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

Goal state before:
pp: α✝ : Type u
β : Type v
α : Type u_1
inst✝¹ : Max α
inst✝ : Min α
sup_comm : ∀ (a b : α), a ⊔ b = b ⊔ a
sup_assoc : ∀ (a b c : α), a ⊔ b ⊔ c = a ⊔ (b ⊔ c)
inf_comm : ∀ (a b : α), a ⊓ b = b ⊓ a
inf_assoc : ∀ (a b c : α), a ⊓ b ⊓ c = a ⊓ (b ⊓ c)
sup_inf_self : ∀ (a b : α), a ⊔ a ⊓ b = a
inf_sup_self : ∀ (a b : α), a ⊓ (a ⊔ b) = a
sup_idem : ∀ (b : α), b ⊔ b = b
b : α
⊢ b = b

Tactic:
]

Goal state after:
Module: Mathlib.Order.Lattice

Goal state before:
pp: α✝ : Type u
β : Type v
α : Type u_1
inst✝¹ : Max α
inst✝ : Min α
sup_comm : ∀ (a b : α), a ⊔ b = b ⊔ a
sup_assoc : ∀ (a b c : α), a ⊔ b ⊔ c = a ⊔ (b ⊔ c)
inf_comm : ∀ (a b : α), a ⊓ b = b ⊓ a
inf_assoc : ∀ (a b c : α), a ⊓ b ⊓ c = a ⊓ (b ⊓ c)
sup_inf_self : ∀ (a b : α), a ⊔ a ⊓ b = a
inf_sup_self : ∀ (a b : α), a ⊓ (a ⊔ b) = a
sup_idem : ∀ (b : α), b ⊔ b = b
inf_idem : ∀ (b : α), b ⊓ b = b
semilatt_inf_inst : SemilatticeInf α := SemilatticeInf.mk' inf_comm inf_assoc inf_idem
semilatt_sup_inst : SemilatticeSup α := SemilatticeSup.mk' sup_comm sup_assoc sup_idem
partial_order_eq : semilatt_sup_inst.toPartialOrder = semilatt_inf_inst.toPartialOrder
a b : α
⊢ SemilatticeInf.inf a b ≤ a

Tactic:
]

Goal state after:
α✝ : Type u
β : Type v
α : Type u_1
inst✝¹ : Max α
inst✝ : Min α
sup_comm : ∀ (a b : α), a ⊔ b = b ⊔ a
sup_assoc : ∀ (a b c : α), a ⊔ b ⊔ c = a ⊔ (b ⊔ c)
inf_comm : ∀ (a b : α), a ⊓ b = b ⊓ a
inf_assoc : ∀ (a b c : α), a ⊓ b ⊓ c = a ⊓ (b ⊓ c)
sup_inf_self : ∀ (a b : α), a ⊔ a ⊓ b = a
inf_sup_self : ∀ (a b : α), a ⊓ (a ⊔ b) = a
sup_idem : ∀ (b : α), b ⊔ b = b
inf_idem : ∀ (b : α), b ⊓ b = b
semilatt_inf_inst : SemilatticeInf α := SemilatticeInf.mk' inf_comm inf_assoc inf_idem
semilatt_sup_inst : SemilatticeSup α := SemilatticeSup.mk' sup_comm sup_assoc sup_idem
partial_order_eq : semilatt_sup_inst.toPartialOrder = semilatt_inf_inst.toPartialOrder
a b : α
⊢ SemilatticeInf.inf a b ≤ a
Module: Mathlib.Order.Lattice

Goal state before:
pp: α✝ : Type u
β : Type v
α : Type u_1
inst✝¹ : Max α
inst✝ : Min α
sup_comm : ∀ (a b : α), a ⊔ b = b ⊔ a
sup_assoc : ∀ (a b c : α), a ⊔ b ⊔ c = a ⊔ (b ⊔ c)
inf_comm : ∀ (a b : α), a ⊓ b = b ⊓ a
inf_assoc : ∀ (a b c : α), a ⊓ b ⊓ c = a ⊓ (b ⊓ c)
sup_inf_self : ∀ (a b : α), a ⊔ a ⊓ b = a
inf_sup_self : ∀ (a b : α), a ⊓ (a ⊔ b) = a
sup_idem : ∀ (b : α), b ⊔ b = b
inf_idem : ∀ (b : α), b ⊓ b = b
semilatt_inf_inst : SemilatticeInf α := SemilatticeInf.mk' inf_comm inf_assoc inf_idem
semilatt_sup_inst : SemilatticeSup α := SemilatticeSup.mk' sup_comm sup_assoc sup_idem
partial_order_eq : semilatt_sup_inst.toPartialOrder = semilatt_inf_inst.toPartialOrder
a b : α
⊢ SemilatticeInf.inf a b ≤ b

Tactic:
]

