Theorem: exists_sum_eq_one_iff_pairwise_coprime
File: Mathlib/RingTheory/Coprime/Lemmas.lean

tactic: simp [exists_apply_eq, Pairwise, Function.onFun]
state_before: case singleton
R : Type u
I : Type v
inst✝¹ : CommSemiring R
x y z : R
s : I → R
t : Finset I
inst✝ : DecidableEq I
a✝ : I
⊢ (∃ μ, ∑ i ∈ {a✝}, μ i * ∏ j ∈ {a✝} \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i => s ↑i)
state_after: no goals

tactic: rw [pairwise_cons']
state_before: case cons
R : Type u
I : Type v
inst✝¹ : CommSemiring R
x y z : R
s : I → R
t✝ : Finset I
inst✝ : DecidableEq I
a : I
t : Finset I
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i => s ↑i)
⊢ (∃ μ, ∑ i ∈ cons a t hat, μ i * ∏ j ∈ cons a t hat \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i => s ↑i)
state_after: case cons
R : Type u
I : Type v
inst✝¹ : CommSemiring R
x y z : R
s : I → R
t✝ : Finset I
inst✝ : DecidableEq I
a : I
t : Finset I
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i => s ↑i)
⊢ (∃ μ, ∑ i ∈ cons a t hat, μ i * ∏ j ∈ cons a t hat \ {i}, s j = 1) ↔
    Pairwise (IsCoprime on fun a => s ↑a) ∧ ∀ b ∈ t, IsCoprime (s a) (s b) ∧ IsCoprime (s b) (s a)

tactic: have mem : ∀ x ∈ t, a ∈ insert a t \ {x} := fun x hx ↦ by
  rw [mem_sdiff, mem_singleton]
  exact ⟨mem_insert_self _ _, fun ha ↦ hat (ha ▸ hx)⟩
state_before: case cons
R : Type u
I : Type v
inst✝¹ : CommSemiring R
x y z : R
s : I → R
t✝ : Finset I
inst✝ : DecidableEq I
a : I
t : Finset I
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i => s ↑i)
⊢ (∃ μ, ∑ i ∈ cons a t hat, μ i * ∏ j ∈ cons a t hat \ {i}, s j = 1) ↔
    Pairwise (IsCoprime on fun a => s ↑a) ∧ ∀ b ∈ t, IsCoprime (s a) (s b) ∧ IsCoprime (s b) (s a)
state_after: case cons
R : Type u
I : Type v
inst✝¹ : CommSemiring R
x y z : R
s : I → R
t✝ : Finset I
inst✝ : DecidableEq I
a : I
t : Finset I
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i => s ↑i)
mem : ∀ x ∈ t, a ∈ insert a t \ {x}
⊢ (∃ μ, ∑ i ∈ cons a t hat, μ i * ∏ j ∈ cons a t hat \ {i}, s j = 1) ↔
    Pairwise (IsCoprime on fun a => s ↑a) ∧ ∀ b ∈ t, IsCoprime (s a) (s b) ∧ IsCoprime (s b) (s a)

tactic: constructor
state_before: case cons
R : Type u
I : Type v
inst✝¹ : CommSemiring R
x y z : R
s : I → R
t✝ : Finset I
inst✝ : DecidableEq I
a : I
t : Finset I
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i => s ↑i)
mem : ∀ x ∈ t, a ∈ insert a t \ {x}
⊢ (∃ μ, ∑ i ∈ cons a t hat, μ i * ∏ j ∈ cons a t hat \ {i}, s j = 1) ↔
    Pairwise (IsCoprime on fun a => s ↑a) ∧ ∀ b ∈ t, IsCoprime (s a) (s b) ∧ IsCoprime (s b) (s a)
state_after: case cons.mp
R : Type u
I : Type v
inst✝¹ : CommSemiring R
x y z : R
s : I → R
t✝ : Finset I
inst✝ : DecidableEq I
a : I
t : Finset I
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i => s ↑i)
mem : ∀ x ∈ t, a ∈ insert a t \ {x}
⊢ (∃ μ, ∑ i ∈ cons a t hat, μ i * ∏ j ∈ cons a t hat \ {i}, s j = 1) →
    Pairwise (IsCoprime on fun a => s ↑a) ∧ ∀ b ∈ t, IsCoprime (s a) (s b) ∧ IsCoprime (s b) (s a)

case cons.mpr
R : Type u
I : Type v
inst✝¹ : CommSemiring R
x y z : R
s : I → R
t✝ : Finset I
inst✝ : DecidableEq I
a : I
t : Finset I
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i => s ↑i)
mem : ∀ x ∈ t, a ∈ insert a t \ {x}
⊢ (Pairwise (IsCoprime on fun a => s ↑a) ∧ ∀ b ∈ t, IsCoprime (s a) (s b) ∧ IsCoprime (s b) (s a)) →
    ∃ μ, ∑ i ∈ cons a t hat, μ i * ∏ j ∈ cons a t hat \ {i}, s j = 1

tactic: rw [mem_sdiff, mem_singleton]
state_before: R : Type u
I : Type v
inst✝¹ : CommSemiring R
x✝ y z : R
s : I → R
t✝ : Finset I
inst✝ : DecidableEq I
a : I
t : Finset I
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i => s ↑i)
x : I
hx : x ∈ t
⊢ a ∈ insert a t \ {x}
state_after: R : Type u
I : Type v
inst✝¹ : CommSemiring R
x✝ y z : R
s : I → R
t✝ : Finset I
inst✝ : DecidableEq I
a : I
t : Finset I
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i => s ↑i)
x : I
hx : x ∈ t
⊢ a ∈ insert a t ∧ ¬a = x

tactic: exact ⟨mem_insert_self _ _, fun ha ↦ hat (ha ▸ hx)⟩
state_before: R : Type u
I : Type v
inst✝¹ : CommSemiring R
x✝ y z : R
s : I → R
t✝ : Finset I
inst✝ : DecidableEq I
a : I
t : Finset I
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i => s ↑i)
x : I
hx : x ∈ t
⊢ a ∈ insert a t ∧ ¬a = x
state_after: no goals

tactic: rintro ⟨μ, hμ⟩
state_before: case cons.mp
R : Type u
I : Type v
inst✝¹ : CommSemiring R
x y z : R
s : I → R
t✝ : Finset I
inst✝ : DecidableEq I
a : I
t : Finset I
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i => s ↑i)
mem : ∀ x ∈ t, a ∈ insert a t \ {x}
⊢ (∃ μ, ∑ i ∈ cons a t hat, μ i * ∏ j ∈ cons a t hat \ {i}, s j = 1) →
    Pairwise (IsCoprime on fun a => s ↑a) ∧ ∀ b ∈ t, IsCoprime (s a) (s b) ∧ IsCoprime (s b) (s a)
state_after: case cons.mp.intro
R : Type u
I : Type v
inst✝¹ : CommSemiring R
x y z : R
s : I → R
t✝ : Finset I
inst✝ : DecidableEq I
a : I
t : Finset I
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i => s ↑i)
mem : ∀ x ∈ t, a ∈ insert a t \ {x}
μ : I → R
hμ : ∑ i ∈ cons a t hat, μ i * ∏ j ∈ cons a t hat \ {i}, s j = 1
⊢ Pairwise (IsCoprime on fun a => s ↑a) ∧ ∀ b ∈ t, IsCoprime (s a) (s b) ∧ IsCoprime (s b) (s a)

tactic: rw [sum_cons, cons_eq_insert, sdiff_singleton_eq_erase, erase_insert hat] at hμ
state_before: case cons.mp.intro
R : Type u
I : Type v
inst✝¹ : CommSemiring R
x y z : R
s : I → R
t✝ : Finset I
inst✝ : DecidableEq I
a : I
t : Finset I
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i => s ↑i)
mem : ∀ x ∈ t, a ∈ insert a t \ {x}
μ : I → R
hμ : ∑ i ∈ cons a t hat, μ i * ∏ j ∈ cons a t hat \ {i}, s j = 1
⊢ Pairwise (IsCoprime on fun a => s ↑a) ∧ ∀ b ∈ t, IsCoprime (s a) (s b) ∧ IsCoprime (s b) (s a)
state_after: case cons.mp.intro
R : Type u
I : Type v
inst✝¹ : CommSemiring R
x y z : R
s : I → R
t✝ : Finset I
inst✝ : DecidableEq I
a : I
t : Finset I
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i => s ↑i)
mem : ∀ x ∈ t, a ∈ insert a t \ {x}
μ : I → R
hμ : μ a * ∏ j ∈ t, s j + ∑ x ∈ t, μ x * ∏ j ∈ insert a t \ {x}, s j = 1
⊢ Pairwise (IsCoprime on fun a => s ↑a) ∧ ∀ b ∈ t, IsCoprime (s a) (s b) ∧ IsCoprime (s b) (s a)

tactic: refine ⟨ih.mp ⟨Pi.single h.choose (μ a * s h.choose) + μ * fun _ ↦ s a, ?_⟩, fun b hb ↦ ?_⟩
state_before: case cons.mp.intro
R : Type u
I : Type v
inst✝¹ : CommSemiring R
x y z : R
s : I → R
t✝ : Finset I
inst✝ : DecidableEq I
a : I
t : Finset I
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i => s ↑i)
mem : ∀ x ∈ t, a ∈ insert a t \ {x}
μ : I → R
hμ : μ a * ∏ j ∈ t, s j + ∑ x ∈ t, μ x * ∏ j ∈ insert a t \ {x}, s j = 1
⊢ Pairwise (IsCoprime on fun a => s ↑a) ∧ ∀ b ∈ t, IsCoprime (s a) (s b) ∧ IsCoprime (s b) (s a)
state_after: case cons.mp.intro.refine_1
R : Type u
I : Type v
inst✝¹ : CommSemiring R
x y z : R
s : I → R
t✝ : Finset I
inst✝ : DecidableEq I
a : I
t : Finset I
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i => s ↑i)
mem : ∀ x ∈ t, a ∈ insert a t \ {x}
μ : I → R
hμ : μ a * ∏ j ∈ t, s j + ∑ x ∈ t, μ x * ∏ j ∈ insert a t \ {x}, s j = 1
⊢ ∑ i ∈ t, (Pi.single (Exists.choose h) (μ a * s (Exists.choose h)) + μ * fun x => s a) i * ∏ j ∈ t \ {i}, s j = 1

