mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.RingTheory.HahnSeries.Basic

Goal state before:
pp: Γ : Type u_1
R : Type u_3
inst✝¹ : PartialOrder Γ
inst✝ : Zero R
a : Γ
r : R
h : r ≠ 0
⊢ ((single a) r).support = {a}

Tactic:
exact Pi.support_single_of_ne h

Goal state after:
Module: Mathlib.RingTheory.HahnSeries.Basic

Goal state before:
pp: case inr
Γ' : Type u_2
R : Type u_3
inst✝² : Zero R
inst✝¹ : PartialOrder Γ'
Γ : Type u_5
inst✝ : LinearOrder Γ
f : Γ ↪o Γ'
x : R⟦Γ⟧
hx : x ≠ 0
⊢ (embDomain f x).orderTop = WithTop.map (⇑f) x.orderTop

Tactic:
rw [← WithTop.coe_untop x.orderTop (by simpa using hx), WithTop.map_coe]

Goal state after:
case inr
Γ' : Type u_2
R : Type u_3
inst✝² : Zero R
inst✝¹ : PartialOrder Γ'
Γ : Type u_5
inst✝ : LinearOrder Γ
f : Γ ↪o Γ'
x : R⟦Γ⟧
hx : x ≠ 0
⊢ (embDomain f x).orderTop = ↑(f (x.orderTop.untop ⋯))
Module: Mathlib.RingTheory.HahnSeries.Basic

Goal state before:
pp: Γ : Type u_1
R : Type u_3
inst✝³ : Zero R
inst✝² : LinearOrder Γ
inst✝¹ : LocallyFiniteOrder Γ
inst✝ : Zero Γ
f : Γ → R
hf : f ≠ 0
n : Γ
hn : ∀ m < n, f m = 0
⊢ ofSuppBddBelow f ⋯ ≠ 0

Tactic:
simpa

Goal state after:
Module: Mathlib.RingTheory.HahnSeries.Addition

Goal state before:
pp: Γ : Type u_1
R : Type u_3
inst✝² : PartialOrder Γ
inst✝¹ : AddGroup R
inst✝ : Zero Γ
f : R⟦Γ⟧
⊢ (-f).order = f.order

Tactic:
classical
  by_cases hf : f = 0
  · simp only [hf, neg_zero]
  simp only [order, support_neg, neg_eq_zero]

Goal state after:
Module: Mathlib.RingTheory.HahnSeries.Addition

Goal state before:
pp: Γ : Type u_1
R : Type u_3
inst✝¹ : PartialOrder Γ
inst✝ : AddGroup R
x : R⟦Γ⟧
⊢ (-x).leadingCoeff = -x.leadingCoeff

Tactic:
by
  obtain rfl | hx := eq_or_ne x 0 <;> simp [leadingCoeff_of_ne_zero, *]

Goal state after:
Module: Mathlib.RingTheory.HahnSeries.Lex

Goal state before:
pp: Γ : Type u_1
R : Type u_2
inst✝² : LinearOrder Γ
inst✝¹ : Zero R
inst✝ : LinearOrder R
x : Lex R⟦Γ⟧
hpos : 0 < (ofLex x).leadingCoeff
hne : ofLex x ≠ 0
htop : (ofLex x).orderTop ≠ ⊤
⊢ (ofLex 0).coeff ((ofLex x).orderTop.untop htop) < (ofLex x).coeff ((ofLex x).orderTop.untop htop)

Tactic:
by simpa [coeff_untop_eq_leadingCoeff] using hpos

Goal state after:
Module: Mathlib.RingTheory.HahnSeries.Lex

Goal state before:
pp: Γ : Type u_1
R : Type u_2
inst✝³ : LinearOrder Γ
inst✝² : LinearOrder R
inst✝¹ : AddCommGroup R
inst✝ : IsOrderedAddMonoid R
x y : Lex R⟦Γ⟧
h : (ofLex y).orderTop < (ofLex x).orderTop
⊢ |x| < |y|

Tactic:
rw [← orderTop_abs x, ← orderTop_abs y] at h

Goal state after:
Γ : Type u_1
R : Type u_2
inst✝³ : LinearOrder Γ
inst✝² : LinearOrder R
inst✝¹ : AddCommGroup R
inst✝ : IsOrderedAddMonoid R
x y : Lex R⟦Γ⟧
h : (ofLex |y|).orderTop < (ofLex |x|).orderTop
⊢ |x| < |y|
Module: Mathlib.Algebra.Order.Monoid.LocallyFiniteOrder

Goal state before:
pp: M : Type u_1
G : Type u_2
inst✝⁷ : AddCancelCommMonoid M
inst✝⁶ : LinearOrder M
inst✝⁵ : IsOrderedAddMonoid M
inst✝⁴ : LocallyFiniteOrder M
inst✝³ : AddCommGroup G
inst✝² : LinearOrder G
inst✝¹ : IsOrderedAddMonoid G
inst✝ : LocallyFiniteOrder G
a b : G
hab : 0 ≤ b
this : 0 ≤ (addMonoidHom G) b
⊢ (↑(addMonoidHom G)).toFun a ≤ (↑(addMonoidHom G)).toFun ((fun x ↦ a + x) b)

