mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Order.Antichain

Goal state before:
pp: α : Type u_1
β : Type u_2
r : α → α → Prop
r' : β → β → Prop
s : Set α
hs : IsAntichain r s
φ : r ↪r r'
a : α
has : a ∈ s
a' : α
has' : a' ∈ s
h₁ : φ a ≠ φ a'
h₂ : r' (φ a) (φ a')
haa' : a = a'
⊢ φ a' = φ a'

Tactic:
]

Goal state after:
Module: Mathlib.Order.Preorder.Chain

Goal state before:
pp: α : Type u_1
r : α → α → Prop
s : Set α
hs : IsChain r s
h : ¬(IsChain r s ∧ ∃ x, SuperChain r s x)
⊢ IsChain r (SuccChain r s)

Tactic:
simpa [SuccChain, dif_neg, h] using hs

Goal state after:
Module: Mathlib.Order.Interval.Set.OrdConnected

Goal state before:
pp: α : Type u_1
inst✝ : PartialOrder α
s : Set α
hs : IsAntichain (fun x1 x2 ↦ x1 ≤ x2) s
x : α
hx : x ∈ s
z : α
hy : x ∈ s
hz : z = x
⊢ z ∈ s

Tactic:
]

Goal state after:
α : Type u_1
inst✝ : PartialOrder α
s : Set α
hs : IsAntichain (fun x1 x2 ↦ x1 ≤ x2) s
x : α
hx : x ∈ s
z : α
hy : x ∈ s
hz : z = x
⊢ z ∈ s
Module: Mathlib.Order.Interval.Set.OrdConnected

Goal state before:
pp: α : Type u_1
inst✝ : PartialOrder α
s : Set α
hs : IsAntichain (fun x1 x2 ↦ x1 ≤ x2) s
x : α
hx : x ∈ s
z : α
hy : x ∈ s
hz : z = x
⊢ x ∈ s

Tactic:
]

Goal state after:
α : Type u_1
inst✝ : PartialOrder α
s : Set α
hs : IsAntichain (fun x1 x2 ↦ x1 ≤ x2) s
x : α
hx : x ∈ s
z : α
hy : x ∈ s
hz : z = x
⊢ x ∈ s
Module: Mathlib.Order.Antichain

Goal state before:
pp: α : Type u_1
r : α → α → Prop
s t : Set α
⊢ (s.Pairwise rᶜ ∧ t.Pairwise rᶜ ∧ ∀ a ∈ s, ∀ b ∈ t, a ≠ b → rᶜ a b ∧ rᶜ b a) ↔
    s.Pairwise rᶜ ∧ t.Pairwise rᶜ ∧ ∀ a ∈ s, ∀ b ∈ t, a ≠ b → rᶜ a b ∧ rᶜ b a

Tactic:
]

Goal state after:
Module: Mathlib.Order.Antichain

Goal state before:
pp: α : Type u_1
r : α → α → Prop
s : Set α
hs : IsChain r s
ht : IsAntichain r s
⊢ s.Subsingleton

Tactic:
by
  intro x hx y hy
  by_contra! hne
  cases hs hx hy hne with
  | inl h => exact ht hx hy hne h
  | inr h => exact ht hy hx hne.symm h

Goal state after:
Module: Mathlib.Order.Interval.Set.OrdConnected

Goal state before:
pp: α : Type u_1
inst✝ : LinearOrder α
s : Set α
x : α
h : ∀ y ∈ s, [[x, y]] ⊆ s
y : α
hy : y ∈ s
z : α
hz : z ∈ s
⊢ [[x, y]] ∪ [[x, z]] = [[x, y]] ∪ [[x, z]]

Tactic:
]

Goal state after:
Module: Mathlib.Order.Antichain

Goal state before:
pp: α : Type u_1
s t : Set α
inst✝ : Preorder α
ht : IsAntichain (fun x1 x2 ↦ x1 ≤ x2) t
h : ∀ (x : α), Maximal (fun x ↦ x ∈ s) x → x ∈ t
hs : ∀ a ∈ t, ∃ b ≤ a, Maximal (fun x ↦ x ∈ s) b
x : α
hx : x ∈ t
y : α
hyx : y ≤ x
hy : Maximal (fun x ↦ x ∈ s) y
⊢ y ∈ {x | Maximal (fun x ↦ x ∈ s) x}

Tactic:
]

Goal state after:
α : Type u_1
s t : Set α
inst✝ : Preorder α
ht : IsAntichain (fun x1 x2 ↦ x1 ≤ x2) t
h : ∀ (x : α), Maximal (fun x ↦ x ∈ s) x → x ∈ t
hs : ∀ a ∈ t, ∃ b ≤ a, Maximal (fun x ↦ x ∈ s) b
x : α
hx : x ∈ t
y : α
hyx : y ≤ x
hy : Maximal (fun x ↦ x ∈ s) y
⊢ y ∈ {x | Maximal (fun x ↦ x ∈ s) x}
Module: Mathlib.Order.Antichain

Goal state before:
pp: α : Type u_1
β : Type u_2
r : α → α → Prop
s : β → β → Prop
e : r ≃r s
c : Set α
hc : IsMaxAntichain r c
t : Set β
ht : IsAntichain s t
hf : ⇑e '' c ⊆ t
⊢ c = ⇑e.symm '' t

