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✝ : Frame β
completeLattice : CompleteLattice α := e.completeLattice
heytingAlgebra : HeytingAlgebra α := e.heytingAlgebra
x✝ y✝ : α
⊢ e x✝ ≤ e y✝ ↔ x✝ ≤ y✝

pp: case lt
α : Type u
β : Type v
ι : Sort w
κ : ι → Sort w'
e : α ≃ β
inst✝ : Frame β
completeLattice : CompleteLattice α := e.completeLattice
heytingAlgebra : HeytingAlgebra α := e.heytingAlgebra
x✝ y✝ : α
⊢ e x✝ < e y✝ ↔ x✝ < y✝

pp: case map_sup
α : Type u
β : Type v
ι : Sort w
κ : ι → Sort w'
e : α ≃ β
inst✝ : Frame β
completeLattice : CompleteLattice α := e.completeLattice
heytingAlgebra : HeytingAlgebra α := e.heytingAlgebra
a✝ b✝ : α
⊢ e (a✝ ⊔ b✝) = e a✝ ⊔ e b✝

pp: case map_inf
α : Type u
β : Type v
ι : Sort w
κ : ι → Sort w'
e : α ≃ β
inst✝ : Frame β
completeLattice : CompleteLattice α := e.completeLattice
heytingAlgebra : HeytingAlgebra α := e.heytingAlgebra
a✝ b✝ : α
⊢ e (a✝ ⊓ b✝) = e a✝ ⊓ e b✝

pp: case map_sSup
α : Type u
β : Type v
ι : Sort w
κ : ι → Sort w'
e : α ≃ β
inst✝ : Frame β
completeLattice : CompleteLattice α := e.completeLattice
heytingAlgebra : HeytingAlgebra α := e.heytingAlgebra
s✝ : Set α
⊢ e (sSup s✝) = ⨆ a ∈ s✝, e a

pp: case map_sInf
α : Type u
β : Type v
ι : Sort w
κ : ι → Sort w'
e : α ≃ β
inst✝ : Frame β
completeLattice : CompleteLattice α := e.completeLattice
heytingAlgebra : HeytingAlgebra α := e.heytingAlgebra
s✝ : Set α
⊢ e (sInf s✝) = ⨅ a ∈ s✝, e a

pp: case map_top
α : Type u
β : Type v
ι : Sort w
κ : ι → Sort w'
e : α ≃ β
inst✝ : Frame β
completeLattice : CompleteLattice α := ⋯
heytingAlgebra : HeytingAlgebra α := ⋯
⊢ e ⊤ = ⊤

pp: case map_bot
α : Type u
β : Type v
ι : Sort w
κ : ι → Sort w'
e : α ≃ β
inst✝ : Frame β
completeLattice : CompleteLattice α := ⋯
heytingAlgebra : HeytingAlgebra α := ⋯
⊢ e ⊥ = ⊥

pp: case map_compl
α : Type u
β : Type v
ι : Sort w
κ : ι → Sort w'
e : α ≃ β
inst✝ : Frame β
completeLattice : CompleteLattice α := e.completeLattice
heytingAlgebra : HeytingAlgebra α := e.heytingAlgebra
a✝ : α
⊢ e a✝ᶜ = (e a✝)ᶜ

pp: case map_himp
α : Type u
β : Type v
ι : Sort w
κ : ι → Sort w'
e : α ≃ β
inst✝ : Frame β
completeLattice : CompleteLattice α := e.completeLattice
heytingAlgebra : HeytingAlgebra α := e.heytingAlgebra
a✝ b✝ : α
⊢ e (a✝ ⇨ b✝) = e a✝ ⇨ e b✝

Tactic:
<;>

Goal state after:
case le
α : Type u
β : Type v
ι : Sort w
κ : ι → Sort w'
e : α ≃ β
inst✝ : Frame β
completeLattice : CompleteLattice α := e.completeLattice
heytingAlgebra : HeytingAlgebra α := e.heytingAlgebra
x✝ y✝ : α
⊢ e x✝ ≤ e y✝ ↔ x✝ ≤ y✝

case lt
α : Type u
β : Type v
ι : Sort w
κ : ι → Sort w'
e : α ≃ β
inst✝ : Frame β
completeLattice : CompleteLattice α := e.completeLattice
heytingAlgebra : HeytingAlgebra α := e.heytingAlgebra
x✝ y✝ : α
⊢ e x✝ < e y✝ ↔ x✝ < y✝

case map_sup
α : Type u
β : Type v
ι : Sort w
κ : ι → Sort w'
e : α ≃ β
inst✝ : Frame β
completeLattice : CompleteLattice α := e.completeLattice
heytingAlgebra : HeytingAlgebra α := e.heytingAlgebra
a✝ b✝ : α
⊢ e (a✝ ⊔ b✝) = e a✝ ⊔ e b✝

case map_inf
α : Type u
β : Type v
ι : Sort w
κ : ι → Sort w'
e : α ≃ β
inst✝ : Frame β
completeLattice : CompleteLattice α := e.completeLattice
heytingAlgebra : HeytingAlgebra α := e.heytingAlgebra
a✝ b✝ : α
⊢ e (a✝ ⊓ b✝) = e a✝ ⊓ e b✝

case map_sSup
α : Type u
β : Type v
ι : Sort w
κ : ι → Sort w'
e : α ≃ β
inst✝ : Frame β
completeLattice : CompleteLattice α := e.completeLattice
heytingAlgebra : HeytingAlgebra α := e.heytingAlgebra
s✝ : Set α
⊢ e (sSup s✝) = ⨆ a ∈ s✝, e a

case map_sInf
α : Type u
β : Type v
ι : Sort w
κ : ι → Sort w'
e : α ≃ β
inst✝ : Frame β
completeLattice : CompleteLattice α := e.completeLattice
heytingAlgebra : HeytingAlgebra α := e.heytingAlgebra
s✝ : Set α
⊢ e (sInf s✝) = ⨅ a ∈ s✝, e a

case map_top
α : Type u
β : Type v
ι : Sort w
κ : ι → Sort w'
e : α ≃ β
inst✝ : Frame β
completeLattice : CompleteLattice α := ⋯
heytingAlgebra : HeytingAlgebra α := ⋯
⊢ e ⊤ = ⊤

case map_bot
α : Type u
β : Type v
ι : Sort w
κ : ι → Sort w'
e : α ≃ β
inst✝ : Frame β
completeLattice : CompleteLattice α := ⋯
heytingAlgebra : HeytingAlgebra α := ⋯
⊢ e ⊥ = ⊥

case map_compl
α : Type u
β : Type v
ι : Sort w
κ : ι → Sort w'
e : α ≃ β
inst✝ : Frame β
completeLattice : CompleteLattice α := e.completeLattice
heytingAlgebra : HeytingAlgebra α := e.heytingAlgebra
a✝ : α
⊢ e a✝ᶜ = (e a✝)ᶜ

case map_himp
α : Type u
β : Type v
ι : Sort w
κ : ι → Sort w'
e : α ≃ β
inst✝ : Frame β
completeLattice : CompleteLattice α := e.completeLattice
heytingAlgebra : HeytingAlgebra α := e.heytingAlgebra
a✝ b✝ : α
⊢ e (a✝ ⇨ b✝) = e a✝ ⇨ e b✝