Tactic:
simpa

Goal state after:
Module: Mathlib.Algebra.Order.Monoid.LocallyFiniteOrder

Goal state before:
pp: M : Type u_1
G : Type u_2
inst✝⁸ : AddCancelCommMonoid M
inst✝⁷ : LinearOrder M
inst✝⁶ : IsOrderedAddMonoid M
inst✝⁵ : LocallyFiniteOrder M
inst✝⁴ : AddCommGroup G
inst✝³ : LinearOrder G
inst✝² : IsOrderedAddMonoid G
inst✝¹ : LocallyFiniteOrder G
inst✝ : Nontrivial G
a b : G
this : 0 ≤ (orderAddMonoidHom G) b ↔ 0 ≤ b
⊢ (↑(orderAddMonoidHom G).toAddMonoidHom).toFun a ≤ (↑(orderAddMonoidHom G).toAddMonoidHom).toFun ((fun x ↦ a + x) b) ↔
    a ≤ (fun x ↦ a + x) b

Tactic:
simpa

Goal state after:
Module: Mathlib.RingTheory.HahnSeries.Multiplication

Goal state before:
pp: Γ : Type u_1
Γ' : Type u_2
R : Type u_3
V : Type u_5
inst✝⁶ : PartialOrder Γ
inst✝⁵ : PartialOrder Γ'
inst✝⁴ : VAdd Γ Γ'
inst✝³ : IsOrderedCancelVAdd Γ Γ'
inst✝² : AddCommMonoid V
inst✝¹ : Zero R
inst✝ : DistribSMul R V
x : R⟦Γ⟧
y z : HahnModule Γ' R V
k : Γ'
hwf : (((of R).symm y).support ∪ ((of R).symm z).support).IsPWO
⊢ ((of R).symm (x • (y + z))).coeff k = ((of R).symm (x • y + x • z)).coeff k

Tactic:
rw [coeff_smul_right hwf, of_symm_add]

Goal state after:
Γ : Type u_1
Γ' : Type u_2
R : Type u_3
V : Type u_5
inst✝⁶ : PartialOrder Γ
inst✝⁵ : PartialOrder Γ'
inst✝⁴ : VAdd Γ Γ'
inst✝³ : IsOrderedCancelVAdd Γ Γ'
inst✝² : AddCommMonoid V
inst✝¹ : Zero R
inst✝ : DistribSMul R V
x : R⟦Γ⟧
y z : HahnModule Γ' R V
k : Γ'
hwf : (((of R).symm y).support ∪ ((of R).symm z).support).IsPWO
⊢ ∑ ij ∈ VAddAntidiagonal k ⋯, x.coeff ij.1 • ((of R).symm y + (of R).symm z).coeff ij.2 =
    ((of R).symm (x • y + x • z)).coeff k

Γ : Type u_1
Γ' : Type u_2
R : Type u_3
V : Type u_5
inst✝⁶ : PartialOrder Γ
inst✝⁵ : PartialOrder Γ'
inst✝⁴ : VAdd Γ Γ'
inst✝³ : IsOrderedCancelVAdd Γ Γ'
inst✝² : AddCommMonoid V
inst✝¹ : Zero R
inst✝ : DistribSMul R V
x : R⟦Γ⟧
y z : HahnModule Γ' R V
k : Γ'
hwf : (((of R).symm y).support ∪ ((of R).symm z).support).IsPWO
⊢ ((of R).symm (y + z)).support ⊆ ((of R).symm y).support ∪ ((of R).symm z).support
Module: Mathlib.RingTheory.HahnSeries.Lex

Goal state before:
pp: case ha
Γ : Type u_1
R : Type u_2
inst✝⁶ : LinearOrder Γ
inst✝⁵ : LinearOrder R
inst✝⁴ : Ring R
inst✝³ : AddCommMonoid Γ
inst✝² : IsOrderedCancelAddMonoid Γ
inst✝¹ : IsOrderedRing R
inst✝ : NoZeroDivisors R
a : Lex R⟦Γ⟧
ha : 0 ≤ a
b c : Lex R⟦Γ⟧
hbc : 0 ≤ c - b
⊢ 0 ≤ (ofLex a).leadingCoeff

Tactic:
simpa

Goal state after:
Module: Mathlib.RingTheory.HahnSeries.Lex

Goal state before:
pp: case hb
Γ : Type u_1
R : Type u_2
inst✝⁶ : LinearOrder Γ
inst✝⁵ : LinearOrder R
inst✝⁴ : Ring R
inst✝³ : AddCommMonoid Γ
inst✝² : IsOrderedCancelAddMonoid Γ
inst✝¹ : IsOrderedRing R
inst✝ : NoZeroDivisors R
a : Lex R⟦Γ⟧
ha : 0 ≤ a
b c : Lex R⟦Γ⟧
hbc : 0 ≤ c - b
⊢ 0 ≤ (ofLex a).leadingCoeff