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
r : α → α → Prop
s : β → β → Prop
e : r ≃r s
c : Set α
hc : IsMaxAntichain r c
t : Set β
ht : IsAntichain s t
hf : ⇑e '' c ⊆ t
⊢ c = ⇑e.symm '' t
Module: Mathlib.Algebra.Order.Ring.InjSurj

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁴ : Semiring R
inst✝³ : PartialOrder R
inst✝² : IsOrderedRing R
inst✝¹ : Semiring S
inst✝ : PartialOrder S
f : S → R
zero : f 0 = 0
one : f 1 = 1
add : ∀ (x y : S), f (x + y) = f x + f y
mul : ∀ (x y : S), f (x * y) = f x * f y
le : ∀ {x y : S}, f x ≤ f y ↔ x ≤ y
a : S
ha : 0 ≤ a
b c : S
hbc : b ≤ c
⊢ f a * f b ≤ f a * f c

Tactic:
]

Goal state after:
R : Type u_1
S : Type u_2
inst✝⁴ : Semiring R
inst✝³ : PartialOrder R
inst✝² : IsOrderedRing R
inst✝¹ : Semiring S
inst✝ : PartialOrder S
f : S → R
zero : f 0 = 0
one : f 1 = 1
add : ∀ (x y : S), f (x + y) = f x + f y
mul : ∀ (x y : S), f (x * y) = f x * f y
le : ∀ {x y : S}, f x ≤ f y ↔ x ≤ y
a : S
ha : 0 ≤ a
b c : S
hbc : b ≤ c
⊢ f a * f b ≤ f a * f c
Module: Mathlib.Algebra.Order.Ring.InjSurj

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁴ : Semiring R
inst✝³ : PartialOrder R
inst✝² : IsOrderedRing R
inst✝¹ : Semiring S
inst✝ : PartialOrder S
f : S → R
zero : f 0 = 0
one : f 1 = 1
add : ∀ (x y : S), f (x + y) = f x + f y
mul : ∀ (x y : S), f (x * y) = f x * f y
le : ∀ {x y : S}, f x ≤ f y ↔ x ≤ y
a : S
ha : 0 ≤ a
b c : S
hbc : b ≤ c
⊢ 0 ≤ a

Tactic:
]

Goal state after:
R : Type u_1
S : Type u_2
inst✝⁴ : Semiring R
inst✝³ : PartialOrder R
inst✝² : IsOrderedRing R
inst✝¹ : Semiring S
inst✝ : PartialOrder S
f : S → R
zero : f 0 = 0
one : f 1 = 1
add : ∀ (x y : S), f (x + y) = f x + f y
mul : ∀ (x y : S), f (x * y) = f x * f y
le : ∀ {x y : S}, f x ≤ f y ↔ x ≤ y
a : S
ha : 0 ≤ a
b c : S
hbc : b ≤ c
⊢ 0 ≤ a
Module: Mathlib.Algebra.Order.Ring.InjSurj

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁴ : Semiring R
inst✝³ : PartialOrder R
inst✝² : IsOrderedRing R
inst✝¹ : Semiring S
inst✝ : PartialOrder S
f : S → R
zero : f 0 = 0
one : f 1 = 1
add : ∀ (x y : S), f (x + y) = f x + f y
mul : ∀ (x y : S), f (x * y) = f x * f y
le : ∀ {x y : S}, f x ≤ f y ↔ x ≤ y
a : S
ha : 0 ≤ a
b c : S
hbc : b ≤ c
⊢ f b * f a ≤ f c * f a

Tactic:
]

Goal state after:
R : Type u_1
S : Type u_2
inst✝⁴ : Semiring R
inst✝³ : PartialOrder R
inst✝² : IsOrderedRing R
inst✝¹ : Semiring S
inst✝ : PartialOrder S
f : S → R
zero : f 0 = 0
one : f 1 = 1
add : ∀ (x y : S), f (x + y) = f x + f y
mul : ∀ (x y : S), f (x * y) = f x * f y
le : ∀ {x y : S}, f x ≤ f y ↔ x ≤ y
a : S
ha : 0 ≤ a
b c : S
hbc : b ≤ c
⊢ f b * f a ≤ f c * f a
Module: Mathlib.Algebra.Order.Monoid.Basic

Goal state before:
pp: α : Type u
β : Type u_1
inst✝⁴ : CommMonoid α
inst✝³ : Preorder α
inst✝² : IsOrderedMonoid α
inst✝¹ : CommMonoid β
inst✝ : Preorder β
f : β → α
mul : ∀ (x y : β), f (x * y) = f x * f y
le : ∀ {x y : β}, f x ≤ f y ↔ x ≤ y
a b : β
ab : a ≤ b
c : β
⊢ f a * f c ≤ f b * f c

Tactic:
]

Goal state after:
α : Type u
β : Type u_1
inst✝⁴ : CommMonoid α
inst✝³ : Preorder α
inst✝² : IsOrderedMonoid α
inst✝¹ : CommMonoid β
inst✝ : Preorder β
f : β → α
mul : ∀ (x y : β), f (x * y) = f x * f y
le : ∀ {x y : β}, f x ≤ f y ↔ x ≤ y
a b : β
ab : a ≤ b
c : β
⊢ f a * f c ≤ f b * f c
Module: Mathlib.Algebra.Order.Monoid.Basic

