mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Algebra.Module.Submodule.Finsupp

Goal state before:
pp: case mp
R : Type u_2
M : Type u_3
inst✝³ : Semiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
sR : Set R
N : Submodule R M
x : M
inst✝ : SMulCommClass R R ↥N
⊢ x ∈ sR • N → ∃ c, ↑c.support ⊆ sR ∧ x = ↑(c.sum fun r m ↦ r • m)

pp: case mpr
R : Type u_2
M : Type u_3
inst✝³ : Semiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
sR : Set R
N : Submodule R M
x : M
inst✝ : SMulCommClass R R ↥N
⊢ (∃ c, ↑c.support ⊆ sR ∧ x = ↑(c.sum fun r m ↦ r • m)) → x ∈ sR • N

Tactic:
·

Goal state after:
case mp
R : Type u_2
M : Type u_3
inst✝³ : Semiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
sR : Set R
N : Submodule R M
x : M
inst✝ : SMulCommClass R R ↥N
⊢ x ∈ sR • N → ∃ c, ↑c.support ⊆ sR ∧ x = ↑(c.sum fun r m ↦ r • m)
Module: Mathlib.LinearAlgebra.TensorProduct.Defs

Goal state before:
pp: case refine_1
R : Type u_1
inst✝⁴ : CommSemiring R
M : Type u_7
N : Type u_8
inst✝³ : AddCommMonoid M
inst✝² : AddCommMonoid N
inst✝¹ : Module R M
inst✝ : Module R N
t : M ⊗[R] N
⊢ 0 ∈ Submodule.span R {t | ∃ m n, m ⊗ₜ[R] n = t}

pp: case refine_2
R : Type u_1
inst✝⁴ : CommSemiring R
M : Type u_7
N : Type u_8
inst✝³ : AddCommMonoid M
inst✝² : AddCommMonoid N
inst✝¹ : Module R M
inst✝ : Module R N
t : M ⊗[R] N
⊢ ∀ (x : M) (y : N), x ⊗ₜ[R] y ∈ Submodule.span R {t | ∃ m n, m ⊗ₜ[R] n = t}

pp: case refine_3
R : Type u_1
inst✝⁴ : CommSemiring R
M : Type u_7
N : Type u_8
inst✝³ : AddCommMonoid M
inst✝² : AddCommMonoid N
inst✝¹ : Module R M
inst✝ : Module R N
t : M ⊗[R] N
⊢ ∀ (x y : M ⊗[R] N),
    x ∈ Submodule.span R {t | ∃ m n, m ⊗ₜ[R] n = t} →
      y ∈ Submodule.span R {t | ∃ m n, m ⊗ₜ[R] n = t} → x + y ∈ Submodule.span R {t | ∃ m n, m ⊗ₜ[R] n = t}

Tactic:
·

Goal state after:
case refine_1
R : Type u_1
inst✝⁴ : CommSemiring R
M : Type u_7
N : Type u_8
inst✝³ : AddCommMonoid M
inst✝² : AddCommMonoid N
inst✝¹ : Module R M
inst✝ : Module R N
t : M ⊗[R] N
⊢ 0 ∈ Submodule.span R {t | ∃ m n, m ⊗ₜ[R] n = t}
Module: Mathlib.LinearAlgebra.TensorProduct.Defs

Goal state before:
pp: case refine_2
R : Type u_1
inst✝⁴ : CommSemiring R
M : Type u_7
N : Type u_8
inst✝³ : AddCommMonoid M
inst✝² : AddCommMonoid N
inst✝¹ : Module R M
inst✝ : Module R N
t : M ⊗[R] N
⊢ ∀ (x : M) (y : N), x ⊗ₜ[R] y ∈ Submodule.span R {t | ∃ m n, m ⊗ₜ[R] n = t}

pp: case refine_3
R : Type u_1
inst✝⁴ : CommSemiring R
M : Type u_7
N : Type u_8
inst✝³ : AddCommMonoid M
inst✝² : AddCommMonoid N
inst✝¹ : Module R M
inst✝ : Module R N
t : M ⊗[R] N
⊢ ∀ (x y : M ⊗[R] N),
    x ∈ Submodule.span R {t | ∃ m n, m ⊗ₜ[R] n = t} →
      y ∈ Submodule.span R {t | ∃ m n, m ⊗ₜ[R] n = t} → x + y ∈ Submodule.span R {t | ∃ m n, m ⊗ₜ[R] n = t}

