statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
value | commit stringclasses 1
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|---|---|---|---|---|---|---|---|---|---|---|
const (b : β) : inf_hom α β | ⟨λ _, b, λ _ _, inf_idem.symm⟩ | def | inf_hom.const | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"inf_hom"
] | The constant function as an `inf_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_inf (f g : inf_hom α β) : ⇑(f ⊓ g) = f ⊓ g | rfl | lemma | inf_hom.coe_inf | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"inf_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_bot [has_bot β] : ⇑(⊥ : inf_hom α β) = ⊥ | rfl | lemma | inf_hom.coe_bot | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"has_bot",
"inf_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_top [has_top β] : ⇑(⊤ : inf_hom α β) = ⊤ | rfl | lemma | inf_hom.coe_top | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"has_top",
"inf_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inf_apply (f g : inf_hom α β) (a : α) : (f ⊓ g) a = f a ⊓ g a | rfl | lemma | inf_hom.inf_apply | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"inf_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bot_apply [has_bot β] (a : α) : (⊥ : inf_hom α β) a = ⊥ | rfl | lemma | inf_hom.bot_apply | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"has_bot",
"inf_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
top_apply [has_top β] (a : α) : (⊤ : inf_hom α β) a = ⊤ | rfl | lemma | inf_hom.top_apply | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"has_top",
"inf_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_bot_hom (f : sup_bot_hom α β) : bot_hom α β | { ..f } | def | sup_bot_hom.to_bot_hom | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"bot_hom",
"sup_bot_hom"
] | Reinterpret a `sup_bot_hom` as a `bot_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_fun_eq_coe {f : sup_bot_hom α β} : f.to_fun = (f : α → β) | rfl | lemma | sup_bot_hom.to_fun_eq_coe | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"sup_bot_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext {f g : sup_bot_hom α β} (h : ∀ a, f a = g a) : f = g | fun_like.ext f g h | lemma | sup_bot_hom.ext | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"fun_like.ext",
"sup_bot_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
copy (f : sup_bot_hom α β) (f' : α → β) (h : f' = f) : sup_bot_hom α β | { to_sup_hom := f.to_sup_hom.copy f' h, ..f.to_bot_hom.copy f' h } | def | sup_bot_hom.copy | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"sup_bot_hom"
] | Copy of a `sup_bot_hom` with a new `to_fun` equal to the old one. Useful to fix definitional
equalities. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_copy (f : sup_bot_hom α β) (f' : α → β) (h : f' = f) : ⇑(f.copy f' h) = f' | rfl | lemma | sup_bot_hom.coe_copy | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"sup_bot_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
copy_eq (f : sup_bot_hom α β) (f' : α → β) (h : f' = f) : f.copy f' h = f | fun_like.ext' h | lemma | sup_bot_hom.copy_eq | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"fun_like.ext'",
"sup_bot_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id : sup_bot_hom α α | ⟨sup_hom.id α, rfl⟩ | def | sup_bot_hom.id | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"sup_bot_hom"
] | `id` as a `sup_bot_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_id : ⇑(sup_bot_hom.id α) = id | rfl | lemma | sup_bot_hom.coe_id | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"sup_bot_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_apply (a : α) : sup_bot_hom.id α a = a | rfl | lemma | sup_bot_hom.id_apply | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"sup_bot_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp (f : sup_bot_hom β γ) (g : sup_bot_hom α β) : sup_bot_hom α γ | { ..f.to_sup_hom.comp g.to_sup_hom, ..f.to_bot_hom.comp g.to_bot_hom } | def | sup_bot_hom.comp | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"sup_bot_hom"
] | Composition of `sup_bot_hom`s as a `sup_bot_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_comp (f : sup_bot_hom β γ) (g : sup_bot_hom α β) : (f.comp g : α → γ) = f ∘ g | rfl | lemma | sup_bot_hom.coe_comp | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"sup_bot_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_apply (f : sup_bot_hom β γ) (g : sup_bot_hom α β) (a : α) :
(f.comp g) a = f (g a) | rfl | lemma | sup_bot_hom.comp_apply | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"sup_bot_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_assoc (f : sup_bot_hom γ δ) (g : sup_bot_hom β γ) (h : sup_bot_hom α β) :
