums (ZMod n)",
"after_state": "n : ℕ\nn2 : 2 ≤ n\nthis : Fintype (ZMod n)\n⊢ ¬UniqueSums (ZMod n)"
},
{
"line": "refine\n no_implicit_lambda%\n haveI : Fintype (ZMod n) := @ZMod.fintype n ⟨(zero_lt_two.trans_le n2).ne'⟩;\n ?_",
"before_state": "n : ℕ\nn2 : 2 ≤ n\n⊢ ¬UniqueSums (ZMod n)",
"after_state": "n : ℕ\nn2 : 2 ≤ n\nthis : Fintype (ZMod n)\n⊢ ¬UniqueSums (ZMod n)"
},
{
"line": "rotate_right",
"before_state": "n : ℕ\nn2 : 2 ≤ n\nthis : Fintype (ZMod n)\n⊢ ¬UniqueSums (ZMod n)",
"after_state": "n : ℕ\nn2 : 2 ≤ n\nthis : Fintype (ZMod n)\n⊢ ¬UniqueSums (ZMod n)"
},
{
"line": "haveI : Nontrivial (ZMod n) := CharP.nontrivial_of_char_ne_one (one_lt_two.trans_le n2).ne'",
"before_state": "n : ℕ\nn2 : 2 ≤ n\nthis : Fintype (ZMod n)\n⊢ ¬UniqueSums (ZMod n)",
"after_state": "n : ℕ\nn2 : 2 ≤ n\nthis✝ : Fintype (ZMod n)\nthis : Nontrivial (ZMod n)\n⊢ ¬UniqueSums (ZMod n)"
},
{
"line": "refine_lift\n haveI : Nontrivial (ZMod n) := CharP.nontrivial_of_char_ne_one (one_lt_two.trans_le n2).ne';\n ?_",
"before_state": "n : ℕ\nn2 : 2 ≤ n\nthis : Fintype (ZMod n)\n⊢ ¬UniqueSums (ZMod n)",
"after_state": "n : ℕ\nn2 : 2 ≤ n\nthis✝ : Fintype (ZMod n)\nthis : Nontrivial (ZMod n)\n⊢ ¬UniqueSums (ZMod n)"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n haveI : Nontrivial (ZMod n) := CharP.nontrivial_of_char_ne_one (one_lt_two.trans_le n2).ne';\n ?_;\n rotate_right)",
"before_state": "n : ℕ\nn2 : 2 ≤ n\nthis : Fintype (ZMod n)\n⊢ ¬UniqueSums (ZMod n)",
"after_state": "n : ℕ\nn2 : 2 ≤ n\nthis✝ : Fintype (ZMod n)\nthis : Nontrivial (ZMod n)\n⊢ ¬UniqueSums (ZMod n)"
},
{
"line": "refine\n no_implicit_lambda%\n haveI : Nontrivial (ZMod n) := CharP.nontrivial_of_char_ne_one (one_lt_two.trans_le n2).ne';\n ?_",
"before_state": "n : ℕ\nn2 : 2 ≤ n\nthis : Fintype (ZMod n)\n⊢ ¬UniqueSums (ZMod n)",
"after_state": "n : ℕ\nn2 : 2 ≤ n\nthis✝ : Fintype (ZMod n)\nthis : Nontrivial (ZMod n)\n⊢ ¬UniqueSums (ZMod n)"
},
{
"line": "rotate_right",
"before_state": "n : ℕ\nn2 : 2 ≤ n\nthis✝ : Fintype (ZMod n)\nthis : Nontrivial (ZMod n)\n⊢ ¬UniqueSums (ZMod n)",
"after_state": "n : ℕ\nn2 : 2 ≤ n\nthis✝ : Fintype (ZMod n)\nthis : Nontrivial (ZMod n)\n⊢ ¬UniqueSums (ZMod n)"
},
{
"line": "rintro ⟨h⟩",
"before_state": "n : ℕ\nn2 : 2 ≤ n\nthis✝ : Fintype (ZMod n)\nthis : Nontrivial (ZMod n)\n⊢ ¬UniqueSums (ZMod n)",
"after_state": "case mk\nn : ℕ\nn2 : 2 ≤ n\nthis✝ : Fintype (ZMod n)\nthis : Nontrivial (ZMod n)\nh : ∀ {A B : Finset (ZMod n)}, A.Nonempty → B.Nonempty → ∃ a0 ∈ A, ∃ b0 ∈ B, UniqueAdd A B a0 b0\n⊢ False"
},
{
"line": "refine not_not.mpr (h Finset.univ_nonempty Finset.univ_nonempty) ?_",
"before_state": "case mk\nn : ℕ\nn2 : 2 ≤ n\nthis✝ : Fintype (ZMod n)\nthis : Nontrivial (ZMod n)\nh : ∀ {A B : Finset (ZMod n)}, A.Nonempty → B.Nonempty → ∃ a0 ∈ A, ∃ b0 ∈ B, UniqueAdd A B a0 b0\n⊢ False",
"after_state": "case mk\nn : ℕ\nn2 : 2 ≤ n\nthis✝ : Fintype (ZMod n)\nthis : Nontrivial (ZMod n)\nh : ∀ {A B : Finset (ZMod n)}, A.Nonempty → B.Nonempty → ∃ a0 ∈ A, ∃ b0 ∈ B, UniqueAdd A B a0 b0\n⊢ ¬∃ a0 ∈ Finset.univ, ∃ b0 ∈ Finset.univ, UniqueAdd Finset.univ Finset.univ a0 b0"
},
{
"line": "suffices ∀ x y : ZMod n, ∃ x' y' : ZMod n, x' + y' = x + y ∧ (x' = x → ¬y' = y) by simpa [UniqueAdd]",
"before_state": "case mk\nn : ℕ\nn2 : 2 ≤ n\nthis✝ : Fintype (ZMod n)\nthis : Nontrivial (ZMod n)\nh : ∀ {A B : Finset (ZMod n)}, A.Nonempty → B.Nonempty → ∃ a0 ∈ A, ∃ b0 ∈ B, UniqueAdd A B a0 b0\n⊢ ¬∃ a0 ∈ Finset.univ, ∃ b0 ∈ Finset.univ, UniqueAdd Finset.univ Finset.univ a0 b0",
"after_state": "case mk\nn : ℕ\nn2 : 2 ≤ n\nthis✝ : Fintype (ZMod n)\nthis : Nontrivial (ZMod n)\nh : ∀ {A B : Finset (ZMod n)}, A.Nonempty → B.Nonempty → ∃ a0 ∈ A, ∃ b0 ∈ B, UniqueAdd A B a0 b0\n⊢ ∀ (x y : ZMod n), ∃ x' y', x' + y' = x + y ∧ (x' = x → ¬y' = y)"
},
{
"line": "refine_lift\n suffices ∀ x y : ZMod n, ∃ x' y' : ZMod n, x' + y' = x + y ∧ (x' = x → ¬y' = y) by simpa [UniqueAdd];\n ?_",
"before_state": "case mk\nn : ℕ\nn2 : 2 ≤ n\nthis✝ : Fintype (ZMod n)\nthis : Nontrivial (ZMod n)\nh : ∀ {A B : Finset (ZMod n)}, A.Nonempty → B.Nonempty → ∃ a0 ∈ A, ∃ b0 ∈ B, UniqueAdd A B a0 b0\n⊢ ¬∃ a0 ∈ Finset.univ, ∃ b0 ∈ Finset.univ, UniqueAdd Finset.univ Finset.univ a0 b0",
"after_state": "case mk\nn : ℕ\nn2 : 2 ≤ n\nthis✝ : Fintype (ZMod n)\nthis : Nontrivial (ZMod n)\nh : ∀ {A B : Finset (ZMod n)}, A.Nonempty → B.Nonempty → ∃ a0 ∈ A, ∃ b0 ∈ B, UniqueAdd A B a0 b0\n⊢ ∀ (x y : ZMod n), ∃ x' y', x' + y' = x + y ∧ (x' = x → ¬y' = y)"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (suffices ∀ x y : ZMod n, ∃ x' y' : ZMod n, x' + y' = x + y ∧ (x' = x → ¬y' = y) by simpa [UniqueAdd];\n ?_);\n rotate_right)",
"before_state": "case mk\nn : ℕ\nn2 : 2 ≤ n\nthis✝ : Fintype (ZMod n)\nthis : Nontrivial (ZMod n)\nh : ∀ {A B : Finset (ZMod n)}, A.Nonempty → B.Nonempty → ∃ a0 ∈ A, ∃ b0 ∈ B, UniqueAdd A B a0 b0\n⊢ ¬∃ a0 ∈ Finset.univ, ∃ b0 ∈ Finset.univ, UniqueAdd Finset.univ Finset.univ a0 b0",
"after_state": "case mk\nn : ℕ\nn2 : 2 ≤ n\nthis✝ : Fintype (ZMod n)\nthis : Nontrivial (ZMod n)\nh : ∀ {A B : Finset (ZMod n)}, A.Nonempty → B.Nonempty → ∃ a0 ∈ A, ∃ b0 ∈ B, UniqueAdd A B a0 b0\n⊢ ∀ (x y : ZMod n), ∃ x' y', x' + y' = x + y ∧ (x' = x → ¬y' = y)"
},
{
"line": "refine\n no_implicit_lambda%\n (suffices ∀ x y : ZMod n, ∃ x' y' : ZMod n, x' + y' = x + y ∧ (x' = x → ¬y' = y) by simpa [UniqueAdd];\n ?_)",
"before_state": "case mk\nn : ℕ\nn2 : 2 ≤ n\nthis✝ : Fintype (ZMod n)\nthis : Nontrivial (ZMod n)\nh : ∀ {A B : Finset (ZMod n)}, A.Nonempty → B.Nonempty → ∃ a0 ∈ A, ∃ b0 ∈ B, UniqueAdd A B a0 b0\n⊢ ¬∃ a0 ∈ Finset.univ, ∃ b0 ∈ Finset.univ, UniqueAdd Finset.univ Finset.univ a0 b0",
"after_state": "case mk\nn : ℕ\nn2 : 2 ≤ n\nthis✝ : Fintype (ZMod n)\nthis : Nontrivial (ZMod n)\nh : ∀ {A B : Finset (ZMod n)}, A.Nonempty → B.Nonempty → ∃ a0 ∈ A, ∃ b0 ∈ B, UniqueAdd A B a0 b0\n⊢ ∀ (x y : ZMod n), ∃ x' y', x' + y' = x + y ∧ (x' = x → ¬y' = y)"
},
{
"line": "simpa [UniqueAdd]",
"before_state": "n : ℕ\nn2 : 2 ≤ n\nthis✝¹ : Fintype (ZMod n)\nthis✝ : Nontrivial (ZMod n)\nh : ∀ {A B : Finset (ZMod n)}, A.Nonempty → B.Nonempty → ∃ a0 ∈ A, ∃ b0 ∈ B, UniqueAdd A B a0 b0\nthis : ∀ (x y : ZMod n), ∃ x' y', x' + y' = x + y ∧ (x' = x → ¬y' = y)\n⊢ ¬∃ a0 ∈ Finset.univ, ∃ b0 ∈ Finset.univ, UniqueAdd Finset.univ Finset.univ a0 b0",
"after_state": "No Goals!"
},
{
"line": "rotate_right",
"before_state": "case mk\nn : ℕ\nn2 : 2 ≤ n\nthis✝ : Fintype (ZMod n)\nthis : Nontrivial (ZMod n)\nh : ∀ {A B : Finset (ZMod n)}, A.Nonempty → B.Nonempty → ∃ a0 ∈ A, ∃ b0 ∈ B, UniqueAdd A B a0 b0\n⊢ ∀ (x y : ZMod n), ∃ x' y', x' + y' = x + y ∧ (x' = x → ¬y' = y)",
"after_state": "case mk\nn : ℕ\nn2 : 2 ≤ n\nthis✝ : Fintype (ZMod n)\nthis : Nontrivial (ZMod n)\nh : ∀ {A B : Finset (ZMod n)}, A.Nonempty → B.Nonempty → ∃ a0 ∈ A, ∃ b0 ∈ B, UniqueAdd A B a0 b0\n⊢ ∀ (x y : ZMod n), ∃ x' y', x' + y' = x + y ∧ (x' = x → ¬y' = y)"
},
{
"line": "exact fun x y => ⟨x - 1, y + 1, sub_add_add_cancel _ _ _, by simp⟩",
"before_state": "case mk\nn : ℕ\nn2 : 2 ≤ n\nthis✝ : Fintype (ZMod n)\nthis : Nontrivial (ZMod n)\nh : ∀ {A B : Finset (ZMod n)}, A.Nonempty → B.Nonempty → ∃ a0 ∈ A, ∃ b0 ∈ B, UniqueAdd A B a0 b0\n⊢ ∀ (x y : ZMod n), ∃ x' y', x' + y' = x + y ∧ (x' = x → ¬y' = y)",
"after_state": "No Goals!"
},
{
"line": "simp",
"before_state": "n : ℕ\nn2 : 2 ≤ n\nthis✝ : Fintype (ZMod n)\nthis : Nontrivial (ZMod n)\nh : ∀ {A B : Finset (ZMod n)}, A.Nonempty → B.Nonempty → ∃ a0 ∈ A, ∃ b0 ∈ B, UniqueAdd A B a0 b0\nx y : ZMod n\n⊢ x - 1 = x → ¬y + 1 = y",
"after_state": "No Goals!"
}
] |
theorem infinite_of_charZero (R A : Type*) [CommRing R] [Ring A] [Algebra R A]
[CharZero A] : { x : A | IsAlgebraic R x }.Infinite := by
letI := MulActionWithZero.nontrivial R A
exact infinite_of_injective_forall_mem Nat.cast_injective isAlgebraic_nat
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/AlgebraicCard.lean
|
{
"open": [
"Cardinal Polynomial Set",
"Cardinal Polynomial"
],
"variables": []
}
|
[
{
"line": "letI := MulActionWithZero.nontrivial R A",
"before_state": "R : Type u_1\nA : Type u_2\ninst✝³ : CommRing R\ninst✝² : Ring A\ninst✝¹ : Algebra R A\ninst✝ : CharZero A\n⊢ {x | IsAlgebraic R x}.Infinite",
"after_state": "R : Type u_1\nA : Type u_2\ninst✝³ : CommRing R\ninst✝² : Ring A\ninst✝¹ : Algebra R A\ninst✝ : CharZero A\nthis : Nontrivial R := MulActionWithZero.nontrivial R A\n⊢ {x | IsAlgebraic R x}.Infinite"
},
{
"line": "refine_lift\n letI := MulActionWithZero.nontrivial R A;\n ?_",
"before_state": "R : Type u_1\nA : Type u_2\ninst✝³ : CommRing R\ninst✝² : Ring A\ninst✝¹ : Algebra R A\ninst✝ : CharZero A\n⊢ {x | IsAlgebraic R x}.Infinite",
"after_state": "R : Type u_1\nA : Type u_2\ninst✝³ : CommRing R\ninst✝² : Ring A\ninst✝¹ : Algebra R A\ninst✝ : CharZero A\nthis : Nontrivial R := MulActionWithZero.nontrivial R A\n⊢ {x | IsAlgebraic R x}.Infinite"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n letI := MulActionWithZero.nontrivial R A;\n ?_;\n rotate_right)",
"before_state": "R : Type u_1\nA : Type u_2\ninst✝³ : CommRing R\ninst✝² : Ring A\ninst✝¹ : Algebra R A\ninst✝ : CharZero A\n⊢ {x | IsAlgebraic R x}.Infinite",
"after_state": "R : Type u_1\nA : Type u_2\ninst✝³ : CommRing R\ninst✝² : Ring A\ninst✝¹ : Algebra R A\ninst✝ : CharZero A\nthis : Nontrivial R := MulActionWithZero.nontrivial R A\n⊢ {x | IsAlgebraic R x}.Infinite"
},
{
"line": "refine\n no_implicit_lambda%\n letI := MulActionWithZero.nontrivial R A;\n ?_",
"before_state": "R : Type u_1\nA : Type u_2\ninst✝³ : CommRing R\ninst✝² : Ring A\ninst✝¹ : Algebra R A\ninst✝ : CharZero A\n⊢ {x | IsAlgebraic R x}.Infinite",
"after_state": "R : Type u_1\nA : Type u_2\ninst✝³ : CommRing R\ninst✝² : Ring A\ninst✝¹ : Algebra R A\ninst✝ : CharZero A\nthis : Nontrivial R := MulActionWithZero.nontrivial R A\n⊢ {x | IsAlgebraic R x}.Infinite"
},
{
"line": "rotate_right",
"before_state": "R : Type u_1\nA : Type u_2\ninst✝³ : CommRing R\ninst✝² : Ring A\ninst✝¹ : Algebra R A\ninst✝ : CharZero A\nthis : Nontrivial R := MulActionWithZero.nontrivial R A\n⊢ {x | IsAlgebraic R x}.Infinite",
"after_state": "R : Type u_1\nA : Type u_2\ninst✝³ : CommRing R\ninst✝² : Ring A\ninst✝¹ : Algebra R A\ninst✝ : CharZero A\nthis : Nontrivial R := MulActionWithZero.nontrivial R A\n⊢ {x | IsAlgebraic R x}.Infinite"
},
{
"line": "exact infinite_of_injective_forall_mem Nat.cast_injective isAlgebraic_nat",
"before_state": "R : Type u_1\nA : Type u_2\ninst✝³ : CommRing R\ninst✝² : Ring A\ninst✝¹ : Algebra R A\ninst✝ : CharZero A\nthis : Nontrivial R := MulActionWithZero.nontrivial R A\n⊢ {x | IsAlgebraic R x}.Infinite",
"after_state": "No Goals!"
}
] |
theorem cardinalMk_le_max : #{ x : A // IsAlgebraic R x } ≤ max #R ℵ₀ := by
rw [← lift_id #_]
rw [← lift_id #R]
exact cardinalMk_lift_le_max R A
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/AlgebraicCard.lean
|
{
"open": [
"Cardinal Polynomial Set",
"Cardinal Polynomial"
],
"variables": [
"(R : Type u) (A : Type v) [CommRing R] [CommRing A] [IsDomain A] [Algebra R A]",
"[Countable R]",
"(R A : Type u) [CommRing R] [CommRing A] [IsDomain A] [Algebra R A]"
]
}
|
[
{
"line": "rw [← lift_id #_]",
"before_state": "R A : Type u\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : Algebra R A\n⊢ #{ x // IsAlgebraic R x } ≤ max #R ℵ₀",
"after_state": "R A : Type u\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : Algebra R A\n⊢ lift.{u, u} #{ x // IsAlgebraic R x } ≤ max #R ℵ₀"
},
{
"line": "rewrite [← lift_id #_]",
"before_state": "R A : Type u\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : Algebra R A\n⊢ #{ x // IsAlgebraic R x } ≤ max #R ℵ₀",
"after_state": "R A : Type u\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : Algebra R A\n⊢ lift.{u, u} #{ x // IsAlgebraic R x } ≤ max #R ℵ₀"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "R A : Type u\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : Algebra R A\n⊢ lift.{u, u} #{ x // IsAlgebraic R x } ≤ max #R ℵ₀",
"after_state": "R A : Type u\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : Algebra R A\n⊢ lift.{u, u} #{ x // IsAlgebraic R x } ≤ max #R ℵ₀"
},
{
"line": "try (with_reducible rfl)",
"before_state": "R A : Type u\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : Algebra R A\n⊢ lift.{u, u} #{ x // IsAlgebraic R x } ≤ max #R ℵ₀",
"after_state": "R A : Type u\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : Algebra R A\n⊢ lift.{u, u} #{ x // IsAlgebraic R x } ≤ max #R ℵ₀"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "R A : Type u\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : Algebra R A\n⊢ lift.{u, u} #{ x // IsAlgebraic R x } ≤ max #R ℵ₀",
"after_state": "R A : Type u\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : Algebra R A\n⊢ lift.{u, u} #{ x // IsAlgebraic R x } ≤ max #R ℵ₀"
},
{
"line": "with_reducible rfl",
"before_state": "R A : Type u\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : Algebra R A\n⊢ lift.{u, u} #{ x // IsAlgebraic R x } ≤ max #R ℵ₀",
"after_state": "R A : Type u\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : Algebra R A\n⊢ lift.{u, u} #{ x // IsAlgebraic R x } ≤ max #R ℵ₀"
},
{
"line": "rfl",
"before_state": "R A : Type u\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : Algebra R A\n⊢ lift.{u, u} #{ x // IsAlgebraic R x } ≤ max #R ℵ₀",
"after_state": "R A : Type u\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : Algebra R A\n⊢ lift.{u, u} #{ x // IsAlgebraic R x } ≤ max #R ℵ₀"
},
{
"line": "apply_rfl",
"before_state": "R A : Type u\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : Algebra R A\n⊢ lift.{u, u} #{ x // IsAlgebraic R x } ≤ max #R ℵ₀",
"after_state": "R A : Type u\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : Algebra R A\n⊢ lift.{u, u} #{ x // IsAlgebraic R x } ≤ max #R ℵ₀"
},
{
"line": "skip",
"before_state": "R A : Type u\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : Algebra R A\n⊢ lift.{u, u} #{ x // IsAlgebraic R x } ≤ max #R ℵ₀",
"after_state": "R A : Type u\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : Algebra R A\n⊢ lift.{u, u} #{ x // IsAlgebraic R x } ≤ max #R ℵ₀"
},
{
"line": "rw [← lift_id #R]",
"before_state": "R A : Type u\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : Algebra R A\n⊢ lift.{u, u} #{ x // IsAlgebraic R x } ≤ max #R ℵ₀",
"after_state": "R A : Type u\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : Algebra R A\n⊢ lift.{u, u} #{ x // IsAlgebraic R x } ≤ max (lift.{u, u} #R) ℵ₀"
},
{
"line": "rewrite [← lift_id #R]",
"before_state": "R A : Type u\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : Algebra R A\n⊢ lift.{u, u} #{ x // IsAlgebraic R x } ≤ max #R ℵ₀",
"after_state": "R A : Type u\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : Algebra R A\n⊢ lift.{u, u} #{ x // IsAlgebraic R x } ≤ max (lift.{u, u} #R) ℵ₀"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "R A : Type u\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : Algebra R A\n⊢ lift.{u, u} #{ x // IsAlgebraic R x } ≤ max (lift.{u, u} #R) ℵ₀",
"after_state": "R A : Type u\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : Algebra R A\n⊢ lift.{u, u} #{ x // IsAlgebraic R x } ≤ max (lift.{u, u} #R) ℵ₀"
},
{
"line": "try (with_reducible rfl)",
"before_state": "R A : Type u\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : Algebra R A\n⊢ lift.{u, u} #{ x // IsAlgebraic R x } ≤ max (lift.{u, u} #R) ℵ₀",
"after_state": "R A : Type u\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : Algebra R A\n⊢ lift.{u, u} #{ x // IsAlgebraic R x } ≤ max (lift.{u, u} #R) ℵ₀"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "R A : Type u\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : Algebra R A\n⊢ lift.{u, u} #{ x // IsAlgebraic R x } ≤ max (lift.{u, u} #R) ℵ₀",
"after_state": "R A : Type u\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : Algebra R A\n⊢ lift.{u, u} #{ x // IsAlgebraic R x } ≤ max (lift.{u, u} #R) ℵ₀"
},
{
"line": "with_reducible rfl",
"before_state": "R A : Type u\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : Algebra R A\n⊢ lift.{u, u} #{ x // IsAlgebraic R x } ≤ max (lift.{u, u} #R) ℵ₀",
"after_state": "R A : Type u\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : Algebra R A\n⊢ lift.{u, u} #{ x // IsAlgebraic R x } ≤ max (lift.{u, u} #R) ℵ₀"
},
{
"line": "rfl",
"before_state": "R A : Type u\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : Algebra R A\n⊢ lift.{u, u} #{ x // IsAlgebraic R x } ≤ max (lift.{u, u} #R) ℵ₀",
"after_state": "R A : Type u\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : Algebra R A\n⊢ lift.{u, u} #{ x // IsAlgebraic R x } ≤ max (lift.{u, u} #R) ℵ₀"
},
{
"line": "apply_rfl",
"before_state": "R A : Type u\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : Algebra R A\n⊢ lift.{u, u} #{ x // IsAlgebraic R x } ≤ max (lift.{u, u} #R) ℵ₀",
"after_state": "R A : Type u\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : Algebra R A\n⊢ lift.{u, u} #{ x // IsAlgebraic R x } ≤ max (lift.{u, u} #R) ℵ₀"
},
{
"line": "skip",
"before_state": "R A : Type u\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : Algebra R A\n⊢ lift.{u, u} #{ x // IsAlgebraic R x } ≤ max (lift.{u, u} #R) ℵ₀",
"after_state": "R A : Type u\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : Algebra R A\n⊢ lift.{u, u} #{ x // IsAlgebraic R x } ≤ max (lift.{u, u} #R) ℵ₀"
},
{
"line": "exact cardinalMk_lift_le_max R A",
"before_state": "R A : Type u\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : Algebra R A\n⊢ lift.{u, u} #{ x // IsAlgebraic R x } ≤ max (lift.{u, u} #R) ℵ₀",
"after_state": "No Goals!"
}
] |
lemma injective_of_surjective_of_injective_of_injective (hi₁ : Function.Surjective i₁)
(hi₂ : Function.Injective i₂) (hi₄ : Function.Injective i₄) : Function.Injective i₃ := by
rw [injective_iff_map_eq_zero]
intro m hm
obtain ⟨x, rfl⟩ := (hf₂ m).mp <| by
suffices h : i₄ (f₃ m) = 0 by rwa [map_eq_zero_iff _ hi₄] at h
simp [← show g₃ (i₃ m) = i₄ (f₃ m) by simpa using DFunLike.congr_fun hc₃ m, hm]
obtain ⟨y, hy⟩ := (hg₁ _).mp <| by
rwa [show g₂ (i₂ x) = i₃ (f₂ x) by simpa using DFunLike.congr_fun hc₂ x]
obtain ⟨a, rfl⟩ := hi₁ y
rw [show g₁ (i₁ a) = i₂ (f₁ a) by simpa using DFunLike.congr_fun hc₁ a] at hy
apply hi₂ at hy
subst hy
rw [hf₁.apply_apply_eq_zero]
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/FiveLemma.lean
|
{
"open": [],
"variables": [
"{M₁ M₂ M₃ M₄ M₅ N₁ N₂ N₃ N₄ N₅ : Type*}",
"[AddGroup M₁] [AddGroup M₂] [AddGroup M₃] [AddGroup M₄] [AddGroup M₅]",
"[AddGroup N₁] [AddGroup N₂] [AddGroup N₃] [AddGroup N₄] [AddGroup N₅]",
"(f₁ : M₁ →+ M₂) (f₂ : M₂ →+ M₃) (f₃ : M₃ →+ M₄) (f₄ : M₄ →+ M₅)",
"(g₁ : N₁ →+ N₂) (g₂ : N₂ →+ N₃) (g₃ : N₃ →+ N₄) (g₄ : N₄ →+ N₅)",
"(i₁ : M₁ →+ N₁) (i₂ : M₂ →+ N₂) (i₃ : M₃ →+ N₃) (i₄ : M₄ →+ N₄)",
"(hc₁ : g₁.comp i₁ = i₂.comp f₁) (hc₂ : g₂.comp i₂ = i₃.comp f₂)",
"(hf₁ : Function.Exact f₁ f₂) (hf₂ : Function.Exact f₂ f₃) (hf₃ : Function.Exact f₃ f₄)",
"(hg₁ : Function.Exact g₁ g₂) (hg₂ : Function.Exact g₂ g₃) (hg₃ : Function.Exact g₃ g₄)"
]
}
|
[
{
"line": "rw [injective_iff_map_eq_zero]",
"before_state": "M₁ : Type u_1\nM₂ : Type u_2\nM₃ : Type u_3\nM₄ : Type u_4\nN₁ : Type u_6\nN₂ : Type u_7\nN₃ : Type u_8\nN₄ : Type u_9\ninst✝⁷ : AddGroup M₁\ninst✝⁶ : AddGroup M₂\ninst✝⁵ : AddGroup M₃\ninst✝⁴ : AddGroup M₄\ninst✝³ : AddGroup N₁\ninst✝² : AddGroup N₂\ninst✝¹ : AddGroup N₃\ninst✝ : AddGroup N₄\ni₁ : M₁ →+ N₁\ni₂ : M₂ →+ N₂\ni₃ : M₃ →+ N₃\ni₄ : M₄ →+ N₄\nhi₁ : Function.Surjective ⇑i₁\nhi₂ : Function.Injective ⇑i₂\nhi₄ : Function.Injective ⇑i₄\n⊢ Function.Injective ⇑i₃",
"after_state": "M₁ : Type u_1\nM₂ : Type u_2\nM₃ : Type u_3\nM₄ : Type u_4\nN₁ : Type u_6\nN₂ : Type u_7\nN₃ : Type u_8\nN₄ : Type u_9\ninst✝⁷ : AddGroup M₁\ninst✝⁶ : AddGroup M₂\ninst✝⁵ : AddGroup M₃\ninst✝⁴ : AddGroup M₄\ninst✝³ : AddGroup N₁\ninst✝² : AddGroup N₂\ninst✝¹ : AddGroup N₃\ninst✝ : AddGroup N₄\ni₁ : M₁ →+ N₁\ni₂ : M₂ →+ N₂\ni₃ : M₃ →+ N₃\ni₄ : M₄ →+ N₄\nhi₁ : Function.Surjective ⇑i₁\nhi₂ : Function.Injective ⇑i₂\nhi₄ : Function.Injective ⇑i₄\n⊢ ∀ (a : M₃), i₃ a = 0 → a = 0"
},
{
"line": "rewrite [injective_iff_map_eq_zero]",
"before_state": "M₁ : Type u_1\nM₂ : Type u_2\nM₃ : Type u_3\nM₄ : Type u_4\nN₁ : Type u_6\nN₂ : Type u_7\nN₃ : Type u_8\nN₄ : Type u_9\ninst✝⁷ : AddGroup M₁\ninst✝⁶ : AddGroup M₂\ninst✝⁵ : AddGroup M₃\ninst✝⁴ : AddGroup M₄\ninst✝³ : AddGroup N₁\ninst✝² : AddGroup N₂\ninst✝¹ : AddGroup N₃\ninst✝ : AddGroup N₄\ni₁ : M₁ →+ N₁\ni₂ : M₂ →+ N₂\ni₃ : M₃ →+ N₃\ni₄ : M₄ →+ N₄\nhi₁ : Function.Surjective ⇑i₁\nhi₂ : Function.Injective ⇑i₂\nhi₄ : Function.Injective ⇑i₄\n⊢ Function.Injective ⇑i₃",
"after_state": "M₁ : Type u_1\nM₂ : Type u_2\nM₃ : Type u_3\nM₄ : Type u_4\nN₁ : Type u_6\nN₂ : Type u_7\nN₃ : Type u_8\nN₄ : Type u_9\ninst✝⁷ : AddGroup M₁\ninst✝⁶ : AddGroup M₂\ninst✝⁵ : AddGroup M₃\ninst✝⁴ : AddGroup M₄\ninst✝³ : AddGroup N₁\ninst✝² : AddGroup N₂\ninst✝¹ : AddGroup N₃\ninst✝ : AddGroup N₄\ni₁ : M₁ →+ N₁\ni₂ : M₂ →+ N₂\ni₃ : M₃ →+ N₃\ni₄ : M₄ →+ N₄\nhi₁ : Function.Surjective ⇑i₁\nhi₂ : Function.Injective ⇑i₂\nhi₄ : Function.Injective ⇑i₄\n⊢ ∀ (a : M₃), i₃ a = 0 → a = 0"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "M₁ : Type u_1\nM₂ : Type u_2\nM₃ : Type u_3\nM₄ : Type u_4\nN₁ : Type u_6\nN₂ : Type u_7\nN₃ : Type u_8\nN₄ : Type u_9\ninst✝⁷ : AddGroup M₁\ninst✝⁶ : AddGroup M₂\ninst✝⁵ : AddGroup M₃\ninst✝⁴ : AddGroup M₄\ninst✝³ : AddGroup N₁\ninst✝² : AddGroup N₂\ninst✝¹ : AddGroup N₃\ninst✝ : AddGroup N₄\ni₁ : M₁ →+ N₁\ni₂ : M₂ →+ N₂\ni₃ : M₃ →+ N₃\ni₄ : M₄ →+ N₄\nhi₁ : Function.Surjective ⇑i₁\nhi₂ : Function.Injective ⇑i₂\nhi₄ : Function.Injective ⇑i₄\n⊢ ∀ (a : M₃), i₃ a = 0 → a = 0",
"after_state": "M₁ : Type u_1\nM₂ : Type u_2\nM₃ : Type u_3\nM₄ : Type u_4\nN₁ : Type u_6\nN₂ : Type u_7\nN₃ : Type u_8\nN₄ : Type u_9\ninst✝⁷ : AddGroup M₁\ninst✝⁶ : AddGroup M₂\ninst✝⁵ : AddGroup M₃\ninst✝⁴ : AddGroup M₄\ninst✝³ : AddGroup N₁\ninst✝² : AddGroup N₂\ninst✝¹ : AddGroup N₃\ninst✝ : AddGroup N₄\ni₁ : M₁ →+ N₁\ni₂ : M₂ →+ N₂\ni₃ : M₃ →+ N₃\ni₄ : M₄ →+ N₄\nhi₁ : Function.Surjective ⇑i₁\nhi₂ : Function.Injective ⇑i₂\nhi₄ : Function.Injective ⇑i₄\n⊢ ∀ (a : M₃), i₃ a = 0 → a = 0"
},
{
"line": "try (with_reducible rfl)",
"before_state": "M₁ : Type u_1\nM₂ : Type u_2\nM₃ : Type u_3\nM₄ : Type u_4\nN₁ : Type u_6\nN₂ : Type u_7\nN₃ : Type u_8\nN₄ : Type u_9\ninst✝⁷ : AddGroup M₁\ninst✝⁶ : AddGroup M₂\ninst✝⁵ : AddGroup M₃\ninst✝⁴ : AddGroup M₄\ninst✝³ : AddGroup N₁\ninst✝² : AddGroup N₂\ninst✝¹ : AddGroup N₃\ninst✝ : AddGroup N₄\ni₁ : M₁ →+ N₁\ni₂ : M₂ →+ N₂\ni₃ : M₃ →+ N₃\ni₄ : M₄ →+ N₄\nhi₁ : Function.Surjective ⇑i₁\nhi₂ : Function.Injective ⇑i₂\nhi₄ : Function.Injective ⇑i₄\n⊢ ∀ (a : M₃), i₃ a = 0 → a = 0",
"after_state": "M₁ : Type u_1\nM₂ : Type u_2\nM₃ : Type u_3\nM₄ : Type u_4\nN₁ : Type u_6\nN₂ : Type u_7\nN₃ : Type u_8\nN₄ : Type u_9\ninst✝⁷ : AddGroup M₁\ninst✝⁶ : AddGroup M₂\ninst✝⁵ : AddGroup M₃\ninst✝⁴ : AddGroup M₄\ninst✝³ : AddGroup N₁\ninst✝² : AddGroup N₂\ninst✝¹ : AddGroup N₃\ninst✝ : AddGroup N₄\ni₁ : M₁ →+ N₁\ni₂ : M₂ →+ N₂\ni₃ : M₃ →+ N₃\ni₄ : M₄ →+ N₄\nhi₁ : Function.Surjective ⇑i₁\nhi₂ : Function.Injective ⇑i₂\nhi₄ : Function.Injective ⇑i₄\n⊢ ∀ (a : M₃), i₃ a = 0 → a = 0"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "M₁ : Type u_1\nM₂ : Type u_2\nM₃ : Type u_3\nM₄ : Type u_4\nN₁ : Type u_6\nN₂ : Type u_7\nN₃ : Type u_8\nN₄ : Type u_9\ninst✝⁷ : AddGroup M₁\ninst✝⁶ : AddGroup M₂\ninst✝⁵ : AddGroup M₃\ninst✝⁴ : AddGroup M₄\ninst✝³ : AddGroup N₁\ninst✝² : AddGroup N₂\ninst✝¹ : AddGroup N₃\ninst✝ : AddGroup N₄\ni₁ : M₁ →+ N₁\ni₂ : M₂ →+ N₂\ni₃ : M₃ →+ N₃\ni₄ : M₄ →+ N₄\nhi₁ : Function.Surjective ⇑i₁\nhi₂ : Function.Injective ⇑i₂\nhi₄ : Function.Injective ⇑i₄\n⊢ ∀ (a : M₃), i₃ a = 0 → a = 0",
"after_state": "M₁ : Type u_1\nM₂ : Type u_2\nM₃ : Type u_3\nM₄ : Type u_4\nN₁ : Type u_6\nN₂ : Type u_7\nN₃ : Type u_8\nN₄ : Type u_9\ninst✝⁷ : AddGroup M₁\ninst✝⁶ : AddGroup M₂\ninst✝⁵ : AddGroup M₃\ninst✝⁴ : AddGroup M₄\ninst✝³ : AddGroup N₁\ninst✝² : AddGroup N₂\ninst✝¹ : AddGroup N₃\ninst✝ : AddGroup N₄\ni₁ : M₁ →+ N₁\ni₂ : M₂ →+ N₂\ni₃ : M₃ →+ N₃\ni₄ : M₄ →+ N₄\nhi₁ : Function.Surjective ⇑i₁\nhi₂ : Function.Injective ⇑i₂\nhi₄ : Function.Injective ⇑i₄\n⊢ ∀ (a : M₃), i₃ a = 0 → a = 0"
},
{
"line": "with_reducible rfl",
"before_state": "M₁ : Type u_1\nM₂ : Type u_2\nM₃ : Type u_3\nM₄ : Type u_4\nN₁ : Type u_6\nN₂ : Type u_7\nN₃ : Type u_8\nN₄ : Type u_9\ninst✝⁷ : AddGroup M₁\ninst✝⁶ : AddGroup M₂\ninst✝⁵ : AddGroup M₃\ninst✝⁴ : AddGroup M₄\ninst✝³ : AddGroup N₁\ninst✝² : AddGroup N₂\ninst✝¹ : AddGroup N₃\ninst✝ : AddGroup N₄\ni₁ : M₁ →+ N₁\ni₂ : M₂ →+ N₂\ni₃ : M₃ →+ N₃\ni₄ : M₄ →+ N₄\nhi₁ : Function.Surjective ⇑i₁\nhi₂ : Function.Injective ⇑i₂\nhi₄ : Function.Injective ⇑i₄\n⊢ ∀ (a : M₃), i₃ a = 0 → a = 0",
"after_state": "M₁ : Type u_1\nM₂ : Type u_2\nM₃ : Type u_3\nM₄ : Type u_4\nN₁ : Type u_6\nN₂ : Type u_7\nN₃ : Type u_8\nN₄ : Type u_9\ninst✝⁷ : AddGroup M₁\ninst✝⁶ : AddGroup M₂\ninst✝⁵ : AddGroup M₃\ninst✝⁴ : AddGroup M₄\ninst✝³ : AddGroup N₁\ninst✝² : AddGroup N₂\ninst✝¹ : AddGroup N₃\ninst✝ : AddGroup N₄\ni₁ : M₁ →+ N₁\ni₂ : M₂ →+ N₂\ni₃ : M₃ →+ N₃\ni₄ : M₄ →+ N₄\nhi₁ : Function.Surjective ⇑i₁\nhi₂ : Function.Injective ⇑i₂\nhi₄ : Function.Injective ⇑i₄\n⊢ ∀ (a : M₃), i₃ a = 0 → a = 0"
},
{
"line": "rfl",
"before_state": "M₁ : Type u_1\nM₂ : Type u_2\nM₃ : Type u_3\nM₄ : Type u_4\nN₁ : Type u_6\nN₂ : Type u_7\nN₃ : Type u_8\nN₄ : Type u_9\ninst✝⁷ : AddGroup M₁\ninst✝⁶ : AddGroup M₂\ninst✝⁵ : AddGroup M₃\ninst✝⁴ : AddGroup M₄\ninst✝³ : AddGroup N₁\ninst✝² : AddGroup N₂\ninst✝¹ : AddGroup N₃\ninst✝ : AddGroup N₄\ni₁ : M₁ →+ N₁\ni₂ : M₂ →+ N₂\ni₃ : M₃ →+ N₃\ni₄ : M₄ →+ N₄\nhi₁ : Function.Surjective ⇑i₁\nhi₂ : Function.Injective ⇑i₂\nhi₄ : Function.Injective ⇑i₄\n⊢ ∀ (a : M₃), i₃ a = 0 → a = 0",
"after_state": "M₁ : Type u_1\nM₂ : Type u_2\nM₃ : Type u_3\nM₄ : Type u_4\nN₁ : Type u_6\nN₂ : Type u_7\nN₃ : Type u_8\nN₄ : Type u_9\ninst✝⁷ : AddGroup M₁\ninst✝⁶ : AddGroup M₂\ninst✝⁵ : AddGroup M₃\ninst✝⁴ : AddGroup M₄\ninst✝³ : AddGroup N₁\ninst✝² : AddGroup N₂\ninst✝¹ : AddGroup N₃\ninst✝ : AddGroup N₄\ni₁ : M₁ →+ N₁\ni₂ : M₂ →+ N₂\ni₃ : M₃ →+ N₃\ni₄ : M₄ →+ N₄\nhi₁ : Function.Surjective ⇑i₁\nhi₂ : Function.Injective ⇑i₂\nhi₄ : Function.Injective ⇑i₄\n⊢ ∀ (a : M₃), i₃ a = 0 → a = 0"
},
{
"line": "apply_rfl",
"before_state": "M₁ : Type u_1\nM₂ : Type u_2\nM₃ : Type u_3\nM₄ : Type u_4\nN₁ : Type u_6\nN₂ : Type u_7\nN₃ : Type u_8\nN₄ : Type u_9\ninst✝⁷ : AddGroup M₁\ninst✝⁶ : AddGroup M₂\ninst✝⁵ : AddGroup M₃\ninst✝⁴ : AddGroup M₄\ninst✝³ : AddGroup N₁\ninst✝² : AddGroup N₂\ninst✝¹ : AddGroup N₃\ninst✝ : AddGroup N₄\ni₁ : M₁ →+ N₁\ni₂ : M₂ →+ N₂\ni₃ : M₃ →+ N₃\ni₄ : M₄ →+ N₄\nhi₁ : Function.Surjective ⇑i₁\nhi₂ : Function.Injective ⇑i₂\nhi₄ : Function.Injective ⇑i₄\n⊢ ∀ (a : M₃), i₃ a = 0 → a = 0",
"after_state": "M₁ : Type u_1\nM₂ : Type u_2\nM₃ : Type u_3\nM₄ : Type u_4\nN₁ : Type u_6\nN₂ : Type u_7\nN₃ : Type u_8\nN₄ : Type u_9\ninst✝⁷ : AddGroup M₁\ninst✝⁶ : AddGroup M₂\ninst✝⁵ : AddGroup M₃\ninst✝⁴ : AddGroup M₄\ninst✝³ : AddGroup N₁\ninst✝² : AddGroup N₂\ninst✝¹ : AddGroup N₃\ninst✝ : AddGroup N₄\ni₁ : M₁ →+ N₁\ni₂ : M₂ →+ N₂\ni₃ : M₃ →+ N₃\ni₄ : M₄ →+ N₄\nhi₁ : Function.Surjective ⇑i₁\nhi₂ : Function.Injective ⇑i₂\nhi₄ : Function.Injective ⇑i₄\n⊢ ∀ (a : M₃), i₃ a = 0 → a = 0"
},
{
"line": "skip",
"before_state": "M₁ : Type u_1\nM₂ : Type u_2\nM₃ : Type u_3\nM₄ : Type u_4\nN₁ : Type u_6\nN₂ : Type u_7\nN₃ : Type u_8\nN₄ : Type u_9\ninst✝⁷ : AddGroup M₁\ninst✝⁶ : AddGroup M₂\ninst✝⁵ : AddGroup M₃\ninst✝⁴ : AddGroup M₄\ninst✝³ : AddGroup N₁\ninst✝² : AddGroup N₂\ninst✝¹ : AddGroup N₃\ninst✝ : AddGroup N₄\ni₁ : M₁ →+ N₁\ni₂ : M₂ →+ N₂\ni₃ : M₃ →+ N₃\ni₄ : M₄ →+ N₄\nhi₁ : Function.Surjective ⇑i₁\nhi₂ : Function.Injective ⇑i₂\nhi₄ : Function.Injective ⇑i₄\n⊢ ∀ (a : M₃), i₃ a = 0 → a = 0",
"after_state": "M₁ : Type u_1\nM₂ : Type u_2\nM₃ : Type u_3\nM₄ : Type u_4\nN₁ : Type u_6\nN₂ : Type u_7\nN₃ : Type u_8\nN₄ : Type u_9\ninst✝⁷ : AddGroup M₁\ninst✝⁶ : AddGroup M₂\ninst✝⁵ : AddGroup M₃\ninst✝⁴ : AddGroup M₄\ninst✝³ : AddGroup N₁\ninst✝² : AddGroup N₂\ninst✝¹ : AddGroup N₃\ninst✝ : AddGroup N₄\ni₁ : M₁ →+ N₁\ni₂ : M₂ →+ N₂\ni₃ : M₃ →+ N₃\ni₄ : M₄ →+ N₄\nhi₁ : Function.Surjective ⇑i₁\nhi₂ : Function.Injective ⇑i₂\nhi₄ : Function.Injective ⇑i₄\n⊢ ∀ (a : M₃), i₃ a = 0 → a = 0"
},
{
"line": "intro m hm",
"before_state": "M₁ : Type u_1\nM₂ : Type u_2\nM₃ : Type u_3\nM₄ : Type u_4\nN₁ : Type u_6\nN₂ : Type u_7\nN₃ : Type u_8\nN₄ : Type u_9\ninst✝⁷ : AddGroup M₁\ninst✝⁶ : AddGroup M₂\ninst✝⁵ : AddGroup M₃\ninst✝⁴ : AddGroup M₄\ninst✝³ : AddGroup N₁\ninst✝² : AddGroup N₂\ninst✝¹ : AddGroup N₃\ninst✝ : AddGroup N₄\ni₁ : M₁ →+ N₁\ni₂ : M₂ →+ N₂\ni₃ : M₃ →+ N₃\ni₄ : M₄ →+ N₄\nhi₁ : Function.Surjective ⇑i₁\nhi₂ : Function.Injective ⇑i₂\nhi₄ : Function.Injective ⇑i₄\n⊢ ∀ (a : M₃), i₃ a = 0 → a = 0",
"after_state": "M₁ : Type u_1\nM₂ : Type u_2\nM₃ : Type u_3\nM₄ : Type u_4\nN₁ : Type u_6\nN₂ : Type u_7\nN₃ : Type u_8\nN₄ : Type u_9\ninst✝⁷ : AddGroup M₁\ninst✝⁶ : AddGroup M₂\ninst✝⁵ : AddGroup M₃\ninst✝⁴ : AddGroup M₄\ninst✝³ : AddGroup N₁\ninst✝² : AddGroup N₂\ninst✝¹ : AddGroup N₃\ninst✝ : AddGroup N₄\ni₁ : M₁ →+ N₁\ni₂ : M₂ →+ N₂\ni₃ : M₃ →+ N₃\ni₄ : M₄ →+ N₄\nhi₁ : Function.Surjective ⇑i₁\nhi₂ : Function.Injective ⇑i₂\nhi₄ : Function.Injective ⇑i₄\nm : M₃\nhm : i₃ m = 0\n⊢ m = 0"
},
{
"line": "intro m;\n intro hm",
"before_state": "M₁ : Type u_1\nM₂ : Type u_2\nM₃ : Type u_3\nM₄ : Type u_4\nN₁ : Type u_6\nN₂ : Type u_7\nN₃ : Type u_8\nN₄ : Type u_9\ninst✝⁷ : AddGroup M₁\ninst✝⁶ : AddGroup M₂\ninst✝⁵ : AddGroup M₃\ninst✝⁴ : AddGroup M₄\ninst✝³ : AddGroup N₁\ninst✝² : AddGroup N₂\ninst✝¹ : AddGroup N₃\ninst✝ : AddGroup N₄\ni₁ : M₁ →+ N₁\ni₂ : M₂ →+ N₂\ni₃ : M₃ →+ N₃\ni₄ : M₄ →+ N₄\nhi₁ : Function.Surjective ⇑i₁\nhi₂ : Function.Injective ⇑i₂\nhi₄ : Function.Injective ⇑i₄\n⊢ ∀ (a : M₃), i₃ a = 0 → a = 0",
"after_state": "M₁ : Type u_1\nM₂ : Type u_2\nM₃ : Type u_3\nM₄ : Type u_4\nN₁ : Type u_6\nN₂ : Type u_7\nN₃ : Type u_8\nN₄ : Type u_9\ninst✝⁷ : AddGroup M₁\ninst✝⁶ : AddGroup M₂\ninst✝⁵ : AddGroup M₃\ninst✝⁴ : AddGroup M₄\ninst✝³ : AddGroup N₁\ninst✝² : AddGroup N₂\ninst✝¹ : AddGroup N₃\ninst✝ : AddGroup N₄\ni₁ : M₁ →+ N₁\ni₂ : M₂ →+ N₂\ni₃ : M₃ →+ N₃\ni₄ : M₄ →+ N₄\nhi₁ : Function.Surjective ⇑i₁\nhi₂ : Function.Injective ⇑i₂\nhi₄ : Function.Injective ⇑i₄\nm : M₃\nhm : i₃ m = 0\n⊢ m = 0"
},
{
"line": "intro m",
"before_state": "M₁ : Type u_1\nM₂ : Type u_2\nM₃ : Type u_3\nM₄ : Type u_4\nN₁ : Type u_6\nN₂ : Type u_7\nN₃ : Type u_8\nN₄ : Type u_9\ninst✝⁷ : AddGroup M₁\ninst✝⁶ : AddGroup M₂\ninst✝⁵ : AddGroup M₃\ninst✝⁴ : AddGroup M₄\ninst✝³ : AddGroup N₁\ninst✝² : AddGroup N₂\ninst✝¹ : AddGroup N₃\ninst✝ : AddGroup N₄\ni₁ : M₁ →+ N₁\ni₂ : M₂ →+ N₂\ni₃ : M₃ →+ N₃\ni₄ : M₄ →+ N₄\nhi₁ : Function.Surjective ⇑i₁\nhi₂ : Function.Injective ⇑i₂\nhi₄ : Function.Injective ⇑i₄\n⊢ ∀ (a : M₃), i₃ a = 0 → a = 0",
"after_state": "M₁ : Type u_1\nM₂ : Type u_2\nM₃ : Type u_3\nM₄ : Type u_4\nN₁ : Type u_6\nN₂ : Type u_7\nN₃ : Type u_8\nN₄ : Type u_9\ninst✝⁷ : AddGroup M₁\ninst✝⁶ : AddGroup M₂\ninst✝⁵ : AddGroup M₃\ninst✝⁴ : AddGroup M₄\ninst✝³ : AddGroup N₁\ninst✝² : AddGroup N₂\ninst✝¹ : AddGroup N₃\ninst✝ : AddGroup N₄\ni₁ : M₁ →+ N₁\ni₂ : M₂ →+ N₂\ni₃ : M₃ →+ N₃\ni₄ : M₄ →+ N₄\nhi₁ : Function.Surjective ⇑i₁\nhi₂ : Function.Injective ⇑i₂\nhi₄ : Function.Injective ⇑i₄\nm : M₃\n⊢ i₃ m = 0 → m = 0"
},
{
"line": "intro hm",
"before_state": "M₁ : Type u_1\nM₂ : Type u_2\nM₃ : Type u_3\nM₄ : Type u_4\nN₁ : Type u_6\nN₂ : Type u_7\nN₃ : Type u_8\nN₄ : Type u_9\ninst✝⁷ : AddGroup M₁\ninst✝⁶ : AddGroup M₂\ninst✝⁵ : AddGroup M₃\ninst✝⁴ : AddGroup M₄\ninst✝³ : AddGroup N₁\ninst✝² : AddGroup N₂\ninst✝¹ : AddGroup N₃\ninst✝ : AddGroup N₄\ni₁ : M₁ →+ N₁\ni₂ : M₂ →+ N₂\ni₃ : M₃ →+ N₃\ni₄ : M₄ →+ N₄\nhi₁ : Function.Surjective ⇑i₁\nhi₂ : Function.Injective ⇑i₂\nhi₄ : Function.Injective ⇑i₄\nm : M₃\n⊢ i₃ m = 0 → m = 0",
"after_state": "M₁ : Type u_1\nM₂ : Type u_2\nM₃ : Type u_3\nM₄ : Type u_4\nN₁ : Type u_6\nN₂ : Type u_7\nN₃ : Type u_8\nN₄ : Type u_9\ninst✝⁷ : AddGroup M₁\ninst✝⁶ : AddGroup M₂\ninst✝⁵ : AddGroup M₃\ninst✝⁴ : AddGroup M₄\ninst✝³ : AddGroup N₁\ninst✝² : AddGroup N₂\ninst✝¹ : AddGroup N₃\ninst✝ : AddGroup N₄\ni₁ : M₁ →+ N₁\ni₂ : M₂ →+ N₂\ni₃ : M₃ →+ N₃\ni₄ : M₄ →+ N₄\nhi₁ : Function.Surjective ⇑i₁\nhi₂ : Function.Injective ⇑i₂\nhi₄ : Function.Injective ⇑i₄\nm : M₃\nhm : i₃ m = 0\n⊢ m = 0"
},
{
"line": "obtain ⟨x, rfl⟩ :=\n (hf₂ m).mp <| by\n suffices h : i₄ (f₃ m) = 0 by rwa [map_eq_zero_iff _ hi₄] at h\n simp [← show g₃ (i₃ m) = i₄ (f₃ m) by simpa using DFunLike.congr_fun hc₃ m, hm]",
"before_state": "M₁ : Type u_1\nM₂ : Type u_2\nM₃ : Type u_3\nM₄ : Type u_4\nN₁ : Type u_6\nN₂ : Type u_7\nN₃ : Type u_8\nN₄ : Type u_9\ninst✝⁷ : AddGroup M₁\ninst✝⁶ : AddGroup M₂\ninst✝⁵ : AddGroup M₃\ninst✝⁴ : AddGroup M₄\ninst✝³ : AddGroup N₁\ninst✝² : AddGroup N₂\ninst✝¹ : AddGroup N₃\ninst✝ : AddGroup N₄\ni₁ : M₁ →+ N₁\ni₂ : M₂ →+ N₂\ni₃ : M₃ →+ N₃\ni₄ : M₄ →+ N₄\nhi₁ : Function.Surjective ⇑i₁\nhi₂ : Function.Injective ⇑i₂\nhi₄ : Function.Injective ⇑i₄\nm : M₃\nhm : i₃ m = 0\n⊢ m = 0",
"after_state": "No Goals!"