case cons.mp.intro.refine_2
R : Type u
I : Type v
inst✝¹ : CommSemiring R
x y z : R
s : I → R
t✝ : Finset I
inst✝ : DecidableEq I
a : I
t : Finset I
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i => s ↑i)
mem : ∀ x ∈ t, a ∈ insert a t \ {x}
μ : I → R
hμ : μ a * ∏ j ∈ t, s j + ∑ x ∈ t, μ x * ∏ j ∈ insert a t \ {x}, s j = 1
b : I
hb : b ∈ t
⊢ IsCoprime (s a) (s b) ∧ IsCoprime (s b) (s a)

tactic: rw [prod_eq_mul_prod_diff_singleton h.choose_spec, ← mul_assoc, ←
  @if_pos _ _ h.choose_spec R (_ * _) 0, ← sum_pi_single', ← sum_add_distrib] at hμ
state_before: case cons.mp.intro.refine_1
R : Type u
I : Type v
inst✝¹ : CommSemiring R
x y z : R
s : I → R
t✝ : Finset I
inst✝ : DecidableEq I
a : I
t : Finset I
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i => s ↑i)
mem : ∀ x ∈ t, a ∈ insert a t \ {x}
μ : I → R
hμ : μ a * ∏ j ∈ t, s j + ∑ x ∈ t, μ x * ∏ j ∈ insert a t \ {x}, s j = 1
⊢ ∑ i ∈ t, (Pi.single (Exists.choose h) (μ a * s (Exists.choose h)) + μ * fun x => s a) i * ∏ j ∈ t \ {i}, s j = 1
state_after: case cons.mp.intro.refine_1
R : Type u
I : Type v
inst✝¹ : CommSemiring R
x y z : R
s : I → R
t✝ : Finset I
inst✝ : DecidableEq I
a : I
t : Finset I
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i => s ↑i)
mem : ∀ x ∈ t, a ∈ insert a t \ {x}
μ : I → R
hμ✝ : μ a * s (Exists.choose h) * ∏ x ∈ t \ {Exists.choose h}, s x + ∑ x ∈ t, μ x * ∏ j ∈ insert a t \ {x}, s j = 1
hμ :
  ∑ x ∈ t,
      (Pi.single (Exists.choose h) (μ a * s (Exists.choose h) * ∏ x ∈ t \ {Exists.choose h}, s x) x +
        μ x * ∏ j ∈ insert a t \ {x}, s j) =
    1
⊢ ∑ i ∈ t, (Pi.single (Exists.choose h) (μ a * s (Exists.choose h)) + μ * fun x => s a) i * ∏ j ∈ t \ {i}, s j = 1

tactic: rw [← hμ, sum_congr rfl]
state_before: case cons.mp.intro.refine_1
R : Type u
I : Type v
inst✝¹ : CommSemiring R
x y z : R
s : I → R
t✝ : Finset I
inst✝ : DecidableEq I
a : I
t : Finset I
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i => s ↑i)
mem : ∀ x ∈ t, a ∈ insert a t \ {x}
μ : I → R
hμ✝ : μ a * s (Exists.choose h) * ∏ x ∈ t \ {Exists.choose h}, s x + ∑ x ∈ t, μ x * ∏ j ∈ insert a t \ {x}, s j = 1
hμ :
  ∑ x ∈ t,
      (Pi.single (Exists.choose h) (μ a * s (Exists.choose h) * ∏ x ∈ t \ {Exists.choose h}, s x) x +
        μ x * ∏ j ∈ insert a t \ {x}, s j) =
    1
⊢ ∑ i ∈ t, (Pi.single (Exists.choose h) (μ a * s (Exists.choose h)) + μ * fun x => s a) i * ∏ j ∈ t \ {i}, s j = 1
state_after: case cons.mp.intro.refine_1
R : Type u
I : Type v
inst✝¹ : CommSemiring R
x y z : R
s : I → R
t✝ : Finset I
inst✝ : DecidableEq I
a : I
t : Finset I
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i => s ↑i)
mem : ∀ x ∈ t, a ∈ insert a t \ {x}
μ : I → R
hμ✝ : μ a * s (Exists.choose h) * ∏ x ∈ t \ {Exists.choose h}, s x + ∑ x ∈ t, μ x * ∏ j ∈ insert a t \ {x}, s j = 1
hμ :
  ∑ x ∈ t,
      (Pi.single (Exists.choose h) (μ a * s (Exists.choose h) * ∏ x ∈ t \ {Exists.choose h}, s x) x +
        μ x * ∏ j ∈ insert a t \ {x}, s j) =
    1
⊢ ∀ x ∈ t,
    (Pi.single (Exists.choose h) (μ a * s (Exists.choose h)) + μ * fun x => s a) x * ∏ j ∈ t \ {x}, s j =
      Pi.single (Exists.choose h) (μ a * s (Exists.choose h) * ∏ x ∈ t \ {Exists.choose h}, s x) x +
        μ x * ∏ j ∈ insert a t \ {x}, s j

tactic: intro x hx
state_before: case cons.mp.intro.refine_1
R : Type u
I : Type v
inst✝¹ : CommSemiring R
x y z : R
s : I → R
t✝ : Finset I
inst✝ : DecidableEq I
a : I
t : Finset I
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i => s ↑i)
mem : ∀ x ∈ t, a ∈ insert a t \ {x}
μ : I → R
hμ✝ : μ a * s (Exists.choose h) * ∏ x ∈ t \ {Exists.choose h}, s x + ∑ x ∈ t, μ x * ∏ j ∈ insert a t \ {x}, s j = 1
hμ :
  ∑ x ∈ t,
      (Pi.single (Exists.choose h) (μ a * s (Exists.choose h) * ∏ x ∈ t \ {Exists.choose h}, s x) x +
        μ x * ∏ j ∈ insert a t \ {x}, s j) =
    1
⊢ ∀ x ∈ t,
    (Pi.single (Exists.choose h) (μ a * s (Exists.choose h)) + μ * fun x => s a) x * ∏ j ∈ t \ {x}, s j =
      Pi.single (Exists.choose h) (μ a * s (Exists.choose h) * ∏ x ∈ t \ {Exists.choose h}, s x) x +
        μ x * ∏ j ∈ insert a t \ {x}, s j
state_after: case cons.mp.intro.refine_1
R : Type u
I : Type v
inst✝¹ : CommSemiring R
x✝ y z : R
s : I → R
t✝ : Finset I
inst✝ : DecidableEq I
a : I
t : Finset I
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i => s ↑i)
mem : ∀ x ∈ t, a ∈ insert a t \ {x}
μ : I → R
hμ✝ : μ a * s (Exists.choose h) * ∏ x ∈ t \ {Exists.choose h}, s x + ∑ x ∈ t, μ x * ∏ j ∈ insert a t \ {x}, s j = 1
hμ :
  ∑ x ∈ t,
      (Pi.single (Exists.choose h) (μ a * s (Exists.choose h) * ∏ x ∈ t \ {Exists.choose h}, s x) x +
        μ x * ∏ j ∈ insert a t \ {x}, s j) =
    1
x : I
hx : x ∈ t
⊢ (Pi.single (Exists.choose h) (μ a * s (Exists.choose h)) + μ * fun x => s a) x * ∏ j ∈ t \ {x}, s j =
    Pi.single (Exists.choose h) (μ a * s (Exists.choose h) * ∏ x ∈ t \ {Exists.choose h}, s x) x +
      μ x * ∏ j ∈ insert a t \ {x}, s j

tactic: dsimp
state_before: case cons.mp.intro.refine_1
R : Type u
I : Type v
inst✝¹ : CommSemiring R
x✝ y z : R
s : I → R
t✝ : Finset I
inst✝ : DecidableEq I
a : I
t : Finset I
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i => s ↑i)
mem : ∀ x ∈ t, a ∈ insert a t \ {x}
μ : I → R
hμ✝ : μ a * s (Exists.choose h) * ∏ x ∈ t \ {Exists.choose h}, s x + ∑ x ∈ t, μ x * ∏ j ∈ insert a t \ {x}, s j = 1
hμ :
  ∑ x ∈ t,
      (Pi.single (Exists.choose h) (μ a * s (Exists.choose h) * ∏ x ∈ t \ {Exists.choose h}, s x) x +
        μ x * ∏ j ∈ insert a t \ {x}, s j) =
    1
x : I
hx : x ∈ t
⊢ (Pi.single (Exists.choose h) (μ a * s (Exists.choose h)) + μ * fun x => s a) x * ∏ j ∈ t \ {x}, s j =
    Pi.single (Exists.choose h) (μ a * s (Exists.choose h) * ∏ x ∈ t \ {Exists.choose h}, s x) x +
      μ x * ∏ j ∈ insert a t \ {x}, s j
state_after: case cons.mp.intro.refine_1
R : Type u
I : Type v
inst✝¹ : CommSemiring R
x✝ y z : R
s : I → R
t✝ : Finset I
inst✝ : DecidableEq I
a : I
t : Finset I
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i => s ↑i)
mem : ∀ x ∈ t, a ∈ insert a t \ {x}
μ : I → R
hμ✝ : μ a * s (Exists.choose h) * ∏ x ∈ t \ {Exists.choose h}, s x + ∑ x ∈ t, μ x * ∏ j ∈ insert a t \ {x}, s j = 1
hμ :
  ∑ x ∈ t,
      (Pi.single (Exists.choose h) (μ a * s (Exists.choose h) * ∏ x ∈ t \ {Exists.choose h}, s x) x +
        μ x * ∏ j ∈ insert a t \ {x}, s j) =
    1
x : I
hx : x ∈ t
⊢ (Pi.single (Exists.choose h) (μ a * s (Exists.choose h)) x + μ x * s a) * ∏ j ∈ t \ {x}, s j =
    Pi.single (Exists.choose h) (μ a * s (Exists.choose h) * ∏ x ∈ t \ {Exists.choose h}, s x) x +
      μ x * ∏ j ∈ insert a t \ {x}, s j