Goal state after:
α✝ : Type u
β : Type v
α : Type u_1
inst✝¹ : Max α
inst✝ : Min α
sup_comm : ∀ (a b : α), a ⊔ b = b ⊔ a
sup_assoc : ∀ (a b c : α), a ⊔ b ⊔ c = a ⊔ (b ⊔ c)
inf_comm : ∀ (a b : α), a ⊓ b = b ⊓ a
inf_assoc : ∀ (a b c : α), a ⊓ b ⊓ c = a ⊓ (b ⊓ c)
sup_inf_self : ∀ (a b : α), a ⊔ a ⊓ b = a
inf_sup_self : ∀ (a b : α), a ⊓ (a ⊔ b) = a
sup_idem : ∀ (b : α), b ⊔ b = b
inf_idem : ∀ (b : α), b ⊓ b = b
semilatt_inf_inst : SemilatticeInf α := SemilatticeInf.mk' inf_comm inf_assoc inf_idem
semilatt_sup_inst : SemilatticeSup α := SemilatticeSup.mk' sup_comm sup_assoc sup_idem
partial_order_eq : semilatt_sup_inst.toPartialOrder = semilatt_inf_inst.toPartialOrder
a b : α
⊢ SemilatticeInf.inf a b ≤ b
Module: Mathlib.Order.Lattice

Goal state before:
pp: α✝ : Type u
β : Type v
α : Type u_1
inst✝¹ : Max α
inst✝ : Min α
sup_comm : ∀ (a b : α), a ⊔ b = b ⊔ a
sup_assoc : ∀ (a b c : α), a ⊔ b ⊔ c = a ⊔ (b ⊔ c)
inf_comm : ∀ (a b : α), a ⊓ b = b ⊓ a
inf_assoc : ∀ (a b c : α), a ⊓ b ⊓ c = a ⊓ (b ⊓ c)
sup_inf_self : ∀ (a b : α), a ⊔ a ⊓ b = a
inf_sup_self : ∀ (a b : α), a ⊓ (a ⊔ b) = a
sup_idem : ∀ (b : α), b ⊔ b = b
inf_idem : ∀ (b : α), b ⊓ b = b
semilatt_inf_inst : SemilatticeInf α := SemilatticeInf.mk' inf_comm inf_assoc inf_idem
semilatt_sup_inst : SemilatticeSup α := SemilatticeSup.mk' sup_comm sup_assoc sup_idem
partial_order_eq : semilatt_sup_inst.toPartialOrder = semilatt_inf_inst.toPartialOrder
a b c : α
⊢ a ≤ b → a ≤ c → a ≤ SemilatticeInf.inf b c

Tactic:
]

Goal state after:
α✝ : Type u
β : Type v
α : Type u_1
inst✝¹ : Max α
inst✝ : Min α
sup_comm : ∀ (a b : α), a ⊔ b = b ⊔ a
sup_assoc : ∀ (a b c : α), a ⊔ b ⊔ c = a ⊔ (b ⊔ c)
inf_comm : ∀ (a b : α), a ⊓ b = b ⊓ a
inf_assoc : ∀ (a b c : α), a ⊓ b ⊓ c = a ⊓ (b ⊓ c)
sup_inf_self : ∀ (a b : α), a ⊔ a ⊓ b = a
inf_sup_self : ∀ (a b : α), a ⊓ (a ⊔ b) = a
sup_idem : ∀ (b : α), b ⊔ b = b
inf_idem : ∀ (b : α), b ⊓ b = b
semilatt_inf_inst : SemilatticeInf α := SemilatticeInf.mk' inf_comm inf_assoc inf_idem
semilatt_sup_inst : SemilatticeSup α := SemilatticeSup.mk' sup_comm sup_assoc sup_idem
partial_order_eq : semilatt_sup_inst.toPartialOrder = semilatt_inf_inst.toPartialOrder
a b c : α
⊢ a ≤ b → a ≤ c → a ≤ SemilatticeInf.inf b c
Module: Mathlib.Order.Lattice

Goal state before:
pp: α : Type u
inst✝ : Lattice α
a b : α
⊢ a = b ↔ a = b

Tactic:
]

Goal state after:
Module: Mathlib.Order.Lattice

Goal state before:
pp: α : Type u
inst✝ : Lattice α
a b : α
⊢ ¬a = b ↔ a ≠ b

Tactic:
]

Goal state after:
Module: Mathlib.Order.Lattice

Goal state before:
pp: α : Type u
inst✝ : Lattice α
a b : α
⊢ a ⊓ b = a ↔ a ⊓ b = a

Tactic:
]

Goal state after:
Module: Mathlib.Order.Lattice

Goal state before:
pp: α : Type u
inst✝ : DistribLattice α
a b c : α
⊢ a ⊓ (b ⊔ c) = a ⊓ (b ⊔ c)

Tactic:
]

Goal state after:
Module: Mathlib.Order.Lattice

Goal state before:
pp: α : Type u
inst✝ : DistribLattice α
a b c : α
⊢ a ⊓ (a ⊓ b ⊔ c) = a ⊓ (a ⊓ b ⊔ c)

Tactic:
]

