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

Goal state before:
pp: case map_sInf
α : Type u
β : Type v
ι : Sort w
κ : ι → Sort w'
e : α ≃ β
inst✝ : Coframe β
completeLattice : CompleteLattice α := e.completeLattice
coheytingAlgebra : CoheytingAlgebra α := e.coheytingAlgebra
s✝ : Set α
⊢ e (sInf s✝) = ⨅ a ∈ s✝, e a

Tactic:
first | rfl | exact e.apply_symm_apply _

Goal state after:
Module: Mathlib.Order.CompleteBooleanAlgebra

Goal state before:
pp: case map_top
α : Type u
β : Type v
ι : Sort w
κ : ι → Sort w'
e : α ≃ β
inst✝ : Coframe β
completeLattice : CompleteLattice α := ⋯
coheytingAlgebra : CoheytingAlgebra α := ⋯
⊢ e ⊤ = ⊤

Tactic:
first | rfl | exact e.apply_symm_apply _

Goal state after:
Module: Mathlib.Order.CompleteBooleanAlgebra

Goal state before:
pp: case map_bot
α : Type u
β : Type v
ι : Sort w
κ : ι → Sort w'
e : α ≃ β
inst✝ : Coframe β
completeLattice : CompleteLattice α := ⋯
coheytingAlgebra : CoheytingAlgebra α := ⋯
⊢ e ⊥ = ⊥

Tactic:
first | rfl | exact e.apply_symm_apply _

Goal state after:
Module: Mathlib.Order.CompleteBooleanAlgebra

Goal state before:
pp: case map_hnot
α : Type u
β : Type v
ι : Sort w
κ : ι → Sort w'
e : α ≃ β
inst✝ : Coframe β
completeLattice : CompleteLattice α := e.completeLattice
coheytingAlgebra : CoheytingAlgebra α := e.coheytingAlgebra
a✝ : α
⊢ e (￢a✝) = ￢e a✝

Tactic:
first | rfl | exact e.apply_symm_apply _

Goal state after:
Module: Mathlib.Order.CompleteBooleanAlgebra

Goal state before:
pp: case map_sdiff
α : Type u
β : Type v
ι : Sort w
κ : ι → Sort w'
e : α ≃ β
inst✝ : Coframe β
completeLattice : CompleteLattice α := e.completeLattice
coheytingAlgebra : CoheytingAlgebra α := e.coheytingAlgebra
a✝ b✝ : α
⊢ e (a✝ \ b✝) = e a✝ \ e b✝

Tactic:
first | rfl | exact e.apply_symm_apply _

Goal state after:
Module: Mathlib.Order.ConditionallyCompleteLattice.Basic

Goal state before:
pp: case inl
α✝ : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
α : Type u_5
inst✝ : ConditionallyCompleteLattice α
S : Set (WithTop (WithBot α))
a : WithTop (WithBot α)
ha : a ∈ upperBounds S
h : S = ∅
⊢ sSup S ≤ a

pp: case inr
α✝ : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
α : Type u_5
inst✝ : ConditionallyCompleteLattice α
S : Set (WithTop (WithBot α))
a : WithTop (WithBot α)
ha : a ∈ upperBounds S
h : S.Nonempty
⊢ sSup S ≤ a

Tactic:
· change ite _ _ _ ≤ a
      simp [h]

Goal state after:
case inr
α✝ : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
α : Type u_5
inst✝ : ConditionallyCompleteLattice α
S : Set (WithTop (WithBot α))
a : WithTop (WithBot α)
ha : a ∈ upperBounds S
h : S.Nonempty
⊢ sSup S ≤ a
Module: Mathlib.Order.CompleteBooleanAlgebra

Goal state before:
pp: case le
α : Type u
β : Type v
ι : Sort w
κ : ι → Sort w'
e : α ≃ β
inst✝ : CompleteDistribLattice β
completeLattice : CompleteLattice α := e.completeLattice
biheytingAlgebra : BiheytingAlgebra α := e.biheytingAlgebra
x✝ y✝ : α
⊢ e x✝ ≤ e y✝ ↔ x✝ ≤ y✝