Tactic:
·

Goal state after:
case refine_2
R : Type u_1
inst✝⁴ : CommSemiring R
M : Type u_7
N : Type u_8
inst✝³ : AddCommMonoid M
inst✝² : AddCommMonoid N
inst✝¹ : Module R M
inst✝ : Module R N
t : M ⊗[R] N
⊢ ∀ (x : M) (y : N), x ⊗ₜ[R] y ∈ Submodule.span R {t | ∃ m n, m ⊗ₜ[R] n = t}
Module: Mathlib.Algebra.Module.Submodule.Finsupp

Goal state before:
pp: case mpr
R : Type u_2
M : Type u_3
inst✝³ : Semiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
sR : Set R
N : Submodule R M
x : M
inst✝ : SMulCommClass R R ↥N
⊢ (∃ c, ↑c.support ⊆ sR ∧ x = ↑(c.sum fun r m ↦ r • m)) → x ∈ sR • N

Tactic:
·

Goal state after:
case mpr
R : Type u_2
M : Type u_3
inst✝³ : Semiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
sR : Set R
N : Submodule R M
x : M
inst✝ : SMulCommClass R R ↥N
⊢ (∃ c, ↑c.support ⊆ sR ∧ x = ↑(c.sum fun r m ↦ r • m)) → x ∈ sR • N
Module: Mathlib.LinearAlgebra.TensorProduct.Defs

Goal state before:
pp: case refine_3
R : Type u_1
inst✝⁴ : CommSemiring R
M : Type u_7
N : Type u_8
inst✝³ : AddCommMonoid M
inst✝² : AddCommMonoid N
inst✝¹ : Module R M
inst✝ : Module R N
t : M ⊗[R] N
⊢ ∀ (x y : M ⊗[R] N),
    x ∈ Submodule.span R {t | ∃ m n, m ⊗ₜ[R] n = t} →
      y ∈ Submodule.span R {t | ∃ m n, m ⊗ₜ[R] n = t} → x + y ∈ Submodule.span R {t | ∃ m n, m ⊗ₜ[R] n = t}

Tactic:
·

Goal state after:
case refine_3
R : Type u_1
inst✝⁴ : CommSemiring R
M : Type u_7
N : Type u_8
inst✝³ : AddCommMonoid M
inst✝² : AddCommMonoid N
inst✝¹ : Module R M
inst✝ : Module R N
t : M ⊗[R] N
⊢ ∀ (x y : M ⊗[R] N),
    x ∈ Submodule.span R {t | ∃ m n, m ⊗ₜ[R] n = t} →
      y ∈ Submodule.span R {t | ∃ m n, m ⊗ₜ[R] n = t} → x + y ∈ Submodule.span R {t | ∃ m n, m ⊗ₜ[R] n = t}
Module: Mathlib.Algebra.Ring.NonZeroDivisors

Goal state before:
pp: case inl
R : Type u_1
inst✝¹ : Monoid R
r : R
inst✝ : IsMulTorsionFree R
hx : IsLeftRegular r
hx' : r ≠ 1
n m : ℕ
hnm : (fun n ↦ r ^ n) n = (fun n ↦ r ^ n) m
main : ∀ {n m : ℕ}, n ≤ m → r ^ n = r ^ m → n = m
h : n ≤ m
⊢ n = m

pp: case inr
R : Type u_1
inst✝¹ : Monoid R
r : R
inst✝ : IsMulTorsionFree R
hx : IsLeftRegular r
hx' : r ≠ 1
n m : ℕ
hnm : (fun n ↦ r ^ n) n = (fun n ↦ r ^ n) m
main : ∀ {n m : ℕ}, n ≤ m → r ^ n = r ^ m → n = m
h : m ≤ n
⊢ n = m

Tactic:
·

Goal state after:
case inl
R : Type u_1
inst✝¹ : Monoid R
r : R
inst✝ : IsMulTorsionFree R
hx : IsLeftRegular r
hx' : r ≠ 1
n m : ℕ
hnm : (fun n ↦ r ^ n) n = (fun n ↦ r ^ n) m
main : ∀ {n m : ℕ}, n ≤ m → r ^ n = r ^ m → n = m
h : n ≤ m
⊢ n = m
Module: Mathlib.Algebra.Ring.NonZeroDivisors