tactic: rw [add_mul]
state_before: case cons.mp.intro.refine_1
R : Type u
I : Type v
inst✝¹ : CommSemiring R
x✝ y z : R
s : I → R
t✝ : Finset I
inst✝ : DecidableEq I
a : I
t : Finset I
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i => s ↑i)
mem : ∀ x ∈ t, a ∈ insert a t \ {x}
μ : I → R
hμ✝ : μ a * s (Exists.choose h) * ∏ x ∈ t \ {Exists.choose h}, s x + ∑ x ∈ t, μ x * ∏ j ∈ insert a t \ {x}, s j = 1
hμ :
  ∑ x ∈ t,
      (Pi.single (Exists.choose h) (μ a * s (Exists.choose h) * ∏ x ∈ t \ {Exists.choose h}, s x) x +
        μ x * ∏ j ∈ insert a t \ {x}, s j) =
    1
x : I
hx : x ∈ t
⊢ (Pi.single (Exists.choose h) (μ a * s (Exists.choose h)) x + μ x * s a) * ∏ j ∈ t \ {x}, s j =
    Pi.single (Exists.choose h) (μ a * s (Exists.choose h) * ∏ x ∈ t \ {Exists.choose h}, s x) x +
      μ x * ∏ j ∈ insert a t \ {x}, s j
state_after: case cons.mp.intro.refine_1
R : Type u
I : Type v
inst✝¹ : CommSemiring R
x✝ y z : R
s : I → R
t✝ : Finset I
inst✝ : DecidableEq I
a : I
t : Finset I
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i => s ↑i)
mem : ∀ x ∈ t, a ∈ insert a t \ {x}
μ : I → R
hμ✝ : μ a * s (Exists.choose h) * ∏ x ∈ t \ {Exists.choose h}, s x + ∑ x ∈ t, μ x * ∏ j ∈ insert a t \ {x}, s j = 1
hμ :
  ∑ x ∈ t,
      (Pi.single (Exists.choose h) (μ a * s (Exists.choose h) * ∏ x ∈ t \ {Exists.choose h}, s x) x +
        μ x * ∏ j ∈ insert a t \ {x}, s j) =
    1
x : I
hx : x ∈ t
⊢ Pi.single (Exists.choose h) (μ a * s (Exists.choose h)) x * ∏ j ∈ t \ {x}, s j + μ x * s a * ∏ j ∈ t \ {x}, s j =
    Pi.single (Exists.choose h) (μ a * s (Exists.choose h) * ∏ x ∈ t \ {Exists.choose h}, s x) x +
      μ x * ∏ j ∈ insert a t \ {x}, s j

tactic: congr 1
state_before: case cons.mp.intro.refine_1
R : Type u
I : Type v
inst✝¹ : CommSemiring R
x✝ y z : R
s : I → R
t✝ : Finset I
inst✝ : DecidableEq I
a : I
t : Finset I
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i => s ↑i)
mem : ∀ x ∈ t, a ∈ insert a t \ {x}
μ : I → R
hμ✝ : μ a * s (Exists.choose h) * ∏ x ∈ t \ {Exists.choose h}, s x + ∑ x ∈ t, μ x * ∏ j ∈ insert a t \ {x}, s j = 1
hμ :
  ∑ x ∈ t,
      (Pi.single (Exists.choose h) (μ a * s (Exists.choose h) * ∏ x ∈ t \ {Exists.choose h}, s x) x +
        μ x * ∏ j ∈ insert a t \ {x}, s j) =
    1
x : I
hx : x ∈ t
⊢ Pi.single (Exists.choose h) (μ a * s (Exists.choose h)) x * ∏ j ∈ t \ {x}, s j + μ x * s a * ∏ j ∈ t \ {x}, s j =
    Pi.single (Exists.choose h) (μ a * s (Exists.choose h) * ∏ x ∈ t \ {Exists.choose h}, s x) x +
      μ x * ∏ j ∈ insert a t \ {x}, s j
state_after: case cons.mp.intro.refine_1.e_a
R : Type u
I : Type v
inst✝¹ : CommSemiring R
x✝ y z : R
s : I → R
t✝ : Finset I
inst✝ : DecidableEq I
a : I
t : Finset I
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i => s ↑i)
mem : ∀ x ∈ t, a ∈ insert a t \ {x}
μ : I → R
hμ✝ : μ a * s (Exists.choose h) * ∏ x ∈ t \ {Exists.choose h}, s x + ∑ x ∈ t, μ x * ∏ j ∈ insert a t \ {x}, s j = 1
hμ :
  ∑ x ∈ t,
      (Pi.single (Exists.choose h) (μ a * s (Exists.choose h) * ∏ x ∈ t \ {Exists.choose h}, s x) x +
        μ x * ∏ j ∈ insert a t \ {x}, s j) =
    1
x : I
hx : x ∈ t
⊢ Pi.single (Exists.choose h) (μ a * s (Exists.choose h)) x * ∏ j ∈ t \ {x}, s j =
    Pi.single (Exists.choose h) (μ a * s (Exists.choose h) * ∏ x ∈ t \ {Exists.choose h}, s x) x

case cons.mp.intro.refine_1.e_a
R : Type u
I : Type v
inst✝¹ : CommSemiring R
x✝ y z : R
s : I → R
t✝ : Finset I
inst✝ : DecidableEq I
a : I
t : Finset I
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i => s ↑i)
mem : ∀ x ∈ t, a ∈ insert a t \ {x}
μ : I → R
hμ✝ : μ a * s (Exists.choose h) * ∏ x ∈ t \ {Exists.choose h}, s x + ∑ x ∈ t, μ x * ∏ j ∈ insert a t \ {x}, s j = 1
hμ :
  ∑ x ∈ t,
      (Pi.single (Exists.choose h) (μ a * s (Exists.choose h) * ∏ x ∈ t \ {Exists.choose h}, s x) x +
        μ x * ∏ j ∈ insert a t \ {x}, s j) =
    1
x : I
hx : x ∈ t
⊢ μ x * s a * ∏ j ∈ t \ {x}, s j = μ x * ∏ j ∈ insert a t \ {x}, s j

tactic: by_cases hx : x = h.choose
state_before: case cons.mp.intro.refine_1.e_a
R : Type u
I : Type v
inst✝¹ : CommSemiring R
x✝ y z : R
s : I → R
t✝ : Finset I
inst✝ : DecidableEq I
a : I
t : Finset I
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i => s ↑i)
mem : ∀ x ∈ t, a ∈ insert a t \ {x}
μ : I → R
hμ✝ : μ a * s (Exists.choose h) * ∏ x ∈ t \ {Exists.choose h}, s x + ∑ x ∈ t, μ x * ∏ j ∈ insert a t \ {x}, s j = 1
hμ :
  ∑ x ∈ t,
      (Pi.single (Exists.choose h) (μ a * s (Exists.choose h) * ∏ x ∈ t \ {Exists.choose h}, s x) x +
        μ x * ∏ j ∈ insert a t \ {x}, s j) =
    1
x : I
hx : x ∈ t
⊢ Pi.single (Exists.choose h) (μ a * s (Exists.choose h)) x * ∏ j ∈ t \ {x}, s j =
    Pi.single (Exists.choose h) (μ a * s (Exists.choose h) * ∏ x ∈ t \ {Exists.choose h}, s x) x
state_after: case pos
R : Type u
I : Type v
inst✝¹ : CommSemiring R
x✝ y z : R
s : I → R
t✝ : Finset I
inst✝ : DecidableEq I
a : I
t : Finset I
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i => s ↑i)
mem : ∀ x ∈ t, a ∈ insert a t \ {x}
μ : I → R
hμ✝ : μ a * s (Exists.choose h) * ∏ x ∈ t \ {Exists.choose h}, s x + ∑ x ∈ t, μ x * ∏ j ∈ insert a t \ {x}, s j = 1
hμ :
  ∑ x ∈ t,
      (Pi.single (Exists.choose h) (μ a * s (Exists.choose h) * ∏ x ∈ t \ {Exists.choose h}, s x) x +
        μ x * ∏ j ∈ insert a t \ {x}, s j) =
    1
x : I
hx✝ : x ∈ t
hx : x = Exists.choose h
⊢ Pi.single (Exists.choose h) (μ a * s (Exists.choose h)) x * ∏ j ∈ t \ {x}, s j =
    Pi.single (Exists.choose h) (μ a * s (Exists.choose h) * ∏ x ∈ t \ {Exists.choose h}, s x) x