Tactic:
simpa

Goal state after:
Module: Mathlib.Algebra.Order.Module.HahnEmbedding

Goal state before:
pp: K : Type u_1
inst✝¹¹ : DivisionRing K
inst✝¹⁰ : LinearOrder K
inst✝⁹ : IsOrderedRing K
inst✝⁸ : Archimedean K
M : Type u_2
inst✝⁷ : AddCommGroup M
inst✝⁶ : LinearOrder M
inst✝⁵ : IsOrderedAddMonoid M
inst✝⁴ : Module K M
inst✝³ : IsOrderedModule K M
R : Type u_3
inst✝² : AddCommGroup R
inst✝¹ : LinearOrder R
inst✝ : Module K R
seed : Seed K M R
x : ↥seed.baseEmbedding.domain
hx : 0 < x
hmem : ↑x ∈ seed.baseDomain
f : Π₀ (i : FiniteArchimedeanClass M), ↥(seed.stratum i)
hf : (f.sum fun x xi ↦ ↑xi) = ↑x
hfpos : 0 < f.sum fun x x_1 ↦ ↑x_1
hsupport : f.support.Nonempty
j : FiniteArchimedeanClass M
hj : j ∈ f.support
⊢ ↑(f.support.min' hsupport) ≤ ↑j

Tactic:
Subtype.coe_mk

Goal state after:
K : Type u_1
inst✝¹¹ : DivisionRing K
inst✝¹⁰ : LinearOrder K
inst✝⁹ : IsOrderedRing K
inst✝⁸ : Archimedean K
M : Type u_2
inst✝⁷ : AddCommGroup M
inst✝⁶ : LinearOrder M
inst✝⁵ : IsOrderedAddMonoid M
inst✝⁴ : Module K M
inst✝³ : IsOrderedModule K M
R : Type u_3
inst✝² : AddCommGroup R
inst✝¹ : LinearOrder R
inst✝ : Module K R
seed : Seed K M R
x : ↥seed.baseEmbedding.domain
hx : 0 < x
hmem : ↑x ∈ seed.baseDomain
f : Π₀ (i : FiniteArchimedeanClass M), ↥(seed.stratum i)
hf : (f.sum fun x xi ↦ ↑xi) = ↑x
hfpos : 0 < f.sum fun x x_1 ↦ ↑x_1
hsupport : f.support.Nonempty
j : FiniteArchimedeanClass M
hj : j ∈ f.support
⊢ ↑(f.support.min' hsupport) ≤ ↑j
Module: Mathlib.Algebra.Order.Ring.IsNonarchimedean

Goal state before:
pp: case succ
R : Type u_1
inst✝⁵ : Semiring R
inst✝⁴ : LinearOrder R
F : Type u_2
α : Type u_3
inst✝³ : AddMonoid α
inst✝² : FunLike F α R
inst✝¹ : ZeroHomClass F α R
inst✝ : NonnegHomClass F α R
f : F
hna : IsNonarchimedean ⇑f
a : α
n : ℕ
a✝ : f (n • a) ≤ f a
⊢ max (f (n • a)) (f (1 • a)) ≤ f a

Tactic:
simpa

Goal state after:
Module: Mathlib.Algebra.Order.Ring.IsNonarchimedean

Goal state before:
pp: R : Type u_1
inst✝⁴ : Semiring R
inst✝³ : LinearOrder R
F : Type u_2
α : Type u_3
inst✝² : AddGroup α
inst✝¹ : FunLike F α R
inst✝ : AddGroupSeminormClass F α R
f : F
hna : IsNonarchimedean ⇑f
x y : α
hne : f x ≠ f y
⊢ f (x + y) = max (f x) (f y)

Tactic:
rcases hne.lt_or_gt with h_lt | h_lt

Goal state after:
case inl
R : Type u_1
inst✝⁴ : Semiring R
inst✝³ : LinearOrder R
F : Type u_2
α : Type u_3
inst✝² : AddGroup α
inst✝¹ : FunLike F α R
inst✝ : AddGroupSeminormClass F α R
f : F
hna : IsNonarchimedean ⇑f
x y : α
hne : f x ≠ f y
h_lt : f x < f y
⊢ f (x + y) = max (f x) (f y)

case inr
R : Type u_1
inst✝⁴ : Semiring R
inst✝³ : LinearOrder R
F : Type u_2
α : Type u_3
inst✝² : AddGroup α
inst✝¹ : FunLike F α R
inst✝ : AddGroupSeminormClass F α R
f : F
hna : IsNonarchimedean ⇑f
x y : α
hne : f x ≠ f y
h_lt : f y < f x
⊢ f (x + y) = max (f x) (f y)
Module: Mathlib.Algebra.Order.Ring.IsNonarchimedean

Goal state before:
pp: case inl
R : Type u_1
inst✝⁴ : Semiring R
inst✝³ : LinearOrder R
F : Type u_2
α : Type u_3
inst✝² : AddGroup α
inst✝¹ : FunLike F α R
inst✝ : AddGroupSeminormClass F α R
f : F
hna : IsNonarchimedean ⇑f
x y : α
hne : f x ≠ f y
h_lt : f x < f y
⊢ f (x + y) = max (f x) (f y)