}
] |
theorem commute_eps_left [Semiring R] (x : DualNumber R) : Commute ε x := by
ext <;> simp
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/DualNumber.lean
|
{
"open": [
"DualNumber",
"TrivSqZeroExt"
],
"variables": [
"{R A B : Type*}"
]
}
|
[
{
"line": "focus\n ext\n with_annotate_state\"<;>\" skip\n all_goals simp",
"before_state": "R : Type u_1\ninst✝ : Semiring R\nx : R[ε]\n⊢ Commute ε x",
"after_state": "No Goals!"
},
{
"line": "ext",
"before_state": "R : Type u_1\ninst✝ : Semiring R\nx : R[ε]\n⊢ Commute ε x",
"after_state": "case h1\nR : Type u_1\ninst✝ : Semiring R\nx : R[ε]\n⊢ fst (ε * x) = fst (x * ε)\n---\ncase h2\nR : Type u_1\ninst✝ : Semiring R\nx : R[ε]\n⊢ snd (ε * x) = snd (x * ε)"
},
{
"line": "with_annotate_state\"<;>\" skip",
"before_state": "case h1\nR : Type u_1\ninst✝ : Semiring R\nx : R[ε]\n⊢ fst (ε * x) = fst (x * ε)\n---\ncase h2\nR : Type u_1\ninst✝ : Semiring R\nx : R[ε]\n⊢ snd (ε * x) = snd (x * ε)",
"after_state": "case h1\nR : Type u_1\ninst✝ : Semiring R\nx : R[ε]\n⊢ fst (ε * x) = fst (x * ε)\n---\ncase h2\nR : Type u_1\ninst✝ : Semiring R\nx : R[ε]\n⊢ snd (ε * x) = snd (x * ε)"
},
{
"line": "skip",
"before_state": "case h1\nR : Type u_1\ninst✝ : Semiring R\nx : R[ε]\n⊢ fst (ε * x) = fst (x * ε)\n---\ncase h2\nR : Type u_1\ninst✝ : Semiring R\nx : R[ε]\n⊢ snd (ε * x) = snd (x * ε)",
"after_state": "case h1\nR : Type u_1\ninst✝ : Semiring R\nx : R[ε]\n⊢ fst (ε * x) = fst (x * ε)\n---\ncase h2\nR : Type u_1\ninst✝ : Semiring R\nx : R[ε]\n⊢ snd (ε * x) = snd (x * ε)"
},
{
"line": "all_goals simp",
"before_state": "case h1\nR : Type u_1\ninst✝ : Semiring R\nx : R[ε]\n⊢ fst (ε * x) = fst (x * ε)\n---\ncase h2\nR : Type u_1\ninst✝ : Semiring R\nx : R[ε]\n⊢ snd (ε * x) = snd (x * ε)",
"after_state": "No Goals!"
},
{
"line": "simp",
"before_state": "case h1\nR : Type u_1\ninst✝ : Semiring R\nx : R[ε]\n⊢ fst (ε * x) = fst (x * ε)",
"after_state": "No Goals!"
},
{
"line": "simp",
"before_state": "case h2\nR : Type u_1\ninst✝ : Semiring R\nx : R[ε]\n⊢ snd (ε * x) = snd (x * ε)",
"after_state": "No Goals!"
}
] |
lemma exact_zero_iff_injective {M N : Type*} (P : Type*)
[AddCommGroup M] [AddCommGroup N] [AddCommMonoid P] [Module R N] [Module R M]
[Module R P] (f : M →ₗ[R] N) :
Function.Exact (0 : P →ₗ[R] M) f ↔ Function.Injective f := by
simp [← ker_eq_bot, exact_iff]
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/Exact.lean
|
{
"open": [
"Function",
"AddMonoidHom",
"Function"
],
"variables": [
"{R M M' N N' P P' : Type*}",
"(f : M → N) (g : N → P) (g' : P → P')",
"{f g}",
"[AddGroup M] [AddGroup N] [AddGroup P] {f : M →+ N} {g : N →+ P}",
"{X₁ X₂ X₃ Y₁ Y₂ Y₃ : Type*} [AddCommMonoid X₁] [AddCommMonoid X₂] [AddCommMonoid X₃]",
"[Semiring R] [AddCommMonoid M] [AddCommMonoid M'] [AddCommMonoid N]",
"{f : M →ₗ[R] N} {g : N →ₗ[R] P}",
"{R M N P : Type*} [Ring R]"
]
}
|
[
{
"line": "simp [← ker_eq_bot, exact_iff]",
"before_state": "R : Type u_14\ninst✝⁶ : Ring R\nM : Type u_18\nN : Type u_19\nP : Type u_20\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommMonoid P\ninst✝² : Module R N\ninst✝¹ : Module R M\ninst✝ : Module R P\nf : M →ₗ[R] N\n⊢ Exact ⇑0 ⇑f ↔ Injective ⇑f",
"after_state": "No Goals!"
}
] |
lemma exact_zero_iff_surjective {M N : Type*} (P : Type*)
[AddCommGroup M] [AddCommGroup N] [AddCommMonoid P] [Module R N] [Module R M]
[Module R P] (f : M →ₗ[R] N) :
Function.Exact f (0 : N →ₗ[R] P) ↔ Function.Surjective f := by
simp [← range_eq_top, exact_iff, eq_comm]
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/Exact.lean
|
{
"open": [
"Function",
"AddMonoidHom",
"Function"
],
"variables": [
"{R M M' N N' P P' : Type*}",
"(f : M → N) (g : N → P) (g' : P → P')",
"{f g}",
"[AddGroup M] [AddGroup N] [AddGroup P] {f : M →+ N} {g : N →+ P}",
"{X₁ X₂ X₃ Y₁ Y₂ Y₃ : Type*} [AddCommMonoid X₁] [AddCommMonoid X₂] [AddCommMonoid X₃]",
"[Semiring R] [AddCommMonoid M] [AddCommMonoid M'] [AddCommMonoid N]",
"{f : M →ₗ[R] N} {g : N →ₗ[R] P}",
"{R M N P : Type*} [Ring R]"
]
}
|
[
{
"line": "simp [← range_eq_top, exact_iff, eq_comm]",
"before_state": "R : Type u_14\ninst✝⁶ : Ring R\nM : Type u_18\nN : Type u_19\nP : Type u_20\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommMonoid P\ninst✝² : Module R N\ninst✝¹ : Module R M\ninst✝ : Module R P\nf : M →ₗ[R] N\n⊢ Exact ⇑f ⇑0 ↔ Surjective ⇑f",
"after_state": "No Goals!"
}
] |
theorem Exact.split_tfae' (h : Function.Exact f g) :
List.TFAE [
Function.Injective f ∧ ∃ l, g ∘ₗ l = LinearMap.id,
Function.Surjective g ∧ ∃ l, l ∘ₗ f = LinearMap.id,
∃ e : N ≃ₗ[R] M × P, f = e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ e] := by
tfae_have 1 → 3
| ⟨hf, l, hl⟩ => ⟨_, (h.splitSurjectiveEquiv hf ⟨l, hl⟩).2⟩
tfae_have 2 → 3
| ⟨hg, l, hl⟩ => ⟨_, (h.splitInjectiveEquiv hg ⟨l, hl⟩).2⟩
tfae_have 3 → 1
| ⟨e, e₁, e₂⟩ => by
have : Function.Injective f := e₁ ▸ e.symm.injective.comp LinearMap.inl_injective
exact ⟨this, ⟨_, ((h.splitSurjectiveEquiv this).symm ⟨e, e₁, e₂⟩).2⟩⟩
tfae_have 3 → 2
| ⟨e, e₁, e₂⟩ => by
have : Function.Surjective g := e₂ ▸ Prod.snd_surjective.comp e.surjective
exact ⟨this, ⟨_, ((h.splitInjectiveEquiv this).symm ⟨e, e₁, e₂⟩).2⟩⟩
tfae_finish
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/Exact.lean
|
{
"open": [
"Function",
"AddMonoidHom",
"Function",
"LinearMap",
"LinearMap"
],
"variables": [
"{R M M' N N' P P' : Type*}",
"(f : M → N) (g : N → P) (g' : P → P')",
"{f g}",
"[AddGroup M] [AddGroup N] [AddGroup P] {f : M →+ N} {g : N →+ P}",
"{X₁ X₂ X₃ Y₁ Y₂ Y₃ : Type*} [AddCommMonoid X₁] [AddCommMonoid X₂] [AddCommMonoid X₃]",
"[Semiring R] [AddCommMonoid M] [AddCommMonoid M'] [AddCommMonoid N]",
"{f : M →ₗ[R] N} {g : N →ₗ[R] P}",
"{R M N P : Type*} [Ring R]",
"(f g) in",
"",
"[Semiring R]",
"[AddCommGroup M] [AddCommGroup N] [AddCommGroup P] [Module R M] [Module R N] [Module R P]",
"{f : M →ₗ[R] N} {g : N →ₗ[R] P}"
]
}
|
[
{
"line": "tfae_have 1 → 3\n | ⟨hf, l, hl⟩ => ⟨_, (h.splitSurjectiveEquiv hf ⟨l, hl⟩).2⟩",
"before_state": "R : Type u_14\nM : Type u_15\nN : Type u_16\nP : Type u_17\ninst✝⁷ : Ring R\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\n⊢ [Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id, Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id,\n ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e].TFAE",
"after_state": "R : Type u_14\nM : Type u_15\nN : Type u_16\nP : Type u_17\ninst✝⁷ : Ring R\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\ntfae_1_to_3 :\n (Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\n⊢ [Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id, Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id,\n ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e].TFAE"
},
{
"line": "tfae_have tfae_1_to_3 : 1 → 3\n | ⟨hf, l, hl⟩ => ⟨_, (h.splitSurjectiveEquiv hf ⟨l, hl⟩).2⟩",
"before_state": "R : Type u_14\nM : Type u_15\nN : Type u_16\nP : Type u_17\ninst✝⁷ : Ring R\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\n⊢ [Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id, Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id,\n ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e].TFAE",
"after_state": "R : Type u_14\nM : Type u_15\nN : Type u_16\nP : Type u_17\ninst✝⁷ : Ring R\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\ntfae_1_to_3 :\n (Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\n⊢ [Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id, Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id,\n ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e].TFAE"
},
{
"line": "have tfae_1_to_3 : ?m✝\n | ⟨hf, l, hl⟩ => ⟨_, (h.splitSurjectiveEquiv hf ⟨l, hl⟩).2⟩",
"before_state": "R : Type u_14\nM : Type u_15\nN : Type u_16\nP : Type u_17\ninst✝⁷ : Ring R\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\n⊢ [Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id, Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id,\n ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e].TFAE",
"after_state": "R : Type u_14\nM : Type u_15\nN : Type u_16\nP : Type u_17\ninst✝⁷ : Ring R\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\ntfae_1_to_3 :\n (Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\n⊢ [Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id, Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id,\n ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e].TFAE"
},
{
"line": "refine_lift\n have tfae_1_to_3 : ?m✝\n | ⟨hf, l, hl⟩ => ⟨_, (h.splitSurjectiveEquiv hf ⟨l, hl⟩).2⟩;\n ?_",
"before_state": "R : Type u_14\nM : Type u_15\nN : Type u_16\nP : Type u_17\ninst✝⁷ : Ring R\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\n⊢ [Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id, Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id,\n ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e].TFAE",
"after_state": "R : Type u_14\nM : Type u_15\nN : Type u_16\nP : Type u_17\ninst✝⁷ : Ring R\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\ntfae_1_to_3 :\n (Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\n⊢ [Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id, Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id,\n ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e].TFAE"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (have tfae_1_to_3 : ?m✝\n | ⟨hf, l, hl⟩ => ⟨_, (h.splitSurjectiveEquiv hf ⟨l, hl⟩).2⟩;\n ?_);\n rotate_right)",
"before_state": "R : Type u_14\nM : Type u_15\nN : Type u_16\nP : Type u_17\ninst✝⁷ : Ring R\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\n⊢ [Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id, Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id,\n ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e].TFAE",
"after_state": "R : Type u_14\nM : Type u_15\nN : Type u_16\nP : Type u_17\ninst✝⁷ : Ring R\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\ntfae_1_to_3 :\n (Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\n⊢ [Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id, Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id,\n ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e].TFAE"
},
{
"line": "refine\n no_implicit_lambda%\n (have tfae_1_to_3 : ?m✝\n | ⟨hf, l, hl⟩ => ⟨_, (h.splitSurjectiveEquiv hf ⟨l, hl⟩).2⟩;\n ?_)",
"before_state": "R : Type u_14\nM : Type u_15\nN : Type u_16\nP : Type u_17\ninst✝⁷ : Ring R\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\n⊢ [Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id, Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id,\n ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e].TFAE",
"after_state": "R : Type u_14\nM : Type u_15\nN : Type u_16\nP : Type u_17\ninst✝⁷ : Ring R\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\ntfae_1_to_3 :\n (Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\n⊢ [Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id, Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id,\n ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e].TFAE"
},
{
"line": "rotate_right",
"before_state": "R : Type u_14\nM : Type u_15\nN : Type u_16\nP : Type u_17\ninst✝⁷ : Ring R\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\ntfae_1_to_3 :\n (Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\n⊢ [Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id, Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id,\n ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e].TFAE",
"after_state": "R : Type u_14\nM : Type u_15\nN : Type u_16\nP : Type u_17\ninst✝⁷ : Ring R\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\ntfae_1_to_3 :\n (Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\n⊢ [Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id, Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id,\n ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e].TFAE"
},
{
"line": "tfae_have 2 → 3\n | ⟨hg, l, hl⟩ => ⟨_, (h.splitInjectiveEquiv hg ⟨l, hl⟩).2⟩",
"before_state": "R : Type u_14\nM : Type u_15\nN : Type u_16\nP : Type u_17\ninst✝⁷ : Ring R\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\ntfae_1_to_3 :\n (Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\n⊢ [Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id, Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id,\n ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e].TFAE",
"after_state": "R : Type u_14\nM : Type u_15\nN : Type u_16\nP : Type u_17\ninst✝⁷ : Ring R\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\ntfae_1_to_3 :\n (Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ntfae_2_to_3 :\n (Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\n⊢ [Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id, Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id,\n ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e].TFAE"
},
{
"line": "tfae_have tfae_2_to_3 : 2 → 3\n | ⟨hg, l, hl⟩ => ⟨_, (h.splitInjectiveEquiv hg ⟨l, hl⟩).2⟩",
"before_state": "R : Type u_14\nM : Type u_15\nN : Type u_16\nP : Type u_17\ninst✝⁷ : Ring R\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\ntfae_1_to_3 :\n (Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\n⊢ [Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id, Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id,\n ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e].TFAE",
"after_state": "R : Type u_14\nM : Type u_15\nN : Type u_16\nP : Type u_17\ninst✝⁷ : Ring R\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\ntfae_1_to_3 :\n (Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ntfae_2_to_3 :\n (Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\n⊢ [Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id, Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id,\n ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e].TFAE"
},
{
"line": "have tfae_2_to_3 : ?m✝\n | ⟨hg, l, hl⟩ => ⟨_, (h.splitInjectiveEquiv hg ⟨l, hl⟩).2⟩",
"before_state": "R : Type u_14\nM : Type u_15\nN : Type u_16\nP : Type u_17\ninst✝⁷ : Ring R\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\ntfae_1_to_3 :\n (Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\n⊢ [Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id, Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id,\n ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e].TFAE",
"after_state": "R : Type u_14\nM : Type u_15\nN : Type u_16\nP : Type u_17\ninst✝⁷ : Ring R\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\ntfae_1_to_3 :\n (Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ntfae_2_to_3 :\n (Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\n⊢ [Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id, Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id,\n ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e].TFAE"
},
{
"line": "refine_lift\n have tfae_2_to_3 : ?m✝\n | ⟨hg, l, hl⟩ => ⟨_, (h.splitInjectiveEquiv hg ⟨l, hl⟩).2⟩;\n ?_",
"before_state": "R : Type u_14\nM : Type u_15\nN : Type u_16\nP : Type u_17\ninst✝⁷ : Ring R\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\ntfae_1_to_3 :\n (Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\n⊢ [Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id, Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id,\n ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e].TFAE",
"after_state": "R : Type u_14\nM : Type u_15\nN : Type u_16\nP : Type u_17\ninst✝⁷ : Ring R\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\ntfae_1_to_3 :\n (Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ntfae_2_to_3 :\n (Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\n⊢ [Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id, Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id,\n ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e].TFAE"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (have tfae_2_to_3 : ?m✝\n | ⟨hg, l, hl⟩ => ⟨_, (h.splitInjectiveEquiv hg ⟨l, hl⟩).2⟩;\n ?_);\n rotate_right)",
"before_state": "R : Type u_14\nM : Type u_15\nN : Type u_16\nP : Type u_17\ninst✝⁷ : Ring R\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\ntfae_1_to_3 :\n (Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\n⊢ [Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id, Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id,\n ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e].TFAE",
"after_state": "R : Type u_14\nM : Type u_15\nN : Type u_16\nP : Type u_17\ninst✝⁷ : Ring R\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\ntfae_1_to_3 :\n (Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ntfae_2_to_3 :\n (Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\n⊢ [Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id, Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id,\n ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e].TFAE"
},
{
"line": "refine\n no_implicit_lambda%\n (have tfae_2_to_3 : ?m✝\n | ⟨hg, l, hl⟩ => ⟨_, (h.splitInjectiveEquiv hg ⟨l, hl⟩).2⟩;\n ?_)",
"before_state": "R : Type u_14\nM : Type u_15\nN : Type u_16\nP : Type u_17\ninst✝⁷ : Ring R\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\ntfae_1_to_3 :\n (Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\n⊢ [Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id, Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id,\n ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e].TFAE",
"after_state": "R : Type u_14\nM : Type u_15\nN : Type u_16\nP : Type u_17\ninst✝⁷ : Ring R\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\ntfae_1_to_3 :\n (Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ntfae_2_to_3 :\n (Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\n⊢ [Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id, Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id,\n ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e].TFAE"
},
{
"line": "rotate_right",
"before_state": "R : Type u_14\nM : Type u_15\nN : Type u_16\nP : Type u_17\ninst✝⁷ : Ring R\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\ntfae_1_to_3 :\n (Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ntfae_2_to_3 :\n (Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\n⊢ [Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id, Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id,\n ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e].TFAE",
"after_state": "R : Type u_14\nM : Type u_15\nN : Type u_16\nP : Type u_17\ninst✝⁷ : Ring R\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\ntfae_1_to_3 :\n (Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ntfae_2_to_3 :\n (Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\n⊢ [Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id, Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id,\n ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e].TFAE"
},
{
"line": "tfae_have 3 → 1\n | ⟨e, e₁, e₂⟩ => by\n have : Function.Injective f := e₁ ▸ e.symm.injective.comp LinearMap.inl_injective\n exact ⟨this, ⟨_, ((h.splitSurjectiveEquiv this).symm ⟨e, e₁, e₂⟩).2⟩⟩",
"before_state": "R : Type u_14\nM : Type u_15\nN : Type u_16\nP : Type u_17\ninst✝⁷ : Ring R\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\ntfae_1_to_3 :\n (Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ntfae_2_to_3 :\n (Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\n⊢ [Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id, Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id,\n ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e].TFAE",
"after_state": "R : Type u_14\nM : Type u_15\nN : Type u_16\nP : Type u_17\ninst✝⁷ : Ring R\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\ntfae_1_to_3 :\n (Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ntfae_2_to_3 :\n (Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ntfae_3_to_1 :\n (∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e) → Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id\n⊢ [Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id, Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id,\n ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e].TFAE"
},
{
"line": "tfae_have tfae_3_to_1 : 3 → 1\n | ⟨e, e₁, e₂⟩ => by\n have : Function.Injective f := e₁ ▸ e.symm.injective.comp LinearMap.inl_injective\n exact ⟨this, ⟨_, ((h.splitSurjectiveEquiv this).symm ⟨e, e₁, e₂⟩).2⟩⟩",
"before_state": "R : Type u_14\nM : Type u_15\nN : Type u_16\nP : Type u_17\ninst✝⁷ : Ring R\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\ntfae_1_to_3 :\n (Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ntfae_2_to_3 :\n (Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\n⊢ [Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id, Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id,\n ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e].TFAE",
"after_state": "R : Type u_14\nM : Type u_15\nN : Type u_16\nP : Type u_17\ninst✝⁷ : Ring R\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\ntfae_1_to_3 :\n (Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ntfae_2_to_3 :\n (Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ntfae_3_to_1 :\n (∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e) → Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id\n⊢ [Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id, Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id,\n ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e].TFAE"
},
{
"line": "have tfae_3_to_1 : ?m✝\n | ⟨e, e₁, e₂⟩ => by\n have : Function.Injective f := e₁ ▸ e.symm.injective.comp LinearMap.inl_injective\n exact ⟨this, ⟨_, ((h.splitSurjectiveEquiv this).symm ⟨e, e₁, e₂⟩).2⟩⟩",
"before_state": "R : Type u_14\nM : Type u_15\nN : Type u_16\nP : Type u_17\ninst✝⁷ : Ring R\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\ntfae_1_to_3 :\n (Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ntfae_2_to_3 :\n (Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\n⊢ [Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id, Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id,\n ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e].TFAE",
"after_state": "R : Type u_14\nM : Type u_15\nN : Type u_16\nP : Type u_17\ninst✝⁷ : Ring R\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\ntfae_1_to_3 :\n (Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ntfae_2_to_3 :\n (Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ntfae_3_to_1 :\n (∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e) → Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id\n⊢ [Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id, Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id,\n ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e].TFAE"
},
{
"line": "refine_lift\n have tfae_3_to_1 : ?m✝\n | ⟨e, e₁, e₂⟩ => by\n have : Function.Injective f := e₁ ▸ e.symm.injective.comp LinearMap.inl_injective\n exact ⟨this, ⟨_, ((h.splitSurjectiveEquiv this).symm ⟨e, e₁, e₂⟩).2⟩⟩;\n ?_",
"before_state": "R : Type u_14\nM : Type u_15\nN : Type u_16\nP : Type u_17\ninst✝⁷ : Ring R\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\ntfae_1_to_3 :\n (Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ntfae_2_to_3 :\n (Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\n⊢ [Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id, Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id,\n ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e].TFAE",
"after_state": "R : Type u_14\nM : Type u_15\nN : Type u_16\nP : Type u_17\ninst✝⁷ : Ring R\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\ntfae_1_to_3 :\n (Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ntfae_2_to_3 :\n (Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ntfae_3_to_1 :\n (∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e) → Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id\n⊢ [Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id, Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id,\n ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e].TFAE"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (have tfae_3_to_1 : ?m✝\n | ⟨e, e₁, e₂⟩ =>\n by\n have : Function.Injective f := e₁ ▸ e.symm.injective.comp LinearMap.inl_injective\n exact ⟨this, ⟨_, ((h.splitSurjectiveEquiv this).symm ⟨e, e₁, e₂⟩).2⟩⟩;\n ?_);\n rotate_right)",
"before_state": "R : Type u_14\nM : Type u_15\nN : Type u_16\nP : Type u_17\ninst✝⁷ : Ring R\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\ntfae_1_to_3 :\n (Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ntfae_2_to_3 :\n (Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\n⊢ [Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id, Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id,\n ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e].TFAE",
"after_state": "R : Type u_14\nM : Type u_15\nN : Type u_16\nP : Type u_17\ninst✝⁷ : Ring R\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\ntfae_1_to_3 :\n (Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ntfae_2_to_3 :\n (Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ntfae_3_to_1 :\n (∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e) → Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id\n⊢ [Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id, Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id,\n ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e].TFAE"
},
{
"line": "refine\n no_implicit_lambda%\n (have tfae_3_to_1 : ?m✝\n | ⟨e, e₁, e₂⟩ =>\n by\n have : Function.Injective f := e₁ ▸ e.symm.injective.comp LinearMap.inl_injective\n exact ⟨this, ⟨_, ((h.splitSurjectiveEquiv this).symm ⟨e, e₁, e₂⟩).2⟩⟩;\n ?_)",
"before_state": "R : Type u_14\nM : Type u_15\nN : Type u_16\nP : Type u_17\ninst✝⁷ : Ring R\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\ntfae_1_to_3 :\n (Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ntfae_2_to_3 :\n (Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\n⊢ [Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id, Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id,\n ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e].TFAE",
"after_state": "R : Type u_14\nM : Type u_15\nN : Type u_16\nP : Type u_17\ninst✝⁷ : Ring R\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\ntfae_1_to_3 :\n (Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ntfae_2_to_3 :\n (Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ntfae_3_to_1 :\n (∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e) → Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id\n⊢ [Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id, Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id,\n ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e].TFAE"
},
{
"line": "have : Function.Injective f := e₁ ▸ e.symm.injective.comp LinearMap.inl_injective",
"before_state": "R : Type u_14\nM : Type u_15\nN : Type u_16\nP : Type u_17\ninst✝⁷ : Ring R\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\ntfae_1_to_3 :\n (Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ntfae_2_to_3 :\n (Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ne : N ≃ₗ[R] M × P\ne₁ : f = ↑e.symm ∘ₗ LinearMap.inl R M P\ne₂ : g = LinearMap.snd R M P ∘ₗ ↑e\n⊢ Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id",
"after_state": "R : Type u_14\nM : Type u_15\nN : Type u_16\nP : Type u_17\ninst✝⁷ : Ring R\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\ntfae_1_to_3 :\n (Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ntfae_2_to_3 :\n (Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ne : N ≃ₗ[R] M × P\ne₁ : f = ↑e.symm ∘ₗ LinearMap.inl R M P\ne₂ : g = LinearMap.snd R M P ∘ₗ ↑e\nthis : Injective ⇑f\n⊢ Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id"
},
{
"line": "refine_lift\n have : Function.Injective f := e₁ ▸ e.symm.injective.comp LinearMap.inl_injective;\n ?_",
"before_state": "R : Type u_14\nM : Type u_15\nN : Type u_16\nP : Type u_17\ninst✝⁷ : Ring R\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\ntfae_1_to_3 :\n (Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ntfae_2_to_3 :\n (Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ne : N ≃ₗ[R] M × P\ne₁ : f = ↑e.symm ∘ₗ LinearMap.inl R M P\ne₂ : g = LinearMap.snd R M P ∘ₗ ↑e\n⊢ Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id",
"after_state": "R : Type u_14\nM : Type u_15\nN : Type u_16\nP : Type u_17\ninst✝⁷ : Ring R\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\ntfae_1_to_3 :\n (Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ntfae_2_to_3 :\n (Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ne : N ≃ₗ[R] M × P\ne₁ : f = ↑e.symm ∘ₗ LinearMap.inl R M P\ne₂ : g = LinearMap.snd R M P ∘ₗ ↑e\nthis : Injective ⇑f\n⊢ Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (have : Function.Injective f := e₁ ▸ e.symm.injective.comp LinearMap.inl_injective;\n ?_);\n rotate_right)",
"before_state": "R : Type u_14\nM : Type u_15\nN : Type u_16\nP : Type u_17\ninst✝⁷ : Ring R\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\ntfae_1_to_3 :\n (Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ntfae_2_to_3 :\n (Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ne : N ≃ₗ[R] M × P\ne₁ : f = ↑e.symm ∘ₗ LinearMap.inl R M P\ne₂ : g = LinearMap.snd R M P ∘ₗ ↑e\n⊢ Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id",
"after_state": "R : Type u_14\nM : Type u_15\nN : Type u_16\nP : Type u_17\ninst✝⁷ : Ring R\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\ntfae_1_to_3 :\n (Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ntfae_2_to_3 :\n (Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ne : N ≃ₗ[R] M × P\ne₁ : f = ↑e.symm ∘ₗ LinearMap.inl R M P\ne₂ : g = LinearMap.snd R M P ∘ₗ ↑e\nthis : Injective ⇑f\n⊢ Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id"
},
{
"line": "refine\n no_implicit_lambda%\n (have : Function.Injective f := e₁ ▸ e.symm.injective.comp LinearMap.inl_injective;\n ?_)",
"before_state": "R : Type u_14\nM : Type u_15\nN : Type u_16\nP : Type u_17\ninst✝⁷ : Ring R\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\ntfae_1_to_3 :\n (Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ntfae_2_to_3 :\n (Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ne : N ≃ₗ[R] M × P\ne₁ : f = ↑e.symm ∘ₗ LinearMap.inl R M P\ne₂ : g = LinearMap.snd R M P ∘ₗ ↑e\n⊢ Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id",
"after_state": "R : Type u_14\nM : Type u_15\nN : Type u_16\nP : Type u_17\ninst✝⁷ : Ring R\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\ntfae_1_to_3 :\n (Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ntfae_2_to_3 :\n (Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ne : N ≃ₗ[R] M × P\ne₁ : f = ↑e.symm ∘ₗ LinearMap.inl R M P\ne₂ : g = LinearMap.snd R M P ∘ₗ ↑e\nthis : Injective ⇑f\n⊢ Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id"
},
{
"line": "rotate_right",
"before_state": "R : Type u_14\nM : Type u_15\nN : Type u_16\nP : Type u_17\ninst✝⁷ : Ring R\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\ntfae_1_to_3 :\n (Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ntfae_2_to_3 :\n (Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ne : N ≃ₗ[R] M × P\ne₁ : f = ↑e.symm ∘ₗ LinearMap.inl R M P\ne₂ : g = LinearMap.snd R M P ∘ₗ ↑e\nthis : Injective ⇑f\n⊢ Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id",
"after_state": "R : Type u_14\nM : Type u_15\nN : Type u_16\nP : Type u_17\ninst✝⁷ : Ring R\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\ntfae_1_to_3 :\n (Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ntfae_2_to_3 :\n (Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ne : N ≃ₗ[R] M × P\ne₁ : f = ↑e.symm ∘ₗ LinearMap.inl R M P\ne₂ : g = LinearMap.snd R M P ∘ₗ ↑e\nthis : Injective ⇑f\n⊢ Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id"
},
{
"line": "exact ⟨this, ⟨_, ((h.splitSurjectiveEquiv this).symm ⟨e, e₁, e₂⟩).2⟩⟩",
"before_state": "R : Type u_14\nM : Type u_15\nN : Type u_16\nP : Type u_17\ninst✝⁷ : Ring R\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\ntfae_1_to_3 :\n (Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ntfae_2_to_3 :\n (Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ne : N ≃ₗ[R] M × P\ne₁ : f = ↑e.symm ∘ₗ LinearMap.inl R M P\ne₂ : g = LinearMap.snd R M P ∘ₗ ↑e\nthis : Injective ⇑f\n⊢ Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id",
"after_state": "No Goals!"
},
{
"line": "rotate_right",
"before_state": "R : Type u_14\nM : Type u_15\nN : Type u_16\nP : Type u_17\ninst✝⁷ : Ring R\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\ntfae_1_to_3 :\n (Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ntfae_2_to_3 :\n (Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ntfae_3_to_1 :\n (∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e) → Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id\n⊢ [Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id, Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id,\n ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e].TFAE",
"after_state": "R : Type u_14\nM : Type u_15\nN : Type u_16\nP : Type u_17\ninst✝⁷ : Ring R\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\ntfae_1_to_3 :\n (Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ntfae_2_to_3 :\n (Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ntfae_3_to_1 :\n (∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e) → Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id\n⊢ [Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id, Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id,\n ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e].TFAE"
},
{
"line": "tfae_have 3 → 2\n | ⟨e, e₁, e₂⟩ => by\n have : Function.Surjective g := e₂ ▸ Prod.snd_surjective.comp e.surjective\n exact ⟨this, ⟨_, ((h.splitInjectiveEquiv this).symm ⟨e, e₁, e₂⟩).2⟩⟩",
"before_state": "R : Type u_14\nM : Type u_15\nN : Type u_16\nP : Type u_17\ninst✝⁷ : Ring R\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\ntfae_1_to_3 :\n (Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ntfae_2_to_3 :\n (Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ntfae_3_to_1 :\n (∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e) → Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id\n⊢ [Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id, Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id,\n ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e].TFAE",
"after_state": "R : Type u_14\nM : Type u_15\nN : Type u_16\nP : Type u_17\ninst✝⁷ : Ring R\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\ntfae_1_to_3 :\n (Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ntfae_2_to_3 :\n (Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ntfae_3_to_1 :\n (∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e) → Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id\ntfae_3_to_2 :\n (∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e) → Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id\n⊢ [Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id, Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id,\n ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e].TFAE"
},
{
"line": "tfae_have tfae_3_to_2 : 3 → 2\n | ⟨e, e₁, e₂⟩ => by\n have : Function.Surjective g := e₂ ▸ Prod.snd_surjective.comp e.surjective\n exact ⟨this, ⟨_, ((h.splitInjectiveEquiv this).symm ⟨e, e₁, e₂⟩).2⟩⟩",
"before_state": "R : Type u_14\nM : Type u_15\nN : Type u_16\nP : Type u_17\ninst✝⁷ : Ring R\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\ntfae_1_to_3 :\n (Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ntfae_2_to_3 :\n (Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ntfae_3_to_1 :\n (∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e) → Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id\n⊢ [Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id, Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id,\n ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e].TFAE",
"after_state": "R : Type u_14\nM : Type u_15\nN : Type u_16\nP : Type u_17\ninst✝⁷ : Ring R\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\ntfae_1_to_3 :\n (Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ntfae_2_to_3 :\n (Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ntfae_3_to_1 :\n (∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e) → Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id\ntfae_3_to_2 :\n (∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e) → Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id\n⊢ [Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id, Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id,\n ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e].TFAE"
},
{
"line": "have tfae_3_to_2 : ?m✝\n | ⟨e, e₁, e₂⟩ => by\n have : Function.Surjective g := e₂ ▸ Prod.snd_surjective.comp e.surjective\n exact ⟨this, ⟨_, ((h.splitInjectiveEquiv this).symm ⟨e, e₁, e₂⟩).2⟩⟩",
"before_state": "R : Type u_14\nM : Type u_15\nN : Type u_16\nP : Type u_17\ninst✝⁷ : Ring R\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\ntfae_1_to_3 :\n (Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ntfae_2_to_3 :\n (Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ntfae_3_to_1 :\n (∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e) → Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id\n⊢ [Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id, Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id,\n ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e].TFAE",
"after_state": "R : Type u_14\nM : Type u_15\nN : Type u_16\nP : Type u_17\ninst✝⁷ : Ring R\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\ntfae_1_to_3 :\n (Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ntfae_2_to_3 :\n (Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ntfae_3_to_1 :\n (∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e) → Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id\ntfae_3_to_2 :\n (∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e) → Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id\n⊢ [Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id, Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id,\n ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e].TFAE"
},
{
"line": "refine_lift\n have tfae_3_to_2 : ?m✝\n | ⟨e, e₁, e₂⟩ => by\n have : Function.Surjective g := e₂ ▸ Prod.snd_surjective.comp e.surjective\n exact ⟨this, ⟨_, ((h.splitInjectiveEquiv this).symm ⟨e, e₁, e₂⟩).2⟩⟩;\n ?_",
"before_state": "R : Type u_14\nM : Type u_15\nN : Type u_16\nP : Type u_17\ninst✝⁷ : Ring R\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\ntfae_1_to_3 :\n (Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ntfae_2_to_3 :\n (Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ntfae_3_to_1 :\n (∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e) → Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id\n⊢ [Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id, Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id,\n ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e].TFAE",
"after_state": "R : Type u_14\nM : Type u_15\nN : Type u_16\nP : Type u_17\ninst✝⁷ : Ring R\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\ntfae_1_to_3 :\n (Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ntfae_2_to_3 :\n (Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ntfae_3_to_1 :\n (∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e) → Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id\ntfae_3_to_2 :\n (∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e) → Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id\n⊢ [Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id, Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id,\n ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e].TFAE"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (have tfae_3_to_2 : ?m✝\n | ⟨e, e₁, e₂⟩ => by\n have : Function.Surjective g := e₂ ▸ Prod.snd_surjective.comp e.surjective\n exact ⟨this, ⟨_, ((h.splitInjectiveEquiv this).symm ⟨e, e₁, e₂⟩).2⟩⟩;\n ?_);\n rotate_right)",
"before_state": "R : Type u_14\nM : Type u_15\nN : Type u_16\nP : Type u_17\ninst✝⁷ : Ring R\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\ntfae_1_to_3 :\n (Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ntfae_2_to_3 :\n (Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ntfae_3_to_1 :\n (∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e) → Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id\n⊢ [Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id, Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id,\n ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e].TFAE",
"after_state": "R : Type u_14\nM : Type u_15\nN : Type u_16\nP : Type u_17\ninst✝⁷ : Ring R\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\ntfae_1_to_3 :\n (Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ntfae_2_to_3 :\n (Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ntfae_3_to_1 :\n (∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e) → Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id\ntfae_3_to_2 :\n (∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e) → Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id\n⊢ [Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id, Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id,\n ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e].TFAE"
},
{
"line": "refine\n no_implicit_lambda%\n (have tfae_3_to_2 : ?m✝\n | ⟨e, e₁, e₂⟩ => by\n have : Function.Surjective g := e₂ ▸ Prod.snd_surjective.comp e.surjective\n exact ⟨this, ⟨_, ((h.splitInjectiveEquiv this).symm ⟨e, e₁, e₂⟩).2⟩⟩;\n ?_)",
"before_state": "R : Type u_14\nM : Type u_15\nN : Type u_16\nP : Type u_17\ninst✝⁷ : Ring R\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\ntfae_1_to_3 :\n (Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ntfae_2_to_3 :\n (Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ntfae_3_to_1 :\n (∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e) → Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id\n⊢ [Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id, Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id,\n ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e].TFAE",
"after_state": "R : Type u_14\nM : Type u_15\nN : Type u_16\nP : Type u_17\ninst✝⁷ : Ring R\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\ntfae_1_to_3 :\n (Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ntfae_2_to_3 :\n (Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ntfae_3_to_1 :\n (∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e) → Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id\ntfae_3_to_2 :\n (∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e) → Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id\n⊢ [Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id, Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id,\n ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e].TFAE"
},
{
"line": "have : Function.Surjective g := e₂ ▸ Prod.snd_surjective.comp e.surjective",
"before_state": "R : Type u_14\nM : Type u_15\nN : Type u_16\nP : Type u_17\ninst✝⁷ : Ring R\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\ntfae_1_to_3 :\n (Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ntfae_2_to_3 :\n (Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ntfae_3_to_1 :\n (∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e) → Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id\ne : N ≃ₗ[R] M × P\ne₁ : f = ↑e.symm ∘ₗ LinearMap.inl R M P\ne₂ : g = LinearMap.snd R M P ∘ₗ ↑e\n⊢ Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id",
"after_state": "R : Type u_14\nM : Type u_15\nN : Type u_16\nP : Type u_17\ninst✝⁷ : Ring R\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\ntfae_1_to_3 :\n (Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ntfae_2_to_3 :\n (Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ntfae_3_to_1 :\n (∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e) → Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id\ne : N ≃ₗ[R] M × P\ne₁ : f = ↑e.symm ∘ₗ LinearMap.inl R M P\ne₂ : g = LinearMap.snd R M P ∘ₗ ↑e\nthis : Surjective ⇑g\n⊢ Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id"
},
{
"line": "refine_lift\n have : Function.Surjective g := e₂ ▸ Prod.snd_surjective.comp e.surjective;\n ?_",
"before_state": "R : Type u_14\nM : Type u_15\nN : Type u_16\nP : Type u_17\ninst✝⁷ : Ring R\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\ntfae_1_to_3 :\n (Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ntfae_2_to_3 :\n (Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ntfae_3_to_1 :\n (∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e) → Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id\ne : N ≃ₗ[R] M × P\ne₁ : f = ↑e.symm ∘ₗ LinearMap.inl R M P\ne₂ : g = LinearMap.snd R M P ∘ₗ ↑e\n⊢ Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id",
"after_state": "R : Type u_14\nM : Type u_15\nN : Type u_16\nP : Type u_17\ninst✝⁷ : Ring R\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\ntfae_1_to_3 :\n (Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ntfae_2_to_3 :\n (Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ntfae_3_to_1 :\n (∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e) → Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id\ne : N ≃ₗ[R] M × P\ne₁ : f = ↑e.symm ∘ₗ LinearMap.inl R M P\ne₂ : g = LinearMap.snd R M P ∘ₗ ↑e\nthis : Surjective ⇑g\n⊢ Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (have : Function.Surjective g := e₂ ▸ Prod.snd_surjective.comp e.surjective;\n ?_);\n rotate_right)",
"before_state": "R : Type u_14\nM : Type u_15\nN : Type u_16\nP : Type u_17\ninst✝⁷ : Ring R\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\ntfae_1_to_3 :\n (Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ntfae_2_to_3 :\n (Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ntfae_3_to_1 :\n (∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e) → Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id\ne : N ≃ₗ[R] M × P\ne₁ : f = ↑e.symm ∘ₗ LinearMap.inl R M P\ne₂ : g = LinearMap.snd R M P ∘ₗ ↑e\n⊢ Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id",
"after_state": "R : Type u_14\nM : Type u_15\nN : Type u_16\nP : Type u_17\ninst✝⁷ : Ring R\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\ntfae_1_to_3 :\n (Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ntfae_2_to_3 :\n (Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ntfae_3_to_1 :\n (∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e) → Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id\ne : N ≃ₗ[R] M × P\ne₁ : f = ↑e.symm ∘ₗ LinearMap.inl R M P\ne₂ : g = LinearMap.snd R M P ∘ₗ ↑e\nthis : Surjective ⇑g\n⊢ Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id"
},
{
"line": "refine\n no_implicit_lambda%\n (have : Function.Surjective g := e₂ ▸ Prod.snd_surjective.comp e.surjective;\n ?_)",
"before_state": "R : Type u_14\nM : Type u_15\nN : Type u_16\nP : Type u_17\ninst✝⁷ : Ring R\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\ntfae_1_to_3 :\n (Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ntfae_2_to_3 :\n (Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ntfae_3_to_1 :\n (∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e) → Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id\ne : N ≃ₗ[R] M × P\ne₁ : f = ↑e.symm ∘ₗ LinearMap.inl R M P\ne₂ : g = LinearMap.snd R M P ∘ₗ ↑e\n⊢ Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id",
"after_state": "R : Type u_14\nM : Type u_15\nN : Type u_16\nP : Type u_17\ninst✝⁷ : Ring R\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\ntfae_1_to_3 :\n (Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ntfae_2_to_3 :\n (Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ntfae_3_to_1 :\n (∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e) → Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id\ne : N ≃ₗ[R] M × P\ne₁ : f = ↑e.symm ∘ₗ LinearMap.inl R M P\ne₂ : g = LinearMap.snd R M P ∘ₗ ↑e\nthis : Surjective ⇑g\n⊢ Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id"
},
{
"line": "rotate_right",
"before_state": "R : Type u_14\nM : Type u_15\nN : Type u_16\nP : Type u_17\ninst✝⁷ : Ring R\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\ntfae_1_to_3 :\n (Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ntfae_2_to_3 :\n (Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ntfae_3_to_1 :\n (∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e) → Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id\ne : N ≃ₗ[R] M × P\ne₁ : f = ↑e.symm ∘ₗ LinearMap.inl R M P\ne₂ : g = LinearMap.snd R M P ∘ₗ ↑e\nthis : Surjective ⇑g\n⊢ Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id",
"after_state": "R : Type u_14\nM : Type u_15\nN : Type u_16\nP : Type u_17\ninst✝⁷ : Ring R\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\ntfae_1_to_3 :\n (Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ntfae_2_to_3 :\n (Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ntfae_3_to_1 :\n (∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e) → Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id\ne : N ≃ₗ[R] M × P\ne₁ : f = ↑e.symm ∘ₗ LinearMap.inl R M P\ne₂ : g = LinearMap.snd R M P ∘ₗ ↑e\nthis : Surjective ⇑g\n⊢ Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id"
},
{
"line": "exact ⟨this, ⟨_, ((h.splitInjectiveEquiv this).symm ⟨e, e₁, e₂⟩).2⟩⟩",
"before_state": "R : Type u_14\nM : Type u_15\nN : Type u_16\nP : Type u_17\ninst✝⁷ : Ring R\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\ntfae_1_to_3 :\n (Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ntfae_2_to_3 :\n (Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ntfae_3_to_1 :\n (∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e) → Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id\ne : N ≃ₗ[R] M × P\ne₁ : f = ↑e.symm ∘ₗ LinearMap.inl R M P\ne₂ : g = LinearMap.snd R M P ∘ₗ ↑e\nthis : Surjective ⇑g\n⊢ Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id",
"after_state": "No Goals!"
},
{
"line": "rotate_right",
"before_state": "R : Type u_14\nM : Type u_15\nN : Type u_16\nP : Type u_17\ninst✝⁷ : Ring R\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\ntfae_1_to_3 :\n (Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ntfae_2_to_3 :\n (Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ntfae_3_to_1 :\n (∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e) → Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id\ntfae_3_to_2 :\n (∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e) → Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id\n⊢ [Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id, Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id,\n ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e].TFAE",
"after_state": "R : Type u_14\nM : Type u_15\nN : Type u_16\nP : Type u_17\ninst✝⁷ : Ring R\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\ntfae_1_to_3 :\n (Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ntfae_2_to_3 :\n (Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ntfae_3_to_1 :\n (∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e) → Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id\ntfae_3_to_2 :\n (∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e) → Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id\n⊢ [Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id, Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id,\n ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e].TFAE"
},
{
"line": "tfae_finish",
"before_state": "R : Type u_14\nM : Type u_15\nN : Type u_16\nP : Type u_17\ninst✝⁷ : Ring R\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\ntfae_1_to_3 :\n (Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ntfae_2_to_3 :\n (Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id) → ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ntfae_3_to_1 :\n (∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e) → Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id\ntfae_3_to_2 :\n (∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e) → Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id\n⊢ [Injective ⇑f ∧ ∃ l, g ∘ₗ l = LinearMap.id, Surjective ⇑g ∧ ∃ l, l ∘ₗ f = LinearMap.id,\n ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e].TFAE",
"after_state": "No Goals!"