Goal state before:
pp: case inr
R : Type u_1
inst✝¹ : Monoid R
r : R
inst✝ : IsMulTorsionFree R
hx : IsLeftRegular r
hx' : r ≠ 1
n m : ℕ
hnm : (fun n ↦ r ^ n) n = (fun n ↦ r ^ n) m
main : ∀ {n m : ℕ}, n ≤ m → r ^ n = r ^ m → n = m
h : m ≤ n
⊢ n = m

Tactic:
·

Goal state after:
case inr
R : Type u_1
inst✝¹ : Monoid R
r : R
inst✝ : IsMulTorsionFree R
hx : IsLeftRegular r
hx' : r ≠ 1
n m : ℕ
hnm : (fun n ↦ r ^ n) n = (fun n ↦ r ^ n) m
main : ∀ {n m : ℕ}, n ≤ m → r ^ n = r ^ m → n = m
h : m ≤ n
⊢ n = m
Module: Mathlib.Algebra.Ring.Submonoid.Pointwise

Goal state before:
pp: case refine_1
R : Type u_2
A : Type u_3
inst✝² : AddMonoid R
inst✝¹ : AddMonoid A
inst✝ : DistribSMul R A
M : AddSubmonoid R
N P : AddSubmonoid A
m : R
hm : m ∈ M
np : A
hnp : np ∈ N ⊔ P
⊢ ∀ x ∈ ↑N ∪ ↑P, m • x ∈ M • N ⊔ M • P

pp: case refine_2
R : Type u_2
A : Type u_3
inst✝² : AddMonoid R
inst✝¹ : AddMonoid A
inst✝ : DistribSMul R A
M : AddSubmonoid R
N P : AddSubmonoid A
m : R
hm : m ∈ M
np : A
hnp : np ∈ N ⊔ P
⊢ m • 0 ∈ M • N ⊔ M • P

pp: case refine_3
R : Type u_2
A : Type u_3
inst✝² : AddMonoid R
inst✝¹ : AddMonoid A
inst✝ : DistribSMul R A
M : AddSubmonoid R
N P : AddSubmonoid A
m : R
hm : m ∈ M
np : A
hnp : np ∈ N ⊔ P
⊢ ∀ (x y : A),
    x ∈ closure (↑N ∪ ↑P) →
      y ∈ closure (↑N ∪ ↑P) → m • x ∈ M • N ⊔ M • P → m • y ∈ M • N ⊔ M • P → m • (x + y) ∈ M • N ⊔ M • P

Tactic:
·

Goal state after:
case refine_1
R : Type u_2
A : Type u_3
inst✝² : AddMonoid R
inst✝¹ : AddMonoid A
inst✝ : DistribSMul R A
M : AddSubmonoid R
N P : AddSubmonoid A
m : R
hm : m ∈ M
np : A
hnp : np ∈ N ⊔ P
⊢ ∀ x ∈ ↑N ∪ ↑P, m • x ∈ M • N ⊔ M • P
Module: Mathlib.Algebra.Ring.Submonoid.Pointwise

Goal state before:
pp: case refine_2
R : Type u_2
A : Type u_3
inst✝² : AddMonoid R
inst✝¹ : AddMonoid A
inst✝ : DistribSMul R A
M : AddSubmonoid R
N P : AddSubmonoid A
m : R
hm : m ∈ M
np : A
hnp : np ∈ N ⊔ P
⊢ m • 0 ∈ M • N ⊔ M • P

pp: case refine_3
R : Type u_2
A : Type u_3
inst✝² : AddMonoid R
inst✝¹ : AddMonoid A
inst✝ : DistribSMul R A
M : AddSubmonoid R
N P : AddSubmonoid A
m : R
hm : m ∈ M
np : A
hnp : np ∈ N ⊔ P
⊢ ∀ (x y : A),
    x ∈ closure (↑N ∪ ↑P) →
      y ∈ closure (↑N ∪ ↑P) → m • x ∈ M • N ⊔ M • P → m • y ∈ M • N ⊔ M • P → m • (x + y) ∈ M • N ⊔ M • P

Tactic:
·

Goal state after:
case refine_2
R : Type u_2
A : Type u_3
inst✝² : AddMonoid R
inst✝¹ : AddMonoid A
inst✝ : DistribSMul R A
M : AddSubmonoid R
N P : AddSubmonoid A
m : R
hm : m ∈ M
np : A
hnp : np ∈ N ⊔ P
⊢ m • 0 ∈ M • N ⊔ M • P
Module: Mathlib.Algebra.Ring.Submonoid.Pointwise