Tactic:
first | rfl | exact e.apply_symm_apply _

Goal state after:
Module: Mathlib.Order.CompleteBooleanAlgebra

Goal state before:
pp: case lt
α : Type u
β : Type v
ι : Sort w
κ : ι → Sort w'
e : α ≃ β
inst✝ : CompleteDistribLattice β
completeLattice : CompleteLattice α := e.completeLattice
biheytingAlgebra : BiheytingAlgebra α := e.biheytingAlgebra
x✝ y✝ : α
⊢ e x✝ < e y✝ ↔ x✝ < y✝

Tactic:
first | rfl | exact e.apply_symm_apply _

Goal state after:
Module: Mathlib.Order.CompleteBooleanAlgebra

Goal state before:
pp: case map_sup
α : Type u
β : Type v
ι : Sort w
κ : ι → Sort w'
e : α ≃ β
inst✝ : CompleteDistribLattice β
completeLattice : CompleteLattice α := e.completeLattice
biheytingAlgebra : BiheytingAlgebra α := e.biheytingAlgebra
a✝ b✝ : α
⊢ e (a✝ ⊔ b✝) = e a✝ ⊔ e b✝

Tactic:
first | rfl | exact e.apply_symm_apply _

Goal state after:
Module: Mathlib.Order.CompleteBooleanAlgebra

Goal state before:
pp: case map_inf
α : Type u
β : Type v
ι : Sort w
κ : ι → Sort w'
e : α ≃ β
inst✝ : CompleteDistribLattice β
completeLattice : CompleteLattice α := e.completeLattice
biheytingAlgebra : BiheytingAlgebra α := e.biheytingAlgebra
a✝ b✝ : α
⊢ e (a✝ ⊓ b✝) = e a✝ ⊓ e b✝

Tactic:
first | rfl | exact e.apply_symm_apply _

Goal state after:
Module: Mathlib.Order.CompleteBooleanAlgebra

Goal state before:
pp: case map_sSup
α : Type u
β : Type v
ι : Sort w
κ : ι → Sort w'
e : α ≃ β
inst✝ : CompleteDistribLattice β
completeLattice : CompleteLattice α := e.completeLattice
biheytingAlgebra : BiheytingAlgebra α := e.biheytingAlgebra
s✝ : Set α
⊢ e (sSup s✝) = ⨆ a ∈ s✝, e a

Tactic:
first | rfl | exact e.apply_symm_apply _

Goal state after:
Module: Mathlib.Order.CompleteBooleanAlgebra

Goal state before:
pp: case map_sInf
α : Type u
β : Type v
ι : Sort w
κ : ι → Sort w'
e : α ≃ β
inst✝ : CompleteDistribLattice β
completeLattice : CompleteLattice α := e.completeLattice
biheytingAlgebra : BiheytingAlgebra α := e.biheytingAlgebra
s✝ : Set α
⊢ e (sInf s✝) = ⨅ a ∈ s✝, e a

Tactic:
first | rfl | exact e.apply_symm_apply _

Goal state after:
Module: Mathlib.Order.CompleteBooleanAlgebra

Goal state before:
pp: case map_top
α : Type u
β : Type v
ι : Sort w
κ : ι → Sort w'
e : α ≃ β
inst✝ : CompleteDistribLattice β
completeLattice : CompleteLattice α := ⋯
biheytingAlgebra : BiheytingAlgebra α := ⋯
⊢ e ⊤ = ⊤

Tactic:
first | rfl | exact e.apply_symm_apply _

Goal state after:
Module: Mathlib.Order.CompleteBooleanAlgebra

Goal state before:
pp: case map_bot
α : Type u
β : Type v
ι : Sort w
κ : ι → Sort w'
e : α ≃ β
inst✝ : CompleteDistribLattice β
completeLattice : CompleteLattice α := ⋯
biheytingAlgebra : BiheytingAlgebra α := ⋯
⊢ e ⊥ = ⊥

Tactic:
first | rfl | exact e.apply_symm_apply _

Goal state after:
Module: Mathlib.Order.CompleteBooleanAlgebra

Goal state before:
pp: case map_compl
α : Type u
β : Type v
ι : Sort w
κ : ι → Sort w'
e : α ≃ β
inst✝ : CompleteDistribLattice β
completeLattice : CompleteLattice α := e.completeLattice
biheytingAlgebra : BiheytingAlgebra α := e.biheytingAlgebra
a✝ : α
⊢ e a✝ᶜ = (e a✝)ᶜ