}
] |
theorem Exact.split_tfae
{R M N P} [Semiring R] [AddCommGroup M] [AddCommGroup N]
[AddCommGroup P] [Module R M] [Module R N] [Module R P] {f : M →ₗ[R] N} {g : N →ₗ[R] P}
(h : Function.Exact f g) (hf : Function.Injective f) (hg : Function.Surjective g) :
List.TFAE [
∃ l, g ∘ₗ l = LinearMap.id,
∃ l, l ∘ₗ f = LinearMap.id,
∃ e : N ≃ₗ[R] M × P, f = e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ e] := by
tfae_have 1 ↔ 3 := by
simpa using (h.splitSurjectiveEquiv hf).nonempty_congr
tfae_have 2 ↔ 3 := by
simpa using (h.splitInjectiveEquiv hg).nonempty_congr
tfae_finish
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/Exact.lean
|
{
"open": [
"Function",
"AddMonoidHom",
"Function",
"LinearMap",
"LinearMap"
],
"variables": [
"{R M M' N N' P P' : Type*}",
"(f : M → N) (g : N → P) (g' : P → P')",
"{f g}",
"[AddGroup M] [AddGroup N] [AddGroup P] {f : M →+ N} {g : N →+ P}",
"{X₁ X₂ X₃ Y₁ Y₂ Y₃ : Type*} [AddCommMonoid X₁] [AddCommMonoid X₂] [AddCommMonoid X₃]",
"[Semiring R] [AddCommMonoid M] [AddCommMonoid M'] [AddCommMonoid N]",
"{f : M →ₗ[R] N} {g : N →ₗ[R] P}",
"{R M N P : Type*} [Ring R]",
"(f g) in",
"",
"[Semiring R]",
"[AddCommGroup M] [AddCommGroup N] [AddCommGroup P] [Module R M] [Module R N] [Module R P]",
"{f : M →ₗ[R] N} {g : N →ₗ[R] P}"
]
}
|
[
{
"line": "tfae_have 1 ↔ 3 := by simpa using (h.splitSurjectiveEquiv hf).nonempty_congr",
"before_state": "R : Type u_18\nM : Type u_19\nN : Type u_20\nP : Type u_21\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\nhf : Injective ⇑f\nhg : Surjective ⇑g\n⊢ [∃ l, g ∘ₗ l = LinearMap.id, ∃ l, l ∘ₗ f = LinearMap.id,\n ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e].TFAE",
"after_state": "R : Type u_18\nM : Type u_19\nN : Type u_20\nP : Type u_21\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\nhf : Injective ⇑f\nhg : Surjective ⇑g\ntfae_1_iff_3 : (∃ l, g ∘ₗ l = LinearMap.id) ↔ ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\n⊢ [∃ l, g ∘ₗ l = LinearMap.id, ∃ l, l ∘ₗ f = LinearMap.id,\n ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e].TFAE"
},
{
"line": "tfae_have tfae_1_iff_3 : 1 ↔ 3 := by simpa using (h.splitSurjectiveEquiv hf).nonempty_congr",
"before_state": "R : Type u_18\nM : Type u_19\nN : Type u_20\nP : Type u_21\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\nhf : Injective ⇑f\nhg : Surjective ⇑g\n⊢ [∃ l, g ∘ₗ l = LinearMap.id, ∃ l, l ∘ₗ f = LinearMap.id,\n ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e].TFAE",
"after_state": "R : Type u_18\nM : Type u_19\nN : Type u_20\nP : Type u_21\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\nhf : Injective ⇑f\nhg : Surjective ⇑g\ntfae_1_iff_3 : (∃ l, g ∘ₗ l = LinearMap.id) ↔ ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\n⊢ [∃ l, g ∘ₗ l = LinearMap.id, ∃ l, l ∘ₗ f = LinearMap.id,\n ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e].TFAE"
},
{
"line": "have tfae_1_iff_3 : ?m✝ := by simpa using (h.splitSurjectiveEquiv hf).nonempty_congr",
"before_state": "R : Type u_18\nM : Type u_19\nN : Type u_20\nP : Type u_21\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\nhf : Injective ⇑f\nhg : Surjective ⇑g\n⊢ [∃ l, g ∘ₗ l = LinearMap.id, ∃ l, l ∘ₗ f = LinearMap.id,\n ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e].TFAE",
"after_state": "R : Type u_18\nM : Type u_19\nN : Type u_20\nP : Type u_21\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\nhf : Injective ⇑f\nhg : Surjective ⇑g\ntfae_1_iff_3 : (∃ l, g ∘ₗ l = LinearMap.id) ↔ ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\n⊢ [∃ l, g ∘ₗ l = LinearMap.id, ∃ l, l ∘ₗ f = LinearMap.id,\n ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e].TFAE"
},
{
"line": "focus\n refine\n no_implicit_lambda%\n (have tfae_1_iff_3 : ?m✝ := ?body✝;\n ?_)\n case body✝ => with_annotate_state\"by\" (simpa using (h.splitSurjectiveEquiv hf).nonempty_congr)",
"before_state": "R : Type u_18\nM : Type u_19\nN : Type u_20\nP : Type u_21\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\nhf : Injective ⇑f\nhg : Surjective ⇑g\n⊢ [∃ l, g ∘ₗ l = LinearMap.id, ∃ l, l ∘ₗ f = LinearMap.id,\n ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e].TFAE",
"after_state": "R : Type u_18\nM : Type u_19\nN : Type u_20\nP : Type u_21\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\nhf : Injective ⇑f\nhg : Surjective ⇑g\ntfae_1_iff_3 : (∃ l, g ∘ₗ l = LinearMap.id) ↔ ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\n⊢ [∃ l, g ∘ₗ l = LinearMap.id, ∃ l, l ∘ₗ f = LinearMap.id,\n ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e].TFAE"
},
{
"line": "refine\n no_implicit_lambda%\n (have tfae_1_iff_3 : ?m✝ := ?body✝;\n ?_)",
"before_state": "R : Type u_18\nM : Type u_19\nN : Type u_20\nP : Type u_21\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\nhf : Injective ⇑f\nhg : Surjective ⇑g\n⊢ [∃ l, g ∘ₗ l = LinearMap.id, ∃ l, l ∘ₗ f = LinearMap.id,\n ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e].TFAE",
"after_state": "case body\nR : Type u_18\nM : Type u_19\nN : Type u_20\nP : Type u_21\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\nhf : Injective ⇑f\nhg : Surjective ⇑g\n⊢ (∃ l, g ∘ₗ l = LinearMap.id) ↔ ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\n---\nR : Type u_18\nM : Type u_19\nN : Type u_20\nP : Type u_21\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\nhf : Injective ⇑f\nhg : Surjective ⇑g\ntfae_1_iff_3 : (∃ l, g ∘ₗ l = LinearMap.id) ↔ ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\n⊢ [∃ l, g ∘ₗ l = LinearMap.id, ∃ l, l ∘ₗ f = LinearMap.id,\n ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e].TFAE"
},
{
"line": "case body✝ => with_annotate_state\"by\" (simpa using (h.splitSurjectiveEquiv hf).nonempty_congr)",
"before_state": "case body\nR : Type u_18\nM : Type u_19\nN : Type u_20\nP : Type u_21\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\nhf : Injective ⇑f\nhg : Surjective ⇑g\n⊢ (∃ l, g ∘ₗ l = LinearMap.id) ↔ ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\n---\nR : Type u_18\nM : Type u_19\nN : Type u_20\nP : Type u_21\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\nhf : Injective ⇑f\nhg : Surjective ⇑g\ntfae_1_iff_3 : (∃ l, g ∘ₗ l = LinearMap.id) ↔ ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\n⊢ [∃ l, g ∘ₗ l = LinearMap.id, ∃ l, l ∘ₗ f = LinearMap.id,\n ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e].TFAE",
"after_state": "R : Type u_18\nM : Type u_19\nN : Type u_20\nP : Type u_21\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\nhf : Injective ⇑f\nhg : Surjective ⇑g\ntfae_1_iff_3 : (∃ l, g ∘ₗ l = LinearMap.id) ↔ ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\n⊢ [∃ l, g ∘ₗ l = LinearMap.id, ∃ l, l ∘ₗ f = LinearMap.id,\n ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e].TFAE"
},
{
"line": "with_annotate_state\"by\" (simpa using (h.splitSurjectiveEquiv hf).nonempty_congr)",
"before_state": "R : Type u_18\nM : Type u_19\nN : Type u_20\nP : Type u_21\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\nhf : Injective ⇑f\nhg : Surjective ⇑g\n⊢ (∃ l, g ∘ₗ l = LinearMap.id) ↔ ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e",
"after_state": "No Goals!"
},
{
"line": "simpa using (h.splitSurjectiveEquiv hf).nonempty_congr",
"before_state": "R : Type u_18\nM : Type u_19\nN : Type u_20\nP : Type u_21\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\nhf : Injective ⇑f\nhg : Surjective ⇑g\n⊢ (∃ l, g ∘ₗ l = LinearMap.id) ↔ ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e",
"after_state": "No Goals!"
},
{
"line": "tfae_have 2 ↔ 3 := by simpa using (h.splitInjectiveEquiv hg).nonempty_congr",
"before_state": "R : Type u_18\nM : Type u_19\nN : Type u_20\nP : Type u_21\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\nhf : Injective ⇑f\nhg : Surjective ⇑g\ntfae_1_iff_3 : (∃ l, g ∘ₗ l = LinearMap.id) ↔ ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\n⊢ [∃ l, g ∘ₗ l = LinearMap.id, ∃ l, l ∘ₗ f = LinearMap.id,\n ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e].TFAE",
"after_state": "R : Type u_18\nM : Type u_19\nN : Type u_20\nP : Type u_21\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\nhf : Injective ⇑f\nhg : Surjective ⇑g\ntfae_1_iff_3 : (∃ l, g ∘ₗ l = LinearMap.id) ↔ ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ntfae_2_iff_3 : (∃ l, l ∘ₗ f = LinearMap.id) ↔ ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\n⊢ [∃ l, g ∘ₗ l = LinearMap.id, ∃ l, l ∘ₗ f = LinearMap.id,\n ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e].TFAE"
},
{
"line": "tfae_have tfae_2_iff_3 : 2 ↔ 3 := by simpa using (h.splitInjectiveEquiv hg).nonempty_congr",
"before_state": "R : Type u_18\nM : Type u_19\nN : Type u_20\nP : Type u_21\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\nhf : Injective ⇑f\nhg : Surjective ⇑g\ntfae_1_iff_3 : (∃ l, g ∘ₗ l = LinearMap.id) ↔ ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\n⊢ [∃ l, g ∘ₗ l = LinearMap.id, ∃ l, l ∘ₗ f = LinearMap.id,\n ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e].TFAE",
"after_state": "R : Type u_18\nM : Type u_19\nN : Type u_20\nP : Type u_21\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\nhf : Injective ⇑f\nhg : Surjective ⇑g\ntfae_1_iff_3 : (∃ l, g ∘ₗ l = LinearMap.id) ↔ ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ntfae_2_iff_3 : (∃ l, l ∘ₗ f = LinearMap.id) ↔ ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\n⊢ [∃ l, g ∘ₗ l = LinearMap.id, ∃ l, l ∘ₗ f = LinearMap.id,\n ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e].TFAE"
},
{
"line": "have tfae_2_iff_3 : ?m✝ := by simpa using (h.splitInjectiveEquiv hg).nonempty_congr",
"before_state": "R : Type u_18\nM : Type u_19\nN : Type u_20\nP : Type u_21\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\nhf : Injective ⇑f\nhg : Surjective ⇑g\ntfae_1_iff_3 : (∃ l, g ∘ₗ l = LinearMap.id) ↔ ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\n⊢ [∃ l, g ∘ₗ l = LinearMap.id, ∃ l, l ∘ₗ f = LinearMap.id,\n ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e].TFAE",
"after_state": "R : Type u_18\nM : Type u_19\nN : Type u_20\nP : Type u_21\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\nhf : Injective ⇑f\nhg : Surjective ⇑g\ntfae_1_iff_3 : (∃ l, g ∘ₗ l = LinearMap.id) ↔ ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ntfae_2_iff_3 : (∃ l, l ∘ₗ f = LinearMap.id) ↔ ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\n⊢ [∃ l, g ∘ₗ l = LinearMap.id, ∃ l, l ∘ₗ f = LinearMap.id,\n ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e].TFAE"
},
{
"line": "focus\n refine\n no_implicit_lambda%\n (have tfae_2_iff_3 : ?m✝ := ?body✝;\n ?_)\n case body✝ => with_annotate_state\"by\" (simpa using (h.splitInjectiveEquiv hg).nonempty_congr)",
"before_state": "R : Type u_18\nM : Type u_19\nN : Type u_20\nP : Type u_21\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\nhf : Injective ⇑f\nhg : Surjective ⇑g\ntfae_1_iff_3 : (∃ l, g ∘ₗ l = LinearMap.id) ↔ ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\n⊢ [∃ l, g ∘ₗ l = LinearMap.id, ∃ l, l ∘ₗ f = LinearMap.id,\n ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e].TFAE",
"after_state": "R : Type u_18\nM : Type u_19\nN : Type u_20\nP : Type u_21\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\nhf : Injective ⇑f\nhg : Surjective ⇑g\ntfae_1_iff_3 : (∃ l, g ∘ₗ l = LinearMap.id) ↔ ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ntfae_2_iff_3 : (∃ l, l ∘ₗ f = LinearMap.id) ↔ ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\n⊢ [∃ l, g ∘ₗ l = LinearMap.id, ∃ l, l ∘ₗ f = LinearMap.id,\n ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e].TFAE"
},
{
"line": "refine\n no_implicit_lambda%\n (have tfae_2_iff_3 : ?m✝ := ?body✝;\n ?_)",
"before_state": "R : Type u_18\nM : Type u_19\nN : Type u_20\nP : Type u_21\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\nhf : Injective ⇑f\nhg : Surjective ⇑g\ntfae_1_iff_3 : (∃ l, g ∘ₗ l = LinearMap.id) ↔ ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\n⊢ [∃ l, g ∘ₗ l = LinearMap.id, ∃ l, l ∘ₗ f = LinearMap.id,\n ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e].TFAE",
"after_state": "case body\nR : Type u_18\nM : Type u_19\nN : Type u_20\nP : Type u_21\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\nhf : Injective ⇑f\nhg : Surjective ⇑g\ntfae_1_iff_3 : (∃ l, g ∘ₗ l = LinearMap.id) ↔ ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\n⊢ (∃ l, l ∘ₗ f = LinearMap.id) ↔ ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\n---\nR : Type u_18\nM : Type u_19\nN : Type u_20\nP : Type u_21\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\nhf : Injective ⇑f\nhg : Surjective ⇑g\ntfae_1_iff_3 : (∃ l, g ∘ₗ l = LinearMap.id) ↔ ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ntfae_2_iff_3 : (∃ l, l ∘ₗ f = LinearMap.id) ↔ ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\n⊢ [∃ l, g ∘ₗ l = LinearMap.id, ∃ l, l ∘ₗ f = LinearMap.id,\n ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e].TFAE"
},
{
"line": "case body✝ => with_annotate_state\"by\" (simpa using (h.splitInjectiveEquiv hg).nonempty_congr)",
"before_state": "case body\nR : Type u_18\nM : Type u_19\nN : Type u_20\nP : Type u_21\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\nhf : Injective ⇑f\nhg : Surjective ⇑g\ntfae_1_iff_3 : (∃ l, g ∘ₗ l = LinearMap.id) ↔ ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\n⊢ (∃ l, l ∘ₗ f = LinearMap.id) ↔ ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\n---\nR : Type u_18\nM : Type u_19\nN : Type u_20\nP : Type u_21\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\nhf : Injective ⇑f\nhg : Surjective ⇑g\ntfae_1_iff_3 : (∃ l, g ∘ₗ l = LinearMap.id) ↔ ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ntfae_2_iff_3 : (∃ l, l ∘ₗ f = LinearMap.id) ↔ ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\n⊢ [∃ l, g ∘ₗ l = LinearMap.id, ∃ l, l ∘ₗ f = LinearMap.id,\n ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e].TFAE",
"after_state": "R : Type u_18\nM : Type u_19\nN : Type u_20\nP : Type u_21\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\nhf : Injective ⇑f\nhg : Surjective ⇑g\ntfae_1_iff_3 : (∃ l, g ∘ₗ l = LinearMap.id) ↔ ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ntfae_2_iff_3 : (∃ l, l ∘ₗ f = LinearMap.id) ↔ ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\n⊢ [∃ l, g ∘ₗ l = LinearMap.id, ∃ l, l ∘ₗ f = LinearMap.id,\n ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e].TFAE"
},
{
"line": "with_annotate_state\"by\" (simpa using (h.splitInjectiveEquiv hg).nonempty_congr)",
"before_state": "R : Type u_18\nM : Type u_19\nN : Type u_20\nP : Type u_21\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\nhf : Injective ⇑f\nhg : Surjective ⇑g\ntfae_1_iff_3 : (∃ l, g ∘ₗ l = LinearMap.id) ↔ ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\n⊢ (∃ l, l ∘ₗ f = LinearMap.id) ↔ ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e",
"after_state": "No Goals!"
},
{
"line": "simpa using (h.splitInjectiveEquiv hg).nonempty_congr",
"before_state": "R : Type u_18\nM : Type u_19\nN : Type u_20\nP : Type u_21\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\nhf : Injective ⇑f\nhg : Surjective ⇑g\ntfae_1_iff_3 : (∃ l, g ∘ₗ l = LinearMap.id) ↔ ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\n⊢ (∃ l, l ∘ₗ f = LinearMap.id) ↔ ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e",
"after_state": "No Goals!"
},
{
"line": "tfae_finish",
"before_state": "R : Type u_18\nM : Type u_19\nN : Type u_20\nP : Type u_21\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh : Exact ⇑f ⇑g\nhf : Injective ⇑f\nhg : Surjective ⇑g\ntfae_1_iff_3 : (∃ l, g ∘ₗ l = LinearMap.id) ↔ ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\ntfae_2_iff_3 : (∃ l, l ∘ₗ f = LinearMap.id) ↔ ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e\n⊢ [∃ l, g ∘ₗ l = LinearMap.id, ∃ l, l ∘ₗ f = LinearMap.id,\n ∃ e, f = ↑e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ ↑e].TFAE",
"after_state": "No Goals!"
}
] |
lemma exact_iff_of_surjective_of_bijective_of_injective
{M₁ M₂ M₃ N₁ N₂ N₃ : Type*} [AddCommMonoid M₁] [AddCommMonoid M₂] [AddCommMonoid M₃]
[AddCommMonoid N₁] [AddCommMonoid N₂] [AddCommMonoid N₃]
[Module R M₁] [Module R M₂] [Module R M₃]
[Module R N₁] [Module R N₂] [Module R N₃]
(f : M₁ →ₗ[R] M₂) (g : M₂ →ₗ[R] M₃) (f' : N₁ →ₗ[R] N₂) (g' : N₂ →ₗ[R] N₃)
(τ₁ : M₁ →ₗ[R] N₁) (τ₂ : M₂ →ₗ[R] N₂) (τ₃ : M₃ →ₗ[R] N₃)
(comm₁₂ : f'.comp τ₁ = τ₂.comp f) (comm₂₃ : g'.comp τ₂ = τ₃.comp g)
(h₁ : Function.Surjective τ₁) (h₂ : Function.Bijective τ₂) (h₃ : Function.Injective τ₃) :
Function.Exact f g ↔ Function.Exact f' g' :=
AddMonoidHom.exact_iff_of_surjective_of_bijective_of_injective
f.toAddMonoidHom g.toAddMonoidHom f'.toAddMonoidHom g'.toAddMonoidHom
τ₁.toAddMonoidHom τ₂.toAddMonoidHom τ₃.toAddMonoidHom
(by ext; apply DFunLike.congr_fun comm₁₂) (by ext; apply DFunLike.congr_fun comm₂₃) h₁ h₂ h₃
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/Exact.lean
|
{
"open": [
"Function",
"AddMonoidHom",
"Function",
"LinearMap",
"LinearMap",
"LinearMap Submodule"
],
"variables": [
"{R M M' N N' P P' : Type*}",
"(f : M → N) (g : N → P) (g' : P → P')",
"{f g}",
"[AddGroup M] [AddGroup N] [AddGroup P] {f : M →+ N} {g : N →+ P}",
"{X₁ X₂ X₃ Y₁ Y₂ Y₃ : Type*} [AddCommMonoid X₁] [AddCommMonoid X₂] [AddCommMonoid X₃]",
"[Semiring R] [AddCommMonoid M] [AddCommMonoid M'] [AddCommMonoid N]",
"{f : M →ₗ[R] N} {g : N →ₗ[R] P}",
"{R M N P : Type*} [Ring R]",
"(f g) in",
"",
"[Semiring R]",
"[AddCommGroup M] [AddCommGroup N] [AddCommGroup P] [Module R M] [Module R N] [Module R P]",
"{f : M →ₗ[R] N} {g : N →ₗ[R] P}",
"[Semiring R] [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N]",
"[Ring R] [AddCommGroup M] [AddCommGroup N] [AddCommGroup P]"
]
}
|
[
{
"line": "ext",
"before_state": "R : Type u_14\ninst✝¹⁵ : Ring R\ninst✝¹⁴ inst✝¹³ : Semiring R\ninst✝¹² : Ring R\nM₁ : Type u_18\nM₂ : Type u_19\nM₃ : Type u_20\nN₁ : Type u_21\nN₂ : Type u_22\nN₃ : Type u_23\ninst✝¹¹ : AddCommMonoid M₁\ninst✝¹⁰ : AddCommMonoid M₂\ninst✝⁹ : AddCommMonoid M₃\ninst✝⁸ : AddCommMonoid N₁\ninst✝⁷ : AddCommMonoid N₂\ninst✝⁶ : AddCommMonoid N₃\ninst✝⁵ : Module R M₁\ninst✝⁴ : Module R M₂\ninst✝³ : Module R M₃\ninst✝² : Module R N₁\ninst✝¹ : Module R N₂\ninst✝ : Module R N₃\nf : M₁ →ₗ[R] M₂\ng : M₂ →ₗ[R] M₃\nf' : N₁ →ₗ[R] N₂\ng' : N₂ →ₗ[R] N₃\nτ₁ : M₁ →ₗ[R] N₁\nτ₂ : M₂ →ₗ[R] N₂\nτ₃ : M₃ →ₗ[R] N₃\ncomm₁₂ : f' ∘ₗ τ₁ = τ₂ ∘ₗ f\ncomm₂₃ : g' ∘ₗ τ₂ = τ₃ ∘ₗ g\nh₁ : Surjective ⇑τ₁\nh₂ : Bijective ⇑τ₂\nh₃ : Injective ⇑τ₃\n⊢ f'.toAddMonoidHom.comp τ₁.toAddMonoidHom = τ₂.toAddMonoidHom.comp f.toAddMonoidHom",
"after_state": "case h\nR : Type u_14\ninst✝¹⁵ : Ring R\ninst✝¹⁴ inst✝¹³ : Semiring R\ninst✝¹² : Ring R\nM₁ : Type u_18\nM₂ : Type u_19\nM₃ : Type u_20\nN₁ : Type u_21\nN₂ : Type u_22\nN₃ : Type u_23\ninst✝¹¹ : AddCommMonoid M₁\ninst✝¹⁰ : AddCommMonoid M₂\ninst✝⁹ : AddCommMonoid M₃\ninst✝⁸ : AddCommMonoid N₁\ninst✝⁷ : AddCommMonoid N₂\ninst✝⁶ : AddCommMonoid N₃\ninst✝⁵ : Module R M₁\ninst✝⁴ : Module R M₂\ninst✝³ : Module R M₃\ninst✝² : Module R N₁\ninst✝¹ : Module R N₂\ninst✝ : Module R N₃\nf : M₁ →ₗ[R] M₂\ng : M₂ →ₗ[R] M₃\nf' : N₁ →ₗ[R] N₂\ng' : N₂ →ₗ[R] N₃\nτ₁ : M₁ →ₗ[R] N₁\nτ₂ : M₂ →ₗ[R] N₂\nτ₃ : M₃ →ₗ[R] N₃\ncomm₁₂ : f' ∘ₗ τ₁ = τ₂ ∘ₗ f\ncomm₂₃ : g' ∘ₗ τ₂ = τ₃ ∘ₗ g\nh₁ : Surjective ⇑τ₁\nh₂ : Bijective ⇑τ₂\nh₃ : Injective ⇑τ₃\nx✝ : M₁\n⊢ (f'.toAddMonoidHom.comp τ₁.toAddMonoidHom) x✝ = (τ₂.toAddMonoidHom.comp f.toAddMonoidHom) x✝"
},
{
"line": "apply DFunLike.congr_fun comm₁₂",
"before_state": "case h\nR : Type u_14\ninst✝¹⁵ : Ring R\ninst✝¹⁴ inst✝¹³ : Semiring R\ninst✝¹² : Ring R\nM₁ : Type u_18\nM₂ : Type u_19\nM₃ : Type u_20\nN₁ : Type u_21\nN₂ : Type u_22\nN₃ : Type u_23\ninst✝¹¹ : AddCommMonoid M₁\ninst✝¹⁰ : AddCommMonoid M₂\ninst✝⁹ : AddCommMonoid M₃\ninst✝⁸ : AddCommMonoid N₁\ninst✝⁷ : AddCommMonoid N₂\ninst✝⁶ : AddCommMonoid N₃\ninst✝⁵ : Module R M₁\ninst✝⁴ : Module R M₂\ninst✝³ : Module R M₃\ninst✝² : Module R N₁\ninst✝¹ : Module R N₂\ninst✝ : Module R N₃\nf : M₁ →ₗ[R] M₂\ng : M₂ →ₗ[R] M₃\nf' : N₁ →ₗ[R] N₂\ng' : N₂ →ₗ[R] N₃\nτ₁ : M₁ →ₗ[R] N₁\nτ₂ : M₂ →ₗ[R] N₂\nτ₃ : M₃ →ₗ[R] N₃\ncomm₁₂ : f' ∘ₗ τ₁ = τ₂ ∘ₗ f\ncomm₂₃ : g' ∘ₗ τ₂ = τ₃ ∘ₗ g\nh₁ : Surjective ⇑τ₁\nh₂ : Bijective ⇑τ₂\nh₃ : Injective ⇑τ₃\nx✝ : M₁\n⊢ (f'.toAddMonoidHom.comp τ₁.toAddMonoidHom) x✝ = (τ₂.toAddMonoidHom.comp f.toAddMonoidHom) x✝",
"after_state": "No Goals!"
},
{
"line": "ext",
"before_state": "R : Type u_14\ninst✝¹⁵ : Ring R\ninst✝¹⁴ inst✝¹³ : Semiring R\ninst✝¹² : Ring R\nM₁ : Type u_18\nM₂ : Type u_19\nM₃ : Type u_20\nN₁ : Type u_21\nN₂ : Type u_22\nN₃ : Type u_23\ninst✝¹¹ : AddCommMonoid M₁\ninst✝¹⁰ : AddCommMonoid M₂\ninst✝⁹ : AddCommMonoid M₃\ninst✝⁸ : AddCommMonoid N₁\ninst✝⁷ : AddCommMonoid N₂\ninst✝⁶ : AddCommMonoid N₃\ninst✝⁵ : Module R M₁\ninst✝⁴ : Module R M₂\ninst✝³ : Module R M₃\ninst✝² : Module R N₁\ninst✝¹ : Module R N₂\ninst✝ : Module R N₃\nf : M₁ →ₗ[R] M₂\ng : M₂ →ₗ[R] M₃\nf' : N₁ →ₗ[R] N₂\ng' : N₂ →ₗ[R] N₃\nτ₁ : M₁ →ₗ[R] N₁\nτ₂ : M₂ →ₗ[R] N₂\nτ₃ : M₃ →ₗ[R] N₃\ncomm₁₂ : f' ∘ₗ τ₁ = τ₂ ∘ₗ f\ncomm₂₃ : g' ∘ₗ τ₂ = τ₃ ∘ₗ g\nh₁ : Surjective ⇑τ₁\nh₂ : Bijective ⇑τ₂\nh₃ : Injective ⇑τ₃\n⊢ g'.toAddMonoidHom.comp τ₂.toAddMonoidHom = τ₃.toAddMonoidHom.comp g.toAddMonoidHom",
"after_state": "case h\nR : Type u_14\ninst✝¹⁵ : Ring R\ninst✝¹⁴ inst✝¹³ : Semiring R\ninst✝¹² : Ring R\nM₁ : Type u_18\nM₂ : Type u_19\nM₃ : Type u_20\nN₁ : Type u_21\nN₂ : Type u_22\nN₃ : Type u_23\ninst✝¹¹ : AddCommMonoid M₁\ninst✝¹⁰ : AddCommMonoid M₂\ninst✝⁹ : AddCommMonoid M₃\ninst✝⁸ : AddCommMonoid N₁\ninst✝⁷ : AddCommMonoid N₂\ninst✝⁶ : AddCommMonoid N₃\ninst✝⁵ : Module R M₁\ninst✝⁴ : Module R M₂\ninst✝³ : Module R M₃\ninst✝² : Module R N₁\ninst✝¹ : Module R N₂\ninst✝ : Module R N₃\nf : M₁ →ₗ[R] M₂\ng : M₂ →ₗ[R] M₃\nf' : N₁ →ₗ[R] N₂\ng' : N₂ →ₗ[R] N₃\nτ₁ : M₁ →ₗ[R] N₁\nτ₂ : M₂ →ₗ[R] N₂\nτ₃ : M₃ →ₗ[R] N₃\ncomm₁₂ : f' ∘ₗ τ₁ = τ₂ ∘ₗ f\ncomm₂₃ : g' ∘ₗ τ₂ = τ₃ ∘ₗ g\nh₁ : Surjective ⇑τ₁\nh₂ : Bijective ⇑τ₂\nh₃ : Injective ⇑τ₃\nx✝ : M₂\n⊢ (g'.toAddMonoidHom.comp τ₂.toAddMonoidHom) x✝ = (τ₃.toAddMonoidHom.comp g.toAddMonoidHom) x✝"
},
{
"line": "apply DFunLike.congr_fun comm₂₃",
"before_state": "case h\nR : Type u_14\ninst✝¹⁵ : Ring R\ninst✝¹⁴ inst✝¹³ : Semiring R\ninst✝¹² : Ring R\nM₁ : Type u_18\nM₂ : Type u_19\nM₃ : Type u_20\nN₁ : Type u_21\nN₂ : Type u_22\nN₃ : Type u_23\ninst✝¹¹ : AddCommMonoid M₁\ninst✝¹⁰ : AddCommMonoid M₂\ninst✝⁹ : AddCommMonoid M₃\ninst✝⁸ : AddCommMonoid N₁\ninst✝⁷ : AddCommMonoid N₂\ninst✝⁶ : AddCommMonoid N₃\ninst✝⁵ : Module R M₁\ninst✝⁴ : Module R M₂\ninst✝³ : Module R M₃\ninst✝² : Module R N₁\ninst✝¹ : Module R N₂\ninst✝ : Module R N₃\nf : M₁ →ₗ[R] M₂\ng : M₂ →ₗ[R] M₃\nf' : N₁ →ₗ[R] N₂\ng' : N₂ →ₗ[R] N₃\nτ₁ : M₁ →ₗ[R] N₁\nτ₂ : M₂ →ₗ[R] N₂\nτ₃ : M₃ →ₗ[R] N₃\ncomm₁₂ : f' ∘ₗ τ₁ = τ₂ ∘ₗ f\ncomm₂₃ : g' ∘ₗ τ₂ = τ₃ ∘ₗ g\nh₁ : Surjective ⇑τ₁\nh₂ : Bijective ⇑τ₂\nh₃ : Injective ⇑τ₃\nx✝ : M₂\n⊢ (g'.toAddMonoidHom.comp τ₂.toAddMonoidHom) x✝ = (τ₃.toAddMonoidHom.comp g.toAddMonoidHom) x✝",
"after_state": "No Goals!"
}
] |
theorem length_mul (x y : FreeSemigroup α) : (x * y).length = x.length + y.length := by
simp [length, Nat.add_right_comm, List.length, List.length_append]
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/Free.lean
|
{
"open": [],
"variables": [
"{α : Type u}",
"{α : Type u} {β : Type v} [Mul β] (f : α → β)",
"{α : Type u} {β : Type v} (f : α → β)",
"{α β : Type u}",
"{α : Type u}",
"{β : Type u}",
"{m : Type u → Type u} [Applicative m] (F : α → m β)",
"{α : Type u} [Mul α]",
"{β : Type v} [Semigroup β] (f : α →ₙ* β)",
"{β : Type v} [Mul β] (f : α →ₙ* β)",
"{α : Type u}"
]
}
|
[
{
"line": "simp [length, Nat.add_right_comm, List.length, List.length_append]",
"before_state": "α : Type u\nx y : FreeSemigroup α\n⊢ (x * y).length = x.length + y.length",
"after_state": "No Goals!"
}
] |
theorem hom_ext {β : Type v} [Mul β] {f g : FreeSemigroup α →ₙ* β} (h : f ∘ of = g ∘ of) : f = g :=
(DFunLike.ext _ _) fun x ↦
FreeSemigroup.recOnMul x (congr_fun h) fun x y hx hy ↦ by simp only [map_mul, *]
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/Free.lean
|
{
"open": [],
"variables": [
"{α : Type u}",
"{α : Type u} {β : Type v} [Mul β] (f : α → β)",
"{α : Type u} {β : Type v} (f : α → β)",
"{α β : Type u}",
"{α : Type u}",
"{β : Type u}",
"{m : Type u → Type u} [Applicative m] (F : α → m β)",
"{α : Type u} [Mul α]",
"{β : Type v} [Semigroup β] (f : α →ₙ* β)",
"{β : Type v} [Mul β] (f : α →ₙ* β)",
"{α : Type u}"
]
}
|
[
{
"line": "simp only [map_mul, *]",
"before_state": "α : Type u\nα✝ : Sort u_1\nof : α✝ → FreeSemigroup α\nβ : Type v\ninst✝ : Mul β\nf g : FreeSemigroup α →ₙ* β\nh : ⇑f ∘ of = ⇑g ∘ of\nx✝ : FreeSemigroup α\nx : α\ny : FreeSemigroup α\nhx : f (FreeSemigroup.of x) = g (FreeSemigroup.of x)\nhy : f y = g y\n⊢ f (FreeSemigroup.of x * y) = g (FreeSemigroup.of x * y)",
"after_state": "No Goals!"
}
] |
theorem pow_mem_graded (n : ℕ) {r : R} {i : ι} (h : r ∈ A i) : r ^ n ∈ A (n • i) := by
match n with
| 0 =>
rw [pow_zero]
rw [zero_nsmul]
exact one_mem_graded _
| n+1 =>
rw [pow_succ']
rw [succ_nsmul']
exact mul_mem_graded h (pow_mem_graded n h)
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/GradedMonoid.lean
|
{
"open": [],
"variables": [
"{ι : Type*}",
"{α β} {A : ι → Type*}",
"(A : ι → Type*)",
"{A}",
"[AddMonoid ι] [GMul A] [GOne A]",
"(A : ι → Type*)",
"[Zero ι] [GOne A]",
"[AddZeroClass ι] [GMul A]",
"{A}",
"[AddMonoid ι] [GMonoid A]",
"{A} in",
"[AddCommMonoid ι] [GCommMonoid A]",
"[AddMonoid ι] [GMonoid A]",
"{α : Type*} {A : ι → Type*} [AddMonoid ι] [GradedMonoid.GMonoid A]",
"(ι) {R : Type*}",
"{R : Type*}",
"{S : Type*} [SetLike S R] [Monoid R] [AddMonoid ι]",
"{A : ι → S} [SetLike.GradedMonoid A]",
"(A) in"
]
}
|
[
{
"line": "match n with\n| 0 =>\n rw [pow_zero]\n rw [zero_nsmul]\n exact one_mem_graded _\n| n + 1 =>\n rw [pow_succ']\n rw [succ_nsmul']\n exact mul_mem_graded h (pow_mem_graded n h)",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn : ℕ\nr : R\ni : ι\nh : r ∈ A i\n⊢ r ^ n ∈ A (n • i)",
"after_state": "No Goals!"
},
{
"line": "refine\n no_implicit_lambda%\n (match n with\n | 0 => ?rhs✝\n | n + 1 => ?rhs✝¹)",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn : ℕ\nr : R\ni : ι\nh : r ∈ A i\n⊢ r ^ n ∈ A (n • i)",
"after_state": "case rhs\nι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn : ℕ\nr : R\ni : ι\nh : r ∈ A i\n⊢ r ^ 0 ∈ A (0 • i)\n---\ncase rhs\nι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn✝ : ℕ\nr : R\ni : ι\nh : r ∈ A i\nn : ℕ\n⊢ r ^ (n + 1) ∈ A ((n + 1) • i)"
},
{
"line": "case rhs✝ =>\n with_annotate_state[\"|\" \"=>\"] skip\n rw [pow_zero]\n rw [zero_nsmul]\n exact one_mem_graded _",
"before_state": "case rhs\nι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn : ℕ\nr : R\ni : ι\nh : r ∈ A i\n⊢ r ^ 0 ∈ A (0 • i)\n---\ncase rhs\nι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn✝ : ℕ\nr : R\ni : ι\nh : r ∈ A i\nn : ℕ\n⊢ r ^ (n + 1) ∈ A ((n + 1) • i)",
"after_state": "case rhs\nι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn✝ : ℕ\nr : R\ni : ι\nh : r ∈ A i\nn : ℕ\n⊢ r ^ (n + 1) ∈ A ((n + 1) • i)"
},
{
"line": "with_annotate_state[\"|\" \"=>\"] skip",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn : ℕ\nr : R\ni : ι\nh : r ∈ A i\n⊢ r ^ 0 ∈ A (0 • i)",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn : ℕ\nr : R\ni : ι\nh : r ∈ A i\n⊢ r ^ 0 ∈ A (0 • i)"
},
{
"line": "skip",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn : ℕ\nr : R\ni : ι\nh : r ∈ A i\n⊢ r ^ 0 ∈ A (0 • i)",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn : ℕ\nr : R\ni : ι\nh : r ∈ A i\n⊢ r ^ 0 ∈ A (0 • i)"
},
{
"line": "rw [pow_zero]",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn : ℕ\nr : R\ni : ι\nh : r ∈ A i\n⊢ r ^ 0 ∈ A (0 • i)",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn : ℕ\nr : R\ni : ι\nh : r ∈ A i\n⊢ 1 ∈ A (0 • i)"
},
{
"line": "rewrite [pow_zero]",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn : ℕ\nr : R\ni : ι\nh : r ∈ A i\n⊢ r ^ 0 ∈ A (0 • i)",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn : ℕ\nr : R\ni : ι\nh : r ∈ A i\n⊢ 1 ∈ A (0 • i)"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn : ℕ\nr : R\ni : ι\nh : r ∈ A i\n⊢ 1 ∈ A (0 • i)",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn : ℕ\nr : R\ni : ι\nh : r ∈ A i\n⊢ 1 ∈ A (0 • i)"
},
{
"line": "try (with_reducible rfl)",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn : ℕ\nr : R\ni : ι\nh : r ∈ A i\n⊢ 1 ∈ A (0 • i)",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn : ℕ\nr : R\ni : ι\nh : r ∈ A i\n⊢ 1 ∈ A (0 • i)"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn : ℕ\nr : R\ni : ι\nh : r ∈ A i\n⊢ 1 ∈ A (0 • i)",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn : ℕ\nr : R\ni : ι\nh : r ∈ A i\n⊢ 1 ∈ A (0 • i)"
},
{
"line": "with_reducible rfl",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn : ℕ\nr : R\ni : ι\nh : r ∈ A i\n⊢ 1 ∈ A (0 • i)",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn : ℕ\nr : R\ni : ι\nh : r ∈ A i\n⊢ 1 ∈ A (0 • i)"
},
{
"line": "rfl",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn : ℕ\nr : R\ni : ι\nh : r ∈ A i\n⊢ 1 ∈ A (0 • i)",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn : ℕ\nr : R\ni : ι\nh : r ∈ A i\n⊢ 1 ∈ A (0 • i)"
},
{
"line": "apply_rfl",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn : ℕ\nr : R\ni : ι\nh : r ∈ A i\n⊢ 1 ∈ A (0 • i)",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn : ℕ\nr : R\ni : ι\nh : r ∈ A i\n⊢ 1 ∈ A (0 • i)"
},
{
"line": "skip",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn : ℕ\nr : R\ni : ι\nh : r ∈ A i\n⊢ 1 ∈ A (0 • i)",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn : ℕ\nr : R\ni : ι\nh : r ∈ A i\n⊢ 1 ∈ A (0 • i)"
},
{
"line": "rw [zero_nsmul]",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn : ℕ\nr : R\ni : ι\nh : r ∈ A i\n⊢ 1 ∈ A (0 • i)",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn : ℕ\nr : R\ni : ι\nh : r ∈ A i\n⊢ 1 ∈ A 0"
},
{
"line": "rewrite [zero_nsmul]",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn : ℕ\nr : R\ni : ι\nh : r ∈ A i\n⊢ 1 ∈ A (0 • i)",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn : ℕ\nr : R\ni : ι\nh : r ∈ A i\n⊢ 1 ∈ A 0"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn : ℕ\nr : R\ni : ι\nh : r ∈ A i\n⊢ 1 ∈ A 0",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn : ℕ\nr : R\ni : ι\nh : r ∈ A i\n⊢ 1 ∈ A 0"
},
{
"line": "try (with_reducible rfl)",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn : ℕ\nr : R\ni : ι\nh : r ∈ A i\n⊢ 1 ∈ A 0",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn : ℕ\nr : R\ni : ι\nh : r ∈ A i\n⊢ 1 ∈ A 0"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn : ℕ\nr : R\ni : ι\nh : r ∈ A i\n⊢ 1 ∈ A 0",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn : ℕ\nr : R\ni : ι\nh : r ∈ A i\n⊢ 1 ∈ A 0"
},
{
"line": "with_reducible rfl",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn : ℕ\nr : R\ni : ι\nh : r ∈ A i\n⊢ 1 ∈ A 0",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn : ℕ\nr : R\ni : ι\nh : r ∈ A i\n⊢ 1 ∈ A 0"
},
{
"line": "rfl",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn : ℕ\nr : R\ni : ι\nh : r ∈ A i\n⊢ 1 ∈ A 0",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn : ℕ\nr : R\ni : ι\nh : r ∈ A i\n⊢ 1 ∈ A 0"
},
{
"line": "apply_rfl",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn : ℕ\nr : R\ni : ι\nh : r ∈ A i\n⊢ 1 ∈ A 0",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn : ℕ\nr : R\ni : ι\nh : r ∈ A i\n⊢ 1 ∈ A 0"
},
{
"line": "skip",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn : ℕ\nr : R\ni : ι\nh : r ∈ A i\n⊢ 1 ∈ A 0",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn : ℕ\nr : R\ni : ι\nh : r ∈ A i\n⊢ 1 ∈ A 0"
},
{
"line": "exact one_mem_graded _",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn : ℕ\nr : R\ni : ι\nh : r ∈ A i\n⊢ 1 ∈ A 0",
"after_state": "No Goals!"
},
{
"line": "case rhs✝ =>\n with_annotate_state[\"|\" \"=>\"] skip\n rw [pow_succ']\n rw [succ_nsmul']\n exact mul_mem_graded h (pow_mem_graded n h)",
"before_state": "case rhs\nι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn✝ : ℕ\nr : R\ni : ι\nh : r ∈ A i\nn : ℕ\n⊢ r ^ (n + 1) ∈ A ((n + 1) • i)",
"after_state": "No Goals!"
},
{
"line": "with_annotate_state[\"|\" \"=>\"] skip",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn✝ : ℕ\nr : R\ni : ι\nh : r ∈ A i\nn : ℕ\n⊢ r ^ (n + 1) ∈ A ((n + 1) • i)",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn✝ : ℕ\nr : R\ni : ι\nh : r ∈ A i\nn : ℕ\n⊢ r ^ (n + 1) ∈ A ((n + 1) • i)"
},
{
"line": "skip",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn✝ : ℕ\nr : R\ni : ι\nh : r ∈ A i\nn : ℕ\n⊢ r ^ (n + 1) ∈ A ((n + 1) • i)",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn✝ : ℕ\nr : R\ni : ι\nh : r ∈ A i\nn : ℕ\n⊢ r ^ (n + 1) ∈ A ((n + 1) • i)"
},
{
"line": "rw [pow_succ']",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn✝ : ℕ\nr : R\ni : ι\nh : r ∈ A i\nn : ℕ\n⊢ r ^ (n + 1) ∈ A ((n + 1) • i)",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn✝ : ℕ\nr : R\ni : ι\nh : r ∈ A i\nn : ℕ\n⊢ r * r ^ n ∈ A ((n + 1) • i)"
},
{
"line": "rewrite [pow_succ']",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn✝ : ℕ\nr : R\ni : ι\nh : r ∈ A i\nn : ℕ\n⊢ r ^ (n + 1) ∈ A ((n + 1) • i)",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn✝ : ℕ\nr : R\ni : ι\nh : r ∈ A i\nn : ℕ\n⊢ r * r ^ n ∈ A ((n + 1) • i)"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn✝ : ℕ\nr : R\ni : ι\nh : r ∈ A i\nn : ℕ\n⊢ r * r ^ n ∈ A ((n + 1) • i)",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn✝ : ℕ\nr : R\ni : ι\nh : r ∈ A i\nn : ℕ\n⊢ r * r ^ n ∈ A ((n + 1) • i)"
},
{
"line": "try (with_reducible rfl)",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn✝ : ℕ\nr : R\ni : ι\nh : r ∈ A i\nn : ℕ\n⊢ r * r ^ n ∈ A ((n + 1) • i)",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn✝ : ℕ\nr : R\ni : ι\nh : r ∈ A i\nn : ℕ\n⊢ r * r ^ n ∈ A ((n + 1) • i)"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn✝ : ℕ\nr : R\ni : ι\nh : r ∈ A i\nn : ℕ\n⊢ r * r ^ n ∈ A ((n + 1) • i)",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn✝ : ℕ\nr : R\ni : ι\nh : r ∈ A i\nn : ℕ\n⊢ r * r ^ n ∈ A ((n + 1) • i)"
},
{
"line": "with_reducible rfl",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn✝ : ℕ\nr : R\ni : ι\nh : r ∈ A i\nn : ℕ\n⊢ r * r ^ n ∈ A ((n + 1) • i)",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn✝ : ℕ\nr : R\ni : ι\nh : r ∈ A i\nn : ℕ\n⊢ r * r ^ n ∈ A ((n + 1) • i)"
},
{
"line": "rfl",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn✝ : ℕ\nr : R\ni : ι\nh : r ∈ A i\nn : ℕ\n⊢ r * r ^ n ∈ A ((n + 1) • i)",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn✝ : ℕ\nr : R\ni : ι\nh : r ∈ A i\nn : ℕ\n⊢ r * r ^ n ∈ A ((n + 1) • i)"
},
{
"line": "apply_rfl",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn✝ : ℕ\nr : R\ni : ι\nh : r ∈ A i\nn : ℕ\n⊢ r * r ^ n ∈ A ((n + 1) • i)",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn✝ : ℕ\nr : R\ni : ι\nh : r ∈ A i\nn : ℕ\n⊢ r * r ^ n ∈ A ((n + 1) • i)"
},
{
"line": "skip",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn✝ : ℕ\nr : R\ni : ι\nh : r ∈ A i\nn : ℕ\n⊢ r * r ^ n ∈ A ((n + 1) • i)",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn✝ : ℕ\nr : R\ni : ι\nh : r ∈ A i\nn : ℕ\n⊢ r * r ^ n ∈ A ((n + 1) • i)"
},
{
"line": "rw [succ_nsmul']",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn✝ : ℕ\nr : R\ni : ι\nh : r ∈ A i\nn : ℕ\n⊢ r * r ^ n ∈ A ((n + 1) • i)",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn✝ : ℕ\nr : R\ni : ι\nh : r ∈ A i\nn : ℕ\n⊢ r * r ^ n ∈ A (i + n • i)"
},
{
"line": "rewrite [succ_nsmul']",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn✝ : ℕ\nr : R\ni : ι\nh : r ∈ A i\nn : ℕ\n⊢ r * r ^ n ∈ A ((n + 1) • i)",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn✝ : ℕ\nr : R\ni : ι\nh : r ∈ A i\nn : ℕ\n⊢ r * r ^ n ∈ A (i + n • i)"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn✝ : ℕ\nr : R\ni : ι\nh : r ∈ A i\nn : ℕ\n⊢ r * r ^ n ∈ A (i + n • i)",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn✝ : ℕ\nr : R\ni : ι\nh : r ∈ A i\nn : ℕ\n⊢ r * r ^ n ∈ A (i + n • i)"
},
{
"line": "try (with_reducible rfl)",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn✝ : ℕ\nr : R\ni : ι\nh : r ∈ A i\nn : ℕ\n⊢ r * r ^ n ∈ A (i + n • i)",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn✝ : ℕ\nr : R\ni : ι\nh : r ∈ A i\nn : ℕ\n⊢ r * r ^ n ∈ A (i + n • i)"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn✝ : ℕ\nr : R\ni : ι\nh : r ∈ A i\nn : ℕ\n⊢ r * r ^ n ∈ A (i + n • i)",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn✝ : ℕ\nr : R\ni : ι\nh : r ∈ A i\nn : ℕ\n⊢ r * r ^ n ∈ A (i + n • i)"
},
{
"line": "with_reducible rfl",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn✝ : ℕ\nr : R\ni : ι\nh : r ∈ A i\nn : ℕ\n⊢ r * r ^ n ∈ A (i + n • i)",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn✝ : ℕ\nr : R\ni : ι\nh : r ∈ A i\nn : ℕ\n⊢ r * r ^ n ∈ A (i + n • i)"
},
{
"line": "rfl",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn✝ : ℕ\nr : R\ni : ι\nh : r ∈ A i\nn : ℕ\n⊢ r * r ^ n ∈ A (i + n • i)",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn✝ : ℕ\nr : R\ni : ι\nh : r ∈ A i\nn : ℕ\n⊢ r * r ^ n ∈ A (i + n • i)"
},
{
"line": "apply_rfl",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn✝ : ℕ\nr : R\ni : ι\nh : r ∈ A i\nn : ℕ\n⊢ r * r ^ n ∈ A (i + n • i)",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn✝ : ℕ\nr : R\ni : ι\nh : r ∈ A i\nn : ℕ\n⊢ r * r ^ n ∈ A (i + n • i)"
},
{
"line": "skip",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn✝ : ℕ\nr : R\ni : ι\nh : r ∈ A i\nn : ℕ\n⊢ r * r ^ n ∈ A (i + n • i)",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn✝ : ℕ\nr : R\ni : ι\nh : r ∈ A i\nn : ℕ\n⊢ r * r ^ n ∈ A (i + n • i)"
},
{
"line": "exact mul_mem_graded h (pow_mem_graded n h)",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn✝ : ℕ\nr : R\ni : ι\nh : r ∈ A i\nn : ℕ\n⊢ r * r ^ n ∈ A (i + n • i)",
"after_state": "No Goals!"
}
] |
theorem list_prod_map_mem_graded {ι'} (l : List ι') (i : ι' → ι) (r : ι' → R)
(h : ∀ j ∈ l, r j ∈ A (i j)) : (l.map r).prod ∈ A (l.map i).sum := by
match l with
| [] =>
rw [List.map_nil]
rw [List.map_nil]
rw [List.prod_nil]
rw [List.sum_nil]
exact one_mem_graded _
| head::tail =>
rw [List.map_cons]
rw [List.map_cons]
rw [List.prod_cons]
rw [List.sum_cons]
exact
mul_mem_graded (h _ List.mem_cons_self)
(list_prod_map_mem_graded tail _ _ fun j hj => h _ <| List.mem_cons_of_mem _ hj)
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/GradedMonoid.lean
|
{
"open": [],
"variables": [
"{ι : Type*}",
"{α β} {A : ι → Type*}",
"(A : ι → Type*)",
"{A}",
"[AddMonoid ι] [GMul A] [GOne A]",
"(A : ι → Type*)",
"[Zero ι] [GOne A]",
"[AddZeroClass ι] [GMul A]",
"{A}",
"[AddMonoid ι] [GMonoid A]",
"{A} in",
"[AddCommMonoid ι] [GCommMonoid A]",
"[AddMonoid ι] [GMonoid A]",
"{α : Type*} {A : ι → Type*} [AddMonoid ι] [GradedMonoid.GMonoid A]",
"(ι) {R : Type*}",
"{R : Type*}",
"{S : Type*} [SetLike S R] [Monoid R] [AddMonoid ι]",
"{A : ι → S} [SetLike.GradedMonoid A]",
"(A) in"
]
}
|
[
{
"line": "match l with\n| [] =>\n rw [List.map_nil]\n rw [List.map_nil]\n rw [List.prod_nil]\n rw [List.sum_nil]\n exact one_mem_graded _\n| head :: tail =>\n rw [List.map_cons]\n rw [List.map_cons]\n rw [List.prod_cons]\n rw [List.sum_cons]\n exact\n mul_mem_graded (h _ List.mem_cons_self)\n (list_prod_map_mem_graded tail _ _ fun j hj => h _ <| List.mem_cons_of_mem _ hj)",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nh : ∀ j ∈ l, r j ∈ A (i j)\n⊢ (List.map r l).prod ∈ A (List.map i l).sum",
"after_state": "No Goals!"