Goal state before:
pp: α : Type u
β : Type u_1
inst✝⁴ : CommMonoid α
inst✝³ : Preorder α
inst✝² : IsOrderedMonoid α
inst✝¹ : CommMonoid β
inst✝ : Preorder β
f : β → α
mul : ∀ (x y : β), f (x * y) = f x * f y
le : ∀ {x y : β}, f x ≤ f y ↔ x ≤ y
a b : β
ab : a ≤ b
c : β
⊢ f b * f c ≤ f b * f c

Tactic:
]

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

Goal state before:
pp: α : Type u
β : Type u_1
inst✝⁴ : CommMonoid α
inst✝³ : Preorder α
inst✝² : IsOrderedCancelMonoid α
inst✝¹ : CommMonoid β
inst✝ : Preorder β
f : β → α
mul : ∀ (x y : β), f (x * y) = f x * f y
le : ∀ {x y : β}, f x ≤ f y ↔ x ≤ y
a b c : β
bc : a * b ≤ a * c
⊢ a * b ≤ a * c

Tactic:
]

Goal state after:
α : Type u
β : Type u_1
inst✝⁴ : CommMonoid α
inst✝³ : Preorder α
inst✝² : IsOrderedCancelMonoid α
inst✝¹ : CommMonoid β
inst✝ : Preorder β
f : β → α
mul : ∀ (x y : β), f (x * y) = f x * f y
le : ∀ {x y : β}, f x ≤ f y ↔ x ≤ y
a b c : β
bc : a * b ≤ a * c
⊢ a * b ≤ a * c
Module: Mathlib.Algebra.Order.Ring.InjSurj

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁴ : Semiring R
inst✝³ : PartialOrder R
inst✝² : IsStrictOrderedRing R
inst✝¹ : Semiring S
inst✝ : PartialOrder S
f : S → R
zero : f 0 = 0
one : f 1 = 1
add : ∀ (x y : S), f (x + y) = f x + f y
mul : ∀ (x y : S), f (x * y) = f x * f y
le : ∀ {x y : S}, f x ≤ f y ↔ x ≤ y
lt : ∀ {x y : S}, f x < f y ↔ x < y
a : S
ha : 0 < a
b c : S
hbc : b < c
⊢ f a * f b < f a * f c

Tactic:
]

Goal state after:
R : Type u_1
S : Type u_2
inst✝⁴ : Semiring R
inst✝³ : PartialOrder R
inst✝² : IsStrictOrderedRing R
inst✝¹ : Semiring S
inst✝ : PartialOrder S
f : S → R
zero : f 0 = 0
one : f 1 = 1
add : ∀ (x y : S), f (x + y) = f x + f y
mul : ∀ (x y : S), f (x * y) = f x * f y
le : ∀ {x y : S}, f x ≤ f y ↔ x ≤ y
lt : ∀ {x y : S}, f x < f y ↔ x < y
a : S
ha : 0 < a
b c : S
hbc : b < c
⊢ f a * f b < f a * f c
Module: Mathlib.Algebra.Order.Ring.InjSurj

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁴ : Semiring R
inst✝³ : PartialOrder R
inst✝² : IsStrictOrderedRing R
inst✝¹ : Semiring S
inst✝ : PartialOrder S
f : S → R
zero : f 0 = 0
one : f 1 = 1
add : ∀ (x y : S), f (x + y) = f x + f y
mul : ∀ (x y : S), f (x * y) = f x * f y
le : ∀ {x y : S}, f x ≤ f y ↔ x ≤ y
lt : ∀ {x y : S}, f x < f y ↔ x < y
a : S
ha : 0 < a
b c : S
hbc : b < c
⊢ 0 < a

Tactic:
]

Goal state after:
R : Type u_1
S : Type u_2
inst✝⁴ : Semiring R
inst✝³ : PartialOrder R
inst✝² : IsStrictOrderedRing R
inst✝¹ : Semiring S
inst✝ : PartialOrder S
f : S → R
zero : f 0 = 0
one : f 1 = 1
add : ∀ (x y : S), f (x + y) = f x + f y
mul : ∀ (x y : S), f (x * y) = f x * f y
le : ∀ {x y : S}, f x ≤ f y ↔ x ≤ y
lt : ∀ {x y : S}, f x < f y ↔ x < y
a : S
ha : 0 < a
b c : S
hbc : b < c
⊢ 0 < a
Module: Mathlib.Algebra.Order.Ring.InjSurj

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁴ : Semiring R
inst✝³ : PartialOrder R
inst✝² : IsStrictOrderedRing R
inst✝¹ : Semiring S
inst✝ : PartialOrder S
f : S → R
zero : f 0 = 0
one : f 1 = 1
add : ∀ (x y : S), f (x + y) = f x + f y
mul : ∀ (x y : S), f (x * y) = f x * f y
le : ∀ {x y : S}, f x ≤ f y ↔ x ≤ y
lt : ∀ {x y : S}, f x < f y ↔ x < y
a : S
ha : 0 < a
b c : S
hbc : b < c
⊢ f b * f a < f c * f a