Goal state before:
pp: case refine_3
R : Type u_2
A : Type u_3
inst✝² : AddMonoid R
inst✝¹ : AddMonoid A
inst✝ : DistribSMul R A
M : AddSubmonoid R
N P : AddSubmonoid A
m : R
hm : m ∈ M
np : A
hnp : np ∈ N ⊔ P
⊢ ∀ (x y : A),
    x ∈ closure (↑N ∪ ↑P) →
      y ∈ closure (↑N ∪ ↑P) → m • x ∈ M • N ⊔ M • P → m • y ∈ M • N ⊔ M • P → m • (x + y) ∈ M • N ⊔ M • P

Tactic:
·

Goal state after:
case refine_3
R : Type u_2
A : Type u_3
inst✝² : AddMonoid R
inst✝¹ : AddMonoid A
inst✝ : DistribSMul R A
M : AddSubmonoid R
N P : AddSubmonoid A
m : R
hm : m ∈ M
np : A
hnp : np ∈ N ⊔ P
⊢ ∀ (x y : A),
    x ∈ closure (↑N ∪ ↑P) →
      y ∈ closure (↑N ∪ ↑P) → m • x ∈ M • N ⊔ M • P → m • y ∈ M • N ⊔ M • P → m • (x + y) ∈ M • N ⊔ M • P
Module: Mathlib.Algebra.Ring.Submonoid.Pointwise

Goal state before:
pp: case a
R : Type u_2
inst✝ : NonUnitalNonAssocSemiring R
S T : Set R
⊢ closure S * closure T ≤ closure (S * T)

pp: case a
R : Type u_2
inst✝ : NonUnitalNonAssocSemiring R
S T : Set R
⊢ closure (S * T) ≤ closure S * closure T

Tactic:
·

Goal state after:
case a
R : Type u_2
inst✝ : NonUnitalNonAssocSemiring R
S T : Set R
⊢ closure S * closure T ≤ closure (S * T)
Module: Mathlib.Algebra.Ring.Submonoid.Pointwise

Goal state before:
pp: case a
R : Type u_2
inst✝ : NonUnitalNonAssocSemiring R
S T : Set R
⊢ closure (S * T) ≤ closure S * closure T

Tactic:
·

Goal state after:
case a
R : Type u_2
inst✝ : NonUnitalNonAssocSemiring R
S T : Set R
⊢ closure (S * T) ≤ closure S * closure T
Module: Mathlib.LinearAlgebra.TensorProduct.Map

Goal state before:
pp: R : Type u_1
inst✝⁶ : CommSemiring R
M : Type u_2
N : Type u_3
P : Type u_4
inst✝⁵ : AddCommGroup M
inst✝⁴ : AddCommMonoid N
inst✝³ : AddCommGroup P
inst✝² : Module R M
inst✝¹ : Module R N
inst✝ : Module R P
f : N →ₗ[R] P
⊢ lTensor M (-f) = -lTensor M f

Tactic:
simp only [← coe_lTensorHom]

Goal state after:
R : Type u_1
inst✝⁶ : CommSemiring R
M : Type u_2
N : Type u_3
P : Type u_4
inst✝⁵ : AddCommGroup M
inst✝⁴ : AddCommMonoid N
inst✝³ : AddCommGroup P
inst✝² : Module R M
inst✝¹ : Module R N
inst✝ : Module R P
f : N →ₗ[R] P
⊢ (lTensorHom M) (-f) = -(lTensorHom M) f
Module: Mathlib.Algebra.Ring.Submonoid.Pointwise

Goal state before:
pp: case refine_1
M : Type u_1
R : Type u_2
A : Type u_3
inst✝ : NonUnitalNonAssocRing R
x y : AddSubmonoid R
m n : R
hn : n ∈ y
hm : -m ∈ x
⊢ -m * n ∈ x * y

pp: case refine_2
M : Type u_1
R : Type u_2
A : Type u_3
inst✝ : NonUnitalNonAssocRing R
x y : AddSubmonoid R
m : R
hm : m ∈ x
n : R
hn : n ∈ y
⊢ -m * n ∈ -x * y