},
{
"line": "refine\n no_implicit_lambda%\n (match l with\n | [] => ?rhs✝\n | head :: tail => ?rhs✝¹)",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nh : ∀ j ∈ l, r j ∈ A (i j)\n⊢ (List.map r l).prod ∈ A (List.map i l).sum",
"after_state": "case rhs\nι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nh : ∀ j ∈ [], r j ∈ A (i j)\n⊢ (List.map r []).prod ∈ A (List.map i []).sum\n---\ncase rhs\nι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nhead : ι'\ntail : List ι'\nh : ∀ j ∈ head :: tail, r j ∈ A (i j)\n⊢ (List.map r (head :: tail)).prod ∈ A (List.map i (head :: tail)).sum"
},
{
"line": "case rhs✝ =>\n with_annotate_state[\"|\" \"=>\"] skip\n rw [List.map_nil]\n rw [List.map_nil]\n rw [List.prod_nil]\n rw [List.sum_nil]\n exact one_mem_graded _",
"before_state": "case rhs\nι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nh : ∀ j ∈ [], r j ∈ A (i j)\n⊢ (List.map r []).prod ∈ A (List.map i []).sum\n---\ncase rhs\nι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nhead : ι'\ntail : List ι'\nh : ∀ j ∈ head :: tail, r j ∈ A (i j)\n⊢ (List.map r (head :: tail)).prod ∈ A (List.map i (head :: tail)).sum",
"after_state": "case rhs\nι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nhead : ι'\ntail : List ι'\nh : ∀ j ∈ head :: tail, r j ∈ A (i j)\n⊢ (List.map r (head :: tail)).prod ∈ A (List.map i (head :: tail)).sum"
},
{
"line": "with_annotate_state[\"|\" \"=>\"] skip",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nh : ∀ j ∈ [], r j ∈ A (i j)\n⊢ (List.map r []).prod ∈ A (List.map i []).sum",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nh : ∀ j ∈ [], r j ∈ A (i j)\n⊢ (List.map r []).prod ∈ A (List.map i []).sum"
},
{
"line": "skip",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nh : ∀ j ∈ [], r j ∈ A (i j)\n⊢ (List.map r []).prod ∈ A (List.map i []).sum",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nh : ∀ j ∈ [], r j ∈ A (i j)\n⊢ (List.map r []).prod ∈ A (List.map i []).sum"
},
{
"line": "rw [List.map_nil]",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nh : ∀ j ∈ [], r j ∈ A (i j)\n⊢ (List.map r []).prod ∈ A (List.map i []).sum",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nh : ∀ j ∈ [], r j ∈ A (i j)\n⊢ (List.map r []).prod ∈ A [].sum"
},
{
"line": "rewrite [List.map_nil]",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nh : ∀ j ∈ [], r j ∈ A (i j)\n⊢ (List.map r []).prod ∈ A (List.map i []).sum",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nh : ∀ j ∈ [], r j ∈ A (i j)\n⊢ (List.map r []).prod ∈ A [].sum"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nh : ∀ j ∈ [], r j ∈ A (i j)\n⊢ (List.map r []).prod ∈ A [].sum",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nh : ∀ j ∈ [], r j ∈ A (i j)\n⊢ (List.map r []).prod ∈ A [].sum"
},
{
"line": "try (with_reducible rfl)",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nh : ∀ j ∈ [], r j ∈ A (i j)\n⊢ (List.map r []).prod ∈ A [].sum",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nh : ∀ j ∈ [], r j ∈ A (i j)\n⊢ (List.map r []).prod ∈ A [].sum"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nh : ∀ j ∈ [], r j ∈ A (i j)\n⊢ (List.map r []).prod ∈ A [].sum",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nh : ∀ j ∈ [], r j ∈ A (i j)\n⊢ (List.map r []).prod ∈ A [].sum"
},
{
"line": "with_reducible rfl",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nh : ∀ j ∈ [], r j ∈ A (i j)\n⊢ (List.map r []).prod ∈ A [].sum",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nh : ∀ j ∈ [], r j ∈ A (i j)\n⊢ (List.map r []).prod ∈ A [].sum"
},
{
"line": "rfl",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nh : ∀ j ∈ [], r j ∈ A (i j)\n⊢ (List.map r []).prod ∈ A [].sum",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nh : ∀ j ∈ [], r j ∈ A (i j)\n⊢ (List.map r []).prod ∈ A [].sum"
},
{
"line": "apply_rfl",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nh : ∀ j ∈ [], r j ∈ A (i j)\n⊢ (List.map r []).prod ∈ A [].sum",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nh : ∀ j ∈ [], r j ∈ A (i j)\n⊢ (List.map r []).prod ∈ A [].sum"
},
{
"line": "skip",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nh : ∀ j ∈ [], r j ∈ A (i j)\n⊢ (List.map r []).prod ∈ A [].sum",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nh : ∀ j ∈ [], r j ∈ A (i j)\n⊢ (List.map r []).prod ∈ A [].sum"
},
{
"line": "rw [List.map_nil]",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nh : ∀ j ∈ [], r j ∈ A (i j)\n⊢ (List.map r []).prod ∈ A [].sum",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nh : ∀ j ∈ [], r j ∈ A (i j)\n⊢ [].prod ∈ A [].sum"
},
{
"line": "rewrite [List.map_nil]",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nh : ∀ j ∈ [], r j ∈ A (i j)\n⊢ (List.map r []).prod ∈ A [].sum",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nh : ∀ j ∈ [], r j ∈ A (i j)\n⊢ [].prod ∈ A [].sum"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nh : ∀ j ∈ [], r j ∈ A (i j)\n⊢ [].prod ∈ A [].sum",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nh : ∀ j ∈ [], r j ∈ A (i j)\n⊢ [].prod ∈ A [].sum"
},
{
"line": "try (with_reducible rfl)",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nh : ∀ j ∈ [], r j ∈ A (i j)\n⊢ [].prod ∈ A [].sum",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nh : ∀ j ∈ [], r j ∈ A (i j)\n⊢ [].prod ∈ A [].sum"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nh : ∀ j ∈ [], r j ∈ A (i j)\n⊢ [].prod ∈ A [].sum",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nh : ∀ j ∈ [], r j ∈ A (i j)\n⊢ [].prod ∈ A [].sum"
},
{
"line": "with_reducible rfl",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nh : ∀ j ∈ [], r j ∈ A (i j)\n⊢ [].prod ∈ A [].sum",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nh : ∀ j ∈ [], r j ∈ A (i j)\n⊢ [].prod ∈ A [].sum"
},
{
"line": "rfl",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nh : ∀ j ∈ [], r j ∈ A (i j)\n⊢ [].prod ∈ A [].sum",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nh : ∀ j ∈ [], r j ∈ A (i j)\n⊢ [].prod ∈ A [].sum"
},
{
"line": "apply_rfl",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nh : ∀ j ∈ [], r j ∈ A (i j)\n⊢ [].prod ∈ A [].sum",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nh : ∀ j ∈ [], r j ∈ A (i j)\n⊢ [].prod ∈ A [].sum"
},
{
"line": "skip",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nh : ∀ j ∈ [], r j ∈ A (i j)\n⊢ [].prod ∈ A [].sum",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nh : ∀ j ∈ [], r j ∈ A (i j)\n⊢ [].prod ∈ A [].sum"
},
{
"line": "rw [List.prod_nil]",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nh : ∀ j ∈ [], r j ∈ A (i j)\n⊢ [].prod ∈ A [].sum",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nh : ∀ j ∈ [], r j ∈ A (i j)\n⊢ 1 ∈ A [].sum"
},
{
"line": "rewrite [List.prod_nil]",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nh : ∀ j ∈ [], r j ∈ A (i j)\n⊢ [].prod ∈ A [].sum",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nh : ∀ j ∈ [], r j ∈ A (i j)\n⊢ 1 ∈ A [].sum"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nh : ∀ j ∈ [], r j ∈ A (i j)\n⊢ 1 ∈ A [].sum",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nh : ∀ j ∈ [], r j ∈ A (i j)\n⊢ 1 ∈ A [].sum"
},
{
"line": "try (with_reducible rfl)",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nh : ∀ j ∈ [], r j ∈ A (i j)\n⊢ 1 ∈ A [].sum",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nh : ∀ j ∈ [], r j ∈ A (i j)\n⊢ 1 ∈ A [].sum"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nh : ∀ j ∈ [], r j ∈ A (i j)\n⊢ 1 ∈ A [].sum",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nh : ∀ j ∈ [], r j ∈ A (i j)\n⊢ 1 ∈ A [].sum"
},
{
"line": "with_reducible rfl",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nh : ∀ j ∈ [], r j ∈ A (i j)\n⊢ 1 ∈ A [].sum",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nh : ∀ j ∈ [], r j ∈ A (i j)\n⊢ 1 ∈ A [].sum"
},
{
"line": "rfl",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nh : ∀ j ∈ [], r j ∈ A (i j)\n⊢ 1 ∈ A [].sum",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nh : ∀ j ∈ [], r j ∈ A (i j)\n⊢ 1 ∈ A [].sum"
},
{
"line": "apply_rfl",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nh : ∀ j ∈ [], r j ∈ A (i j)\n⊢ 1 ∈ A [].sum",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nh : ∀ j ∈ [], r j ∈ A (i j)\n⊢ 1 ∈ A [].sum"
},
{
"line": "skip",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nh : ∀ j ∈ [], r j ∈ A (i j)\n⊢ 1 ∈ A [].sum",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nh : ∀ j ∈ [], r j ∈ A (i j)\n⊢ 1 ∈ A [].sum"
},
{
"line": "rw [List.sum_nil]",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nh : ∀ j ∈ [], r j ∈ A (i j)\n⊢ 1 ∈ A [].sum",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nh : ∀ j ∈ [], r j ∈ A (i j)\n⊢ 1 ∈ A 0"
},
{
"line": "rewrite [List.sum_nil]",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nh : ∀ j ∈ [], r j ∈ A (i j)\n⊢ 1 ∈ A [].sum",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nh : ∀ j ∈ [], r j ∈ A (i j)\n⊢ 1 ∈ A 0"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nh : ∀ j ∈ [], r j ∈ A (i j)\n⊢ 1 ∈ A 0",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nh : ∀ j ∈ [], r j ∈ A (i j)\n⊢ 1 ∈ A 0"
},
{
"line": "try (with_reducible rfl)",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nh : ∀ j ∈ [], r j ∈ A (i j)\n⊢ 1 ∈ A 0",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nh : ∀ j ∈ [], r j ∈ A (i j)\n⊢ 1 ∈ A 0"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nh : ∀ j ∈ [], r j ∈ A (i j)\n⊢ 1 ∈ A 0",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nh : ∀ j ∈ [], r j ∈ A (i j)\n⊢ 1 ∈ A 0"
},
{
"line": "with_reducible rfl",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nh : ∀ j ∈ [], r j ∈ A (i j)\n⊢ 1 ∈ A 0",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nh : ∀ j ∈ [], r j ∈ A (i j)\n⊢ 1 ∈ A 0"
},
{
"line": "rfl",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nh : ∀ j ∈ [], r j ∈ A (i j)\n⊢ 1 ∈ A 0",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nh : ∀ j ∈ [], r j ∈ A (i j)\n⊢ 1 ∈ A 0"
},
{
"line": "apply_rfl",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nh : ∀ j ∈ [], r j ∈ A (i j)\n⊢ 1 ∈ A 0",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nh : ∀ j ∈ [], r j ∈ A (i j)\n⊢ 1 ∈ A 0"
},
{
"line": "skip",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nh : ∀ j ∈ [], r j ∈ A (i j)\n⊢ 1 ∈ A 0",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nh : ∀ j ∈ [], r j ∈ A (i j)\n⊢ 1 ∈ A 0"
},
{
"line": "exact one_mem_graded _",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nh : ∀ j ∈ [], r j ∈ A (i j)\n⊢ 1 ∈ A 0",
"after_state": "No Goals!"
},
{
"line": "case rhs✝ =>\n with_annotate_state[\"|\" \"=>\"] skip\n rw [List.map_cons]\n rw [List.map_cons]\n rw [List.prod_cons]\n rw [List.sum_cons]\n exact\n mul_mem_graded (h _ List.mem_cons_self)\n (list_prod_map_mem_graded tail _ _ fun j hj => h _ <| List.mem_cons_of_mem _ hj)",
"before_state": "case rhs\nι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nhead : ι'\ntail : List ι'\nh : ∀ j ∈ head :: tail, r j ∈ A (i j)\n⊢ (List.map r (head :: tail)).prod ∈ A (List.map i (head :: tail)).sum",
"after_state": "No Goals!"
},
{
"line": "with_annotate_state[\"|\" \"=>\"] skip",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nhead : ι'\ntail : List ι'\nh : ∀ j ∈ head :: tail, r j ∈ A (i j)\n⊢ (List.map r (head :: tail)).prod ∈ A (List.map i (head :: tail)).sum",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nhead : ι'\ntail : List ι'\nh : ∀ j ∈ head :: tail, r j ∈ A (i j)\n⊢ (List.map r (head :: tail)).prod ∈ A (List.map i (head :: tail)).sum"
},
{
"line": "skip",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nhead : ι'\ntail : List ι'\nh : ∀ j ∈ head :: tail, r j ∈ A (i j)\n⊢ (List.map r (head :: tail)).prod ∈ A (List.map i (head :: tail)).sum",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nhead : ι'\ntail : List ι'\nh : ∀ j ∈ head :: tail, r j ∈ A (i j)\n⊢ (List.map r (head :: tail)).prod ∈ A (List.map i (head :: tail)).sum"
},
{
"line": "rw [List.map_cons]",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nhead : ι'\ntail : List ι'\nh : ∀ j ∈ head :: tail, r j ∈ A (i j)\n⊢ (List.map r (head :: tail)).prod ∈ A (List.map i (head :: tail)).sum",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nhead : ι'\ntail : List ι'\nh : ∀ j ∈ head :: tail, r j ∈ A (i j)\n⊢ (List.map r (head :: tail)).prod ∈ A (i head :: List.map i tail).sum"
},
{
"line": "rewrite [List.map_cons]",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nhead : ι'\ntail : List ι'\nh : ∀ j ∈ head :: tail, r j ∈ A (i j)\n⊢ (List.map r (head :: tail)).prod ∈ A (List.map i (head :: tail)).sum",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nhead : ι'\ntail : List ι'\nh : ∀ j ∈ head :: tail, r j ∈ A (i j)\n⊢ (List.map r (head :: tail)).prod ∈ A (i head :: List.map i tail).sum"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nhead : ι'\ntail : List ι'\nh : ∀ j ∈ head :: tail, r j ∈ A (i j)\n⊢ (List.map r (head :: tail)).prod ∈ A (i head :: List.map i tail).sum",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nhead : ι'\ntail : List ι'\nh : ∀ j ∈ head :: tail, r j ∈ A (i j)\n⊢ (List.map r (head :: tail)).prod ∈ A (i head :: List.map i tail).sum"
},
{
"line": "try (with_reducible rfl)",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nhead : ι'\ntail : List ι'\nh : ∀ j ∈ head :: tail, r j ∈ A (i j)\n⊢ (List.map r (head :: tail)).prod ∈ A (i head :: List.map i tail).sum",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nhead : ι'\ntail : List ι'\nh : ∀ j ∈ head :: tail, r j ∈ A (i j)\n⊢ (List.map r (head :: tail)).prod ∈ A (i head :: List.map i tail).sum"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nhead : ι'\ntail : List ι'\nh : ∀ j ∈ head :: tail, r j ∈ A (i j)\n⊢ (List.map r (head :: tail)).prod ∈ A (i head :: List.map i tail).sum",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nhead : ι'\ntail : List ι'\nh : ∀ j ∈ head :: tail, r j ∈ A (i j)\n⊢ (List.map r (head :: tail)).prod ∈ A (i head :: List.map i tail).sum"
},
{
"line": "with_reducible rfl",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nhead : ι'\ntail : List ι'\nh : ∀ j ∈ head :: tail, r j ∈ A (i j)\n⊢ (List.map r (head :: tail)).prod ∈ A (i head :: List.map i tail).sum",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nhead : ι'\ntail : List ι'\nh : ∀ j ∈ head :: tail, r j ∈ A (i j)\n⊢ (List.map r (head :: tail)).prod ∈ A (i head :: List.map i tail).sum"
},
{
"line": "rfl",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nhead : ι'\ntail : List ι'\nh : ∀ j ∈ head :: tail, r j ∈ A (i j)\n⊢ (List.map r (head :: tail)).prod ∈ A (i head :: List.map i tail).sum",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nhead : ι'\ntail : List ι'\nh : ∀ j ∈ head :: tail, r j ∈ A (i j)\n⊢ (List.map r (head :: tail)).prod ∈ A (i head :: List.map i tail).sum"
},
{
"line": "apply_rfl",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nhead : ι'\ntail : List ι'\nh : ∀ j ∈ head :: tail, r j ∈ A (i j)\n⊢ (List.map r (head :: tail)).prod ∈ A (i head :: List.map i tail).sum",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nhead : ι'\ntail : List ι'\nh : ∀ j ∈ head :: tail, r j ∈ A (i j)\n⊢ (List.map r (head :: tail)).prod ∈ A (i head :: List.map i tail).sum"
},
{
"line": "skip",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nhead : ι'\ntail : List ι'\nh : ∀ j ∈ head :: tail, r j ∈ A (i j)\n⊢ (List.map r (head :: tail)).prod ∈ A (i head :: List.map i tail).sum",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nhead : ι'\ntail : List ι'\nh : ∀ j ∈ head :: tail, r j ∈ A (i j)\n⊢ (List.map r (head :: tail)).prod ∈ A (i head :: List.map i tail).sum"
},
{
"line": "rw [List.map_cons]",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nhead : ι'\ntail : List ι'\nh : ∀ j ∈ head :: tail, r j ∈ A (i j)\n⊢ (List.map r (head :: tail)).prod ∈ A (i head :: List.map i tail).sum",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nhead : ι'\ntail : List ι'\nh : ∀ j ∈ head :: tail, r j ∈ A (i j)\n⊢ (r head :: List.map r tail).prod ∈ A (i head :: List.map i tail).sum"
},
{
"line": "rewrite [List.map_cons]",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nhead : ι'\ntail : List ι'\nh : ∀ j ∈ head :: tail, r j ∈ A (i j)\n⊢ (List.map r (head :: tail)).prod ∈ A (i head :: List.map i tail).sum",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nhead : ι'\ntail : List ι'\nh : ∀ j ∈ head :: tail, r j ∈ A (i j)\n⊢ (r head :: List.map r tail).prod ∈ A (i head :: List.map i tail).sum"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nhead : ι'\ntail : List ι'\nh : ∀ j ∈ head :: tail, r j ∈ A (i j)\n⊢ (r head :: List.map r tail).prod ∈ A (i head :: List.map i tail).sum",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nhead : ι'\ntail : List ι'\nh : ∀ j ∈ head :: tail, r j ∈ A (i j)\n⊢ (r head :: List.map r tail).prod ∈ A (i head :: List.map i tail).sum"
},
{
"line": "try (with_reducible rfl)",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nhead : ι'\ntail : List ι'\nh : ∀ j ∈ head :: tail, r j ∈ A (i j)\n⊢ (r head :: List.map r tail).prod ∈ A (i head :: List.map i tail).sum",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nhead : ι'\ntail : List ι'\nh : ∀ j ∈ head :: tail, r j ∈ A (i j)\n⊢ (r head :: List.map r tail).prod ∈ A (i head :: List.map i tail).sum"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nhead : ι'\ntail : List ι'\nh : ∀ j ∈ head :: tail, r j ∈ A (i j)\n⊢ (r head :: List.map r tail).prod ∈ A (i head :: List.map i tail).sum",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nhead : ι'\ntail : List ι'\nh : ∀ j ∈ head :: tail, r j ∈ A (i j)\n⊢ (r head :: List.map r tail).prod ∈ A (i head :: List.map i tail).sum"
},
{
"line": "with_reducible rfl",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nhead : ι'\ntail : List ι'\nh : ∀ j ∈ head :: tail, r j ∈ A (i j)\n⊢ (r head :: List.map r tail).prod ∈ A (i head :: List.map i tail).sum",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nhead : ι'\ntail : List ι'\nh : ∀ j ∈ head :: tail, r j ∈ A (i j)\n⊢ (r head :: List.map r tail).prod ∈ A (i head :: List.map i tail).sum"
},
{
"line": "rfl",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nhead : ι'\ntail : List ι'\nh : ∀ j ∈ head :: tail, r j ∈ A (i j)\n⊢ (r head :: List.map r tail).prod ∈ A (i head :: List.map i tail).sum",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nhead : ι'\ntail : List ι'\nh : ∀ j ∈ head :: tail, r j ∈ A (i j)\n⊢ (r head :: List.map r tail).prod ∈ A (i head :: List.map i tail).sum"
},
{
"line": "apply_rfl",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nhead : ι'\ntail : List ι'\nh : ∀ j ∈ head :: tail, r j ∈ A (i j)\n⊢ (r head :: List.map r tail).prod ∈ A (i head :: List.map i tail).sum",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nhead : ι'\ntail : List ι'\nh : ∀ j ∈ head :: tail, r j ∈ A (i j)\n⊢ (r head :: List.map r tail).prod ∈ A (i head :: List.map i tail).sum"
},
{
"line": "skip",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nhead : ι'\ntail : List ι'\nh : ∀ j ∈ head :: tail, r j ∈ A (i j)\n⊢ (r head :: List.map r tail).prod ∈ A (i head :: List.map i tail).sum",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nhead : ι'\ntail : List ι'\nh : ∀ j ∈ head :: tail, r j ∈ A (i j)\n⊢ (r head :: List.map r tail).prod ∈ A (i head :: List.map i tail).sum"
},
{
"line": "rw [List.prod_cons]",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nhead : ι'\ntail : List ι'\nh : ∀ j ∈ head :: tail, r j ∈ A (i j)\n⊢ (r head :: List.map r tail).prod ∈ A (i head :: List.map i tail).sum",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nhead : ι'\ntail : List ι'\nh : ∀ j ∈ head :: tail, r j ∈ A (i j)\n⊢ r head * (List.map r tail).prod ∈ A (i head :: List.map i tail).sum"
},
{
"line": "rewrite [List.prod_cons]",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nhead : ι'\ntail : List ι'\nh : ∀ j ∈ head :: tail, r j ∈ A (i j)\n⊢ (r head :: List.map r tail).prod ∈ A (i head :: List.map i tail).sum",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nhead : ι'\ntail : List ι'\nh : ∀ j ∈ head :: tail, r j ∈ A (i j)\n⊢ r head * (List.map r tail).prod ∈ A (i head :: List.map i tail).sum"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nhead : ι'\ntail : List ι'\nh : ∀ j ∈ head :: tail, r j ∈ A (i j)\n⊢ r head * (List.map r tail).prod ∈ A (i head :: List.map i tail).sum",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nhead : ι'\ntail : List ι'\nh : ∀ j ∈ head :: tail, r j ∈ A (i j)\n⊢ r head * (List.map r tail).prod ∈ A (i head :: List.map i tail).sum"
},
{
"line": "try (with_reducible rfl)",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nhead : ι'\ntail : List ι'\nh : ∀ j ∈ head :: tail, r j ∈ A (i j)\n⊢ r head * (List.map r tail).prod ∈ A (i head :: List.map i tail).sum",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nhead : ι'\ntail : List ι'\nh : ∀ j ∈ head :: tail, r j ∈ A (i j)\n⊢ r head * (List.map r tail).prod ∈ A (i head :: List.map i tail).sum"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nhead : ι'\ntail : List ι'\nh : ∀ j ∈ head :: tail, r j ∈ A (i j)\n⊢ r head * (List.map r tail).prod ∈ A (i head :: List.map i tail).sum",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nhead : ι'\ntail : List ι'\nh : ∀ j ∈ head :: tail, r j ∈ A (i j)\n⊢ r head * (List.map r tail).prod ∈ A (i head :: List.map i tail).sum"
},
{
"line": "with_reducible rfl",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nhead : ι'\ntail : List ι'\nh : ∀ j ∈ head :: tail, r j ∈ A (i j)\n⊢ r head * (List.map r tail).prod ∈ A (i head :: List.map i tail).sum",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nhead : ι'\ntail : List ι'\nh : ∀ j ∈ head :: tail, r j ∈ A (i j)\n⊢ r head * (List.map r tail).prod ∈ A (i head :: List.map i tail).sum"
},
{
"line": "rfl",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nhead : ι'\ntail : List ι'\nh : ∀ j ∈ head :: tail, r j ∈ A (i j)\n⊢ r head * (List.map r tail).prod ∈ A (i head :: List.map i tail).sum",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nhead : ι'\ntail : List ι'\nh : ∀ j ∈ head :: tail, r j ∈ A (i j)\n⊢ r head * (List.map r tail).prod ∈ A (i head :: List.map i tail).sum"
},
{
"line": "apply_rfl",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nhead : ι'\ntail : List ι'\nh : ∀ j ∈ head :: tail, r j ∈ A (i j)\n⊢ r head * (List.map r tail).prod ∈ A (i head :: List.map i tail).sum",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nhead : ι'\ntail : List ι'\nh : ∀ j ∈ head :: tail, r j ∈ A (i j)\n⊢ r head * (List.map r tail).prod ∈ A (i head :: List.map i tail).sum"
},
{
"line": "skip",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nhead : ι'\ntail : List ι'\nh : ∀ j ∈ head :: tail, r j ∈ A (i j)\n⊢ r head * (List.map r tail).prod ∈ A (i head :: List.map i tail).sum",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nhead : ι'\ntail : List ι'\nh : ∀ j ∈ head :: tail, r j ∈ A (i j)\n⊢ r head * (List.map r tail).prod ∈ A (i head :: List.map i tail).sum"
},
{
"line": "rw [List.sum_cons]",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nhead : ι'\ntail : List ι'\nh : ∀ j ∈ head :: tail, r j ∈ A (i j)\n⊢ r head * (List.map r tail).prod ∈ A (i head :: List.map i tail).sum",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nhead : ι'\ntail : List ι'\nh : ∀ j ∈ head :: tail, r j ∈ A (i j)\n⊢ r head * (List.map r tail).prod ∈ A (i head + (List.map i tail).sum)"
},
{
"line": "rewrite [List.sum_cons]",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nhead : ι'\ntail : List ι'\nh : ∀ j ∈ head :: tail, r j ∈ A (i j)\n⊢ r head * (List.map r tail).prod ∈ A (i head :: List.map i tail).sum",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nhead : ι'\ntail : List ι'\nh : ∀ j ∈ head :: tail, r j ∈ A (i j)\n⊢ r head * (List.map r tail).prod ∈ A (i head + (List.map i tail).sum)"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nhead : ι'\ntail : List ι'\nh : ∀ j ∈ head :: tail, r j ∈ A (i j)\n⊢ r head * (List.map r tail).prod ∈ A (i head + (List.map i tail).sum)",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nhead : ι'\ntail : List ι'\nh : ∀ j ∈ head :: tail, r j ∈ A (i j)\n⊢ r head * (List.map r tail).prod ∈ A (i head + (List.map i tail).sum)"
},
{
"line": "try (with_reducible rfl)",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nhead : ι'\ntail : List ι'\nh : ∀ j ∈ head :: tail, r j ∈ A (i j)\n⊢ r head * (List.map r tail).prod ∈ A (i head + (List.map i tail).sum)",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nhead : ι'\ntail : List ι'\nh : ∀ j ∈ head :: tail, r j ∈ A (i j)\n⊢ r head * (List.map r tail).prod ∈ A (i head + (List.map i tail).sum)"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nhead : ι'\ntail : List ι'\nh : ∀ j ∈ head :: tail, r j ∈ A (i j)\n⊢ r head * (List.map r tail).prod ∈ A (i head + (List.map i tail).sum)",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nhead : ι'\ntail : List ι'\nh : ∀ j ∈ head :: tail, r j ∈ A (i j)\n⊢ r head * (List.map r tail).prod ∈ A (i head + (List.map i tail).sum)"
},
{
"line": "with_reducible rfl",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nhead : ι'\ntail : List ι'\nh : ∀ j ∈ head :: tail, r j ∈ A (i j)\n⊢ r head * (List.map r tail).prod ∈ A (i head + (List.map i tail).sum)",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nhead : ι'\ntail : List ι'\nh : ∀ j ∈ head :: tail, r j ∈ A (i j)\n⊢ r head * (List.map r tail).prod ∈ A (i head + (List.map i tail).sum)"
},
{
"line": "rfl",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nhead : ι'\ntail : List ι'\nh : ∀ j ∈ head :: tail, r j ∈ A (i j)\n⊢ r head * (List.map r tail).prod ∈ A (i head + (List.map i tail).sum)",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nhead : ι'\ntail : List ι'\nh : ∀ j ∈ head :: tail, r j ∈ A (i j)\n⊢ r head * (List.map r tail).prod ∈ A (i head + (List.map i tail).sum)"
},
{
"line": "apply_rfl",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nhead : ι'\ntail : List ι'\nh : ∀ j ∈ head :: tail, r j ∈ A (i j)\n⊢ r head * (List.map r tail).prod ∈ A (i head + (List.map i tail).sum)",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nhead : ι'\ntail : List ι'\nh : ∀ j ∈ head :: tail, r j ∈ A (i j)\n⊢ r head * (List.map r tail).prod ∈ A (i head + (List.map i tail).sum)"
},
{
"line": "skip",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nhead : ι'\ntail : List ι'\nh : ∀ j ∈ head :: tail, r j ∈ A (i j)\n⊢ r head * (List.map r tail).prod ∈ A (i head + (List.map i tail).sum)",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nhead : ι'\ntail : List ι'\nh : ∀ j ∈ head :: tail, r j ∈ A (i j)\n⊢ r head * (List.map r tail).prod ∈ A (i head + (List.map i tail).sum)"
},
{
"line": "exact\n mul_mem_graded (h _ List.mem_cons_self)\n (list_prod_map_mem_graded tail _ _ fun j hj => h _ <| List.mem_cons_of_mem _ hj)",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nι' : Type u_10\nl : List ι'\ni : ι' → ι\nr : ι' → R\nhead : ι'\ntail : List ι'\nh : ∀ j ∈ head :: tail, r j ∈ A (i j)\n⊢ r head * (List.map r tail).prod ∈ A (i head + (List.map i tail).sum)",
"after_state": "No Goals!"
}
] |
theorem list_prod_ofFn_mem_graded {n} (i : Fin n → ι) (r : Fin n → R) (h : ∀ j, r j ∈ A (i j)) :
(List.ofFn r).prod ∈ A (List.ofFn i).sum := by
rw [List.ofFn_eq_map]
rw [List.ofFn_eq_map]
exact list_prod_map_mem_graded _ _ _ fun _ _ => h _
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/GradedMonoid.lean
|
{
"open": [],
"variables": [
"{ι : Type*}",
"{α β} {A : ι → Type*}",
"(A : ι → Type*)",
"{A}",
"[AddMonoid ι] [GMul A] [GOne A]",
"(A : ι → Type*)",
"[Zero ι] [GOne A]",
"[AddZeroClass ι] [GMul A]",
"{A}",
"[AddMonoid ι] [GMonoid A]",
"{A} in",
"[AddCommMonoid ι] [GCommMonoid A]",
"[AddMonoid ι] [GMonoid A]",
"{α : Type*} {A : ι → Type*} [AddMonoid ι] [GradedMonoid.GMonoid A]",
"(ι) {R : Type*}",
"{R : Type*}",
"{S : Type*} [SetLike S R] [Monoid R] [AddMonoid ι]",
"{A : ι → S} [SetLike.GradedMonoid A]",
"(A) in"
]
}
|
[
{
"line": "rw [List.ofFn_eq_map]",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn : ℕ\ni : Fin n → ι\nr : Fin n → R\nh : ∀ (j : Fin n), r j ∈ A (i j)\n⊢ (List.ofFn r).prod ∈ A (List.ofFn i).sum",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn : ℕ\ni : Fin n → ι\nr : Fin n → R\nh : ∀ (j : Fin n), r j ∈ A (i j)\n⊢ (List.ofFn r).prod ∈ A (List.map i (List.finRange n)).sum"
},
{
"line": "rewrite [List.ofFn_eq_map]",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn : ℕ\ni : Fin n → ι\nr : Fin n → R\nh : ∀ (j : Fin n), r j ∈ A (i j)\n⊢ (List.ofFn r).prod ∈ A (List.ofFn i).sum",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn : ℕ\ni : Fin n → ι\nr : Fin n → R\nh : ∀ (j : Fin n), r j ∈ A (i j)\n⊢ (List.ofFn r).prod ∈ A (List.map i (List.finRange n)).sum"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn : ℕ\ni : Fin n → ι\nr : Fin n → R\nh : ∀ (j : Fin n), r j ∈ A (i j)\n⊢ (List.ofFn r).prod ∈ A (List.map i (List.finRange n)).sum",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn : ℕ\ni : Fin n → ι\nr : Fin n → R\nh : ∀ (j : Fin n), r j ∈ A (i j)\n⊢ (List.ofFn r).prod ∈ A (List.map i (List.finRange n)).sum"
},
{
"line": "try (with_reducible rfl)",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn : ℕ\ni : Fin n → ι\nr : Fin n → R\nh : ∀ (j : Fin n), r j ∈ A (i j)\n⊢ (List.ofFn r).prod ∈ A (List.map i (List.finRange n)).sum",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn : ℕ\ni : Fin n → ι\nr : Fin n → R\nh : ∀ (j : Fin n), r j ∈ A (i j)\n⊢ (List.ofFn r).prod ∈ A (List.map i (List.finRange n)).sum"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn : ℕ\ni : Fin n → ι\nr : Fin n → R\nh : ∀ (j : Fin n), r j ∈ A (i j)\n⊢ (List.ofFn r).prod ∈ A (List.map i (List.finRange n)).sum",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn : ℕ\ni : Fin n → ι\nr : Fin n → R\nh : ∀ (j : Fin n), r j ∈ A (i j)\n⊢ (List.ofFn r).prod ∈ A (List.map i (List.finRange n)).sum"
},
{
"line": "with_reducible rfl",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn : ℕ\ni : Fin n → ι\nr : Fin n → R\nh : ∀ (j : Fin n), r j ∈ A (i j)\n⊢ (List.ofFn r).prod ∈ A (List.map i (List.finRange n)).sum",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn : ℕ\ni : Fin n → ι\nr : Fin n → R\nh : ∀ (j : Fin n), r j ∈ A (i j)\n⊢ (List.ofFn r).prod ∈ A (List.map i (List.finRange n)).sum"
},
{
"line": "rfl",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn : ℕ\ni : Fin n → ι\nr : Fin n → R\nh : ∀ (j : Fin n), r j ∈ A (i j)\n⊢ (List.ofFn r).prod ∈ A (List.map i (List.finRange n)).sum",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn : ℕ\ni : Fin n → ι\nr : Fin n → R\nh : ∀ (j : Fin n), r j ∈ A (i j)\n⊢ (List.ofFn r).prod ∈ A (List.map i (List.finRange n)).sum"
},
{
"line": "apply_rfl",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn : ℕ\ni : Fin n → ι\nr : Fin n → R\nh : ∀ (j : Fin n), r j ∈ A (i j)\n⊢ (List.ofFn r).prod ∈ A (List.map i (List.finRange n)).sum",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn : ℕ\ni : Fin n → ι\nr : Fin n → R\nh : ∀ (j : Fin n), r j ∈ A (i j)\n⊢ (List.ofFn r).prod ∈ A (List.map i (List.finRange n)).sum"
},
{
"line": "skip",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn : ℕ\ni : Fin n → ι\nr : Fin n → R\nh : ∀ (j : Fin n), r j ∈ A (i j)\n⊢ (List.ofFn r).prod ∈ A (List.map i (List.finRange n)).sum",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn : ℕ\ni : Fin n → ι\nr : Fin n → R\nh : ∀ (j : Fin n), r j ∈ A (i j)\n⊢ (List.ofFn r).prod ∈ A (List.map i (List.finRange n)).sum"
},
{
"line": "rw [List.ofFn_eq_map]",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn : ℕ\ni : Fin n → ι\nr : Fin n → R\nh : ∀ (j : Fin n), r j ∈ A (i j)\n⊢ (List.ofFn r).prod ∈ A (List.map i (List.finRange n)).sum",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn : ℕ\ni : Fin n → ι\nr : Fin n → R\nh : ∀ (j : Fin n), r j ∈ A (i j)\n⊢ (List.map r (List.finRange n)).prod ∈ A (List.map i (List.finRange n)).sum"
},
{
"line": "rewrite [List.ofFn_eq_map]",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn : ℕ\ni : Fin n → ι\nr : Fin n → R\nh : ∀ (j : Fin n), r j ∈ A (i j)\n⊢ (List.ofFn r).prod ∈ A (List.map i (List.finRange n)).sum",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn : ℕ\ni : Fin n → ι\nr : Fin n → R\nh : ∀ (j : Fin n), r j ∈ A (i j)\n⊢ (List.map r (List.finRange n)).prod ∈ A (List.map i (List.finRange n)).sum"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn : ℕ\ni : Fin n → ι\nr : Fin n → R\nh : ∀ (j : Fin n), r j ∈ A (i j)\n⊢ (List.map r (List.finRange n)).prod ∈ A (List.map i (List.finRange n)).sum",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn : ℕ\ni : Fin n → ι\nr : Fin n → R\nh : ∀ (j : Fin n), r j ∈ A (i j)\n⊢ (List.map r (List.finRange n)).prod ∈ A (List.map i (List.finRange n)).sum"
},
{
"line": "try (with_reducible rfl)",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn : ℕ\ni : Fin n → ι\nr : Fin n → R\nh : ∀ (j : Fin n), r j ∈ A (i j)\n⊢ (List.map r (List.finRange n)).prod ∈ A (List.map i (List.finRange n)).sum",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn : ℕ\ni : Fin n → ι\nr : Fin n → R\nh : ∀ (j : Fin n), r j ∈ A (i j)\n⊢ (List.map r (List.finRange n)).prod ∈ A (List.map i (List.finRange n)).sum"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn : ℕ\ni : Fin n → ι\nr : Fin n → R\nh : ∀ (j : Fin n), r j ∈ A (i j)\n⊢ (List.map r (List.finRange n)).prod ∈ A (List.map i (List.finRange n)).sum",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn : ℕ\ni : Fin n → ι\nr : Fin n → R\nh : ∀ (j : Fin n), r j ∈ A (i j)\n⊢ (List.map r (List.finRange n)).prod ∈ A (List.map i (List.finRange n)).sum"
},
{
"line": "with_reducible rfl",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn : ℕ\ni : Fin n → ι\nr : Fin n → R\nh : ∀ (j : Fin n), r j ∈ A (i j)\n⊢ (List.map r (List.finRange n)).prod ∈ A (List.map i (List.finRange n)).sum",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn : ℕ\ni : Fin n → ι\nr : Fin n → R\nh : ∀ (j : Fin n), r j ∈ A (i j)\n⊢ (List.map r (List.finRange n)).prod ∈ A (List.map i (List.finRange n)).sum"
},
{
"line": "rfl",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn : ℕ\ni : Fin n → ι\nr : Fin n → R\nh : ∀ (j : Fin n), r j ∈ A (i j)\n⊢ (List.map r (List.finRange n)).prod ∈ A (List.map i (List.finRange n)).sum",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn : ℕ\ni : Fin n → ι\nr : Fin n → R\nh : ∀ (j : Fin n), r j ∈ A (i j)\n⊢ (List.map r (List.finRange n)).prod ∈ A (List.map i (List.finRange n)).sum"
},
{
"line": "apply_rfl",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn : ℕ\ni : Fin n → ι\nr : Fin n → R\nh : ∀ (j : Fin n), r j ∈ A (i j)\n⊢ (List.map r (List.finRange n)).prod ∈ A (List.map i (List.finRange n)).sum",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn : ℕ\ni : Fin n → ι\nr : Fin n → R\nh : ∀ (j : Fin n), r j ∈ A (i j)\n⊢ (List.map r (List.finRange n)).prod ∈ A (List.map i (List.finRange n)).sum"
},
{
"line": "skip",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn : ℕ\ni : Fin n → ι\nr : Fin n → R\nh : ∀ (j : Fin n), r j ∈ A (i j)\n⊢ (List.map r (List.finRange n)).prod ∈ A (List.map i (List.finRange n)).sum",
"after_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn : ℕ\ni : Fin n → ι\nr : Fin n → R\nh : ∀ (j : Fin n), r j ∈ A (i j)\n⊢ (List.map r (List.finRange n)).prod ∈ A (List.map i (List.finRange n)).sum"
},
{
"line": "exact list_prod_map_mem_graded _ _ _ fun _ _ => h _",
"before_state": "ι : Type u_1\ninst✝⁴ : AddMonoid ι\nR : Type u_8\nS : Type u_9\ninst✝³ : SetLike S R\ninst✝² : Monoid R\ninst✝¹ : AddMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nn : ℕ\ni : Fin n → ι\nr : Fin n → R\nh : ∀ (j : Fin n), r j ∈ A (i j)\n⊢ (List.map r (List.finRange n)).prod ∈ A (List.map i (List.finRange n)).sum",
"after_state": "No Goals!"
}
] |
theorem prod_mem_graded (hF : ∀ k ∈ F, g k ∈ A (i k)) : ∏ k ∈ F, g k ∈ A (∑ k ∈ F, i k) := by
classical
induction F using Finset.induction_on
· simp [GradedOne.one_mem]
· case insert j F' hF2 h3 =>
rw [Finset.prod_insert hF2]
rw [Finset.sum_insert hF2]
apply SetLike.mul_mem_graded (hF j <| Finset.mem_insert_self j F')
apply h3
intro k hk
apply hF k
exact Finset.mem_insert_of_mem hk
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/GradedMonoid.lean
|
{
"open": [
"SetLike SetLike.GradedMonoid"
],
"variables": [
"{ι : Type*}",
"{α β} {A : ι → Type*}",
"(A : ι → Type*)",
"{A}",
"[AddMonoid ι] [GMul A] [GOne A]",
"(A : ι → Type*)",
"[Zero ι] [GOne A]",
"[AddZeroClass ι] [GMul A]",
"{A}",
"[AddMonoid ι] [GMonoid A]",
"{A} in",
"[AddCommMonoid ι] [GCommMonoid A]",
"[AddMonoid ι] [GMonoid A]",
"{α : Type*} {A : ι → Type*} [AddMonoid ι] [GradedMonoid.GMonoid A]",
"(ι) {R : Type*}",
"{R : Type*}",
"{S : Type*} [SetLike S R] [Monoid R] [AddMonoid ι]",
"{A : ι → S} [SetLike.GradedMonoid A]",
"(A) in",
"{α S : Type*} [SetLike S R] [Monoid R] [AddMonoid ι]",
"{R S : Type*} [SetLike S R]",
"{ι R S : Type*} [SetLike S R] [CommMonoid R] [AddCommMonoid ι]",
"(A : ι → S) [SetLike.GradedMonoid A]",
"{κ : Type*} (i : κ → ι) (g : κ → R) {F : Finset κ}"
]
}
|
[
{
"line": "classical\ninduction F using Finset.induction_on\n· simp [GradedOne.one_mem]\n·\n case insert j F' hF2 h3 =>\n rw [Finset.prod_insert hF2]\n rw [Finset.sum_insert hF2]\n apply SetLike.mul_mem_graded (hF j <| Finset.mem_insert_self j F')\n apply h3\n intro k hk\n apply hF k\n exact Finset.mem_insert_of_mem hk",
"before_state": "ι : Type u_12\nR : Type u_13\nS : Type u_14\ninst✝³ : SetLike S R\ninst✝² : CommMonoid R\ninst✝¹ : AddCommMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nκ : Type u_15\ni : κ → ι\ng : κ → R\nF : Finset κ\nhF : ∀ k ∈ F, g k ∈ A (i k)\n⊢ ∏ k ∈ F, g k ∈ A (∑ k ∈ F, i k)",
"after_state": "No Goals!"
},
{
"line": "induction F using Finset.induction_on",
"before_state": "ι : Type u_12\nR : Type u_13\nS : Type u_14\ninst✝³ : SetLike S R\ninst✝² : CommMonoid R\ninst✝¹ : AddCommMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nκ : Type u_15\ni : κ → ι\ng : κ → R\nF : Finset κ\nhF : ∀ k ∈ F, g k ∈ A (i k)\n⊢ ∏ k ∈ F, g k ∈ A (∑ k ∈ F, i k)",
"after_state": "case empty\nι : Type u_12\nR : Type u_13\nS : Type u_14\ninst✝³ : SetLike S R\ninst✝² : CommMonoid R\ninst✝¹ : AddCommMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nκ : Type u_15\ni : κ → ι\ng : κ → R\nF : Finset κ\nhF : ∀ k ∈ ∅, g k ∈ A (i k)\n⊢ ∏ k ∈ ∅, g k ∈ A (∑ k ∈ ∅, i k)\n---\ncase insert\nι : Type u_12\nR : Type u_13\nS : Type u_14\ninst✝³ : SetLike S R\ninst✝² : CommMonoid R\ninst✝¹ : AddCommMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nκ : Type u_15\ni : κ → ι\ng : κ → R\nF : Finset κ\na✝² : κ\ns✝ : Finset κ\na✝¹ : a✝² ∉ s✝\na✝ : (∀ k ∈ s✝, g k ∈ A (i k)) → ∏ k ∈ s✝, g k ∈ A (∑ k ∈ s✝, i k)\nhF : ∀ k ∈ insert a✝² s✝, g k ∈ A (i k)\n⊢ ∏ k ∈ insert a✝² s✝, g k ∈ A (∑ k ∈ insert a✝² s✝, i k)"
},
{
"line": "simp [GradedOne.one_mem]",
"before_state": "case empty\nι : Type u_12\nR : Type u_13\nS : Type u_14\ninst✝³ : SetLike S R\ninst✝² : CommMonoid R\ninst✝¹ : AddCommMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nκ : Type u_15\ni : κ → ι\ng : κ → R\nF : Finset κ\nhF : ∀ k ∈ ∅, g k ∈ A (i k)\n⊢ ∏ k ∈ ∅, g k ∈ A (∑ k ∈ ∅, i k)",
"after_state": "No Goals!"
},
{
"line": "case insert j F' hF2 h3 =>\n rw [Finset.prod_insert hF2]\n rw [Finset.sum_insert hF2]\n apply SetLike.mul_mem_graded (hF j <| Finset.mem_insert_self j F')\n apply h3\n intro k hk\n apply hF k\n exact Finset.mem_insert_of_mem hk",
"before_state": "case insert\nι : Type u_12\nR : Type u_13\nS : Type u_14\ninst✝³ : SetLike S R\ninst✝² : CommMonoid R\ninst✝¹ : AddCommMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nκ : Type u_15\ni : κ → ι\ng : κ → R\nF : Finset κ\na✝² : κ\ns✝ : Finset κ\na✝¹ : a✝² ∉ s✝\na✝ : (∀ k ∈ s✝, g k ∈ A (i k)) → ∏ k ∈ s✝, g k ∈ A (∑ k ∈ s✝, i k)\nhF : ∀ k ∈ insert a✝² s✝, g k ∈ A (i k)\n⊢ ∏ k ∈ insert a✝² s✝, g k ∈ A (∑ k ∈ insert a✝² s✝, i k)",
"after_state": "No Goals!"