Tactic:
]

Goal state after:
R : Type u_1
S : Type u_2
inst✝⁴ : Semiring R
inst✝³ : PartialOrder R
inst✝² : IsStrictOrderedRing R
inst✝¹ : Semiring S
inst✝ : PartialOrder S
f : S → R
zero : f 0 = 0
one : f 1 = 1
add : ∀ (x y : S), f (x + y) = f x + f y
mul : ∀ (x y : S), f (x * y) = f x * f y
le : ∀ {x y : S}, f x ≤ f y ↔ x ≤ y
lt : ∀ {x y : S}, f x < f y ↔ x < y
a : S
ha : 0 < a
b c : S
hbc : b < c
⊢ f b * f a < f c * f a
Module: Mathlib.Algebra.Order.Nonneg.Lattice

Goal state before:
pp: α : Type u_1
inst✝ : ConditionallyCompleteLinearOrder α
a : α
⊢ (fun t ↦
        if ht : t.Nonempty ∧ BddAbove t ∧ sSup (Subtype.val '' t) ∈ Ici a then ⟨sSup (Subtype.val '' t), ⋯⟩
        else default)
      ∅ =
    ⊥

Tactic:
]

Goal state after:
α : Type u_1
inst✝ : ConditionallyCompleteLinearOrder α
a : α
⊢ (fun t ↦
        if ht : t.Nonempty ∧ BddAbove t ∧ sSup (Subtype.val '' t) ∈ Ici a then ⟨sSup (Subtype.val '' t), ⋯⟩
        else default)
      ∅ =
    ⊥
Module: Mathlib.Algebra.Order.Nonneg.Ring

Goal state before:
pp: α : Type u_1
inst✝³ : Semiring α
inst✝² : PartialOrder α
inst✝¹ : IsStrictOrderedRing α
inst✝ : ExistsAddOfLE α
a b : { x // 0 ≤ x }
h : ↑a ≤ ↑b
⊢ ∃ c, b = a + c

Tactic:
]

Goal state after:
α : Type u_1
inst✝³ : Semiring α
inst✝² : PartialOrder α
inst✝¹ : IsStrictOrderedRing α
inst✝ : ExistsAddOfLE α
a b : { x // 0 ≤ x }
h : ↑a ≤ ↑b
⊢ ∃ c, b = a + c
Module: Mathlib.Algebra.Order.Nonneg.Ring

Goal state before:
pp: α : Type u_1
inst✝³ : Semiring α
inst✝² : PartialOrder α
inst✝¹ : IsStrictOrderedRing α
inst✝ : ExistsAddOfLE α
a b : { x // 0 ≤ x }
c : α
h : ↑a + 0 ≤ ↑a + c
hc : ↑b = ↑a + c
⊢ 0 ≤ c

Tactic:
]

Goal state after:
α : Type u_1
inst✝³ : Semiring α
inst✝² : PartialOrder α
inst✝¹ : IsStrictOrderedRing α
inst✝ : ExistsAddOfLE α
a b : { x // 0 ≤ x }
c : α
h : ↑a + 0 ≤ ↑a + c
hc : ↑b = ↑a + c
⊢ 0 ≤ c
Module: Mathlib.Order.CompleteLatticeIntervals

Goal state before:
pp: ι : Sort u_1
α : Type u_2
s : Set α
inst✝¹ : ConditionallyCompleteLinearOrder α
inst✝ : Inhabited ↑s
h_Sup : ∀ {t : Set ↑s}, t.Nonempty → BddAbove t → sSup (Subtype.val '' t) ∈ s
h_Inf : ∀ {t : Set ↑s}, t.Nonempty → BddBelow t → sInf (Subtype.val '' t) ∈ s
t : Set ↑s
ht : t.Nonempty
h_bdd : BddAbove t
⊢ IsLUB ((fun {y} ↦ ↑y) '' t) (sSup (Subtype.val '' t))

Tactic:
]

Goal state after:
ι : Sort u_1
α : Type u_2
s : Set α
inst✝¹ : ConditionallyCompleteLinearOrder α
inst✝ : Inhabited ↑s
h_Sup : ∀ {t : Set ↑s}, t.Nonempty → BddAbove t → sSup (Subtype.val '' t) ∈ s
h_Inf : ∀ {t : Set ↑s}, t.Nonempty → BddBelow t → sInf (Subtype.val '' t) ∈ s
t : Set ↑s
ht : t.Nonempty
h_bdd : BddAbove t
⊢ IsLUB ((fun {y} ↦ ↑y) '' t) (sSup (Subtype.val '' t))
Module: Mathlib.Order.CompleteLatticeIntervals

Goal state before:
pp: ι : Sort u_1
α : Type u_2
s : Set α
inst✝¹ : ConditionallyCompleteLinearOrder α
inst✝ : Inhabited ↑s
h_Sup : ∀ {t : Set ↑s}, t.Nonempty → BddAbove t → sSup (Subtype.val '' t) ∈ s
h_Inf : ∀ {t : Set ↑s}, t.Nonempty → BddBelow t → sInf (Subtype.val '' t) ∈ s
t : Set ↑s
ht : t.Nonempty
h_bdd : BddBelow t
⊢ IsGLB ((fun {x} ↦ ↑x) '' t) (sInf (Subtype.val '' t))

