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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Ici_supr (f : ι → α) : Ici (⨆ i, f i) = ⨆ i, Ici (f i) | set_like.ext $ λ c, by simp only [mem_Ici_iff, mem_supr_iff, supr_le_iff] | lemma | upper_set.Ici_supr | order.upper_lower | src/order/upper_lower/basic.lean | [
"data.set_like.basic",
"data.set.intervals.ord_connected",
"data.set.intervals.order_iso",
"tactic.by_contra"
] | [
"set_like.ext",
"supr_le_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Ici_supr₂ (f : Π i, κ i → α) : Ici (⨆ i j, f i j) = ⨆ i j, Ici (f i j) | by simp_rw Ici_supr | lemma | upper_set.Ici_supr₂ | order.upper_lower | src/order/upper_lower/basic.lean | [
"data.set_like.basic",
"data.set.intervals.ord_connected",
"data.set.intervals.order_iso",
"tactic.by_contra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Iic (a : α) : lower_set α | ⟨Iic a, is_lower_set_Iic a⟩ | def | lower_set.Iic | order.upper_lower | src/order/upper_lower/basic.lean | [
"data.set_like.basic",
"data.set.intervals.ord_connected",
"data.set.intervals.order_iso",
"tactic.by_contra"
] | [
"is_lower_set_Iic",
"lower_set"
] | Principal lower set. `set.Iic` as a lower set. The smallest lower set containing a given
element. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Iio (a : α) : lower_set α | ⟨Iio a, is_lower_set_Iio a⟩ | def | lower_set.Iio | order.upper_lower | src/order/upper_lower/basic.lean | [
"data.set_like.basic",
"data.set.intervals.ord_connected",
"data.set.intervals.order_iso",
"tactic.by_contra"
] | [
"is_lower_set_Iio",
"lower_set"
] | Strict principal lower set. `set.Iio` as a lower set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_Iic (a : α) : ↑(Iic a) = set.Iic a | rfl | lemma | lower_set.coe_Iic | order.upper_lower | src/order/upper_lower/basic.lean | [
"data.set_like.basic",
"data.set.intervals.ord_connected",
"data.set.intervals.order_iso",
"tactic.by_contra"
] | [
"set.Iic"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_Iio (a : α) : ↑(Iio a) = set.Iio a | rfl | lemma | lower_set.coe_Iio | order.upper_lower | src/order/upper_lower/basic.lean | [
"data.set_like.basic",
"data.set.intervals.ord_connected",
"data.set.intervals.order_iso",
"tactic.by_contra"
] | [
"set.Iio"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_Iic_iff : b ∈ Iic a ↔ b ≤ a | iff.rfl | lemma | lower_set.mem_Iic_iff | order.upper_lower | src/order/upper_lower/basic.lean | [
"data.set_like.basic",
"data.set.intervals.ord_connected",
"data.set.intervals.order_iso",
"tactic.by_contra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_Iio_iff : b ∈ Iio a ↔ b < a | iff.rfl | lemma | lower_set.mem_Iio_iff | order.upper_lower | src/order/upper_lower/basic.lean | [
"data.set_like.basic",
"data.set.intervals.ord_connected",
"data.set.intervals.order_iso",
"tactic.by_contra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_Iic (f : α ≃o β) (a : α) : map f (Iic a) = Iic (f a) | by { ext, simp } | lemma | lower_set.map_Iic | order.upper_lower | src/order/upper_lower/basic.lean | [
"data.set_like.basic",
"data.set.intervals.ord_connected",
"data.set.intervals.order_iso",
"tactic.by_contra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_Iio (f : α ≃o β) (a : α) : map f (Iio a) = Iio (f a) | by { ext, simp } | lemma | lower_set.map_Iio | order.upper_lower | src/order/upper_lower/basic.lean | [
"data.set_like.basic",
"data.set.intervals.ord_connected",
"data.set.intervals.order_iso",
"tactic.by_contra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Ioi_le_Ici (a : α) : Ioi a ≤ Ici a | Ioi_subset_Ici_self | lemma | lower_set.Ioi_le_Ici | order.upper_lower | src/order/upper_lower/basic.lean | [
"data.set_like.basic",
"data.set.intervals.ord_connected",
"data.set.intervals.order_iso",
"tactic.by_contra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Iic_top [order_top α] : Iic (⊤ : α) = ⊤ | set_like.coe_injective Iic_top | lemma | lower_set.Iic_top | order.upper_lower | src/order/upper_lower/basic.lean | [
"data.set_like.basic",
"data.set.intervals.ord_connected",
"data.set.intervals.order_iso",
"tactic.by_contra"
] | [
"order_top",
"set_like.coe_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Iio_bot [order_bot α] : Iio (⊥ : α) = ⊥ | set_like.coe_injective Iio_bot | lemma | lower_set.Iio_bot | order.upper_lower | src/order/upper_lower/basic.lean | [
"data.set_like.basic",
"data.set.intervals.ord_connected",
"data.set.intervals.order_iso",
"tactic.by_contra"
] | [
"order_bot",
"set_like.coe_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Iic_inf [semilattice_inf α] (a b : α) : Iic (a ⊓ b) = Iic a ⊓ Iic b | set_like.coe_injective Iic_inter_Iic.symm | lemma | lower_set.Iic_inf | order.upper_lower | src/order/upper_lower/basic.lean | [
"data.set_like.basic",
"data.set.intervals.ord_connected",
"data.set.intervals.order_iso",
"tactic.by_contra"
] | [
"semilattice_inf",