Tactic:
first | rfl | exact e.apply_symm_apply _

Goal state after:
Module: Mathlib.Order.CompleteBooleanAlgebra

Goal state before:
pp: case map_himp
α : Type u
β : Type v
ι : Sort w
κ : ι → Sort w'
e : α ≃ β
inst✝ : CompleteDistribLattice β
completeLattice : CompleteLattice α := e.completeLattice
biheytingAlgebra : BiheytingAlgebra α := e.biheytingAlgebra
a✝ b✝ : α
⊢ e (a✝ ⇨ b✝) = e a✝ ⇨ e b✝

Tactic:
first | rfl | exact e.apply_symm_apply _

Goal state after:
Module: Mathlib.Order.CompleteBooleanAlgebra

Goal state before:
pp: case map_hnot
α : Type u
β : Type v
ι : Sort w
κ : ι → Sort w'
e : α ≃ β
inst✝ : CompleteDistribLattice β
completeLattice : CompleteLattice α := e.completeLattice
biheytingAlgebra : BiheytingAlgebra α := e.biheytingAlgebra
a✝ : α
⊢ e (￢a✝) = ￢e a✝

Tactic:
first | rfl | exact e.apply_symm_apply _

Goal state after:
Module: Mathlib.Order.CompleteBooleanAlgebra

Goal state before:
pp: case map_sdiff
α : Type u
β : Type v
ι : Sort w
κ : ι → Sort w'
e : α ≃ β
inst✝ : CompleteDistribLattice β
completeLattice : CompleteLattice α := e.completeLattice
biheytingAlgebra : BiheytingAlgebra α := e.biheytingAlgebra
a✝ b✝ : α
⊢ e (a✝ \ b✝) = e a✝ \ e b✝

Tactic:
first | rfl | exact e.apply_symm_apply _

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

Goal state before:
pp: α : Type u_1
inst✝ : Lattice α
a b : α
⊢ [[a, b]] = [[b, a]]

Tactic:
by simp_rw [uIcc, inf_comm, sup_comm]

Goal state after:
Module: Mathlib.Order.CompleteBooleanAlgebra

Goal state before:
pp: case le
α : Type u
β : Type v
ι : Sort w
κ : ι → Sort w'
e : α ≃ β
inst✝ : CompletelyDistribLattice β
completeDistribLattice : CompleteDistribLattice α := e.completeDistribLattice
x✝ y✝ : α
⊢ e x✝ ≤ e y✝ ↔ x✝ ≤ y✝

Tactic:
first | rfl | exact e.apply_symm_apply _

Goal state after:
Module: Mathlib.Order.CompleteBooleanAlgebra

Goal state before:
pp: case lt
α : Type u
β : Type v
ι : Sort w
κ : ι → Sort w'
e : α ≃ β
inst✝ : CompletelyDistribLattice β
completeDistribLattice : CompleteDistribLattice α := e.completeDistribLattice
x✝ y✝ : α
⊢ e x✝ < e y✝ ↔ x✝ < y✝

Tactic:
first | rfl | exact e.apply_symm_apply _

Goal state after:
Module: Mathlib.Data.Set.Pairwise.Basic

Goal state before:
pp: α : Type u_1
ι : Type u_4
r : α → α → Prop
f : ι → α
s : Set ι
h : InjOn f s
⊢ (f '' s).Pairwise r ↔ s.Pairwise (r on f)

Tactic:
by
  simp +contextual [h.eq_iff, Set.Pairwise]

Goal state after:
Module: Mathlib.Order.CompleteBooleanAlgebra

Goal state before:
pp: case map_sup
α : Type u
β : Type v
ι : Sort w
κ : ι → Sort w'
e : α ≃ β
inst✝ : CompletelyDistribLattice β
completeDistribLattice : CompleteDistribLattice α := e.completeDistribLattice
a✝ b✝ : α
⊢ e (a✝ ⊔ b✝) = e a✝ ⊔ e b✝

Tactic:
first | rfl | exact e.apply_symm_apply _

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

Goal state before:
pp: α : Type u_1
inst✝ : LinearOrder α
a b : α
⊢ [[a, b]] = Icc a b ∪ Icc b a

Tactic:
Icc_union_Icc',

Goal state after:
α : Type u_1
inst✝ : LinearOrder α
a b : α
⊢ [[a, b]] = Icc (min a b) (max b a)

case h₁
α : Type u_1
inst✝ : LinearOrder α
a b : α
⊢ b ≤ b

case h₂
α : Type u_1
inst✝ : LinearOrder α
a b : α
⊢ a ≤ a
Module: Mathlib.Order.CompleteBooleanAlgebra