Tactic:
]

Goal state after:
ι : Sort u_1
α : Type u_2
s : Set α
inst✝¹ : ConditionallyCompleteLinearOrder α
inst✝ : Inhabited ↑s
h_Sup : ∀ {t : Set ↑s}, t.Nonempty → BddAbove t → sSup (Subtype.val '' t) ∈ s
h_Inf : ∀ {t : Set ↑s}, t.Nonempty → BddBelow t → sInf (Subtype.val '' t) ∈ s
t : Set ↑s
ht : t.Nonempty
h_bdd : BddBelow t
⊢ IsGLB ((fun {x} ↦ ↑x) '' t) (sInf (Subtype.val '' t))
Module: Mathlib.Order.CompleteLatticeIntervals

Goal state before:
pp: ι : Sort u_1
α : Type u_2
s : Set α
inst✝¹ : ConditionallyCompleteLattice α
a b : α
inst✝ : Fact (a ≤ b)
S : Set ↑(Icc a b)
hS : S.Nonempty
⊢ sSup (Subtype.val '' S) ∈ Icc a b

Tactic:
]

Goal state after:
ι : Sort u_1
α : Type u_2
s : Set α
inst✝¹ : ConditionallyCompleteLattice α
a b : α
inst✝ : Fact (a ≤ b)
S : Set ↑(Icc a b)
hS : S.Nonempty
⊢ sSup (Subtype.val '' S) ∈ Icc a b
Module: Mathlib.Order.CompleteLatticeIntervals

Goal state before:
pp: ι : Sort u_1
α : Type u_2
s : Set α
inst✝¹ : ConditionallyCompleteLattice α
a b : α
inst✝ : Fact (a ≤ b)
S : Set ↑(Icc a b)
hS : S.Nonempty
⊢ sInf (Subtype.val '' S) ∈ Icc a b

Tactic:
]

Goal state after:
ι : Sort u_1
α : Type u_2
s : Set α
inst✝¹ : ConditionallyCompleteLattice α
a b : α
inst✝ : Fact (a ≤ b)
S : Set ↑(Icc a b)
hS : S.Nonempty
⊢ sInf (Subtype.val '' S) ∈ Icc a b
Module: Mathlib.Order.CompleteLatticeIntervals

Goal state before:
pp: ι : Sort u_1
α : Type u_2
inst✝ : CompleteLattice α
a : α
f : ι → ↑(Iic a)
⊢ sSup (Subtype.val '' range fun i ↦ f i) = ⨆ i, ↑(f i)

Tactic:
]

Goal state after:
ι : Sort u_1
α : Type u_2
inst✝ : CompleteLattice α
a : α
f : ι → ↑(Iic a)
⊢ sSup (Subtype.val '' range fun i ↦ f i) = ⨆ i, ↑(f i)
Module: Mathlib.Order.CompleteLatticeIntervals

Goal state before:
pp: ι : Sort u_1
α : Type u_2
inst✝ : CompleteLattice α
a : α
f : ι → ↑(Iic a)
⊢ a ⊓ sInf (Subtype.val '' range fun i ↦ f i) = a ⊓ ⨅ i, ↑(f i)

Tactic:
]

Goal state after:
ι : Sort u_1
α : Type u_2
inst✝ : CompleteLattice α
a : α
f : ι → ↑(Iic a)
⊢ a ⊓ sInf (Subtype.val '' range fun i ↦ f i) = a ⊓ ⨅ i, ↑(f i)
Module: Mathlib.Algebra.Order.Nonneg.Field

Goal state before:
pp: α : Type u_1
inst✝² : Semifield α
inst✝¹ : LinearOrder α
inst✝ : IsStrictOrderedRing α
q : ℚ≥0
⊢ 0 ≤ ↑q.num / ↑q.den

Tactic:
]

Goal state after:
α : Type u_1
inst✝² : Semifield α
inst✝¹ : LinearOrder α
inst✝ : IsStrictOrderedRing α
q : ℚ≥0
⊢ 0 ≤ ↑q.num / ↑q.den
Module: Mathlib.Algebra.Order.Nonneg.Field

Goal state before:
pp: α : Type u_1
inst✝² : Semifield α
inst✝¹ : LinearOrder α
inst✝ : IsStrictOrderedRing α
a : α
q : ℚ≥0
ha : 0 ≤ a
⊢ 0 ≤ ↑q * a

Tactic:
]

Goal state after:
α : Type u_1
inst✝² : Semifield α
inst✝¹ : LinearOrder α
inst✝ : IsStrictOrderedRing α
a : α
q : ℚ≥0
ha : 0 ≤ a
⊢ 0 ≤ ↑q * a
Module: Mathlib.Algebra.Order.Nonneg.Field

Goal state before:
pp: α : Type u_1
inst✝² : Semifield α
inst✝¹ : LinearOrder α
inst✝ : IsStrictOrderedRing α
x y : α
q : ℚ≥0
a : { x // 0 ≤ x }
⊢ 0 ≤ ↑q * ↑a

Tactic:
]