case neg
R : Type u
I : Type v
inst✝¹ : CommSemiring R
x✝ y z : R
s : I → R
t✝ : Finset I
inst✝ : DecidableEq I
a : I
t : Finset I
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i => s ↑i)
mem : ∀ x ∈ t, a ∈ insert a t \ {x}
μ : I → R
hμ✝ : μ a * s (Exists.choose h) * ∏ x ∈ t \ {Exists.choose h}, s x + ∑ x ∈ t, μ x * ∏ j ∈ insert a t \ {x}, s j = 1
hμ :
  ∑ x ∈ t,
      (Pi.single (Exists.choose h) (μ a * s (Exists.choose h) * ∏ x ∈ t \ {Exists.choose h}, s x) x +
        μ x * ∏ j ∈ insert a t \ {x}, s j) =
    1
x : I
hx✝ : x ∈ t
hx : ¬x = Exists.choose h
⊢ Pi.single (Exists.choose h) (μ a * s (Exists.choose h)) x * ∏ j ∈ t \ {x}, s j =
    Pi.single (Exists.choose h) (μ a * s (Exists.choose h) * ∏ x ∈ t \ {Exists.choose h}, s x) x

tactic: rw [hx, Pi.single_eq_same, Pi.single_eq_same]
state_before: case pos
R : Type u
I : Type v
inst✝¹ : CommSemiring R
x✝ y z : R
s : I → R
t✝ : Finset I
inst✝ : DecidableEq I
a : I
t : Finset I
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i => s ↑i)
mem : ∀ x ∈ t, a ∈ insert a t \ {x}
μ : I → R
hμ✝ : μ a * s (Exists.choose h) * ∏ x ∈ t \ {Exists.choose h}, s x + ∑ x ∈ t, μ x * ∏ j ∈ insert a t \ {x}, s j = 1
hμ :
  ∑ x ∈ t,
      (Pi.single (Exists.choose h) (μ a * s (Exists.choose h) * ∏ x ∈ t \ {Exists.choose h}, s x) x +
        μ x * ∏ j ∈ insert a t \ {x}, s j) =
    1
x : I
hx✝ : x ∈ t
hx : x = Exists.choose h
⊢ Pi.single (Exists.choose h) (μ a * s (Exists.choose h)) x * ∏ j ∈ t \ {x}, s j =
    Pi.single (Exists.choose h) (μ a * s (Exists.choose h) * ∏ x ∈ t \ {Exists.choose h}, s x) x
state_after: no goals

tactic: rw [Pi.single_eq_of_ne hx, Pi.single_eq_of_ne hx, zero_mul]
state_before: case neg
R : Type u
I : Type v
inst✝¹ : CommSemiring R
x✝ y z : R
s : I → R
t✝ : Finset I
inst✝ : DecidableEq I
a : I
t : Finset I
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i => s ↑i)
mem : ∀ x ∈ t, a ∈ insert a t \ {x}
μ : I → R
hμ✝ : μ a * s (Exists.choose h) * ∏ x ∈ t \ {Exists.choose h}, s x + ∑ x ∈ t, μ x * ∏ j ∈ insert a t \ {x}, s j = 1
hμ :
  ∑ x ∈ t,
      (Pi.single (Exists.choose h) (μ a * s (Exists.choose h) * ∏ x ∈ t \ {Exists.choose h}, s x) x +
        μ x * ∏ j ∈ insert a t \ {x}, s j) =
    1
x : I
hx✝ : x ∈ t
hx : ¬x = Exists.choose h
⊢ Pi.single (Exists.choose h) (μ a * s (Exists.choose h)) x * ∏ j ∈ t \ {x}, s j =
    Pi.single (Exists.choose h) (μ a * s (Exists.choose h) * ∏ x ∈ t \ {Exists.choose h}, s x) x
state_after: no goals

tactic: rw [mul_assoc]
state_before: case cons.mp.intro.refine_1.e_a
R : Type u
I : Type v
inst✝¹ : CommSemiring R
x✝ y z : R
s : I → R
t✝ : Finset I
inst✝ : DecidableEq I
a : I
t : Finset I
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i => s ↑i)
mem : ∀ x ∈ t, a ∈ insert a t \ {x}
μ : I → R
hμ✝ : μ a * s (Exists.choose h) * ∏ x ∈ t \ {Exists.choose h}, s x + ∑ x ∈ t, μ x * ∏ j ∈ insert a t \ {x}, s j = 1
hμ :
  ∑ x ∈ t,
      (Pi.single (Exists.choose h) (μ a * s (Exists.choose h) * ∏ x ∈ t \ {Exists.choose h}, s x) x +
        μ x * ∏ j ∈ insert a t \ {x}, s j) =
    1
x : I
hx : x ∈ t
⊢ μ x * s a * ∏ j ∈ t \ {x}, s j = μ x * ∏ j ∈ insert a t \ {x}, s j
state_after: case cons.mp.intro.refine_1.e_a
R : Type u
I : Type v
inst✝¹ : CommSemiring R
x✝ y z : R
s : I → R
t✝ : Finset I
inst✝ : DecidableEq I
a : I
t : Finset I
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i => s ↑i)
mem : ∀ x ∈ t, a ∈ insert a t \ {x}
μ : I → R
hμ✝ : μ a * s (Exists.choose h) * ∏ x ∈ t \ {Exists.choose h}, s x + ∑ x ∈ t, μ x * ∏ j ∈ insert a t \ {x}, s j = 1
hμ :
  ∑ x ∈ t,
      (Pi.single (Exists.choose h) (μ a * s (Exists.choose h) * ∏ x ∈ t \ {Exists.choose h}, s x) x +
        μ x * ∏ j ∈ insert a t \ {x}, s j) =
    1
x : I
hx : x ∈ t
⊢ μ x * (s a * ∏ j ∈ t \ {x}, s j) = μ x * ∏ j ∈ insert a t \ {x}, s j

tactic: congr
state_before: case cons.mp.intro.refine_1.e_a
R : Type u
I : Type v
inst✝¹ : CommSemiring R
x✝ y z : R
s : I → R
t✝ : Finset I
inst✝ : DecidableEq I
a : I
t : Finset I
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i => s ↑i)
mem : ∀ x ∈ t, a ∈ insert a t \ {x}
μ : I → R
hμ✝ : μ a * s (Exists.choose h) * ∏ x ∈ t \ {Exists.choose h}, s x + ∑ x ∈ t, μ x * ∏ j ∈ insert a t \ {x}, s j = 1
hμ :
  ∑ x ∈ t,
      (Pi.single (Exists.choose h) (μ a * s (Exists.choose h) * ∏ x ∈ t \ {Exists.choose h}, s x) x +
        μ x * ∏ j ∈ insert a t \ {x}, s j) =
    1
x : I
hx : x ∈ t
⊢ μ x * (s a * ∏ j ∈ t \ {x}, s j) = μ x * ∏ j ∈ insert a t \ {x}, s j
state_after: case cons.mp.intro.refine_1.e_a.e_a
R : Type u
I : Type v
inst✝¹ : CommSemiring R
x✝ y z : R
s : I → R
t✝ : Finset I
inst✝ : DecidableEq I
a : I
t : Finset I
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i => s ↑i)
mem : ∀ x ∈ t, a ∈ insert a t \ {x}
μ : I → R
hμ✝ : μ a * s (Exists.choose h) * ∏ x ∈ t \ {Exists.choose h}, s x + ∑ x ∈ t, μ x * ∏ j ∈ insert a t \ {x}, s j = 1
hμ :
  ∑ x ∈ t,
      (Pi.single (Exists.choose h) (μ a * s (Exists.choose h) * ∏ x ∈ t \ {Exists.choose h}, s x) x +
        μ x * ∏ j ∈ insert a t \ {x}, s j) =
    1
x : I
hx : x ∈ t
⊢ s a * ∏ j ∈ t \ {x}, s j = ∏ j ∈ insert a t \ {x}, s j

tactic: rw [prod_eq_prod_diff_singleton_mul (mem x hx) _, mul_comm]
state_before: case cons.mp.intro.refine_1.e_a.e_a
R : Type u
I : Type v
inst✝¹ : CommSemiring R
x✝ y z : R
s : I → R
t✝ : Finset I
inst✝ : DecidableEq I
a : I
t : Finset I
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i => s ↑i)
mem : ∀ x ∈ t, a ∈ insert a t \ {x}
μ : I → R
hμ✝ : μ a * s (Exists.choose h) * ∏ x ∈ t \ {Exists.choose h}, s x + ∑ x ∈ t, μ x * ∏ j ∈ insert a t \ {x}, s j = 1
hμ :
  ∑ x ∈ t,
      (Pi.single (Exists.choose h) (μ a * s (Exists.choose h) * ∏ x ∈ t \ {Exists.choose h}, s x) x +
        μ x * ∏ j ∈ insert a t \ {x}, s j) =
    1
x : I
hx : x ∈ t
⊢ s a * ∏ j ∈ t \ {x}, s j = ∏ j ∈ insert a t \ {x}, s j
state_after: case cons.mp.intro.refine_1.e_a.e_a
R : Type u
I : Type v
inst✝¹ : CommSemiring R
x✝ y z : R
s : I → R
t✝ : Finset I
inst✝ : DecidableEq I
a : I
t : Finset I
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i => s ↑i)
mem : ∀ x ∈ t, a ∈ insert a t \ {x}
μ : I → R
hμ✝ : μ a * s (Exists.choose h) * ∏ x ∈ t \ {Exists.choose h}, s x + ∑ x ∈ t, μ x * ∏ j ∈ insert a t \ {x}, s j = 1
hμ :
  ∑ x ∈ t,
      (Pi.single (Exists.choose h) (μ a * s (Exists.choose h) * ∏ x ∈ t \ {Exists.choose h}, s x) x +
        μ x * ∏ j ∈ insert a t \ {x}, s j) =
    1
x : I
hx : x ∈ t
⊢ (∏ j ∈ t \ {x}, s j) * s a = (∏ x ∈ (insert a t \ {x}) \ {a}, s x) * s a