Tactic:
rw [add_eq_right_of_lt hna h_lt]
    exact (max_eq_right_of_lt h_lt).symm

Goal state after:
Module: Mathlib.Algebra.Order.Ring.Archimedean

Goal state before:
pp: R : Type u_1
inst✝⁶ : LinearOrder R
inst✝⁵ : CommRing R
inst✝⁴ : IsStrictOrderedRing R
S : Type u_3
inst✝³ : LinearOrder S
inst✝² : CommRing S
inst✝¹ : IsStrictOrderedRing S
inst✝ : Archimedean S
f : S →+*o R
x : R
y : S
⊢ mk x ≤ mk x + mk (f y)

Tactic:
add_comm

Goal state after:
R : Type u_1
inst✝⁶ : LinearOrder R
inst✝⁵ : CommRing R
inst✝⁴ : IsStrictOrderedRing R
S : Type u_3
inst✝³ : LinearOrder S
inst✝² : CommRing S
inst✝¹ : IsStrictOrderedRing S
inst✝ : Archimedean S
f : S →+*o R
x : R
y : S
⊢ mk x ≤ mk (f y) + mk x
Module: Mathlib.Algebra.Order.Ring.IsNonarchimedean

Goal state before:
pp: R : Type u_1
inst✝⁵ : Semiring R
inst✝⁴ : LinearOrder R
inst✝³ : IsStrictOrderedRing R
F : Type u_2
α : Type u_3
inst✝² : CommRing α
inst✝¹ : FunLike F α R
inst✝ : AddGroupSeminormClass F α R
f : F
hf_na : IsNonarchimedean ⇑f
s : Multiset α
m : ℕ
g : Multiset α → α := fun t ↦ t.prod
b : Multiset α
hb_in : powersetCard (s.card - m) s ≠ 0 → b ∈ powersetCard (s.card - m) s
hb_le : f (map g (powersetCard (s.card - m) s)).sum ≤ f (g b)
⊢ b ≤ s ∧ b.card = s.card - m

Tactic:
rw [← mem_powersetCard]

Goal state after:
R : Type u_1
inst✝⁵ : Semiring R
inst✝⁴ : LinearOrder R
inst✝³ : IsStrictOrderedRing R
F : Type u_2
α : Type u_3
inst✝² : CommRing α
inst✝¹ : FunLike F α R
inst✝ : AddGroupSeminormClass F α R
f : F
hf_na : IsNonarchimedean ⇑f
s : Multiset α
m : ℕ
g : Multiset α → α := fun t ↦ t.prod
b : Multiset α
hb_in : powersetCard (s.card - m) s ≠ 0 → b ∈ powersetCard (s.card - m) s
hb_le : f (map g (powersetCard (s.card - m) s)).sum ≤ f (g b)
⊢ b ∈ powersetCard (s.card - m) s
Module: Mathlib.RingTheory.HahnSeries.Multiplication

Goal state before:
pp: Γ : Type u_1
Γ' : Type u_2
R : Type u_3
S : Type u_4
V : Type u_5
inst✝³ : AddCommMonoid Γ
inst✝² : PartialOrder Γ
inst✝¹ : IsOrderedCancelAddMonoid Γ
inst✝ : NonUnitalNonAssocSemiring R
x✝ : R⟦Γ⟧
⊢ 0 * x✝ = 0

Tactic:
ext
    simp [coeff_mul]

Goal state after:
Module: Mathlib.RingTheory.HahnSeries.Multiplication

Goal state before:
pp: Γ : Type u_1
Γ' : Type u_2
R : Type u_3
S : Type u_4
V : Type u_5
inst✝³ : AddCommMonoid Γ
inst✝² : PartialOrder Γ
inst✝¹ : IsOrderedCancelAddMonoid Γ
inst✝ : NonUnitalNonAssocSemiring R
x✝ : R⟦Γ⟧
⊢ x✝ * 0 = 0

Tactic:
ext
    simp [coeff_mul]

Goal state after:
Module: Mathlib.RingTheory.HahnSeries.Multiplication

Goal state before:
pp: case inr
Γ : Type u_1
R : Type u_3
inst✝³ : AddCommMonoid Γ
inst✝² : LinearOrder Γ
inst✝¹ : IsOrderedCancelAddMonoid Γ
inst✝ : NonUnitalNonAssocSemiring R
g : Γ
r : R
hr : IsRegular r
x : R⟦Γ⟧
hx : x ≠ 0
h✝ : Nontrivial R
⊢ ((single g) r * x).order = g + x.order

Tactic:
have hrx : ((single g) r).leadingCoeff * x.leadingCoeff ≠ 0 := by
    rwa [leadingCoeff_of_single, ne_eq, hr.left.mul_left_eq_zero_iff, leadingCoeff_eq_zero]