"set_like.coe_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Iic_Inf (S : set α) : Iic (Inf S) = ⨅ a ∈ S, Iic a | set_like.ext $ λ c, by simp only [mem_Iic_iff, mem_infi₂_iff, le_Inf_iff] | lemma | lower_set.Iic_Inf | order.upper_lower | src/order/upper_lower/basic.lean | [
"data.set_like.basic",
"data.set.intervals.ord_connected",
"data.set.intervals.order_iso",
"tactic.by_contra"
] | [
"le_Inf_iff",
"set_like.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Iic_infi (f : ι → α) : Iic (⨅ i, f i) = ⨅ i, Iic (f i) | set_like.ext $ λ c, by simp only [mem_Iic_iff, mem_infi_iff, le_infi_iff] | lemma | lower_set.Iic_infi | order.upper_lower | src/order/upper_lower/basic.lean | [
"data.set_like.basic",
"data.set.intervals.ord_connected",
"data.set.intervals.order_iso",
"tactic.by_contra"
] | [
"le_infi_iff",
"set_like.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Iic_infi₂ (f : Π i, κ i → α) : Iic (⨅ i j, f i j) = ⨅ i j, Iic (f i j) | by simp_rw Iic_infi | lemma | lower_set.Iic_infi₂ | order.upper_lower | src/order/upper_lower/basic.lean | [
"data.set_like.basic",
"data.set.intervals.ord_connected",
"data.set.intervals.order_iso",
"tactic.by_contra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
upper_closure (s : set α) : upper_set α | ⟨{x | ∃ a ∈ s, a ≤ x}, λ x y h, Exists₂.imp $ λ a _, h.trans'⟩ | def | upper_closure | order.upper_lower | src/order/upper_lower/basic.lean | [
"data.set_like.basic",
"data.set.intervals.ord_connected",
"data.set.intervals.order_iso",
"tactic.by_contra"
] | [
"Exists₂.imp",
"upper_set"
] | The greatest upper set containing a given set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lower_closure (s : set α) : lower_set α | ⟨{x | ∃ a ∈ s, x ≤ a}, λ x y h, Exists₂.imp $ λ a _, h.trans⟩ | def | lower_closure | order.upper_lower | src/order/upper_lower/basic.lean | [
"data.set_like.basic",
"data.set.intervals.ord_connected",
"data.set.intervals.order_iso",
"tactic.by_contra"
] | [
"Exists₂.imp",
"lower_set"
] | The least lower set containing a given set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_upper_closure : x ∈ upper_closure s ↔ ∃ a ∈ s, a ≤ x | iff.rfl | lemma | mem_upper_closure | order.upper_lower | src/order/upper_lower/basic.lean | [
"data.set_like.basic",
"data.set.intervals.ord_connected",
"data.set.intervals.order_iso",
"tactic.by_contra"
] | [
"upper_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_lower_closure : x ∈ lower_closure s ↔ ∃ a ∈ s, x ≤ a | iff.rfl | lemma | mem_lower_closure | order.upper_lower | src/order/upper_lower/basic.lean | [
"data.set_like.basic",
"data.set.intervals.ord_connected",
"data.set.intervals.order_iso",
"tactic.by_contra"
] | [
"lower_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_upper_closure (s : set α) : ↑(upper_closure s) = ⋃ a ∈ s, Ici a | by { ext, simp } | lemma | coe_upper_closure | order.upper_lower | src/order/upper_lower/basic.lean | [
"data.set_like.basic",
"data.set.intervals.ord_connected",
"data.set.intervals.order_iso",
"tactic.by_contra"
] | [
"upper_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_lower_closure (s : set α) : ↑(lower_closure s) = ⋃ a ∈ s, Iic a | by { ext, simp } | lemma | coe_lower_closure | order.upper_lower | src/order/upper_lower/basic.lean | [
"data.set_like.basic",
"data.set.intervals.ord_connected",
"data.set.intervals.order_iso",
"tactic.by_contra"
] | [
"lower_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subset_upper_closure : s ⊆ upper_closure s | λ x hx, ⟨x, hx, le_rfl⟩ | lemma | subset_upper_closure | order.upper_lower | src/order/upper_lower/basic.lean | [
"data.set_like.basic",
"data.set.intervals.ord_connected",
"data.set.intervals.order_iso",
"tactic.by_contra"
] | [
"upper_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subset_lower_closure : s ⊆ lower_closure s | λ x hx, ⟨x, hx, le_rfl⟩ | lemma | subset_lower_closure | order.upper_lower | src/order/upper_lower/basic.lean | [
"data.set_like.basic",
"data.set.intervals.ord_connected",
"data.set.intervals.order_iso",
"tactic.by_contra"
] | [
"lower_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
upper_closure_min (h : s ⊆ t) (ht : is_upper_set t) : ↑(upper_closure s) ⊆ t | λ a ⟨b, hb, hba⟩, ht hba $ h hb | lemma | upper_closure_min | order.upper_lower | src/order/upper_lower/basic.lean | [
"data.set_like.basic",
"data.set.intervals.ord_connected",
"data.set.intervals.order_iso",
"tactic.by_contra"
] | [
"is_upper_set",
"upper_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lower_closure_min (h : s ⊆ t) (ht : is_lower_set t) : ↑(lower_closure s) ⊆ t | λ a ⟨b, hb, hab⟩, ht hab $ h hb | lemma | lower_closure_min | order.upper_lower | src/order/upper_lower/basic.lean | [
"data.set_like.basic",
"data.set.intervals.ord_connected",
"data.set.intervals.order_iso",
"tactic.by_contra"
] | [
"is_lower_set",
"lower_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_upper_set.upper_closure (hs : is_upper_set s) : ↑(upper_closure s) = s | (upper_closure_min subset.rfl hs).antisymm subset_upper_closure | lemma | is_upper_set.upper_closure | order.upper_lower | src/order/upper_lower/basic.lean | [