Goal state after:
Module: Mathlib.Order.Lattice

Goal state before:
pp: α : Type u
inst✝ : DistribLattice α
a b c : α
⊢ (a ⊓ b ⊔ a) ⊓ (a ⊓ b ⊔ c) = (a ⊓ b ⊔ a) ⊓ (a ⊓ b ⊔ c)

Tactic:
]

Goal state after:
Module: Mathlib.Order.Lattice

Goal state before:
pp: α : Type u
inst✝ : DistribLattice α
x y z : α
⊢ x ⊓ y ⊔ x ⊓ z ≤ x ⊓ y ⊔ x ⊓ z

Tactic:
]

Goal state after:
Module: Mathlib.Order.Lattice

Goal state before:
pp: α : Type u
inst✝ : DistribLattice α
x y z : α
h₁ : x ⊓ z ≤ y ⊓ z
h₂ : x ⊔ z ≤ y ⊔ z
⊢ (y ⊔ x) ⊓ (z ⊔ x) = (y ⊔ x) ⊓ (z ⊔ x)

Tactic:
]

Goal state after:
Module: Mathlib.Order.Lattice

Goal state before:
pp: α : Type u
inst✝ : DistribLattice α
x y z : α
h₁ : x ⊓ z ≤ y ⊓ z
h₂ : x ⊔ z ≤ y ⊔ z
⊢ y ⊔ x ⊓ z = y ⊔ x ⊓ z

Tactic:
]

Goal state after:
Module: Mathlib.Order.Lattice

Goal state before:
pp: α : Type u
inst✝ : LinearOrder α
a b : α
p : α → Prop
ha : p a
hb : p b
h : a ≤ b
⊢ p b

Tactic:
]

Goal state after:
α : Type u
inst✝ : LinearOrder α
a b : α
p : α → Prop
ha : p a
hb : p b
h : a ≤ b
⊢ p b
Module: Mathlib.Order.Lattice

Goal state before:
pp: α : Type u
inst✝ : LinearOrder α
a b : α
p : α → Prop
ha : p a
hb : p b
h : b ≤ a
⊢ p a

Tactic:
]

Goal state after:
α : Type u
inst✝ : LinearOrder α
a b : α
p : α → Prop
ha : p a
hb : p b
h : b ≤ a
⊢ p a
Module: Mathlib.Order.Lattice

Goal state before:
pp: α : Type u
β : Type v
inst✝¹ : SemilatticeSup α
inst✝ : Preorder β
f : α → β
h : ∀ (x y : α), f x ≤ f (x ⊔ y)
x y : α
hxy : x ≤ y
⊢ f x ≤ f (x ⊔ y)

Tactic:
]

Goal state after:
α : Type u
β : Type v
inst✝¹ : SemilatticeSup α
inst✝ : Preorder β
f : α → β
h : ∀ (x y : α), f x ≤ f (x ⊔ y)
x y : α
hxy : x ≤ y
⊢ f x ≤ f (x ⊔ y)
Module: Mathlib.Order.Lattice

Goal state before:
pp: α : Type u
β : Type v
f : α → β
s : Set α
inst✝¹ : SemilatticeSup α
inst✝ : SemilatticeSup β
h : ∀ (x : α), x ∈ s → ∀ (y : α), y ∈ s → f (x ⊔ y) = f x ⊔ f y
x : α
hx : x ∈ s
y : α
hy : y ∈ s
hxy : x ≤ y
⊢ f y = f y

Tactic:
]

Goal state after:
Module: Mathlib.Order.Lattice

Goal state before:
pp: α : Type u
β : Type v
f : α → β
s : Set α
inst✝¹ : SemilatticeInf α
inst✝ : SemilatticeSup β
h : ∀ (x : α), x ∈ s → ∀ (y : α), y ∈ s → f (x ⊓ y) = f x ⊔ f y
x : α
hx : x ∈ s
y : α
hy : y ∈ s
hxy : x ≤ y
⊢ f x = f x

Tactic:
]

Goal state after:
Module: Mathlib.Order.Lattice

Goal state before:
pp: α : Type u
β : Type v
inst✝³ : Max α
inst✝² : LE α
inst✝¹ : LT α
inst✝ : SemilatticeSup β
f : α → β
hf_inj : Injective f
le : ∀ {x y : α}, f x ≤ f y ↔ x ≤ y
lt : ∀ {x y : α}, f x < f y ↔ x < y
map_sup : ∀ (a b : α), f (a ⊔ b) = f a ⊔ f b
a b : α
⊢ f a ≤ f a ⊔ f b

Tactic:
]

Goal state after:
α : Type u
β : Type v
inst✝³ : Max α
inst✝² : LE α
inst✝¹ : LT α
inst✝ : SemilatticeSup β
f : α → β
hf_inj : Injective f
le : ∀ {x y : α}, f x ≤ f y ↔ x ≤ y
lt : ∀ {x y : α}, f x < f y ↔ x < y
map_sup : ∀ (a b : α), f (a ⊔ b) = f a ⊔ f b
a b : α
⊢ f a ≤ f a ⊔ f b
Module: Mathlib.Order.Lattice