Goal state before:
pp: case map_inf
α : Type u
β : Type v
ι : Sort w
κ : ι → Sort w'
e : α ≃ β
inst✝ : CompletelyDistribLattice β
completeDistribLattice : CompleteDistribLattice α := e.completeDistribLattice
a✝ b✝ : α
⊢ e (a✝ ⊓ b✝) = e a✝ ⊓ e b✝

Tactic:
first | rfl | exact e.apply_symm_apply _

Goal state after:
Module: Mathlib.Order.CompleteBooleanAlgebra

Goal state before:
pp: case map_sSup
α : Type u
β : Type v
ι : Sort w
κ : ι → Sort w'
e : α ≃ β
inst✝ : CompletelyDistribLattice β
completeDistribLattice : CompleteDistribLattice α := e.completeDistribLattice
s✝ : Set α
⊢ e (sSup s✝) = ⨆ a ∈ s✝, e a

Tactic:
first | rfl | exact e.apply_symm_apply _

Goal state after:
Module: Mathlib.Order.CompleteBooleanAlgebra

Goal state before:
pp: case map_sInf
α : Type u
β : Type v
ι : Sort w
κ : ι → Sort w'
e : α ≃ β
inst✝ : CompletelyDistribLattice β
completeDistribLattice : CompleteDistribLattice α := e.completeDistribLattice
s✝ : Set α
⊢ e (sInf s✝) = ⨅ a ∈ s✝, e a

Tactic:
first | rfl | exact e.apply_symm_apply _

Goal state after:
Module: Mathlib.Order.CompleteBooleanAlgebra

Goal state before:
pp: case map_top
α : Type u
β : Type v
ι : Sort w
κ : ι → Sort w'
e : α ≃ β
inst✝ : CompletelyDistribLattice β
completeDistribLattice : CompleteDistribLattice α := ⋯
⊢ e ⊤ = ⊤

Tactic:
first | rfl | exact e.apply_symm_apply _

Goal state after:
Module: Mathlib.Order.CompleteBooleanAlgebra

Goal state before:
pp: case map_bot
α : Type u
β : Type v
ι : Sort w
κ : ι → Sort w'
e : α ≃ β
inst✝ : CompletelyDistribLattice β
completeDistribLattice : CompleteDistribLattice α := ⋯
⊢ e ⊥ = ⊥

Tactic:
first | rfl | exact e.apply_symm_apply _

Goal state after:
Module: Mathlib.Order.CompleteBooleanAlgebra

Goal state before:
pp: case map_compl
α : Type u
β : Type v
ι : Sort w
κ : ι → Sort w'
e : α ≃ β
inst✝ : CompletelyDistribLattice β
completeDistribLattice : CompleteDistribLattice α := e.completeDistribLattice
a✝ : α
⊢ e a✝ᶜ = (e a✝)ᶜ

Tactic:
first | rfl | exact e.apply_symm_apply _

Goal state after:
Module: Mathlib.Order.CompleteBooleanAlgebra

Goal state before:
pp: case map_himp
α : Type u
β : Type v
ι : Sort w
κ : ι → Sort w'
e : α ≃ β
inst✝ : CompletelyDistribLattice β
completeDistribLattice : CompleteDistribLattice α := e.completeDistribLattice
a✝ b✝ : α
⊢ e (a✝ ⇨ b✝) = e a✝ ⇨ e b✝

Tactic:
first | rfl | exact e.apply_symm_apply _

Goal state after:
Module: Mathlib.Order.CompleteBooleanAlgebra

Goal state before:
pp: case map_hnot
α : Type u
β : Type v
ι : Sort w
κ : ι → Sort w'
e : α ≃ β
inst✝ : CompletelyDistribLattice β
completeDistribLattice : CompleteDistribLattice α := e.completeDistribLattice
a✝ : α
⊢ e (￢a✝) = ￢e a✝

Tactic:
first | rfl | exact e.apply_symm_apply _

Goal state after:
Module: Mathlib.Order.CompleteBooleanAlgebra

Goal state before:
pp: case map_sdiff
α : Type u
β : Type v
ι : Sort w
κ : ι → Sort w'
e : α ≃ β
inst✝ : CompletelyDistribLattice β
completeDistribLattice : CompleteDistribLattice α := e.completeDistribLattice
a✝ b✝ : α
⊢ e (a✝ \ b✝) = e a✝ \ e b✝