"data.set_like.basic",
"data.set.intervals.ord_connected",
"data.set.intervals.order_iso",
"tactic.by_contra"
] | [
"is_upper_set",
"subset_upper_closure",
"upper_closure",
"upper_closure_min"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_lower_set.lower_closure (hs : is_lower_set s) : ↑(lower_closure s) = s | (lower_closure_min subset.rfl hs).antisymm subset_lower_closure | lemma | is_lower_set.lower_closure | order.upper_lower | src/order/upper_lower/basic.lean | [
"data.set_like.basic",
"data.set.intervals.ord_connected",
"data.set.intervals.order_iso",
"tactic.by_contra"
] | [
"is_lower_set",
"lower_closure",
"lower_closure_min",
"subset_lower_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
upper_set.upper_closure (s : upper_set α) : upper_closure (s : set α) = s | set_like.coe_injective s.2.upper_closure | lemma | upper_set.upper_closure | order.upper_lower | src/order/upper_lower/basic.lean | [
"data.set_like.basic",
"data.set.intervals.ord_connected",
"data.set.intervals.order_iso",
"tactic.by_contra"
] | [
"set_like.coe_injective",
"upper_closure",
"upper_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lower_set.lower_closure (s : lower_set α) : lower_closure (s : set α) = s | set_like.coe_injective s.2.lower_closure | lemma | lower_set.lower_closure | order.upper_lower | src/order/upper_lower/basic.lean | [
"data.set_like.basic",
"data.set.intervals.ord_connected",
"data.set.intervals.order_iso",
"tactic.by_contra"
] | [
"lower_closure",
"lower_set",
"set_like.coe_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
upper_closure_image (f : α ≃o β) :
upper_closure (f '' s) = upper_set.map f (upper_closure s) | begin
rw [←f.symm_symm, ←upper_set.symm_map, f.symm_symm],
ext,
simp [-upper_set.symm_map, upper_set.map, order_iso.symm, ←f.le_symm_apply],
end | lemma | upper_closure_image | order.upper_lower | src/order/upper_lower/basic.lean | [
"data.set_like.basic",
"data.set.intervals.ord_connected",
"data.set.intervals.order_iso",
"tactic.by_contra"
] | [
"order_iso.symm",
"upper_closure",
"upper_set.map",
"upper_set.symm_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lower_closure_image (f : α ≃o β) :
lower_closure (f '' s) = lower_set.map f (lower_closure s) | begin
rw [←f.symm_symm, ←lower_set.symm_map, f.symm_symm],
ext,
simp [-lower_set.symm_map, lower_set.map, order_iso.symm, ←f.symm_apply_le],
end | lemma | lower_closure_image | order.upper_lower | src/order/upper_lower/basic.lean | [
"data.set_like.basic",
"data.set.intervals.ord_connected",
"data.set.intervals.order_iso",
"tactic.by_contra"
] | [
"lower_closure",
"lower_set.map",
"lower_set.symm_map",
"order_iso.symm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
upper_set.infi_Ici (s : set α) : (⨅ a ∈ s, upper_set.Ici a) = upper_closure s | by { ext, simp } | lemma | upper_set.infi_Ici | order.upper_lower | src/order/upper_lower/basic.lean | [
"data.set_like.basic",
"data.set.intervals.ord_connected",
"data.set.intervals.order_iso",
"tactic.by_contra"
] | [
"upper_closure",
"upper_set.Ici"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lower_set.supr_Iic (s : set α) : (⨆ a ∈ s, lower_set.Iic a) = lower_closure s | by { ext, simp } | lemma | lower_set.supr_Iic | order.upper_lower | src/order/upper_lower/basic.lean | [
"data.set_like.basic",
"data.set.intervals.ord_connected",
"data.set.intervals.order_iso",
"tactic.by_contra"
] | [
"lower_closure",
"lower_set.Iic"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gc_upper_closure_coe :
galois_connection (to_dual ∘ upper_closure : set α → (upper_set α)ᵒᵈ) (coe ∘ of_dual) | λ s t, ⟨λ h, subset_upper_closure.trans $ upper_set.coe_subset_coe.2 h,
λ h, upper_closure_min h t.upper⟩ | lemma | gc_upper_closure_coe | order.upper_lower | src/order/upper_lower/basic.lean | [
"data.set_like.basic",
"data.set.intervals.ord_connected",
"data.set.intervals.order_iso",
"tactic.by_contra"
] | [
"galois_connection",
"upper_closure",
"upper_closure_min",
"upper_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gc_lower_closure_coe : galois_connection (lower_closure : set α → lower_set α) coe | λ s t, ⟨λ h, subset_lower_closure.trans $ lower_set.coe_subset_coe.2 h,
λ h, lower_closure_min h t.lower⟩ | lemma | gc_lower_closure_coe | order.upper_lower | src/order/upper_lower/basic.lean | [
"data.set_like.basic",
"data.set.intervals.ord_connected",
"data.set.intervals.order_iso",
"tactic.by_contra"
] | [
"galois_connection",
"lower_closure",
"lower_closure_min",
"lower_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gi_upper_closure_coe :
galois_insertion (to_dual ∘ upper_closure : set α → (upper_set α)ᵒᵈ) (coe ∘ of_dual) | { choice := λ s hs, to_dual (⟨s, λ a b hab ha, hs ⟨a, ha, hab⟩⟩ : upper_set α),
gc := gc_upper_closure_coe,
le_l_u := λ _, subset_upper_closure,