Goal state before:
pp: α : Type u
β : Type v
inst✝³ : Max α
inst✝² : LE α
inst✝¹ : LT α
inst✝ : SemilatticeSup β
f : α → β
hf_inj : Injective f
le : ∀ {x y : α}, f x ≤ f y ↔ x ≤ y
lt : ∀ {x y : α}, f x < f y ↔ x < y
map_sup : ∀ (a b : α), f (a ⊔ b) = f a ⊔ f b
a b : α
⊢ f b ≤ f a ⊔ f b

Tactic:
]

Goal state after:
α : Type u
β : Type v
inst✝³ : Max α
inst✝² : LE α
inst✝¹ : LT α
inst✝ : SemilatticeSup β
f : α → β
hf_inj : Injective f
le : ∀ {x y : α}, f x ≤ f y ↔ x ≤ y
lt : ∀ {x y : α}, f x < f y ↔ x < y
map_sup : ∀ (a b : α), f (a ⊔ b) = f a ⊔ f b
a b : α
⊢ f b ≤ f a ⊔ f b
Module: Mathlib.Order.Lattice

Goal state before:
pp: α : Type u
β : Type v
inst✝³ : Max α
inst✝² : LE α
inst✝¹ : LT α
inst✝ : SemilatticeSup β
f : α → β
hf_inj : Injective f
le : ∀ {x y : α}, f x ≤ f y ↔ x ≤ y
lt : ∀ {x y : α}, f x < f y ↔ x < y
map_sup : ∀ (a b : α), f (a ⊔ b) = f a ⊔ f b
a b c : α
ha : f a ≤ f c
hb : f b ≤ f c
⊢ f (a ⊔ b) ≤ f c

Tactic:
]

Goal state after:
α : Type u
β : Type v
inst✝³ : Max α
inst✝² : LE α
inst✝¹ : LT α
inst✝ : SemilatticeSup β
f : α → β
hf_inj : Injective f
le : ∀ {x y : α}, f x ≤ f y ↔ x ≤ y
lt : ∀ {x y : α}, f x < f y ↔ x < y
map_sup : ∀ (a b : α), f (a ⊔ b) = f a ⊔ f b
a b c : α
ha : f a ≤ f c
hb : f b ≤ f c
⊢ f (a ⊔ b) ≤ f c
Module: Mathlib.Order.Lattice

Goal state before:
pp: α : Type u
β : Type v
inst✝³ : Max α
inst✝² : LE α
inst✝¹ : LT α
inst✝ : SemilatticeSup β
f : α → β
hf_inj : Injective f
le : ∀ {x y : α}, f x ≤ f y ↔ x ≤ y
lt : ∀ {x y : α}, f x < f y ↔ x < y
map_sup : ∀ (a b : α), f (a ⊔ b) = f a ⊔ f b
a b c : α
ha : f a ≤ f c
hb : f b ≤ f c
⊢ f a ⊔ f b ≤ f c

Tactic:
]

Goal state after:
α : Type u
β : Type v
inst✝³ : Max α
inst✝² : LE α
inst✝¹ : LT α
inst✝ : SemilatticeSup β
f : α → β
hf_inj : Injective f
le : ∀ {x y : α}, f x ≤ f y ↔ x ≤ y
lt : ∀ {x y : α}, f x < f y ↔ x < y
map_sup : ∀ (a b : α), f (a ⊔ b) = f a ⊔ f b
a b c : α
ha : f a ≤ f c
hb : f b ≤ f c
⊢ f a ⊔ f b ≤ f c
Module: Mathlib.Data.Set.Subsingleton

Goal state before:
pp: α : Type u
a : α
s✝ t : Set α
inst✝ : Subsingleton α
s : Set α
⊢ s.Subsingleton

Tactic:
]

Goal state after:
α : Type u
a : α
s✝ t : Set α
inst✝ : Subsingleton α
s : Set α
⊢ s.Subsingleton
Module: Mathlib.Data.Set.Subsingleton

Goal state before:
pp: α : Type u
s : Set α
z : α
H : ∀ (x : α), x ∈ s → x = z
x : α
hx : x ∈ s
y : α
hy : y ∈ s
hxy : z ≠ z
⊢ False

Tactic:
]

Goal state after:
α : Type u
s : Set α
z : α
H : ∀ (x : α), x ∈ s → x = z
x : α
hx : x ∈ s
y : α
hy : y ∈ s
hxy : z ≠ z
⊢ False
Module: Mathlib.Data.Set.Subsingleton

Goal state before:
pp: α : Type u
x : α
H : ∃ y, y ∈ {x} ∧ y ≠ x
⊢ False

Tactic:
]

Goal state after:
α : Type u
x : α
H : ∃ y, y ∈ {x} ∧ y ≠ x
⊢ False
Module: Mathlib.Data.Set.Subsingleton

Goal state before:
pp: α : Type u
s : Set α
x : α
hs : {x}.Nontrivial
H : s = {x}
⊢ False

Tactic:
]