Tactic:
first | rfl | exact e.apply_symm_apply _

Goal state after:
Module: Mathlib.Order.CompleteBooleanAlgebra

Goal state before:
pp: case le
α : Type u
β : Type v
ι : Sort w
κ : ι → Sort w'
e : α ≃ β
inst✝ : CompleteBooleanAlgebra β
completeLattice : CompleteLattice α := e.completeLattice
booleanAlgebra : BooleanAlgebra α := e.booleanAlgebra
x✝ y✝ : α
⊢ e x✝ ≤ e y✝ ↔ x✝ ≤ y✝

Tactic:
first | rfl | exact e.apply_symm_apply _

Goal state after:
Module: Mathlib.Order.CompleteBooleanAlgebra

Goal state before:
pp: case lt
α : Type u
β : Type v
ι : Sort w
κ : ι → Sort w'
e : α ≃ β
inst✝ : CompleteBooleanAlgebra β
completeLattice : CompleteLattice α := e.completeLattice
booleanAlgebra : BooleanAlgebra α := e.booleanAlgebra
x✝ y✝ : α
⊢ e x✝ < e y✝ ↔ x✝ < y✝

Tactic:
first | rfl | exact e.apply_symm_apply _

Goal state after:
Module: Mathlib.Order.CompleteBooleanAlgebra

Goal state before:
pp: case map_sup
α : Type u
β : Type v
ι : Sort w
κ : ι → Sort w'
e : α ≃ β
inst✝ : CompleteBooleanAlgebra β
completeLattice : CompleteLattice α := e.completeLattice
booleanAlgebra : BooleanAlgebra α := e.booleanAlgebra
a✝ b✝ : α
⊢ e (a✝ ⊔ b✝) = e a✝ ⊔ e b✝

Tactic:
first | rfl | exact e.apply_symm_apply _

Goal state after:
Module: Mathlib.Order.CompleteBooleanAlgebra

Goal state before:
pp: case map_inf
α : Type u
β : Type v
ι : Sort w
κ : ι → Sort w'
e : α ≃ β
inst✝ : CompleteBooleanAlgebra β
completeLattice : CompleteLattice α := e.completeLattice
booleanAlgebra : BooleanAlgebra α := e.booleanAlgebra
a✝ b✝ : α
⊢ e (a✝ ⊓ b✝) = e a✝ ⊓ e b✝

Tactic:
first | rfl | exact e.apply_symm_apply _

Goal state after:
Module: Mathlib.Order.CompleteBooleanAlgebra

Goal state before:
pp: case map_sSup
α : Type u
β : Type v
ι : Sort w
κ : ι → Sort w'
e : α ≃ β
inst✝ : CompleteBooleanAlgebra β
completeLattice : CompleteLattice α := e.completeLattice
booleanAlgebra : BooleanAlgebra α := e.booleanAlgebra
s✝ : Set α
⊢ e (sSup s✝) = ⨆ a ∈ s✝, e a

Tactic:
first | rfl | exact e.apply_symm_apply _

Goal state after:
Module: Mathlib.Order.CompleteBooleanAlgebra

Goal state before:
pp: case map_sInf
α : Type u
β : Type v
ι : Sort w
κ : ι → Sort w'
e : α ≃ β
inst✝ : CompleteBooleanAlgebra β
completeLattice : CompleteLattice α := e.completeLattice
booleanAlgebra : BooleanAlgebra α := e.booleanAlgebra
s✝ : Set α
⊢ e (sInf s✝) = ⨅ a ∈ s✝, e a

Tactic:
first | rfl | exact e.apply_symm_apply _

Goal state after:
Module: Mathlib.Order.CompleteBooleanAlgebra

Goal state before:
pp: case map_top
α : Type u
β : Type v
ι : Sort w
κ : ι → Sort w'
e : α ≃ β
inst✝ : CompleteBooleanAlgebra β
completeLattice : CompleteLattice α := ⋯
booleanAlgebra : BooleanAlgebra α := ⋯
⊢ e ⊤ = ⊤