Goal state after:
case inr
Γ : Type u_1
R : Type u_3
inst✝³ : AddCommMonoid Γ
inst✝² : LinearOrder Γ
inst✝¹ : IsOrderedCancelAddMonoid Γ
inst✝ : NonUnitalNonAssocSemiring R
g : Γ
r : R
hr : IsRegular r
x : R⟦Γ⟧
hx : x ≠ 0
h✝ : Nontrivial R
hrx : ((single g) r).leadingCoeff * x.leadingCoeff ≠ 0
⊢ ((single g) r * x).order = g + x.order
Module: Mathlib.RingTheory.HahnSeries.Multiplication

Goal state before:
pp: Γ : Type u_1
Γ' : Type u_2
R : Type u_3
V : Type u_5
inst✝⁸ : AddCommMonoid Γ
inst✝⁷ : PartialOrder Γ
inst✝⁶ : IsOrderedCancelAddMonoid Γ
inst✝⁵ : PartialOrder Γ'
inst✝⁴ : AddAction Γ Γ'
inst✝³ : IsOrderedCancelVAdd Γ Γ'
inst✝² : AddCommMonoid V
inst✝¹ : Semiring R
inst✝ : Module R V
x y : R⟦Γ⟧
z : HahnModule Γ' R V
b : Γ'
⊢ ∑ ij ∈ VAddAntidiagonal b ⋯, (x * y).coeff ij.1 • ((of R).symm z).coeff ij.2 = ((of R).symm (x • y • z)).coeff b

Tactic:
coeff_smul_right
    (y.isPWO_support.vadd ((of R).symm z).isPWO_support) support_smul_subset_vadd_support

Goal state after:
Γ : Type u_1
Γ' : Type u_2
R : Type u_3
V : Type u_5
inst✝⁸ : AddCommMonoid Γ
inst✝⁷ : PartialOrder Γ
inst✝⁶ : IsOrderedCancelAddMonoid Γ
inst✝⁵ : PartialOrder Γ'
inst✝⁴ : AddAction Γ Γ'
inst✝³ : IsOrderedCancelVAdd Γ Γ'
inst✝² : AddCommMonoid V
inst✝¹ : Semiring R
inst✝ : Module R V
x y : R⟦Γ⟧
z : HahnModule Γ' R V
b : Γ'
⊢ ∑ ij ∈ VAddAntidiagonal b ⋯, (x * y).coeff ij.1 • ((of R).symm z).coeff ij.2 =
    ∑ ij ∈ VAddAntidiagonal b ⋯, x.coeff ij.1 • ((of R).symm (y • z)).coeff ij.2
Module: Mathlib.Order.Quotient

Goal state before:
pp: case inl.inr
α : Type u_1
s : Setoid α
inst✝ : LinearOrder α
H : ∀ (x : Quotient s), (Quotient.mk s ⁻¹' {x}).OrdConnected
x y z : α
h₁ : x ≤ y
h₂ : y ≈ z
⊢ x ≤ z ∨ x ≈ z

Tactic:
or_iff_not_imp_left,

Goal state after:
case inl.inr
α : Type u_1
s : Setoid α
inst✝ : LinearOrder α
H : ∀ (x : Quotient s), (Quotient.mk s ⁻¹' {x}).OrdConnected
x y z : α
h₁ : x ≤ y
h₂ : y ≈ z
⊢ ¬x ≤ z → x ≈ z
Module: Mathlib.Order.Quotient

Goal state before:
pp: case inr.inl
α : Type u_1
s : Setoid α
inst✝ : LinearOrder α
H : ∀ (x : Quotient s), (Quotient.mk s ⁻¹' {x}).OrdConnected
x y z : α
h₁ : x ≈ y
h₂ : y ≤ z
⊢ x ≤ z ∨ x ≈ z

Tactic:
or_iff_not_imp_left,

Goal state after:
case inr.inl
α : Type u_1
s : Setoid α
inst✝ : LinearOrder α
H : ∀ (x : Quotient s), (Quotient.mk s ⁻¹' {x}).OrdConnected
x y z : α
h₁ : x ≈ y
h₂ : y ≤ z
⊢ ¬x ≤ z → x ≈ z
Module: Mathlib.RingTheory.HahnSeries.Multiplication

Goal state before:
pp: case a
Γ : Type u_1
R : Type u_3
inst✝⁴ : AddCommMonoid Γ
inst✝³ : LinearOrder Γ
inst✝² : IsOrderedCancelAddMonoid Γ
inst✝¹ : NonUnitalNonAssocSemiring R
inst✝ : NoZeroDivisors R
x y : R⟦Γ⟧
hx : x ≠ 0
hy : y ≠ 0
⊢ (x * y).order ≤ x.order + y.order

Tactic:
apply order_le_of_coeff_ne_zero

Goal state after:
case a
Γ : Type u_1
R : Type u_3
inst✝⁴ : AddCommMonoid Γ
inst✝³ : LinearOrder Γ
inst✝² : IsOrderedCancelAddMonoid Γ
inst✝¹ : NonUnitalNonAssocSemiring R
inst✝ : NoZeroDivisors R
x y : R⟦Γ⟧
hx : x ≠ 0
hy : y ≠ 0
⊢ (x * y).coeff (x.order + y.order) ≠ 0
Module: Mathlib.Algebra.Order.Module.HahnEmbedding