Goal state after:
α : Type u
s : Set α
x : α
hs : {x}.Nontrivial
H : s = {x}
⊢ False
Module: Mathlib.Data.Set.Subsingleton

Goal state before:
pp: α : Type u
a : α
s : Set α
ha : a ∈ s
⊢ s.Subsingleton ∨ s.Nontrivial

Tactic:
]

Goal state after:
α : Type u
a : α
s : Set α
ha : a ∈ s
⊢ s.Subsingleton ∨ s.Nontrivial
Module: Mathlib.Data.Set.Subsingleton

Goal state before:
pp: x : Prop
⊢ x = True ∨ x = False

Tactic:
]

Goal state after:
x : Prop
⊢ x = True ∨ x = False
Module: Mathlib.Order.Lattice

Goal state before:
pp: α : Type u
β : Type v
inst✝⁴ : Max α
inst✝³ : Min α
inst✝² : LE α
inst✝¹ : LT α
inst✝ : DistribLattice β
f : α → β
hf_inj : Injective f
le : ∀ {x y : α}, f x ≤ f y ↔ x ≤ y
lt : ∀ {x y : α}, f x < f y ↔ x < y
map_sup : ∀ (a b : α), f (a ⊔ b) = f a ⊔ f b
map_inf : ∀ (a b : α), f (a ⊓ b) = f a ⊓ f b
a b c : α
⊢ (f a ⊔ f b) ⊓ (f a ⊔ f c) ≤ f a ⊔ f b ⊓ f c

Tactic:
]

Goal state after:
α : Type u
β : Type v
inst✝⁴ : Max α
inst✝³ : Min α
inst✝² : LE α
inst✝¹ : LT α
inst✝ : DistribLattice β
f : α → β
hf_inj : Injective f
le : ∀ {x y : α}, f x ≤ f y ↔ x ≤ y
lt : ∀ {x y : α}, f x < f y ↔ x < y
map_sup : ∀ (a b : α), f (a ⊔ b) = f a ⊔ f b
map_inf : ∀ (a b : α), f (a ⊓ b) = f a ⊓ f b
a b c : α
⊢ (f a ⊔ f b) ⊓ (f a ⊔ f c) ≤ f a ⊔ f b ⊓ f c
Module: Mathlib.Order.GaloisConnection.Defs

Goal state before:
pp: α : Type u
β : Type v
inst✝¹ : Preorder α
inst✝ : Preorder β
l : α → β
u : β → α
⊢ (∀ (b : α) (a : β), u a ≤ b ↔ a ≤ l b) = ∀ (a : α) (b : β), b ≤ l a ↔ u b ≤ a

Tactic:
]

Goal state after:
α : Type u
β : Type v
inst✝¹ : Preorder α
inst✝ : Preorder β
l : α → β
u : β → α
⊢ (∀ (b : α) (a : β), u a ≤ b ↔ a ≤ l b) = ∀ (a : α) (b : β), b ≤ l a ↔ u b ≤ a
Module: Mathlib.Order.Heyting.Basic

Goal state before:
pp: α : Type u_2
inst✝ : GeneralizedCoheytingAlgebra α
a b c : α
⊢ a ≤ c ⊔ b ↔ a ≤ c ⊔ b

Tactic:
]

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

Goal state before:
pp: α : Type u_2
inst✝ : GeneralizedHeytingAlgebra α
a b c : α
⊢ b ⊓ a ≤ c ↔ b ⊓ a ≤ c

Tactic:
]

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

Goal state before:
pp: α : Type u_2
inst✝ : GeneralizedHeytingAlgebra α
a b c : α
⊢ a ⊓ b ≤ c ↔ a ⊓ b ≤ c

Tactic:
]

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

Goal state before:
pp: α : Type u_2
inst✝ : GeneralizedHeytingAlgebra α
a b : α
⊢ a ≤ b ↔ a ≤ b

Tactic:
]

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

Goal state before:
pp: α : Type u_2
inst✝ : GeneralizedHeytingAlgebra α
a b : α
⊢ a ⇨ b ≤ a ⇨ b

Tactic:
]

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

Goal state before:
pp: α : Type u_2
inst✝ : GeneralizedHeytingAlgebra α
a b : α
⊢ b ⊓ a = b ⊓ a

Tactic:
]

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

Goal state before:
pp: α : Type u_2
inst✝ : GeneralizedHeytingAlgebra α
a b : α
⊢ b ≤ a ↔ b ≤ a

Tactic:
]

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

Goal state before:
pp: α : Type u_2
inst✝ : GeneralizedHeytingAlgebra α
a b c : α
⊢ c ⊓ (b ⊓ a) ≤ c

Tactic:
]

Goal state after:
α : Type u_2
inst✝ : GeneralizedHeytingAlgebra α
a b c : α
⊢ c ⊓ (b ⊓ a) ≤ c
Module: Mathlib.Order.Heyting.Basic

Goal state before:
pp: α : Type u_2
inst✝ : GeneralizedHeytingAlgebra α
a b : α
⊢ b ⇨ a = b ⇨ a