Tactic:
first | rfl | exact e.apply_symm_apply _

Goal state after:
Module: Mathlib.Order.CompleteBooleanAlgebra

Goal state before:
pp: case map_bot
α : Type u
β : Type v
ι : Sort w
κ : ι → Sort w'
e : α ≃ β
inst✝ : CompleteBooleanAlgebra β
completeLattice : CompleteLattice α := ⋯
booleanAlgebra : BooleanAlgebra α := ⋯
⊢ e ⊥ = ⊥

Tactic:
first | rfl | exact e.apply_symm_apply _

Goal state after:
Module: Mathlib.Order.CompleteBooleanAlgebra

Goal state before:
pp: case map_compl
α : Type u
β : Type v
ι : Sort w
κ : ι → Sort w'
e : α ≃ β
inst✝ : CompleteBooleanAlgebra β
completeLattice : CompleteLattice α := e.completeLattice
booleanAlgebra : BooleanAlgebra α := e.booleanAlgebra
a✝ : α
⊢ e a✝ᶜ = (e a✝)ᶜ

Tactic:
first | rfl | exact e.apply_symm_apply _

Goal state after:
Module: Mathlib.Order.CompleteBooleanAlgebra

Goal state before:
pp: case map_himp
α : Type u
β : Type v
ι : Sort w
κ : ι → Sort w'
e : α ≃ β
inst✝ : CompleteBooleanAlgebra β
completeLattice : CompleteLattice α := e.completeLattice
booleanAlgebra : BooleanAlgebra α := e.booleanAlgebra
a✝ b✝ : α
⊢ e (a✝ ⇨ b✝) = e a✝ ⇨ e b✝

Tactic:
first | rfl | exact e.apply_symm_apply _

Goal state after:
Module: Mathlib.Order.CompleteBooleanAlgebra

Goal state before:
pp: case map_sdiff
α : Type u
β : Type v
ι : Sort w
κ : ι → Sort w'
e : α ≃ β
inst✝ : CompleteBooleanAlgebra β
completeLattice : CompleteLattice α := e.completeLattice
booleanAlgebra : BooleanAlgebra α := e.booleanAlgebra
a✝ b✝ : α
⊢ e (a✝ \ b✝) = e a✝ \ e b✝

Tactic:
first | rfl | exact e.apply_symm_apply _

Goal state after:
Module: Mathlib.Order.CompleteBooleanAlgebra

Goal state before:
pp: case le
α : Type u
β : Type v
ι : Sort w
κ : ι → Sort w'
e : α ≃ β
inst✝ : CompleteAtomicBooleanAlgebra β
completeBooleanAlgebra : CompleteBooleanAlgebra α := e.completeBooleanAlgebra
x✝ y✝ : α
⊢ e x✝ ≤ e y✝ ↔ x✝ ≤ y✝

Tactic:
first | rfl | exact e.apply_symm_apply _

Goal state after:
Module: Mathlib.Order.CompleteBooleanAlgebra

Goal state before:
pp: case lt
α : Type u
β : Type v
ι : Sort w
κ : ι → Sort w'
e : α ≃ β
inst✝ : CompleteAtomicBooleanAlgebra β
completeBooleanAlgebra : CompleteBooleanAlgebra α := e.completeBooleanAlgebra
x✝ y✝ : α
⊢ e x✝ < e y✝ ↔ x✝ < y✝

Tactic:
first | rfl | exact e.apply_symm_apply _

Goal state after:
Module: Mathlib.Order.CompleteBooleanAlgebra

Goal state before:
pp: case map_sup
α : Type u
β : Type v
ι : Sort w
κ : ι → Sort w'
e : α ≃ β
inst✝ : CompleteAtomicBooleanAlgebra β
completeBooleanAlgebra : CompleteBooleanAlgebra α := e.completeBooleanAlgebra
a✝ b✝ : α
⊢ e (a✝ ⊔ b✝) = e a✝ ⊔ e b✝

Tactic:
first | rfl | exact e.apply_symm_apply _

Goal state after:
Module: Mathlib.Order.CompleteBooleanAlgebra

Goal state before:
pp: case map_inf
α : Type u
β : Type v
ι : Sort w
κ : ι → Sort w'
e : α ≃ β
inst✝ : CompleteAtomicBooleanAlgebra β
completeBooleanAlgebra : CompleteBooleanAlgebra α := e.completeBooleanAlgebra
a✝ b✝ : α
⊢ e (a✝ ⊓ b✝) = e a✝ ⊓ e b✝

