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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|---|---|---|---|---|---|---|---|---|---|---|
coe_eq_univ : (s : set α) = univ ↔ s = ⊥ | by simp [set_like.ext'_iff] | lemma | upper_set.coe_eq_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"
] | [
"set_like.ext'_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_eq_empty : (s : set α) = ∅ ↔ s = ⊤ | by simp [set_like.ext'_iff] | lemma | upper_set.coe_eq_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"
] | [
"set_like.ext'_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_sup (s t : upper_set α) : (↑(s ⊔ t) : set α) = s ∩ t | rfl | lemma | upper_set.coe_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"
] | [
"upper_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_inf (s t : upper_set α) : (↑(s ⊓ t) : set α) = s ∪ t | rfl | lemma | upper_set.coe_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"
] | [
"upper_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_Sup (S : set (upper_set α)) : (↑(Sup S) : set α) = ⋂ s ∈ S, ↑s | rfl | lemma | upper_set.coe_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"
] | [
"upper_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_Inf (S : set (upper_set α)) : (↑(Inf S) : set α) = ⋃ s ∈ S, ↑s | rfl | lemma | upper_set.coe_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"
] | [
"upper_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_supr (f : ι → upper_set α) : (↑(⨆ i, f i) : set α) = ⋂ i, f i | by simp [supr] | lemma | upper_set.coe_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"
] | [
"supr",
"upper_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_infi (f : ι → upper_set α) : (↑(⨅ i, f i) : set α) = ⋃ i, f i | by simp [infi] | lemma | upper_set.coe_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"
] | [
"infi",
"upper_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_supr₂ (f : Π i, κ i → upper_set α) :
(↑(⨆ i j, f i j) : set α) = ⋂ i j, f i j | by simp_rw coe_supr | lemma | upper_set.coe_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"
] | [
"upper_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_infi₂ (f : Π i, κ i → upper_set α) :
(↑(⨅ i j, f i j) : set α) = ⋃ i j, f i j | by simp_rw coe_infi | lemma | upper_set.coe_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"
] | [
"upper_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
not_mem_top : a ∉ (⊤ : upper_set α) | id | lemma | upper_set.not_mem_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 | |
mem_bot : a ∈ (⊥ : upper_set α) | trivial | lemma | upper_set.mem_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 | |
mem_sup_iff : a ∈ s ⊔ t ↔ a ∈ s ∧ a ∈ t | iff.rfl | lemma | upper_set.mem_sup_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_inf_iff : a ∈ s ⊓ t ↔ a ∈ s ∨ a ∈ t | iff.rfl | lemma | upper_set.mem_inf_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_Sup_iff : a ∈ Sup S ↔ ∀ s ∈ S, a ∈ s | mem_Inter₂ | lemma | upper_set.mem_Sup_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_Inf_iff : a ∈ Inf S ↔ ∃ s ∈ S, a ∈ s | mem_Union₂ | lemma | upper_set.mem_Inf_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_supr_iff {f : ι → upper_set α} : a ∈ (⨆ i, f i) ↔ ∀ i, a ∈ f i | by { rw [←set_like.mem_coe, coe_supr], exact mem_Inter } | lemma | upper_set.mem_supr_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_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_infi_iff {f : ι → upper_set α} : a ∈ (⨅ i, f i) ↔ ∃ i, a ∈ f i | by { rw [←set_like.mem_coe, coe_infi], exact mem_Union } | lemma | upper_set.mem_infi_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_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_supr₂_iff {f : Π i, κ i → upper_set α} : a ∈ (⨆ i j, f i j) ↔ ∀ i j, a ∈ f i j | by simp_rw mem_supr_iff | lemma | upper_set.mem_supr₂_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_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_infi₂_iff {f : Π i, κ i → upper_set α} : a ∈ (⨅ i j, f i j) ↔ ∃ i j, a ∈ f i j | by simp_rw mem_infi_iff | lemma | upper_set.mem_infi₂_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_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