tactic: congr 2
state_before: case cons.mp.intro.refine_1.e_a.e_a
R : Type u
I : Type v
inst✝¹ : CommSemiring R
x✝ y z : R
s : I → R
t✝ : Finset I
inst✝ : DecidableEq I
a : I
t : Finset I
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i => s ↑i)
mem : ∀ x ∈ t, a ∈ insert a t \ {x}
μ : I → R
hμ✝ : μ a * s (Exists.choose h) * ∏ x ∈ t \ {Exists.choose h}, s x + ∑ x ∈ t, μ x * ∏ j ∈ insert a t \ {x}, s j = 1
hμ :
  ∑ x ∈ t,
      (Pi.single (Exists.choose h) (μ a * s (Exists.choose h) * ∏ x ∈ t \ {Exists.choose h}, s x) x +
        μ x * ∏ j ∈ insert a t \ {x}, s j) =
    1
x : I
hx : x ∈ t
⊢ (∏ j ∈ t \ {x}, s j) * s a = (∏ x ∈ (insert a t \ {x}) \ {a}, s x) * s a
state_after: case cons.mp.intro.refine_1.e_a.e_a.e_a.e_s
R : Type u
I : Type v
inst✝¹ : CommSemiring R
x✝ y z : R
s : I → R
t✝ : Finset I
inst✝ : DecidableEq I
a : I
t : Finset I
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i => s ↑i)
mem : ∀ x ∈ t, a ∈ insert a t \ {x}
μ : I → R
hμ✝ : μ a * s (Exists.choose h) * ∏ x ∈ t \ {Exists.choose h}, s x + ∑ x ∈ t, μ x * ∏ j ∈ insert a t \ {x}, s j = 1
hμ :
  ∑ x ∈ t,
      (Pi.single (Exists.choose h) (μ a * s (Exists.choose h) * ∏ x ∈ t \ {Exists.choose h}, s x) x +
        μ x * ∏ j ∈ insert a t \ {x}, s j) =
    1
x : I
hx : x ∈ t
⊢ t \ {x} = (insert a t \ {x}) \ {a}

tactic: rw [sdiff_sdiff_comm, sdiff_singleton_eq_erase a, erase_insert hat]
state_before: case cons.mp.intro.refine_1.e_a.e_a.e_a.e_s
R : Type u
I : Type v
inst✝¹ : CommSemiring R
x✝ y z : R
s : I → R
t✝ : Finset I
inst✝ : DecidableEq I
a : I
t : Finset I
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i => s ↑i)
mem : ∀ x ∈ t, a ∈ insert a t \ {x}
μ : I → R
hμ✝ : μ a * s (Exists.choose h) * ∏ x ∈ t \ {Exists.choose h}, s x + ∑ x ∈ t, μ x * ∏ j ∈ insert a t \ {x}, s j = 1
hμ :
  ∑ x ∈ t,
      (Pi.single (Exists.choose h) (μ a * s (Exists.choose h) * ∏ x ∈ t \ {Exists.choose h}, s x) x +
        μ x * ∏ j ∈ insert a t \ {x}, s j) =
    1
x : I
hx : x ∈ t
⊢ t \ {x} = (insert a t \ {x}) \ {a}
state_after: no goals

tactic: have : IsCoprime (s b) (s a) :=
  ⟨μ a * ∏ i ∈ t \ {b}, s i, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j, ?_⟩
state_before: case cons.mp.intro.refine_2
R : Type u
I : Type v
inst✝¹ : CommSemiring R
x y z : R
s : I → R
t✝ : Finset I
inst✝ : DecidableEq I
a : I
t : Finset I
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i => s ↑i)
mem : ∀ x ∈ t, a ∈ insert a t \ {x}
μ : I → R
hμ : μ a * ∏ j ∈ t, s j + ∑ x ∈ t, μ x * ∏ j ∈ insert a t \ {x}, s j = 1
b : I
hb : b ∈ t
⊢ IsCoprime (s a) (s b) ∧ IsCoprime (s b) (s a)
state_after: case cons.mp.intro.refine_2.refine_2
R : Type u
I : Type v
inst✝¹ : CommSemiring R
x y z : R
s : I → R
t✝ : Finset I
inst✝ : DecidableEq I
a : I
t : Finset I
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i => s ↑i)
mem : ∀ x ∈ t, a ∈ insert a t \ {x}
μ : I → R
hμ : μ a * ∏ j ∈ t, s j + ∑ x ∈ t, μ x * ∏ j ∈ insert a t \ {x}, s j = 1
b : I
hb : b ∈ t
this : IsCoprime (s b) (s a)
⊢ IsCoprime (s a) (s b) ∧ IsCoprime (s b) (s a)

case cons.mp.intro.refine_2.refine_1
R : Type u
I : Type v
inst✝¹ : CommSemiring R
x y z : R
s : I → R
t✝ : Finset I
inst✝ : DecidableEq I
a : I
t : Finset I
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i => s ↑i)
mem : ∀ x ∈ t, a ∈ insert a t \ {x}
μ : I → R
hμ : μ a * ∏ j ∈ t, s j + ∑ x ∈ t, μ x * ∏ j ∈ insert a t \ {x}, s j = 1
b : I
hb : b ∈ t
⊢ (μ a * ∏ i ∈ t \ {b}, s i) * s b + (∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j) * s a = 1

tactic: rw [mul_assoc, ← prod_eq_prod_diff_singleton_mul hb, sum_mul, ← hμ, sum_congr rfl]
state_before: case cons.mp.intro.refine_2.refine_1
R : Type u
I : Type v
inst✝¹ : CommSemiring R
x y z : R
s : I → R
t✝ : Finset I
inst✝ : DecidableEq I
a : I
t : Finset I
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i => s ↑i)
mem : ∀ x ∈ t, a ∈ insert a t \ {x}
μ : I → R
hμ : μ a * ∏ j ∈ t, s j + ∑ x ∈ t, μ x * ∏ j ∈ insert a t \ {x}, s j = 1
b : I
hb : b ∈ t
⊢ (μ a * ∏ i ∈ t \ {b}, s i) * s b + (∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j) * s a = 1
state_after: case cons.mp.intro.refine_2.refine_1
R : Type u
I : Type v
inst✝¹ : CommSemiring R
x y z : R
s : I → R
t✝ : Finset I
inst✝ : DecidableEq I
a : I
t : Finset I
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i => s ↑i)
mem : ∀ x ∈ t, a ∈ insert a t \ {x}
μ : I → R
hμ : μ a * ∏ j ∈ t, s j + ∑ x ∈ t, μ x * ∏ j ∈ insert a t \ {x}, s j = 1
b : I
hb : b ∈ t
⊢ ∀ x ∈ t, (μ x * ∏ j ∈ t \ {x}, s j) * s a = μ x * ∏ j ∈ insert a t \ {x}, s j

tactic: intro x hx
state_before: case cons.mp.intro.refine_2.refine_1
R : Type u
I : Type v
inst✝¹ : CommSemiring R
x y z : R
s : I → R
t✝ : Finset I
inst✝ : DecidableEq I
a : I
t : Finset I
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i => s ↑i)
mem : ∀ x ∈ t, a ∈ insert a t \ {x}
μ : I → R
hμ : μ a * ∏ j ∈ t, s j + ∑ x ∈ t, μ x * ∏ j ∈ insert a t \ {x}, s j = 1
b : I
hb : b ∈ t
⊢ ∀ x ∈ t, (μ x * ∏ j ∈ t \ {x}, s j) * s a = μ x * ∏ j ∈ insert a t \ {x}, s j
state_after: case cons.mp.intro.refine_2.refine_1
R : Type u
I : Type v
inst✝¹ : CommSemiring R
x✝ y z : R
s : I → R
t✝ : Finset I
inst✝ : DecidableEq I
a : I
t : Finset I
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i => s ↑i)
mem : ∀ x ∈ t, a ∈ insert a t \ {x}
μ : I → R
hμ : μ a * ∏ j ∈ t, s j + ∑ x ∈ t, μ x * ∏ j ∈ insert a t \ {x}, s j = 1
b : I
hb : b ∈ t
x : I
hx : x ∈ t
⊢ (μ x * ∏ j ∈ t \ {x}, s j) * s a = μ x * ∏ j ∈ insert a t \ {x}, s j

tactic: rw [mul_assoc]
state_before: case cons.mp.intro.refine_2.refine_1
R : Type u
I : Type v
inst✝¹ : CommSemiring R
x✝ y z : R
s : I → R
t✝ : Finset I
inst✝ : DecidableEq I
a : I
t : Finset I
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i => s ↑i)
mem : ∀ x ∈ t, a ∈ insert a t \ {x}
μ : I → R
hμ : μ a * ∏ j ∈ t, s j + ∑ x ∈ t, μ x * ∏ j ∈ insert a t \ {x}, s j = 1
b : I
hb : b ∈ t
x : I
hx : x ∈ t
⊢ (μ x * ∏ j ∈ t \ {x}, s j) * s a = μ x * ∏ j ∈ insert a t \ {x}, s j
state_after: case cons.mp.intro.refine_2.refine_1
R : Type u
I : Type v
inst✝¹ : CommSemiring R
x✝ y z : R
s : I → R
t✝ : Finset I
inst✝ : DecidableEq I
a : I
t : Finset I
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i => s ↑i)
mem : ∀ x ∈ t, a ∈ insert a t \ {x}
μ : I → R
hμ : μ a * ∏ j ∈ t, s j + ∑ x ∈ t, μ x * ∏ j ∈ insert a t \ {x}, s j = 1
b : I
hb : b ∈ t
x : I
hx : x ∈ t
⊢ μ x * ((∏ j ∈ t \ {x}, s j) * s a) = μ x * ∏ j ∈ insert a t \ {x}, s j