Goal state after:
α : Type u_1
inst✝² : Semifield α
inst✝¹ : LinearOrder α
inst✝ : IsStrictOrderedRing α
x y : α
q : ℚ≥0
a : { x // 0 ≤ x }
⊢ 0 ≤ ↑q * ↑a
Module: Mathlib.Algebra.Order.Nonneg.Field

Goal state before:
pp: α : Type u_1
inst✝² : Semifield α
inst✝¹ : LinearOrder α
inst✝ : IsStrictOrderedRing α
x y : α
q : ℚ≥0
a : α
ha : 0 ≤ a
⊢ 0 ≤ ↑q * a

Tactic:
]

Goal state after:
α : Type u_1
inst✝² : Semifield α
inst✝¹ : LinearOrder α
inst✝ : IsStrictOrderedRing α
x y : α
q : ℚ≥0
a : α
ha : 0 ≤ a
⊢ 0 ≤ ↑q * a
Module: Mathlib.Tactic.NormNum.Abs

Goal state before:
pp: α : Type u_1
inst✝² : Ring α
inst✝¹ : Lattice α
inst✝ : IsOrderedRing α
a : α
na : ℕ
pa : IsNat a na
⊢ IsNat (↑na) na

Tactic:
]

Goal state after:
α : Type u_1
inst✝² : Ring α
inst✝¹ : Lattice α
inst✝ : IsOrderedRing α
a : α
na : ℕ
pa : IsNat a na
⊢ IsNat (↑na) na
Module: Mathlib.Tactic.NormNum.Abs

Goal state before:
pp: α : Type u_1
inst✝² : Ring α
inst✝¹ : Lattice α
inst✝ : IsOrderedRing α
a : α
na : ℕ
pa : IsInt a (Int.negOfNat na)
⊢ IsNat |↑(Int.negOfNat na)| na

Tactic:
]

Goal state after:
α : Type u_1
inst✝² : Ring α
inst✝¹ : Lattice α
inst✝ : IsOrderedRing α
a : α
na : ℕ
pa : IsInt a (Int.negOfNat na)
⊢ IsNat |↑(Int.negOfNat na)| na
Module: Mathlib.Tactic.NormNum.Eq

Goal state before:
pp: α : Type u_1
inst✝² : Semiring α
n₁ n₂ : ℕ
a₁ a₂ : α
inst✝¹ : Invertible a₁
inst✝ : Invertible a₂
⊢ ↑n₁ * a₂ = ↑n₂ * a₁ ↔ ↑n₁ * a₂ = ↑n₂ * a₁

Tactic:
]

Goal state after:
Module: Mathlib.Tactic.NormNum.Eq

Goal state before:
pp: α : Type u_1
inst✝¹ : Semiring α
inst✝ : CharZero α
num✝¹ num✝ denom✝¹ denom✝ : ℕ
inv✝¹ : Invertible ↑denom✝¹
inv✝ : Invertible ↑denom✝
h : decide (num✝¹.mul denom✝ = num✝.mul denom✝¹) = false
⊢ ¬↑num✝¹ * ↑denom✝ = ↑num✝ * ↑denom✝¹

Tactic:
]

Goal state after:
α : Type u_1
inst✝¹ : Semiring α
inst✝ : CharZero α
num✝¹ num✝ denom✝¹ denom✝ : ℕ
inv✝¹ : Invertible ↑denom✝¹
inv✝ : Invertible ↑denom✝
h : decide (num✝¹.mul denom✝ = num✝.mul denom✝¹) = false
⊢ ¬↑num✝¹ * ↑denom✝ = ↑num✝ * ↑denom✝¹
Module: Mathlib.Tactic.NormNum.Eq

Goal state before:
pp: α : Type u_1
inst✝² : Ring α
n₁ n₂ : ℤ
a₁ a₂ : α
inst✝¹ : Invertible a₁
inst✝ : Invertible a₂
⊢ ↑n₁ * a₂ = ↑n₂ * a₁ ↔ ↑n₁ * a₂ = ↑n₂ * a₁

Tactic:
]

Goal state after:
Module: Mathlib.Tactic.NormNum.Eq

Goal state before:
pp: α : Type u_1
inst✝¹ : Ring α
inst✝ : CharZero α
num✝¹ num✝ : ℤ
denom✝¹ denom✝ : ℕ
inv✝¹ : Invertible ↑denom✝¹
inv✝ : Invertible ↑denom✝
h : decide (num✝¹.mul (Int.ofNat denom✝) = num✝.mul (Int.ofNat denom✝¹)) = false
⊢ ¬↑num✝¹ * ↑denom✝ = ↑num✝ * ↑denom✝¹

Tactic:
]

Goal state after:
α : Type u_1
inst✝¹ : Ring α
inst✝ : CharZero α
num✝¹ num✝ : ℤ
denom✝¹ denom✝ : ℕ
inv✝¹ : Invertible ↑denom✝¹
inv✝ : Invertible ↑denom✝
h : decide (num✝¹.mul (Int.ofNat denom✝) = num✝.mul (Int.ofNat denom✝¹)) = false
⊢ ¬↑num✝¹ * ↑denom✝ = ↑num✝ * ↑denom✝¹
Module: Mathlib.Tactic.NormNum.DivMod