codisjoint_coe : codisjoint (s : set α) t ↔ disjoint s t | by simp [disjoint_iff, codisjoint_iff, set_like.ext'_iff] | lemma | upper_set.codisjoint_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"
] | [
"codisjoint",
"codisjoint_iff",
"disjoint",
"disjoint_iff",
"set_like.ext'_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_subset_coe : (s : set α) ⊆ t ↔ s ≤ t | iff.rfl | lemma | lower_set.coe_subset_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_top : ((⊤ : lower_set α) : set α) = univ | rfl | lemma | lower_set.coe_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 | |
coe_bot : ((⊥ : lower_set α) : set α) = ∅ | rfl | lemma | lower_set.coe_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 | |
coe_eq_univ : (s : set α) = univ ↔ s = ⊤ | by simp [set_like.ext'_iff] | lemma | lower_set.coe_eq_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"
] | [
"set_like.ext'_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_eq_empty : (s : set α) = ∅ ↔ s = ⊥ | by simp [set_like.ext'_iff] | lemma | lower_set.coe_eq_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"
] | [
"set_like.ext'_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_sup (s t : lower_set α) : (↑(s ⊔ t) : set α) = s ∪ t | rfl | lemma | lower_set.coe_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"
] | [
"lower_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_inf (s t : lower_set α) : (↑(s ⊓ t) : set α) = s ∩ t | rfl | lemma | lower_set.coe_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"
] | [
"lower_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_Sup (S : set (lower_set α)) : (↑(Sup S) : set α) = ⋃ s ∈ S, ↑s | rfl | lemma | lower_set.coe_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"
] | [
"lower_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_Inf (S : set (lower_set α)) : (↑(Inf S) : set α) = ⋂ s ∈ S, ↑s | rfl | lemma | lower_set.coe_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"
] | [
"lower_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_supr (f : ι → lower_set α) : (↑(⨆ i, f i) : set α) = ⋃ i, f i | by simp_rw [supr, coe_Sup, mem_range, Union_exists, Union_Union_eq'] | lemma | lower_set.coe_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"
] | [
"lower_set",
"supr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_infi (f : ι → lower_set α) : (↑(⨅ i, f i) : set α) = ⋂ i, f i | by simp_rw [infi, coe_Inf, mem_range, Inter_exists, Inter_Inter_eq'] | lemma | lower_set.coe_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"
] | [
"infi",
"lower_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_supr₂ (f : Π i, κ i → lower_set α) :
(↑(⨆ i j, f i j) : set α) = ⋃ i j, f i j | by simp_rw coe_supr | lemma | lower_set.coe_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"
] | [
"lower_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_infi₂ (f : Π i, κ i → lower_set α) :
(↑(⨅ i j, f i j) : set α) = ⋂ i j, f i j | by simp_rw coe_infi | lemma | lower_set.coe_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"
] | [
"lower_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_top : a ∈ (⊤ : lower_set α) | trivial | lemma | lower_set.mem_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 | |
not_mem_bot : a ∉ (⊥ : lower_set α) | id | lemma | lower_set.not_mem_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 | |