Tactic:
·

Goal state after:
case refine_1
M : Type u_1
R : Type u_2
A : Type u_3
inst✝ : NonUnitalNonAssocRing R
x y : AddSubmonoid R
m n : R
hn : n ∈ y
hm : -m ∈ x
⊢ -m * n ∈ x * y
Module: Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise

Goal state before:
pp: case mp
R : Type u_1
M : Type u_2
inst✝⁴ : Monoid R
inst✝³ : MulAction R M
inst✝² : MulOneClass M
inst✝¹ : IsScalarTower R M M
inst✝ : SMulCommClass R M M
a : SubMulAction R M
x : M
⊢ (∃ y ∈ a, ∃ a, a • y = x) → x ∈ a

pp: case mpr
R : Type u_1
M : Type u_2
inst✝⁴ : Monoid R
inst✝³ : MulAction R M
inst✝² : MulOneClass M
inst✝¹ : IsScalarTower R M M
inst✝ : SMulCommClass R M M
a : SubMulAction R M
x : M
⊢ x ∈ a → ∃ y ∈ a, ∃ a, a • y = x

Tactic:
·

Goal state after:
case mp
R : Type u_1
M : Type u_2
inst✝⁴ : Monoid R
inst✝³ : MulAction R M
inst✝² : MulOneClass M
inst✝¹ : IsScalarTower R M M
inst✝ : SMulCommClass R M M
a : SubMulAction R M
x : M
⊢ (∃ y ∈ a, ∃ a, a • y = x) → x ∈ a
Module: Mathlib.Algebra.Ring.Submonoid.Pointwise

Goal state before:
pp: case refine_2
M : Type u_1
R : Type u_2
A : Type u_3
inst✝ : NonUnitalNonAssocRing R
x y : AddSubmonoid R
m : R
hm : m ∈ x
n : R
hn : n ∈ y
⊢ -m * n ∈ -x * y

Tactic:
·

Goal state after:
case refine_2
M : Type u_1
R : Type u_2
A : Type u_3
inst✝ : NonUnitalNonAssocRing R
x y : AddSubmonoid R
m : R
hm : m ∈ x
n : R
hn : n ∈ y
⊢ -m * n ∈ -x * y
Module: Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise

Goal state before:
pp: case mpr
R : Type u_1
M : Type u_2
inst✝⁴ : Monoid R
inst✝³ : MulAction R M
inst✝² : MulOneClass M
inst✝¹ : IsScalarTower R M M
inst✝ : SMulCommClass R M M
a : SubMulAction R M
x : M
⊢ x ∈ a → ∃ y ∈ a, ∃ a, a • y = x

Tactic:
·

Goal state after:
case mpr
R : Type u_1
M : Type u_2
inst✝⁴ : Monoid R
inst✝³ : MulAction R M
inst✝² : MulOneClass M
inst✝¹ : IsScalarTower R M M
inst✝ : SMulCommClass R M M
a : SubMulAction R M
x : M
⊢ x ∈ a → ∃ y ∈ a, ∃ a, a • y = x
Module: Mathlib.Algebra.Ring.Submonoid.Pointwise

Goal state before:
pp: case refine_1
M : Type u_1
R : Type u_2
A : Type u_3
inst✝ : NonUnitalNonAssocRing R
x y : AddSubmonoid R
m : R
hm : m ∈ x
n : R
hn : -n ∈ y
⊢ m * -n ∈ x * y

pp: case refine_2
M : Type u_1
R : Type u_2
A : Type u_3
inst✝ : NonUnitalNonAssocRing R
x y : AddSubmonoid R
m : R
hm : m ∈ x
n : R
hn : n ∈ y
⊢ m * -n ∈ x * -y

Tactic:
·

Goal state after:
case refine_1
M : Type u_1
R : Type u_2
A : Type u_3
inst✝ : NonUnitalNonAssocRing R
x y : AddSubmonoid R
m : R
hm : m ∈ x
n : R
hn : -n ∈ y
⊢ m * -n ∈ x * y
Module: Mathlib.Algebra.Ring.Submonoid.Pointwise

Goal state before:
pp: case refine_2
M : Type u_1
R : Type u_2
A : Type u_3
inst✝ : NonUnitalNonAssocRing R
x y : AddSubmonoid R
m : R
hm : m ∈ x
n : R
hn : n ∈ y
⊢ m * -n ∈ x * -y