choice_eq := λ s hs,
of_dual.injective $ set_like.coe_injective $ subset_upper_closure.antisymm hs } | def | gi_upper_closure_coe | order.upper_lower | src/order/upper_lower/basic.lean | [
"data.set_like.basic",
"data.set.intervals.ord_connected",
"data.set.intervals.order_iso",
"tactic.by_contra"
] | [
"galois_insertion",
"gc_upper_closure_coe",
"set_like.coe_injective",
"subset_upper_closure",
"upper_closure",
"upper_set"
] | `upper_closure` forms a reversed Galois insertion with the coercion from upper sets to sets. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
gi_lower_closure_coe : galois_insertion (lower_closure : set α → lower_set α) coe | { choice := λ s hs, ⟨s, λ a b hba ha, hs ⟨a, ha, hba⟩⟩,
gc := gc_lower_closure_coe,
le_l_u := λ _, subset_lower_closure,
choice_eq := λ s hs, set_like.coe_injective $ subset_lower_closure.antisymm hs } | def | gi_lower_closure_coe | order.upper_lower | src/order/upper_lower/basic.lean | [
"data.set_like.basic",
"data.set.intervals.ord_connected",
"data.set.intervals.order_iso",
"tactic.by_contra"
] | [
"galois_insertion",
"gc_lower_closure_coe",
"lower_closure",
"lower_set",
"set_like.coe_injective",
"subset_lower_closure"
] | `lower_closure` forms a Galois insertion with the coercion from lower sets to sets. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
upper_closure_anti : antitone (upper_closure : set α → upper_set α) | gc_upper_closure_coe.monotone_l | lemma | upper_closure_anti | order.upper_lower | src/order/upper_lower/basic.lean | [
"data.set_like.basic",
"data.set.intervals.ord_connected",
"data.set.intervals.order_iso",
"tactic.by_contra"
] | [
"antitone",
"upper_closure",
"upper_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lower_closure_mono : monotone (lower_closure : set α → lower_set α) | gc_lower_closure_coe.monotone_l | lemma | lower_closure_mono | order.upper_lower | src/order/upper_lower/basic.lean | [
"data.set_like.basic",
"data.set.intervals.ord_connected",
"data.set.intervals.order_iso",
"tactic.by_contra"
] | [
"lower_closure",
"lower_set",
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
upper_closure_empty : upper_closure (∅ : set α) = ⊤ | by { ext, simp } | lemma | upper_closure_empty | order.upper_lower | src/order/upper_lower/basic.lean | [
"data.set_like.basic",
"data.set.intervals.ord_connected",
"data.set.intervals.order_iso",
"tactic.by_contra"
] | [
"upper_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lower_closure_empty : lower_closure (∅ : set α) = ⊥ | by { ext, simp } | lemma | lower_closure_empty | order.upper_lower | src/order/upper_lower/basic.lean | [
"data.set_like.basic",
"data.set.intervals.ord_connected",
"data.set.intervals.order_iso",
"tactic.by_contra"
] | [
"lower_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
upper_closure_singleton (a : α) : upper_closure ({a} : set α) = upper_set.Ici a | by { ext, simp } | lemma | upper_closure_singleton | order.upper_lower | src/order/upper_lower/basic.lean | [
"data.set_like.basic",
"data.set.intervals.ord_connected",
"data.set.intervals.order_iso",
"tactic.by_contra"
] | [
"upper_closure",
"upper_set.Ici"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lower_closure_singleton (a : α) : lower_closure ({a} : set α) = lower_set.Iic a | by { ext, simp } | lemma | lower_closure_singleton | order.upper_lower | src/order/upper_lower/basic.lean | [
"data.set_like.basic",
"data.set.intervals.ord_connected",
"data.set.intervals.order_iso",
"tactic.by_contra"
] | [
"lower_closure",
"lower_set.Iic"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
upper_closure_univ : upper_closure (univ : set α) = ⊥ | le_bot_iff.1 subset_upper_closure | lemma | upper_closure_univ | order.upper_lower | src/order/upper_lower/basic.lean | [
"data.set_like.basic",
"data.set.intervals.ord_connected",
"data.set.intervals.order_iso",
"tactic.by_contra"
] | [
"subset_upper_closure",
"upper_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lower_closure_univ : lower_closure (univ : set α) = ⊤ | top_le_iff.1 subset_lower_closure | lemma | lower_closure_univ | order.upper_lower | src/order/upper_lower/basic.lean | [
"data.set_like.basic",
"data.set.intervals.ord_connected",
"data.set.intervals.order_iso",
"tactic.by_contra"
] | [
"lower_closure",
"subset_lower_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
upper_closure_eq_top_iff : upper_closure s = ⊤ ↔ s = ∅ | ⟨λ h, subset_empty_iff.1 $ subset_upper_closure.trans (congr_arg coe h).subset,
by { rintro rfl, exact upper_closure_empty }⟩ | lemma | upper_closure_eq_top_iff | order.upper_lower | src/order/upper_lower/basic.lean | [
"data.set_like.basic",
"data.set.intervals.ord_connected",
"data.set.intervals.order_iso",
"tactic.by_contra"
] | [
"upper_closure",
"upper_closure_empty"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lower_closure_eq_bot_iff : lower_closure s = ⊥ ↔ s = ∅ | ⟨λ h, subset_empty_iff.1 $ subset_lower_closure.trans (congr_arg coe h).subset,