mem_sup_iff : a ∈ s ⊔ t ↔ a ∈ s ∨ a ∈ t | iff.rfl | lemma | lower_set.mem_sup_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_inf_iff : a ∈ s ⊓ t ↔ a ∈ s ∧ a ∈ t | iff.rfl | lemma | lower_set.mem_inf_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_Sup_iff : a ∈ Sup S ↔ ∃ s ∈ S, a ∈ s | mem_Union₂ | lemma | lower_set.mem_Sup_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_Inf_iff : a ∈ Inf S ↔ ∀ s ∈ S, a ∈ s | mem_Inter₂ | lemma | lower_set.mem_Inf_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_supr_iff {f : ι → lower_set α} : a ∈ (⨆ i, f i) ↔ ∃ i, a ∈ f i | by { rw [←set_like.mem_coe, coe_supr], exact mem_Union } | lemma | lower_set.mem_supr_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_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_infi_iff {f : ι → lower_set α} : a ∈ (⨅ i, f i) ↔ ∀ i, a ∈ f i | by { rw [←set_like.mem_coe, coe_infi], exact mem_Inter } | lemma | lower_set.mem_infi_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_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_supr₂_iff {f : Π i, κ i → lower_set α} : a ∈ (⨆ i j, f i j) ↔ ∃ i j, a ∈ f i j | by simp_rw mem_supr_iff | lemma | lower_set.mem_supr₂_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_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_infi₂_iff {f : Π i, κ i → lower_set α} : a ∈ (⨅ i j, f i j) ↔ ∀ i j, a ∈ f i j | by simp_rw mem_infi_iff | lemma | lower_set.mem_infi₂_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_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
disjoint_coe : disjoint (s : set α) t ↔ disjoint s t | by simp [disjoint_iff, set_like.ext'_iff] | lemma | lower_set.disjoint_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"
] | [
"disjoint",
"disjoint_iff",
"set_like.ext'_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
upper_set.compl (s : upper_set α) : lower_set α | ⟨sᶜ, s.upper.compl⟩ | def | upper_set.compl | 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",
"upper_set"
] | The complement of a lower set as an upper set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lower_set.compl (s : lower_set α) : upper_set α | ⟨sᶜ, s.lower.compl⟩ | def | lower_set.compl | 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",
"upper_set"
] | The complement of a lower set as an upper set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_compl (s : upper_set α) : (s.compl : set α) = sᶜ | rfl | lemma | upper_set.coe_compl | 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 | |
mem_compl_iff : a ∈ s.compl ↔ a ∉ s | iff.rfl | lemma | upper_set.mem_compl_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 | |
compl_compl (s : upper_set α) : s.compl.compl = s | upper_set.ext $ compl_compl _ | lemma | upper_set.compl_compl | 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"
] | [
"compl_compl",
"upper_set",
"upper_set.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
compl_le_compl : s.compl ≤ t.compl ↔ s ≤ t | compl_subset_compl | lemma | upper_set.compl_le_compl | 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"
] | [
"compl_le_compl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
compl_sup (s t : upper_set α) : (s ⊔ t).compl = s.compl ⊔ t.compl | lower_set.ext compl_inf | lemma | upper_set.compl_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"
] | [
"compl_inf",
"compl_sup",
"lower_set.ext",
"upper_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
compl_inf (s t : upper_set α) : (s ⊓ t).compl = s.compl ⊓ t.compl | lower_set.ext compl_sup | lemma | upper_set.compl_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"
] | [
"compl_inf",
"compl_sup",
"lower_set.ext",
"upper_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
compl_top : (⊤ : upper_set α).compl = ⊤ | lower_set.ext compl_empty | lemma | upper_set.compl_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"
] | [
"compl_top",
"lower_set.ext",
"upper_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
compl_bot : (⊥ : upper_set α).compl = ⊥ | lower_set.ext compl_univ | lemma | upper_set.compl_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"
] | [
"compl_bot",
"lower_set.ext",
"upper_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
compl_Sup (S : set (upper_set α)) :
(Sup S).compl = ⨆ s ∈ S, upper_set.compl s | lower_set.ext $ by simp only [coe_compl, coe_Sup, compl_Inter₂, lower_set.coe_supr₂] | lemma | upper_set.compl_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"
] | [
"compl_Sup",
"lower_set.coe_supr₂",
"lower_set.ext",
"upper_set",
"upper_set.compl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
compl_Inf (S : set (upper_set α)) :