Goal state before:
pp: a q m : ℤ
b' r : ℕ
hm : q * ↑b' = m
h : ↑r + m = a
h₂ : r < b'
⊢ ↑r / ↑b' + q = q

Tactic:
]

Goal state after:
a q m : ℤ
b' r : ℕ
hm : q * ↑b' = m
h : ↑r + m = a
h₂ : r < b'
⊢ ↑r / ↑b' + q = q
Module: Mathlib.Tactic.NormNum.DivMod

Goal state before:
pp: a q m : ℤ
b' r : ℕ
hm : q * ↑b' = m
h : ↑r + m = a
h₂ : r < b'
⊢ q = q

pp: case H1
a q m : ℤ
b' r : ℕ
hm : q * ↑b' = m
h : ↑r + m = a
h₂ : r < b'
⊢ 0 ≤ ↑r

pp: case H2
a q m : ℤ
b' r : ℕ
hm : q * ↑b' = m
h : ↑r + m = a
h₂ : r < b'
⊢ ↑r < ↑b'

Tactic:
]

Goal state after:
case H1
a q m : ℤ
b' r : ℕ
hm : q * ↑b' = m
h : ↑r + m = a
h₂ : r < b'
⊢ 0 ≤ ↑r

case H2
a q m : ℤ
b' r : ℕ
hm : q * ↑b' = m
h : ↑r + m = a
h₂ : r < b'
⊢ ↑r < ↑b'
Module: Mathlib.Tactic.NormNum.DivMod

Goal state before:
pp: a b q q' : ℤ
h : IsInt (a / -b) q
hq : -q = q'
⊢ a / b = a / b

Tactic:
]

Goal state after:
Module: Mathlib.Tactic.NormNum.DivMod

Goal state before:
pp: a q m : ℤ
b' r : ℕ
hm : q * ↑b' = m
h₂ : r.blt b' = true
h : ↑r + m = a
⊢ ↑r = ↑r

pp: case H1
a q m : ℤ
b' r : ℕ
hm : q * ↑b' = m
h₂ : r.blt b' = true
h : ↑r + m = a
⊢ 0 ≤ ↑r

pp: case H2
a q m : ℤ
b' r : ℕ
hm : q * ↑b' = m
h₂ : r.blt b' = true
h : ↑r + m = a
⊢ ↑r < ↑b'

Tactic:
]

Goal state after:
case H1
a q m : ℤ
b' r : ℕ
hm : q * ↑b' = m
h₂ : r.blt b' = true
h : ↑r + m = a
⊢ 0 ≤ ↑r

case H2
a q m : ℤ
b' r : ℕ
hm : q * ↑b' = m
h₂ : r.blt b' = true
h : ↑r + m = a
⊢ ↑r < ↑b'
Module: Mathlib.Tactic.NormNum.DivMod

Goal state before:
pp: a b : ℤ
r : ℕ
h : IsNat (a % -b) r
⊢ ↑r = ↑r

Tactic:
]

Goal state after:
Module: Mathlib.Data.Rat.Cast.Order

Goal state before:
pp: q : ℚ
K : Type u_5
inst✝² : Field K
inst✝¹ : LinearOrder K
inst✝ : IsStrictOrderedRing K
hq : 0 < q
⊢ 0 < ↑q.num / ↑q.den

Tactic:
]

Goal state after:
q : ℚ
K : Type u_5
inst✝² : Field K
inst✝¹ : LinearOrder K
inst✝ : IsStrictOrderedRing K
hq : 0 < q
⊢ 0 < ↑q.num / ↑q.den
Module: Mathlib.Data.Rat.Cast.Order

Goal state before:
pp: K : Type u_5
inst✝² : Field K
inst✝¹ : LinearOrder K
inst✝ : IsStrictOrderedRing K
m : ℚ
n : ℕ
⊢ ↑m ≤ ↑n ↔ ↑m ≤ ↑n

Tactic:
]

Goal state after:
Module: Mathlib.Data.Rat.Cast.Order

Goal state before:
pp: K : Type u_5
inst✝² : Field K
inst✝¹ : LinearOrder K
inst✝ : IsStrictOrderedRing K
m : ℕ
n : ℚ
⊢ ↑m ≤ ↑n ↔ ↑m ≤ ↑n

Tactic:
]

Goal state after:
Module: Mathlib.Data.Rat.Cast.Order

Goal state before:
pp: K : Type u_5
inst✝² : Field K
inst✝¹ : LinearOrder K
inst✝ : IsStrictOrderedRing K
m : ℚ
n : ℤ
⊢ ↑m ≤ ↑n ↔ ↑m ≤ ↑n

Tactic:
]

Goal state after:
Module: Mathlib.Data.Rat.Cast.Order

Goal state before:
pp: K : Type u_5
inst✝² : Field K
inst✝¹ : LinearOrder K
inst✝ : IsStrictOrderedRing K
m : ℤ
n : ℚ
⊢ ↑m ≤ ↑n ↔ ↑m ≤ ↑n

Tactic:
]

Goal state after:
Module: Mathlib.Data.Rat.Cast.Order

Goal state before:
pp: K : Type u_5
inst✝² : Field K
inst✝¹ : LinearOrder K
inst✝ : IsStrictOrderedRing K
m : ℚ
n : ℕ
⊢ ↑m < ↑n ↔ ↑m < ↑n