},
{
"line": "rw [Finset.prod_insert hF2]",
"before_state": "ι : Type u_12\nR : Type u_13\nS : Type u_14\ninst✝³ : SetLike S R\ninst✝² : CommMonoid R\ninst✝¹ : AddCommMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nκ : Type u_15\ni : κ → ι\ng : κ → R\nF : Finset κ\nj : κ\nF' : Finset κ\nhF2 : j ∉ F'\nh3 : (∀ k ∈ F', g k ∈ A (i k)) → ∏ k ∈ F', g k ∈ A (∑ k ∈ F', i k)\nhF : ∀ k ∈ insert j F', g k ∈ A (i k)\n⊢ ∏ k ∈ insert j F', g k ∈ A (∑ k ∈ insert j F', i k)",
"after_state": "ι : Type u_12\nR : Type u_13\nS : Type u_14\ninst✝³ : SetLike S R\ninst✝² : CommMonoid R\ninst✝¹ : AddCommMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nκ : Type u_15\ni : κ → ι\ng : κ → R\nF : Finset κ\nj : κ\nF' : Finset κ\nhF2 : j ∉ F'\nh3 : (∀ k ∈ F', g k ∈ A (i k)) → ∏ k ∈ F', g k ∈ A (∑ k ∈ F', i k)\nhF : ∀ k ∈ insert j F', g k ∈ A (i k)\n⊢ g j * ∏ x ∈ F', g x ∈ A (∑ k ∈ insert j F', i k)"
},
{
"line": "rewrite [Finset.prod_insert hF2]",
"before_state": "ι : Type u_12\nR : Type u_13\nS : Type u_14\ninst✝³ : SetLike S R\ninst✝² : CommMonoid R\ninst✝¹ : AddCommMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nκ : Type u_15\ni : κ → ι\ng : κ → R\nF : Finset κ\nj : κ\nF' : Finset κ\nhF2 : j ∉ F'\nh3 : (∀ k ∈ F', g k ∈ A (i k)) → ∏ k ∈ F', g k ∈ A (∑ k ∈ F', i k)\nhF : ∀ k ∈ insert j F', g k ∈ A (i k)\n⊢ ∏ k ∈ insert j F', g k ∈ A (∑ k ∈ insert j F', i k)",
"after_state": "ι : Type u_12\nR : Type u_13\nS : Type u_14\ninst✝³ : SetLike S R\ninst✝² : CommMonoid R\ninst✝¹ : AddCommMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nκ : Type u_15\ni : κ → ι\ng : κ → R\nF : Finset κ\nj : κ\nF' : Finset κ\nhF2 : j ∉ F'\nh3 : (∀ k ∈ F', g k ∈ A (i k)) → ∏ k ∈ F', g k ∈ A (∑ k ∈ F', i k)\nhF : ∀ k ∈ insert j F', g k ∈ A (i k)\n⊢ g j * ∏ x ∈ F', g x ∈ A (∑ k ∈ insert j F', i k)"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "ι : Type u_12\nR : Type u_13\nS : Type u_14\ninst✝³ : SetLike S R\ninst✝² : CommMonoid R\ninst✝¹ : AddCommMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nκ : Type u_15\ni : κ → ι\ng : κ → R\nF : Finset κ\nj : κ\nF' : Finset κ\nhF2 : j ∉ F'\nh3 : (∀ k ∈ F', g k ∈ A (i k)) → ∏ k ∈ F', g k ∈ A (∑ k ∈ F', i k)\nhF : ∀ k ∈ insert j F', g k ∈ A (i k)\n⊢ g j * ∏ x ∈ F', g x ∈ A (∑ k ∈ insert j F', i k)",
"after_state": "ι : Type u_12\nR : Type u_13\nS : Type u_14\ninst✝³ : SetLike S R\ninst✝² : CommMonoid R\ninst✝¹ : AddCommMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nκ : Type u_15\ni : κ → ι\ng : κ → R\nF : Finset κ\nj : κ\nF' : Finset κ\nhF2 : j ∉ F'\nh3 : (∀ k ∈ F', g k ∈ A (i k)) → ∏ k ∈ F', g k ∈ A (∑ k ∈ F', i k)\nhF : ∀ k ∈ insert j F', g k ∈ A (i k)\n⊢ g j * ∏ x ∈ F', g x ∈ A (∑ k ∈ insert j F', i k)"
},
{
"line": "try (with_reducible rfl)",
"before_state": "ι : Type u_12\nR : Type u_13\nS : Type u_14\ninst✝³ : SetLike S R\ninst✝² : CommMonoid R\ninst✝¹ : AddCommMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nκ : Type u_15\ni : κ → ι\ng : κ → R\nF : Finset κ\nj : κ\nF' : Finset κ\nhF2 : j ∉ F'\nh3 : (∀ k ∈ F', g k ∈ A (i k)) → ∏ k ∈ F', g k ∈ A (∑ k ∈ F', i k)\nhF : ∀ k ∈ insert j F', g k ∈ A (i k)\n⊢ g j * ∏ x ∈ F', g x ∈ A (∑ k ∈ insert j F', i k)",
"after_state": "ι : Type u_12\nR : Type u_13\nS : Type u_14\ninst✝³ : SetLike S R\ninst✝² : CommMonoid R\ninst✝¹ : AddCommMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nκ : Type u_15\ni : κ → ι\ng : κ → R\nF : Finset κ\nj : κ\nF' : Finset κ\nhF2 : j ∉ F'\nh3 : (∀ k ∈ F', g k ∈ A (i k)) → ∏ k ∈ F', g k ∈ A (∑ k ∈ F', i k)\nhF : ∀ k ∈ insert j F', g k ∈ A (i k)\n⊢ g j * ∏ x ∈ F', g x ∈ A (∑ k ∈ insert j F', i k)"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "ι : Type u_12\nR : Type u_13\nS : Type u_14\ninst✝³ : SetLike S R\ninst✝² : CommMonoid R\ninst✝¹ : AddCommMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nκ : Type u_15\ni : κ → ι\ng : κ → R\nF : Finset κ\nj : κ\nF' : Finset κ\nhF2 : j ∉ F'\nh3 : (∀ k ∈ F', g k ∈ A (i k)) → ∏ k ∈ F', g k ∈ A (∑ k ∈ F', i k)\nhF : ∀ k ∈ insert j F', g k ∈ A (i k)\n⊢ g j * ∏ x ∈ F', g x ∈ A (∑ k ∈ insert j F', i k)",
"after_state": "ι : Type u_12\nR : Type u_13\nS : Type u_14\ninst✝³ : SetLike S R\ninst✝² : CommMonoid R\ninst✝¹ : AddCommMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nκ : Type u_15\ni : κ → ι\ng : κ → R\nF : Finset κ\nj : κ\nF' : Finset κ\nhF2 : j ∉ F'\nh3 : (∀ k ∈ F', g k ∈ A (i k)) → ∏ k ∈ F', g k ∈ A (∑ k ∈ F', i k)\nhF : ∀ k ∈ insert j F', g k ∈ A (i k)\n⊢ g j * ∏ x ∈ F', g x ∈ A (∑ k ∈ insert j F', i k)"
},
{
"line": "with_reducible rfl",
"before_state": "ι : Type u_12\nR : Type u_13\nS : Type u_14\ninst✝³ : SetLike S R\ninst✝² : CommMonoid R\ninst✝¹ : AddCommMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nκ : Type u_15\ni : κ → ι\ng : κ → R\nF : Finset κ\nj : κ\nF' : Finset κ\nhF2 : j ∉ F'\nh3 : (∀ k ∈ F', g k ∈ A (i k)) → ∏ k ∈ F', g k ∈ A (∑ k ∈ F', i k)\nhF : ∀ k ∈ insert j F', g k ∈ A (i k)\n⊢ g j * ∏ x ∈ F', g x ∈ A (∑ k ∈ insert j F', i k)",
"after_state": "ι : Type u_12\nR : Type u_13\nS : Type u_14\ninst✝³ : SetLike S R\ninst✝² : CommMonoid R\ninst✝¹ : AddCommMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nκ : Type u_15\ni : κ → ι\ng : κ → R\nF : Finset κ\nj : κ\nF' : Finset κ\nhF2 : j ∉ F'\nh3 : (∀ k ∈ F', g k ∈ A (i k)) → ∏ k ∈ F', g k ∈ A (∑ k ∈ F', i k)\nhF : ∀ k ∈ insert j F', g k ∈ A (i k)\n⊢ g j * ∏ x ∈ F', g x ∈ A (∑ k ∈ insert j F', i k)"
},
{
"line": "rfl",
"before_state": "ι : Type u_12\nR : Type u_13\nS : Type u_14\ninst✝³ : SetLike S R\ninst✝² : CommMonoid R\ninst✝¹ : AddCommMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nκ : Type u_15\ni : κ → ι\ng : κ → R\nF : Finset κ\nj : κ\nF' : Finset κ\nhF2 : j ∉ F'\nh3 : (∀ k ∈ F', g k ∈ A (i k)) → ∏ k ∈ F', g k ∈ A (∑ k ∈ F', i k)\nhF : ∀ k ∈ insert j F', g k ∈ A (i k)\n⊢ g j * ∏ x ∈ F', g x ∈ A (∑ k ∈ insert j F', i k)",
"after_state": "ι : Type u_12\nR : Type u_13\nS : Type u_14\ninst✝³ : SetLike S R\ninst✝² : CommMonoid R\ninst✝¹ : AddCommMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nκ : Type u_15\ni : κ → ι\ng : κ → R\nF : Finset κ\nj : κ\nF' : Finset κ\nhF2 : j ∉ F'\nh3 : (∀ k ∈ F', g k ∈ A (i k)) → ∏ k ∈ F', g k ∈ A (∑ k ∈ F', i k)\nhF : ∀ k ∈ insert j F', g k ∈ A (i k)\n⊢ g j * ∏ x ∈ F', g x ∈ A (∑ k ∈ insert j F', i k)"
},
{
"line": "apply_rfl",
"before_state": "ι : Type u_12\nR : Type u_13\nS : Type u_14\ninst✝³ : SetLike S R\ninst✝² : CommMonoid R\ninst✝¹ : AddCommMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nκ : Type u_15\ni : κ → ι\ng : κ → R\nF : Finset κ\nj : κ\nF' : Finset κ\nhF2 : j ∉ F'\nh3 : (∀ k ∈ F', g k ∈ A (i k)) → ∏ k ∈ F', g k ∈ A (∑ k ∈ F', i k)\nhF : ∀ k ∈ insert j F', g k ∈ A (i k)\n⊢ g j * ∏ x ∈ F', g x ∈ A (∑ k ∈ insert j F', i k)",
"after_state": "ι : Type u_12\nR : Type u_13\nS : Type u_14\ninst✝³ : SetLike S R\ninst✝² : CommMonoid R\ninst✝¹ : AddCommMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nκ : Type u_15\ni : κ → ι\ng : κ → R\nF : Finset κ\nj : κ\nF' : Finset κ\nhF2 : j ∉ F'\nh3 : (∀ k ∈ F', g k ∈ A (i k)) → ∏ k ∈ F', g k ∈ A (∑ k ∈ F', i k)\nhF : ∀ k ∈ insert j F', g k ∈ A (i k)\n⊢ g j * ∏ x ∈ F', g x ∈ A (∑ k ∈ insert j F', i k)"
},
{
"line": "skip",
"before_state": "ι : Type u_12\nR : Type u_13\nS : Type u_14\ninst✝³ : SetLike S R\ninst✝² : CommMonoid R\ninst✝¹ : AddCommMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nκ : Type u_15\ni : κ → ι\ng : κ → R\nF : Finset κ\nj : κ\nF' : Finset κ\nhF2 : j ∉ F'\nh3 : (∀ k ∈ F', g k ∈ A (i k)) → ∏ k ∈ F', g k ∈ A (∑ k ∈ F', i k)\nhF : ∀ k ∈ insert j F', g k ∈ A (i k)\n⊢ g j * ∏ x ∈ F', g x ∈ A (∑ k ∈ insert j F', i k)",
"after_state": "ι : Type u_12\nR : Type u_13\nS : Type u_14\ninst✝³ : SetLike S R\ninst✝² : CommMonoid R\ninst✝¹ : AddCommMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nκ : Type u_15\ni : κ → ι\ng : κ → R\nF : Finset κ\nj : κ\nF' : Finset κ\nhF2 : j ∉ F'\nh3 : (∀ k ∈ F', g k ∈ A (i k)) → ∏ k ∈ F', g k ∈ A (∑ k ∈ F', i k)\nhF : ∀ k ∈ insert j F', g k ∈ A (i k)\n⊢ g j * ∏ x ∈ F', g x ∈ A (∑ k ∈ insert j F', i k)"
},
{
"line": "rw [Finset.sum_insert hF2]",
"before_state": "ι : Type u_12\nR : Type u_13\nS : Type u_14\ninst✝³ : SetLike S R\ninst✝² : CommMonoid R\ninst✝¹ : AddCommMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nκ : Type u_15\ni : κ → ι\ng : κ → R\nF : Finset κ\nj : κ\nF' : Finset κ\nhF2 : j ∉ F'\nh3 : (∀ k ∈ F', g k ∈ A (i k)) → ∏ k ∈ F', g k ∈ A (∑ k ∈ F', i k)\nhF : ∀ k ∈ insert j F', g k ∈ A (i k)\n⊢ g j * ∏ x ∈ F', g x ∈ A (∑ k ∈ insert j F', i k)",
"after_state": "ι : Type u_12\nR : Type u_13\nS : Type u_14\ninst✝³ : SetLike S R\ninst✝² : CommMonoid R\ninst✝¹ : AddCommMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nκ : Type u_15\ni : κ → ι\ng : κ → R\nF : Finset κ\nj : κ\nF' : Finset κ\nhF2 : j ∉ F'\nh3 : (∀ k ∈ F', g k ∈ A (i k)) → ∏ k ∈ F', g k ∈ A (∑ k ∈ F', i k)\nhF : ∀ k ∈ insert j F', g k ∈ A (i k)\n⊢ g j * ∏ x ∈ F', g x ∈ A (i j + ∑ x ∈ F', i x)"
},
{
"line": "rewrite [Finset.sum_insert hF2]",
"before_state": "ι : Type u_12\nR : Type u_13\nS : Type u_14\ninst✝³ : SetLike S R\ninst✝² : CommMonoid R\ninst✝¹ : AddCommMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nκ : Type u_15\ni : κ → ι\ng : κ → R\nF : Finset κ\nj : κ\nF' : Finset κ\nhF2 : j ∉ F'\nh3 : (∀ k ∈ F', g k ∈ A (i k)) → ∏ k ∈ F', g k ∈ A (∑ k ∈ F', i k)\nhF : ∀ k ∈ insert j F', g k ∈ A (i k)\n⊢ g j * ∏ x ∈ F', g x ∈ A (∑ k ∈ insert j F', i k)",
"after_state": "ι : Type u_12\nR : Type u_13\nS : Type u_14\ninst✝³ : SetLike S R\ninst✝² : CommMonoid R\ninst✝¹ : AddCommMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nκ : Type u_15\ni : κ → ι\ng : κ → R\nF : Finset κ\nj : κ\nF' : Finset κ\nhF2 : j ∉ F'\nh3 : (∀ k ∈ F', g k ∈ A (i k)) → ∏ k ∈ F', g k ∈ A (∑ k ∈ F', i k)\nhF : ∀ k ∈ insert j F', g k ∈ A (i k)\n⊢ g j * ∏ x ∈ F', g x ∈ A (i j + ∑ x ∈ F', i x)"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "ι : Type u_12\nR : Type u_13\nS : Type u_14\ninst✝³ : SetLike S R\ninst✝² : CommMonoid R\ninst✝¹ : AddCommMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nκ : Type u_15\ni : κ → ι\ng : κ → R\nF : Finset κ\nj : κ\nF' : Finset κ\nhF2 : j ∉ F'\nh3 : (∀ k ∈ F', g k ∈ A (i k)) → ∏ k ∈ F', g k ∈ A (∑ k ∈ F', i k)\nhF : ∀ k ∈ insert j F', g k ∈ A (i k)\n⊢ g j * ∏ x ∈ F', g x ∈ A (i j + ∑ x ∈ F', i x)",
"after_state": "ι : Type u_12\nR : Type u_13\nS : Type u_14\ninst✝³ : SetLike S R\ninst✝² : CommMonoid R\ninst✝¹ : AddCommMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nκ : Type u_15\ni : κ → ι\ng : κ → R\nF : Finset κ\nj : κ\nF' : Finset κ\nhF2 : j ∉ F'\nh3 : (∀ k ∈ F', g k ∈ A (i k)) → ∏ k ∈ F', g k ∈ A (∑ k ∈ F', i k)\nhF : ∀ k ∈ insert j F', g k ∈ A (i k)\n⊢ g j * ∏ x ∈ F', g x ∈ A (i j + ∑ x ∈ F', i x)"
},
{
"line": "try (with_reducible rfl)",
"before_state": "ι : Type u_12\nR : Type u_13\nS : Type u_14\ninst✝³ : SetLike S R\ninst✝² : CommMonoid R\ninst✝¹ : AddCommMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nκ : Type u_15\ni : κ → ι\ng : κ → R\nF : Finset κ\nj : κ\nF' : Finset κ\nhF2 : j ∉ F'\nh3 : (∀ k ∈ F', g k ∈ A (i k)) → ∏ k ∈ F', g k ∈ A (∑ k ∈ F', i k)\nhF : ∀ k ∈ insert j F', g k ∈ A (i k)\n⊢ g j * ∏ x ∈ F', g x ∈ A (i j + ∑ x ∈ F', i x)",
"after_state": "ι : Type u_12\nR : Type u_13\nS : Type u_14\ninst✝³ : SetLike S R\ninst✝² : CommMonoid R\ninst✝¹ : AddCommMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nκ : Type u_15\ni : κ → ι\ng : κ → R\nF : Finset κ\nj : κ\nF' : Finset κ\nhF2 : j ∉ F'\nh3 : (∀ k ∈ F', g k ∈ A (i k)) → ∏ k ∈ F', g k ∈ A (∑ k ∈ F', i k)\nhF : ∀ k ∈ insert j F', g k ∈ A (i k)\n⊢ g j * ∏ x ∈ F', g x ∈ A (i j + ∑ x ∈ F', i x)"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "ι : Type u_12\nR : Type u_13\nS : Type u_14\ninst✝³ : SetLike S R\ninst✝² : CommMonoid R\ninst✝¹ : AddCommMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nκ : Type u_15\ni : κ → ι\ng : κ → R\nF : Finset κ\nj : κ\nF' : Finset κ\nhF2 : j ∉ F'\nh3 : (∀ k ∈ F', g k ∈ A (i k)) → ∏ k ∈ F', g k ∈ A (∑ k ∈ F', i k)\nhF : ∀ k ∈ insert j F', g k ∈ A (i k)\n⊢ g j * ∏ x ∈ F', g x ∈ A (i j + ∑ x ∈ F', i x)",
"after_state": "ι : Type u_12\nR : Type u_13\nS : Type u_14\ninst✝³ : SetLike S R\ninst✝² : CommMonoid R\ninst✝¹ : AddCommMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nκ : Type u_15\ni : κ → ι\ng : κ → R\nF : Finset κ\nj : κ\nF' : Finset κ\nhF2 : j ∉ F'\nh3 : (∀ k ∈ F', g k ∈ A (i k)) → ∏ k ∈ F', g k ∈ A (∑ k ∈ F', i k)\nhF : ∀ k ∈ insert j F', g k ∈ A (i k)\n⊢ g j * ∏ x ∈ F', g x ∈ A (i j + ∑ x ∈ F', i x)"
},
{
"line": "with_reducible rfl",
"before_state": "ι : Type u_12\nR : Type u_13\nS : Type u_14\ninst✝³ : SetLike S R\ninst✝² : CommMonoid R\ninst✝¹ : AddCommMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nκ : Type u_15\ni : κ → ι\ng : κ → R\nF : Finset κ\nj : κ\nF' : Finset κ\nhF2 : j ∉ F'\nh3 : (∀ k ∈ F', g k ∈ A (i k)) → ∏ k ∈ F', g k ∈ A (∑ k ∈ F', i k)\nhF : ∀ k ∈ insert j F', g k ∈ A (i k)\n⊢ g j * ∏ x ∈ F', g x ∈ A (i j + ∑ x ∈ F', i x)",
"after_state": "ι : Type u_12\nR : Type u_13\nS : Type u_14\ninst✝³ : SetLike S R\ninst✝² : CommMonoid R\ninst✝¹ : AddCommMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nκ : Type u_15\ni : κ → ι\ng : κ → R\nF : Finset κ\nj : κ\nF' : Finset κ\nhF2 : j ∉ F'\nh3 : (∀ k ∈ F', g k ∈ A (i k)) → ∏ k ∈ F', g k ∈ A (∑ k ∈ F', i k)\nhF : ∀ k ∈ insert j F', g k ∈ A (i k)\n⊢ g j * ∏ x ∈ F', g x ∈ A (i j + ∑ x ∈ F', i x)"
},
{
"line": "rfl",
"before_state": "ι : Type u_12\nR : Type u_13\nS : Type u_14\ninst✝³ : SetLike S R\ninst✝² : CommMonoid R\ninst✝¹ : AddCommMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nκ : Type u_15\ni : κ → ι\ng : κ → R\nF : Finset κ\nj : κ\nF' : Finset κ\nhF2 : j ∉ F'\nh3 : (∀ k ∈ F', g k ∈ A (i k)) → ∏ k ∈ F', g k ∈ A (∑ k ∈ F', i k)\nhF : ∀ k ∈ insert j F', g k ∈ A (i k)\n⊢ g j * ∏ x ∈ F', g x ∈ A (i j + ∑ x ∈ F', i x)",
"after_state": "ι : Type u_12\nR : Type u_13\nS : Type u_14\ninst✝³ : SetLike S R\ninst✝² : CommMonoid R\ninst✝¹ : AddCommMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nκ : Type u_15\ni : κ → ι\ng : κ → R\nF : Finset κ\nj : κ\nF' : Finset κ\nhF2 : j ∉ F'\nh3 : (∀ k ∈ F', g k ∈ A (i k)) → ∏ k ∈ F', g k ∈ A (∑ k ∈ F', i k)\nhF : ∀ k ∈ insert j F', g k ∈ A (i k)\n⊢ g j * ∏ x ∈ F', g x ∈ A (i j + ∑ x ∈ F', i x)"
},
{
"line": "apply_rfl",
"before_state": "ι : Type u_12\nR : Type u_13\nS : Type u_14\ninst✝³ : SetLike S R\ninst✝² : CommMonoid R\ninst✝¹ : AddCommMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nκ : Type u_15\ni : κ → ι\ng : κ → R\nF : Finset κ\nj : κ\nF' : Finset κ\nhF2 : j ∉ F'\nh3 : (∀ k ∈ F', g k ∈ A (i k)) → ∏ k ∈ F', g k ∈ A (∑ k ∈ F', i k)\nhF : ∀ k ∈ insert j F', g k ∈ A (i k)\n⊢ g j * ∏ x ∈ F', g x ∈ A (i j + ∑ x ∈ F', i x)",
"after_state": "ι : Type u_12\nR : Type u_13\nS : Type u_14\ninst✝³ : SetLike S R\ninst✝² : CommMonoid R\ninst✝¹ : AddCommMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nκ : Type u_15\ni : κ → ι\ng : κ → R\nF : Finset κ\nj : κ\nF' : Finset κ\nhF2 : j ∉ F'\nh3 : (∀ k ∈ F', g k ∈ A (i k)) → ∏ k ∈ F', g k ∈ A (∑ k ∈ F', i k)\nhF : ∀ k ∈ insert j F', g k ∈ A (i k)\n⊢ g j * ∏ x ∈ F', g x ∈ A (i j + ∑ x ∈ F', i x)"
},
{
"line": "skip",
"before_state": "ι : Type u_12\nR : Type u_13\nS : Type u_14\ninst✝³ : SetLike S R\ninst✝² : CommMonoid R\ninst✝¹ : AddCommMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nκ : Type u_15\ni : κ → ι\ng : κ → R\nF : Finset κ\nj : κ\nF' : Finset κ\nhF2 : j ∉ F'\nh3 : (∀ k ∈ F', g k ∈ A (i k)) → ∏ k ∈ F', g k ∈ A (∑ k ∈ F', i k)\nhF : ∀ k ∈ insert j F', g k ∈ A (i k)\n⊢ g j * ∏ x ∈ F', g x ∈ A (i j + ∑ x ∈ F', i x)",
"after_state": "ι : Type u_12\nR : Type u_13\nS : Type u_14\ninst✝³ : SetLike S R\ninst✝² : CommMonoid R\ninst✝¹ : AddCommMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nκ : Type u_15\ni : κ → ι\ng : κ → R\nF : Finset κ\nj : κ\nF' : Finset κ\nhF2 : j ∉ F'\nh3 : (∀ k ∈ F', g k ∈ A (i k)) → ∏ k ∈ F', g k ∈ A (∑ k ∈ F', i k)\nhF : ∀ k ∈ insert j F', g k ∈ A (i k)\n⊢ g j * ∏ x ∈ F', g x ∈ A (i j + ∑ x ∈ F', i x)"
},
{
"line": "apply SetLike.mul_mem_graded (hF j <| Finset.mem_insert_self j F')",
"before_state": "ι : Type u_12\nR : Type u_13\nS : Type u_14\ninst✝³ : SetLike S R\ninst✝² : CommMonoid R\ninst✝¹ : AddCommMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nκ : Type u_15\ni : κ → ι\ng : κ → R\nF : Finset κ\nj : κ\nF' : Finset κ\nhF2 : j ∉ F'\nh3 : (∀ k ∈ F', g k ∈ A (i k)) → ∏ k ∈ F', g k ∈ A (∑ k ∈ F', i k)\nhF : ∀ k ∈ insert j F', g k ∈ A (i k)\n⊢ g j * ∏ x ∈ F', g x ∈ A (i j + ∑ x ∈ F', i x)",
"after_state": "ι : Type u_12\nR : Type u_13\nS : Type u_14\ninst✝³ : SetLike S R\ninst✝² : CommMonoid R\ninst✝¹ : AddCommMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nκ : Type u_15\ni : κ → ι\ng : κ → R\nF : Finset κ\nj : κ\nF' : Finset κ\nhF2 : j ∉ F'\nh3 : (∀ k ∈ F', g k ∈ A (i k)) → ∏ k ∈ F', g k ∈ A (∑ k ∈ F', i k)\nhF : ∀ k ∈ insert j F', g k ∈ A (i k)\n⊢ ∏ x ∈ F', g x ∈ A (∑ x ∈ F', i x)"
},
{
"line": "apply h3",
"before_state": "ι : Type u_12\nR : Type u_13\nS : Type u_14\ninst✝³ : SetLike S R\ninst✝² : CommMonoid R\ninst✝¹ : AddCommMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nκ : Type u_15\ni : κ → ι\ng : κ → R\nF : Finset κ\nj : κ\nF' : Finset κ\nhF2 : j ∉ F'\nh3 : (∀ k ∈ F', g k ∈ A (i k)) → ∏ k ∈ F', g k ∈ A (∑ k ∈ F', i k)\nhF : ∀ k ∈ insert j F', g k ∈ A (i k)\n⊢ ∏ x ∈ F', g x ∈ A (∑ x ∈ F', i x)",
"after_state": "ι : Type u_12\nR : Type u_13\nS : Type u_14\ninst✝³ : SetLike S R\ninst✝² : CommMonoid R\ninst✝¹ : AddCommMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nκ : Type u_15\ni : κ → ι\ng : κ → R\nF : Finset κ\nj : κ\nF' : Finset κ\nhF2 : j ∉ F'\nh3 : (∀ k ∈ F', g k ∈ A (i k)) → ∏ k ∈ F', g k ∈ A (∑ k ∈ F', i k)\nhF : ∀ k ∈ insert j F', g k ∈ A (i k)\n⊢ ∀ k ∈ F', g k ∈ A (i k)"
},
{
"line": "intro k hk",
"before_state": "ι : Type u_12\nR : Type u_13\nS : Type u_14\ninst✝³ : SetLike S R\ninst✝² : CommMonoid R\ninst✝¹ : AddCommMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nκ : Type u_15\ni : κ → ι\ng : κ → R\nF : Finset κ\nj : κ\nF' : Finset κ\nhF2 : j ∉ F'\nh3 : (∀ k ∈ F', g k ∈ A (i k)) → ∏ k ∈ F', g k ∈ A (∑ k ∈ F', i k)\nhF : ∀ k ∈ insert j F', g k ∈ A (i k)\n⊢ ∀ k ∈ F', g k ∈ A (i k)",
"after_state": "ι : Type u_12\nR : Type u_13\nS : Type u_14\ninst✝³ : SetLike S R\ninst✝² : CommMonoid R\ninst✝¹ : AddCommMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nκ : Type u_15\ni : κ → ι\ng : κ → R\nF : Finset κ\nj : κ\nF' : Finset κ\nhF2 : j ∉ F'\nh3 : (∀ k ∈ F', g k ∈ A (i k)) → ∏ k ∈ F', g k ∈ A (∑ k ∈ F', i k)\nhF : ∀ k ∈ insert j F', g k ∈ A (i k)\nk : κ\nhk : k ∈ F'\n⊢ g k ∈ A (i k)"
},
{
"line": "intro k;\n intro hk",
"before_state": "ι : Type u_12\nR : Type u_13\nS : Type u_14\ninst✝³ : SetLike S R\ninst✝² : CommMonoid R\ninst✝¹ : AddCommMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nκ : Type u_15\ni : κ → ι\ng : κ → R\nF : Finset κ\nj : κ\nF' : Finset κ\nhF2 : j ∉ F'\nh3 : (∀ k ∈ F', g k ∈ A (i k)) → ∏ k ∈ F', g k ∈ A (∑ k ∈ F', i k)\nhF : ∀ k ∈ insert j F', g k ∈ A (i k)\n⊢ ∀ k ∈ F', g k ∈ A (i k)",
"after_state": "ι : Type u_12\nR : Type u_13\nS : Type u_14\ninst✝³ : SetLike S R\ninst✝² : CommMonoid R\ninst✝¹ : AddCommMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nκ : Type u_15\ni : κ → ι\ng : κ → R\nF : Finset κ\nj : κ\nF' : Finset κ\nhF2 : j ∉ F'\nh3 : (∀ k ∈ F', g k ∈ A (i k)) → ∏ k ∈ F', g k ∈ A (∑ k ∈ F', i k)\nhF : ∀ k ∈ insert j F', g k ∈ A (i k)\nk : κ\nhk : k ∈ F'\n⊢ g k ∈ A (i k)"
},
{
"line": "intro k",
"before_state": "ι : Type u_12\nR : Type u_13\nS : Type u_14\ninst✝³ : SetLike S R\ninst✝² : CommMonoid R\ninst✝¹ : AddCommMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nκ : Type u_15\ni : κ → ι\ng : κ → R\nF : Finset κ\nj : κ\nF' : Finset κ\nhF2 : j ∉ F'\nh3 : (∀ k ∈ F', g k ∈ A (i k)) → ∏ k ∈ F', g k ∈ A (∑ k ∈ F', i k)\nhF : ∀ k ∈ insert j F', g k ∈ A (i k)\n⊢ ∀ k ∈ F', g k ∈ A (i k)",
"after_state": "ι : Type u_12\nR : Type u_13\nS : Type u_14\ninst✝³ : SetLike S R\ninst✝² : CommMonoid R\ninst✝¹ : AddCommMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nκ : Type u_15\ni : κ → ι\ng : κ → R\nF : Finset κ\nj : κ\nF' : Finset κ\nhF2 : j ∉ F'\nh3 : (∀ k ∈ F', g k ∈ A (i k)) → ∏ k ∈ F', g k ∈ A (∑ k ∈ F', i k)\nhF : ∀ k ∈ insert j F', g k ∈ A (i k)\nk : κ\n⊢ k ∈ F' → g k ∈ A (i k)"
},
{
"line": "intro hk",
"before_state": "ι : Type u_12\nR : Type u_13\nS : Type u_14\ninst✝³ : SetLike S R\ninst✝² : CommMonoid R\ninst✝¹ : AddCommMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nκ : Type u_15\ni : κ → ι\ng : κ → R\nF : Finset κ\nj : κ\nF' : Finset κ\nhF2 : j ∉ F'\nh3 : (∀ k ∈ F', g k ∈ A (i k)) → ∏ k ∈ F', g k ∈ A (∑ k ∈ F', i k)\nhF : ∀ k ∈ insert j F', g k ∈ A (i k)\nk : κ\n⊢ k ∈ F' → g k ∈ A (i k)",
"after_state": "ι : Type u_12\nR : Type u_13\nS : Type u_14\ninst✝³ : SetLike S R\ninst✝² : CommMonoid R\ninst✝¹ : AddCommMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nκ : Type u_15\ni : κ → ι\ng : κ → R\nF : Finset κ\nj : κ\nF' : Finset κ\nhF2 : j ∉ F'\nh3 : (∀ k ∈ F', g k ∈ A (i k)) → ∏ k ∈ F', g k ∈ A (∑ k ∈ F', i k)\nhF : ∀ k ∈ insert j F', g k ∈ A (i k)\nk : κ\nhk : k ∈ F'\n⊢ g k ∈ A (i k)"
},
{
"line": "apply hF k",
"before_state": "ι : Type u_12\nR : Type u_13\nS : Type u_14\ninst✝³ : SetLike S R\ninst✝² : CommMonoid R\ninst✝¹ : AddCommMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nκ : Type u_15\ni : κ → ι\ng : κ → R\nF : Finset κ\nj : κ\nF' : Finset κ\nhF2 : j ∉ F'\nh3 : (∀ k ∈ F', g k ∈ A (i k)) → ∏ k ∈ F', g k ∈ A (∑ k ∈ F', i k)\nhF : ∀ k ∈ insert j F', g k ∈ A (i k)\nk : κ\nhk : k ∈ F'\n⊢ g k ∈ A (i k)",
"after_state": "ι : Type u_12\nR : Type u_13\nS : Type u_14\ninst✝³ : SetLike S R\ninst✝² : CommMonoid R\ninst✝¹ : AddCommMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nκ : Type u_15\ni : κ → ι\ng : κ → R\nF : Finset κ\nj : κ\nF' : Finset κ\nhF2 : j ∉ F'\nh3 : (∀ k ∈ F', g k ∈ A (i k)) → ∏ k ∈ F', g k ∈ A (∑ k ∈ F', i k)\nhF : ∀ k ∈ insert j F', g k ∈ A (i k)\nk : κ\nhk : k ∈ F'\n⊢ k ∈ insert j F'"
},
{
"line": "exact Finset.mem_insert_of_mem hk",
"before_state": "ι : Type u_12\nR : Type u_13\nS : Type u_14\ninst✝³ : SetLike S R\ninst✝² : CommMonoid R\ninst✝¹ : AddCommMonoid ι\nA : ι → S\ninst✝ : SetLike.GradedMonoid A\nκ : Type u_15\ni : κ → ι\ng : κ → R\nF : Finset κ\nj : κ\nF' : Finset κ\nhF2 : j ∉ F'\nh3 : (∀ k ∈ F', g k ∈ A (i k)) → ∏ k ∈ F', g k ∈ A (∑ k ∈ F', i k)\nhF : ∀ k ∈ insert j F', g k ∈ A (i k)\nk : κ\nhk : k ∈ F'\n⊢ k ∈ insert j F'",
"after_state": "No Goals!"
}
] |
theorem sol_eq_of_eq_init (u v : ℕ → R) (hu : E.IsSolution u) (hv : E.IsSolution v) :
u = v ↔ Set.EqOn u v ↑(range E.order) := by
refine Iff.intro (fun h x _ ↦ h ▸ rfl) ?_
intro h
set u' : ↥E.solSpace := ⟨u, hu⟩
set v' : ↥E.solSpace := ⟨v, hv⟩
change u'.val = v'.val
suffices h' : u' = v' from h' ▸ rfl
rw [← E.toInit.toEquiv.apply_eq_iff_eq]
rw [LinearEquiv.coe_toEquiv]
ext x
exact mod_cast h (mem_range.mpr x.2)
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/LinearRecurrence.lean
|
{
"open": [
"Finset",
"Polynomial"
],
"variables": [
"{R : Type*} [CommSemiring R] (E : LinearRecurrence R)"
]
}
|
[
{
"line": "refine Iff.intro (fun h x _ ↦ h ▸ rfl) ?_",
"before_state": "R : Type u_1\ninst✝ : CommSemiring R\nE : LinearRecurrence R\nu v : ℕ → R\nhu : E.IsSolution u\nhv : E.IsSolution v\n⊢ u = v ↔ Set.EqOn u v ↑(range E.order)",
"after_state": "R : Type u_1\ninst✝ : CommSemiring R\nE : LinearRecurrence R\nu v : ℕ → R\nhu : E.IsSolution u\nhv : E.IsSolution v\n⊢ Set.EqOn u v ↑(range E.order) → u = v"
},
{
"line": "intro h",
"before_state": "R : Type u_1\ninst✝ : CommSemiring R\nE : LinearRecurrence R\nu v : ℕ → R\nhu : E.IsSolution u\nhv : E.IsSolution v\n⊢ Set.EqOn u v ↑(range E.order) → u = v",
"after_state": "R : Type u_1\ninst✝ : CommSemiring R\nE : LinearRecurrence R\nu v : ℕ → R\nhu : E.IsSolution u\nhv : E.IsSolution v\nh : Set.EqOn u v ↑(range E.order)\n⊢ u = v"
},
{
"line": "set u' : ↥E.solSpace := ⟨u, hu⟩",
"before_state": "R : Type u_1\ninst✝ : CommSemiring R\nE : LinearRecurrence R\nu v : ℕ → R\nhu : E.IsSolution u\nhv : E.IsSolution v\nh : Set.EqOn u v ↑(range E.order)\n⊢ u = v",
"after_state": "R : Type u_1\ninst✝ : CommSemiring R\nE : LinearRecurrence R\nu v : ℕ → R\nhu : E.IsSolution u\nhv : E.IsSolution v\nh : Set.EqOn u v ↑(range E.order)\nu' : ↥E.solSpace := ⟨u, hu⟩\n⊢ u = v"
},
{
"line": "try rewrite [show ?m✝ = u' from rfl✝] at *",
"before_state": "R : Type u_1\ninst✝ : CommSemiring R\nE : LinearRecurrence R\nu v : ℕ → R\nhu : E.IsSolution u\nhv : E.IsSolution v\nh : Set.EqOn u v ↑(range E.order)\nu' : ↥E.solSpace := ⟨u, hu⟩\n⊢ u = v",
"after_state": "R : Type u_1\ninst✝ : CommSemiring R\nE : LinearRecurrence R\nu v : ℕ → R\nhu : E.IsSolution u\nhv : E.IsSolution v\nh : Set.EqOn u v ↑(range E.order)\nu' : ↥E.solSpace := ⟨u, hu⟩\n⊢ u = v"
},
{
"line": "first\n| rewrite [show ?m✝ = u' from rfl✝] at *\n| skip",
"before_state": "R : Type u_1\ninst✝ : CommSemiring R\nE : LinearRecurrence R\nu v : ℕ → R\nhu : E.IsSolution u\nhv : E.IsSolution v\nh : Set.EqOn u v ↑(range E.order)\nu' : ↥E.solSpace := ⟨u, hu⟩\n⊢ u = v",
"after_state": "R : Type u_1\ninst✝ : CommSemiring R\nE : LinearRecurrence R\nu v : ℕ → R\nhu : E.IsSolution u\nhv : E.IsSolution v\nh : Set.EqOn u v ↑(range E.order)\nu' : ↥E.solSpace := ⟨u, hu⟩\n⊢ u = v"
},
{
"line": "rewrite [show ?m✝ = u' from rfl✝] at *",
"before_state": "R : Type u_1\ninst✝ : CommSemiring R\nE : LinearRecurrence R\nu v : ℕ → R\nhu : E.IsSolution u\nhv : E.IsSolution v\nh : Set.EqOn u v ↑(range E.order)\nu' : ↥E.solSpace := ⟨u, hu⟩\n⊢ u = v",
"after_state": "R : Type u_1\ninst✝ : CommSemiring R\nE : LinearRecurrence R\nu v : ℕ → R\nhu : E.IsSolution u\nhv : E.IsSolution v\nh : Set.EqOn u v ↑(range E.order)\nu' : ↥E.solSpace := ⟨u, hu⟩\n⊢ u = v"
},
{
"line": "skip",
"before_state": "R : Type u_1\ninst✝ : CommSemiring R\nE : LinearRecurrence R\nu v : ℕ → R\nhu : E.IsSolution u\nhv : E.IsSolution v\nh : Set.EqOn u v ↑(range E.order)\nu' : ↥E.solSpace := ⟨u, hu⟩\n⊢ u = v",
"after_state": "R : Type u_1\ninst✝ : CommSemiring R\nE : LinearRecurrence R\nu v : ℕ → R\nhu : E.IsSolution u\nhv : E.IsSolution v\nh : Set.EqOn u v ↑(range E.order)\nu' : ↥E.solSpace := ⟨u, hu⟩\n⊢ u = v"
},
{
"line": "set v' : ↥E.solSpace := ⟨v, hv⟩",
"before_state": "R : Type u_1\ninst✝ : CommSemiring R\nE : LinearRecurrence R\nu v : ℕ → R\nhu : E.IsSolution u\nhv : E.IsSolution v\nh : Set.EqOn u v ↑(range E.order)\nu' : ↥E.solSpace := ⟨u, hu⟩\n⊢ u = v",
"after_state": "R : Type u_1\ninst✝ : CommSemiring R\nE : LinearRecurrence R\nu v : ℕ → R\nhu : E.IsSolution u\nhv : E.IsSolution v\nh : Set.EqOn u v ↑(range E.order)\nu' : ↥E.solSpace := ⟨u, hu⟩\nv' : ↥E.solSpace := ⟨v, hv⟩\n⊢ u = v"
},
{
"line": "try rewrite [show ?m✝ = v' from rfl✝] at *",
"before_state": "R : Type u_1\ninst✝ : CommSemiring R\nE : LinearRecurrence R\nu v : ℕ → R\nhu : E.IsSolution u\nhv : E.IsSolution v\nh : Set.EqOn u v ↑(range E.order)\nu' : ↥E.solSpace := ⟨u, hu⟩\nv' : ↥E.solSpace := ⟨v, hv⟩\n⊢ u = v",
"after_state": "R : Type u_1\ninst✝ : CommSemiring R\nE : LinearRecurrence R\nu v : ℕ → R\nhu : E.IsSolution u\nhv : E.IsSolution v\nh : Set.EqOn u v ↑(range E.order)\nu' : ↥E.solSpace := ⟨u, hu⟩\nv' : ↥E.solSpace := ⟨v, hv⟩\n⊢ u = v"
},
{
"line": "first\n| rewrite [show ?m✝ = v' from rfl✝] at *\n| skip",
"before_state": "R : Type u_1\ninst✝ : CommSemiring R\nE : LinearRecurrence R\nu v : ℕ → R\nhu : E.IsSolution u\nhv : E.IsSolution v\nh : Set.EqOn u v ↑(range E.order)\nu' : ↥E.solSpace := ⟨u, hu⟩\nv' : ↥E.solSpace := ⟨v, hv⟩\n⊢ u = v",
"after_state": "R : Type u_1\ninst✝ : CommSemiring R\nE : LinearRecurrence R\nu v : ℕ → R\nhu : E.IsSolution u\nhv : E.IsSolution v\nh : Set.EqOn u v ↑(range E.order)\nu' : ↥E.solSpace := ⟨u, hu⟩\nv' : ↥E.solSpace := ⟨v, hv⟩\n⊢ u = v"
},
{
"line": "rewrite [show ?m✝ = v' from rfl✝] at *",
"before_state": "R : Type u_1\ninst✝ : CommSemiring R\nE : LinearRecurrence R\nu v : ℕ → R\nhu : E.IsSolution u\nhv : E.IsSolution v\nh : Set.EqOn u v ↑(range E.order)\nu' : ↥E.solSpace := ⟨u, hu⟩\nv' : ↥E.solSpace := ⟨v, hv⟩\n⊢ u = v",
"after_state": "R : Type u_1\ninst✝ : CommSemiring R\nE : LinearRecurrence R\nu v : ℕ → R\nhu : E.IsSolution u\nhv : E.IsSolution v\nh : Set.EqOn u v ↑(range E.order)\nu' : ↥E.solSpace := ⟨u, hu⟩\nv' : ↥E.solSpace := ⟨v, hv⟩\n⊢ u = v"
},
{
"line": "skip",
"before_state": "R : Type u_1\ninst✝ : CommSemiring R\nE : LinearRecurrence R\nu v : ℕ → R\nhu : E.IsSolution u\nhv : E.IsSolution v\nh : Set.EqOn u v ↑(range E.order)\nu' : ↥E.solSpace := ⟨u, hu⟩\nv' : ↥E.solSpace := ⟨v, hv⟩\n⊢ u = v",
"after_state": "R : Type u_1\ninst✝ : CommSemiring R\nE : LinearRecurrence R\nu v : ℕ → R\nhu : E.IsSolution u\nhv : E.IsSolution v\nh : Set.EqOn u v ↑(range E.order)\nu' : ↥E.solSpace := ⟨u, hu⟩\nv' : ↥E.solSpace := ⟨v, hv⟩\n⊢ u = v"
},
{
"line": "change u'.val = v'.val",
"before_state": "R : Type u_1\ninst✝ : CommSemiring R\nE : LinearRecurrence R\nu v : ℕ → R\nhu : E.IsSolution u\nhv : E.IsSolution v\nh : Set.EqOn u v ↑(range E.order)\nu' : ↥E.solSpace := ⟨u, hu⟩\nv' : ↥E.solSpace := ⟨v, hv⟩\n⊢ u = v",
"after_state": "R : Type u_1\ninst✝ : CommSemiring R\nE : LinearRecurrence R\nu v : ℕ → R\nhu : E.IsSolution u\nhv : E.IsSolution v\nh : Set.EqOn u v ↑(range E.order)\nu' : ↥E.solSpace := ⟨u, hu⟩\nv' : ↥E.solSpace := ⟨v, hv⟩\n⊢ ↑u' = ↑v'"
},
{
"line": "suffices h' : u' = v' from h' ▸ rfl",
"before_state": "R : Type u_1\ninst✝ : CommSemiring R\nE : LinearRecurrence R\nu v : ℕ → R\nhu : E.IsSolution u\nhv : E.IsSolution v\nh : Set.EqOn u v ↑(range E.order)\nu' : ↥E.solSpace := ⟨u, hu⟩\nv' : ↥E.solSpace := ⟨v, hv⟩\n⊢ ↑u' = ↑v'",
"after_state": "R : Type u_1\ninst✝ : CommSemiring R\nE : LinearRecurrence R\nu v : ℕ → R\nhu : E.IsSolution u\nhv : E.IsSolution v\nh : Set.EqOn u v ↑(range E.order)\nu' : ↥E.solSpace := ⟨u, hu⟩\nv' : ↥E.solSpace := ⟨v, hv⟩\n⊢ u' = v'"
},
{
"line": "refine_lift\n suffices h' : u' = v' from h' ▸ rfl;\n ?_",
"before_state": "R : Type u_1\ninst✝ : CommSemiring R\nE : LinearRecurrence R\nu v : ℕ → R\nhu : E.IsSolution u\nhv : E.IsSolution v\nh : Set.EqOn u v ↑(range E.order)\nu' : ↥E.solSpace := ⟨u, hu⟩\nv' : ↥E.solSpace := ⟨v, hv⟩\n⊢ ↑u' = ↑v'",
"after_state": "R : Type u_1\ninst✝ : CommSemiring R\nE : LinearRecurrence R\nu v : ℕ → R\nhu : E.IsSolution u\nhv : E.IsSolution v\nh : Set.EqOn u v ↑(range E.order)\nu' : ↥E.solSpace := ⟨u, hu⟩\nv' : ↥E.solSpace := ⟨v, hv⟩\n⊢ u' = v'"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (suffices h' : u' = v' from h' ▸ rfl;\n ?_);\n rotate_right)",
"before_state": "R : Type u_1\ninst✝ : CommSemiring R\nE : LinearRecurrence R\nu v : ℕ → R\nhu : E.IsSolution u\nhv : E.IsSolution v\nh : Set.EqOn u v ↑(range E.order)\nu' : ↥E.solSpace := ⟨u, hu⟩\nv' : ↥E.solSpace := ⟨v, hv⟩\n⊢ ↑u' = ↑v'",
"after_state": "R : Type u_1\ninst✝ : CommSemiring R\nE : LinearRecurrence R\nu v : ℕ → R\nhu : E.IsSolution u\nhv : E.IsSolution v\nh : Set.EqOn u v ↑(range E.order)\nu' : ↥E.solSpace := ⟨u, hu⟩\nv' : ↥E.solSpace := ⟨v, hv⟩\n⊢ u' = v'"
},
{
"line": "refine\n no_implicit_lambda%\n (suffices h' : u' = v' from h' ▸ rfl;\n ?_)",
"before_state": "R : Type u_1\ninst✝ : CommSemiring R\nE : LinearRecurrence R\nu v : ℕ → R\nhu : E.IsSolution u\nhv : E.IsSolution v\nh : Set.EqOn u v ↑(range E.order)\nu' : ↥E.solSpace := ⟨u, hu⟩\nv' : ↥E.solSpace := ⟨v, hv⟩\n⊢ ↑u' = ↑v'",
"after_state": "R : Type u_1\ninst✝ : CommSemiring R\nE : LinearRecurrence R\nu v : ℕ → R\nhu : E.IsSolution u\nhv : E.IsSolution v\nh : Set.EqOn u v ↑(range E.order)\nu' : ↥E.solSpace := ⟨u, hu⟩\nv' : ↥E.solSpace := ⟨v, hv⟩\n⊢ u' = v'"
},
{
"line": "rotate_right",
"before_state": "R : Type u_1\ninst✝ : CommSemiring R\nE : LinearRecurrence R\nu v : ℕ → R\nhu : E.IsSolution u\nhv : E.IsSolution v\nh : Set.EqOn u v ↑(range E.order)\nu' : ↥E.solSpace := ⟨u, hu⟩\nv' : ↥E.solSpace := ⟨v, hv⟩\n⊢ u' = v'",
"after_state": "R : Type u_1\ninst✝ : CommSemiring R\nE : LinearRecurrence R\nu v : ℕ → R\nhu : E.IsSolution u\nhv : E.IsSolution v\nh : Set.EqOn u v ↑(range E.order)\nu' : ↥E.solSpace := ⟨u, hu⟩\nv' : ↥E.solSpace := ⟨v, hv⟩\n⊢ u' = v'"
},
{
"line": "rw [← E.toInit.toEquiv.apply_eq_iff_eq]",
"before_state": "R : Type u_1\ninst✝ : CommSemiring R\nE : LinearRecurrence R\nu v : ℕ → R\nhu : E.IsSolution u\nhv : E.IsSolution v\nh : Set.EqOn u v ↑(range E.order)\nu' : ↥E.solSpace := ⟨u, hu⟩\nv' : ↥E.solSpace := ⟨v, hv⟩\n⊢ u' = v'",
"after_state": "R : Type u_1\ninst✝ : CommSemiring R\nE : LinearRecurrence R\nu v : ℕ → R\nhu : E.IsSolution u\nhv : E.IsSolution v\nh : Set.EqOn u v ↑(range E.order)\nu' : ↥E.solSpace := ⟨u, hu⟩\nv' : ↥E.solSpace := ⟨v, hv⟩\n⊢ E.toInit.toEquiv u' = E.toInit.toEquiv v'"
},
{
"line": "rewrite [← E.toInit.toEquiv.apply_eq_iff_eq]",
"before_state": "R : Type u_1\ninst✝ : CommSemiring R\nE : LinearRecurrence R\nu v : ℕ → R\nhu : E.IsSolution u\nhv : E.IsSolution v\nh : Set.EqOn u v ↑(range E.order)\nu' : ↥E.solSpace := ⟨u, hu⟩\nv' : ↥E.solSpace := ⟨v, hv⟩\n⊢ u' = v'",
"after_state": "R : Type u_1\ninst✝ : CommSemiring R\nE : LinearRecurrence R\nu v : ℕ → R\nhu : E.IsSolution u\nhv : E.IsSolution v\nh : Set.EqOn u v ↑(range E.order)\nu' : ↥E.solSpace := ⟨u, hu⟩\nv' : ↥E.solSpace := ⟨v, hv⟩\n⊢ E.toInit.toEquiv u' = E.toInit.toEquiv v'"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "R : Type u_1\ninst✝ : CommSemiring R\nE : LinearRecurrence R\nu v : ℕ → R\nhu : E.IsSolution u\nhv : E.IsSolution v\nh : Set.EqOn u v ↑(range E.order)\nu' : ↥E.solSpace := ⟨u, hu⟩\nv' : ↥E.solSpace := ⟨v, hv⟩\n⊢ E.toInit.toEquiv u' = E.toInit.toEquiv v'",
"after_state": "R : Type u_1\ninst✝ : CommSemiring R\nE : LinearRecurrence R\nu v : ℕ → R\nhu : E.IsSolution u\nhv : E.IsSolution v\nh : Set.EqOn u v ↑(range E.order)\nu' : ↥E.solSpace := ⟨u, hu⟩\nv' : ↥E.solSpace := ⟨v, hv⟩\n⊢ E.toInit.toEquiv u' = E.toInit.toEquiv v'"
},
{
"line": "try (with_reducible rfl)",
"before_state": "R : Type u_1\ninst✝ : CommSemiring R\nE : LinearRecurrence R\nu v : ℕ → R\nhu : E.IsSolution u\nhv : E.IsSolution v\nh : Set.EqOn u v ↑(range E.order)\nu' : ↥E.solSpace := ⟨u, hu⟩\nv' : ↥E.solSpace := ⟨v, hv⟩\n⊢ E.toInit.toEquiv u' = E.toInit.toEquiv v'",
"after_state": "R : Type u_1\ninst✝ : CommSemiring R\nE : LinearRecurrence R\nu v : ℕ → R\nhu : E.IsSolution u\nhv : E.IsSolution v\nh : Set.EqOn u v ↑(range E.order)\nu' : ↥E.solSpace := ⟨u, hu⟩\nv' : ↥E.solSpace := ⟨v, hv⟩\n⊢ E.toInit.toEquiv u' = E.toInit.toEquiv v'"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "R : Type u_1\ninst✝ : CommSemiring R\nE : LinearRecurrence R\nu v : ℕ → R\nhu : E.IsSolution u\nhv : E.IsSolution v\nh : Set.EqOn u v ↑(range E.order)\nu' : ↥E.solSpace := ⟨u, hu⟩\nv' : ↥E.solSpace := ⟨v, hv⟩\n⊢ E.toInit.toEquiv u' = E.toInit.toEquiv v'",
"after_state": "R : Type u_1\ninst✝ : CommSemiring R\nE : LinearRecurrence R\nu v : ℕ → R\nhu : E.IsSolution u\nhv : E.IsSolution v\nh : Set.EqOn u v ↑(range E.order)\nu' : ↥E.solSpace := ⟨u, hu⟩\nv' : ↥E.solSpace := ⟨v, hv⟩\n⊢ E.toInit.toEquiv u' = E.toInit.toEquiv v'"
},
{
"line": "with_reducible rfl",
"before_state": "R : Type u_1\ninst✝ : CommSemiring R\nE : LinearRecurrence R\nu v : ℕ → R\nhu : E.IsSolution u\nhv : E.IsSolution v\nh : Set.EqOn u v ↑(range E.order)\nu' : ↥E.solSpace := ⟨u, hu⟩\nv' : ↥E.solSpace := ⟨v, hv⟩\n⊢ E.toInit.toEquiv u' = E.toInit.toEquiv v'",
"after_state": "R : Type u_1\ninst✝ : CommSemiring R\nE : LinearRecurrence R\nu v : ℕ → R\nhu : E.IsSolution u\nhv : E.IsSolution v\nh : Set.EqOn u v ↑(range E.order)\nu' : ↥E.solSpace := ⟨u, hu⟩\nv' : ↥E.solSpace := ⟨v, hv⟩\n⊢ E.toInit.toEquiv u' = E.toInit.toEquiv v'"
},
{
"line": "rfl",
"before_state": "R : Type u_1\ninst✝ : CommSemiring R\nE : LinearRecurrence R\nu v : ℕ → R\nhu : E.IsSolution u\nhv : E.IsSolution v\nh : Set.EqOn u v ↑(range E.order)\nu' : ↥E.solSpace := ⟨u, hu⟩\nv' : ↥E.solSpace := ⟨v, hv⟩\n⊢ E.toInit.toEquiv u' = E.toInit.toEquiv v'",
"after_state": "R : Type u_1\ninst✝ : CommSemiring R\nE : LinearRecurrence R\nu v : ℕ → R\nhu : E.IsSolution u\nhv : E.IsSolution v\nh : Set.EqOn u v ↑(range E.order)\nu' : ↥E.solSpace := ⟨u, hu⟩\nv' : ↥E.solSpace := ⟨v, hv⟩\n⊢ E.toInit.toEquiv u' = E.toInit.toEquiv v'"
},
{
"line": "apply_rfl",
"before_state": "R : Type u_1\ninst✝ : CommSemiring R\nE : LinearRecurrence R\nu v : ℕ → R\nhu : E.IsSolution u\nhv : E.IsSolution v\nh : Set.EqOn u v ↑(range E.order)\nu' : ↥E.solSpace := ⟨u, hu⟩\nv' : ↥E.solSpace := ⟨v, hv⟩\n⊢ E.toInit.toEquiv u' = E.toInit.toEquiv v'",
"after_state": "R : Type u_1\ninst✝ : CommSemiring R\nE : LinearRecurrence R\nu v : ℕ → R\nhu : E.IsSolution u\nhv : E.IsSolution v\nh : Set.EqOn u v ↑(range E.order)\nu' : ↥E.solSpace := ⟨u, hu⟩\nv' : ↥E.solSpace := ⟨v, hv⟩\n⊢ E.toInit.toEquiv u' = E.toInit.toEquiv v'"
},
{
"line": "skip",
"before_state": "R : Type u_1\ninst✝ : CommSemiring R\nE : LinearRecurrence R\nu v : ℕ → R\nhu : E.IsSolution u\nhv : E.IsSolution v\nh : Set.EqOn u v ↑(range E.order)\nu' : ↥E.solSpace := ⟨u, hu⟩\nv' : ↥E.solSpace := ⟨v, hv⟩\n⊢ E.toInit.toEquiv u' = E.toInit.toEquiv v'",
"after_state": "R : Type u_1\ninst✝ : CommSemiring R\nE : LinearRecurrence R\nu v : ℕ → R\nhu : E.IsSolution u\nhv : E.IsSolution v\nh : Set.EqOn u v ↑(range E.order)\nu' : ↥E.solSpace := ⟨u, hu⟩\nv' : ↥E.solSpace := ⟨v, hv⟩\n⊢ E.toInit.toEquiv u' = E.toInit.toEquiv v'"
},
{
"line": "rw [LinearEquiv.coe_toEquiv]",
"before_state": "R : Type u_1\ninst✝ : CommSemiring R\nE : LinearRecurrence R\nu v : ℕ → R\nhu : E.IsSolution u\nhv : E.IsSolution v\nh : Set.EqOn u v ↑(range E.order)\nu' : ↥E.solSpace := ⟨u, hu⟩\nv' : ↥E.solSpace := ⟨v, hv⟩\n⊢ E.toInit.toEquiv u' = E.toInit.toEquiv v'",
"after_state": "R : Type u_1\ninst✝ : CommSemiring R\nE : LinearRecurrence R\nu v : ℕ → R\nhu : E.IsSolution u\nhv : E.IsSolution v\nh : Set.EqOn u v ↑(range E.order)\nu' : ↥E.solSpace := ⟨u, hu⟩\nv' : ↥E.solSpace := ⟨v, hv⟩\n⊢ E.toInit u' = E.toInit v'"
},
{
"line": "rewrite [LinearEquiv.coe_toEquiv]",
"before_state": "R : Type u_1\ninst✝ : CommSemiring R\nE : LinearRecurrence R\nu v : ℕ → R\nhu : E.IsSolution u\nhv : E.IsSolution v\nh : Set.EqOn u v ↑(range E.order)\nu' : ↥E.solSpace := ⟨u, hu⟩\nv' : ↥E.solSpace := ⟨v, hv⟩\n⊢ E.toInit.toEquiv u' = E.toInit.toEquiv v'",
"after_state": "R : Type u_1\ninst✝ : CommSemiring R\nE : LinearRecurrence R\nu v : ℕ → R\nhu : E.IsSolution u\nhv : E.IsSolution v\nh : Set.EqOn u v ↑(range E.order)\nu' : ↥E.solSpace := ⟨u, hu⟩\nv' : ↥E.solSpace := ⟨v, hv⟩\n⊢ E.toInit u' = E.toInit v'"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "R : Type u_1\ninst✝ : CommSemiring R\nE : LinearRecurrence R\nu v : ℕ → R\nhu : E.IsSolution u\nhv : E.IsSolution v\nh : Set.EqOn u v ↑(range E.order)\nu' : ↥E.solSpace := ⟨u, hu⟩\nv' : ↥E.solSpace := ⟨v, hv⟩\n⊢ E.toInit u' = E.toInit v'",
"after_state": "R : Type u_1\ninst✝ : CommSemiring R\nE : LinearRecurrence R\nu v : ℕ → R\nhu : E.IsSolution u\nhv : E.IsSolution v\nh : Set.EqOn u v ↑(range E.order)\nu' : ↥E.solSpace := ⟨u, hu⟩\nv' : ↥E.solSpace := ⟨v, hv⟩\n⊢ E.toInit u' = E.toInit v'"
},
{
"line": "try (with_reducible rfl)",
"before_state": "R : Type u_1\ninst✝ : CommSemiring R\nE : LinearRecurrence R\nu v : ℕ → R\nhu : E.IsSolution u\nhv : E.IsSolution v\nh : Set.EqOn u v ↑(range E.order)\nu' : ↥E.solSpace := ⟨u, hu⟩\nv' : ↥E.solSpace := ⟨v, hv⟩\n⊢ E.toInit u' = E.toInit v'",
"after_state": "R : Type u_1\ninst✝ : CommSemiring R\nE : LinearRecurrence R\nu v : ℕ → R\nhu : E.IsSolution u\nhv : E.IsSolution v\nh : Set.EqOn u v ↑(range E.order)\nu' : ↥E.solSpace := ⟨u, hu⟩\nv' : ↥E.solSpace := ⟨v, hv⟩\n⊢ E.toInit u' = E.toInit v'"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "R : Type u_1\ninst✝ : CommSemiring R\nE : LinearRecurrence R\nu v : ℕ → R\nhu : E.IsSolution u\nhv : E.IsSolution v\nh : Set.EqOn u v ↑(range E.order)\nu' : ↥E.solSpace := ⟨u, hu⟩\nv' : ↥E.solSpace := ⟨v, hv⟩\n⊢ E.toInit u' = E.toInit v'",
"after_state": "R : Type u_1\ninst✝ : CommSemiring R\nE : LinearRecurrence R\nu v : ℕ → R\nhu : E.IsSolution u\nhv : E.IsSolution v\nh : Set.EqOn u v ↑(range E.order)\nu' : ↥E.solSpace := ⟨u, hu⟩\nv' : ↥E.solSpace := ⟨v, hv⟩\n⊢ E.toInit u' = E.toInit v'"
},
{
"line": "with_reducible rfl",
"before_state": "R : Type u_1\ninst✝ : CommSemiring R\nE : LinearRecurrence R\nu v : ℕ → R\nhu : E.IsSolution u\nhv : E.IsSolution v\nh : Set.EqOn u v ↑(range E.order)\nu' : ↥E.solSpace := ⟨u, hu⟩\nv' : ↥E.solSpace := ⟨v, hv⟩\n⊢ E.toInit u' = E.toInit v'",
"after_state": "R : Type u_1\ninst✝ : CommSemiring R\nE : LinearRecurrence R\nu v : ℕ → R\nhu : E.IsSolution u\nhv : E.IsSolution v\nh : Set.EqOn u v ↑(range E.order)\nu' : ↥E.solSpace := ⟨u, hu⟩\nv' : ↥E.solSpace := ⟨v, hv⟩\n⊢ E.toInit u' = E.toInit v'"
},
{
"line": "rfl",
"before_state": "R : Type u_1\ninst✝ : CommSemiring R\nE : LinearRecurrence R\nu v : ℕ → R\nhu : E.IsSolution u\nhv : E.IsSolution v\nh : Set.EqOn u v ↑(range E.order)\nu' : ↥E.solSpace := ⟨u, hu⟩\nv' : ↥E.solSpace := ⟨v, hv⟩\n⊢ E.toInit u' = E.toInit v'",
"after_state": "R : Type u_1\ninst✝ : CommSemiring R\nE : LinearRecurrence R\nu v : ℕ → R\nhu : E.IsSolution u\nhv : E.IsSolution v\nh : Set.EqOn u v ↑(range E.order)\nu' : ↥E.solSpace := ⟨u, hu⟩\nv' : ↥E.solSpace := ⟨v, hv⟩\n⊢ E.toInit u' = E.toInit v'"
},
{
"line": "apply_rfl",
"before_state": "R : Type u_1\ninst✝ : CommSemiring R\nE : LinearRecurrence R\nu v : ℕ → R\nhu : E.IsSolution u\nhv : E.IsSolution v\nh : Set.EqOn u v ↑(range E.order)\nu' : ↥E.solSpace := ⟨u, hu⟩\nv' : ↥E.solSpace := ⟨v, hv⟩\n⊢ E.toInit u' = E.toInit v'",
"after_state": "R : Type u_1\ninst✝ : CommSemiring R\nE : LinearRecurrence R\nu v : ℕ → R\nhu : E.IsSolution u\nhv : E.IsSolution v\nh : Set.EqOn u v ↑(range E.order)\nu' : ↥E.solSpace := ⟨u, hu⟩\nv' : ↥E.solSpace := ⟨v, hv⟩\n⊢ E.toInit u' = E.toInit v'"
},
{
"line": "skip",
"before_state": "R : Type u_1\ninst✝ : CommSemiring R\nE : LinearRecurrence R\nu v : ℕ → R\nhu : E.IsSolution u\nhv : E.IsSolution v\nh : Set.EqOn u v ↑(range E.order)\nu' : ↥E.solSpace := ⟨u, hu⟩\nv' : ↥E.solSpace := ⟨v, hv⟩\n⊢ E.toInit u' = E.toInit v'",
"after_state": "R : Type u_1\ninst✝ : CommSemiring R\nE : LinearRecurrence R\nu v : ℕ → R\nhu : E.IsSolution u\nhv : E.IsSolution v\nh : Set.EqOn u v ↑(range E.order)\nu' : ↥E.solSpace := ⟨u, hu⟩\nv' : ↥E.solSpace := ⟨v, hv⟩\n⊢ E.toInit u' = E.toInit v'"
},
{
"line": "ext x",
"before_state": "R : Type u_1\ninst✝ : CommSemiring R\nE : LinearRecurrence R\nu v : ℕ → R\nhu : E.IsSolution u\nhv : E.IsSolution v\nh : Set.EqOn u v ↑(range E.order)\nu' : ↥E.solSpace := ⟨u, hu⟩\nv' : ↥E.solSpace := ⟨v, hv⟩\n⊢ E.toInit u' = E.toInit v'",
"after_state": "case h\nR : Type u_1\ninst✝ : CommSemiring R\nE : LinearRecurrence R\nu v : ℕ → R\nhu : E.IsSolution u\nhv : E.IsSolution v\nh : Set.EqOn u v ↑(range E.order)\nu' : ↥E.solSpace := ⟨u, hu⟩\nv' : ↥E.solSpace := ⟨v, hv⟩\nx : Fin E.order\n⊢ E.toInit u' x = E.toInit v' x"
},
{
"line": "exact mod_cast h (mem_range.mpr x.2)",
"before_state": "case h\nR : Type u_1\ninst✝ : CommSemiring R\nE : LinearRecurrence R\nu v : ℕ → R\nhu : E.IsSolution u\nhv : E.IsSolution v\nh : Set.EqOn u v ↑(range E.order)\nu' : ↥E.solSpace := ⟨u, hu⟩\nv' : ↥E.solSpace := ⟨v, hv⟩\nx : Fin E.order\n⊢ E.toInit u' x = E.toInit v' x",
"after_state": "No Goals!"