(f.comp g).comp h = f.comp (g.comp h) | rfl | lemma | sup_bot_hom.comp_assoc | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"sup_bot_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_id (f : sup_bot_hom α β) : f.comp (sup_bot_hom.id α) = f | ext $ λ a, rfl | lemma | sup_bot_hom.comp_id | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"sup_bot_hom",
"sup_bot_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_comp (f : sup_bot_hom α β) : (sup_bot_hom.id β).comp f = f | ext $ λ a, rfl | lemma | sup_bot_hom.id_comp | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"sup_bot_hom",
"sup_bot_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cancel_right {g₁ g₂ : sup_bot_hom β γ} {f : sup_bot_hom α β} (hf : surjective f) :
g₁.comp f = g₂.comp f ↔ g₁ = g₂ | ⟨λ h, ext $ hf.forall.2 $ fun_like.ext_iff.1 h, congr_arg _⟩ | lemma | sup_bot_hom.cancel_right | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"sup_bot_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cancel_left {g : sup_bot_hom β γ} {f₁ f₂ : sup_bot_hom α β} (hg : injective g) :
g.comp f₁ = g.comp f₂ ↔ f₁ = f₂ | ⟨λ h, sup_bot_hom.ext $ λ a, hg $
by rw [←comp_apply, h, comp_apply], congr_arg _⟩ | lemma | sup_bot_hom.cancel_left | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"sup_bot_hom",
"sup_bot_hom.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_sup (f g : sup_bot_hom α β) : ⇑(f ⊔ g) = f ⊔ g | rfl | lemma | sup_bot_hom.coe_sup | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"sup_bot_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_bot : ⇑(⊥ : sup_bot_hom α β) = ⊥ | rfl | lemma | sup_bot_hom.coe_bot | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"sup_bot_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sup_apply (f g : sup_bot_hom α β) (a : α) : (f ⊔ g) a = f a ⊔ g a | rfl | lemma | sup_bot_hom.sup_apply | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"sup_bot_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bot_apply (a : α) : (⊥ : sup_bot_hom α β) a = ⊥ | rfl | lemma | sup_bot_hom.bot_apply | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"sup_bot_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_top_hom (f : inf_top_hom α β) : top_hom α β | { ..f } | def | inf_top_hom.to_top_hom | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"inf_top_hom",
"top_hom"
] | Reinterpret an `inf_top_hom` as a `top_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_fun_eq_coe {f : inf_top_hom α β} : f.to_fun = (f : α → β) | rfl | lemma | inf_top_hom.to_fun_eq_coe | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"inf_top_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext {f g : inf_top_hom α β} (h : ∀ a, f a = g a) : f = g | fun_like.ext f g h | lemma | inf_top_hom.ext | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"fun_like.ext",
"inf_top_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
copy (f : inf_top_hom α β) (f' : α → β) (h : f' = f) : inf_top_hom α β | { to_inf_hom := f.to_inf_hom.copy f' h, ..f.to_top_hom.copy f' h } | def | inf_top_hom.copy | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"inf_top_hom"
] | Copy of an `inf_top_hom` with a new `to_fun` equal to the old one. Useful to fix definitional
equalities. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_copy (f : inf_top_hom α β) (f' : α → β) (h : f' = f) : ⇑(f.copy f' h) = f' | rfl | lemma | inf_top_hom.coe_copy | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"inf_top_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
copy_eq (f : inf_top_hom α β) (f' : α → β) (h : f' = f) : f.copy f' h = f | fun_like.ext' h | lemma | inf_top_hom.copy_eq | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"fun_like.ext'",
"inf_top_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id : inf_top_hom α α | ⟨inf_hom.id α, rfl⟩ | def | inf_top_hom.id | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"inf_top_hom"
] | `id` as an `inf_top_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_id : ⇑(inf_top_hom.id α) = id | rfl | lemma | inf_top_hom.coe_id | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"inf_top_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_apply (a : α) : inf_top_hom.id α a = a | rfl | lemma | inf_top_hom.id_apply | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"inf_top_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp (f : inf_top_hom β γ) (g : inf_top_hom α β) : inf_top_hom α γ | { ..f.to_inf_hom.comp g.to_inf_hom, ..f.to_top_hom.comp g.to_top_hom } | def | inf_top_hom.comp | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"inf_top_hom"
] | Composition of `inf_top_hom`s as an `inf_top_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_comp (f : inf_top_hom β γ) (g : inf_top_hom α β) : (f.comp g : α → γ) = f ∘ g | rfl | lemma | inf_top_hom.coe_comp | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"inf_top_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_apply (f : inf_top_hom β γ) (g : inf_top_hom α β) (a : α) :
(f.comp g) a = f (g a) | rfl | lemma | inf_top_hom.comp_apply | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"inf_top_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_assoc (f : inf_top_hom γ δ) (g : inf_top_hom β γ) (h : inf_top_hom α β) :
(f.comp g).comp h = f.comp (g.comp h) | rfl | lemma | inf_top_hom.comp_assoc | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"inf_top_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_id (f : inf_top_hom α β) : f.comp (inf_top_hom.id α) = f | ext $ λ a, rfl | lemma | inf_top_hom.comp_id | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"inf_top_hom",
"inf_top_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_comp (f : inf_top_hom α β) : (inf_top_hom.id β).comp f = f | ext $ λ a, rfl | lemma | inf_top_hom.id_comp | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"inf_top_hom",
"inf_top_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cancel_right {g₁ g₂ : inf_top_hom β γ} {f : inf_top_hom α β} (hf : surjective f) :
g₁.comp f = g₂.comp f ↔ g₁ = g₂ | ⟨λ h, ext $ hf.forall.2 $ fun_like.ext_iff.1 h, congr_arg _⟩ | lemma | inf_top_hom.cancel_right | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"inf_top_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cancel_left {g : inf_top_hom β γ} {f₁ f₂ : inf_top_hom α β} (hg : injective g) :
g.comp f₁ = g.comp f₂ ↔ f₁ = f₂ | ⟨λ h, inf_top_hom.ext $ λ a, hg $
by rw [←comp_apply, h, comp_apply], congr_arg _⟩ | lemma | inf_top_hom.cancel_left | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"inf_top_hom",
"inf_top_hom.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_inf (f g : inf_top_hom α β) : ⇑(f ⊓ g) = f ⊓ g | rfl | lemma | inf_top_hom.coe_inf | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"inf_top_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_top : ⇑(⊤ : inf_top_hom α β) = ⊤ | rfl | lemma | inf_top_hom.coe_top | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"inf_top_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inf_apply (f g : inf_top_hom α β) (a : α) : (f ⊓ g) a = f a ⊓ g a | rfl | lemma | inf_top_hom.inf_apply | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"inf_top_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
top_apply (a : α) : (⊤ : inf_top_hom α β) a = ⊤ | rfl | lemma | inf_top_hom.top_apply | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"inf_top_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_inf_hom (f : lattice_hom α β) : inf_hom α β | { ..f } | def | lattice_hom.to_inf_hom | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"inf_hom",
"lattice_hom"
] | Reinterpret a `lattice_hom` as an `inf_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_fun_eq_coe {f : lattice_hom α β} : f.to_fun = (f : α → β) | rfl | lemma | lattice_hom.to_fun_eq_coe | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"lattice_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext {f g : lattice_hom α β} (h : ∀ a, f a = g a) : f = g | fun_like.ext f g h | lemma | lattice_hom.ext | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"fun_like.ext",
"lattice_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
copy (f : lattice_hom α β) (f' : α → β) (h : f' = f) : lattice_hom α β | { .. f.to_sup_hom.copy f' h, .. f.to_inf_hom.copy f' h } | def | lattice_hom.copy | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"lattice_hom"
] | Copy of a `lattice_hom` with a new `to_fun` equal to the old one. Useful to fix definitional
equalities. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_copy (f : lattice_hom α β) (f' : α → β) (h : f' = f) : ⇑(f.copy f' h) = f' | rfl | lemma | lattice_hom.coe_copy | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"lattice_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
copy_eq (f : lattice_hom α β) (f' : α → β) (h : f' = f) : f.copy f' h = f | fun_like.ext' h | lemma | lattice_hom.copy_eq | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"fun_like.ext'",
"lattice_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id : lattice_hom α α | { to_fun := id,
map_sup' := λ _ _, rfl,
map_inf' := λ _ _, rfl } | def | lattice_hom.id | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"lattice_hom"
] | `id` as a `lattice_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_id : ⇑(lattice_hom.id α) = id | rfl | lemma | lattice_hom.coe_id | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"lattice_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_apply (a : α) : lattice_hom.id α a = a | rfl | lemma | lattice_hom.id_apply | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"lattice_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp (f : lattice_hom β γ) (g : lattice_hom α β) : lattice_hom α γ | { ..f.to_sup_hom.comp g.to_sup_hom, ..f.to_inf_hom.comp g.to_inf_hom } | def | lattice_hom.comp | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"lattice_hom"
] | Composition of `lattice_hom`s as a `lattice_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_comp (f : lattice_hom β γ) (g : lattice_hom α β) : (f.comp g : α → γ) = f ∘ g | rfl | lemma | lattice_hom.coe_comp | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"lattice_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_apply (f : lattice_hom β γ) (g : lattice_hom α β) (a : α) :
(f.comp g) a = f (g a) | rfl | lemma | lattice_hom.comp_apply | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"lattice_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_comp_sup_hom (f : lattice_hom β γ) (g : lattice_hom α β) :
(f.comp g : sup_hom α γ) = (f : sup_hom β γ).comp g | rfl | lemma | lattice_hom.coe_comp_sup_hom | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"lattice_hom",
"sup_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_comp_inf_hom (f : lattice_hom β γ) (g : lattice_hom α β) :
(f.comp g : inf_hom α γ) = (f : inf_hom β γ).comp g | rfl | lemma | lattice_hom.coe_comp_inf_hom | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"inf_hom",
"lattice_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_assoc (f : lattice_hom γ δ) (g : lattice_hom β γ) (h : lattice_hom α β) :
(f.comp g).comp h = f.comp (g.comp h) | rfl | lemma | lattice_hom.comp_assoc | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"lattice_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_id (f : lattice_hom α β) : f.comp (lattice_hom.id α) = f | lattice_hom.ext $ λ a, rfl | lemma | lattice_hom.comp_id | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"lattice_hom",
"lattice_hom.ext",
"lattice_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_comp (f : lattice_hom α β) : (lattice_hom.id β).comp f = f | lattice_hom.ext $ λ a, rfl | lemma | lattice_hom.id_comp | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"lattice_hom",
"lattice_hom.ext",
"lattice_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cancel_right {g₁ g₂ : lattice_hom β γ} {f : lattice_hom α β} (hf : surjective f) :
g₁.comp f = g₂.comp f ↔ g₁ = g₂ | ⟨λ h, lattice_hom.ext $ hf.forall.2 $ fun_like.ext_iff.1 h, congr_arg _⟩ | lemma | lattice_hom.cancel_right | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"lattice_hom",
"lattice_hom.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cancel_left {g : lattice_hom β γ} {f₁ f₂ : lattice_hom α β} (hg : injective g) :
g.comp f₁ = g.comp f₂ ↔ f₁ = f₂ | ⟨λ h, lattice_hom.ext $ λ a, hg $
by rw [←lattice_hom.comp_apply, h, lattice_hom.comp_apply], congr_arg _⟩ | lemma | lattice_hom.cancel_left | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"lattice_hom",
"lattice_hom.comp_apply",
"lattice_hom.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_lattice_hom_class : lattice_hom_class F α β | { map_sup := λ f a b, begin
obtain h | h := le_total a b,
{ rw [sup_eq_right.2 h, sup_eq_right.2 (order_hom_class.mono f h : f a ≤ f b)] },
{ rw [sup_eq_left.2 h, sup_eq_left.2 (order_hom_class.mono f h : f b ≤ f a)] }
end,
map_inf := λ f a b, begin
obtain h | h := le_total a b,
{ rw [inf_eq_lef... | def | order_hom_class.to_lattice_hom_class | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"lattice_hom_class",
"order_hom_class.mono"
] | An order homomorphism from a linear order is a lattice homomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_lattice_hom (f : F) : lattice_hom α β | by { haveI : lattice_hom_class F α β := order_hom_class.to_lattice_hom_class α β, exact f } | def | order_hom_class.to_lattice_hom | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"lattice_hom",
"lattice_hom_class",
"order_hom_class.to_lattice_hom_class"
] | Reinterpret an order homomorphism to a linear order as a `lattice_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_to_lattice_hom (f : F) : ⇑(to_lattice_hom α β f) = f | rfl | lemma | order_hom_class.coe_to_lattice_hom | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_lattice_hom_apply (f : F) (a : α) : to_lattice_hom α β f a = f a | rfl | lemma | order_hom_class.to_lattice_hom_apply | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_sup_bot_hom (f : bounded_lattice_hom α β) : sup_bot_hom α β | { ..f } | def | bounded_lattice_hom.to_sup_bot_hom | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"bounded_lattice_hom",
"sup_bot_hom"
] | Reinterpret a `bounded_lattice_hom` as a `sup_bot_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_inf_top_hom (f : bounded_lattice_hom α β) : inf_top_hom α β | { ..f } | def | bounded_lattice_hom.to_inf_top_hom | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"bounded_lattice_hom",
"inf_top_hom"
] | Reinterpret a `bounded_lattice_hom` as an `inf_top_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_bounded_order_hom (f : bounded_lattice_hom α β) : bounded_order_hom α β | { ..f, ..(f.to_lattice_hom : α →o β) } | def | bounded_lattice_hom.to_bounded_order_hom | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"bounded_lattice_hom",
"bounded_order_hom"
] | Reinterpret a `bounded_lattice_hom` as a `bounded_order_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_fun_eq_coe {f : bounded_lattice_hom α β} : f.to_fun = (f : α → β) | rfl | lemma | bounded_lattice_hom.to_fun_eq_coe | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"bounded_lattice_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext {f g : bounded_lattice_hom α β} (h : ∀ a, f a = g a) : f = g | fun_like.ext f g h | lemma | bounded_lattice_hom.ext | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"bounded_lattice_hom",
"fun_like.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
copy (f : bounded_lattice_hom α β) (f' : α → β) (h : f' = f) :
bounded_lattice_hom α β | { .. f.to_lattice_hom.copy f' h, .. f.to_bounded_order_hom.copy f' h } | def | bounded_lattice_hom.copy | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"bounded_lattice_hom"
] | Copy of a `bounded_lattice_hom` with a new `to_fun` equal to the old one. Useful to fix
definitional equalities. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_copy (f : bounded_lattice_hom α β) (f' : α → β) (h : f' = f) :
⇑(f.copy f' h) = f' | rfl | lemma | bounded_lattice_hom.coe_copy | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"bounded_lattice_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
copy_eq (f : bounded_lattice_hom α β) (f' : α → β) (h : f' = f) : f.copy f' h = f | fun_like.ext' h | lemma | bounded_lattice_hom.copy_eq | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"bounded_lattice_hom",
"fun_like.ext'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id : bounded_lattice_hom α α | { ..lattice_hom.id α, ..bounded_order_hom.id α } | def | bounded_lattice_hom.id | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"bounded_lattice_hom",
"bounded_order_hom.id",
"lattice_hom.id"
] | `id` as a `bounded_lattice_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_id : ⇑(bounded_lattice_hom.id α) = id | rfl | lemma | bounded_lattice_hom.coe_id | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"bounded_lattice_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_apply (a : α) : bounded_lattice_hom.id α a = a | rfl | lemma | bounded_lattice_hom.id_apply | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"bounded_lattice_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp (f : bounded_lattice_hom β γ) (g : bounded_lattice_hom α β) : bounded_lattice_hom α γ | { ..f.to_lattice_hom.comp g.to_lattice_hom, ..f.to_bounded_order_hom.comp g.to_bounded_order_hom } | def | bounded_lattice_hom.comp | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"bounded_lattice_hom"
] | Composition of `bounded_lattice_hom`s as a `bounded_lattice_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_comp (f : bounded_lattice_hom β γ) (g : bounded_lattice_hom α β) :
(f.comp g : α → γ) = f ∘ g | rfl | lemma | bounded_lattice_hom.coe_comp | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"bounded_lattice_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_apply (f : bounded_lattice_hom β γ) (g : bounded_lattice_hom α β) (a : α) :
(f.comp g) a = f (g a) | rfl | lemma | bounded_lattice_hom.comp_apply | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"bounded_lattice_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_comp_lattice_hom (f : bounded_lattice_hom β γ) (g : bounded_lattice_hom α β) :
(f.comp g : lattice_hom α γ) = (f : lattice_hom β γ).comp g | rfl | lemma | bounded_lattice_hom.coe_comp_lattice_hom | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"bounded_lattice_hom",
"lattice_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_comp_sup_hom (f : bounded_lattice_hom β γ) (g : bounded_lattice_hom α β) :
(f.comp g : sup_hom α γ) = (f : sup_hom β γ).comp g | rfl | lemma | bounded_lattice_hom.coe_comp_sup_hom | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"bounded_lattice_hom",
"sup_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_comp_inf_hom (f : bounded_lattice_hom β γ) (g : bounded_lattice_hom α β) :
(f.comp g : inf_hom α γ) = (f : inf_hom β γ).comp g | rfl | lemma | bounded_lattice_hom.coe_comp_inf_hom | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"bounded_lattice_hom",
"inf_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_assoc (f : bounded_lattice_hom γ δ) (g : bounded_lattice_hom β γ)
(h : bounded_lattice_hom α β) :
(f.comp g).comp h = f.comp (g.comp h) | rfl | lemma | bounded_lattice_hom.comp_assoc | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"bounded_lattice_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_id (f : bounded_lattice_hom α β) : f.comp (bounded_lattice_hom.id α) = f | bounded_lattice_hom.ext $ λ a, rfl | lemma | bounded_lattice_hom.comp_id | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"bounded_lattice_hom",
"bounded_lattice_hom.ext",
"bounded_lattice_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_comp (f : bounded_lattice_hom α β) : (bounded_lattice_hom.id β).comp f = f | bounded_lattice_hom.ext $ λ a, rfl | lemma | bounded_lattice_hom.id_comp | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"bounded_lattice_hom",
"bounded_lattice_hom.ext",
"bounded_lattice_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cancel_right {g₁ g₂ : bounded_lattice_hom β γ} {f : bounded_lattice_hom α β}
(hf : surjective f) :
g₁.comp f = g₂.comp f ↔ g₁ = g₂ | ⟨λ h, bounded_lattice_hom.ext $ hf.forall.2 $ fun_like.ext_iff.1 h, congr_arg _⟩ | lemma | bounded_lattice_hom.cancel_right | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"bounded_lattice_hom",
"bounded_lattice_hom.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cancel_left {g : bounded_lattice_hom β γ} {f₁ f₂ : bounded_lattice_hom α β}
(hg : injective g) :
g.comp f₁ = g.comp f₂ ↔ f₁ = f₂ | ⟨λ h, ext $ λ a, hg $ by rw [←comp_apply, h, comp_apply], congr_arg _⟩ | lemma | bounded_lattice_hom.cancel_left | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"bounded_lattice_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dual : sup_hom α β ≃ inf_hom αᵒᵈ βᵒᵈ | { to_fun := λ f, ⟨f, f.map_sup'⟩,
inv_fun := λ f, ⟨f, f.map_inf'⟩,
left_inv := λ f, sup_hom.ext $ λ _, rfl,
right_inv := λ f, inf_hom.ext $ λ _, rfl } | def | sup_hom.dual | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"inf_hom",
"inf_hom.ext",
"inv_fun",
"sup_hom",
"sup_hom.ext"
] | Reinterpret a supremum homomorphism as an infimum homomorphism between the dual lattices. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dual_id : (sup_hom.id α).dual = inf_hom.id _ | rfl | lemma | sup_hom.dual_id | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"inf_hom.id",
"sup_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dual_comp (g : sup_hom β γ) (f : sup_hom α β) :
(g.comp f).dual = g.dual.comp f.dual | rfl | lemma | sup_hom.dual_comp | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"sup_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_dual_id : sup_hom.dual.symm (inf_hom.id _) = sup_hom.id α | rfl | lemma | sup_hom.symm_dual_id | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"inf_hom.id",
"sup_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_dual_comp (g : inf_hom βᵒᵈ γᵒᵈ) (f : inf_hom αᵒᵈ βᵒᵈ) :
sup_hom.dual.symm (g.comp f) = (sup_hom.dual.symm g).comp (sup_hom.dual.symm f) | rfl | lemma | sup_hom.symm_dual_comp | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"inf_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dual : inf_hom α β ≃ sup_hom αᵒᵈ βᵒᵈ | { to_fun := λ f, ⟨f, f.map_inf'⟩,
inv_fun := λ f, ⟨f, f.map_sup'⟩,
left_inv := λ f, inf_hom.ext $ λ _, rfl,
right_inv := λ f, sup_hom.ext $ λ _, rfl } | def | inf_hom.dual | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"inf_hom",
"inf_hom.ext",
"inv_fun",
"sup_hom",
"sup_hom.ext"
] | Reinterpret an infimum homomorphism as a supremum homomorphism between the dual lattices. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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