by { rintro rfl, exact lower_closure_empty }⟩ | lemma | lower_closure_eq_bot_iff | order.upper_lower | src/order/upper_lower/basic.lean | [
"data.set_like.basic",
"data.set.intervals.ord_connected",
"data.set.intervals.order_iso",
"tactic.by_contra"
] | [
"lower_closure",
"lower_closure_empty"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
upper_closure_union (s t : set α) :
upper_closure (s ∪ t) = upper_closure s ⊓ upper_closure t | by { ext, simp [or_and_distrib_right, exists_or_distrib] } | lemma | upper_closure_union | order.upper_lower | src/order/upper_lower/basic.lean | [
"data.set_like.basic",
"data.set.intervals.ord_connected",
"data.set.intervals.order_iso",
"tactic.by_contra"
] | [
"exists_or_distrib",
"or_and_distrib_right",
"upper_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lower_closure_union (s t : set α) :
lower_closure (s ∪ t) = lower_closure s ⊔ lower_closure t | by { ext, simp [or_and_distrib_right, exists_or_distrib] } | lemma | lower_closure_union | order.upper_lower | src/order/upper_lower/basic.lean | [
"data.set_like.basic",
"data.set.intervals.ord_connected",
"data.set.intervals.order_iso",
"tactic.by_contra"
] | [
"exists_or_distrib",
"lower_closure",
"or_and_distrib_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
upper_closure_Union (f : ι → set α) :
upper_closure (⋃ i, f i) = ⨅ i, upper_closure (f i) | by { ext, simp [←exists_and_distrib_right, @exists_comm α] } | lemma | upper_closure_Union | order.upper_lower | src/order/upper_lower/basic.lean | [
"data.set_like.basic",
"data.set.intervals.ord_connected",
"data.set.intervals.order_iso",
"tactic.by_contra"
] | [
"exists_comm",
"upper_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lower_closure_Union (f : ι → set α) :
lower_closure (⋃ i, f i) = ⨆ i, lower_closure (f i) | by { ext, simp [←exists_and_distrib_right, @exists_comm α] } | lemma | lower_closure_Union | order.upper_lower | src/order/upper_lower/basic.lean | [
"data.set_like.basic",
"data.set.intervals.ord_connected",
"data.set.intervals.order_iso",
"tactic.by_contra"
] | [
"exists_comm",
"lower_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
upper_closure_sUnion (S : set (set α)) :
upper_closure (⋃₀ S) = ⨅ s ∈ S, upper_closure s | by simp_rw [sUnion_eq_bUnion, upper_closure_Union] | lemma | upper_closure_sUnion | order.upper_lower | src/order/upper_lower/basic.lean | [
"data.set_like.basic",
"data.set.intervals.ord_connected",
"data.set.intervals.order_iso",
"tactic.by_contra"
] | [
"upper_closure",
"upper_closure_Union"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lower_closure_sUnion (S : set (set α)) :
lower_closure (⋃₀ S) = ⨆ s ∈ S, lower_closure s | by simp_rw [sUnion_eq_bUnion, lower_closure_Union] | lemma | lower_closure_sUnion | order.upper_lower | src/order/upper_lower/basic.lean | [
"data.set_like.basic",
"data.set.intervals.ord_connected",
"data.set.intervals.order_iso",
"tactic.by_contra"
] | [
"lower_closure",
"lower_closure_Union"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set.ord_connected.upper_closure_inter_lower_closure (h : s.ord_connected) :
↑(upper_closure s) ∩ ↑(lower_closure s) = s | (subset_inter subset_upper_closure subset_lower_closure).antisymm' $ λ a ⟨⟨b, hb, hba⟩, c, hc, hac⟩,
h.out hb hc ⟨hba, hac⟩ | lemma | set.ord_connected.upper_closure_inter_lower_closure | order.upper_lower | src/order/upper_lower/basic.lean | [
"data.set_like.basic",
"data.set.intervals.ord_connected",
"data.set.intervals.order_iso",
"tactic.by_contra"
] | [
"antisymm'",
"lower_closure",
"subset_lower_closure",
"subset_upper_closure",
"upper_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ord_connected_iff_upper_closure_inter_lower_closure :
s.ord_connected ↔ ↑(upper_closure s) ∩ ↑(lower_closure s) = s | begin
refine ⟨set.ord_connected.upper_closure_inter_lower_closure, λ h, _⟩,
rw ←h,
exact (upper_set.upper _).ord_connected.inter (lower_set.lower _).ord_connected,
end | lemma | ord_connected_iff_upper_closure_inter_lower_closure | order.upper_lower | src/order/upper_lower/basic.lean | [
"data.set_like.basic",
"data.set.intervals.ord_connected",
"data.set.intervals.order_iso",
"tactic.by_contra"
] | [
"lower_closure",
"lower_set.lower",
"upper_closure",
"upper_set.upper"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
upper_bounds_lower_closure :
upper_bounds (lower_closure s : set α) = upper_bounds s | (upper_bounds_mono_set subset_lower_closure).antisymm $ λ a ha b ⟨c, hc, hcb⟩, hcb.trans $ ha hc | lemma | upper_bounds_lower_closure | order.upper_lower | src/order/upper_lower/basic.lean | [
"data.set_like.basic",
"data.set.intervals.ord_connected",
"data.set.intervals.order_iso",
"tactic.by_contra"
] | [
"lower_closure",
"subset_lower_closure",
"upper_bounds",
"upper_bounds_mono_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lower_bounds_upper_closure :