}
] |
theorem modEq_refl (a : α) : a ≡ a [PMOD p] :=
⟨0, by simp⟩
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/ModEq.lean
|
{
"open": [],
"variables": [
"{α : Type*}",
"[AddCommGroup α] {p a a₁ a₂ b b₁ b₂ c : α} {n : ℕ} {z : ℤ}"
]
}
|
[
{
"line": "simp",
"before_state": "α : Type u_1\ninst✝ : AddCommGroup α\np a : α\n⊢ a - a = 0 • p",
"after_state": "No Goals!"
}
] |
theorem ModEq.trans : a ≡ b [PMOD p] → b ≡ c [PMOD p] → a ≡ c [PMOD p] := fun ⟨m, hm⟩ ⟨n, hn⟩ =>
⟨m + n, by simp [add_smul, ← hm, ← hn]⟩
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/ModEq.lean
|
{
"open": [],
"variables": [
"{α : Type*}",
"[AddCommGroup α] {p a a₁ a₂ b b₁ b₂ c : α} {n : ℕ} {z : ℤ}"
]
}
|
[
{
"line": "simp [add_smul, ← hm, ← hn]",
"before_state": "α : Type u_1\ninst✝ : AddCommGroup α\np a b c : α\nx✝¹ : a ≡ b [PMOD p]\nx✝ : b ≡ c [PMOD p]\nm : ℤ\nhm : b - a = m • p\nn : ℤ\nhn : c - b = n • p\n⊢ c - a = (m + n) • p",
"after_state": "No Goals!"
}
] |
theorem modEq_zero : a ≡ b [PMOD 0] ↔ a = b := by simp [ModEq, sub_eq_zero, eq_comm]
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/ModEq.lean
|
{
"open": [],
"variables": [
"{α : Type*}",
"[AddCommGroup α] {p a a₁ a₂ b b₁ b₂ c : α} {n : ℕ} {z : ℤ}"
]
}
|
[
{
"line": "simp [ModEq, sub_eq_zero, eq_comm]",
"before_state": "α : Type u_1\ninst✝ : AddCommGroup α\na b : α\n⊢ a ≡ b [PMOD 0] ↔ a = b",
"after_state": "No Goals!"
}
] |
theorem self_modEq_zero : p ≡ 0 [PMOD p] :=
⟨-1, by simp⟩
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/ModEq.lean
|
{
"open": [],
"variables": [
"{α : Type*}",
"[AddCommGroup α] {p a a₁ a₂ b b₁ b₂ c : α} {n : ℕ} {z : ℤ}"
]
}
|
[
{
"line": "simp",
"before_state": "α : Type u_1\ninst✝ : AddCommGroup α\np : α\n⊢ 0 - p = -1 • p",
"after_state": "No Goals!"
}
] |
theorem zsmul_modEq_zero (z : ℤ) : z • p ≡ 0 [PMOD p] :=
⟨-z, by simp⟩
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/ModEq.lean
|
{
"open": [],
"variables": [
"{α : Type*}",
"[AddCommGroup α] {p a a₁ a₂ b b₁ b₂ c : α} {n : ℕ} {z : ℤ}"
]
}
|
[
{
"line": "simp",
"before_state": "α : Type u_1\ninst✝ : AddCommGroup α\np : α\nz : ℤ\n⊢ 0 - z • p = -z • p",
"after_state": "No Goals!"
}
] |
theorem add_zsmul_modEq (z : ℤ) : a + z • p ≡ a [PMOD p] :=
⟨-z, by simp⟩
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/ModEq.lean
|
{
"open": [],
"variables": [
"{α : Type*}",
"[AddCommGroup α] {p a a₁ a₂ b b₁ b₂ c : α} {n : ℕ} {z : ℤ}"
]
}
|
[
{
"line": "simp",
"before_state": "α : Type u_1\ninst✝ : AddCommGroup α\np a : α\nz : ℤ\n⊢ a - (a + z • p) = -z • p",
"after_state": "No Goals!"
}
] |
theorem zsmul_add_modEq (z : ℤ) : z • p + a ≡ a [PMOD p] :=
⟨-z, by simp [← sub_sub]⟩
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/ModEq.lean
|
{
"open": [],
"variables": [
"{α : Type*}",
"[AddCommGroup α] {p a a₁ a₂ b b₁ b₂ c : α} {n : ℕ} {z : ℤ}"
]
}
|
[
{
"line": "simp [← sub_sub]",
"before_state": "α : Type u_1\ninst✝ : AddCommGroup α\np a : α\nz : ℤ\n⊢ a - (z • p + a) = -z • p",
"after_state": "No Goals!"
}
] |
theorem add_nsmul_modEq (n : ℕ) : a + n • p ≡ a [PMOD p] :=
⟨-n, by simp⟩
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/ModEq.lean
|
{
"open": [],
"variables": [
"{α : Type*}",
"[AddCommGroup α] {p a a₁ a₂ b b₁ b₂ c : α} {n : ℕ} {z : ℤ}"
]
}
|
[
{
"line": "simp",
"before_state": "α : Type u_1\ninst✝ : AddCommGroup α\np a : α\nn : ℕ\n⊢ a - (a + n • p) = -↑n • p",
"after_state": "No Goals!"
}
] |
theorem nsmul_add_modEq (n : ℕ) : n • p + a ≡ a [PMOD p] :=
⟨-n, by simp [← sub_sub]⟩
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/ModEq.lean
|
{
"open": [],
"variables": [
"{α : Type*}",
"[AddCommGroup α] {p a a₁ a₂ b b₁ b₂ c : α} {n : ℕ} {z : ℤ}"
]
}
|
[
{
"line": "simp [← sub_sub]",
"before_state": "α : Type u_1\ninst✝ : AddCommGroup α\np a : α\nn : ℕ\n⊢ a - (n • p + a) = -↑n • p",
"after_state": "No Goals!"
}
] |
theorem zsmul_modEq_zsmul [NoZeroSMulDivisors ℤ α] (hn : z ≠ 0) :
z • a ≡ z • b [PMOD z • p] ↔ a ≡ b [PMOD p] :=
exists_congr fun m => by rw [← smul_sub, smul_comm, smul_right_inj hn]
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/ModEq.lean
|
{
"open": [],
"variables": [
"{α : Type*}",
"[AddCommGroup α] {p a a₁ a₂ b b₁ b₂ c : α} {n : ℕ} {z : ℤ}"
]
}
|
[
{
"line": "rw [← smul_sub, smul_comm, smul_right_inj hn]",
"before_state": "α : Type u_1\ninst✝¹ : AddCommGroup α\np a b : α\nz : ℤ\ninst✝ : NoZeroSMulDivisors ℤ α\nhn : z ≠ 0\nm : ℤ\n⊢ z • b - z • a = m • z • p ↔ b - a = m • p",
"after_state": "No Goals!"
},
{
"line": "rewrite [← smul_sub, smul_comm, smul_right_inj hn]",
"before_state": "α : Type u_1\ninst✝¹ : AddCommGroup α\np a b : α\nz : ℤ\ninst✝ : NoZeroSMulDivisors ℤ α\nhn : z ≠ 0\nm : ℤ\n⊢ z • b - z • a = m • z • p ↔ b - a = m • p",
"after_state": "α : Type u_1\ninst✝¹ : AddCommGroup α\np a b : α\nz : ℤ\ninst✝ : NoZeroSMulDivisors ℤ α\nhn : z ≠ 0\nm : ℤ\n⊢ b - a = m • p ↔ b - a = m • p"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "α : Type u_1\ninst✝¹ : AddCommGroup α\np a b : α\nz : ℤ\ninst✝ : NoZeroSMulDivisors ℤ α\nhn : z ≠ 0\nm : ℤ\n⊢ b - a = m • p ↔ b - a = m • p",
"after_state": "No Goals!"
},
{
"line": "try (with_reducible rfl)",
"before_state": "α : Type u_1\ninst✝¹ : AddCommGroup α\np a b : α\nz : ℤ\ninst✝ : NoZeroSMulDivisors ℤ α\nhn : z ≠ 0\nm : ℤ\n⊢ b - a = m • p ↔ b - a = m • p",
"after_state": "No Goals!"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "α : Type u_1\ninst✝¹ : AddCommGroup α\np a b : α\nz : ℤ\ninst✝ : NoZeroSMulDivisors ℤ α\nhn : z ≠ 0\nm : ℤ\n⊢ b - a = m • p ↔ b - a = m • p",
"after_state": "No Goals!"
},
{
"line": "with_reducible rfl",
"before_state": "α : Type u_1\ninst✝¹ : AddCommGroup α\np a b : α\nz : ℤ\ninst✝ : NoZeroSMulDivisors ℤ α\nhn : z ≠ 0\nm : ℤ\n⊢ b - a = m • p ↔ b - a = m • p",
"after_state": "No Goals!"
},
{
"line": "rfl",
"before_state": "α : Type u_1\ninst✝¹ : AddCommGroup α\np a b : α\nz : ℤ\ninst✝ : NoZeroSMulDivisors ℤ α\nhn : z ≠ 0\nm : ℤ\n⊢ b - a = m • p ↔ b - a = m • p",
"after_state": "No Goals!"
},
{
"line": "exact Iff.rfl✝",
"before_state": "α : Type u_1\ninst✝¹ : AddCommGroup α\np a b : α\nz : ℤ\ninst✝ : NoZeroSMulDivisors ℤ α\nhn : z ≠ 0\nm : ℤ\n⊢ b - a = m • p ↔ b - a = m • p",
"after_state": "No Goals!"
}
] |
theorem nsmul_modEq_nsmul [NoZeroSMulDivisors ℕ α] (hn : n ≠ 0) :
n • a ≡ n • b [PMOD n • p] ↔ a ≡ b [PMOD p] :=
exists_congr fun m => by rw [← smul_sub, smul_comm, smul_right_inj hn]
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/ModEq.lean
|
{
"open": [],
"variables": [
"{α : Type*}",
"[AddCommGroup α] {p a a₁ a₂ b b₁ b₂ c : α} {n : ℕ} {z : ℤ}"
]
}
|
[
{
"line": "rw [← smul_sub, smul_comm, smul_right_inj hn]",
"before_state": "α : Type u_1\ninst✝¹ : AddCommGroup α\np a b : α\nn : ℕ\ninst✝ : NoZeroSMulDivisors ℕ α\nhn : n ≠ 0\nm : ℤ\n⊢ n • b - n • a = m • n • p ↔ b - a = m • p",
"after_state": "No Goals!"
},
{
"line": "rewrite [← smul_sub, smul_comm, smul_right_inj hn]",
"before_state": "α : Type u_1\ninst✝¹ : AddCommGroup α\np a b : α\nn : ℕ\ninst✝ : NoZeroSMulDivisors ℕ α\nhn : n ≠ 0\nm : ℤ\n⊢ n • b - n • a = m • n • p ↔ b - a = m • p",
"after_state": "α : Type u_1\ninst✝¹ : AddCommGroup α\np a b : α\nn : ℕ\ninst✝ : NoZeroSMulDivisors ℕ α\nhn : n ≠ 0\nm : ℤ\n⊢ b - a = m • p ↔ b - a = m • p"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "α : Type u_1\ninst✝¹ : AddCommGroup α\np a b : α\nn : ℕ\ninst✝ : NoZeroSMulDivisors ℕ α\nhn : n ≠ 0\nm : ℤ\n⊢ b - a = m • p ↔ b - a = m • p",
"after_state": "No Goals!"
},
{
"line": "try (with_reducible rfl)",
"before_state": "α : Type u_1\ninst✝¹ : AddCommGroup α\np a b : α\nn : ℕ\ninst✝ : NoZeroSMulDivisors ℕ α\nhn : n ≠ 0\nm : ℤ\n⊢ b - a = m • p ↔ b - a = m • p",
"after_state": "No Goals!"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "α : Type u_1\ninst✝¹ : AddCommGroup α\np a b : α\nn : ℕ\ninst✝ : NoZeroSMulDivisors ℕ α\nhn : n ≠ 0\nm : ℤ\n⊢ b - a = m • p ↔ b - a = m • p",
"after_state": "No Goals!"
},
{
"line": "with_reducible rfl",
"before_state": "α : Type u_1\ninst✝¹ : AddCommGroup α\np a b : α\nn : ℕ\ninst✝ : NoZeroSMulDivisors ℕ α\nhn : n ≠ 0\nm : ℤ\n⊢ b - a = m • p ↔ b - a = m • p",
"after_state": "No Goals!"
},
{
"line": "rfl",
"before_state": "α : Type u_1\ninst✝¹ : AddCommGroup α\np a b : α\nn : ℕ\ninst✝ : NoZeroSMulDivisors ℕ α\nhn : n ≠ 0\nm : ℤ\n⊢ b - a = m • p ↔ b - a = m • p",
"after_state": "No Goals!"
},
{
"line": "exact Iff.rfl✝",
"before_state": "α : Type u_1\ninst✝¹ : AddCommGroup α\np a b : α\nn : ℕ\ninst✝ : NoZeroSMulDivisors ℕ α\nhn : n ≠ 0\nm : ℤ\n⊢ b - a = m • p ↔ b - a = m • p",
"after_state": "No Goals!"
}
] |
theorem sub_modEq_zero : a - b ≡ 0 [PMOD p] ↔ a ≡ b [PMOD p] := by simp [sub_modEq_iff_modEq_add]
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/ModEq.lean
|
{
"open": [],
"variables": [
"{α : Type*}",
"[AddCommGroup α] {p a a₁ a₂ b b₁ b₂ c : α} {n : ℕ} {z : ℤ}"
]
}
|
[
{
"line": "simp [sub_modEq_iff_modEq_add]",
"before_state": "α : Type u_1\ninst✝ : AddCommGroup α\np a b : α\n⊢ a - b ≡ 0 [PMOD p] ↔ a ≡ b [PMOD p]",
"after_state": "No Goals!"
}
] |
theorem add_modEq_left : a + b ≡ a [PMOD p] ↔ b ≡ 0 [PMOD p] := by simp [← modEq_sub_iff_add_modEq']
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/ModEq.lean
|
{
"open": [],
"variables": [
"{α : Type*}",
"[AddCommGroup α] {p a a₁ a₂ b b₁ b₂ c : α} {n : ℕ} {z : ℤ}"
]
}
|
[
{
"line": "simp [← modEq_sub_iff_add_modEq']",
"before_state": "α : Type u_1\ninst✝ : AddCommGroup α\np a b : α\n⊢ a + b ≡ a [PMOD p] ↔ b ≡ 0 [PMOD p]",
"after_state": "No Goals!"
}
] |
theorem add_modEq_right : a + b ≡ b [PMOD p] ↔ a ≡ 0 [PMOD p] := by simp [← modEq_sub_iff_add_modEq]
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/ModEq.lean
|
{
"open": [],
"variables": [
"{α : Type*}",
"[AddCommGroup α] {p a a₁ a₂ b b₁ b₂ c : α} {n : ℕ} {z : ℤ}"
]
}
|
[
{
"line": "simp [← modEq_sub_iff_add_modEq]",
"before_state": "α : Type u_1\ninst✝ : AddCommGroup α\np a b : α\n⊢ a + b ≡ b [PMOD p] ↔ a ≡ 0 [PMOD p]",
"after_state": "No Goals!"
}
] |
theorem modEq_iff_int_modEq {a b z : ℤ} : a ≡ b [PMOD z] ↔ a ≡ b [ZMOD z] := by
simp [ModEq, dvd_iff_exists_eq_mul_left, Int.modEq_iff_dvd]
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/ModEq.lean
|
{
"open": [],
"variables": [
"{α : Type*}",
"[AddCommGroup α] {p a a₁ a₂ b b₁ b₂ c : α} {n : ℕ} {z : ℤ}"
]
}
|
[
{
"line": "simp [ModEq, dvd_iff_exists_eq_mul_left, Int.modEq_iff_dvd]",
"before_state": "a b z : ℤ\n⊢ a ≡ b [PMOD z] ↔ a ≡ b [ZMOD z]",
"after_state": "No Goals!"
}
] |
lemma intCast_modEq_intCast' {a b : ℤ} {n : ℕ} : a ≡ b [PMOD (n : α)] ↔ a ≡ b [PMOD (n : ℤ)] := by
simpa using intCast_modEq_intCast (α := α) (z := n)
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/ModEq.lean
|
{
"open": [],
"variables": [
"{α : Type*}",
"[AddCommGroup α] {p a a₁ a₂ b b₁ b₂ c : α} {n : ℕ} {z : ℤ}",
"[AddCommGroupWithOne α] [CharZero α]"
]
}
|
[
{
"line": "simpa using intCast_modEq_intCast (α := α) (z := n)",
"before_state": "α : Type u_1\ninst✝² : AddCommGroup α\ninst✝¹ : AddCommGroupWithOne α\ninst✝ : CharZero α\na b : ℤ\nn : ℕ\n⊢ ↑a ≡ ↑b [PMOD ↑n] ↔ a ≡ b [PMOD ↑n]",
"after_state": "No Goals!"
}
] |
theorem left_cancel (x : R) {y y' : R} : x ◃ y = x ◃ y' ↔ y = y' := by
constructor
· apply (act' x).injective
rintro rfl
rfl
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/Quandle.lean
|
{
"open": [
"MulOpposite",
"Quandles",
"Shelf"
],
"variables": [
"{S : Type*} [UnitalShelf S]",
"{R : Type*} [Rack R]"
]
}
|
[
{
"line": "constructor",
"before_state": "R : Type u_2\ninst✝ : Rack R\nx y y' : R\n⊢ x ◃ y = x ◃ y' ↔ y = y'",
"after_state": "case mp\nR : Type u_2\ninst✝ : Rack R\nx y y' : R\n⊢ x ◃ y = x ◃ y' → y = y'\n---\ncase mpr\nR : Type u_2\ninst✝ : Rack R\nx y y' : R\n⊢ y = y' → x ◃ y = x ◃ y'"
},
{
"line": "apply (act' x).injective",
"before_state": "case mp\nR : Type u_2\ninst✝ : Rack R\nx y y' : R\n⊢ x ◃ y = x ◃ y' → y = y'",
"after_state": "No Goals!"
},
{
"line": "rintro rfl",
"before_state": "case mpr\nR : Type u_2\ninst✝ : Rack R\nx y y' : R\n⊢ y = y' → x ◃ y = x ◃ y'",
"after_state": "case mpr\nR : Type u_2\ninst✝ : Rack R\nx y : R\n⊢ x ◃ y = x ◃ y"
},
{
"line": "rfl",
"before_state": "case mpr\nR : Type u_2\ninst✝ : Rack R\nx y : R\n⊢ x ◃ y = x ◃ y",
"after_state": "No Goals!"
},
{
"line": "eq_refl",
"before_state": "case mpr\nR : Type u_2\ninst✝ : Rack R\nx y : R\n⊢ x ◃ y = x ◃ y",
"after_state": "No Goals!"
}
] |
theorem left_cancel_inv (x : R) {y y' : R} : x ◃⁻¹ y = x ◃⁻¹ y' ↔ y = y' := by
constructor
· apply (act' x).symm.injective
rintro rfl
rfl
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/Quandle.lean
|
{
"open": [
"MulOpposite",
"Quandles",
"Shelf"
],
"variables": [
"{S : Type*} [UnitalShelf S]",
"{R : Type*} [Rack R]"
]
}
|
[
{
"line": "constructor",
"before_state": "R : Type u_2\ninst✝ : Rack R\nx y y' : R\n⊢ x ◃⁻¹ y = x ◃⁻¹ y' ↔ y = y'",
"after_state": "case mp\nR : Type u_2\ninst✝ : Rack R\nx y y' : R\n⊢ x ◃⁻¹ y = x ◃⁻¹ y' → y = y'\n---\ncase mpr\nR : Type u_2\ninst✝ : Rack R\nx y y' : R\n⊢ y = y' → x ◃⁻¹ y = x ◃⁻¹ y'"
},
{
"line": "apply (act' x).symm.injective",
"before_state": "case mp\nR : Type u_2\ninst✝ : Rack R\nx y y' : R\n⊢ x ◃⁻¹ y = x ◃⁻¹ y' → y = y'",
"after_state": "No Goals!"
},
{
"line": "rintro rfl",
"before_state": "case mpr\nR : Type u_2\ninst✝ : Rack R\nx y y' : R\n⊢ y = y' → x ◃⁻¹ y = x ◃⁻¹ y'",
"after_state": "case mpr\nR : Type u_2\ninst✝ : Rack R\nx y : R\n⊢ x ◃⁻¹ y = x ◃⁻¹ y"
},
{
"line": "rfl",
"before_state": "case mpr\nR : Type u_2\ninst✝ : Rack R\nx y : R\n⊢ x ◃⁻¹ y = x ◃⁻¹ y",
"after_state": "No Goals!"
},
{
"line": "eq_refl",
"before_state": "case mpr\nR : Type u_2\ninst✝ : Rack R\nx y : R\n⊢ x ◃⁻¹ y = x ◃⁻¹ y",
"after_state": "No Goals!"
}
] |
theorem fix_inv {x : Q} : x ◃⁻¹ x = x := by
rw [← left_cancel x]
simp
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/Quandle.lean
|
{
"open": [
"MulOpposite",
"Quandles",
"Shelf",
"Rack"
],
"variables": [
"{S : Type*} [UnitalShelf S]",
"{R : Type*} [Rack R]",
"{S₁ : Type*} {S₂ : Type*} {S₃ : Type*} [Shelf S₁] [Shelf S₂] [Shelf S₃]",
"{Q : Type*} [Quandle Q]"
]
}
|
[
{
"line": "rw [← left_cancel x]",
"before_state": "Q : Type u_6\ninst✝ : Quandle Q\nx : Q\n⊢ x ◃⁻¹ x = x",
"after_state": "Q : Type u_6\ninst✝ : Quandle Q\nx : Q\n⊢ x ◃ x ◃⁻¹ x = x ◃ x"
},
{
"line": "rewrite [← left_cancel x]",
"before_state": "Q : Type u_6\ninst✝ : Quandle Q\nx : Q\n⊢ x ◃⁻¹ x = x",
"after_state": "Q : Type u_6\ninst✝ : Quandle Q\nx : Q\n⊢ x ◃ x ◃⁻¹ x = x ◃ x"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "Q : Type u_6\ninst✝ : Quandle Q\nx : Q\n⊢ x ◃ x ◃⁻¹ x = x ◃ x",
"after_state": "Q : Type u_6\ninst✝ : Quandle Q\nx : Q\n⊢ x ◃ x ◃⁻¹ x = x ◃ x"
},
{
"line": "try (with_reducible rfl)",
"before_state": "Q : Type u_6\ninst✝ : Quandle Q\nx : Q\n⊢ x ◃ x ◃⁻¹ x = x ◃ x",
"after_state": "Q : Type u_6\ninst✝ : Quandle Q\nx : Q\n⊢ x ◃ x ◃⁻¹ x = x ◃ x"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "Q : Type u_6\ninst✝ : Quandle Q\nx : Q\n⊢ x ◃ x ◃⁻¹ x = x ◃ x",
"after_state": "Q : Type u_6\ninst✝ : Quandle Q\nx : Q\n⊢ x ◃ x ◃⁻¹ x = x ◃ x"
},
{
"line": "with_reducible rfl",
"before_state": "Q : Type u_6\ninst✝ : Quandle Q\nx : Q\n⊢ x ◃ x ◃⁻¹ x = x ◃ x",
"after_state": "Q : Type u_6\ninst✝ : Quandle Q\nx : Q\n⊢ x ◃ x ◃⁻¹ x = x ◃ x"
},
{
"line": "rfl",
"before_state": "Q : Type u_6\ninst✝ : Quandle Q\nx : Q\n⊢ x ◃ x ◃⁻¹ x = x ◃ x",
"after_state": "Q : Type u_6\ninst✝ : Quandle Q\nx : Q\n⊢ x ◃ x ◃⁻¹ x = x ◃ x"
},
{
"line": "apply_rfl",
"before_state": "Q : Type u_6\ninst✝ : Quandle Q\nx : Q\n⊢ x ◃ x ◃⁻¹ x = x ◃ x",
"after_state": "Q : Type u_6\ninst✝ : Quandle Q\nx : Q\n⊢ x ◃ x ◃⁻¹ x = x ◃ x"
},
{
"line": "skip",
"before_state": "Q : Type u_6\ninst✝ : Quandle Q\nx : Q\n⊢ x ◃ x ◃⁻¹ x = x ◃ x",
"after_state": "Q : Type u_6\ninst✝ : Quandle Q\nx : Q\n⊢ x ◃ x ◃⁻¹ x = x ◃ x"
},
{
"line": "simp",
"before_state": "Q : Type u_6\ninst✝ : Quandle Q\nx : Q\n⊢ x ◃ x ◃⁻¹ x = x ◃ x",
"after_state": "No Goals!"
}
] |
theorem well_def {R : Type*} [Rack R] {G : Type*} [Group G] (f : R →◃ Quandle.Conj G) :
∀ {a b : PreEnvelGroup R},
PreEnvelGroupRel' R a b → toEnvelGroup.mapAux f a = toEnvelGroup.mapAux f b
| _, _, PreEnvelGroupRel'.refl => rfl
| _, _, PreEnvelGroupRel'.symm h => (well_def f h).symm
| _, _, PreEnvelGroupRel'.trans hac hcb => Eq.trans (well_def f hac) (well_def f hcb)
| _, _, PreEnvelGroupRel'.congr_mul ha hb => by
simp [toEnvelGroup.mapAux, well_def f ha, well_def f hb]
| _, _, congr_inv ha => by simp [toEnvelGroup.mapAux, well_def f ha]
| _, _, assoc a b c => by apply mul_assoc
| _, _, PreEnvelGroupRel'.one_mul a => by simp [toEnvelGroup.mapAux]
| _, _, PreEnvelGroupRel'.mul_one a => by simp [toEnvelGroup.mapAux]
| _, _, PreEnvelGroupRel'.inv_mul_cancel a => by simp [toEnvelGroup.mapAux]
| _, _, act_incl x y => by simp [toEnvelGroup.mapAux]
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/Quandle.lean
|
{
"open": [
"MulOpposite",
"Quandles",
"Shelf",
"Rack",
"PreEnvelGroup",
"PreEnvelGroupRel'"
],
"variables": [
"{S : Type*} [UnitalShelf S]",
"{R : Type*} [Rack R]",
"{S₁ : Type*} {S₂ : Type*} {S₃ : Type*} [Shelf S₁] [Shelf S₂] [Shelf S₃]",
"{Q : Type*} [Quandle Q]"
]
}
|
[
{
"line": "simp [toEnvelGroup.mapAux, well_def f ha, well_def f hb]",
"before_state": "R : Type u_7\ninst✝¹ : Rack R\nG : Type u_8\ninst✝ : Group G\nf : R →◃ Quandle.Conj G\na✝ b✝ a'✝ b'✝ : PreEnvelGroup R\nha : PreEnvelGroupRel' R a✝ a'✝\nhb : PreEnvelGroupRel' R b✝ b'✝\n⊢ toEnvelGroup.mapAux f (a✝.mul b✝) = toEnvelGroup.mapAux f (a'✝.mul b'✝)",
"after_state": "No Goals!"
},
{
"line": "simp [toEnvelGroup.mapAux, well_def f ha]",
"before_state": "R : Type u_7\ninst✝¹ : Rack R\nG : Type u_8\ninst✝ : Group G\nf : R →◃ Quandle.Conj G\na✝ a'✝ : PreEnvelGroup R\nha : PreEnvelGroupRel' R a✝ a'✝\n⊢ toEnvelGroup.mapAux f a✝.inv = toEnvelGroup.mapAux f a'✝.inv",
"after_state": "No Goals!"
},
{
"line": "apply mul_assoc",
"before_state": "R : Type u_7\ninst✝¹ : Rack R\nG : Type u_8\ninst✝ : Group G\nf : R →◃ Quandle.Conj G\na b c : PreEnvelGroup R\n⊢ toEnvelGroup.mapAux f ((a.mul b).mul c) = toEnvelGroup.mapAux f (a.mul (b.mul c))",
"after_state": "No Goals!"
},
{
"line": "simp [toEnvelGroup.mapAux]",
"before_state": "R : Type u_7\ninst✝¹ : Rack R\nG : Type u_8\ninst✝ : Group G\nf : R →◃ Quandle.Conj G\na : PreEnvelGroup R\n⊢ toEnvelGroup.mapAux f (unit.mul a) = toEnvelGroup.mapAux f a",
"after_state": "No Goals!"
},
{
"line": "simp [toEnvelGroup.mapAux]",
"before_state": "R : Type u_7\ninst✝¹ : Rack R\nG : Type u_8\ninst✝ : Group G\nf : R →◃ Quandle.Conj G\na : PreEnvelGroup R\n⊢ toEnvelGroup.mapAux f (a.mul unit) = toEnvelGroup.mapAux f a",
"after_state": "No Goals!"
},
{
"line": "simp [toEnvelGroup.mapAux]",
"before_state": "R : Type u_7\ninst✝¹ : Rack R\nG : Type u_8\ninst✝ : Group G\nf : R →◃ Quandle.Conj G\na : PreEnvelGroup R\n⊢ toEnvelGroup.mapAux f (a.inv.mul a) = toEnvelGroup.mapAux f unit",
"after_state": "No Goals!"
},
{
"line": "simp [toEnvelGroup.mapAux]",
"before_state": "R : Type u_7\ninst✝¹ : Rack R\nG : Type u_8\ninst✝ : Group G\nf : R →◃ Quandle.Conj G\nx y : R\n⊢ toEnvelGroup.mapAux f (((incl x).mul (incl y)).mul (incl x).inv) = toEnvelGroup.mapAux f (incl (x ◃ y))",
"after_state": "No Goals!"
}
] |
theorem mul_re : (a * b).re = a.re * b.re - a.imI * b.imI - a.imJ * b.imJ - a.imK * b.imK :=
(QuaternionAlgebra.mul_re a b).trans <| by simp [one_mul, neg_mul, sub_eq_add_neg, neg_neg]
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/Quaternion.lean
|
{
"open": [
"Quaternion",
"MulOpposite",
"Quaternion"
],
"variables": [
"{S T R : Type*} {c₁ c₂ c₃ : R} (r x y : R) (a b : ℍ[R,c₁,c₂,c₃])",
"[Zero R]",
"[One R]",
"[Add R]",
"[AddZeroClass R]",
"[Neg R]",
"[AddGroup R]",
"[Ring R]",
"[SMul S R] [SMul T R] (s : S)",
"[AddCommGroupWithOne R]",
"[CommRing R]",
"(c₁ c₂ c₃)",
"[NoZeroDivisors R] [CharZero R]",
"{S T R : Type*} [CommRing R] (r x y : R) (a b : ℍ[R])"
]
}
|
[
{
"line": "simp [one_mul, neg_mul, sub_eq_add_neg, neg_neg]",
"before_state": "R : Type u_6\ninst✝ : CommRing R\na b : ℍ[R]\n⊢ a.re * b.re + -1 * a.imI * b.imI + -1 * a.imJ * b.imJ + 0 * -1 * a.imJ * b.imK - -1 * -1 * a.imK * b.imK =\n a.re * b.re - a.imI * b.imI - a.imJ * b.imJ - a.imK * b.imK",
"after_state": "No Goals!"
}
] |
theorem mul_imI : (a * b).imI = a.re * b.imI + a.imI * b.re + a.imJ * b.imK - a.imK * b.imJ :=
(QuaternionAlgebra.mul_imI a b).trans <| by ring
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/Quaternion.lean
|
{
"open": [
"Quaternion",
"MulOpposite",
"Quaternion"
],
"variables": [
"{S T R : Type*} {c₁ c₂ c₃ : R} (r x y : R) (a b : ℍ[R,c₁,c₂,c₃])",
"[Zero R]",
"[One R]",
"[Add R]",
"[AddZeroClass R]",
"[Neg R]",
"[AddGroup R]",
"[Ring R]",
"[SMul S R] [SMul T R] (s : S)",
"[AddCommGroupWithOne R]",
"[CommRing R]",
"(c₁ c₂ c₃)",
"[NoZeroDivisors R] [CharZero R]",
"{S T R : Type*} [CommRing R] (r x y : R) (a b : ℍ[R])"
]
}
|
[
{
"line": "ring",
"before_state": "R : Type u_6\ninst✝ : CommRing R\na b : ℍ[R]\n⊢ a.re * b.imI + a.imI * b.re + 0 * a.imI * b.imI - -1 * a.imJ * b.imK + -1 * a.imK * b.imJ =\n a.re * b.imI + a.imI * b.re + a.imJ * b.imK - a.imK * b.imJ",
"after_state": "No Goals!"
},
{
"line": "first\n| ring1\n|\n try_this ring_nf\"\\n\\nThe `ring` tactic failed to close the goal. Use `ring_nf` to obtain a normal form.\n \\nNote that `ring` works primarily in *commutative* rings. \\\n If you have a noncommutative ring, abelian group or module, consider using \\\n `noncomm_ring`, `abel` or `module` instead.\"",
"before_state": "R : Type u_6\ninst✝ : CommRing R\na b : ℍ[R]\n⊢ a.re * b.imI + a.imI * b.re + 0 * a.imI * b.imI - -1 * a.imJ * b.imK + -1 * a.imK * b.imJ =\n a.re * b.imI + a.imI * b.re + a.imJ * b.imK - a.imK * b.imJ",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "R : Type u_6\ninst✝ : CommRing R\na b : ℍ[R]\n⊢ a.re * b.imI + a.imI * b.re + 0 * a.imI * b.imI - -1 * a.imJ * b.imK + -1 * a.imK * b.imJ =\n a.re * b.imI + a.imI * b.re + a.imJ * b.imK - a.imK * b.imJ",
"after_state": "No Goals!"
}
] |
theorem mul_imJ : (a * b).imJ = a.re * b.imJ - a.imI * b.imK + a.imJ * b.re + a.imK * b.imI :=
(QuaternionAlgebra.mul_imJ a b).trans <| by ring
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/Quaternion.lean
|
{
"open": [
"Quaternion",
"MulOpposite",
"Quaternion"
],
"variables": [
"{S T R : Type*} {c₁ c₂ c₃ : R} (r x y : R) (a b : ℍ[R,c₁,c₂,c₃])",
"[Zero R]",
"[One R]",
"[Add R]",
"[AddZeroClass R]",
"[Neg R]",
"[AddGroup R]",
"[Ring R]",
"[SMul S R] [SMul T R] (s : S)",
"[AddCommGroupWithOne R]",
"[CommRing R]",
"(c₁ c₂ c₃)",
"[NoZeroDivisors R] [CharZero R]",
"{S T R : Type*} [CommRing R] (r x y : R) (a b : ℍ[R])"
]
}
|
[
{
"line": "ring",
"before_state": "R : Type u_6\ninst✝ : CommRing R\na b : ℍ[R]\n⊢ a.re * b.imJ + -1 * a.imI * b.imK + a.imJ * b.re + 0 * a.imJ * b.imI - -1 * a.imK * b.imI =\n a.re * b.imJ - a.imI * b.imK + a.imJ * b.re + a.imK * b.imI",
"after_state": "No Goals!"
},
{
"line": "first\n| ring1\n|\n try_this ring_nf\"\\n\\nThe `ring` tactic failed to close the goal. Use `ring_nf` to obtain a normal form.\n \\nNote that `ring` works primarily in *commutative* rings. \\\n If you have a noncommutative ring, abelian group or module, consider using \\\n `noncomm_ring`, `abel` or `module` instead.\"",
"before_state": "R : Type u_6\ninst✝ : CommRing R\na b : ℍ[R]\n⊢ a.re * b.imJ + -1 * a.imI * b.imK + a.imJ * b.re + 0 * a.imJ * b.imI - -1 * a.imK * b.imI =\n a.re * b.imJ - a.imI * b.imK + a.imJ * b.re + a.imK * b.imI",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "R : Type u_6\ninst✝ : CommRing R\na b : ℍ[R]\n⊢ a.re * b.imJ + -1 * a.imI * b.imK + a.imJ * b.re + 0 * a.imJ * b.imI - -1 * a.imK * b.imI =\n a.re * b.imJ - a.imI * b.imK + a.imJ * b.re + a.imK * b.imI",
"after_state": "No Goals!"
}
] |
theorem mul_imK : (a * b).imK = a.re * b.imK + a.imI * b.imJ - a.imJ * b.imI + a.imK * b.re :=
(QuaternionAlgebra.mul_imK a b).trans <| by ring
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/Quaternion.lean
|
{
"open": [
"Quaternion",
"MulOpposite",
"Quaternion"
],
"variables": [
"{S T R : Type*} {c₁ c₂ c₃ : R} (r x y : R) (a b : ℍ[R,c₁,c₂,c₃])",
"[Zero R]",
"[One R]",
"[Add R]",
"[AddZeroClass R]",
"[Neg R]",
"[AddGroup R]",
"[Ring R]",
"[SMul S R] [SMul T R] (s : S)",
"[AddCommGroupWithOne R]",
"[CommRing R]",
"(c₁ c₂ c₃)",
"[NoZeroDivisors R] [CharZero R]",
"{S T R : Type*} [CommRing R] (r x y : R) (a b : ℍ[R])"
]
}
|
[
{
"line": "ring",
"before_state": "R : Type u_6\ninst✝ : CommRing R\na b : ℍ[R]\n⊢ a.re * b.imK + a.imI * b.imJ + 0 * a.imI * b.imK - a.imJ * b.imI + a.imK * b.re =\n a.re * b.imK + a.imI * b.imJ - a.imJ * b.imI + a.imK * b.re",
"after_state": "No Goals!"