tactic: congr
state_before: case cons.mp.intro.refine_2.refine_1
R : Type u
I : Type v
inst✝¹ : CommSemiring R
x✝ y z : R
s : I → R
t✝ : Finset I
inst✝ : DecidableEq I
a : I
t : Finset I
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i => s ↑i)
mem : ∀ x ∈ t, a ∈ insert a t \ {x}
μ : I → R
hμ : μ a * ∏ j ∈ t, s j + ∑ x ∈ t, μ x * ∏ j ∈ insert a t \ {x}, s j = 1
b : I
hb : b ∈ t
x : I
hx : x ∈ t
⊢ μ x * ((∏ j ∈ t \ {x}, s j) * s a) = μ x * ∏ j ∈ insert a t \ {x}, s j
state_after: case cons.mp.intro.refine_2.refine_1.e_a
R : Type u
I : Type v
inst✝¹ : CommSemiring R
x✝ y z : R
s : I → R
t✝ : Finset I
inst✝ : DecidableEq I
a : I
t : Finset I
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i => s ↑i)
mem : ∀ x ∈ t, a ∈ insert a t \ {x}
μ : I → R
hμ : μ a * ∏ j ∈ t, s j + ∑ x ∈ t, μ x * ∏ j ∈ insert a t \ {x}, s j = 1
b : I
hb : b ∈ t
x : I
hx : x ∈ t
⊢ (∏ j ∈ t \ {x}, s j) * s a = ∏ j ∈ insert a t \ {x}, s j

tactic: rw [prod_eq_prod_diff_singleton_mul (mem x hx) _]
state_before: case cons.mp.intro.refine_2.refine_1.e_a
R : Type u
I : Type v
inst✝¹ : CommSemiring R
x✝ y z : R
s : I → R
t✝ : Finset I
inst✝ : DecidableEq I
a : I
t : Finset I
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i => s ↑i)
mem : ∀ x ∈ t, a ∈ insert a t \ {x}
μ : I → R
hμ : μ a * ∏ j ∈ t, s j + ∑ x ∈ t, μ x * ∏ j ∈ insert a t \ {x}, s j = 1
b : I
hb : b ∈ t
x : I
hx : x ∈ t
⊢ (∏ j ∈ t \ {x}, s j) * s a = ∏ j ∈ insert a t \ {x}, s j
state_after: case cons.mp.intro.refine_2.refine_1.e_a
R : Type u
I : Type v
inst✝¹ : CommSemiring R
x✝ y z : R
s : I → R
t✝ : Finset I
inst✝ : DecidableEq I
a : I
t : Finset I
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i => s ↑i)
mem : ∀ x ∈ t, a ∈ insert a t \ {x}
μ : I → R
hμ : μ a * ∏ j ∈ t, s j + ∑ x ∈ t, μ x * ∏ j ∈ insert a t \ {x}, s j = 1
b : I
hb : b ∈ t
x : I
hx : x ∈ t
⊢ (∏ j ∈ t \ {x}, s j) * s a = (∏ x ∈ (insert a t \ {x}) \ {a}, s x) * s a

tactic: congr 2
state_before: case cons.mp.intro.refine_2.refine_1.e_a
R : Type u
I : Type v
inst✝¹ : CommSemiring R
x✝ y z : R
s : I → R
t✝ : Finset I
inst✝ : DecidableEq I
a : I
t : Finset I
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i => s ↑i)
mem : ∀ x ∈ t, a ∈ insert a t \ {x}
μ : I → R
hμ : μ a * ∏ j ∈ t, s j + ∑ x ∈ t, μ x * ∏ j ∈ insert a t \ {x}, s j = 1
b : I
hb : b ∈ t
x : I
hx : x ∈ t
⊢ (∏ j ∈ t \ {x}, s j) * s a = (∏ x ∈ (insert a t \ {x}) \ {a}, s x) * s a
state_after: case cons.mp.intro.refine_2.refine_1.e_a.e_a.e_s
R : Type u
I : Type v
inst✝¹ : CommSemiring R
x✝ y z : R
s : I → R
t✝ : Finset I
inst✝ : DecidableEq I
a : I
t : Finset I
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i => s ↑i)
mem : ∀ x ∈ t, a ∈ insert a t \ {x}
μ : I → R
hμ : μ a * ∏ j ∈ t, s j + ∑ x ∈ t, μ x * ∏ j ∈ insert a t \ {x}, s j = 1
b : I
hb : b ∈ t
x : I
hx : x ∈ t
⊢ t \ {x} = (insert a t \ {x}) \ {a}

tactic: rw [sdiff_sdiff_comm, sdiff_singleton_eq_erase a, erase_insert hat]
state_before: case cons.mp.intro.refine_2.refine_1.e_a.e_a.e_s
R : Type u
I : Type v
inst✝¹ : CommSemiring R
x✝ y z : R
s : I → R
t✝ : Finset I
inst✝ : DecidableEq I
a : I
t : Finset I
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i => s ↑i)
mem : ∀ x ∈ t, a ∈ insert a t \ {x}
μ : I → R
hμ : μ a * ∏ j ∈ t, s j + ∑ x ∈ t, μ x * ∏ j ∈ insert a t \ {x}, s j = 1
b : I
hb : b ∈ t
x : I
hx : x ∈ t
⊢ t \ {x} = (insert a t \ {x}) \ {a}
state_after: no goals

tactic: exact ⟨this.symm, this⟩
state_before: case cons.mp.intro.refine_2.refine_2
R : Type u
I : Type v
inst✝¹ : CommSemiring R
x y z : R
s : I → R
t✝ : Finset I
inst✝ : DecidableEq I
a : I
t : Finset I
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i => s ↑i)
mem : ∀ x ∈ t, a ∈ insert a t \ {x}
μ : I → R
hμ : μ a * ∏ j ∈ t, s j + ∑ x ∈ t, μ x * ∏ j ∈ insert a t \ {x}, s j = 1
b : I
hb : b ∈ t
this : IsCoprime (s b) (s a)
⊢ IsCoprime (s a) (s b) ∧ IsCoprime (s b) (s a)
state_after: no goals

tactic: rintro ⟨hs, Hb⟩
state_before: case cons.mpr
R : Type u
I : Type v
inst✝¹ : CommSemiring R
x y z : R
s : I → R
t✝ : Finset I
inst✝ : DecidableEq I
a : I
t : Finset I
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i => s ↑i)
mem : ∀ x ∈ t, a ∈ insert a t \ {x}
⊢ (Pairwise (IsCoprime on fun a => s ↑a) ∧ ∀ b ∈ t, IsCoprime (s a) (s b) ∧ IsCoprime (s b) (s a)) →
    ∃ μ, ∑ i ∈ cons a t hat, μ i * ∏ j ∈ cons a t hat \ {i}, s j = 1
state_after: case cons.mpr.intro
R : Type u
I : Type v
inst✝¹ : CommSemiring R
x y z : R
s : I → R
t✝ : Finset I
inst✝ : DecidableEq I
a : I
t : Finset I
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i => s ↑i)
mem : ∀ x ∈ t, a ∈ insert a t \ {x}
hs : Pairwise (IsCoprime on fun a => s ↑a)
Hb : ∀ b ∈ t, IsCoprime (s a) (s b) ∧ IsCoprime (s b) (s a)
⊢ ∃ μ, ∑ i ∈ cons a t hat, μ i * ∏ j ∈ cons a t hat \ {i}, s j = 1

tactic: obtain ⟨μ, hμ⟩ := ih.mpr hs
state_before: case cons.mpr.intro
R : Type u
I : Type v
inst✝¹ : CommSemiring R
x y z : R
s : I → R
t✝ : Finset I
inst✝ : DecidableEq I
a : I
t : Finset I
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i => s ↑i)
mem : ∀ x ∈ t, a ∈ insert a t \ {x}
hs : Pairwise (IsCoprime on fun a => s ↑a)
Hb : ∀ b ∈ t, IsCoprime (s a) (s b) ∧ IsCoprime (s b) (s a)
⊢ ∃ μ, ∑ i ∈ cons a t hat, μ i * ∏ j ∈ cons a t hat \ {i}, s j = 1
state_after: case cons.mpr.intro.intro
R : Type u
I : Type v
inst✝¹ : CommSemiring R
x y z : R
s : I → R
t✝ : Finset I
inst✝ : DecidableEq I
a : I
t : Finset I
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i => s ↑i)
mem : ∀ x ∈ t, a ∈ insert a t \ {x}
hs : Pairwise (IsCoprime on fun a => s ↑a)
Hb : ∀ b ∈ t, IsCoprime (s a) (s b) ∧ IsCoprime (s b) (s a)
μ : I → R
hμ : ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1
⊢ ∃ μ, ∑ i ∈ cons a t hat, μ i * ∏ j ∈ cons a t hat \ {i}, s j = 1

tactic: obtain ⟨u, v, huv⟩ := IsCoprime.prod_left fun b hb ↦ (Hb b hb).right
state_before: case cons.mpr.intro.intro
R : Type u
I : Type v
inst✝¹ : CommSemiring R
x y z : R
s : I → R
t✝ : Finset I
inst✝ : DecidableEq I
a : I
t : Finset I
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i => s ↑i)
mem : ∀ x ∈ t, a ∈ insert a t \ {x}
hs : Pairwise (IsCoprime on fun a => s ↑a)
Hb : ∀ b ∈ t, IsCoprime (s a) (s b) ∧ IsCoprime (s b) (s a)
μ : I → R
hμ : ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1
⊢ ∃ μ, ∑ i ∈ cons a t hat, μ i * ∏ j ∈ cons a t hat \ {i}, s j = 1
state_after: case cons.mpr.intro.intro.intro.intro
R : Type u
I : Type v
inst✝¹ : CommSemiring R
x y z : R
s : I → R
t✝ : Finset I
inst✝ : DecidableEq I
a : I
t : Finset I
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i => s ↑i)
mem : ∀ x ∈ t, a ∈ insert a t \ {x}
hs : Pairwise (IsCoprime on fun a => s ↑a)
Hb : ∀ b ∈ t, IsCoprime (s a) (s b) ∧ IsCoprime (s b) (s a)
μ : I → R
hμ : ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1
u v : R
huv : u * ∏ i ∈ t, s i + v * s a = 1
⊢ ∃ μ, ∑ i ∈ cons a t hat, μ i * ∏ j ∈ cons a t hat \ {i}, s j = 1