lower_bounds (upper_closure s : set α) = lower_bounds s | (lower_bounds_mono_set subset_upper_closure).antisymm $ λ a ha b ⟨c, hc, hcb⟩, (ha hc).trans hcb | lemma | lower_bounds_upper_closure | order.upper_lower | src/order/upper_lower/basic.lean | [
"data.set_like.basic",
"data.set.intervals.ord_connected",
"data.set.intervals.order_iso",
"tactic.by_contra"
] | [
"lower_bounds",
"lower_bounds_mono_set",
"subset_upper_closure",
"upper_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bdd_above_lower_closure : bdd_above (lower_closure s : set α) ↔ bdd_above s | by simp_rw [bdd_above, upper_bounds_lower_closure] | lemma | bdd_above_lower_closure | order.upper_lower | src/order/upper_lower/basic.lean | [
"data.set_like.basic",
"data.set.intervals.ord_connected",
"data.set.intervals.order_iso",
"tactic.by_contra"
] | [
"bdd_above",
"lower_closure",
"upper_bounds_lower_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bdd_below_upper_closure : bdd_below (upper_closure s : set α) ↔ bdd_below s | by simp_rw [bdd_below, lower_bounds_upper_closure] | lemma | bdd_below_upper_closure | order.upper_lower | src/order/upper_lower/basic.lean | [
"data.set_like.basic",
"data.set.intervals.ord_connected",
"data.set.intervals.order_iso",
"tactic.by_contra"
] | [
"bdd_below",
"lower_bounds_upper_closure",
"upper_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_upper_set.prod (hs : is_upper_set s) (ht : is_upper_set t) : is_upper_set (s ×ˢ t) | λ a b h ha, ⟨hs h.1 ha.1, ht h.2 ha.2⟩ | lemma | is_upper_set.prod | order.upper_lower | src/order/upper_lower/basic.lean | [
"data.set_like.basic",
"data.set.intervals.ord_connected",
"data.set.intervals.order_iso",
"tactic.by_contra"
] | [
"is_upper_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_lower_set.prod (hs : is_lower_set s) (ht : is_lower_set t) : is_lower_set (s ×ˢ t) | λ a b h ha, ⟨hs h.1 ha.1, ht h.2 ha.2⟩ | lemma | is_lower_set.prod | order.upper_lower | src/order/upper_lower/basic.lean | [
"data.set_like.basic",
"data.set.intervals.ord_connected",
"data.set.intervals.order_iso",
"tactic.by_contra"
] | [
"is_lower_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod : upper_set (α × β) | ⟨s ×ˢ t, s.2.prod t.2⟩ | def | upper_set.prod | order.upper_lower | src/order/upper_lower/basic.lean | [
"data.set_like.basic",
"data.set.intervals.ord_connected",
"data.set.intervals.order_iso",
"tactic.by_contra"
] | [
"upper_set"
] | The product of two upper sets as an upper set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_prod : (↑(s ×ˢ t) : set (α × β)) = s ×ˢ t | rfl | lemma | upper_set.coe_prod | order.upper_lower | src/order/upper_lower/basic.lean | [
"data.set_like.basic",
"data.set.intervals.ord_connected",
"data.set.intervals.order_iso",
"tactic.by_contra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_prod {s : upper_set α} {t : upper_set β} : x ∈ s ×ˢ t ↔ x.1 ∈ s ∧ x.2 ∈ t | iff.rfl | lemma | upper_set.mem_prod | order.upper_lower | src/order/upper_lower/basic.lean | [
"data.set_like.basic",
"data.set.intervals.ord_connected",
"data.set.intervals.order_iso",
"tactic.by_contra"
] | [
"upper_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Ici_prod (x : α × β) : Ici x = Ici x.1 ×ˢ Ici x.2 | rfl | lemma | upper_set.Ici_prod | order.upper_lower | src/order/upper_lower/basic.lean | [
"data.set_like.basic",
"data.set.intervals.ord_connected",
"data.set.intervals.order_iso",
"tactic.by_contra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_top : s ×ˢ (⊤ : upper_set β) = ⊤ | ext prod_empty | lemma | upper_set.prod_top | order.upper_lower | src/order/upper_lower/basic.lean | [
"data.set_like.basic",
"data.set.intervals.ord_connected",
"data.set.intervals.order_iso",
"tactic.by_contra"
] | [
"upper_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
top_prod : (⊤ : upper_set α) ×ˢ t = ⊤ | ext empty_prod | lemma | upper_set.top_prod | order.upper_lower | src/order/upper_lower/basic.lean | [
"data.set_like.basic",
"data.set.intervals.ord_connected",
"data.set.intervals.order_iso",
"tactic.by_contra"
] | [
"upper_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bot_prod_bot : (⊥ : upper_set α) ×ˢ (⊥ : upper_set β) = ⊥ | ext univ_prod_univ | lemma | upper_set.bot_prod_bot | order.upper_lower | src/order/upper_lower/basic.lean | [
"data.set_like.basic",
"data.set.intervals.ord_connected",
"data.set.intervals.order_iso",
"tactic.by_contra"
] | [
"upper_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sup_prod : (s₁ ⊔ s₂) ×ˢ t = s₁ ×ˢ t ⊔ s₂ ×ˢ t | ext inter_prod | lemma | upper_set.sup_prod | order.upper_lower | src/order/upper_lower/basic.lean | [
"data.set_like.basic",
"data.set.intervals.ord_connected",
"data.set.intervals.order_iso",
"tactic.by_contra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_sup : s ×ˢ (t₁ ⊔ t₂) = s ×ˢ t₁ ⊔ s ×ˢ t₂ | ext prod_inter | lemma | upper_set.prod_sup | order.upper_lower | src/order/upper_lower/basic.lean | [
"data.set_like.basic",
"data.set.intervals.ord_connected",
"data.set.intervals.order_iso",