Tactic:
·

Goal state after:
case refine_2
M : Type u_1
R : Type u_2
A : Type u_3
inst✝ : NonUnitalNonAssocRing R
x y : AddSubmonoid R
m : R
hm : m ∈ x
n : R
hn : n ∈ y
⊢ m * -n ∈ x * -y
Module: Mathlib.Algebra.Ring.Submonoid.Pointwise

Goal state before:
pp: M✝ : Type u_1
R : Type u_2
A : Type u_3
inst✝ : NonAssocSemiring R
M : AddSubmonoid R
⊢ 1 * M = M

Tactic:
one_eq_closure_one_set,

Goal state after:
M✝ : Type u_1
R : Type u_2
A : Type u_3
inst✝ : NonAssocSemiring R
M : AddSubmonoid R
⊢ closure 1 * M = M
Module: Mathlib.Algebra.Ring.Submonoid.Pointwise

Goal state before:
pp: M✝ : Type u_1
R : Type u_2
A : Type u_3
inst✝ : NonAssocSemiring R
M : AddSubmonoid R
⊢ M * 1 = M

Tactic:
one_eq_closure_one_set,

Goal state after:
M✝ : Type u_1
R : Type u_2
A : Type u_3
inst✝ : NonAssocSemiring R
M : AddSubmonoid R
⊢ M * closure 1 = M
Module: Mathlib.RingTheory.Coprime.Basic

Goal state before:
pp: case h₁
R : Type u
inst✝ : CommSemiring R
x y z : R
H1 : x ∣ z
H2 : y ∣ z
a b : R
h : a * x + b * y = 1
⊢ x * y ∣ z * (a * x)

pp: case h₂
R : Type u
inst✝ : CommSemiring R
x y z : R
H1 : x ∣ z
H2 : y ∣ z
a b : R
h : a * x + b * y = 1
⊢ x * y ∣ z * (b * y)

Tactic:
·

Goal state after:
case h₁
R : Type u
inst✝ : CommSemiring R
x y z : R
H1 : x ∣ z
H2 : y ∣ z
a b : R
h : a * x + b * y = 1
⊢ x * y ∣ z * (a * x)
Module: Mathlib.RingTheory.Coprime.Basic

Goal state before:
pp: case h₂
R : Type u
inst✝ : CommSemiring R
x y z : R
H1 : x ∣ z
H2 : y ∣ z
a b : R
h : a * x + b * y = 1
⊢ x * y ∣ z * (b * y)

Tactic:
·

Goal state after:
case h₂
R : Type u
inst✝ : CommSemiring R
x y z : R
H1 : x ∣ z
H2 : y ∣ z
a b : R
h : a * x + b * y = 1
⊢ x * y ∣ z * (b * y)
Module: Mathlib.RingTheory.Coprime.Basic

Goal state before:
pp: case refine_1
m n : ℕ
x✝ : IsCoprime m n
a b : ℕ
H : a * m + b * n = 1
⊢ m = 1 ∨ n = 1

pp: case refine_2
m n : ℕ
h : m = 1 ∨ n = 1
⊢ IsCoprime m n

Tactic:
·

Goal state after:
case refine_1
m n : ℕ
x✝ : IsCoprime m n
a b : ℕ
H : a * m + b * n = 1
⊢ m = 1 ∨ n = 1
Module: Mathlib.RingTheory.Coprime.Basic

Goal state before:
pp: case refine_2
m n : ℕ
h : m = 1 ∨ n = 1
⊢ IsCoprime m n

Tactic:
·

Goal state after:
case refine_2
m n : ℕ
h : m = 1 ∨ n = 1
⊢ IsCoprime m n
Module: Mathlib.RingTheory.Coprime.Basic

Goal state before:
pp: case refine_2.inl
n : ℕ
⊢ IsCoprime 1 n

pp: case refine_2.inr
m : ℕ
⊢ IsCoprime m 1

Tactic:
·

Goal state after:
case refine_2.inl
n : ℕ
⊢ IsCoprime 1 n
Module: Mathlib.RingTheory.Coprime.Basic

Goal state before:
pp: case refine_2.inr
m : ℕ
⊢ IsCoprime m 1

Tactic:
·

Goal state after:
case refine_2.inr
m : ℕ
⊢ IsCoprime m 1
Module: Mathlib.RingTheory.Coprime.Basic