Tactic:
]

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

Goal state before:
pp: α : Type u_2
inst✝ : GeneralizedHeytingAlgebra α
a b c d : α
⊢ a ≤ d ⇨ c ∧ b ≤ d ⇨ c ↔ d ≤ a ⇨ c ∧ d ≤ b ⇨ c

Tactic:
]

Goal state after:
α : Type u_2
inst✝ : GeneralizedHeytingAlgebra α
a b c d : α
⊢ a ≤ d ⇨ c ∧ b ≤ d ⇨ c ↔ d ≤ a ⇨ c ∧ d ≤ b ⇨ c
Module: Mathlib.Order.Heyting.Basic

Goal state before:
pp: α : Type u_2
inst✝ : GeneralizedHeytingAlgebra α
a b : α
⊢ a ⇨ b = a ⇨ b

Tactic:
]

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

Goal state before:
pp: α : Type u_2
inst✝ : GeneralizedHeytingAlgebra α
a b : α
h : Codisjoint a b
⊢ b ⇨ a = b ⇨ a

Tactic:
]

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

Goal state before:
pp: α : Type u_2
inst✝ : GeneralizedHeytingAlgebra α
a b : α
h : Codisjoint a b
⊢ b = b

Tactic:
]

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

Goal state before:
pp: α : Type u_2
inst✝ : GeneralizedHeytingAlgebra α
a b : α
h : Codisjoint a b
⊢ a = a

Tactic:
]

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

Goal state before:
pp: α : Type u_2
inst✝ : GeneralizedHeytingAlgebra α
a b c : α
⊢ (a ⇨ b) ⊓ a ≤ (b ⇨ c) ⇨ c

Tactic:
]

Goal state after:
α : Type u_2
inst✝ : GeneralizedHeytingAlgebra α
a b c : α
⊢ (a ⇨ b) ⊓ a ≤ (b ⇨ c) ⇨ c
Module: Mathlib.Order.Heyting.Basic

Goal state before:
pp: ι : Type u_1
α : Type u_2
β : Type u_3
inst✝ : GeneralizedHeytingAlgebra α
a✝ b✝ c✝ d : α
a b c : αᵒᵈ
⊢ toDual (ofDual b ⇨ ofDual a) ≤ c ↔ a ≤ c ⊔ b

Tactic:
]

Goal state after:
ι : Type u_1
α : Type u_2
β : Type u_3
inst✝ : GeneralizedHeytingAlgebra α
a✝ b✝ c✝ d : α
a b c : αᵒᵈ
⊢ toDual (ofDual b ⇨ ofDual a) ≤ c ↔ a ≤ c ⊔ b
Module: Mathlib.Order.Heyting.Basic

Goal state before:
pp: α : Type u_2
inst✝ : GeneralizedCoheytingAlgebra α
a b c : α
hab : a ≤ b
hbc : b ≤ c
⊢ c = c

Tactic:
]

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

Goal state before:
pp: α : Type u_2
inst✝ : GeneralizedCoheytingAlgebra α
a b : α
h : b ≤ a
⊢ a = a

Tactic:
]

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

Goal state before:
pp: α : Type u_2
inst✝ : GeneralizedCoheytingAlgebra α
a b c : α
hac : c ≤ a
hbc : c ≤ b
h : a \ c = b \ c
⊢ b = b

Tactic:
]

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

Goal state before:
pp: α : Type u_2
inst✝ : GeneralizedCoheytingAlgebra α
a b c : α
⊢ a ≤ a ⊔ (b ⊔ c)

Tactic:
]

Goal state after:
α : Type u_2
inst✝ : GeneralizedCoheytingAlgebra α
a b c : α
⊢ a ≤ a ⊔ (b ⊔ c)
Module: Mathlib.Order.Heyting.Basic

Goal state before:
pp: α : Type u_2
inst✝ : GeneralizedCoheytingAlgebra α
a b : α
⊢ ⊥ = ⊥

Tactic:
]

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

Goal state before:
pp: α : Type u_2
inst✝ : GeneralizedCoheytingAlgebra α
a b : α
⊢ a \ b = a \ b

Tactic:
]

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

Goal state before:
pp: α : Type u_2
inst✝ : GeneralizedCoheytingAlgebra α
a b : α
⊢ b \ a = b \ a

Tactic:
]

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

Goal state before:
pp: α : Type u
inst✝ : GeneralizedBooleanAlgebra α
x y : α
⊢ x = x

Tactic:
]

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

Goal state before:
pp: α : Type u_2
inst✝ : GeneralizedCoheytingAlgebra α
a b c : α
⊢ a \ (a \ b) ≤ c ⊔ b \ c

Tactic:
]

Goal state after:
α : Type u_2
inst✝ : GeneralizedCoheytingAlgebra α
a b c : α
⊢ a \ (a \ b) ≤ c ⊔ b \ c
Module: Mathlib.Order.BooleanAlgebra.Basic

Goal state before:
pp: α : Type u
inst✝ : GeneralizedBooleanAlgebra α
x y : α
⊢ ⊥ = ⊥