},
{
"line": "first\n| ring1\n|\n try_this ring_nf\"\\n\\nThe `ring` tactic failed to close the goal. Use `ring_nf` to obtain a normal form.\n \\nNote that `ring` works primarily in *commutative* rings. \\\n If you have a noncommutative ring, abelian group or module, consider using \\\n `noncomm_ring`, `abel` or `module` instead.\"",
"before_state": "R : Type u_6\ninst✝ : CommRing R\na b : ℍ[R]\n⊢ a.re * b.imK + a.imI * b.imJ + 0 * a.imI * b.imK - a.imJ * b.imI + a.imK * b.re =\n a.re * b.imK + a.imI * b.imJ - a.imJ * b.imI + a.imK * b.re",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "R : Type u_6\ninst✝ : CommRing R\na b : ℍ[R]\n⊢ a.re * b.imK + a.imI * b.imJ + 0 * a.imI * b.imK - a.imJ * b.imI + a.imK * b.re =\n a.re * b.imK + a.imI * b.imJ - a.imJ * b.imI + a.imK * b.re",
"after_state": "No Goals!"
}
] |
theorem im_star : star a.im = -a.im := by ext <;> simp
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/Quaternion.lean
|
{
"open": [
"Quaternion",
"MulOpposite",
"Quaternion"
],
"variables": [
"{S T R : Type*} {c₁ c₂ c₃ : R} (r x y : R) (a b : ℍ[R,c₁,c₂,c₃])",
"[Zero R]",
"[One R]",
"[Add R]",
"[AddZeroClass R]",
"[Neg R]",
"[AddGroup R]",
"[Ring R]",
"[SMul S R] [SMul T R] (s : S)",
"[AddCommGroupWithOne R]",
"[CommRing R]",
"(c₁ c₂ c₃)",
"[NoZeroDivisors R] [CharZero R]",
"{S T R : Type*} [CommRing R] (r x y : R) (a b : ℍ[R])"
]
}
|
[
{
"line": "focus\n ext\n with_annotate_state\"<;>\" skip\n all_goals simp",
"before_state": "R : Type u_6\ninst✝ : CommRing R\na : ℍ[R]\n⊢ star a.im = -a.im",
"after_state": "No Goals!"
},
{
"line": "ext",
"before_state": "R : Type u_6\ninst✝ : CommRing R\na : ℍ[R]\n⊢ star a.im = -a.im",
"after_state": "case a\nR : Type u_6\ninst✝ : CommRing R\na : ℍ[R]\n⊢ (star a.im).re = (-a.im).re\n---\ncase a\nR : Type u_6\ninst✝ : CommRing R\na : ℍ[R]\n⊢ (star a.im).imI = (-a.im).imI\n---\ncase a\nR : Type u_6\ninst✝ : CommRing R\na : ℍ[R]\n⊢ (star a.im).imJ = (-a.im).imJ\n---\ncase a\nR : Type u_6\ninst✝ : CommRing R\na : ℍ[R]\n⊢ (star a.im).imK = (-a.im).imK"
},
{
"line": "with_annotate_state\"<;>\" skip",
"before_state": "case a\nR : Type u_6\ninst✝ : CommRing R\na : ℍ[R]\n⊢ (star a.im).re = (-a.im).re\n---\ncase a\nR : Type u_6\ninst✝ : CommRing R\na : ℍ[R]\n⊢ (star a.im).imI = (-a.im).imI\n---\ncase a\nR : Type u_6\ninst✝ : CommRing R\na : ℍ[R]\n⊢ (star a.im).imJ = (-a.im).imJ\n---\ncase a\nR : Type u_6\ninst✝ : CommRing R\na : ℍ[R]\n⊢ (star a.im).imK = (-a.im).imK",
"after_state": "case a\nR : Type u_6\ninst✝ : CommRing R\na : ℍ[R]\n⊢ (star a.im).re = (-a.im).re\n---\ncase a\nR : Type u_6\ninst✝ : CommRing R\na : ℍ[R]\n⊢ (star a.im).imI = (-a.im).imI\n---\ncase a\nR : Type u_6\ninst✝ : CommRing R\na : ℍ[R]\n⊢ (star a.im).imJ = (-a.im).imJ\n---\ncase a\nR : Type u_6\ninst✝ : CommRing R\na : ℍ[R]\n⊢ (star a.im).imK = (-a.im).imK"
},
{
"line": "skip",
"before_state": "case a\nR : Type u_6\ninst✝ : CommRing R\na : ℍ[R]\n⊢ (star a.im).re = (-a.im).re\n---\ncase a\nR : Type u_6\ninst✝ : CommRing R\na : ℍ[R]\n⊢ (star a.im).imI = (-a.im).imI\n---\ncase a\nR : Type u_6\ninst✝ : CommRing R\na : ℍ[R]\n⊢ (star a.im).imJ = (-a.im).imJ\n---\ncase a\nR : Type u_6\ninst✝ : CommRing R\na : ℍ[R]\n⊢ (star a.im).imK = (-a.im).imK",
"after_state": "case a\nR : Type u_6\ninst✝ : CommRing R\na : ℍ[R]\n⊢ (star a.im).re = (-a.im).re\n---\ncase a\nR : Type u_6\ninst✝ : CommRing R\na : ℍ[R]\n⊢ (star a.im).imI = (-a.im).imI\n---\ncase a\nR : Type u_6\ninst✝ : CommRing R\na : ℍ[R]\n⊢ (star a.im).imJ = (-a.im).imJ\n---\ncase a\nR : Type u_6\ninst✝ : CommRing R\na : ℍ[R]\n⊢ (star a.im).imK = (-a.im).imK"
},
{
"line": "all_goals simp",
"before_state": "case a\nR : Type u_6\ninst✝ : CommRing R\na : ℍ[R]\n⊢ (star a.im).re = (-a.im).re\n---\ncase a\nR : Type u_6\ninst✝ : CommRing R\na : ℍ[R]\n⊢ (star a.im).imI = (-a.im).imI\n---\ncase a\nR : Type u_6\ninst✝ : CommRing R\na : ℍ[R]\n⊢ (star a.im).imJ = (-a.im).imJ\n---\ncase a\nR : Type u_6\ninst✝ : CommRing R\na : ℍ[R]\n⊢ (star a.im).imK = (-a.im).imK",
"after_state": "No Goals!"
},
{
"line": "simp",
"before_state": "case a\nR : Type u_6\ninst✝ : CommRing R\na : ℍ[R]\n⊢ (star a.im).re = (-a.im).re",
"after_state": "No Goals!"
},
{
"line": "simp",
"before_state": "case a\nR : Type u_6\ninst✝ : CommRing R\na : ℍ[R]\n⊢ (star a.im).imI = (-a.im).imI",
"after_state": "No Goals!"
},
{
"line": "simp",
"before_state": "case a\nR : Type u_6\ninst✝ : CommRing R\na : ℍ[R]\n⊢ (star a.im).imJ = (-a.im).imJ",
"after_state": "No Goals!"
},
{
"line": "simp",
"before_state": "case a\nR : Type u_6\ninst✝ : CommRing R\na : ℍ[R]\n⊢ (star a.im).imK = (-a.im).imK",
"after_state": "No Goals!"
}
] |
theorem normSq_coe : normSq (x : ℍ[R]) = x ^ 2 := by
rw [normSq_def]
rw [star_coe]
rw [← coe_mul]
rw [coe_re]
rw [sq]
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/Quaternion.lean
|
{
"open": [
"Quaternion",
"MulOpposite",
"Quaternion",
"MulOpposite"
],
"variables": [
"{S T R : Type*} {c₁ c₂ c₃ : R} (r x y : R) (a b : ℍ[R,c₁,c₂,c₃])",
"[Zero R]",
"[One R]",
"[Add R]",
"[AddZeroClass R]",
"[Neg R]",
"[AddGroup R]",
"[Ring R]",
"[SMul S R] [SMul T R] (s : S)",
"[AddCommGroupWithOne R]",
"[CommRing R]",
"(c₁ c₂ c₃)",
"[NoZeroDivisors R] [CharZero R]",
"{S T R : Type*} [CommRing R] (r x y : R) (a b : ℍ[R])",
"[NoZeroDivisors R] [CharZero R]"
]
}
|
[
{
"line": "rw [normSq_def]",
"before_state": "R : Type u_6\ninst✝² : CommRing R\nx : R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ normSq ↑x = x ^ 2",
"after_state": "R : Type u_6\ninst✝² : CommRing R\nx : R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ (↑x * star ↑x).re = x ^ 2"
},
{
"line": "rewrite [normSq_def]",
"before_state": "R : Type u_6\ninst✝² : CommRing R\nx : R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ normSq ↑x = x ^ 2",
"after_state": "R : Type u_6\ninst✝² : CommRing R\nx : R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ (↑x * star ↑x).re = x ^ 2"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "R : Type u_6\ninst✝² : CommRing R\nx : R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ (↑x * star ↑x).re = x ^ 2",
"after_state": "R : Type u_6\ninst✝² : CommRing R\nx : R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ (↑x * star ↑x).re = x ^ 2"
},
{
"line": "try (with_reducible rfl)",
"before_state": "R : Type u_6\ninst✝² : CommRing R\nx : R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ (↑x * star ↑x).re = x ^ 2",
"after_state": "R : Type u_6\ninst✝² : CommRing R\nx : R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ (↑x * star ↑x).re = x ^ 2"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "R : Type u_6\ninst✝² : CommRing R\nx : R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ (↑x * star ↑x).re = x ^ 2",
"after_state": "R : Type u_6\ninst✝² : CommRing R\nx : R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ (↑x * star ↑x).re = x ^ 2"
},
{
"line": "with_reducible rfl",
"before_state": "R : Type u_6\ninst✝² : CommRing R\nx : R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ (↑x * star ↑x).re = x ^ 2",
"after_state": "R : Type u_6\ninst✝² : CommRing R\nx : R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ (↑x * star ↑x).re = x ^ 2"
},
{
"line": "rfl",
"before_state": "R : Type u_6\ninst✝² : CommRing R\nx : R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ (↑x * star ↑x).re = x ^ 2",
"after_state": "R : Type u_6\ninst✝² : CommRing R\nx : R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ (↑x * star ↑x).re = x ^ 2"
},
{
"line": "apply_rfl",
"before_state": "R : Type u_6\ninst✝² : CommRing R\nx : R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ (↑x * star ↑x).re = x ^ 2",
"after_state": "R : Type u_6\ninst✝² : CommRing R\nx : R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ (↑x * star ↑x).re = x ^ 2"
},
{
"line": "skip",
"before_state": "R : Type u_6\ninst✝² : CommRing R\nx : R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ (↑x * star ↑x).re = x ^ 2",
"after_state": "R : Type u_6\ninst✝² : CommRing R\nx : R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ (↑x * star ↑x).re = x ^ 2"
},
{
"line": "rw [star_coe]",
"before_state": "R : Type u_6\ninst✝² : CommRing R\nx : R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ (↑x * star ↑x).re = x ^ 2",
"after_state": "R : Type u_6\ninst✝² : CommRing R\nx : R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ (↑x * ↑x).re = x ^ 2"
},
{
"line": "rewrite [star_coe]",
"before_state": "R : Type u_6\ninst✝² : CommRing R\nx : R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ (↑x * star ↑x).re = x ^ 2",
"after_state": "R : Type u_6\ninst✝² : CommRing R\nx : R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ (↑x * ↑x).re = x ^ 2"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "R : Type u_6\ninst✝² : CommRing R\nx : R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ (↑x * ↑x).re = x ^ 2",
"after_state": "R : Type u_6\ninst✝² : CommRing R\nx : R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ (↑x * ↑x).re = x ^ 2"
},
{
"line": "try (with_reducible rfl)",
"before_state": "R : Type u_6\ninst✝² : CommRing R\nx : R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ (↑x * ↑x).re = x ^ 2",
"after_state": "R : Type u_6\ninst✝² : CommRing R\nx : R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ (↑x * ↑x).re = x ^ 2"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "R : Type u_6\ninst✝² : CommRing R\nx : R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ (↑x * ↑x).re = x ^ 2",
"after_state": "R : Type u_6\ninst✝² : CommRing R\nx : R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ (↑x * ↑x).re = x ^ 2"
},
{
"line": "with_reducible rfl",
"before_state": "R : Type u_6\ninst✝² : CommRing R\nx : R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ (↑x * ↑x).re = x ^ 2",
"after_state": "R : Type u_6\ninst✝² : CommRing R\nx : R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ (↑x * ↑x).re = x ^ 2"
},
{
"line": "rfl",
"before_state": "R : Type u_6\ninst✝² : CommRing R\nx : R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ (↑x * ↑x).re = x ^ 2",
"after_state": "R : Type u_6\ninst✝² : CommRing R\nx : R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ (↑x * ↑x).re = x ^ 2"
},
{
"line": "apply_rfl",
"before_state": "R : Type u_6\ninst✝² : CommRing R\nx : R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ (↑x * ↑x).re = x ^ 2",
"after_state": "R : Type u_6\ninst✝² : CommRing R\nx : R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ (↑x * ↑x).re = x ^ 2"
},
{
"line": "skip",
"before_state": "R : Type u_6\ninst✝² : CommRing R\nx : R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ (↑x * ↑x).re = x ^ 2",
"after_state": "R : Type u_6\ninst✝² : CommRing R\nx : R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ (↑x * ↑x).re = x ^ 2"
},
{
"line": "rw [← coe_mul]",
"before_state": "R : Type u_6\ninst✝² : CommRing R\nx : R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ (↑x * ↑x).re = x ^ 2",
"after_state": "R : Type u_6\ninst✝² : CommRing R\nx : R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ (↑(x * x)).re = x ^ 2"
},
{
"line": "rewrite [← coe_mul]",
"before_state": "R : Type u_6\ninst✝² : CommRing R\nx : R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ (↑x * ↑x).re = x ^ 2",
"after_state": "R : Type u_6\ninst✝² : CommRing R\nx : R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ (↑(x * x)).re = x ^ 2"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "R : Type u_6\ninst✝² : CommRing R\nx : R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ (↑(x * x)).re = x ^ 2",
"after_state": "R : Type u_6\ninst✝² : CommRing R\nx : R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ (↑(x * x)).re = x ^ 2"
},
{
"line": "try (with_reducible rfl)",
"before_state": "R : Type u_6\ninst✝² : CommRing R\nx : R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ (↑(x * x)).re = x ^ 2",
"after_state": "R : Type u_6\ninst✝² : CommRing R\nx : R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ (↑(x * x)).re = x ^ 2"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "R : Type u_6\ninst✝² : CommRing R\nx : R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ (↑(x * x)).re = x ^ 2",
"after_state": "R : Type u_6\ninst✝² : CommRing R\nx : R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ (↑(x * x)).re = x ^ 2"
},
{
"line": "with_reducible rfl",
"before_state": "R : Type u_6\ninst✝² : CommRing R\nx : R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ (↑(x * x)).re = x ^ 2",
"after_state": "R : Type u_6\ninst✝² : CommRing R\nx : R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ (↑(x * x)).re = x ^ 2"
},
{
"line": "rfl",
"before_state": "R : Type u_6\ninst✝² : CommRing R\nx : R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ (↑(x * x)).re = x ^ 2",
"after_state": "R : Type u_6\ninst✝² : CommRing R\nx : R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ (↑(x * x)).re = x ^ 2"
},
{
"line": "apply_rfl",
"before_state": "R : Type u_6\ninst✝² : CommRing R\nx : R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ (↑(x * x)).re = x ^ 2",
"after_state": "R : Type u_6\ninst✝² : CommRing R\nx : R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ (↑(x * x)).re = x ^ 2"
},
{
"line": "skip",
"before_state": "R : Type u_6\ninst✝² : CommRing R\nx : R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ (↑(x * x)).re = x ^ 2",
"after_state": "R : Type u_6\ninst✝² : CommRing R\nx : R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ (↑(x * x)).re = x ^ 2"
},
{
"line": "rw [coe_re]",
"before_state": "R : Type u_6\ninst✝² : CommRing R\nx : R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ (↑(x * x)).re = x ^ 2",
"after_state": "R : Type u_6\ninst✝² : CommRing R\nx : R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ x * x = x ^ 2"
},
{
"line": "rewrite [coe_re]",
"before_state": "R : Type u_6\ninst✝² : CommRing R\nx : R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ (↑(x * x)).re = x ^ 2",
"after_state": "R : Type u_6\ninst✝² : CommRing R\nx : R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ x * x = x ^ 2"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "R : Type u_6\ninst✝² : CommRing R\nx : R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ x * x = x ^ 2",
"after_state": "R : Type u_6\ninst✝² : CommRing R\nx : R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ x * x = x ^ 2"
},
{
"line": "try (with_reducible rfl)",
"before_state": "R : Type u_6\ninst✝² : CommRing R\nx : R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ x * x = x ^ 2",
"after_state": "R : Type u_6\ninst✝² : CommRing R\nx : R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ x * x = x ^ 2"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "R : Type u_6\ninst✝² : CommRing R\nx : R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ x * x = x ^ 2",
"after_state": "R : Type u_6\ninst✝² : CommRing R\nx : R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ x * x = x ^ 2"
},
{
"line": "with_reducible rfl",
"before_state": "R : Type u_6\ninst✝² : CommRing R\nx : R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ x * x = x ^ 2",
"after_state": "R : Type u_6\ninst✝² : CommRing R\nx : R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ x * x = x ^ 2"
},
{
"line": "rfl",
"before_state": "R : Type u_6\ninst✝² : CommRing R\nx : R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ x * x = x ^ 2",
"after_state": "R : Type u_6\ninst✝² : CommRing R\nx : R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ x * x = x ^ 2"
},
{
"line": "apply_rfl",
"before_state": "R : Type u_6\ninst✝² : CommRing R\nx : R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ x * x = x ^ 2",
"after_state": "R : Type u_6\ninst✝² : CommRing R\nx : R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ x * x = x ^ 2"
},
{
"line": "skip",
"before_state": "R : Type u_6\ninst✝² : CommRing R\nx : R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ x * x = x ^ 2",
"after_state": "R : Type u_6\ninst✝² : CommRing R\nx : R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ x * x = x ^ 2"
},
{
"line": "rw [sq]",
"before_state": "R : Type u_6\ninst✝² : CommRing R\nx : R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ x * x = x ^ 2",
"after_state": "No Goals!"
},
{
"line": "rewrite [sq]",
"before_state": "R : Type u_6\ninst✝² : CommRing R\nx : R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ x * x = x ^ 2",
"after_state": "R : Type u_6\ninst✝² : CommRing R\nx : R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ x * x = x * x"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "R : Type u_6\ninst✝² : CommRing R\nx : R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ x * x = x * x",
"after_state": "No Goals!"
},
{
"line": "try (with_reducible rfl)",
"before_state": "R : Type u_6\ninst✝² : CommRing R\nx : R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ x * x = x * x",
"after_state": "No Goals!"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "R : Type u_6\ninst✝² : CommRing R\nx : R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ x * x = x * x",
"after_state": "No Goals!"
},
{
"line": "with_reducible rfl",
"before_state": "R : Type u_6\ninst✝² : CommRing R\nx : R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ x * x = x * x",
"after_state": "No Goals!"
},
{
"line": "rfl",
"before_state": "R : Type u_6\ninst✝² : CommRing R\nx : R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ x * x = x * x",
"after_state": "No Goals!"
},
{
"line": "eq_refl",
"before_state": "R : Type u_6\ninst✝² : CommRing R\nx : R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ x * x = x * x",
"after_state": "No Goals!"
}
] |
theorem normSq_star : normSq (star a) = normSq a := by simp [normSq_def']
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/Quaternion.lean
|
{
"open": [
"Quaternion",
"MulOpposite",
"Quaternion",
"MulOpposite"
],
"variables": [
"{S T R : Type*} {c₁ c₂ c₃ : R} (r x y : R) (a b : ℍ[R,c₁,c₂,c₃])",
"[Zero R]",
"[One R]",
"[Add R]",
"[AddZeroClass R]",
"[Neg R]",
"[AddGroup R]",
"[Ring R]",
"[SMul S R] [SMul T R] (s : S)",
"[AddCommGroupWithOne R]",
"[CommRing R]",
"(c₁ c₂ c₃)",
"[NoZeroDivisors R] [CharZero R]",
"{S T R : Type*} [CommRing R] (r x y : R) (a b : ℍ[R])",
"[NoZeroDivisors R] [CharZero R]"
]
}
|
[
{
"line": "simp [normSq_def']",
"before_state": "R : Type u_6\ninst✝² : CommRing R\na : ℍ[R]\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ normSq (star a) = normSq a",
"after_state": "No Goals!"
}
] |
theorem normSq_natCast (n : ℕ) : normSq (n : ℍ[R]) = (n : R) ^ 2 := by
rw [← coe_natCast]
rw [normSq_coe]
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/Quaternion.lean
|
{
"open": [
"Quaternion",
"MulOpposite",
"Quaternion",
"MulOpposite"
],
"variables": [
"{S T R : Type*} {c₁ c₂ c₃ : R} (r x y : R) (a b : ℍ[R,c₁,c₂,c₃])",
"[Zero R]",
"[One R]",
"[Add R]",
"[AddZeroClass R]",
"[Neg R]",
"[AddGroup R]",
"[Ring R]",
"[SMul S R] [SMul T R] (s : S)",
"[AddCommGroupWithOne R]",
"[CommRing R]",
"(c₁ c₂ c₃)",
"[NoZeroDivisors R] [CharZero R]",
"{S T R : Type*} [CommRing R] (r x y : R) (a b : ℍ[R])",
"[NoZeroDivisors R] [CharZero R]"
]
}
|
[
{
"line": "rw [← coe_natCast]",
"before_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\nn : ℕ\n⊢ normSq ↑n = ↑n ^ 2",
"after_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\nn : ℕ\n⊢ normSq ↑↑n = ↑n ^ 2"
},
{
"line": "rewrite [← coe_natCast]",
"before_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\nn : ℕ\n⊢ normSq ↑n = ↑n ^ 2",
"after_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\nn : ℕ\n⊢ normSq ↑↑n = ↑n ^ 2"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\nn : ℕ\n⊢ normSq ↑↑n = ↑n ^ 2",
"after_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\nn : ℕ\n⊢ normSq ↑↑n = ↑n ^ 2"
},
{
"line": "try (with_reducible rfl)",
"before_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\nn : ℕ\n⊢ normSq ↑↑n = ↑n ^ 2",
"after_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\nn : ℕ\n⊢ normSq ↑↑n = ↑n ^ 2"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\nn : ℕ\n⊢ normSq ↑↑n = ↑n ^ 2",
"after_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\nn : ℕ\n⊢ normSq ↑↑n = ↑n ^ 2"
},
{
"line": "with_reducible rfl",
"before_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\nn : ℕ\n⊢ normSq ↑↑n = ↑n ^ 2",
"after_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\nn : ℕ\n⊢ normSq ↑↑n = ↑n ^ 2"
},
{
"line": "rfl",
"before_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\nn : ℕ\n⊢ normSq ↑↑n = ↑n ^ 2",
"after_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\nn : ℕ\n⊢ normSq ↑↑n = ↑n ^ 2"
},
{
"line": "apply_rfl",
"before_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\nn : ℕ\n⊢ normSq ↑↑n = ↑n ^ 2",
"after_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\nn : ℕ\n⊢ normSq ↑↑n = ↑n ^ 2"
},
{
"line": "skip",
"before_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\nn : ℕ\n⊢ normSq ↑↑n = ↑n ^ 2",
"after_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\nn : ℕ\n⊢ normSq ↑↑n = ↑n ^ 2"
},
{
"line": "rw [normSq_coe]",
"before_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\nn : ℕ\n⊢ normSq ↑↑n = ↑n ^ 2",
"after_state": "No Goals!"
},
{
"line": "rewrite [normSq_coe]",
"before_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\nn : ℕ\n⊢ normSq ↑↑n = ↑n ^ 2",
"after_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\nn : ℕ\n⊢ ↑n ^ 2 = ↑n ^ 2"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\nn : ℕ\n⊢ ↑n ^ 2 = ↑n ^ 2",
"after_state": "No Goals!"
},
{
"line": "try (with_reducible rfl)",
"before_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\nn : ℕ\n⊢ ↑n ^ 2 = ↑n ^ 2",
"after_state": "No Goals!"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\nn : ℕ\n⊢ ↑n ^ 2 = ↑n ^ 2",
"after_state": "No Goals!"
},
{
"line": "with_reducible rfl",
"before_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\nn : ℕ\n⊢ ↑n ^ 2 = ↑n ^ 2",
"after_state": "No Goals!"
},
{
"line": "rfl",
"before_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\nn : ℕ\n⊢ ↑n ^ 2 = ↑n ^ 2",
"after_state": "No Goals!"
},
{
"line": "eq_refl",
"before_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\nn : ℕ\n⊢ ↑n ^ 2 = ↑n ^ 2",
"after_state": "No Goals!"
}
] |
theorem normSq_intCast (z : ℤ) : normSq (z : ℍ[R]) = (z : R) ^ 2 := by
rw [← coe_intCast]
rw [normSq_coe]
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/Quaternion.lean
|
{
"open": [
"Quaternion",
"MulOpposite",
"Quaternion",
"MulOpposite"
],
"variables": [
"{S T R : Type*} {c₁ c₂ c₃ : R} (r x y : R) (a b : ℍ[R,c₁,c₂,c₃])",
"[Zero R]",
"[One R]",
"[Add R]",
"[AddZeroClass R]",
"[Neg R]",
"[AddGroup R]",
"[Ring R]",
"[SMul S R] [SMul T R] (s : S)",
"[AddCommGroupWithOne R]",
"[CommRing R]",
"(c₁ c₂ c₃)",
"[NoZeroDivisors R] [CharZero R]",
"{S T R : Type*} [CommRing R] (r x y : R) (a b : ℍ[R])",
"[NoZeroDivisors R] [CharZero R]"
]
}
|
[
{
"line": "rw [← coe_intCast]",
"before_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\nz : ℤ\n⊢ normSq ↑z = ↑z ^ 2",
"after_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\nz : ℤ\n⊢ normSq ↑↑z = ↑z ^ 2"
},
{
"line": "rewrite [← coe_intCast]",
"before_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\nz : ℤ\n⊢ normSq ↑z = ↑z ^ 2",
"after_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\nz : ℤ\n⊢ normSq ↑↑z = ↑z ^ 2"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\nz : ℤ\n⊢ normSq ↑↑z = ↑z ^ 2",
"after_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\nz : ℤ\n⊢ normSq ↑↑z = ↑z ^ 2"
},
{
"line": "try (with_reducible rfl)",
"before_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\nz : ℤ\n⊢ normSq ↑↑z = ↑z ^ 2",
"after_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\nz : ℤ\n⊢ normSq ↑↑z = ↑z ^ 2"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\nz : ℤ\n⊢ normSq ↑↑z = ↑z ^ 2",
"after_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\nz : ℤ\n⊢ normSq ↑↑z = ↑z ^ 2"
},
{
"line": "with_reducible rfl",
"before_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\nz : ℤ\n⊢ normSq ↑↑z = ↑z ^ 2",
"after_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\nz : ℤ\n⊢ normSq ↑↑z = ↑z ^ 2"
},
{
"line": "rfl",
"before_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\nz : ℤ\n⊢ normSq ↑↑z = ↑z ^ 2",
"after_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\nz : ℤ\n⊢ normSq ↑↑z = ↑z ^ 2"
},
{
"line": "apply_rfl",
"before_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\nz : ℤ\n⊢ normSq ↑↑z = ↑z ^ 2",
"after_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\nz : ℤ\n⊢ normSq ↑↑z = ↑z ^ 2"
},
{
"line": "skip",
"before_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\nz : ℤ\n⊢ normSq ↑↑z = ↑z ^ 2",
"after_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\nz : ℤ\n⊢ normSq ↑↑z = ↑z ^ 2"
},
{
"line": "rw [normSq_coe]",
"before_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\nz : ℤ\n⊢ normSq ↑↑z = ↑z ^ 2",
"after_state": "No Goals!"
},
{
"line": "rewrite [normSq_coe]",
"before_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\nz : ℤ\n⊢ normSq ↑↑z = ↑z ^ 2",
"after_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\nz : ℤ\n⊢ ↑z ^ 2 = ↑z ^ 2"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\nz : ℤ\n⊢ ↑z ^ 2 = ↑z ^ 2",
"after_state": "No Goals!"
},
{
"line": "try (with_reducible rfl)",
"before_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\nz : ℤ\n⊢ ↑z ^ 2 = ↑z ^ 2",
"after_state": "No Goals!"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\nz : ℤ\n⊢ ↑z ^ 2 = ↑z ^ 2",
"after_state": "No Goals!"
},
{
"line": "with_reducible rfl",
"before_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\nz : ℤ\n⊢ ↑z ^ 2 = ↑z ^ 2",
"after_state": "No Goals!"
},
{
"line": "rfl",
"before_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\nz : ℤ\n⊢ ↑z ^ 2 = ↑z ^ 2",
"after_state": "No Goals!"
},
{
"line": "eq_refl",
"before_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\nz : ℤ\n⊢ ↑z ^ 2 = ↑z ^ 2",
"after_state": "No Goals!"
}
] |
theorem normSq_neg : normSq (-a) = normSq a := by simp only [normSq_def, star_neg, neg_mul_neg]
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/Quaternion.lean
|
{
"open": [
"Quaternion",
"MulOpposite",
"Quaternion",
"MulOpposite"
],
"variables": [
"{S T R : Type*} {c₁ c₂ c₃ : R} (r x y : R) (a b : ℍ[R,c₁,c₂,c₃])",
"[Zero R]",
"[One R]",
"[Add R]",
"[AddZeroClass R]",
"[Neg R]",
"[AddGroup R]",
"[Ring R]",
"[SMul S R] [SMul T R] (s : S)",
"[AddCommGroupWithOne R]",
"[CommRing R]",
"(c₁ c₂ c₃)",
"[NoZeroDivisors R] [CharZero R]",
"{S T R : Type*} [CommRing R] (r x y : R) (a b : ℍ[R])",
"[NoZeroDivisors R] [CharZero R]"
]
}
|
[
{
"line": "simp only [normSq_def, star_neg, neg_mul_neg]",
"before_state": "R : Type u_6\ninst✝² : CommRing R\na : ℍ[R]\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ normSq (-a) = normSq a",
"after_state": "No Goals!"
}
] |
theorem self_mul_star : a * star a = normSq a := by rw [mul_star_eq_coe, normSq_def]
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/Quaternion.lean
|
{
"open": [
"Quaternion",
"MulOpposite",
"Quaternion",
"MulOpposite"
],
"variables": [
"{S T R : Type*} {c₁ c₂ c₃ : R} (r x y : R) (a b : ℍ[R,c₁,c₂,c₃])",
"[Zero R]",
"[One R]",
"[Add R]",
"[AddZeroClass R]",
"[Neg R]",
"[AddGroup R]",
"[Ring R]",
"[SMul S R] [SMul T R] (s : S)",
"[AddCommGroupWithOne R]",
"[CommRing R]",
"(c₁ c₂ c₃)",
"[NoZeroDivisors R] [CharZero R]",
"{S T R : Type*} [CommRing R] (r x y : R) (a b : ℍ[R])",
"[NoZeroDivisors R] [CharZero R]"
]
}
|
[
{
"line": "rw [mul_star_eq_coe, normSq_def]",
"before_state": "R : Type u_6\ninst✝² : CommRing R\na : ℍ[R]\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ a * star a = ↑(normSq a)",
"after_state": "No Goals!"
},
{
"line": "rewrite [mul_star_eq_coe, normSq_def]",
"before_state": "R : Type u_6\ninst✝² : CommRing R\na : ℍ[R]\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ a * star a = ↑(normSq a)",
"after_state": "R : Type u_6\ninst✝² : CommRing R\na : ℍ[R]\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ ↑(a * star a).re = ↑(a * star a).re"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "R : Type u_6\ninst✝² : CommRing R\na : ℍ[R]\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ ↑(a * star a).re = ↑(a * star a).re",
"after_state": "No Goals!"
},
{
"line": "try (with_reducible rfl)",
"before_state": "R : Type u_6\ninst✝² : CommRing R\na : ℍ[R]\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ ↑(a * star a).re = ↑(a * star a).re",
"after_state": "No Goals!"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "R : Type u_6\ninst✝² : CommRing R\na : ℍ[R]\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ ↑(a * star a).re = ↑(a * star a).re",
"after_state": "No Goals!"
},
{
"line": "with_reducible rfl",
"before_state": "R : Type u_6\ninst✝² : CommRing R\na : ℍ[R]\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ ↑(a * star a).re = ↑(a * star a).re",
"after_state": "No Goals!"
},
{
"line": "rfl",
"before_state": "R : Type u_6\ninst✝² : CommRing R\na : ℍ[R]\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ ↑(a * star a).re = ↑(a * star a).re",
"after_state": "No Goals!"
},
{
"line": "eq_refl",
"before_state": "R : Type u_6\ninst✝² : CommRing R\na : ℍ[R]\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ ↑(a * star a).re = ↑(a * star a).re",
"after_state": "No Goals!"
}
] |
theorem star_mul_self : star a * a = normSq a := by rw [star_comm_self, self_mul_star]
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/Quaternion.lean
|
{
"open": [
"Quaternion",
"MulOpposite",
"Quaternion",
"MulOpposite"
],
"variables": [
"{S T R : Type*} {c₁ c₂ c₃ : R} (r x y : R) (a b : ℍ[R,c₁,c₂,c₃])",
"[Zero R]",
"[One R]",
"[Add R]",
"[AddZeroClass R]",
"[Neg R]",
"[AddGroup R]",
"[Ring R]",
"[SMul S R] [SMul T R] (s : S)",
"[AddCommGroupWithOne R]",
"[CommRing R]",
"(c₁ c₂ c₃)",
"[NoZeroDivisors R] [CharZero R]",
"{S T R : Type*} [CommRing R] (r x y : R) (a b : ℍ[R])",
"[NoZeroDivisors R] [CharZero R]"
]
}
|
[
{
"line": "rw [star_comm_self, self_mul_star]",
"before_state": "R : Type u_6\ninst✝² : CommRing R\na : ℍ[R]\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ star a * a = ↑(normSq a)",
"after_state": "No Goals!"
},
{
"line": "rewrite [star_comm_self, self_mul_star]",
"before_state": "R : Type u_6\ninst✝² : CommRing R\na : ℍ[R]\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ star a * a = ↑(normSq a)",
"after_state": "R : Type u_6\ninst✝² : CommRing R\na : ℍ[R]\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ ↑(normSq a) = ↑(normSq a)"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "R : Type u_6\ninst✝² : CommRing R\na : ℍ[R]\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ ↑(normSq a) = ↑(normSq a)",
"after_state": "No Goals!"
},
{
"line": "try (with_reducible rfl)",
"before_state": "R : Type u_6\ninst✝² : CommRing R\na : ℍ[R]\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ ↑(normSq a) = ↑(normSq a)",
"after_state": "No Goals!"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "R : Type u_6\ninst✝² : CommRing R\na : ℍ[R]\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ ↑(normSq a) = ↑(normSq a)",
"after_state": "No Goals!"
},
{
"line": "with_reducible rfl",
"before_state": "R : Type u_6\ninst✝² : CommRing R\na : ℍ[R]\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ ↑(normSq a) = ↑(normSq a)",
"after_state": "No Goals!"
},
{
"line": "rfl",
"before_state": "R : Type u_6\ninst✝² : CommRing R\na : ℍ[R]\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ ↑(normSq a) = ↑(normSq a)",
"after_state": "No Goals!"
},
{
"line": "eq_refl",
"before_state": "R : Type u_6\ninst✝² : CommRing R\na : ℍ[R]\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ ↑(normSq a) = ↑(normSq a)",
"after_state": "No Goals!"
}
] |
theorem im_sq : a.im ^ 2 = -normSq a.im := by
simp_rw [sq, ← star_mul_self, im_star, neg_mul, neg_neg]
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/Quaternion.lean
|
{
"open": [
"Quaternion",
"MulOpposite",
"Quaternion",
"MulOpposite"
],
"variables": [
"{S T R : Type*} {c₁ c₂ c₃ : R} (r x y : R) (a b : ℍ[R,c₁,c₂,c₃])",
"[Zero R]",
"[One R]",
"[Add R]",
"[AddZeroClass R]",
"[Neg R]",
"[AddGroup R]",
"[Ring R]",
"[SMul S R] [SMul T R] (s : S)",
"[AddCommGroupWithOne R]",
"[CommRing R]",
"(c₁ c₂ c₃)",
"[NoZeroDivisors R] [CharZero R]",
"{S T R : Type*} [CommRing R] (r x y : R) (a b : ℍ[R])",
"[NoZeroDivisors R] [CharZero R]"
]
}
|
[
{
"line": "simp_rw [sq, ← star_mul_self, im_star, neg_mul, neg_neg]",
"before_state": "R : Type u_6\ninst✝² : CommRing R\na : ℍ[R]\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ a.im ^ 2 = -↑(normSq a.im)",
"after_state": "No Goals!"
},
{
"line": "simp (failIfUnchanged✝ := false✝) only",
"before_state": "R : Type u_6\ninst✝² : CommRing R\na : ℍ[R]\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ a.im ^ 2 = -↑(normSq a.im)",
"after_state": "R : Type u_6\ninst✝² : CommRing R\na : ℍ[R]\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ a.im ^ 2 = -↑(normSq a.im)"
},
{
"line": "simp only [sq]",
"before_state": "R : Type u_6\ninst✝² : CommRing R\na : ℍ[R]\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ a.im ^ 2 = -↑(normSq a.im)",
"after_state": "R : Type u_6\ninst✝² : CommRing R\na : ℍ[R]\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ a.im * a.im = -↑(normSq a.im)"
},
{
"line": "simp only [← star_mul_self]",
"before_state": "R : Type u_6\ninst✝² : CommRing R\na : ℍ[R]\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ a.im * a.im = -↑(normSq a.im)",
"after_state": "R : Type u_6\ninst✝² : CommRing R\na : ℍ[R]\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ a.im * a.im = -(star a.im * a.im)"
},
{
"line": "simp only [im_star]",
"before_state": "R : Type u_6\ninst✝² : CommRing R\na : ℍ[R]\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ a.im * a.im = -(star a.im * a.im)",
"after_state": "R : Type u_6\ninst✝² : CommRing R\na : ℍ[R]\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ a.im * a.im = -(-a.im * a.im)"
},
{
"line": "simp only [neg_mul]",
"before_state": "R : Type u_6\ninst✝² : CommRing R\na : ℍ[R]\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ a.im * a.im = -(-a.im * a.im)",
"after_state": "R : Type u_6\ninst✝² : CommRing R\na : ℍ[R]\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ a.im * a.im = - -(a.im * a.im)"
},
{
"line": "simp only [neg_neg]",
"before_state": "R : Type u_6\ninst✝² : CommRing R\na : ℍ[R]\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\n⊢ a.im * a.im = - -(a.im * a.im)",
"after_state": "No Goals!"
}
] |
theorem normSq_add (a b : ℍ[R]) : normSq (a + b) = normSq a + normSq b + 2 * (a * star b).re :=
calc
normSq (a + b) = normSq a + (a * star b).re + ((b * star a).re + normSq b) := by
simp_rw [normSq_def, star_add, add_mul, mul_add, add_re]
_ = normSq a + normSq b + ((a * star b).re + (b * star a).re) := by abel
_ = normSq a + normSq b + 2 * (a * star b).re := by
rw [← add_re]
rw [← star_mul_star a b]
rw [self_add_star']
rw [coe_re]
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/Quaternion.lean
|
{
"open": [
"Quaternion",
"MulOpposite",
"Quaternion",
"MulOpposite"
],
"variables": [
"{S T R : Type*} {c₁ c₂ c₃ : R} (r x y : R) (a b : ℍ[R,c₁,c₂,c₃])",
"[Zero R]",
"[One R]",
"[Add R]",
"[AddZeroClass R]",
"[Neg R]",
"[AddGroup R]",
"[Ring R]",
"[SMul S R] [SMul T R] (s : S)",
"[AddCommGroupWithOne R]",
"[CommRing R]",
"(c₁ c₂ c₃)",
"[NoZeroDivisors R] [CharZero R]",
"{S T R : Type*} [CommRing R] (r x y : R) (a b : ℍ[R])",
"[NoZeroDivisors R] [CharZero R]"
]
}
|
[
{
"line": "simp_rw [normSq_def, star_add, add_mul, mul_add, add_re]",
"before_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\na b : ℍ[R]\n⊢ normSq (a + b) = normSq a + (a * star b).re + ((b * star a).re + normSq b)",
"after_state": "No Goals!"
},
{
"line": "simp (failIfUnchanged✝ := false✝) only",
"before_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\na b : ℍ[R]\n⊢ normSq (a + b) = normSq a + (a * star b).re + ((b * star a).re + normSq b)",
"after_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\na b : ℍ[R]\n⊢ normSq (a + b) = normSq a + (a * star b).re + ((b * star a).re + normSq b)"
},
{
"line": "simp only [normSq_def]",
"before_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\na b : ℍ[R]\n⊢ normSq (a + b) = normSq a + (a * star b).re + ((b * star a).re + normSq b)",
"after_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\na b : ℍ[R]\n⊢ ((a + b) * star (a + b)).re = (a * star a).re + (a * star b).re + ((b * star a).re + (b * star b).re)"
},
{
"line": "simp only [star_add]",
"before_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\na b : ℍ[R]\n⊢ ((a + b) * star (a + b)).re = (a * star a).re + (a * star b).re + ((b * star a).re + (b * star b).re)",
"after_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\na b : ℍ[R]\n⊢ ((a + b) * (star a + star b)).re = (a * star a).re + (a * star b).re + ((b * star a).re + (b * star b).re)"
},
{
"line": "simp only [add_mul]",
"before_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\na b : ℍ[R]\n⊢ ((a + b) * (star a + star b)).re = (a * star a).re + (a * star b).re + ((b * star a).re + (b * star b).re)",
"after_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\na b : ℍ[R]\n⊢ (a * (star a + star b) + b * (star a + star b)).re =\n (a * star a).re + (a * star b).re + ((b * star a).re + (b * star b).re)"
},
{
"line": "simp only [mul_add]",
"before_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\na b : ℍ[R]\n⊢ (a * (star a + star b) + b * (star a + star b)).re =\n (a * star a).re + (a * star b).re + ((b * star a).re + (b * star b).re)",
"after_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\na b : ℍ[R]\n⊢ (a * star a + a * star b + (b * star a + b * star b)).re =\n (a * star a).re + (a * star b).re + ((b * star a).re + (b * star b).re)"
},
{
"line": "simp only [add_re]",
"before_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\na b : ℍ[R]\n⊢ (a * star a + a * star b + (b * star a + b * star b)).re =\n (a * star a).re + (a * star b).re + ((b * star a).re + (b * star b).re)",
"after_state": "No Goals!"
},
{
"line": "abel",
"before_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\na b : ℍ[R]\n⊢ normSq a + (a * star b).re + ((b * star a).re + normSq b) = normSq a + normSq b + ((a * star b).re + (b * star a).re)",
"after_state": "No Goals!"
},
{
"line": "first\n| abel1\n| try_this abel_nf",
"before_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\na b : ℍ[R]\n⊢ normSq a + (a * star b).re + ((b * star a).re + normSq b) = normSq a + normSq b + ((a * star b).re + (b * star a).re)",
"after_state": "No Goals!"
},
{
"line": "abel1",
"before_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\na b : ℍ[R]\n⊢ normSq a + (a * star b).re + ((b * star a).re + normSq b) = normSq a + normSq b + ((a * star b).re + (b * star a).re)",
"after_state": "No Goals!"
},
{
"line": "rw [← add_re]",
"before_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\na b : ℍ[R]\n⊢ normSq a + normSq b + ((a * star b).re + (b * star a).re) = normSq a + normSq b + 2 * (a * star b).re",
"after_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\na b : ℍ[R]\n⊢ normSq a + normSq b + (a * star b + b * star a).re = normSq a + normSq b + 2 * (a * star b).re"
},
{
"line": "rewrite [← add_re]",
"before_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\na b : ℍ[R]\n⊢ normSq a + normSq b + ((a * star b).re + (b * star a).re) = normSq a + normSq b + 2 * (a * star b).re",
"after_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\na b : ℍ[R]\n⊢ normSq a + normSq b + (a * star b + b * star a).re = normSq a + normSq b + 2 * (a * star b).re"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\na b : ℍ[R]\n⊢ normSq a + normSq b + (a * star b + b * star a).re = normSq a + normSq b + 2 * (a * star b).re",
"after_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\na b : ℍ[R]\n⊢ normSq a + normSq b + (a * star b + b * star a).re = normSq a + normSq b + 2 * (a * star b).re"
},
{
"line": "try (with_reducible rfl)",
"before_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\na b : ℍ[R]\n⊢ normSq a + normSq b + (a * star b + b * star a).re = normSq a + normSq b + 2 * (a * star b).re",
"after_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\na b : ℍ[R]\n⊢ normSq a + normSq b + (a * star b + b * star a).re = normSq a + normSq b + 2 * (a * star b).re"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\na b : ℍ[R]\n⊢ normSq a + normSq b + (a * star b + b * star a).re = normSq a + normSq b + 2 * (a * star b).re",
"after_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\na b : ℍ[R]\n⊢ normSq a + normSq b + (a * star b + b * star a).re = normSq a + normSq b + 2 * (a * star b).re"
},
{
"line": "with_reducible rfl",
"before_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\na b : ℍ[R]\n⊢ normSq a + normSq b + (a * star b + b * star a).re = normSq a + normSq b + 2 * (a * star b).re",
"after_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\na b : ℍ[R]\n⊢ normSq a + normSq b + (a * star b + b * star a).re = normSq a + normSq b + 2 * (a * star b).re"
},
{
"line": "rfl",
"before_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\na b : ℍ[R]\n⊢ normSq a + normSq b + (a * star b + b * star a).re = normSq a + normSq b + 2 * (a * star b).re",
"after_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\na b : ℍ[R]\n⊢ normSq a + normSq b + (a * star b + b * star a).re = normSq a + normSq b + 2 * (a * star b).re"
},
{
"line": "apply_rfl",
"before_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\na b : ℍ[R]\n⊢ normSq a + normSq b + (a * star b + b * star a).re = normSq a + normSq b + 2 * (a * star b).re",
"after_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\na b : ℍ[R]\n⊢ normSq a + normSq b + (a * star b + b * star a).re = normSq a + normSq b + 2 * (a * star b).re"
},
{
"line": "skip",
"before_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\na b : ℍ[R]\n⊢ normSq a + normSq b + (a * star b + b * star a).re = normSq a + normSq b + 2 * (a * star b).re",
"after_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\na b : ℍ[R]\n⊢ normSq a + normSq b + (a * star b + b * star a).re = normSq a + normSq b + 2 * (a * star b).re"
},
{
"line": "rw [← star_mul_star a b]",
"before_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\na b : ℍ[R]\n⊢ normSq a + normSq b + (a * star b + b * star a).re = normSq a + normSq b + 2 * (a * star b).re",
"after_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\na b : ℍ[R]\n⊢ normSq a + normSq b + (a * star b + star (a * star b)).re = normSq a + normSq b + 2 * (a * star b).re"
},
{
"line": "rewrite [← star_mul_star a b]",
"before_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\na b : ℍ[R]\n⊢ normSq a + normSq b + (a * star b + b * star a).re = normSq a + normSq b + 2 * (a * star b).re",
"after_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\na b : ℍ[R]\n⊢ normSq a + normSq b + (a * star b + star (a * star b)).re = normSq a + normSq b + 2 * (a * star b).re"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\na b : ℍ[R]\n⊢ normSq a + normSq b + (a * star b + star (a * star b)).re = normSq a + normSq b + 2 * (a * star b).re",
"after_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\na b : ℍ[R]\n⊢ normSq a + normSq b + (a * star b + star (a * star b)).re = normSq a + normSq b + 2 * (a * star b).re"
},
{
"line": "try (with_reducible rfl)",
"before_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\na b : ℍ[R]\n⊢ normSq a + normSq b + (a * star b + star (a * star b)).re = normSq a + normSq b + 2 * (a * star b).re",
"after_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\na b : ℍ[R]\n⊢ normSq a + normSq b + (a * star b + star (a * star b)).re = normSq a + normSq b + 2 * (a * star b).re"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\na b : ℍ[R]\n⊢ normSq a + normSq b + (a * star b + star (a * star b)).re = normSq a + normSq b + 2 * (a * star b).re",
"after_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\na b : ℍ[R]\n⊢ normSq a + normSq b + (a * star b + star (a * star b)).re = normSq a + normSq b + 2 * (a * star b).re"
},
{
"line": "with_reducible rfl",
"before_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\na b : ℍ[R]\n⊢ normSq a + normSq b + (a * star b + star (a * star b)).re = normSq a + normSq b + 2 * (a * star b).re",
"after_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\na b : ℍ[R]\n⊢ normSq a + normSq b + (a * star b + star (a * star b)).re = normSq a + normSq b + 2 * (a * star b).re"
},
{
"line": "rfl",
"before_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\na b : ℍ[R]\n⊢ normSq a + normSq b + (a * star b + star (a * star b)).re = normSq a + normSq b + 2 * (a * star b).re",
"after_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\na b : ℍ[R]\n⊢ normSq a + normSq b + (a * star b + star (a * star b)).re = normSq a + normSq b + 2 * (a * star b).re"
},
{
"line": "apply_rfl",
"before_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\na b : ℍ[R]\n⊢ normSq a + normSq b + (a * star b + star (a * star b)).re = normSq a + normSq b + 2 * (a * star b).re",
"after_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\na b : ℍ[R]\n⊢ normSq a + normSq b + (a * star b + star (a * star b)).re = normSq a + normSq b + 2 * (a * star b).re"
},
{
"line": "skip",
"before_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\na b : ℍ[R]\n⊢ normSq a + normSq b + (a * star b + star (a * star b)).re = normSq a + normSq b + 2 * (a * star b).re",
"after_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\na b : ℍ[R]\n⊢ normSq a + normSq b + (a * star b + star (a * star b)).re = normSq a + normSq b + 2 * (a * star b).re"
},
{
"line": "rw [self_add_star']",
"before_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\na b : ℍ[R]\n⊢ normSq a + normSq b + (a * star b + star (a * star b)).re = normSq a + normSq b + 2 * (a * star b).re",
"after_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\na b : ℍ[R]\n⊢ normSq a + normSq b + (↑(2 * (a * star b).re)).re = normSq a + normSq b + 2 * (a * star b).re"
},
{
"line": "rewrite [self_add_star']",
"before_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\na b : ℍ[R]\n⊢ normSq a + normSq b + (a * star b + star (a * star b)).re = normSq a + normSq b + 2 * (a * star b).re",
"after_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\na b : ℍ[R]\n⊢ normSq a + normSq b + (↑(2 * (a * star b).re)).re = normSq a + normSq b + 2 * (a * star b).re"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\na b : ℍ[R]\n⊢ normSq a + normSq b + (↑(2 * (a * star b).re)).re = normSq a + normSq b + 2 * (a * star b).re",
"after_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\na b : ℍ[R]\n⊢ normSq a + normSq b + (↑(2 * (a * star b).re)).re = normSq a + normSq b + 2 * (a * star b).re"
},
{
"line": "try (with_reducible rfl)",
"before_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\na b : ℍ[R]\n⊢ normSq a + normSq b + (↑(2 * (a * star b).re)).re = normSq a + normSq b + 2 * (a * star b).re",
"after_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\na b : ℍ[R]\n⊢ normSq a + normSq b + (↑(2 * (a * star b).re)).re = normSq a + normSq b + 2 * (a * star b).re"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\na b : ℍ[R]\n⊢ normSq a + normSq b + (↑(2 * (a * star b).re)).re = normSq a + normSq b + 2 * (a * star b).re",
"after_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\na b : ℍ[R]\n⊢ normSq a + normSq b + (↑(2 * (a * star b).re)).re = normSq a + normSq b + 2 * (a * star b).re"
},
{
"line": "with_reducible rfl",
"before_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\na b : ℍ[R]\n⊢ normSq a + normSq b + (↑(2 * (a * star b).re)).re = normSq a + normSq b + 2 * (a * star b).re",
"after_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\na b : ℍ[R]\n⊢ normSq a + normSq b + (↑(2 * (a * star b).re)).re = normSq a + normSq b + 2 * (a * star b).re"
},
{
"line": "rfl",
"before_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\na b : ℍ[R]\n⊢ normSq a + normSq b + (↑(2 * (a * star b).re)).re = normSq a + normSq b + 2 * (a * star b).re",
"after_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\na b : ℍ[R]\n⊢ normSq a + normSq b + (↑(2 * (a * star b).re)).re = normSq a + normSq b + 2 * (a * star b).re"
},
{
"line": "apply_rfl",
"before_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\na b : ℍ[R]\n⊢ normSq a + normSq b + (↑(2 * (a * star b).re)).re = normSq a + normSq b + 2 * (a * star b).re",
"after_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\na b : ℍ[R]\n⊢ normSq a + normSq b + (↑(2 * (a * star b).re)).re = normSq a + normSq b + 2 * (a * star b).re"
},
{
"line": "skip",
"before_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\na b : ℍ[R]\n⊢ normSq a + normSq b + (↑(2 * (a * star b).re)).re = normSq a + normSq b + 2 * (a * star b).re",
"after_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\na b : ℍ[R]\n⊢ normSq a + normSq b + (↑(2 * (a * star b).re)).re = normSq a + normSq b + 2 * (a * star b).re"
},
{
"line": "rw [coe_re]",
"before_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\na b : ℍ[R]\n⊢ normSq a + normSq b + (↑(2 * (a * star b).re)).re = normSq a + normSq b + 2 * (a * star b).re",
"after_state": "No Goals!"
},
{
"line": "rewrite [coe_re]",
"before_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\na b : ℍ[R]\n⊢ normSq a + normSq b + (↑(2 * (a * star b).re)).re = normSq a + normSq b + 2 * (a * star b).re",
"after_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\na b : ℍ[R]\n⊢ normSq a + normSq b + 2 * (a * star b).re = normSq a + normSq b + 2 * (a * star b).re"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\na b : ℍ[R]\n⊢ normSq a + normSq b + 2 * (a * star b).re = normSq a + normSq b + 2 * (a * star b).re",
"after_state": "No Goals!"
},
{
"line": "try (with_reducible rfl)",
"before_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\na b : ℍ[R]\n⊢ normSq a + normSq b + 2 * (a * star b).re = normSq a + normSq b + 2 * (a * star b).re",
"after_state": "No Goals!"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\na b : ℍ[R]\n⊢ normSq a + normSq b + 2 * (a * star b).re = normSq a + normSq b + 2 * (a * star b).re",
"after_state": "No Goals!"
},
{
"line": "with_reducible rfl",
"before_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\na b : ℍ[R]\n⊢ normSq a + normSq b + 2 * (a * star b).re = normSq a + normSq b + 2 * (a * star b).re",
"after_state": "No Goals!"
},
{
"line": "rfl",
"before_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\na b : ℍ[R]\n⊢ normSq a + normSq b + 2 * (a * star b).re = normSq a + normSq b + 2 * (a * star b).re",
"after_state": "No Goals!"
},
{
"line": "eq_refl",
"before_state": "R : Type u_6\ninst✝² : CommRing R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\na b : ℍ[R]\n⊢ normSq a + normSq b + 2 * (a * star b).re = normSq a + normSq b + 2 * (a * star b).re",
"after_state": "No Goals!"
}
] |
theorem normSq_nonneg : 0 ≤ normSq a := by
rw [normSq_def']
apply_rules [sq_nonneg, add_nonneg]
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/Quaternion.lean
|
{
"open": [
"Quaternion",
"MulOpposite",
"Quaternion",
"MulOpposite"
],
"variables": [
"{S T R : Type*} {c₁ c₂ c₃ : R} (r x y : R) (a b : ℍ[R,c₁,c₂,c₃])",
"[Zero R]",
"[One R]",
"[Add R]",
"[AddZeroClass R]",
"[Neg R]",
"[AddGroup R]",
"[Ring R]",
"[SMul S R] [SMul T R] (s : S)",
"[AddCommGroupWithOne R]",
"[CommRing R]",
"(c₁ c₂ c₃)",
"[NoZeroDivisors R] [CharZero R]",
"{S T R : Type*} [CommRing R] (r x y : R) (a b : ℍ[R])",
"[NoZeroDivisors R] [CharZero R]",
"{R : Type*}",
"[CommRing R] [LinearOrder R] [IsStrictOrderedRing R] {a : ℍ[R]}"
]
}
|
[
{
"line": "rw [normSq_def']",
"before_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ 0 ≤ normSq a",
"after_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ 0 ≤ a.re ^ 2 + a.imI ^ 2 + a.imJ ^ 2 + a.imK ^ 2"
},
{
"line": "rewrite [normSq_def']",
"before_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ 0 ≤ normSq a",
"after_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ 0 ≤ a.re ^ 2 + a.imI ^ 2 + a.imJ ^ 2 + a.imK ^ 2"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ 0 ≤ a.re ^ 2 + a.imI ^ 2 + a.imJ ^ 2 + a.imK ^ 2",
"after_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ 0 ≤ a.re ^ 2 + a.imI ^ 2 + a.imJ ^ 2 + a.imK ^ 2"
},
{
"line": "try (with_reducible rfl)",
"before_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ 0 ≤ a.re ^ 2 + a.imI ^ 2 + a.imJ ^ 2 + a.imK ^ 2",
"after_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ 0 ≤ a.re ^ 2 + a.imI ^ 2 + a.imJ ^ 2 + a.imK ^ 2"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ 0 ≤ a.re ^ 2 + a.imI ^ 2 + a.imJ ^ 2 + a.imK ^ 2",
"after_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ 0 ≤ a.re ^ 2 + a.imI ^ 2 + a.imJ ^ 2 + a.imK ^ 2"
},
{
"line": "with_reducible rfl",
"before_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ 0 ≤ a.re ^ 2 + a.imI ^ 2 + a.imJ ^ 2 + a.imK ^ 2",
"after_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ 0 ≤ a.re ^ 2 + a.imI ^ 2 + a.imJ ^ 2 + a.imK ^ 2"
},
{
"line": "rfl",
"before_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ 0 ≤ a.re ^ 2 + a.imI ^ 2 + a.imJ ^ 2 + a.imK ^ 2",
"after_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ 0 ≤ a.re ^ 2 + a.imI ^ 2 + a.imJ ^ 2 + a.imK ^ 2"
},
{
"line": "apply_rfl",
"before_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ 0 ≤ a.re ^ 2 + a.imI ^ 2 + a.imJ ^ 2 + a.imK ^ 2",
"after_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ 0 ≤ a.re ^ 2 + a.imI ^ 2 + a.imJ ^ 2 + a.imK ^ 2"
},
{
"line": "skip",
"before_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ 0 ≤ a.re ^ 2 + a.imI ^ 2 + a.imJ ^ 2 + a.imK ^ 2",
"after_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ 0 ≤ a.re ^ 2 + a.imI ^ 2 + a.imJ ^ 2 + a.imK ^ 2"
},
{
"line": "apply_rules [sq_nonneg, add_nonneg]",
"before_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ 0 ≤ a.re ^ 2 + a.imI ^ 2 + a.imJ ^ 2 + a.imK ^ 2",
"after_state": "No Goals!"
}
] |
theorem sq_eq_normSq : a ^ 2 = normSq a ↔ a = a.re := by
rw [← star_eq_self]
rw [← star_mul_self]
rw [sq]
rw [mul_eq_mul_right_iff]
rw [eq_comm]
exact or_iff_left_of_imp fun ha ↦ ha.symm ▸ star_zero _
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/Quaternion.lean
|
{
"open": [
"Quaternion",
"MulOpposite",
"Quaternion",
"MulOpposite"
],
"variables": [
"{S T R : Type*} {c₁ c₂ c₃ : R} (r x y : R) (a b : ℍ[R,c₁,c₂,c₃])",
"[Zero R]",
"[One R]",
"[Add R]",
"[AddZeroClass R]",
"[Neg R]",
"[AddGroup R]",
"[Ring R]",
"[SMul S R] [SMul T R] (s : S)",
"[AddCommGroupWithOne R]",
"[CommRing R]",
"(c₁ c₂ c₃)",
"[NoZeroDivisors R] [CharZero R]",
"{S T R : Type*} [CommRing R] (r x y : R) (a b : ℍ[R])",
"[NoZeroDivisors R] [CharZero R]",
"{R : Type*}",
"[CommRing R] [LinearOrder R] [IsStrictOrderedRing R] {a : ℍ[R]}"
]
}
|
[
{
"line": "rw [← star_eq_self]",
"before_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ a ^ 2 = ↑(normSq a) ↔ a = ↑a.re",
"after_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ a ^ 2 = ↑(normSq a) ↔ star a = a"
},
{
"line": "rewrite [← star_eq_self]",
"before_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ a ^ 2 = ↑(normSq a) ↔ a = ↑a.re",
"after_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ a ^ 2 = ↑(normSq a) ↔ star a = a"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ a ^ 2 = ↑(normSq a) ↔ star a = a",
"after_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ a ^ 2 = ↑(normSq a) ↔ star a = a"
},
{
"line": "try (with_reducible rfl)",
"before_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ a ^ 2 = ↑(normSq a) ↔ star a = a",
"after_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ a ^ 2 = ↑(normSq a) ↔ star a = a"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ a ^ 2 = ↑(normSq a) ↔ star a = a",
"after_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ a ^ 2 = ↑(normSq a) ↔ star a = a"
},
{
"line": "with_reducible rfl",
"before_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ a ^ 2 = ↑(normSq a) ↔ star a = a",
"after_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ a ^ 2 = ↑(normSq a) ↔ star a = a"
},
{
"line": "rfl",
"before_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ a ^ 2 = ↑(normSq a) ↔ star a = a",
"after_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ a ^ 2 = ↑(normSq a) ↔ star a = a"
},
{
"line": "apply_rfl",
"before_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ a ^ 2 = ↑(normSq a) ↔ star a = a",
"after_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ a ^ 2 = ↑(normSq a) ↔ star a = a"
},
{
"line": "skip",
"before_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ a ^ 2 = ↑(normSq a) ↔ star a = a",
"after_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ a ^ 2 = ↑(normSq a) ↔ star a = a"
},
{
"line": "rw [← star_mul_self]",
"before_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ a ^ 2 = ↑(normSq a) ↔ star a = a",
"after_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ a ^ 2 = star a * a ↔ star a = a"
},
{
"line": "rewrite [← star_mul_self]",
"before_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ a ^ 2 = ↑(normSq a) ↔ star a = a",
"after_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ a ^ 2 = star a * a ↔ star a = a"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ a ^ 2 = star a * a ↔ star a = a",
"after_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ a ^ 2 = star a * a ↔ star a = a"
},
{
"line": "try (with_reducible rfl)",
"before_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ a ^ 2 = star a * a ↔ star a = a",
"after_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ a ^ 2 = star a * a ↔ star a = a"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ a ^ 2 = star a * a ↔ star a = a",
"after_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ a ^ 2 = star a * a ↔ star a = a"
},
{
"line": "with_reducible rfl",
"before_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ a ^ 2 = star a * a ↔ star a = a",
"after_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ a ^ 2 = star a * a ↔ star a = a"
},
{
"line": "rfl",
"before_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ a ^ 2 = star a * a ↔ star a = a",
"after_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ a ^ 2 = star a * a ↔ star a = a"
},
{
"line": "apply_rfl",
"before_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ a ^ 2 = star a * a ↔ star a = a",
"after_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ a ^ 2 = star a * a ↔ star a = a"
},
{
"line": "skip",
"before_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ a ^ 2 = star a * a ↔ star a = a",
"after_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ a ^ 2 = star a * a ↔ star a = a"
},
{
"line": "rw [sq]",
"before_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ a ^ 2 = star a * a ↔ star a = a",
"after_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ a * a = star a * a ↔ star a = a"
},
{
"line": "rewrite [sq]",
"before_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ a ^ 2 = star a * a ↔ star a = a",
"after_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ a * a = star a * a ↔ star a = a"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ a * a = star a * a ↔ star a = a",
"after_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ a * a = star a * a ↔ star a = a"
},
{
"line": "try (with_reducible rfl)",
"before_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ a * a = star a * a ↔ star a = a",
"after_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ a * a = star a * a ↔ star a = a"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ a * a = star a * a ↔ star a = a",
"after_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ a * a = star a * a ↔ star a = a"
},
{
"line": "with_reducible rfl",
"before_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ a * a = star a * a ↔ star a = a",
"after_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ a * a = star a * a ↔ star a = a"
},
{
"line": "rfl",
"before_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ a * a = star a * a ↔ star a = a",
"after_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ a * a = star a * a ↔ star a = a"
},
{
"line": "apply_rfl",
"before_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ a * a = star a * a ↔ star a = a",
"after_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ a * a = star a * a ↔ star a = a"
},
{
"line": "skip",
"before_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ a * a = star a * a ↔ star a = a",
"after_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ a * a = star a * a ↔ star a = a"
},
{
"line": "rw [mul_eq_mul_right_iff]",
"before_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ a * a = star a * a ↔ star a = a",
"after_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ a = star a ∨ a = 0 ↔ star a = a"
},
{
"line": "rewrite [mul_eq_mul_right_iff]",
"before_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ a * a = star a * a ↔ star a = a",
"after_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ a = star a ∨ a = 0 ↔ star a = a"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ a = star a ∨ a = 0 ↔ star a = a",
"after_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ a = star a ∨ a = 0 ↔ star a = a"
},
{
"line": "try (with_reducible rfl)",
"before_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ a = star a ∨ a = 0 ↔ star a = a",
"after_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ a = star a ∨ a = 0 ↔ star a = a"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ a = star a ∨ a = 0 ↔ star a = a",
"after_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ a = star a ∨ a = 0 ↔ star a = a"
},
{
"line": "with_reducible rfl",
"before_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ a = star a ∨ a = 0 ↔ star a = a",
"after_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ a = star a ∨ a = 0 ↔ star a = a"
},
{
"line": "rfl",
"before_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ a = star a ∨ a = 0 ↔ star a = a",
"after_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ a = star a ∨ a = 0 ↔ star a = a"
},
{
"line": "apply_rfl",
"before_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ a = star a ∨ a = 0 ↔ star a = a",
"after_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ a = star a ∨ a = 0 ↔ star a = a"
},
{
"line": "skip",
"before_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ a = star a ∨ a = 0 ↔ star a = a",
"after_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ a = star a ∨ a = 0 ↔ star a = a"
},
{
"line": "rw [eq_comm]",
"before_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ a = star a ∨ a = 0 ↔ star a = a",
"after_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ star a = a ∨ a = 0 ↔ star a = a"
},
{
"line": "rewrite [eq_comm]",
"before_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ a = star a ∨ a = 0 ↔ star a = a",
"after_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ star a = a ∨ a = 0 ↔ star a = a"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ star a = a ∨ a = 0 ↔ star a = a",
"after_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ star a = a ∨ a = 0 ↔ star a = a"
},
{
"line": "try (with_reducible rfl)",
"before_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ star a = a ∨ a = 0 ↔ star a = a",
"after_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ star a = a ∨ a = 0 ↔ star a = a"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ star a = a ∨ a = 0 ↔ star a = a",
"after_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ star a = a ∨ a = 0 ↔ star a = a"
},
{
"line": "with_reducible rfl",
"before_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ star a = a ∨ a = 0 ↔ star a = a",
"after_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ star a = a ∨ a = 0 ↔ star a = a"
},
{
"line": "rfl",
"before_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ star a = a ∨ a = 0 ↔ star a = a",
"after_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ star a = a ∨ a = 0 ↔ star a = a"
},
{
"line": "apply_rfl",
"before_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ star a = a ∨ a = 0 ↔ star a = a",
"after_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ star a = a ∨ a = 0 ↔ star a = a"
},
{
"line": "skip",
"before_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ star a = a ∨ a = 0 ↔ star a = a",
"after_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ star a = a ∨ a = 0 ↔ star a = a"
},
{
"line": "exact or_iff_left_of_imp fun ha ↦ ha.symm ▸ star_zero _",
"before_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ star a = a ∨ a = 0 ↔ star a = a",
"after_state": "No Goals!"
}
] |
theorem sq_eq_neg_normSq : a ^ 2 = -normSq a ↔ a.re = 0 := by
simp_rw [← star_eq_neg]
obtain rfl | hq0 := eq_or_ne a 0
· simp
· rw [← star_mul_self, ← mul_neg, ← neg_sq, sq, mul_left_inj' (neg_ne_zero.mpr hq0), eq_comm]
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/Quaternion.lean
|
{
"open": [
"Quaternion",
"MulOpposite",
"Quaternion",
"MulOpposite"
],
"variables": [
"{S T R : Type*} {c₁ c₂ c₃ : R} (r x y : R) (a b : ℍ[R,c₁,c₂,c₃])",
"[Zero R]",
"[One R]",
"[Add R]",
"[AddZeroClass R]",
"[Neg R]",
"[AddGroup R]",
"[Ring R]",
"[SMul S R] [SMul T R] (s : S)",
"[AddCommGroupWithOne R]",
"[CommRing R]",
"(c₁ c₂ c₃)",
"[NoZeroDivisors R] [CharZero R]",
"{S T R : Type*} [CommRing R] (r x y : R) (a b : ℍ[R])",
"[NoZeroDivisors R] [CharZero R]",
"{R : Type*}",
"[CommRing R] [LinearOrder R] [IsStrictOrderedRing R] {a : ℍ[R]}"
]
}
|
[
{
"line": "simp_rw [← star_eq_neg]",
"before_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ a ^ 2 = -↑(normSq a) ↔ a.re = 0",
"after_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ a ^ 2 = -↑(normSq a) ↔ star a = -a"
},
{
"line": "simp (failIfUnchanged✝ := false✝) only",
"before_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ a ^ 2 = -↑(normSq a) ↔ a.re = 0",
"after_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ a ^ 2 = -↑(normSq a) ↔ a.re = 0"
},
{
"line": "simp only [← star_eq_neg]",
"before_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ a ^ 2 = -↑(normSq a) ↔ a.re = 0",
"after_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ a ^ 2 = -↑(normSq a) ↔ star a = -a"
},
{
"line": "obtain rfl | hq0 := eq_or_ne a 0",
"before_state": "R : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ a ^ 2 = -↑(normSq a) ↔ star a = -a",
"after_state": "case inl\nR : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\n⊢ 0 ^ 2 = -↑(normSq 0) ↔ star 0 = -0\n---\ncase inr\nR : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\nhq0 : a ≠ 0\n⊢ a ^ 2 = -↑(normSq a) ↔ star a = -a"
},
{
"line": "simp",
"before_state": "case inl\nR : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\n⊢ 0 ^ 2 = -↑(normSq 0) ↔ star 0 = -0",
"after_state": "No Goals!"
},
{
"line": "rw [← star_mul_self, ← mul_neg, ← neg_sq, sq, mul_left_inj' (neg_ne_zero.mpr hq0), eq_comm]",
"before_state": "case inr\nR : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\nhq0 : a ≠ 0\n⊢ a ^ 2 = -↑(normSq a) ↔ star a = -a",
"after_state": "No Goals!"
},
{
"line": "rewrite [← star_mul_self, ← mul_neg, ← neg_sq, sq, mul_left_inj' (neg_ne_zero.mpr hq0), eq_comm]",
"before_state": "case inr\nR : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\nhq0 : a ≠ 0\n⊢ a ^ 2 = -↑(normSq a) ↔ star a = -a",
"after_state": "case inr\nR : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\nhq0 : a ≠ 0\n⊢ star a = -a ↔ star a = -a"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case inr\nR : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\nhq0 : a ≠ 0\n⊢ star a = -a ↔ star a = -a",
"after_state": "No Goals!"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case inr\nR : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\nhq0 : a ≠ 0\n⊢ star a = -a ↔ star a = -a",
"after_state": "No Goals!"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case inr\nR : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\nhq0 : a ≠ 0\n⊢ star a = -a ↔ star a = -a",
"after_state": "No Goals!"
},
{
"line": "with_reducible rfl",
"before_state": "case inr\nR : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\nhq0 : a ≠ 0\n⊢ star a = -a ↔ star a = -a",
"after_state": "No Goals!"
},
{
"line": "rfl",
"before_state": "case inr\nR : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\nhq0 : a ≠ 0\n⊢ star a = -a ↔ star a = -a",
"after_state": "No Goals!"
},
{
"line": "exact Iff.rfl✝",
"before_state": "case inr\nR : Type u_7\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\nhq0 : a ≠ 0\n⊢ star a = -a ↔ star a = -a",
"after_state": "No Goals!"
}
] |
theorem lift_apply_inl (f : R →ₐ[S] A) (g : M →ₗ[S] A)
(hg : ∀ x y, g x * g y = 0)
(hfg : ∀ r x, g (r •> x) = f r * g x)
(hgf : ∀ r x, g (x <• r) = g x * f r)
(r : R) :
lift f g hg hfg hgf (inl r) = f r :=
show f r + g 0 = f r by rw [map_zero, add_zero]
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/TrivSqZeroExt.lean
|
{
"open": [
"scoped RightActions",
"MulOpposite"
],
"variables": [
"{R : Type u} {M : Type v}",
"(M)",
"(R)",
"{T : Type*} {S : Type*} {R : Type u} {M : Type v}",
"(M)",
"(R)",
"(R M)",
"{R : Type u} {M : Type v}",
"(M)",
"(R)",
"(R M)",
"{R : Type u} {M : Type v}",
"[Neg M] [Inv R] [SMul Rᵐᵒᵖ M] [SMul R M]",
"{R : Type u} {M : Type v}",
"[AddCommGroup M] [Semiring R] [Module Rᵐᵒᵖ M] [Module R M]",
"[SMulCommClass R Rᵐᵒᵖ M]",
"{R : Type u} {M : Type v}",
"[DivisionSemiring R] [AddCommGroup M] [Module Rᵐᵒᵖ M] [Module R M]",
"[SMulCommClass R Rᵐᵒᵖ M]",
"{R : Type u} {M : Type v}",
"[DivisionRing R] [AddCommGroup M] [Module Rᵐᵒᵖ M] [Module R M]",
"(S : Type*) (R R' : Type u) (M : Type v)",
"[CommSemiring S] [Semiring R] [CommSemiring R'] [AddCommMonoid M]",
"[Algebra S R] [Module S M] [Module R M] [Module Rᵐᵒᵖ M] [SMulCommClass R Rᵐᵒᵖ M]",
"[IsScalarTower S R M] [IsScalarTower S Rᵐᵒᵖ M]",
"[Module R' M] [Module R'ᵐᵒᵖ M] [IsCentralScalar R' M]",
"{R R' S M}",
"{A : Type*} [Semiring A] [Algebra S A] [Algebra R' A]"
]
}
|
[
{
"line": "rw [map_zero, add_zero]",
"before_state": "S : Type u_3\nR : Type u\nM : Type v\ninst✝¹¹ : CommSemiring S\ninst✝¹⁰ : Semiring R\ninst✝⁹ : AddCommMonoid M\ninst✝⁸ : Algebra S R\ninst✝⁷ : Module S M\ninst✝⁶ : Module R M\ninst✝⁵ : Module Rᵐᵒᵖ M\ninst✝⁴ : SMulCommClass R Rᵐᵒᵖ M\ninst✝³ : IsScalarTower S R M\ninst✝² : IsScalarTower S Rᵐᵒᵖ M\nA : Type u_4\ninst✝¹ : Semiring A\ninst✝ : Algebra S A\nx✝ : Sort u_5\nlift : x✝\nf : R →ₐ[S] A\ng : M →ₗ[S] A\nhg : ∀ (x y : M), g x * g y = 0\nhfg : ∀ (r : R) (x : M), g (r •> x) = f r * g x\nhgf : ∀ (r : R) (x : M), g (x <• r) = g x * f r\nr : R\n⊢ f r + g 0 = f r",
"after_state": "No Goals!"
},
{
"line": "rewrite [map_zero, add_zero]",
"before_state": "S : Type u_3\nR : Type u\nM : Type v\ninst✝¹¹ : CommSemiring S\ninst✝¹⁰ : Semiring R\ninst✝⁹ : AddCommMonoid M\ninst✝⁸ : Algebra S R\ninst✝⁷ : Module S M\ninst✝⁶ : Module R M\ninst✝⁵ : Module Rᵐᵒᵖ M\ninst✝⁴ : SMulCommClass R Rᵐᵒᵖ M\ninst✝³ : IsScalarTower S R M\ninst✝² : IsScalarTower S Rᵐᵒᵖ M\nA : Type u_4\ninst✝¹ : Semiring A\ninst✝ : Algebra S A\nx✝ : Sort u_5\nlift : x✝\nf : R →ₐ[S] A\ng : M →ₗ[S] A\nhg : ∀ (x y : M), g x * g y = 0\nhfg : ∀ (r : R) (x : M), g (r •> x) = f r * g x\nhgf : ∀ (r : R) (x : M), g (x <• r) = g x * f r\nr : R\n⊢ f r + g 0 = f r",
"after_state": "S : Type u_3\nR : Type u\nM : Type v\ninst✝¹¹ : CommSemiring S\ninst✝¹⁰ : Semiring R\ninst✝⁹ : AddCommMonoid M\ninst✝⁸ : Algebra S R\ninst✝⁷ : Module S M\ninst✝⁶ : Module R M\ninst✝⁵ : Module Rᵐᵒᵖ M\ninst✝⁴ : SMulCommClass R Rᵐᵒᵖ M\ninst✝³ : IsScalarTower S R M\ninst✝² : IsScalarTower S Rᵐᵒᵖ M\nA : Type u_4\ninst✝¹ : Semiring A\ninst✝ : Algebra S A\nx✝ : Sort u_5\nlift : x✝\nf : R →ₐ[S] A\ng : M →ₗ[S] A\nhg : ∀ (x y : M), g x * g y = 0\nhfg : ∀ (r : R) (x : M), g (r •> x) = f r * g x\nhgf : ∀ (r : R) (x : M), g (x <• r) = g x * f r\nr : R\n⊢ f r = f r"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "S : Type u_3\nR : Type u\nM : Type v\ninst✝¹¹ : CommSemiring S\ninst✝¹⁰ : Semiring R\ninst✝⁹ : AddCommMonoid M\ninst✝⁸ : Algebra S R\ninst✝⁷ : Module S M\ninst✝⁶ : Module R M\ninst✝⁵ : Module Rᵐᵒᵖ M\ninst✝⁴ : SMulCommClass R Rᵐᵒᵖ M\ninst✝³ : IsScalarTower S R M\ninst✝² : IsScalarTower S Rᵐᵒᵖ M\nA : Type u_4\ninst✝¹ : Semiring A\ninst✝ : Algebra S A\nx✝ : Sort u_5\nlift : x✝\nf : R →ₐ[S] A\ng : M →ₗ[S] A\nhg : ∀ (x y : M), g x * g y = 0\nhfg : ∀ (r : R) (x : M), g (r •> x) = f r * g x\nhgf : ∀ (r : R) (x : M), g (x <• r) = g x * f r\nr : R\n⊢ f r = f r",
"after_state": "No Goals!"
},
{
"line": "try (with_reducible rfl)",
"before_state": "S : Type u_3\nR : Type u\nM : Type v\ninst✝¹¹ : CommSemiring S\ninst✝¹⁰ : Semiring R\ninst✝⁹ : AddCommMonoid M\ninst✝⁸ : Algebra S R\ninst✝⁷ : Module S M\ninst✝⁶ : Module R M\ninst✝⁵ : Module Rᵐᵒᵖ M\ninst✝⁴ : SMulCommClass R Rᵐᵒᵖ M\ninst✝³ : IsScalarTower S R M\ninst✝² : IsScalarTower S Rᵐᵒᵖ M\nA : Type u_4\ninst✝¹ : Semiring A\ninst✝ : Algebra S A\nx✝ : Sort u_5\nlift : x✝\nf : R →ₐ[S] A\ng : M →ₗ[S] A\nhg : ∀ (x y : M), g x * g y = 0\nhfg : ∀ (r : R) (x : M), g (r •> x) = f r * g x\nhgf : ∀ (r : R) (x : M), g (x <• r) = g x * f r\nr : R\n⊢ f r = f r",
"after_state": "No Goals!"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "S : Type u_3\nR : Type u\nM : Type v\ninst✝¹¹ : CommSemiring S\ninst✝¹⁰ : Semiring R\ninst✝⁹ : AddCommMonoid M\ninst✝⁸ : Algebra S R\ninst✝⁷ : Module S M\ninst✝⁶ : Module R M\ninst✝⁵ : Module Rᵐᵒᵖ M\ninst✝⁴ : SMulCommClass R Rᵐᵒᵖ M\ninst✝³ : IsScalarTower S R M\ninst✝² : IsScalarTower S Rᵐᵒᵖ M\nA : Type u_4\ninst✝¹ : Semiring A\ninst✝ : Algebra S A\nx✝ : Sort u_5\nlift : x✝\nf : R →ₐ[S] A\ng : M →ₗ[S] A\nhg : ∀ (x y : M), g x * g y = 0\nhfg : ∀ (r : R) (x : M), g (r •> x) = f r * g x\nhgf : ∀ (r : R) (x : M), g (x <• r) = g x * f r\nr : R\n⊢ f r = f r",
"after_state": "No Goals!"
},
{
"line": "with_reducible rfl",
"before_state": "S : Type u_3\nR : Type u\nM : Type v\ninst✝¹¹ : CommSemiring S\ninst✝¹⁰ : Semiring R\ninst✝⁹ : AddCommMonoid M\ninst✝⁸ : Algebra S R\ninst✝⁷ : Module S M\ninst✝⁶ : Module R M\ninst✝⁵ : Module Rᵐᵒᵖ M\ninst✝⁴ : SMulCommClass R Rᵐᵒᵖ M\ninst✝³ : IsScalarTower S R M\ninst✝² : IsScalarTower S Rᵐᵒᵖ M\nA : Type u_4\ninst✝¹ : Semiring A\ninst✝ : Algebra S A\nx✝ : Sort u_5\nlift : x✝\nf : R →ₐ[S] A\ng : M →ₗ[S] A\nhg : ∀ (x y : M), g x * g y = 0\nhfg : ∀ (r : R) (x : M), g (r •> x) = f r * g x\nhgf : ∀ (r : R) (x : M), g (x <• r) = g x * f r\nr : R\n⊢ f r = f r",
"after_state": "No Goals!"
},
{
"line": "rfl",
"before_state": "S : Type u_3\nR : Type u\nM : Type v\ninst✝¹¹ : CommSemiring S\ninst✝¹⁰ : Semiring R\ninst✝⁹ : AddCommMonoid M\ninst✝⁸ : Algebra S R\ninst✝⁷ : Module S M\ninst✝⁶ : Module R M\ninst✝⁵ : Module Rᵐᵒᵖ M\ninst✝⁴ : SMulCommClass R Rᵐᵒᵖ M\ninst✝³ : IsScalarTower S R M\ninst✝² : IsScalarTower S Rᵐᵒᵖ M\nA : Type u_4\ninst✝¹ : Semiring A\ninst✝ : Algebra S A\nx✝ : Sort u_5\nlift : x✝\nf : R →ₐ[S] A\ng : M →ₗ[S] A\nhg : ∀ (x y : M), g x * g y = 0\nhfg : ∀ (r : R) (x : M), g (r •> x) = f r * g x\nhgf : ∀ (r : R) (x : M), g (x <• r) = g x * f r\nr : R\n⊢ f r = f r",
"after_state": "No Goals!"
},
{
"line": "eq_refl",
"before_state": "S : Type u_3\nR : Type u\nM : Type v\ninst✝¹¹ : CommSemiring S\ninst✝¹⁰ : Semiring R\ninst✝⁹ : AddCommMonoid M\ninst✝⁸ : Algebra S R\ninst✝⁷ : Module S M\ninst✝⁶ : Module R M\ninst✝⁵ : Module Rᵐᵒᵖ M\ninst✝⁴ : SMulCommClass R Rᵐᵒᵖ M\ninst✝³ : IsScalarTower S R M\ninst✝² : IsScalarTower S Rᵐᵒᵖ M\nA : Type u_4\ninst✝¹ : Semiring A\ninst✝ : Algebra S A\nx✝ : Sort u_5\nlift : x✝\nf : R →ₐ[S] A\ng : M →ₗ[S] A\nhg : ∀ (x y : M), g x * g y = 0\nhfg : ∀ (r : R) (x : M), g (r •> x) = f r * g x\nhgf : ∀ (r : R) (x : M), g (x <• r) = g x * f r\nr : R\n⊢ f r = f r",
"after_state": "No Goals!"
}
] |
theorem lift_apply_inr (f : R →ₐ[S] A) (g : M →ₗ[S] A)
(hg : ∀ x y, g x * g y = 0)
(hfg : ∀ r x, g (r •> x) = f r * g x)
(hgf : ∀ r x, g (x <• r) = g x * f r)
(m : M) :
lift f g hg hfg hgf (inr m) = g m :=
show f 0 + g m = g m by rw [map_zero, zero_add]
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/TrivSqZeroExt.lean
|
{
"open": [
"scoped RightActions",
"MulOpposite"
],
"variables": [
"{R : Type u} {M : Type v}",
"(M)",
"(R)",
"{T : Type*} {S : Type*} {R : Type u} {M : Type v}",
"(M)",
"(R)",
"(R M)",
"{R : Type u} {M : Type v}",
"(M)",
"(R)",
"(R M)",
"{R : Type u} {M : Type v}",
"[Neg M] [Inv R] [SMul Rᵐᵒᵖ M] [SMul R M]",
"{R : Type u} {M : Type v}",
"[AddCommGroup M] [Semiring R] [Module Rᵐᵒᵖ M] [Module R M]",
"[SMulCommClass R Rᵐᵒᵖ M]",
"{R : Type u} {M : Type v}",
"[DivisionSemiring R] [AddCommGroup M] [Module Rᵐᵒᵖ M] [Module R M]",
"[SMulCommClass R Rᵐᵒᵖ M]",
"{R : Type u} {M : Type v}",
"[DivisionRing R] [AddCommGroup M] [Module Rᵐᵒᵖ M] [Module R M]",
"(S : Type*) (R R' : Type u) (M : Type v)",
"[CommSemiring S] [Semiring R] [CommSemiring R'] [AddCommMonoid M]",
"[Algebra S R] [Module S M] [Module R M] [Module Rᵐᵒᵖ M] [SMulCommClass R Rᵐᵒᵖ M]",
"[IsScalarTower S R M] [IsScalarTower S Rᵐᵒᵖ M]",
"[Module R' M] [Module R'ᵐᵒᵖ M] [IsCentralScalar R' M]",
"{R R' S M}",
"{A : Type*} [Semiring A] [Algebra S A] [Algebra R' A]"
]
}
|
[
{
"line": "rw [map_zero, zero_add]",
"before_state": "S : Type u_3\nR : Type u\nM : Type v\ninst✝¹¹ : CommSemiring S\ninst✝¹⁰ : Semiring R\ninst✝⁹ : AddCommMonoid M\ninst✝⁸ : Algebra S R\ninst✝⁷ : Module S M\ninst✝⁶ : Module R M\ninst✝⁵ : Module Rᵐᵒᵖ M\ninst✝⁴ : SMulCommClass R Rᵐᵒᵖ M\ninst✝³ : IsScalarTower S R M\ninst✝² : IsScalarTower S Rᵐᵒᵖ M\nA : Type u_4\ninst✝¹ : Semiring A\ninst✝ : Algebra S A\nx✝ : Sort u_5\nlift : x✝\nf : R →ₐ[S] A\ng : M →ₗ[S] A\nhg : ∀ (x y : M), g x * g y = 0\nhfg : ∀ (r : R) (x : M), g (r •> x) = f r * g x\nhgf : ∀ (r : R) (x : M), g (x <• r) = g x * f r\nm : M\n⊢ f 0 + g m = g m",
"after_state": "No Goals!"
},
{
"line": "rewrite [map_zero, zero_add]",
"before_state": "S : Type u_3\nR : Type u\nM : Type v\ninst✝¹¹ : CommSemiring S\ninst✝¹⁰ : Semiring R\ninst✝⁹ : AddCommMonoid M\ninst✝⁸ : Algebra S R\ninst✝⁷ : Module S M\ninst✝⁶ : Module R M\ninst✝⁵ : Module Rᵐᵒᵖ M\ninst✝⁴ : SMulCommClass R Rᵐᵒᵖ M\ninst✝³ : IsScalarTower S R M\ninst✝² : IsScalarTower S Rᵐᵒᵖ M\nA : Type u_4\ninst✝¹ : Semiring A\ninst✝ : Algebra S A\nx✝ : Sort u_5\nlift : x✝\nf : R →ₐ[S] A\ng : M →ₗ[S] A\nhg : ∀ (x y : M), g x * g y = 0\nhfg : ∀ (r : R) (x : M), g (r •> x) = f r * g x\nhgf : ∀ (r : R) (x : M), g (x <• r) = g x * f r\nm : M\n⊢ f 0 + g m = g m",
"after_state": "S : Type u_3\nR : Type u\nM : Type v\ninst✝¹¹ : CommSemiring S\ninst✝¹⁰ : Semiring R\ninst✝⁹ : AddCommMonoid M\ninst✝⁸ : Algebra S R\ninst✝⁷ : Module S M\ninst✝⁶ : Module R M\ninst✝⁵ : Module Rᵐᵒᵖ M\ninst✝⁴ : SMulCommClass R Rᵐᵒᵖ M\ninst✝³ : IsScalarTower S R M\ninst✝² : IsScalarTower S Rᵐᵒᵖ M\nA : Type u_4\ninst✝¹ : Semiring A\ninst✝ : Algebra S A\nx✝ : Sort u_5\nlift : x✝\nf : R →ₐ[S] A\ng : M →ₗ[S] A\nhg : ∀ (x y : M), g x * g y = 0\nhfg : ∀ (r : R) (x : M), g (r •> x) = f r * g x\nhgf : ∀ (r : R) (x : M), g (x <• r) = g x * f r\nm : M\n⊢ g m = g m"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "S : Type u_3\nR : Type u\nM : Type v\ninst✝¹¹ : CommSemiring S\ninst✝¹⁰ : Semiring R\ninst✝⁹ : AddCommMonoid M\ninst✝⁸ : Algebra S R\ninst✝⁷ : Module S M\ninst✝⁶ : Module R M\ninst✝⁵ : Module Rᵐᵒᵖ M\ninst✝⁴ : SMulCommClass R Rᵐᵒᵖ M\ninst✝³ : IsScalarTower S R M\ninst✝² : IsScalarTower S Rᵐᵒᵖ M\nA : Type u_4\ninst✝¹ : Semiring A\ninst✝ : Algebra S A\nx✝ : Sort u_5\nlift : x✝\nf : R →ₐ[S] A\ng : M →ₗ[S] A\nhg : ∀ (x y : M), g x * g y = 0\nhfg : ∀ (r : R) (x : M), g (r •> x) = f r * g x\nhgf : ∀ (r : R) (x : M), g (x <• r) = g x * f r\nm : M\n⊢ g m = g m",
"after_state": "No Goals!"
},
{
"line": "try (with_reducible rfl)",
"before_state": "S : Type u_3\nR : Type u\nM : Type v\ninst✝¹¹ : CommSemiring S\ninst✝¹⁰ : Semiring R\ninst✝⁹ : AddCommMonoid M\ninst✝⁸ : Algebra S R\ninst✝⁷ : Module S M\ninst✝⁶ : Module R M\ninst✝⁵ : Module Rᵐᵒᵖ M\ninst✝⁴ : SMulCommClass R Rᵐᵒᵖ M\ninst✝³ : IsScalarTower S R M\ninst✝² : IsScalarTower S Rᵐᵒᵖ M\nA : Type u_4\ninst✝¹ : Semiring A\ninst✝ : Algebra S A\nx✝ : Sort u_5\nlift : x✝\nf : R →ₐ[S] A\ng : M →ₗ[S] A\nhg : ∀ (x y : M), g x * g y = 0\nhfg : ∀ (r : R) (x : M), g (r •> x) = f r * g x\nhgf : ∀ (r : R) (x : M), g (x <• r) = g x * f r\nm : M\n⊢ g m = g m",
"after_state": "No Goals!"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "S : Type u_3\nR : Type u\nM : Type v\ninst✝¹¹ : CommSemiring S\ninst✝¹⁰ : Semiring R\ninst✝⁹ : AddCommMonoid M\ninst✝⁸ : Algebra S R\ninst✝⁷ : Module S M\ninst✝⁶ : Module R M\ninst✝⁵ : Module Rᵐᵒᵖ M\ninst✝⁴ : SMulCommClass R Rᵐᵒᵖ M\ninst✝³ : IsScalarTower S R M\ninst✝² : IsScalarTower S Rᵐᵒᵖ M\nA : Type u_4\ninst✝¹ : Semiring A\ninst✝ : Algebra S A\nx✝ : Sort u_5\nlift : x✝\nf : R →ₐ[S] A\ng : M →ₗ[S] A\nhg : ∀ (x y : M), g x * g y = 0\nhfg : ∀ (r : R) (x : M), g (r •> x) = f r * g x\nhgf : ∀ (r : R) (x : M), g (x <• r) = g x * f r\nm : M\n⊢ g m = g m",
"after_state": "No Goals!"
},
{
"line": "with_reducible rfl",
"before_state": "S : Type u_3\nR : Type u\nM : Type v\ninst✝¹¹ : CommSemiring S\ninst✝¹⁰ : Semiring R\ninst✝⁹ : AddCommMonoid M\ninst✝⁸ : Algebra S R\ninst✝⁷ : Module S M\ninst✝⁶ : Module R M\ninst✝⁵ : Module Rᵐᵒᵖ M\ninst✝⁴ : SMulCommClass R Rᵐᵒᵖ M\ninst✝³ : IsScalarTower S R M\ninst✝² : IsScalarTower S Rᵐᵒᵖ M\nA : Type u_4\ninst✝¹ : Semiring A\ninst✝ : Algebra S A\nx✝ : Sort u_5\nlift : x✝\nf : R →ₐ[S] A\ng : M →ₗ[S] A\nhg : ∀ (x y : M), g x * g y = 0\nhfg : ∀ (r : R) (x : M), g (r •> x) = f r * g x\nhgf : ∀ (r : R) (x : M), g (x <• r) = g x * f r\nm : M\n⊢ g m = g m",
"after_state": "No Goals!"
},
{
"line": "rfl",
"before_state": "S : Type u_3\nR : Type u\nM : Type v\ninst✝¹¹ : CommSemiring S\ninst✝¹⁰ : Semiring R\ninst✝⁹ : AddCommMonoid M\ninst✝⁸ : Algebra S R\ninst✝⁷ : Module S M\ninst✝⁶ : Module R M\ninst✝⁵ : Module Rᵐᵒᵖ M\ninst✝⁴ : SMulCommClass R Rᵐᵒᵖ M\ninst✝³ : IsScalarTower S R M\ninst✝² : IsScalarTower S Rᵐᵒᵖ M\nA : Type u_4\ninst✝¹ : Semiring A\ninst✝ : Algebra S A\nx✝ : Sort u_5\nlift : x✝\nf : R →ₐ[S] A\ng : M →ₗ[S] A\nhg : ∀ (x y : M), g x * g y = 0\nhfg : ∀ (r : R) (x : M), g (r •> x) = f r * g x\nhgf : ∀ (r : R) (x : M), g (x <• r) = g x * f r\nm : M\n⊢ g m = g m",
"after_state": "No Goals!"
},
{
"line": "eq_refl",
"before_state": "S : Type u_3\nR : Type u\nM : Type v\ninst✝¹¹ : CommSemiring S\ninst✝¹⁰ : Semiring R\ninst✝⁹ : AddCommMonoid M\ninst✝⁸ : Algebra S R\ninst✝⁷ : Module S M\ninst✝⁶ : Module R M\ninst✝⁵ : Module Rᵐᵒᵖ M\ninst✝⁴ : SMulCommClass R Rᵐᵒᵖ M\ninst✝³ : IsScalarTower S R M\ninst✝² : IsScalarTower S Rᵐᵒᵖ M\nA : Type u_4\ninst✝¹ : Semiring A\ninst✝ : Algebra S A\nx✝ : Sort u_5\nlift : x✝\nf : R →ₐ[S] A\ng : M →ₗ[S] A\nhg : ∀ (x y : M), g x * g y = 0\nhfg : ∀ (r : R) (x : M), g (r •> x) = f r * g x\nhgf : ∀ (r : R) (x : M), g (x <• r) = g x * f r\nm : M\n⊢ g m = g m",
"after_state": "No Goals!"
}
] |
theorem map_add_nsmul [AddMonoid G] [AddMonoid H] [AddConstMapClass F G H a b]
(f : F) (x : G) (n : ℕ) : f (x + n • a) = f x + n • b := by
simpa using (AddConstMapClass.semiconj f).iterate_right n x
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/AddConstMap/Basic.lean
|
{
"open": [
"Function Set"
],
"variables": [
"{F G H : Type*} [FunLike F G H] {a : G} {b : H}"
]
}
|
[
{
"line": "simpa using (AddConstMapClass.semiconj f).iterate_right n x",
"before_state": "F : Type u_1\nG : Type u_2\nH : Type u_3\ninst✝³ : FunLike F G H\na : G\nb : H\ninst✝² : AddMonoid G\ninst✝¹ : AddMonoid H\ninst✝ : AddConstMapClass F G H a b\nf : F\nx : G\nn : ℕ\n⊢ f (x + n • a) = f x + n • b",
"after_state": "No Goals!"
}
] |
theorem map_const [AddZeroClass G] [Add H] [AddConstMapClass F G H a b] (f : F) :
f a = f 0 + b := by
simpa using map_add_const f 0
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/AddConstMap/Basic.lean
|
{
"open": [
"Function Set"
],
"variables": [
"{F G H : Type*} [FunLike F G H] {a : G} {b : H}"
]
}
|
[
{
"line": "simpa using map_add_const f 0",
"before_state": "F : Type u_1\nG : Type u_2\nH : Type u_3\ninst✝³ : FunLike F G H\na : G\nb : H\ninst✝² : AddZeroClass G\ninst✝¹ : Add H\ninst✝ : AddConstMapClass F G H a b\nf : F\n⊢ f a = f 0 + b",
"after_state": "No Goals!"
}
] |
theorem map_nsmul_const [AddMonoid G] [AddMonoid H] [AddConstMapClass F G H a b]
(f : F) (n : ℕ) : f (n • a) = f 0 + n • b := by
simpa using map_add_nsmul f 0 n
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/AddConstMap/Basic.lean
|
{
"open": [
"Function Set"
],
"variables": [
"{F G H : Type*} [FunLike F G H] {a : G} {b : H}"
]
}
|
[
{
"line": "simpa using map_add_nsmul f 0 n",
"before_state": "F : Type u_1\nG : Type u_2\nH : Type u_3\ninst✝³ : FunLike F G H\na : G\nb : H\ninst✝² : AddMonoid G\ninst✝¹ : AddMonoid H\ninst✝ : AddConstMapClass F G H a b\nf : F\nn : ℕ\n⊢ f (n • a) = f 0 + n • b",
"after_state": "No Goals!"
}
] |
theorem map_nat' [AddMonoidWithOne G] [AddMonoid H] [AddConstMapClass F G H 1 b]
(f : F) (n : ℕ) : f n = f 0 + n • b := by
simpa using map_add_nat' f 0 n
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/AddConstMap/Basic.lean
|
{
"open": [
"Function Set"
],
"variables": [
"{F G H : Type*} [FunLike F G H] {a : G} {b : H}"
]
}
|
[
{
"line": "simpa using map_add_nat' f 0 n",
"before_state": "F : Type u_1\nG : Type u_2\nH : Type u_3\ninst✝³ : FunLike F G H\nb : H\ninst✝² : AddMonoidWithOne G\ninst✝¹ : AddMonoid H\ninst✝ : AddConstMapClass F G H 1 b\nf : F\nn : ℕ\n⊢ f ↑n = f 0 + n • b",
"after_state": "No Goals!"
}
] |
theorem map_nat_add' [AddCommMonoidWithOne G] [AddMonoid H] [AddConstMapClass F G H 1 b]
(f : F) (n : ℕ) (x : G) : f (↑n + x) = f x + n • b := by
simpa using map_nsmul_add f n x
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/AddConstMap/Basic.lean
|
{
"open": [
"Function Set"
],
"variables": [
"{F G H : Type*} [FunLike F G H] {a : G} {b : H}"
]
}
|
[
{
"line": "simpa using map_nsmul_add f n x",
"before_state": "F : Type u_1\nG : Type u_2\nH : Type u_3\ninst✝³ : FunLike F G H\nb : H\ninst✝² : AddCommMonoidWithOne G\ninst✝¹ : AddMonoid H\ninst✝ : AddConstMapClass F G H 1 b\nf : F\nn : ℕ\nx : G\n⊢ f (↑n + x) = f x + n • b",
"after_state": "No Goals!"
}
] |
theorem map_sub_const [AddGroup G] [AddGroup H] [AddConstMapClass F G H a b]
(f : F) (x : G) : f (x - a) = f x - b := by
simpa using map_sub_nsmul f x 1
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/AddConstMap/Basic.lean
|
{
"open": [
"Function Set"
],
"variables": [
"{F G H : Type*} [FunLike F G H] {a : G} {b : H}"
]
}
|
[
{
"line": "simpa using map_sub_nsmul f x 1",
"before_state": "F : Type u_1\nG : Type u_2\nH : Type u_3\ninst✝³ : FunLike F G H\na : G\nb : H\ninst✝² : AddGroup G\ninst✝¹ : AddGroup H\ninst✝ : AddConstMapClass F G H a b\nf : F\nx : G\n⊢ f (x - a) = f x - b",
"after_state": "No Goals!"
}
] |
theorem map_sub_nat' [AddGroupWithOne G] [AddGroup H] [AddConstMapClass F G H 1 b]
(f : F) (x : G) (n : ℕ) : f (x - n) = f x - n • b := by
simpa using map_sub_nsmul f x n
|
/root/DuelModelResearch/mathlib4/Mathlib/Algebra/AddConstMap/Basic.lean
|
{
"open": [
"Function Set"
],
"variables": [
"{F G H : Type*} [FunLike F G H] {a : G} {b : H}"
]
}
|
[
{
"line": "simpa using map_sub_nsmul f x n",
"before_state": "F : Type u_1\nG : Type u_2\nH : Type u_3\ninst✝³ : FunLike F G H\nb : H\ninst✝² : AddGroupWithOne G\ninst✝¹ : AddGroup H\ninst✝ : AddConstMapClass F G H 1 b\nf : F\nx : G\nn : ℕ\n⊢ f (x - ↑n) = f x - n • b",
"after_state": "No Goals!"
}
] |
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