Goal state before:
pp: case inl
R : Type u_1
inst✝ : Semifield R
m : R
⊢ IsCoprime m 0 ↔ m ≠ 0 ∨ 0 ≠ 0

pp: case inr
R : Type u_1
inst✝ : Semifield R
m n : R
hn : n ≠ 0
⊢ IsCoprime m n ↔ m ≠ 0 ∨ n ≠ 0

Tactic:
·

Goal state after:
case inl
R : Type u_1
inst✝ : Semifield R
m : R
⊢ IsCoprime m 0 ↔ m ≠ 0 ∨ 0 ≠ 0
Module: Mathlib.RingTheory.Coprime.Lemmas

Goal state before:
pp: case mp
m n : ℤ
⊢ IsCoprime m n → m.gcd n = 1

pp: case mpr
m n : ℤ
⊢ m.gcd n = 1 → IsCoprime m n

Tactic:
·

Goal state after:
case mp
m n : ℤ
⊢ IsCoprime m n → m.gcd n = 1
Module: Mathlib.RingTheory.Coprime.Lemmas

Goal state before:
pp: case mpr
m n : ℤ
⊢ m.gcd n = 1 → IsCoprime m n

Tactic:
·

Goal state after:
case mpr
m n : ℤ
⊢ m.gcd n = 1 → IsCoprime m n
Module: Mathlib.RingTheory.Coprime.Lemmas

Goal state before:
pp: case insert.refine_1
R : Type u
I : Type v
inst✝ : CommSemiring R
z : R
s : I → R
t : Finset I
a : I
r : Finset I
har : a ∉ r
ih : (↑r).Pairwise (IsCoprime on s) → (∀ i ∈ r, s i ∣ z) → ∏ x ∈ r, s x ∣ z
Hs : (↑(insert a r)).Pairwise (IsCoprime on s)
Hs1 : ∀ i ∈ insert a r, s i ∣ z
⊢ IsCoprime (s a) (∏ x ∈ r, s x)

pp: case insert.refine_2
R : Type u
I : Type v
inst✝ : CommSemiring R
z : R
s : I → R
t : Finset I
a : I
r : Finset I
har : a ∉ r
ih : (↑r).Pairwise (IsCoprime on s) → (∀ i ∈ r, s i ∣ z) → ∏ x ∈ r, s x ∣ z
Hs : (↑(insert a r)).Pairwise (IsCoprime on s)
Hs1 : ∀ i ∈ insert a r, s i ∣ z
⊢ s a ∣ z

pp: case insert.refine_3
R : Type u
I : Type v
inst✝ : CommSemiring R
z : R
s : I → R
t : Finset I
a : I
r : Finset I
har : a ∉ r
ih : (↑r).Pairwise (IsCoprime on s) → (∀ i ∈ r, s i ∣ z) → ∏ x ∈ r, s x ∣ z
Hs : (↑(insert a r)).Pairwise (IsCoprime on s)
Hs1 : ∀ i ∈ insert a r, s i ∣ z
⊢ ∏ x ∈ r, s x ∣ z

Tactic:
·

Goal state after:
case insert.refine_1
R : Type u
I : Type v
inst✝ : CommSemiring R
z : R
s : I → R
t : Finset I
a : I
r : Finset I
har : a ∉ r
ih : (↑r).Pairwise (IsCoprime on s) → (∀ i ∈ r, s i ∣ z) → ∏ x ∈ r, s x ∣ z
Hs : (↑(insert a r)).Pairwise (IsCoprime on s)
Hs1 : ∀ i ∈ insert a r, s i ∣ z
⊢ IsCoprime (s a) (∏ x ∈ r, s x)
Module: Mathlib.RingTheory.Coprime.Lemmas

Goal state before:
pp: case insert.refine_2
R : Type u
I : Type v
inst✝ : CommSemiring R
z : R
s : I → R
t : Finset I
a : I
r : Finset I
har : a ∉ r
ih : (↑r).Pairwise (IsCoprime on s) → (∀ i ∈ r, s i ∣ z) → ∏ x ∈ r, s x ∣ z
Hs : (↑(insert a r)).Pairwise (IsCoprime on s)
Hs1 : ∀ i ∈ insert a r, s i ∣ z
⊢ s a ∣ z

pp: case insert.refine_3
R : Type u
I : Type v
inst✝ : CommSemiring R
z : R
s : I → R
t : Finset I
a : I
r : Finset I
har : a ∉ r
ih : (↑r).Pairwise (IsCoprime on s) → (∀ i ∈ r, s i ∣ z) → ∏ x ∈ r, s x ∣ z
Hs : (↑(insert a r)).Pairwise (IsCoprime on s)
Hs1 : ∀ i ∈ insert a r, s i ∣ z
⊢ ∏ x ∈ r, s x ∣ z