(Inf S).compl = ⨅ s ∈ S, upper_set.compl s | lower_set.ext $ by simp only [coe_compl, coe_Inf, compl_Union₂, lower_set.coe_infi₂] | lemma | upper_set.compl_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"
] | [
"compl_Inf",
"lower_set.coe_infi₂",
"lower_set.ext",
"upper_set",
"upper_set.compl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
compl_supr (f : ι → upper_set α) : (⨆ i, f i).compl = ⨆ i, (f i).compl | lower_set.ext $ by simp only [coe_compl, coe_supr, compl_Inter, lower_set.coe_supr] | lemma | upper_set.compl_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"
] | [
"compl_supr",
"lower_set.coe_supr",
"lower_set.ext",
"upper_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
compl_infi (f : ι → upper_set α) : (⨅ i, f i).compl = ⨅ i, (f i).compl | lower_set.ext $ by simp only [coe_compl, coe_infi, compl_Union, lower_set.coe_infi] | lemma | upper_set.compl_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"
] | [
"compl_infi",
"lower_set.coe_infi",
"lower_set.ext",
"upper_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
compl_supr₂ (f : Π i, κ i → upper_set α) :
(⨆ i j, f i j).compl = ⨆ i j, (f i j).compl | by simp_rw upper_set.compl_supr | lemma | upper_set.compl_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"
] | [
"upper_set",
"upper_set.compl_supr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
compl_infi₂ (f : Π i, κ i → upper_set α) :
(⨅ i j, f i j).compl = ⨅ i j, (f i j).compl | by simp_rw upper_set.compl_infi | lemma | upper_set.compl_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"
] | [
"upper_set",
"upper_set.compl_infi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_compl (s : lower_set α) : (s.compl : set α) = sᶜ | rfl | lemma | lower_set.coe_compl | 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 | |
compl_compl (s : lower_set α) : s.compl.compl = s | lower_set.ext $ compl_compl _ | lemma | lower_set.compl_compl | 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"
] | [
"compl_compl",
"lower_set",
"lower_set.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
compl_sup (s t : lower_set α) : (s ⊔ t).compl = s.compl ⊔ t.compl | upper_set.ext compl_sup | lemma | lower_set.compl_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"
] | [
"compl_sup",
"lower_set",
"upper_set.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
compl_inf (s t : lower_set α) : (s ⊓ t).compl = s.compl ⊓ t.compl | upper_set.ext compl_inf | lemma | lower_set.compl_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"
] | [
"compl_inf",
"lower_set",
"upper_set.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
compl_top : (⊤ : lower_set α).compl = ⊤ | upper_set.ext compl_univ | lemma | lower_set.compl_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"
] | [
"compl_top",
"lower_set",
"upper_set.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
compl_bot : (⊥ : lower_set α).compl = ⊥ | upper_set.ext compl_empty | lemma | lower_set.compl_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"
] | [
"compl_bot",
"lower_set",
"upper_set.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
compl_Sup (S : set (lower_set α)) : (Sup S).compl = ⨆ s ∈ S, lower_set.compl s | upper_set.ext $ by simp only [coe_compl, coe_Sup, compl_Union₂, upper_set.coe_supr₂] | lemma | lower_set.compl_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"
] | [
"compl_Sup",
"lower_set",
"lower_set.compl",
"upper_set.coe_supr₂",
"upper_set.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
compl_Inf (S : set (lower_set α)) : (Inf S).compl = ⨅ s ∈ S, lower_set.compl s | upper_set.ext $ by simp only [coe_compl, coe_Inf, compl_Inter₂, upper_set.coe_infi₂] | lemma | lower_set.compl_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"
] | [
"compl_Inf",
"lower_set",
"lower_set.compl",
"upper_set.coe_infi₂",
"upper_set.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
compl_supr (f : ι → lower_set α) : (⨆ i, f i).compl = ⨆ i, (f i).compl | upper_set.ext $ by simp only [coe_compl, coe_supr, compl_Union, upper_set.coe_supr] | lemma | lower_set.compl_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"