Tactic:
]

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

Goal state before:
pp: α : Type u
β : Type u_1
x y z : α
inst✝ : GeneralizedBooleanAlgebra α
a : α
⊢ a ⊓ (a ⊓ a \ a) ≤ a

Tactic:
]

Goal state after:
α : Type u
β : Type u_1
x y z : α
inst✝ : GeneralizedBooleanAlgebra α
a : α
⊢ a ⊓ (a ⊓ a \ a) ≤ a
Module: Mathlib.Order.Heyting.Basic

Goal state before:
pp: α : Type u_2
inst✝ : GeneralizedCoheytingAlgebra α
a b c : α
hinf : a ⊓ c ≤ b
hsup : b ≤ a ⊔ c
⊢ a \ b ≤ a \ (a ⊓ c)

Tactic:
]

Goal state after:
α : Type u_2
inst✝ : GeneralizedCoheytingAlgebra α
a b c : α
hinf : a ⊓ c ≤ b
hsup : b ≤ a ⊔ c
⊢ a \ b ≤ a \ (a ⊓ c)
Module: Mathlib.Order.Heyting.Basic

Goal state before:
pp: α : Type u_2
inst✝ : GeneralizedCoheytingAlgebra α
a b c : α
h : c ⊔ a ≤ c ⊔ b
⊢ (c ⊔ a) \ c ≤ (c ⊔ b) \ c

Tactic:
]

Goal state after:
α : Type u_2
inst✝ : GeneralizedCoheytingAlgebra α
a b c : α
h : c ⊔ a ≤ c ⊔ b
⊢ (c ⊔ a) \ c ≤ (c ⊔ b) \ c
Module: Mathlib.Order.Heyting.Basic

Goal state before:
pp: α : Type u_2
inst✝ : GeneralizedCoheytingAlgebra α
a b c : α
h : a ⊔ c ≤ b ⊔ c
⊢ (a ⊔ c) \ c ≤ (b ⊔ c) \ c

Tactic:
]

Goal state after:
α : Type u_2
inst✝ : GeneralizedCoheytingAlgebra α
a b c : α
h : a ⊔ c ≤ b ⊔ c
⊢ (a ⊔ c) \ c ≤ (b ⊔ c) \ c
Module: Mathlib.Order.BooleanAlgebra.Basic

Goal state before:
pp: α : Type u
x y z : α
inst✝ : GeneralizedBooleanAlgebra α
s : z ⊔ x ⊓ y = x \ y ⊔ x ⊓ y
i : x ⊓ y ⊓ z = ⊥
⊢ x \ y = z

Tactic:
]

Goal state after:
α : Type u
x y z : α
inst✝ : GeneralizedBooleanAlgebra α
s : z ⊔ x ⊓ y = x \ y ⊔ x ⊓ y
i : x ⊓ y ⊓ z = ⊥
⊢ x \ y = z
Module: Mathlib.Order.Heyting.Basic

Goal state before:
pp: ι : Type u_1
α : Type u_2
β : Type u_3
inst✝ : GeneralizedCoheytingAlgebra α
a✝ b✝ c✝ d : α
a b c : αᵒᵈ
⊢ a ≤ toDual (ofDual c \ ofDual b) ↔ b ⊓ a ≤ c

Tactic:
]

Goal state after:
ι : Type u_1
α : Type u_2
β : Type u_3
inst✝ : GeneralizedCoheytingAlgebra α
a✝ b✝ c✝ d : α
a b c : αᵒᵈ
⊢ a ≤ toDual (ofDual c \ ofDual b) ↔ b ⊓ a ≤ c
Module: Mathlib.Order.Heyting.Basic

Goal state before:
pp: α : Type u_2
inst✝ : HeytingAlgebra α
a : α
⊢ a ⇨ ⊥ = a ⇨ ⊥

Tactic:
]

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

Goal state before:
pp: α : Type u_2
inst✝ : HeytingAlgebra α
a b : α
⊢ a ⊓ b ≤ ⊥ ↔ a ⊓ b ≤ ⊥

Tactic:
]

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

Goal state before:
pp: α : Type u_2
inst✝ : HeytingAlgebra α
a b : α
⊢ Disjoint a b ↔ Disjoint a b

Tactic:
]

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

Goal state before:
pp: α : Type u_2
inst✝ : HeytingAlgebra α
a : α
⊢ a = ⊥ ↔ a = ⊥

Tactic:
]

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

Goal state before:
pp: α : Type u_2
inst✝ : HeytingAlgebra α
⊢ ⊤ = ⊤

Tactic:
]

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

Goal state before:
pp: α : Type u
x y z : α
inst✝ : GeneralizedBooleanAlgebra α
s : z ⊔ x ⊓ y = x \ y ⊔ x ⊓ y
i : z ⊓ (x ⊓ y) = x \ y ⊓ (x ⊓ y)
⊢ x \ y = z

Tactic:
]