Tactic:
·

Goal state after:
case insert.refine_2
R : Type u
I : Type v
inst✝ : CommSemiring R
z : R
s : I → R
t : Finset I
a : I
r : Finset I
har : a ∉ r
ih : (↑r).Pairwise (IsCoprime on s) → (∀ i ∈ r, s i ∣ z) → ∏ x ∈ r, s x ∣ z
Hs : (↑(insert a r)).Pairwise (IsCoprime on s)
Hs1 : ∀ i ∈ insert a r, s i ∣ z
⊢ s a ∣ z
Module: Mathlib.RingTheory.Coprime.Lemmas

Goal state before:
pp: case insert.refine_3
R : Type u
I : Type v
inst✝ : CommSemiring R
z : R
s : I → R
t : Finset I
a : I
r : Finset I
har : a ∉ r
ih : (↑r).Pairwise (IsCoprime on s) → (∀ i ∈ r, s i ∣ z) → ∏ x ∈ r, s x ∣ z
Hs : (↑(insert a r)).Pairwise (IsCoprime on s)
Hs1 : ∀ i ∈ insert a r, s i ∣ z
⊢ ∏ x ∈ r, s x ∣ z

Tactic:
·

Goal state after:
case insert.refine_3
R : Type u
I : Type v
inst✝ : CommSemiring R
z : R
s : I → R
t : Finset I
a : I
r : Finset I
har : a ∉ r
ih : (↑r).Pairwise (IsCoprime on s) → (∀ i ∈ r, s i ∣ z) → ∏ x ∈ r, s x ∣ z
Hs : (↑(insert a r)).Pairwise (IsCoprime on s)
Hs1 : ∀ i ∈ insert a r, s i ∣ z
⊢ ∏ x ∈ r, s x ∣ z
Module: Mathlib.RingTheory.Coprime.Lemmas

Goal state before:
pp: case cons.mp
R : Type u
I : Type v
inst✝¹ : CommSemiring 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)

pp: case cons.mpr
R : Type u
I : Type v
inst✝¹ : CommSemiring 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:
·

Goal state after:
case cons.mp
R : Type u
I : Type v
inst✝¹ : CommSemiring 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)
Module: Mathlib.RingTheory.Coprime.Lemmas

Goal state before:
pp: case cons.mp.refine_1
R : Type u
I : Type v
inst✝¹ : CommSemiring 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

pp: case cons.mp.refine_2
R : Type u
I : Type v
inst✝¹ : CommSemiring 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:
·

Goal state after:
case cons.mp.refine_1
R : Type u
I : Type v
inst✝¹ : CommSemiring 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
Module: Mathlib.Tactic.Order.ToInt

Goal state before:
pp: case pos
α : Type u_1
inst✝ : LinearOrder α
n : ℕ
val : Fin n → α
li : List α := List.ofFn val
sli : List α := li.mergeSort fun a b ↦ decide (a ≤ b)
this : ∀ (i : Fin n), ∃ j, sli[j] = val i
i j : Fin n
h_eq : val i = val j
⊢ val i ≤ val j ↔ ↑⋯.choose ≤ ↑⋯.choose

pp: case neg
α : Type u_1
inst✝ : LinearOrder α
n : ℕ
val : Fin n → α
li : List α := List.ofFn val
sli : List α := li.mergeSort fun a b ↦ decide (a ≤ b)
this : ∀ (i : Fin n), ∃ j, sli[j] = val i
i j : Fin n
h_eq : ¬val i = val j
⊢ val i ≤ val j ↔ ↑⋯.choose ≤ ↑⋯.choose

Tactic:
·

Goal state after:
case pos
α : Type u_1
inst✝ : LinearOrder α
n : ℕ
val : Fin n → α
li : List α := List.ofFn val
sli : List α := li.mergeSort fun a b ↦ decide (a ≤ b)
this : ∀ (i : Fin n), ∃ j, sli[j] = val i
i j : Fin n
h_eq : val i = val j
⊢ val i ≤ val j ↔ ↑⋯.choose ≤ ↑⋯.choose