] | [
"compl_supr",
"lower_set",
"upper_set.coe_supr",
"upper_set.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
compl_infi (f : ι → lower_set α) : (⨅ i, f i).compl = ⨅ i, (f i).compl | upper_set.ext $ by simp only [coe_compl, coe_infi, compl_Inter, upper_set.coe_infi] | lemma | lower_set.compl_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"
] | [
"compl_infi",
"lower_set",
"upper_set.coe_infi",
"upper_set.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
compl_supr₂ (f : Π i, κ i → lower_set α) :
(⨆ i j, f i j).compl = ⨆ i j, (f i j).compl | by simp_rw lower_set.compl_supr | lemma | lower_set.compl_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"
] | [
"lower_set",
"lower_set.compl_supr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
compl_infi₂ (f : Π i, κ i → lower_set α) :
(⨅ i j, f i j).compl = ⨅ i j, (f i j).compl | by simp_rw lower_set.compl_infi | lemma | lower_set.compl_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"
] | [
"lower_set",
"lower_set.compl_infi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
upper_set_iso_lower_set : upper_set α ≃o lower_set α | { to_fun := upper_set.compl,
inv_fun := lower_set.compl,
left_inv := upper_set.compl_compl,
right_inv := lower_set.compl_compl,
map_rel_iff' := λ _ _, upper_set.compl_le_compl } | def | upper_set_iso_lower_set | 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"
] | [
"inv_fun",
"lower_set",
"lower_set.compl",
"lower_set.compl_compl",
"upper_set",
"upper_set.compl",
"upper_set.compl_compl",
"upper_set.compl_le_compl"
] | Upper sets are order-isomorphic to lower sets under complementation. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
upper_set.is_total_le : is_total (upper_set α) (≤) | ⟨λ s t, t.upper.total s.upper⟩ | instance | upper_set.is_total_le | 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 | |
lower_set.is_total_le : is_total (lower_set α) (≤) | ⟨λ s t, s.lower.total t.lower⟩ | instance | lower_set.is_total_le | 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 | |
map (f : α ≃o β) : upper_set α ≃o upper_set β | { to_fun := λ s, ⟨f '' s, s.upper.image f⟩,
inv_fun := λ t, ⟨f ⁻¹' t, t.upper.preimage f.monotone⟩,
left_inv := λ _, ext $ f.preimage_image _,
right_inv := λ _, ext $ f.image_preimage _,
map_rel_iff' := λ s t, image_subset_image_iff f.injective } | def | upper_set.map | 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"
] | [
"inv_fun",
"upper_set"
] | An order isomorphism of preorders induces an order isomorphism of their upper sets. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
symm_map (f : α ≃o β) : (map f).symm = map f.symm | fun_like.ext _ _ $ λ s, ext $ set.preimage_equiv_eq_image_symm _ _ | lemma | upper_set.symm_map | 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"
] | [
"fun_like.ext",
"set.preimage_equiv_eq_image_symm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_map : b ∈ map f s ↔ f.symm b ∈ s | by { rw [←f.symm_symm, ←symm_map, f.symm_symm], refl } | lemma | upper_set.mem_map | 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"
] | [
"mem_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_refl : map (order_iso.refl α) = order_iso.refl _ | by { ext, simp } | lemma | upper_set.map_refl | 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.refl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_map (g : β ≃o γ) (f : α ≃o β) : map g (map f s) = map (f.trans g) s | by { ext, simp } | lemma | upper_set.map_map | 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 | |
coe_map : (map f s : set β) = f '' s | rfl | lemma | upper_set.coe_map | 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 (f : α ≃o β) : lower_set α ≃o lower_set β | { to_fun := λ s, ⟨f '' s, s.lower.image f⟩,
inv_fun := λ t, ⟨f ⁻¹' t, t.lower.preimage f.monotone⟩,
left_inv := λ _, set_like.coe_injective $ f.preimage_image _,
right_inv := λ _, set_like.coe_injective $ f.image_preimage _,
map_rel_iff' := λ s t, image_subset_image_iff f.injective } | def | lower_set.map | 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"