Goal state before:
pp: K : Type u_1
inst✝¹³ : DivisionRing K
inst✝¹² : LinearOrder K
inst✝¹¹ : IsOrderedRing K
inst✝¹⁰ : Archimedean K
M : Type u_2
inst✝⁹ : AddCommGroup M
inst✝⁸ : LinearOrder M
inst✝⁷ : IsOrderedAddMonoid M
inst✝⁶ : Module K M
inst✝⁵ : IsOrderedModule K M
R : Type u_3
inst✝⁴ : AddCommGroup R
inst✝³ : LinearOrder R
inst✝² : Module K R
seed : Seed K M R
f : Partial seed
inst✝¹ : IsOrderedAddMonoid R
inst✝ : Archimedean R
x : M
hx : x ∉ (↑f).domain
y : ↥(↑f).domain
this : x - ↑y ≠ 0
xy : FiniteArchimedeanClass M := FiniteArchimedeanClass.mk (x - ↑y) this
u : M
hu : ∀ (h : u ≠ 0), xy < FiniteArchimedeanClass.mk u h
v : M
hv : v ∈ seed.stratum xy
huv : u + v = x - ↑y
huv' : x - ↑y - v = u
hyv : ↑y + v ∈ (↑f).domain
h✝ : f.eval x = ↑↑f y
h : True
⊢ ArchimedeanClass.mk (x - ↑y) < ArchimedeanClass.mk (x - ↑⟨↑y + v, hyv⟩)

Tactic:
simp_rw [← sub_sub, huv']

Goal state after:
K : Type u_1
inst✝¹³ : DivisionRing K
inst✝¹² : LinearOrder K
inst✝¹¹ : IsOrderedRing K
inst✝¹⁰ : Archimedean K
M : Type u_2
inst✝⁹ : AddCommGroup M
inst✝⁸ : LinearOrder M
inst✝⁷ : IsOrderedAddMonoid M
inst✝⁶ : Module K M
inst✝⁵ : IsOrderedModule K M
R : Type u_3
inst✝⁴ : AddCommGroup R
inst✝³ : LinearOrder R
inst✝² : Module K R
seed : Seed K M R
f : Partial seed
inst✝¹ : IsOrderedAddMonoid R
inst✝ : Archimedean R
x : M
hx : x ∉ (↑f).domain
y : ↥(↑f).domain
this : x - ↑y ≠ 0
xy : FiniteArchimedeanClass M := FiniteArchimedeanClass.mk (x - ↑y) this
u : M
hu : ∀ (h : u ≠ 0), xy < FiniteArchimedeanClass.mk u h
v : M
hv : v ∈ seed.stratum xy
huv : u + v = x - ↑y
huv' : x - ↑y - v = u
hyv : ↑y + v ∈ (↑f).domain
h✝ : f.eval x = ↑↑f y
h : True
⊢ ArchimedeanClass.mk (x - ↑y) < ArchimedeanClass.mk u
Module: Mathlib.Algebra.Order.Module.HahnEmbedding

Goal state before:
pp: K : Type u_1
inst✝¹³ : DivisionRing K
inst✝¹² : LinearOrder K
inst✝¹¹ : IsOrderedRing K
inst✝¹⁰ : Archimedean K
M : Type u_2
inst✝⁹ : AddCommGroup M
inst✝⁸ : LinearOrder M
inst✝⁷ : IsOrderedAddMonoid M
inst✝⁶ : Module K M
inst✝⁵ : IsOrderedModule K M
R : Type u_3
inst✝⁴ : AddCommGroup R
inst✝³ : LinearOrder R
inst✝² : Module K R
seed : Seed K M R
f : Partial seed
inst✝¹ : IsOrderedAddMonoid R
inst✝ : Archimedean R
x : M
hx : x ∉ (↑f).domain
y : ↥(↑f).domain
this : x - ↑y ≠ 0
xy : FiniteArchimedeanClass M := FiniteArchimedeanClass.mk (x - ↑y) this
v : M
hv : v ∈ seed.stratum xy
hyv : ↑y + v ∈ (↑f).domain
h✝ : f.eval x = ↑↑f y
h : True
hu : ∀ (h : 0 ≠ 0), xy < FiniteArchimedeanClass.mk 0 h
huv : 0 + v = x - ↑y
huv' : x - ↑y - v = 0
⊢ ArchimedeanClass.mk (x - ↑y) ≠ ⊤

Tactic:
simpa

Goal state after:
Module: Mathlib.Algebra.Order.Sub.Unbundled.Hom

Goal state before:
pp: M : Type u_3
N : Type u_4
inst✝⁷ : Preorder M
inst✝⁶ : Add M
inst✝⁵ : Sub M
inst✝⁴ : OrderedSub M
inst✝³ : PartialOrder N
inst✝² : Add N
inst✝¹ : Sub N
inst✝ : OrderedSub N
e : M ≃o N
h_add : ∀ (a b : M), e (a + b) = e a + e b
a b : M
e_add : M ≃+ N := { toEquiv := e.toEquiv, map_add' := h_add }
this : e (e.symm (e a) - e.symm (e b)) ≤ e (e.symm (e a - e b))
⊢ e (a - b) ≤ e a - e b