Tactic:
first | rfl | exact e.apply_symm_apply _

Goal state after:
Module: Mathlib.Order.CompleteBooleanAlgebra

Goal state before:
pp: case map_sSup
α : Type u
β : Type v
ι : Sort w
κ : ι → Sort w'
e : α ≃ β
inst✝ : CompleteAtomicBooleanAlgebra β
completeBooleanAlgebra : CompleteBooleanAlgebra α := e.completeBooleanAlgebra
s✝ : Set α
⊢ e (sSup s✝) = ⨆ a ∈ s✝, e a

Tactic:
first | rfl | exact e.apply_symm_apply _

Goal state after:
Module: Mathlib.Order.CompleteBooleanAlgebra

Goal state before:
pp: case map_sInf
α : Type u
β : Type v
ι : Sort w
κ : ι → Sort w'
e : α ≃ β
inst✝ : CompleteAtomicBooleanAlgebra β
completeBooleanAlgebra : CompleteBooleanAlgebra α := e.completeBooleanAlgebra
s✝ : Set α
⊢ e (sInf s✝) = ⨅ a ∈ s✝, e a

Tactic:
first | rfl | exact e.apply_symm_apply _

Goal state after:
Module: Mathlib.Order.CompleteBooleanAlgebra

Goal state before:
pp: case map_top
α : Type u
β : Type v
ι : Sort w
κ : ι → Sort w'
e : α ≃ β
inst✝ : CompleteAtomicBooleanAlgebra β
completeBooleanAlgebra : CompleteBooleanAlgebra α := ⋯
⊢ e ⊤ = ⊤

Tactic:
first | rfl | exact e.apply_symm_apply _

Goal state after:
Module: Mathlib.Order.CompleteBooleanAlgebra

Goal state before:
pp: case map_bot
α : Type u
β : Type v
ι : Sort w
κ : ι → Sort w'
e : α ≃ β
inst✝ : CompleteAtomicBooleanAlgebra β
completeBooleanAlgebra : CompleteBooleanAlgebra α := ⋯
⊢ e ⊥ = ⊥

Tactic:
first | rfl | exact e.apply_symm_apply _

Goal state after:
Module: Mathlib.Order.CompleteBooleanAlgebra

Goal state before:
pp: case map_compl
α : Type u
β : Type v
ι : Sort w
κ : ι → Sort w'
e : α ≃ β
inst✝ : CompleteAtomicBooleanAlgebra β
completeBooleanAlgebra : CompleteBooleanAlgebra α := e.completeBooleanAlgebra
a✝ : α
⊢ e a✝ᶜ = (e a✝)ᶜ

Tactic:
first | rfl | exact e.apply_symm_apply _

Goal state after:
Module: Mathlib.Order.CompleteBooleanAlgebra

Goal state before:
pp: case map_himp
α : Type u
β : Type v
ι : Sort w
κ : ι → Sort w'
e : α ≃ β
inst✝ : CompleteAtomicBooleanAlgebra β
completeBooleanAlgebra : CompleteBooleanAlgebra α := e.completeBooleanAlgebra
a✝ b✝ : α
⊢ e (a✝ ⇨ b✝) = e a✝ ⇨ e b✝

Tactic:
first | rfl | exact e.apply_symm_apply _

Goal state after:
Module: Mathlib.Order.CompleteBooleanAlgebra

Goal state before:
pp: case map_hnot
α : Type u
β : Type v
ι : Sort w
κ : ι → Sort w'
e : α ≃ β
inst✝ : CompleteAtomicBooleanAlgebra β
completeBooleanAlgebra : CompleteBooleanAlgebra α := e.completeBooleanAlgebra
a✝ : α
⊢ e (￢a✝) = ￢e a✝

Tactic:
first | rfl | exact e.apply_symm_apply _

Goal state after:
Module: Mathlib.Order.CompleteBooleanAlgebra

Goal state before:
pp: case map_sdiff
α : Type u
β : Type v
ι : Sort w
κ : ι → Sort w'
e : α ≃ β
inst✝ : CompleteAtomicBooleanAlgebra β
completeBooleanAlgebra : CompleteBooleanAlgebra α := e.completeBooleanAlgebra
a✝ b✝ : α
⊢ e (a✝ \ b✝) = e a✝ \ e b✝