tactic: use fun i ↦ if i = a then u else v * μ i
state_before: case cons.mpr.intro.intro.intro.intro
R : Type u
I : Type v
inst✝¹ : CommSemiring R
x y z : R
s : I → R
t✝ : Finset I
inst✝ : DecidableEq I
a : I
t : Finset I
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i => s ↑i)
mem : ∀ x ∈ t, a ∈ insert a t \ {x}
hs : Pairwise (IsCoprime on fun a => s ↑a)
Hb : ∀ b ∈ t, IsCoprime (s a) (s b) ∧ IsCoprime (s b) (s a)
μ : I → R
hμ : ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1
u v : R
huv : u * ∏ i ∈ t, s i + v * s a = 1
⊢ ∃ μ, ∑ i ∈ cons a t hat, μ i * ∏ j ∈ cons a t hat \ {i}, s j = 1
state_after: case h
R : Type u
I : Type v
inst✝¹ : CommSemiring R
x y z : R
s : I → R
t✝ : Finset I
inst✝ : DecidableEq I
a : I
t : Finset I
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i => s ↑i)
mem : ∀ x ∈ t, a ∈ insert a t \ {x}
hs : Pairwise (IsCoprime on fun a => s ↑a)
Hb : ∀ b ∈ t, IsCoprime (s a) (s b) ∧ IsCoprime (s b) (s a)
μ : I → R
hμ : ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1
u v : R
huv : u * ∏ i ∈ t, s i + v * s a = 1
⊢ ∑ i ∈ cons a t hat, (if i = a then u else v * μ i) * ∏ j ∈ cons a t hat \ {i}, s j = 1

tactic: have hμ' : (∑ i ∈ t, v * ((μ i * ∏ j ∈ t \ {i}, s j) * s a)) = v * s a := by
  rw [← mul_sum, ← sum_mul, hμ, one_mul]
state_before: case h
R : Type u
I : Type v
inst✝¹ : CommSemiring R
x y z : R
s : I → R
t✝ : Finset I
inst✝ : DecidableEq I
a : I
t : Finset I
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i => s ↑i)
mem : ∀ x ∈ t, a ∈ insert a t \ {x}
hs : Pairwise (IsCoprime on fun a => s ↑a)
Hb : ∀ b ∈ t, IsCoprime (s a) (s b) ∧ IsCoprime (s b) (s a)
μ : I → R
hμ : ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1
u v : R
huv : u * ∏ i ∈ t, s i + v * s a = 1
⊢ ∑ i ∈ cons a t hat, (if i = a then u else v * μ i) * ∏ j ∈ cons a t hat \ {i}, s j = 1
state_after: case h
R : Type u
I : Type v
inst✝¹ : CommSemiring R
x y z : R
s : I → R
t✝ : Finset I
inst✝ : DecidableEq I
a : I
t : Finset I
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i => s ↑i)
mem : ∀ x ∈ t, a ∈ insert a t \ {x}
hs : Pairwise (IsCoprime on fun a => s ↑a)
Hb : ∀ b ∈ t, IsCoprime (s a) (s b) ∧ IsCoprime (s b) (s a)
μ : I → R
hμ : ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1
u v : R
huv : u * ∏ i ∈ t, s i + v * s a = 1
hμ' : ∑ i ∈ t, v * ((μ i * ∏ j ∈ t \ {i}, s j) * s a) = v * s a
⊢ ∑ i ∈ cons a t hat, (if i = a then u else v * μ i) * ∏ j ∈ cons a t hat \ {i}, s j = 1

tactic: rw [sum_cons, cons_eq_insert, sdiff_singleton_eq_erase, erase_insert hat, if_pos rfl,
  ← huv, ← hμ', sum_congr rfl]
state_before: case h
R : Type u
I : Type v
inst✝¹ : CommSemiring R
x y z : R
s : I → R
t✝ : Finset I
inst✝ : DecidableEq I
a : I
t : Finset I
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i => s ↑i)
mem : ∀ x ∈ t, a ∈ insert a t \ {x}
hs : Pairwise (IsCoprime on fun a => s ↑a)
Hb : ∀ b ∈ t, IsCoprime (s a) (s b) ∧ IsCoprime (s b) (s a)
μ : I → R
hμ : ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1
u v : R
huv : u * ∏ i ∈ t, s i + v * s a = 1
hμ' : ∑ i ∈ t, v * ((μ i * ∏ j ∈ t \ {i}, s j) * s a) = v * s a
⊢ ∑ i ∈ cons a t hat, (if i = a then u else v * μ i) * ∏ j ∈ cons a t hat \ {i}, s j = 1
state_after: case h
R : Type u
I : Type v
inst✝¹ : CommSemiring R
x y z : R
s : I → R
t✝ : Finset I
inst✝ : DecidableEq I
a : I
t : Finset I
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i => s ↑i)
mem : ∀ x ∈ t, a ∈ insert a t \ {x}
hs : Pairwise (IsCoprime on fun a => s ↑a)
Hb : ∀ b ∈ t, IsCoprime (s a) (s b) ∧ IsCoprime (s b) (s a)
μ : I → R
hμ : ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1
u v : R
huv : u * ∏ i ∈ t, s i + v * s a = 1
hμ' : ∑ i ∈ t, v * ((μ i * ∏ j ∈ t \ {i}, s j) * s a) = v * s a
⊢ ∀ x ∈ t, (if x = a then u else v * μ x) * ∏ j ∈ insert a t \ {x}, s j = v * ((μ x * ∏ j ∈ t \ {x}, s j) * s a)

tactic: intro x hx
state_before: case h
R : Type u
I : Type v
inst✝¹ : CommSemiring R
x y z : R
s : I → R
t✝ : Finset I
inst✝ : DecidableEq I
a : I
t : Finset I
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i => s ↑i)
mem : ∀ x ∈ t, a ∈ insert a t \ {x}
hs : Pairwise (IsCoprime on fun a => s ↑a)
Hb : ∀ b ∈ t, IsCoprime (s a) (s b) ∧ IsCoprime (s b) (s a)
μ : I → R
hμ : ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1
u v : R
huv : u * ∏ i ∈ t, s i + v * s a = 1
hμ' : ∑ i ∈ t, v * ((μ i * ∏ j ∈ t \ {i}, s j) * s a) = v * s a
⊢ ∀ x ∈ t, (if x = a then u else v * μ x) * ∏ j ∈ insert a t \ {x}, s j = v * ((μ x * ∏ j ∈ t \ {x}, s j) * s a)
state_after: case h
R : Type u
I : Type v
inst✝¹ : CommSemiring R
x✝ y z : R
s : I → R
t✝ : Finset I
inst✝ : DecidableEq I
a : I
t : Finset I
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i => s ↑i)
mem : ∀ x ∈ t, a ∈ insert a t \ {x}
hs : Pairwise (IsCoprime on fun a => s ↑a)
Hb : ∀ b ∈ t, IsCoprime (s a) (s b) ∧ IsCoprime (s b) (s a)
μ : I → R
hμ : ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1
u v : R
huv : u * ∏ i ∈ t, s i + v * s a = 1
hμ' : ∑ i ∈ t, v * ((μ i * ∏ j ∈ t \ {i}, s j) * s a) = v * s a
x : I
hx : x ∈ t
⊢ (if x = a then u else v * μ x) * ∏ j ∈ insert a t \ {x}, s j = v * ((μ x * ∏ j ∈ t \ {x}, s j) * s a)

tactic: rw [mul_assoc, if_neg fun ha : x = a ↦ hat (ha.casesOn hx)]
state_before: case h
R : Type u
I : Type v
inst✝¹ : CommSemiring R
x✝ y z : R
s : I → R
t✝ : Finset I
inst✝ : DecidableEq I
a : I
t : Finset I
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i => s ↑i)
mem : ∀ x ∈ t, a ∈ insert a t \ {x}
hs : Pairwise (IsCoprime on fun a => s ↑a)
Hb : ∀ b ∈ t, IsCoprime (s a) (s b) ∧ IsCoprime (s b) (s a)
μ : I → R
hμ : ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1
u v : R
huv : u * ∏ i ∈ t, s i + v * s a = 1
hμ' : ∑ i ∈ t, v * ((μ i * ∏ j ∈ t \ {i}, s j) * s a) = v * s a
x : I
hx : x ∈ t
⊢ (if x = a then u else v * μ x) * ∏ j ∈ insert a t \ {x}, s j = v * ((μ x * ∏ j ∈ t \ {x}, s j) * s a)
state_after: case h
R : Type u
I : Type v
inst✝¹ : CommSemiring R
x✝ y z : R
s : I → R
t✝ : Finset I
inst✝ : DecidableEq I
a : I
t : Finset I
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i => s ↑i)
mem : ∀ x ∈ t, a ∈ insert a t \ {x}
hs : Pairwise (IsCoprime on fun a => s ↑a)
Hb : ∀ b ∈ t, IsCoprime (s a) (s b) ∧ IsCoprime (s b) (s a)
μ : I → R
hμ : ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1
u v : R
huv : u * ∏ i ∈ t, s i + v * s a = 1
hμ' : ∑ i ∈ t, v * ((μ i * ∏ j ∈ t \ {i}, s j) * s a) = v * s a
x : I
hx : x ∈ t
⊢ v * μ x * ∏ j ∈ insert a t \ {x}, s j = v * (μ x * ((∏ j ∈ t \ {x}, s j) * s a))