Tactic:
simpa

Goal state after:
Module: Mathlib.Algebra.Order.Ring.StandardPart

Goal state before:
pp: K : Type u_1
inst✝⁶ : LinearOrder K
inst✝⁵ : Field K
inst✝⁴ : IsOrderedRing K
x✝ y✝ : K
R : Type u_2
inst✝³ : LinearOrder R
inst✝² : CommRing R
inst✝¹ : IsStrictOrderedRing R
inst✝ : Archimedean R
x y : FiniteElement K
hy : 0 < mk ↑(y - x)
z : FiniteElement K
hxz : x ≤ z
hzy : z ≤ y
this : x ≤ y
⊢ ↑(z - x) ∈ Set.Ici 0

Tactic:
simpa

Goal state after:
Module: Mathlib.Algebra.Order.Ring.StandardPart

Goal state before:
pp: K : Type u_1
inst✝⁶ : LinearOrder K
inst✝⁵ : Field K
inst✝⁴ : IsOrderedRing K
x✝ y✝ : K
R : Type u_2
inst✝³ : LinearOrder R
inst✝² : CommRing R
inst✝¹ : IsStrictOrderedRing R
inst✝ : Archimedean R
x y : FiniteElement K
hy : 0 < mk ↑(y - x)
z : FiniteElement K
hxz : x ≤ z
hzy : z ≤ y
this : x ≤ y
⊢ ↑(y - x) ∈ Set.Ici 0

Tactic:
simpa

Goal state after:
Module: Mathlib.Algebra.Order.Ring.StandardPart

Goal state before:
pp: K : Type u_1
inst✝⁶ : LinearOrder K
inst✝⁵ : Field K
inst✝⁴ : IsOrderedRing K
x✝ y✝ : K
R : Type u_2
inst✝³ : LinearOrder R
inst✝² : CommRing R
inst✝¹ : IsStrictOrderedRing R
inst✝ : Archimedean R
x y : FiniteElement K
hy : 0 < mk ↑(y - x)
z : FiniteElement K
hxz : x ≤ z
hzy : z ≤ y
this : x ≤ y
⊢ ↑(z - x) ≤ ↑(y - x)

Tactic:
simpa

Goal state after:
Module: Mathlib.Algebra.Order.Ring.StandardPart

Goal state before:
pp: K : Type u_1
inst✝² : LinearOrder K
inst✝¹ : Field K
inst✝ : IsOrderedRing K
x y : FiniteElement K
⊢ mk x ≤ mk y ↔ x ≤ y ∨ mk x = mk y

Tactic:
refine (Quotient.mk_le_mk (H := ordConnected_preimage_mk')).trans ?_
  rw [← Quotient.eq_iff_equiv]
  rfl

Goal state after:
Module: Mathlib.Algebra.Order.Ring.StandardPart

Goal state before:
pp: K : Type u_1
inst✝² : LinearOrder K
inst✝¹ : Field K
inst✝ : IsOrderedRing K
x y z : FiniteResidueField K
h : x ≤ y
hz : 0 ≤ z
⊢ z * x ≤ z * y

Tactic:
induction x with | mk x
  induction y with | mk y
  induction z with | mk z
  rw [← map_mul, ← map_mul]
  rw [← map_zero mk] at hz
  rw [mk_le_mk] at h hz ⊢
  grind [mul_le_mul_of_nonneg_left]

Goal state after:
Module: Mathlib.Algebra.Pointwise.Stabilizer

Goal state before:
pp: case refine_2
G : Type u_1
α : Type u_3
inst✝¹ : Group G
inst✝ : MulAction G α
a : G
s : Set α
h : ∀ (b : α), a • b ∈ s ↔ b ∈ s
⊢ a • s = s

Tactic:
simp_rw [Set.ext_iff, mem_smul_set_iff_inv_smul_mem]

Goal state after:
case refine_2
G : Type u_1
α : Type u_3
inst✝¹ : Group G
inst✝ : MulAction G α
a : G
s : Set α
h : ∀ (b : α), a • b ∈ s ↔ b ∈ s
⊢ ∀ (x : α), a⁻¹ • x ∈ s ↔ x ∈ s
Module: Mathlib.Algebra.Pointwise.Stabilizer

Goal state before:
pp: G : Type u_1
α : Type u_3
inst✝¹ : Group G
inst✝ : MulAction G α
s t : Set α
hdisj : Disjoint s t
hstab : stabilizer G s ≤ stabilizer G t
hstab_union : stabilizer G (s ∪ t) ≤ stabilizer G t
⊢ stabilizer G (s ∪ t) ≤ stabilizer G (s ∪ t) ⊓ stabilizer G t