Tactic:
first | rfl | exact e.apply_symm_apply _

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

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 : IsMaxChain r c
t : Set β
ht : IsChain s t
hf : ⇑e '' c ⊆ t
⊢ c = ⇑e.toEquiv ⁻¹' t

Tactic:
← Equiv.image_symm_eq_preimage

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 : IsMaxChain r c
t : Set β
ht : IsChain s t
hf : ⇑e '' c ⊆ t
⊢ c = ⇑e.symm '' t
Module: Mathlib.Order.Interval.Set.OrdConnected

Goal state before:
pp: α : Type u_1
inst✝ : PartialOrder α
s : Set α
x y : α
hx : x ∈ s
hy : y ∈ s
⊢ ¬(s ∩ Icc x y).OrdConnected ↔ ∃ z ∉ s, z ∈ Ioo x y

Tactic:
by
  simp_rw [ordConnected_inter_Icc_iff hx hy, subset_def, not_forall, exists_prop, and_comm]

Goal state after:
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.toEquiv ⁻¹' t

Tactic:
← Equiv.image_symm_eq_preimage

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.Data.Nat.Factorial.Basic

Goal state before:
pp: m n : ℕ
hn : 1 < n
⊢ n ! = m ! ↔ n = m

Tactic:
refine ⟨fun h => ?_, congr_arg _⟩

Goal state after:
m n : ℕ
hn : 1 < n
h : n ! = m !
⊢ n = m
Module: Mathlib.Data.Nat.Factorial.Basic

Goal state before:
pp: n : ℕ
hi : 3 ≤ n
this : 0 < n
hn : 1 < n.pred
⊢ n.pred.succ < n.pred.succ !

Tactic:
factorial_succ

Goal state after:
n : ℕ
hi : 3 ≤ n
this : 0 < n
hn : 1 < n.pred
⊢ n.pred.succ < (n.pred + 1) * n.pred !
Module: Mathlib.Data.Nat.Factorial.Basic

Goal state before:
pp: k : ℕ
⊢ (k + 1) * k ! = (k + 1)!

Tactic:
factorial_succ

Goal state after:
k : ℕ
⊢ (k + 1) * k ! = (k + 1) * k !
Module: Mathlib.Data.Nat.Factorial.Basic

Goal state before:
pp: n : ℕ
⊢ (n + 1) * n ! = n.succ !

Tactic:
factorial_succ

Goal state after:
n : ℕ
⊢ (n + 1) * n ! = (n + 1) * n !
Module: Mathlib.Data.Nat.Factorial.Basic

Goal state before:
pp: n k : ℕ
⊢ (n.succ + k - 1 - k) * (n.succ + k - 1).descFactorial k = (n + k) * n.ascFactorial k

Tactic:
add_descFactorial_eq_ascFactorial' _ k,

Goal state after:
n k : ℕ
⊢ (n.succ + k - 1 - k) * n.succ.ascFactorial k = (n + k) * n.ascFactorial k
Module: Mathlib.Data.Nat.Factorial.Basic

Goal state before:
pp: n k : ℕ
h : k.succ ≤ n.succ
⊢ (n + 1) * n ! = n.succ !

Tactic:
factorial_succ

Goal state after:
n k : ℕ
h : k.succ ≤ n.succ
⊢ (n + 1) * n ! = (n + 1) * n !
Module: Mathlib.Data.Rat.Cast.Lemmas

Goal state before:
pp: K : Type u_2
inst✝ : DivisionRing K
m : ℕ
s : Bool
e : ℕ
⊢ ↑(OfScientific.ofScientific m s e) = OfScientific.ofScientific m s e

Tactic:
rw [← NNRat.cast_ofScientific (K := K), ← NNRat.cast_ofScientific, cast_nnratCast]

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

Goal state before:
pp: R : Type u_2
inst✝³ : Ring R
inst✝² : LinearOrder R
inst✝¹ : FloorRing R
inst✝ : IsOrderedRing R
a b : R
⊢ fract (a + b) ≤ fract a + fract b

Tactic:
rw [fract, fract, fract, sub_add_sub_comm, sub_le_sub_iff_left, ← Int.cast_add, Int.cast_le]

Goal state after:
R : Type u_2
inst✝³ : Ring R
inst✝² : LinearOrder R
inst✝¹ : FloorRing R
inst✝ : IsOrderedRing R
a b : R
⊢ ⌊a⌋ + ⌊b⌋ ≤ ⌊a + b⌋