Goal state after:
α : Type u
x y z : α
inst✝ : GeneralizedBooleanAlgebra α
s : z ⊔ x ⊓ y = x \ y ⊔ x ⊓ y
i : z ⊓ (x ⊓ y) = x \ y ⊓ (x ⊓ y)
⊢ x \ y = z
Module: Mathlib.Order.Heyting.Basic

Goal state before:
pp: α : Type u_2
inst✝¹ : HeytingAlgebra α
a : α
inst✝ : Nontrivial α
⊢ a ≤ aᶜ ∧ a ≠ aᶜ ↔ a = ⊥

Tactic:
]

Goal state after:
α : Type u_2
inst✝¹ : HeytingAlgebra α
a : α
inst✝ : Nontrivial α
⊢ a ≤ aᶜ ∧ a ≠ aᶜ ↔ a = ⊥
Module: Mathlib.Order.Heyting.Basic

Goal state before:
pp: α : Type u_2
inst✝ : HeytingAlgebra α
a b : α
⊢ Disjoint aᶜᶜ b ↔ Disjoint a b

Tactic:
by
  simp_rw [← le_compl_iff_disjoint_left, compl_compl_compl]

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

Goal state before:
pp: α : Type u
x y : α
inst✝ : GeneralizedBooleanAlgebra α
⊢ y \ x ⊔ x = y \ x ⊔ x

Tactic:
]

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

Goal state before:
pp: α : Type u_2
inst✝ : HeytingAlgebra α
a b : α
⊢ Disjoint (a ⊓ b) (a ⊓ b)ᶜ

Tactic:
]

Goal state after:
α : Type u_2
inst✝ : HeytingAlgebra α
a b : α
⊢ Disjoint (a ⊓ b) (a ⊓ b)ᶜ
Module: Mathlib.Order.Heyting.Basic

Goal state before:
pp: case a
α : Type u_2
inst✝ : HeytingAlgebra α
a b : α
⊢ ((a ⇨ b) ⊓ a)ᶜᶜ ≤ bᶜᶜ

Tactic:
]

Goal state after:
case a
α : Type u_2
inst✝ : HeytingAlgebra α
a b : α
⊢ ((a ⇨ b) ⊓ a)ᶜᶜ ≤ bᶜᶜ
Module: Mathlib.Order.BooleanAlgebra.Basic

Goal state before:
pp: α : Type u
x y : α
inst✝ : GeneralizedBooleanAlgebra α
⊢ y ⊔ x = y ⊔ x

Tactic:
]

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

Goal state before:
pp: case a
α : Type u_2
inst✝ : HeytingAlgebra α
a b : α
⊢ aᶜᶜ ⊓ (aᶜᶜ ⇨ bᶜᶜ) ≤ bᶜᶜ

Tactic:
]

Goal state after:
case a
α : Type u_2
inst✝ : HeytingAlgebra α
a b : α
⊢ aᶜᶜ ⊓ (aᶜᶜ ⇨ bᶜᶜ) ≤ bᶜᶜ
Module: Mathlib.Logic.Equiv.Option

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
e : Option α ≃ Option β
x : α
h : ¬(e (some x)).isSome = true
⊢ e none ≠ none

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
e : Option α ≃ Option β
x : α
h : ¬(e (some x)).isSome = true
⊢ e none ≠ none
Module: Mathlib.Order.Heyting.Basic

Goal state before:
pp: ι : Type u_1
α : Type u_2
β : Type u_3
inst✝ : HeytingAlgebra α
a✝ b✝ : α
a b c : αᵒᵈ
⊢ toDual (ofDual b ⇨ ofDual a) ≤ c ↔ a ≤ c ⊔ b

Tactic:
]

Goal state after:
ι : Type u_1
α : Type u_2
β : Type u_3
inst✝ : HeytingAlgebra α
a✝ b✝ : α
a b c : αᵒᵈ
⊢ toDual (ofDual b ⇨ ofDual a) ≤ c ↔ a ≤ c ⊔ b
Module: Mathlib.Logic.Equiv.Option

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
e : Option α ≃ Option β
x : α
h : e (some x) = e none
hn : e none = none
⊢ False

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
e : Option α ≃ Option β
x : α
h : e (some x) = e none
hn : e none = none
⊢ False
Module: Mathlib.Order.Heyting.Basic

Goal state before:
pp: ι : Type u_1
α✝ : Type u_2
β : Type u_3
inst✝¹ : HeytingAlgebra α✝
a✝ b✝ : α✝
α : Type u_2
inst✝ : CoheytingAlgebra α
a b c : αᵒᵈ
⊢ a ≤ toDual (ofDual c \ ofDual b) ↔ b ⊓ a ≤ c

Tactic:
]

Goal state after:
ι : Type u_1
α✝ : Type u_2
β : Type u_3
inst✝¹ : HeytingAlgebra α✝
a✝ b✝ : α✝
α : Type u_2
inst✝ : CoheytingAlgebra α
a b c : αᵒᵈ
⊢ a ≤ toDual (ofDual c \ ofDual b) ↔ b ⊓ a ≤ c