"tactic.by_contra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inf_prod : (s₁ ⊓ s₂) ×ˢ t = s₁ ×ˢ t ⊓ s₂ ×ˢ t | ext union_prod | lemma | upper_set.inf_prod | order.upper_lower | src/order/upper_lower/basic.lean | [
"data.set_like.basic",
"data.set.intervals.ord_connected",
"data.set.intervals.order_iso",
"tactic.by_contra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_inf : s ×ˢ (t₁ ⊓ t₂) = s ×ˢ t₁ ⊓ s ×ˢ t₂ | ext prod_union | lemma | upper_set.prod_inf | order.upper_lower | src/order/upper_lower/basic.lean | [
"data.set_like.basic",
"data.set.intervals.ord_connected",
"data.set.intervals.order_iso",
"tactic.by_contra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_sup_prod : s₁ ×ˢ t₁ ⊔ s₂ ×ˢ t₂ = (s₁ ⊔ s₂) ×ˢ (t₁ ⊔ t₂) | ext prod_inter_prod | lemma | upper_set.prod_sup_prod | order.upper_lower | src/order/upper_lower/basic.lean | [
"data.set_like.basic",
"data.set.intervals.ord_connected",
"data.set.intervals.order_iso",
"tactic.by_contra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_mono : s₁ ≤ s₂ → t₁ ≤ t₂ → s₁ ×ˢ t₁ ≤ s₂ ×ˢ t₂ | prod_mono | lemma | upper_set.prod_mono | order.upper_lower | src/order/upper_lower/basic.lean | [
"data.set_like.basic",
"data.set.intervals.ord_connected",
"data.set.intervals.order_iso",
"tactic.by_contra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_mono_left : s₁ ≤ s₂ → s₁ ×ˢ t ≤ s₂ ×ˢ t | prod_mono_left | lemma | upper_set.prod_mono_left | order.upper_lower | src/order/upper_lower/basic.lean | [
"data.set_like.basic",
"data.set.intervals.ord_connected",
"data.set.intervals.order_iso",
"tactic.by_contra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_mono_right : t₁ ≤ t₂ → s ×ˢ t₁ ≤ s ×ˢ t₂ | prod_mono_right | lemma | upper_set.prod_mono_right | order.upper_lower | src/order/upper_lower/basic.lean | [
"data.set_like.basic",
"data.set.intervals.ord_connected",
"data.set.intervals.order_iso",
"tactic.by_contra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_self_le_prod_self : s₁ ×ˢ s₁ ≤ s₂ ×ˢ s₂ ↔ s₁ ≤ s₂ | prod_self_subset_prod_self | lemma | upper_set.prod_self_le_prod_self | order.upper_lower | src/order/upper_lower/basic.lean | [
"data.set_like.basic",
"data.set.intervals.ord_connected",
"data.set.intervals.order_iso",
"tactic.by_contra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_self_lt_prod_self : s₁ ×ˢ s₁ < s₂ ×ˢ s₂ ↔ s₁ < s₂ | prod_self_ssubset_prod_self | lemma | upper_set.prod_self_lt_prod_self | order.upper_lower | src/order/upper_lower/basic.lean | [
"data.set_like.basic",
"data.set.intervals.ord_connected",
"data.set.intervals.order_iso",
"tactic.by_contra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_le_prod_iff : s₁ ×ˢ t₁ ≤ s₂ ×ˢ t₂ ↔ s₁ ≤ s₂ ∧ t₁ ≤ t₂ ∨ s₂ = ⊤ ∨ t₂ = ⊤ | prod_subset_prod_iff.trans $ by simp | lemma | upper_set.prod_le_prod_iff | order.upper_lower | src/order/upper_lower/basic.lean | [
"data.set_like.basic",
"data.set.intervals.ord_connected",
"data.set.intervals.order_iso",
"tactic.by_contra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_eq_top : s ×ˢ t = ⊤ ↔ s = ⊤ ∨ t = ⊤ | by { simp_rw set_like.ext'_iff, exact prod_eq_empty_iff } | lemma | upper_set.prod_eq_top | order.upper_lower | src/order/upper_lower/basic.lean | [
"data.set_like.basic",
"data.set.intervals.ord_connected",
"data.set.intervals.order_iso",
"tactic.by_contra"
] | [
"set_like.ext'_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
codisjoint_prod :
codisjoint (s₁ ×ˢ t₁) (s₂ ×ˢ t₂) ↔ codisjoint s₁ s₂ ∨ codisjoint t₁ t₂ | by simp_rw [codisjoint_iff, prod_sup_prod, prod_eq_top] | lemma | upper_set.codisjoint_prod | order.upper_lower | src/order/upper_lower/basic.lean | [
"data.set_like.basic",
"data.set.intervals.ord_connected",
"data.set.intervals.order_iso",
"tactic.by_contra"
] | [
"codisjoint",
"codisjoint_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod : lower_set (α × β) | ⟨s ×ˢ t, s.2.prod t.2⟩ | def | lower_set.prod | order.upper_lower | src/order/upper_lower/basic.lean | [
"data.set_like.basic",
"data.set.intervals.ord_connected",
"data.set.intervals.order_iso",
"tactic.by_contra"
] | [
"lower_set"
] | The product of two lower sets as a lower set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_prod {s : lower_set α} {t : lower_set β} : x ∈ s ×ˢ t ↔ x.1 ∈ s ∧ x.2 ∈ t | iff.rfl | lemma | lower_set.mem_prod | order.upper_lower | src/order/upper_lower/basic.lean | [
"data.set_like.basic",
"data.set.intervals.ord_connected",
"data.set.intervals.order_iso",
"tactic.by_contra"
] | [
"lower_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Iic_prod (x : α × β) : Iic x = Iic x.1 ×ˢ Iic x.2 | rfl | lemma | lower_set.Iic_prod | order.upper_lower | src/order/upper_lower/basic.lean | [
"data.set_like.basic",
"data.set.intervals.ord_connected",
"data.set.intervals.order_iso",
"tactic.by_contra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Ici_prod_Ici (a : α) (b : β) : Iic a ×ˢ Iic b = Iic (a, b) | rfl | lemma | lower_set.Ici_prod_Ici | order.upper_lower | src/order/upper_lower/basic.lean | [