Tactic:
]

Goal state after:
Module: Mathlib.Data.Rat.Cast.Order

Goal state before:
pp: K : Type u_5
inst✝² : Field K
inst✝¹ : LinearOrder K
inst✝ : IsStrictOrderedRing K
m : ℕ
n : ℚ
⊢ ↑m < ↑n ↔ ↑m < ↑n

Tactic:
]

Goal state after:
Module: Mathlib.Data.Rat.Cast.Order

Goal state before:
pp: K : Type u_5
inst✝² : Field K
inst✝¹ : LinearOrder K
inst✝ : IsStrictOrderedRing K
m : ℚ
n : ℤ
⊢ ↑m < ↑n ↔ ↑m < ↑n

Tactic:
]

Goal state after:
Module: Mathlib.Data.Rat.Cast.Order

Goal state before:
pp: K : Type u_5
inst✝² : Field K
inst✝¹ : LinearOrder K
inst✝ : IsStrictOrderedRing K
m : ℤ
n : ℚ
⊢ ↑m < ↑n ↔ ↑m < ↑n

Tactic:
]

Goal state after:
Module: Mathlib.Data.Rat.Cast.Order

Goal state before:
pp: K : Type u_5
inst✝² : Semifield K
inst✝¹ : LinearOrder K
inst✝ : IsStrictOrderedRing K
p q : ℚ≥0
h : p < q
⊢ p < q

pp: case hb
K : Type u_5
inst✝² : Semifield K
inst✝¹ : LinearOrder K
inst✝ : IsStrictOrderedRing K
p q : ℚ≥0
h : p < q
⊢ 0 < ↑p.den

pp: case hd
K : Type u_5
inst✝² : Semifield K
inst✝¹ : LinearOrder K
inst✝ : IsStrictOrderedRing K
p q : ℚ≥0
h : p < q
⊢ 0 < ↑q.den

Tactic:
]

Goal state after:
K : Type u_5
inst✝² : Semifield K
inst✝¹ : LinearOrder K
inst✝ : IsStrictOrderedRing K
p q : ℚ≥0
h : p < q
⊢ p < q

case hb
K : Type u_5
inst✝² : Semifield K
inst✝¹ : LinearOrder K
inst✝ : IsStrictOrderedRing K
p q : ℚ≥0
h : p < q
⊢ 0 < ↑p.den

case hd
K : Type u_5
inst✝² : Semifield K
inst✝¹ : LinearOrder K
inst✝ : IsStrictOrderedRing K
p q : ℚ≥0
h : p < q
⊢ 0 < ↑q.den
Module: Mathlib.Data.Rat.Cast.Order

Goal state before:
pp: K : Type u_5
inst✝² : Semifield K
inst✝¹ : LinearOrder K
inst✝ : IsStrictOrderedRing K
m : ℚ≥0
n : ℕ
⊢ ↑m ≤ ↑n ↔ ↑m ≤ ↑n

Tactic:
]

Goal state after:
Module: Mathlib.Data.Rat.Cast.Order

Goal state before:
pp: K : Type u_5
inst✝² : Semifield K
inst✝¹ : LinearOrder K
inst✝ : IsStrictOrderedRing K
m : ℕ
n : ℚ≥0
⊢ ↑m ≤ ↑n ↔ ↑m ≤ ↑n

Tactic:
]

Goal state after:
Module: Mathlib.Data.Rat.Cast.Order

Goal state before:
pp: K : Type u_5
inst✝² : Semifield K
inst✝¹ : LinearOrder K
inst✝ : IsStrictOrderedRing K
m : ℚ≥0
n : ℕ
⊢ ↑m < ↑n ↔ ↑m < ↑n

Tactic:
]

Goal state after:
Module: Mathlib.Data.Rat.Cast.Order

Goal state before:
pp: K : Type u_5
inst✝² : Semifield K
inst✝¹ : LinearOrder K
inst✝ : IsStrictOrderedRing K
m : ℕ
n : ℚ≥0
⊢ ↑m < ↑n ↔ ↑m < ↑n

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.Notation.Support

Goal state before:
pp: ι : Type u_1
M : Type u_3
inst✝ : One M
f g : ι → M
x✝ : mulSupport f = mulSupport g ∧ ∀ (x : ι), x ∈ mulSupport f → f x = g x
h₁ : mulSupport f = mulSupport g
h₂ : ∀ (x : ι), x ∈ mulSupport f → f x = g x
x : ι
hx : x ∈ mulSupport f
⊢ f x = g x

Tactic:
exact h₂ x hx

Goal state after:
Module: Mathlib.Algebra.Notation.Support

Goal state before:
pp: ι : Type u_1
M : Type u_3
inst✝ : One M
f g : ι → M
x✝ : mulSupport f = mulSupport g ∧ ∀ (x : ι), x ∈ mulSupport f → f x = g x
h₁ : mulSupport f = mulSupport g
h₂ : ∀ (x : ι), x ∈ mulSupport f → f x = g x
x : ι
hx : ¬x ∈ mulSupport f
⊢ 1 = 1

Tactic:
]

Goal state after: