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
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dual_id : (inf_hom.id α).dual = sup_hom.id _ | rfl | lemma | inf_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 : inf_hom β γ) (f : inf_hom α β) :
(g.comp f).dual = g.dual.comp f.dual | rfl | lemma | inf_hom.dual_comp | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"inf_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_dual_id : inf_hom.dual.symm (sup_hom.id _) = inf_hom.id α | rfl | lemma | inf_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 : sup_hom βᵒᵈ γᵒᵈ) (f : sup_hom αᵒᵈ βᵒᵈ) :
inf_hom.dual.symm (g.comp f) = (inf_hom.dual.symm g).comp (inf_hom.dual.symm f) | rfl | lemma | inf_hom.symm_dual_comp | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"sup_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dual : sup_bot_hom α β ≃ inf_top_hom αᵒᵈ βᵒᵈ | { to_fun := λ f, ⟨f.to_sup_hom.dual, f.map_bot'⟩,
inv_fun := λ f, ⟨sup_hom.dual.symm f.to_inf_hom, f.map_top'⟩,
left_inv := λ f, sup_bot_hom.ext $ λ _, rfl,
right_inv := λ f, inf_top_hom.ext $ λ _, rfl } | def | sup_bot_hom.dual | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"inf_top_hom",
"inf_top_hom.ext",
"inv_fun",
"sup_bot_hom",
"sup_bot_hom.ext"
] | Reinterpret a finitary supremum homomorphism as a finitary infimum homomorphism between the dual
lattices. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dual_id : (sup_bot_hom.id α).dual = inf_top_hom.id _ | rfl | lemma | sup_bot_hom.dual_id | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"inf_top_hom.id",
"sup_bot_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dual_comp (g : sup_bot_hom β γ) (f : sup_bot_hom α β) :
(g.comp f).dual = g.dual.comp f.dual | rfl | lemma | sup_bot_hom.dual_comp | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"sup_bot_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_dual_id : sup_bot_hom.dual.symm (inf_top_hom.id _) = sup_bot_hom.id α | rfl | lemma | sup_bot_hom.symm_dual_id | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"inf_top_hom.id",
"sup_bot_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_dual_comp (g : inf_top_hom βᵒᵈ γᵒᵈ) (f : inf_top_hom αᵒᵈ βᵒᵈ) :
sup_bot_hom.dual.symm (g.comp f) = (sup_bot_hom.dual.symm g).comp (sup_bot_hom.dual.symm f) | rfl | lemma | sup_bot_hom.symm_dual_comp | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"inf_top_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dual : inf_top_hom α β ≃ sup_bot_hom αᵒᵈ βᵒᵈ | { to_fun := λ f, ⟨f.to_inf_hom.dual, f.map_top'⟩,
inv_fun := λ f, ⟨inf_hom.dual.symm f.to_sup_hom, f.map_bot'⟩,
left_inv := λ f, inf_top_hom.ext $ λ _, rfl,
right_inv := λ f, sup_bot_hom.ext $ λ _, rfl } | def | inf_top_hom.dual | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"inf_top_hom",
"inf_top_hom.ext",
"inv_fun",
"sup_bot_hom",
"sup_bot_hom.ext"
] | Reinterpret a finitary infimum homomorphism as a finitary supremum homomorphism between the dual
lattices. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dual_id : (inf_top_hom.id α).dual = sup_bot_hom.id _ | rfl | lemma | inf_top_hom.dual_id | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"inf_top_hom.id",
"sup_bot_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dual_comp (g : inf_top_hom β γ) (f : inf_top_hom α β) :
(g.comp f).dual = g.dual.comp f.dual | rfl | lemma | inf_top_hom.dual_comp | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"inf_top_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_dual_id : inf_top_hom.dual.symm (sup_bot_hom.id _) = inf_top_hom.id α | rfl | lemma | inf_top_hom.symm_dual_id | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"inf_top_hom.id",
"sup_bot_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_dual_comp (g : sup_bot_hom βᵒᵈ γᵒᵈ) (f : sup_bot_hom αᵒᵈ βᵒᵈ) :
inf_top_hom.dual.symm (g.comp f) = (inf_top_hom.dual.symm g).comp (inf_top_hom.dual.symm f) | rfl | lemma | inf_top_hom.symm_dual_comp | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"sup_bot_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dual : lattice_hom α β ≃ lattice_hom αᵒᵈ βᵒᵈ | { to_fun := λ f, ⟨f.to_inf_hom.dual, f.map_sup'⟩,
inv_fun := λ f, ⟨f.to_inf_hom.dual, f.map_sup'⟩,
left_inv := λ f, ext $ λ a, rfl,
right_inv := λ f, ext $ λ a, rfl } | def | lattice_hom.dual | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"inv_fun",
"lattice_hom"
] | Reinterpret a lattice homomorphism as a lattice homomorphism between the dual lattices. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dual_id : (lattice_hom.id α).dual = lattice_hom.id _ | rfl | lemma | lattice_hom.dual_id | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"lattice_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dual_comp (g : lattice_hom β γ) (f : lattice_hom α β) :
(g.comp f).dual = g.dual.comp f.dual | rfl | lemma | lattice_hom.dual_comp | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"lattice_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_dual_id : lattice_hom.dual.symm (lattice_hom.id _) = lattice_hom.id α | rfl | lemma | lattice_hom.symm_dual_id | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"lattice_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_dual_comp (g : lattice_hom βᵒᵈ γᵒᵈ) (f : lattice_hom αᵒᵈ βᵒᵈ) :
lattice_hom.dual.symm (g.comp f) = (lattice_hom.dual.symm g).comp (lattice_hom.dual.symm f) | rfl | lemma | lattice_hom.symm_dual_comp | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"lattice_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dual : bounded_lattice_hom α β ≃ bounded_lattice_hom αᵒᵈ βᵒᵈ | { to_fun := λ f, ⟨f.to_lattice_hom.dual, f.map_bot', f.map_top'⟩,
inv_fun := λ f, ⟨lattice_hom.dual.symm f.to_lattice_hom, f.map_bot', f.map_top'⟩,
left_inv := λ f, ext $ λ a, rfl,
right_inv := λ f, ext $ λ a, rfl } | def | bounded_lattice_hom.dual | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"bounded_lattice_hom",
"inv_fun"
] | Reinterpret a bounded lattice homomorphism as a bounded lattice homomorphism between the dual
bounded lattices. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dual_id : (bounded_lattice_hom.id α).dual = bounded_lattice_hom.id _ | rfl | lemma | bounded_lattice_hom.dual_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 | |
dual_comp (g : bounded_lattice_hom β γ) (f : bounded_lattice_hom α β) :
(g.comp f).dual = g.dual.comp f.dual | rfl | lemma | bounded_lattice_hom.dual_comp | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"bounded_lattice_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_dual_id :
bounded_lattice_hom.dual.symm (bounded_lattice_hom.id _) = bounded_lattice_hom.id α | rfl | lemma | bounded_lattice_hom.symm_dual_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 | |
symm_dual_comp (g : bounded_lattice_hom βᵒᵈ γᵒᵈ) (f : bounded_lattice_hom αᵒᵈ βᵒᵈ) :
bounded_lattice_hom.dual.symm (g.comp f) =
(bounded_lattice_hom.dual.symm g).comp (bounded_lattice_hom.dual.symm f) | rfl | lemma | bounded_lattice_hom.symm_dual_comp | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"bounded_lattice_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
with_top (f : sup_hom α β) : sup_hom (with_top α) (with_top β) | { to_fun := option.map f,
map_sup' := λ a b, match a, b with
| ⊤, ⊤ := rfl
| ⊤, (b : α) := rfl
| (a : α), ⊤ := rfl
| (a : α), (b : α) := congr_arg _ (f.map_sup' _ _)
end } | def | sup_hom.with_top | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"sup_hom",
"with_top"
] | Adjoins a `⊤` to the domain and codomain of a `sup_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
with_top_id : (sup_hom.id α).with_top = sup_hom.id _ | fun_like.coe_injective option.map_id | lemma | sup_hom.with_top_id | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"fun_like.coe_injective",
"sup_hom.id",
"with_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
with_top_comp (f : sup_hom β γ) (g : sup_hom α β) :
(f.comp g).with_top = f.with_top.comp g.with_top | fun_like.coe_injective (option.map_comp_map _ _).symm | lemma | sup_hom.with_top_comp | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"fun_like.coe_injective",
"option.map_comp_map",
"sup_hom",
"with_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
with_bot (f : sup_hom α β) : sup_bot_hom (with_bot α) (with_bot β) | { to_fun := option.map f,
map_sup' := λ a b, match a, b with
| ⊥, ⊥ := rfl
| ⊥, (b : α) := rfl
| (a : α), ⊥ := rfl
| (a : α), (b : α) := congr_arg _ (f.map_sup' _ _)
end,
map_bot' := rfl } | def | sup_hom.with_bot | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"sup_bot_hom",
"sup_hom",
"with_bot"
] | Adjoins a `⊥` to the domain and codomain of a `sup_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
with_bot_id : (sup_hom.id α).with_bot = sup_bot_hom.id _ | fun_like.coe_injective option.map_id | lemma | sup_hom.with_bot_id | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"fun_like.coe_injective",
"sup_bot_hom.id",
"sup_hom.id",
"with_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
with_bot_comp (f : sup_hom β γ) (g : sup_hom α β) :
(f.comp g).with_bot = f.with_bot.comp g.with_bot | fun_like.coe_injective (option.map_comp_map _ _).symm | lemma | sup_hom.with_bot_comp | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"fun_like.coe_injective",
"option.map_comp_map",
"sup_hom",
"with_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
with_top' [order_top β] (f : sup_hom α β) : sup_hom (with_top α) β | { to_fun := λ a, a.elim ⊤ f,
map_sup' := λ a b, match a, b with
| ⊤, ⊤ := top_sup_eq.symm
| ⊤, (b : α) := top_sup_eq.symm
| (a : α), ⊤ := sup_top_eq.symm
| (a : α), (b : α) := f.map_sup' _ _
end } | def | sup_hom.with_top' | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"order_top",
"sup_hom",
"with_top"
] | Adjoins a `⊤` to the codomain of a `sup_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
with_bot' [order_bot β] (f : sup_hom α β) : sup_bot_hom (with_bot α) β | { to_fun := λ a, a.elim ⊥ f,
map_sup' := λ a b, match a, b with
| ⊥, ⊥ := bot_sup_eq.symm
| ⊥, (b : α) := bot_sup_eq.symm
| (a : α), ⊥ := sup_bot_eq.symm
| (a : α), (b : α) := f.map_sup' _ _
end,
map_bot' := rfl } | def | sup_hom.with_bot' | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"order_bot",
"sup_bot_hom",
"sup_hom",
"with_bot"
] | Adjoins a `⊥` to the domain of a `sup_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
with_top (f : inf_hom α β) : inf_top_hom (with_top α) (with_top β) | { to_fun := option.map f,
map_inf' := λ a b, match a, b with
| ⊤, ⊤ := rfl
| ⊤, (b : α) := rfl
| (a : α), ⊤ := rfl
| (a : α), (b : α) := congr_arg _ (f.map_inf' _ _)
end,
map_top' := rfl } | def | inf_hom.with_top | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"inf_hom",
"inf_top_hom",
"with_top"
] | Adjoins a `⊤` to the domain and codomain of an `inf_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
with_top_id : (inf_hom.id α).with_top = inf_top_hom.id _ | fun_like.coe_injective option.map_id | lemma | inf_hom.with_top_id | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"fun_like.coe_injective",
"inf_hom.id",
"inf_top_hom.id",
"with_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
with_top_comp (f : inf_hom β γ) (g : inf_hom α β) :
(f.comp g).with_top = f.with_top.comp g.with_top | fun_like.coe_injective (option.map_comp_map _ _).symm | lemma | inf_hom.with_top_comp | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"fun_like.coe_injective",
"inf_hom",
"option.map_comp_map",
"with_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
with_bot (f : inf_hom α β) : inf_hom (with_bot α) (with_bot β) | { to_fun := option.map f,
map_inf' := λ a b, match a, b with
| ⊥, ⊥ := rfl
| ⊥, (b : α) := rfl
| (a : α), ⊥ := rfl
| (a : α), (b : α) := congr_arg _ (f.map_inf' _ _)
end } | def | inf_hom.with_bot | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"inf_hom",
"with_bot"
] | Adjoins a `⊥ to the domain and codomain of an `inf_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
with_bot_id : (inf_hom.id α).with_bot = inf_hom.id _ | fun_like.coe_injective option.map_id | lemma | inf_hom.with_bot_id | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"fun_like.coe_injective",
"inf_hom.id",
"with_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
with_bot_comp (f : inf_hom β γ) (g : inf_hom α β) :
(f.comp g).with_bot = f.with_bot.comp g.with_bot | fun_like.coe_injective (option.map_comp_map _ _).symm | lemma | inf_hom.with_bot_comp | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"fun_like.coe_injective",
"inf_hom",
"option.map_comp_map",
"with_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
with_top' [order_top β] (f : inf_hom α β) : inf_top_hom (with_top α) β | { to_fun := λ a, a.elim ⊤ f,
map_inf' := λ a b, match a, b with
| ⊤, ⊤ := top_inf_eq.symm
| ⊤, (b : α) := top_inf_eq.symm
| (a : α), ⊤ := inf_top_eq.symm
| (a : α), (b : α) := f.map_inf' _ _
end,
map_top' := rfl } | def | inf_hom.with_top' | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"inf_hom",
"inf_top_hom",
"order_top",
"with_top"
] | Adjoins a `⊤` to the codomain of an `inf_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
with_bot' [order_bot β] (f : inf_hom α β) : inf_hom (with_bot α) β | { to_fun := λ a, a.elim ⊥ f,
map_inf' := λ a b, match a, b with
| ⊥, ⊥ := bot_inf_eq.symm
| ⊥, (b : α) := bot_inf_eq.symm
| (a : α), ⊥ := inf_bot_eq.symm
| (a : α), (b : α) := f.map_inf' _ _
end } | def | inf_hom.with_bot' | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"inf_hom",
"order_bot",
"with_bot"
] | Adjoins a `⊥` to the codomain of an `inf_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
with_top (f : lattice_hom α β) : lattice_hom (with_top α) (with_top β) | { to_sup_hom := f.to_sup_hom.with_top, ..f.to_inf_hom.with_top } | def | lattice_hom.with_top | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"lattice_hom",
"with_top"
] | Adjoins a `⊤` to the domain and codomain of a `lattice_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
with_top_id : (lattice_hom.id α).with_top = lattice_hom.id _ | fun_like.coe_injective option.map_id | lemma | lattice_hom.with_top_id | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"fun_like.coe_injective",
"lattice_hom.id",
"with_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
with_top_comp (f : lattice_hom β γ) (g : lattice_hom α β) :
(f.comp g).with_top = f.with_top.comp g.with_top | fun_like.coe_injective (option.map_comp_map _ _).symm | lemma | lattice_hom.with_top_comp | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"fun_like.coe_injective",
"lattice_hom",
"option.map_comp_map",
"with_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
with_bot (f : lattice_hom α β) : lattice_hom (with_bot α) (with_bot β) | { to_sup_hom := f.to_sup_hom.with_bot, ..f.to_inf_hom.with_bot } | def | lattice_hom.with_bot | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"lattice_hom",
"with_bot"
] | Adjoins a `⊥` to the domain and codomain of a `lattice_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
with_bot_id : (lattice_hom.id α).with_bot = lattice_hom.id _ | fun_like.coe_injective option.map_id | lemma | lattice_hom.with_bot_id | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"fun_like.coe_injective",
"lattice_hom.id",
"with_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
with_bot_comp (f : lattice_hom β γ) (g : lattice_hom α β) :
(f.comp g).with_bot = f.with_bot.comp g.with_bot | fun_like.coe_injective (option.map_comp_map _ _).symm | lemma | lattice_hom.with_bot_comp | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"fun_like.coe_injective",
"lattice_hom",
"option.map_comp_map",
"with_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
with_top_with_bot (f : lattice_hom α β) :
bounded_lattice_hom (with_top $ with_bot α) (with_top $ with_bot β) | ⟨f.with_bot.with_top, rfl, rfl⟩ | def | lattice_hom.with_top_with_bot | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"bounded_lattice_hom",
"lattice_hom",
"with_bot",
"with_top"
] | Adjoins a `⊤` and `⊥` to the domain and codomain of a `lattice_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
with_top_with_bot_id :
(lattice_hom.id α).with_top_with_bot = bounded_lattice_hom.id _ | fun_like.coe_injective $ begin
refine (congr_arg option.map _).trans option.map_id,
rw with_bot_id,
refl,
end | lemma | lattice_hom.with_top_with_bot_id | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"bounded_lattice_hom.id",
"fun_like.coe_injective",
"lattice_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
with_top_with_bot_comp (f : lattice_hom β γ) (g : lattice_hom α β) :
(f.comp g).with_top_with_bot = f.with_top_with_bot.comp g.with_top_with_bot | fun_like.coe_injective $ (congr_arg option.map $ (option.map_comp_map _ _).symm).trans
(option.map_comp_map _ _).symm | lemma | lattice_hom.with_top_with_bot_comp | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"fun_like.coe_injective",
"lattice_hom",
"option.map_comp_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
with_top' [order_top β] (f : lattice_hom α β) : lattice_hom (with_top α) β | { ..f.to_sup_hom.with_top', ..f.to_inf_hom.with_top' } | def | lattice_hom.with_top' | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"lattice_hom",
"order_top",
"with_top"
] | Adjoins a `⊥` to the codomain of a `lattice_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
with_bot' [order_bot β] (f : lattice_hom α β) : lattice_hom (with_bot α) β | { ..f.to_sup_hom.with_bot', ..f.to_inf_hom.with_bot' } | def | lattice_hom.with_bot' | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"lattice_hom",
"order_bot",
"with_bot"
] | Adjoins a `⊥` to the domain and codomain of a `lattice_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
with_top_with_bot' [bounded_order β] (f : lattice_hom α β) :
bounded_lattice_hom (with_top $ with_bot α) β | { to_lattice_hom := f.with_bot'.with_top', map_top' := rfl, map_bot' := rfl } | def | lattice_hom.with_top_with_bot' | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"bounded_lattice_hom",
"bounded_order",
"lattice_hom",
"with_bot",
"with_top"
] | Adjoins a `⊤` and `⊥` to the codomain of a `lattice_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Inf_apply [complete_lattice β] (s : set (α →o β)) (x : α) :
Inf s x = ⨅ f ∈ s, (f : _) x | rfl | lemma | order_hom.Inf_apply | order.hom | src/order/hom/order.lean | [
"logic.function.iterate",
"order.galois_connection",
"order.hom.basic"
] | [
"Inf_apply",
"complete_lattice"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
infi_apply {ι : Sort*} [complete_lattice β] (f : ι → α →o β) (x : α) :
(⨅ i, f i) x = ⨅ i, f i x | (Inf_apply _ _).trans infi_range | lemma | order_hom.infi_apply | order.hom | src/order/hom/order.lean | [
"logic.function.iterate",
"order.galois_connection",
"order.hom.basic"
] | [
"Inf_apply",
"complete_lattice",
"infi_apply",
"infi_range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_infi {ι : Sort*} [complete_lattice β] (f : ι → α →o β) :
((⨅ i, f i : α →o β) : α → β) = ⨅ i, f i | funext $ λ x, (infi_apply f x).trans (@_root_.infi_apply _ _ _ _ (λ i, f i) _).symm | lemma | order_hom.coe_infi | order.hom | src/order/hom/order.lean | [
"logic.function.iterate",
"order.galois_connection",
"order.hom.basic"
] | [
"complete_lattice",
"infi_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Sup_apply [complete_lattice β] (s : set (α →o β)) (x : α) :
Sup s x = ⨆ f ∈ s, (f : _) x | rfl | lemma | order_hom.Sup_apply | order.hom | src/order/hom/order.lean | [
"logic.function.iterate",
"order.galois_connection",
"order.hom.basic"
] | [
"Sup_apply",
"complete_lattice"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
supr_apply {ι : Sort*} [complete_lattice β] (f : ι → α →o β) (x : α) :
(⨆ i, f i) x = ⨆ i, f i x | (Sup_apply _ _).trans supr_range | lemma | order_hom.supr_apply | order.hom | src/order/hom/order.lean | [
"logic.function.iterate",
"order.galois_connection",
"order.hom.basic"
] | [
"Sup_apply",
"complete_lattice",
"supr_apply",
"supr_range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_supr {ι : Sort*} [complete_lattice β] (f : ι → α →o β) :
((⨆ i, f i : α →o β) : α → β) = ⨆ i, f i | funext $ λ x, (supr_apply f x).trans (@_root_.supr_apply _ _ _ _ (λ i, f i) _).symm | lemma | order_hom.coe_supr | order.hom | src/order/hom/order.lean | [
"logic.function.iterate",
"order.galois_connection",
"order.hom.basic"
] | [
"complete_lattice",
"supr_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iterate_sup_le_sup_iff {α : Type*} [semilattice_sup α] (f : α →o α) :
(∀ n₁ n₂ a₁ a₂, f^[n₁ + n₂] (a₁ ⊔ a₂) ≤ (f^[n₁] a₁) ⊔ (f^[n₂] a₂)) ↔
(∀ a₁ a₂, f (a₁ ⊔ a₂) ≤ (f a₁) ⊔ a₂) | begin
split; intros h,
{ exact h 1 0, },
{ intros n₁ n₂ a₁ a₂, have h' : ∀ n a₁ a₂, f^[n] (a₁ ⊔ a₂) ≤ (f^[n] a₁) ⊔ a₂,
{ intros n, induction n with n ih; intros a₁ a₂,
{ refl, },
{ calc f^[n + 1] (a₁ ⊔ a₂) = (f^[n] (f (a₁ ⊔ a₂))) : function.iterate_succ_apply f n _
...... | lemma | order_hom.iterate_sup_le_sup_iff | order.hom | src/order/hom/order.lean | [
"logic.function.iterate",
"order.galois_connection",
"order.hom.basic"
] | [
"function.iterate_add_apply",
"function.iterate_succ_apply",
"ih",
"semilattice_sup",
"sup_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
range_eq (e : α ≃o β) : set.range e = set.univ | e.surjective.range_eq | lemma | order_iso.range_eq | order.hom | src/order/hom/set.lean | [
"order.hom.basic",
"logic.equiv.set",
"data.set.image"
] | [
"set.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_image_image (e : α ≃o β) (s : set α) : e.symm '' (e '' s) = s | e.to_equiv.symm_image_image s | lemma | order_iso.symm_image_image | order.hom | src/order/hom/set.lean | [
"order.hom.basic",
"logic.equiv.set",
"data.set.image"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image_symm_image (e : α ≃o β) (s : set β) : e '' (e.symm '' s) = s | e.to_equiv.image_symm_image s | lemma | order_iso.image_symm_image | order.hom | src/order/hom/set.lean | [
"order.hom.basic",
"logic.equiv.set",
"data.set.image"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image_eq_preimage (e : α ≃o β) (s : set α) : e '' s = e.symm ⁻¹' s | e.to_equiv.image_eq_preimage s | lemma | order_iso.image_eq_preimage | order.hom | src/order/hom/set.lean | [
"order.hom.basic",
"logic.equiv.set",
"data.set.image"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preimage_symm_preimage (e : α ≃o β) (s : set α) : e ⁻¹' (e.symm ⁻¹' s) = s | e.to_equiv.preimage_symm_preimage s | lemma | order_iso.preimage_symm_preimage | order.hom | src/order/hom/set.lean | [
"order.hom.basic",
"logic.equiv.set",
"data.set.image"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_preimage_preimage (e : α ≃o β) (s : set β) : e.symm ⁻¹' (e ⁻¹' s) = s | e.to_equiv.symm_preimage_preimage s | lemma | order_iso.symm_preimage_preimage | order.hom | src/order/hom/set.lean | [
"order.hom.basic",
"logic.equiv.set",
"data.set.image"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image_preimage (e : α ≃o β) (s : set β) : e '' (e ⁻¹' s) = s | e.to_equiv.image_preimage s | lemma | order_iso.image_preimage | order.hom | src/order/hom/set.lean | [
"order.hom.basic",
"logic.equiv.set",
"data.set.image"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preimage_image (e : α ≃o β) (s : set α) : e ⁻¹' (e '' s) = s | e.to_equiv.preimage_image s | lemma | order_iso.preimage_image | order.hom | src/order/hom/set.lean | [
"order.hom.basic",
"logic.equiv.set",
"data.set.image"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set_congr (s t : set α) (h : s = t) : s ≃o t | { to_equiv := equiv.set_congr h,
map_rel_iff' := λ x y, iff.rfl } | def | order_iso.set_congr | order.hom | src/order/hom/set.lean | [
"order.hom.basic",
"logic.equiv.set",
"data.set.image"
] | [
"equiv.set_congr"
] | Order isomorphism between two equal sets. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
set.univ : (set.univ : set α) ≃o α | { to_equiv := equiv.set.univ α,
map_rel_iff' := λ x y, iff.rfl } | def | order_iso.set.univ | order.hom | src/order/hom/set.lean | [
"order.hom.basic",
"logic.equiv.set",
"data.set.image"
] | [
"equiv.set.univ"
] | Order isomorphism between `univ : set α` and `α`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strict_mono_on.order_iso {α β} [linear_order α] [preorder β]
(f : α → β) (s : set α) (hf : strict_mono_on f s) :
s ≃o f '' s | { to_equiv := hf.inj_on.bij_on_image.equiv _,
map_rel_iff' := λ x y, hf.le_iff_le x.2 y.2 } | def | strict_mono_on.order_iso | order.hom | src/order/hom/set.lean | [
"order.hom.basic",
"logic.equiv.set",
"data.set.image"
] | [
"strict_mono_on"
] | If a function `f` is strictly monotone on a set `s`, then it defines an order isomorphism
between `s` and its image. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
order_iso : α ≃o set.range f | { to_equiv := equiv.of_injective f h_mono.injective,
map_rel_iff' := λ a b, h_mono.le_iff_le } | def | strict_mono.order_iso | order.hom | src/order/hom/set.lean | [
"order.hom.basic",
"logic.equiv.set",
"data.set.image"
] | [
"equiv.of_injective",
"order_iso",
"set.range"
] | A strictly monotone function from a linear order is an order isomorphism between its domain and
its range. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
order_iso_of_surjective : α ≃o β | (h_mono.order_iso f).trans $ (order_iso.set_congr _ _ h_surj.range_eq).trans order_iso.set.univ | def | strict_mono.order_iso_of_surjective | order.hom | src/order/hom/set.lean | [
"order.hom.basic",
"logic.equiv.set",
"data.set.image"
] | [
"order_iso.set.univ",
"order_iso.set_congr"
] | A strictly monotone surjective function from a linear order is an order isomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_order_iso_of_surjective :
(order_iso_of_surjective f h_mono h_surj : α → β) = f | rfl | lemma | strict_mono.coe_order_iso_of_surjective | order.hom | src/order/hom/set.lean | [
"order.hom.basic",
"logic.equiv.set",
"data.set.image"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
order_iso_of_surjective_symm_apply_self (a : α) :
(order_iso_of_surjective f h_mono h_surj).symm (f a) = a | (order_iso_of_surjective f h_mono h_surj).symm_apply_apply _ | lemma | strict_mono.order_iso_of_surjective_symm_apply_self | order.hom | src/order/hom/set.lean | [
"order.hom.basic",
"logic.equiv.set",
"data.set.image"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
order_iso_of_surjective_self_symm_apply (b : β) :
f ((order_iso_of_surjective f h_mono h_surj).symm b) = b | (order_iso_of_surjective f h_mono h_surj).apply_symm_apply _ | lemma | strict_mono.order_iso_of_surjective_self_symm_apply | order.hom | src/order/hom/set.lean | [
"order.hom.basic",
"logic.equiv.set",
"data.set.image"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
order_iso.compl : α ≃o αᵒᵈ | { to_fun := order_dual.to_dual ∘ compl,
inv_fun := compl ∘ order_dual.of_dual,
left_inv := compl_compl,
right_inv := compl_compl,
map_rel_iff' := λ x y, compl_le_compl_iff_le } | def | order_iso.compl | order.hom | src/order/hom/set.lean | [
"order.hom.basic",
"logic.equiv.set",
"data.set.image"
] | [
"compl_compl",
"compl_le_compl_iff_le",
"inv_fun",
"order_dual.of_dual",
"order_dual.to_dual"
] | Taking complements as an order isomorphism to the order dual. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
compl_strict_anti : strict_anti (compl : α → α) | (order_iso.compl α).strict_mono | theorem | compl_strict_anti | order.hom | src/order/hom/set.lean | [
"order.hom.basic",
"logic.equiv.set",
"data.set.image"
] | [
"order_iso.compl",
"strict_anti",
"strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
compl_antitone : antitone (compl : α → α) | (order_iso.compl α).monotone | theorem | compl_antitone | order.hom | src/order/hom/set.lean | [
"order.hom.basic",
"logic.equiv.set",
"data.set.image"
] | [
"antitone",
"monotone",
"order_iso.compl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monotone (f : α → β) : Prop | ∀ ⦃a b⦄, a ≤ b → f a ≤ f b | def | monotone | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [] | A function `f` is monotone if `a ≤ b` implies `f a ≤ f b`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
antitone (f : α → β) : Prop | ∀ ⦃a b⦄, a ≤ b → f b ≤ f a | def | antitone | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [] | A function `f` is antitone if `a ≤ b` implies `f b ≤ f a`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monotone_on (f : α → β) (s : set α) : Prop | ∀ ⦃a⦄ (ha : a ∈ s) ⦃b⦄ (hb : b ∈ s), a ≤ b → f a ≤ f b | def | monotone_on | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [] | A function `f` is monotone on `s` if, for all `a, b ∈ s`, `a ≤ b` implies `f a ≤ f b`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
antitone_on (f : α → β) (s : set α) : Prop | ∀ ⦃a⦄ (ha : a ∈ s) ⦃b⦄ (hb : b ∈ s), a ≤ b → f b ≤ f a | def | antitone_on | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [] | A function `f` is antitone on `s` if, for all `a, b ∈ s`, `a ≤ b` implies `f b ≤ f a`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strict_mono (f : α → β) : Prop | ∀ ⦃a b⦄, a < b → f a < f b | def | strict_mono | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [] | A function `f` is strictly monotone if `a < b` implies `f a < f b`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strict_anti (f : α → β) : Prop | ∀ ⦃a b⦄, a < b → f b < f a | def | strict_anti | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [] | A function `f` is strictly antitone if `a < b` implies `f b < f a`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strict_mono_on (f : α → β) (s : set α) : Prop | ∀ ⦃a⦄ (ha : a ∈ s) ⦃b⦄ (hb : b ∈ s), a < b → f a < f b | def | strict_mono_on | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [] | A function `f` is strictly monotone on `s` if, for all `a, b ∈ s`, `a < b` implies
`f a < f b`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strict_anti_on (f : α → β) (s : set α) : Prop | ∀ ⦃a⦄ (ha : a ∈ s) ⦃b⦄ (hb : b ∈ s), a < b → f b < f a | def | strict_anti_on | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [] | A function `f` is strictly antitone on `s` if, for all `a, b ∈ s`, `a < b` implies
`f b < f a`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monotone_comp_of_dual_iff : monotone (f ∘ of_dual) ↔ antitone f | forall_swap | lemma | monotone_comp_of_dual_iff | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"antitone",
"forall_swap",
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antitone_comp_of_dual_iff : antitone (f ∘ of_dual) ↔ monotone f | forall_swap | lemma | antitone_comp_of_dual_iff | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"antitone",
"forall_swap",
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monotone_to_dual_comp_iff : monotone (to_dual ∘ f) ↔ antitone f | iff.rfl | lemma | monotone_to_dual_comp_iff | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"antitone",
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antitone_to_dual_comp_iff : antitone (to_dual ∘ f) ↔ monotone f | iff.rfl | lemma | antitone_to_dual_comp_iff | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"antitone",
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monotone_on_comp_of_dual_iff : monotone_on (f ∘ of_dual) s ↔ antitone_on f s | forall₂_swap | lemma | monotone_on_comp_of_dual_iff | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"antitone_on",
"forall₂_swap",
"monotone_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antitone_on_comp_of_dual_iff : antitone_on (f ∘ of_dual) s ↔ monotone_on f s | forall₂_swap | lemma | antitone_on_comp_of_dual_iff | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"antitone_on",
"forall₂_swap",
"monotone_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monotone_on_to_dual_comp_iff : monotone_on (to_dual ∘ f) s ↔ antitone_on f s | iff.rfl | lemma | monotone_on_to_dual_comp_iff | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"antitone_on",
"monotone_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antitone_on_to_dual_comp_iff : antitone_on (to_dual ∘ f) s ↔ monotone_on f s | iff.rfl | lemma | antitone_on_to_dual_comp_iff | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"antitone_on",
"monotone_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_mono_comp_of_dual_iff : strict_mono (f ∘ of_dual) ↔ strict_anti f | forall_swap | lemma | strict_mono_comp_of_dual_iff | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"forall_swap",
"strict_anti",
"strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_anti_comp_of_dual_iff : strict_anti (f ∘ of_dual) ↔ strict_mono f | forall_swap | lemma | strict_anti_comp_of_dual_iff | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"forall_swap",
"strict_anti",
"strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_mono_to_dual_comp_iff : strict_mono (to_dual ∘ f) ↔ strict_anti f | iff.rfl | lemma | strict_mono_to_dual_comp_iff | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"strict_anti",
"strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_anti_to_dual_comp_iff : strict_anti (to_dual ∘ f) ↔ strict_mono f | iff.rfl | lemma | strict_anti_to_dual_comp_iff | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"strict_anti",
"strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_mono_on_comp_of_dual_iff :
strict_mono_on (f ∘ of_dual) s ↔ strict_anti_on f s | forall₂_swap | lemma | strict_mono_on_comp_of_dual_iff | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"forall₂_swap",
"strict_anti_on",
"strict_mono_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_anti_on_comp_of_dual_iff :
strict_anti_on (f ∘ of_dual) s ↔ strict_mono_on f s | forall₂_swap | lemma | strict_anti_on_comp_of_dual_iff | order.monotone | src/order/monotone/basic.lean | [
"order.compare",
"order.max",
"order.rel_classes"
] | [
"forall₂_swap",
"strict_anti_on",
"strict_mono_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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