"data.set_like.basic",
"data.set.intervals.ord_connected",
"data.set.intervals.order_iso",
"tactic.by_contra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_bot : s ×ˢ (⊥ : lower_set β) = ⊥ | ext prod_empty | lemma | lower_set.prod_bot | order.upper_lower | src/order/upper_lower/basic.lean | [
"data.set_like.basic",
"data.set.intervals.ord_connected",
"data.set.intervals.order_iso",
"tactic.by_contra"
] | [
"lower_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bot_prod : (⊥ : lower_set α) ×ˢ t = ⊥ | ext empty_prod | lemma | lower_set.bot_prod | order.upper_lower | src/order/upper_lower/basic.lean | [
"data.set_like.basic",
"data.set.intervals.ord_connected",
"data.set.intervals.order_iso",
"tactic.by_contra"
] | [
"lower_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
top_prod_top : (⊤ : lower_set α) ×ˢ (⊤ : lower_set β) = ⊤ | ext univ_prod_univ | lemma | lower_set.top_prod_top | order.upper_lower | src/order/upper_lower/basic.lean | [
"data.set_like.basic",
"data.set.intervals.ord_connected",
"data.set.intervals.order_iso",
"tactic.by_contra"
] | [
"lower_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inf_prod : (s₁ ⊓ s₂) ×ˢ t = s₁ ×ˢ t ⊓ s₂ ×ˢ t | ext inter_prod | lemma | lower_set.inf_prod | order.upper_lower | src/order/upper_lower/basic.lean | [
"data.set_like.basic",
"data.set.intervals.ord_connected",
"data.set.intervals.order_iso",
"tactic.by_contra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_inf : s ×ˢ (t₁ ⊓ t₂) = s ×ˢ t₁ ⊓ s ×ˢ t₂ | ext prod_inter | lemma | lower_set.prod_inf | order.upper_lower | src/order/upper_lower/basic.lean | [
"data.set_like.basic",
"data.set.intervals.ord_connected",
"data.set.intervals.order_iso",
"tactic.by_contra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sup_prod : (s₁ ⊔ s₂) ×ˢ t = s₁ ×ˢ t ⊔ s₂ ×ˢ t | ext union_prod | lemma | lower_set.sup_prod | order.upper_lower | src/order/upper_lower/basic.lean | [
"data.set_like.basic",
"data.set.intervals.ord_connected",
"data.set.intervals.order_iso",
"tactic.by_contra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_sup : s ×ˢ (t₁ ⊔ t₂) = s ×ˢ t₁ ⊔ s ×ˢ t₂ | ext prod_union | lemma | lower_set.prod_sup | order.upper_lower | src/order/upper_lower/basic.lean | [
"data.set_like.basic",
"data.set.intervals.ord_connected",
"data.set.intervals.order_iso",
"tactic.by_contra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_inf_prod : s₁ ×ˢ t₁ ⊓ s₂ ×ˢ t₂ = (s₁ ⊓ s₂) ×ˢ (t₁ ⊓ t₂) | ext prod_inter_prod | lemma | lower_set.prod_inf_prod | order.upper_lower | src/order/upper_lower/basic.lean | [
"data.set_like.basic",
"data.set.intervals.ord_connected",
"data.set.intervals.order_iso",
"tactic.by_contra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_le_prod_iff : s₁ ×ˢ t₁ ≤ s₂ ×ˢ t₂ ↔ s₁ ≤ s₂ ∧ t₁ ≤ t₂ ∨ s₁ = ⊥ ∨ t₁ = ⊥ | prod_subset_prod_iff.trans $ by simp | lemma | lower_set.prod_le_prod_iff | order.upper_lower | src/order/upper_lower/basic.lean | [
"data.set_like.basic",
"data.set.intervals.ord_connected",
"data.set.intervals.order_iso",
"tactic.by_contra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_eq_bot : s ×ˢ t = ⊥ ↔ s = ⊥ ∨ t = ⊥ | by { simp_rw set_like.ext'_iff, exact prod_eq_empty_iff } | lemma | lower_set.prod_eq_bot | order.upper_lower | src/order/upper_lower/basic.lean | [
"data.set_like.basic",
"data.set.intervals.ord_connected",
"data.set.intervals.order_iso",
"tactic.by_contra"
] | [
"set_like.ext'_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
disjoint_prod : disjoint (s₁ ×ˢ t₁) (s₂ ×ˢ t₂) ↔ disjoint s₁ s₂ ∨ disjoint t₁ t₂ | by simp_rw [disjoint_iff, prod_inf_prod, prod_eq_bot] | lemma | lower_set.disjoint_prod | order.upper_lower | src/order/upper_lower/basic.lean | [
"data.set_like.basic",
"data.set.intervals.ord_connected",
"data.set.intervals.order_iso",
"tactic.by_contra"
] | [
"disjoint",
"disjoint_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
upper_closure_prod (s : set α) (t : set β) :
upper_closure (s ×ˢ t) = upper_closure s ×ˢ upper_closure t | by { ext, simp [prod.le_def, and_and_and_comm _ (_ ∈ t)] } | lemma | upper_closure_prod | order.upper_lower | src/order/upper_lower/basic.lean | [
"data.set_like.basic",
"data.set.intervals.ord_connected",
"data.set.intervals.order_iso",
"tactic.by_contra"
] | [
"and_and_and_comm",
"prod.le_def",
"upper_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lower_closure_prod (s : set α) (t : set β) :
lower_closure (s ×ˢ t) = lower_closure s ×ˢ lower_closure t | by { ext, simp [prod.le_def, and_and_and_comm _ (_ ∈ t)] } | lemma | lower_closure_prod | order.upper_lower | src/order/upper_lower/basic.lean | [
"data.set_like.basic",
"data.set.intervals.ord_connected",
"data.set.intervals.order_iso",
"tactic.by_contra"
] | [
"and_and_and_comm",
"lower_closure",
"prod.le_def"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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