Tactic:
simpa

Goal state after:
Module: Mathlib.Algebra.Pointwise.Stabilizer

Goal state before:
pp: G : Type u_1
α : Type u_3
inst✝¹ : Group G
inst✝ : MulAction G α
s t : Set α
hdisj : Disjoint s t
hstab : stabilizer G s ≤ stabilizer G t
hstab_union : stabilizer G (s ∪ t) ≤ stabilizer G t
⊢ stabilizer G s ≤ stabilizer G s ⊓ stabilizer G t

Tactic:
simpa

Goal state after:
Module: Mathlib.Algebra.Pointwise.Stabilizer

Goal state before:
pp: G : Type u_1
α : Type u_3
inst✝² : Group G
inst✝¹ : MulAction G α
a : G
inst✝ : DecidableEq α
s : Finset α
⊢ a ∈ stabilizer G s ↔ ∀ (b : α), a • b ∈ s ↔ b ∈ s

Tactic:
by
  simp_rw [← stabilizer_coe_finset, mem_stabilizer_set, Finset.mem_coe]

Goal state after:
Module: Mathlib.Algebra.Pointwise.Stabilizer

Goal state before:
pp: G : Type u_1
inst✝ : CommGroup G
s : Set G
a : G ⧸ stabilizer G s
⊢ a ∈ stabilizer (G ⧸ stabilizer G s) (QuotientGroup.mk '' s) ↔ a ∈ ⊥

Tactic:
induction a using QuotientGroup.induction_on with | _ a

Goal state after:
case H
G : Type u_1
inst✝ : CommGroup G
s : Set G
a : G
⊢ ↑a ∈ stabilizer (G ⧸ stabilizer G s) (QuotientGroup.mk '' s) ↔ ↑a ∈ ⊥
Module: Mathlib.Algebra.Order.Ring.StandardPart

Goal state before:
pp: K : Type u_1
inst✝² : LinearOrder K
inst✝¹ : Field K
inst✝ : IsOrderedRing K
x y : K
hy : 0 < mk y
⊢ 0 < mk (-y)

Tactic:
simpa

Goal state after:
Module: Mathlib.Algebra.Order.Ring.StandardPart

Goal state before:
pp: case hx
K : Type u_1
inst✝² : LinearOrder K
inst✝¹ : Field K
inst✝ : IsOrderedRing K
x : K
f : ℝ →+*o K
r : ℝ
hx : 0 ≤ mk x
h : stdPart x < r
⊢ 0 ≤ mk (-x)

Tactic:
simpa

Goal state after:
Module: Mathlib.Algebra.Order.Ring.StandardPart

Goal state before:
pp: case h
K : Type u_1
inst✝² : LinearOrder K
inst✝¹ : Field K
inst✝ : IsOrderedRing K
x : K
f : ℝ →+*o K
r : ℝ
hx : 0 ≤ mk x
h : stdPart x < r
⊢ -r < stdPart (-x)

Tactic:
simpa

Goal state after:
Module: Mathlib.Algebra.Order.Ring.StandardPart

Goal state before:
pp: case inl
K : Type u_1
inst✝² : LinearOrder K
inst✝¹ : Field K
inst✝ : IsOrderedRing K
x : K
f : ℝ →+*o K
r : ℝ
hl : ∀ s < r, f s ≤ x
hr : ∀ s > r, x ≤ f s
hx : 0 ≤ mk x
h : stdPart x < r
⊢ stdPart x = r

pp: case inr.inl
K : Type u_1
inst✝² : LinearOrder K
inst✝¹ : Field K
inst✝ : IsOrderedRing K
x : K
f : ℝ →+*o K
hx : 0 ≤ mk x
hl : ∀ s < stdPart x, f s ≤ x
hr : ∀ s > stdPart x, x ≤ f s
⊢ stdPart x = stdPart x

pp: case inr.inr
K : Type u_1
inst✝² : LinearOrder K
inst✝¹ : Field K
inst✝ : IsOrderedRing K
x : K
f : ℝ →+*o K
r : ℝ
hl : ∀ s < r, f s ≤ x
hr : ∀ s > r, x ≤ f s
hx : 0 ≤ mk x
h : r < stdPart x
⊢ stdPart x = r

Tactic:
· obtain ⟨s, hs, hs'⟩ := exists_between h
    cases (le_stdPart_of_le f hx (hl _ hs')).not_gt hs

Goal state after:
case inr.inl
K : Type u_1
inst✝² : LinearOrder K
inst✝¹ : Field K
inst✝ : IsOrderedRing K
x : K
f : ℝ →+*o K
hx : 0 ≤ mk x
hl : ∀ s < stdPart x, f s ≤ x
hr : ∀ s > stdPart x, x ≤ f s
⊢ stdPart x = stdPart x

case inr.inr
K : Type u_1
inst✝² : LinearOrder K
inst✝¹ : Field K
inst✝ : IsOrderedRing K
x : K
f : ℝ →+*o K
r : ℝ
hl : ∀ s < r, f s ≤ x
hr : ∀ s > r, x ≤ f s
hx : 0 ≤ mk x
h : r < stdPart x
⊢ stdPart x = r