] | [
"inv_fun",
"lower_set",
"set_like.coe_injective"
] | An order isomorphism of preorders induces an order isomorphism of their lower sets. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
symm_map (f : α ≃o β) : (map f).symm = map f.symm | fun_like.ext _ _ $ λ s, set_like.coe_injective $ set.preimage_equiv_eq_image_symm _ _ | lemma | lower_set.symm_map | 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"
] | [
"fun_like.ext",
"set.preimage_equiv_eq_image_symm",
"set_like.coe_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_map {f : α ≃o β} {b : β} : b ∈ map f s ↔ f.symm b ∈ s | by { rw [←f.symm_symm, ←symm_map, f.symm_symm], refl } | lemma | lower_set.mem_map | 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"
] | [
"mem_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
compl_map (f : α ≃o β) (s : upper_set α) :
(map f s).compl = lower_set.map f s.compl | set_like.coe_injective (set.image_compl_eq f.bijective).symm | lemma | upper_set.compl_map | 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.map",
"set.image_compl_eq",
"set_like.coe_injective",
"upper_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
compl_map (f : α ≃o β) (s : lower_set α) :
(map f s).compl = upper_set.map f s.compl | set_like.coe_injective (set.image_compl_eq f.bijective).symm | lemma | lower_set.compl_map | 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",
"set.image_compl_eq",
"set_like.coe_injective",
"upper_set.map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Ici (a : α) : upper_set α | ⟨Ici a, is_upper_set_Ici a⟩ | def | upper_set.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"
] | [
"is_upper_set_Ici",
"upper_set"
] | The smallest upper set containing a given element. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Ioi (a : α) : upper_set α | ⟨Ioi a, is_upper_set_Ioi a⟩ | def | upper_set.Ioi | 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_Ioi",
"upper_set"
] | The smallest upper set containing a given element. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_Ici (a : α) : ↑(Ici a) = set.Ici a | rfl | lemma | upper_set.coe_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"
] | [
"set.Ici"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_Ioi (a : α) : ↑(Ioi a) = set.Ioi a | rfl | lemma | upper_set.coe_Ioi | 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.Ioi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_Ici_iff : b ∈ Ici a ↔ a ≤ b | iff.rfl | lemma | upper_set.mem_Ici_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_Ioi_iff : b ∈ Ioi a ↔ a < b | iff.rfl | lemma | upper_set.mem_Ioi_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_Ici (f : α ≃o β) (a : α) : map f (Ici a) = Ici (f a) | by { ext, simp } | lemma | upper_set.map_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 | |
map_Ioi (f : α ≃o β) (a : α) : map f (Ioi a) = Ioi (f a) | by { ext, simp } | lemma | upper_set.map_Ioi | 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_le_Ioi (a : α) : Ici a ≤ Ioi a | Ioi_subset_Ici_self | lemma | upper_set.Ici_le_Ioi | 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_top [order_top α] : Ioi (⊤ : α) = ⊤ | set_like.coe_injective Ioi_top | lemma | upper_set.Ioi_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 | |
Ici_bot [order_bot α] : Ici (⊥ : α) = ⊥ | set_like.coe_injective Ici_bot | lemma | upper_set.Ici_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 | |
Ici_sup [semilattice_sup α] (a b : α) : Ici (a ⊔ b) = Ici a ⊔ Ici b | ext Ici_inter_Ici.symm | lemma | upper_set.Ici_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"
] | [
"semilattice_sup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Ici_Sup (S : set α) : Ici (Sup S) = ⨆ a ∈ S, Ici a | set_like.ext $ λ c, by simp only [mem_Ici_iff, mem_supr_iff, Sup_le_iff] | lemma | upper_set.Ici_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"
] | [
"Sup_le_iff",
"set_like.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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