tactic: rw [mul_assoc]
state_before: case h
R : Type u
I : Type v
inst✝¹ : CommSemiring R
x✝ y z : R
s : I → R
t✝ : Finset I
inst✝ : DecidableEq I
a : I
t : Finset I
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i => s ↑i)
mem : ∀ x ∈ t, a ∈ insert a t \ {x}
hs : Pairwise (IsCoprime on fun a => s ↑a)
Hb : ∀ b ∈ t, IsCoprime (s a) (s b) ∧ IsCoprime (s b) (s a)
μ : I → R
hμ : ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1
u v : R
huv : u * ∏ i ∈ t, s i + v * s a = 1
hμ' : ∑ i ∈ t, v * ((μ i * ∏ j ∈ t \ {i}, s j) * s a) = v * s a
x : I
hx : x ∈ t
⊢ v * μ x * ∏ j ∈ insert a t \ {x}, s j = v * (μ x * ((∏ j ∈ t \ {x}, s j) * s a))
state_after: case h
R : Type u
I : Type v
inst✝¹ : CommSemiring R
x✝ y z : R
s : I → R
t✝ : Finset I
inst✝ : DecidableEq I
a : I
t : Finset I
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i => s ↑i)
mem : ∀ x ∈ t, a ∈ insert a t \ {x}
hs : Pairwise (IsCoprime on fun a => s ↑a)
Hb : ∀ b ∈ t, IsCoprime (s a) (s b) ∧ IsCoprime (s b) (s a)
μ : I → R
hμ : ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1
u v : R
huv : u * ∏ i ∈ t, s i + v * s a = 1
hμ' : ∑ i ∈ t, v * ((μ i * ∏ j ∈ t \ {i}, s j) * s a) = v * s a
x : I
hx : x ∈ t
⊢ v * (μ x * ∏ j ∈ insert a t \ {x}, s j) = v * (μ x * ((∏ j ∈ t \ {x}, s j) * s a))

tactic: congr
state_before: case h
R : Type u
I : Type v
inst✝¹ : CommSemiring R
x✝ y z : R
s : I → R
t✝ : Finset I
inst✝ : DecidableEq I
a : I
t : Finset I
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i => s ↑i)
mem : ∀ x ∈ t, a ∈ insert a t \ {x}
hs : Pairwise (IsCoprime on fun a => s ↑a)
Hb : ∀ b ∈ t, IsCoprime (s a) (s b) ∧ IsCoprime (s b) (s a)
μ : I → R
hμ : ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1
u v : R
huv : u * ∏ i ∈ t, s i + v * s a = 1
hμ' : ∑ i ∈ t, v * ((μ i * ∏ j ∈ t \ {i}, s j) * s a) = v * s a
x : I
hx : x ∈ t
⊢ v * (μ x * ∏ j ∈ insert a t \ {x}, s j) = v * (μ x * ((∏ j ∈ t \ {x}, s j) * s a))
state_after: case h.e_a.e_a
R : Type u
I : Type v
inst✝¹ : CommSemiring R
x✝ y z : R
s : I → R
t✝ : Finset I
inst✝ : DecidableEq I
a : I
t : Finset I
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i => s ↑i)
mem : ∀ x ∈ t, a ∈ insert a t \ {x}
hs : Pairwise (IsCoprime on fun a => s ↑a)
Hb : ∀ b ∈ t, IsCoprime (s a) (s b) ∧ IsCoprime (s b) (s a)
μ : I → R
hμ : ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1
u v : R
huv : u * ∏ i ∈ t, s i + v * s a = 1
hμ' : ∑ i ∈ t, v * ((μ i * ∏ j ∈ t \ {i}, s j) * s a) = v * s a
x : I
hx : x ∈ t
⊢ ∏ j ∈ insert a t \ {x}, s j = (∏ j ∈ t \ {x}, s j) * s a

tactic: rw [prod_eq_prod_diff_singleton_mul (mem x hx) _]
state_before: case h.e_a.e_a
R : Type u
I : Type v
inst✝¹ : CommSemiring R
x✝ y z : R
s : I → R
t✝ : Finset I
inst✝ : DecidableEq I
a : I
t : Finset I
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i => s ↑i)
mem : ∀ x ∈ t, a ∈ insert a t \ {x}
hs : Pairwise (IsCoprime on fun a => s ↑a)
Hb : ∀ b ∈ t, IsCoprime (s a) (s b) ∧ IsCoprime (s b) (s a)
μ : I → R
hμ : ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1
u v : R
huv : u * ∏ i ∈ t, s i + v * s a = 1
hμ' : ∑ i ∈ t, v * ((μ i * ∏ j ∈ t \ {i}, s j) * s a) = v * s a
x : I
hx : x ∈ t
⊢ ∏ j ∈ insert a t \ {x}, s j = (∏ j ∈ t \ {x}, s j) * s a
state_after: case h.e_a.e_a
R : Type u
I : Type v
inst✝¹ : CommSemiring R
x✝ y z : R
s : I → R
t✝ : Finset I
inst✝ : DecidableEq I
a : I
t : Finset I
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i => s ↑i)
mem : ∀ x ∈ t, a ∈ insert a t \ {x}
hs : Pairwise (IsCoprime on fun a => s ↑a)
Hb : ∀ b ∈ t, IsCoprime (s a) (s b) ∧ IsCoprime (s b) (s a)
μ : I → R
hμ : ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1
u v : R
huv : u * ∏ i ∈ t, s i + v * s a = 1
hμ' : ∑ i ∈ t, v * ((μ i * ∏ j ∈ t \ {i}, s j) * s a) = v * s a
x : I
hx : x ∈ t
⊢ (∏ x ∈ (insert a t \ {x}) \ {a}, s x) * s a = (∏ j ∈ t \ {x}, s j) * s a

tactic: congr 2
state_before: case h.e_a.e_a
R : Type u
I : Type v
inst✝¹ : CommSemiring R
x✝ y z : R
s : I → R
t✝ : Finset I
inst✝ : DecidableEq I
a : I
t : Finset I
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i => s ↑i)
mem : ∀ x ∈ t, a ∈ insert a t \ {x}
hs : Pairwise (IsCoprime on fun a => s ↑a)
Hb : ∀ b ∈ t, IsCoprime (s a) (s b) ∧ IsCoprime (s b) (s a)
μ : I → R
hμ : ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1
u v : R
huv : u * ∏ i ∈ t, s i + v * s a = 1
hμ' : ∑ i ∈ t, v * ((μ i * ∏ j ∈ t \ {i}, s j) * s a) = v * s a
x : I
hx : x ∈ t
⊢ (∏ x ∈ (insert a t \ {x}) \ {a}, s x) * s a = (∏ j ∈ t \ {x}, s j) * s a
state_after: case h.e_a.e_a.e_a.e_s
R : Type u
I : Type v
inst✝¹ : CommSemiring R
x✝ y z : R
s : I → R
t✝ : Finset I
inst✝ : DecidableEq I
a : I
t : Finset I
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i => s ↑i)
mem : ∀ x ∈ t, a ∈ insert a t \ {x}
hs : Pairwise (IsCoprime on fun a => s ↑a)
Hb : ∀ b ∈ t, IsCoprime (s a) (s b) ∧ IsCoprime (s b) (s a)
μ : I → R
hμ : ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1
u v : R
huv : u * ∏ i ∈ t, s i + v * s a = 1
hμ' : ∑ i ∈ t, v * ((μ i * ∏ j ∈ t \ {i}, s j) * s a) = v * s a
x : I
hx : x ∈ t
⊢ (insert a t \ {x}) \ {a} = t \ {x}

tactic: rw [sdiff_sdiff_comm, sdiff_singleton_eq_erase a, erase_insert hat]
state_before: case h.e_a.e_a.e_a.e_s
R : Type u
I : Type v
inst✝¹ : CommSemiring R
x✝ y z : R
s : I → R
t✝ : Finset I
inst✝ : DecidableEq I
a : I
t : Finset I
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i => s ↑i)
mem : ∀ x ∈ t, a ∈ insert a t \ {x}
hs : Pairwise (IsCoprime on fun a => s ↑a)
Hb : ∀ b ∈ t, IsCoprime (s a) (s b) ∧ IsCoprime (s b) (s a)
μ : I → R
hμ : ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1
u v : R
huv : u * ∏ i ∈ t, s i + v * s a = 1
hμ' : ∑ i ∈ t, v * ((μ i * ∏ j ∈ t \ {i}, s j) * s a) = v * s a
x : I
hx : x ∈ t
⊢ (insert a t \ {x}) \ {a} = t \ {x}
state_after: no goals

tactic: rw [← mul_sum, ← sum_mul, hμ, one_mul]
state_before: R : Type u
I : Type v
inst✝¹ : CommSemiring R
x y z : R
s : I → R
t✝ : Finset I
inst✝ : DecidableEq I
a : I
t : Finset I
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i => s ↑i)
mem : ∀ x ∈ t, a ∈ insert a t \ {x}
hs : Pairwise (IsCoprime on fun a => s ↑a)
Hb : ∀ b ∈ t, IsCoprime (s a) (s b) ∧ IsCoprime (s b) (s a)
μ : I → R
hμ : ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1
u v : R
huv : u * ∏ i ∈ t, s i + v * s a = 1
⊢ ∑ i ∈ t, v * ((μ i * ∏ j ∈ t \ {i}, s j) * s a) = v * s a
state_after: no goals