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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|---|---|---|---|---|---|---|---|---|---|---|
to_Z_le_iff (i j : ι) : to_Z i0 i ≤ to_Z i0 j ↔ i ≤ j | ⟨le_of_to_Z_le, to_Z_mono⟩ | lemma | to_Z_le_iff | order.succ_pred | src/order/succ_pred/linear_locally_finite.lean | [
"order.locally_finite",
"order.succ_pred.basic",
"order.hom.basic",
"data.countable.basic",
"logic.encodable.basic"
] | [
"to_Z"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Z_iterate_succ [no_max_order ι] (n : ℕ) : to_Z i0 (succ^[n] i0) = n | to_Z_iterate_succ_of_not_is_max n (not_is_max _) | lemma | to_Z_iterate_succ | order.succ_pred | src/order/succ_pred/linear_locally_finite.lean | [
"order.locally_finite",
"order.succ_pred.basic",
"order.hom.basic",
"data.countable.basic",
"logic.encodable.basic"
] | [
"no_max_order",
"not_is_max",
"to_Z",
"to_Z_iterate_succ_of_not_is_max"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Z_iterate_pred [no_min_order ι] (n : ℕ) : to_Z i0 (pred^[n] i0) = -n | to_Z_iterate_pred_of_not_is_min n (not_is_min _) | lemma | to_Z_iterate_pred | order.succ_pred | src/order/succ_pred/linear_locally_finite.lean | [
"order.locally_finite",
"order.succ_pred.basic",
"order.hom.basic",
"data.countable.basic",
"logic.encodable.basic"
] | [
"no_min_order",
"not_is_min",
"to_Z",
"to_Z_iterate_pred_of_not_is_min"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
injective_to_Z : function.injective (to_Z i0) | λ i j hij, le_antisymm (le_of_to_Z_le hij.le) (le_of_to_Z_le hij.symm.le) | lemma | injective_to_Z | order.succ_pred | src/order/succ_pred/linear_locally_finite.lean | [
"order.locally_finite",
"order.succ_pred.basic",
"order.hom.basic",
"data.countable.basic",
"logic.encodable.basic"
] | [
"le_of_to_Z_le",
"to_Z"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
order_iso_range_to_Z_of_linear_succ_pred_arch [hι : nonempty ι] :
ι ≃o set.range (to_Z hι.some) | { to_equiv := equiv.of_injective _ injective_to_Z,
map_rel_iff' := to_Z_le_iff, } | def | order_iso_range_to_Z_of_linear_succ_pred_arch | order.succ_pred | src/order/succ_pred/linear_locally_finite.lean | [
"order.locally_finite",
"order.succ_pred.basic",
"order.hom.basic",
"data.countable.basic",
"logic.encodable.basic"
] | [
"equiv.of_injective",
"injective_to_Z",
"set.range",
"to_Z",
"to_Z_le_iff"
] | `to_Z` defines an `order_iso` between `ι` and its range. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
countable_of_linear_succ_pred_arch : countable ι | begin
casesI is_empty_or_nonempty ι with _ hι,
{ apply_instance, },
{ exact countable.of_equiv _ (order_iso_range_to_Z_of_linear_succ_pred_arch).symm.to_equiv, },
end | instance | countable_of_linear_succ_pred_arch | order.succ_pred | src/order/succ_pred/linear_locally_finite.lean | [
"order.locally_finite",
"order.succ_pred.basic",
"order.hom.basic",
"data.countable.basic",
"logic.encodable.basic"
] | [
"countable",
"countable.of_equiv",
"is_empty_or_nonempty",
"order_iso_range_to_Z_of_linear_succ_pred_arch"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
order_iso_int_of_linear_succ_pred_arch [no_max_order ι] [no_min_order ι] [hι : nonempty ι] :
ι ≃o ℤ | { to_fun := to_Z hι.some,
inv_fun := λ n, if 0 ≤ n then (succ^[n.to_nat] hι.some) else (pred^[(-n).to_nat] hι.some),
left_inv := λ i,
begin
cases le_or_lt hι.some i with hi hi,
{ have h_nonneg : 0 ≤ to_Z hι.some i := to_Z_nonneg hi,
simp_rw if_pos h_nonneg,
exact iterate_succ_to_Z i hi, },
... | def | order_iso_int_of_linear_succ_pred_arch | order.succ_pred | src/order/succ_pred/linear_locally_finite.lean | [
"order.locally_finite",
"order.succ_pred.basic",
"order.hom.basic",
"data.countable.basic",
"logic.encodable.basic"
] | [
"int.to_nat_of_nonneg",
"inv_fun",
"iterate_pred_to_Z",
"iterate_succ_to_Z",
"no_max_order",
"no_min_order",
"to_Z",
"to_Z_iterate_pred",
"to_Z_iterate_succ",
"to_Z_le_iff",
"to_Z_neg",
"to_Z_nonneg",
"to_nat"
] | If the order has neither bot nor top, `to_Z` defines an `order_iso` between `ι` and `ℤ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
order_iso_nat_of_linear_succ_pred_arch [no_max_order ι] [order_bot ι] :
ι ≃o ℕ | { to_fun := λ i, (to_Z ⊥ i).to_nat,
inv_fun := λ n, succ^[n] ⊥,
left_inv := λ i, by { simp_rw if_pos (to_Z_nonneg bot_le), exact iterate_succ_to_Z i bot_le, },
right_inv := λ n,
begin
simp_rw if_pos bot_le,
rw to_Z_iterate_succ,
exact int.to_nat_coe_nat n,
end,
map_rel_iff' := λ i j,
begin
... | def | order_iso_nat_of_linear_succ_pred_arch | order.succ_pred | src/order/succ_pred/linear_locally_finite.lean | [
"order.locally_finite",
"order.succ_pred.basic",
"order.hom.basic",
"data.countable.basic",
"logic.encodable.basic"
] | [
"bot_le",
"equiv.coe_fn_mk",
"int.to_nat_coe_nat",
"int.to_nat_le",
"int.to_nat_of_nonneg",
"inv_fun",
"iterate_succ_to_Z",
"no_max_order",
"order_bot",
"to_Z",
"to_Z_iterate_succ",
"to_Z_le_iff",
"to_Z_nonneg",
"to_nat"
] | If the order has a bot but no top, `to_Z` defines an `order_iso` between `ι` and `ℕ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
order_iso_range_of_linear_succ_pred_arch [order_bot ι] [order_top ι] :
ι ≃o finset.range ((to_Z ⊥ (⊤ : ι)).to_nat + 1) | { to_fun := λ i, ⟨(to_Z ⊥ i).to_nat,
finset.mem_range_succ_iff.mpr (int.to_nat_le_to_nat ((to_Z_le_iff _ _).mpr le_top))⟩,
inv_fun := λ n, succ^[n] ⊥,
left_inv := λ i, iterate_succ_to_Z i bot_le,
right_inv := λ n, begin
ext1,
simp only [subtype.coe_mk],
refine le_antisymm _ _,
{ rw int.to_nat_... | def | order_iso_range_of_linear_succ_pred_arch | order.succ_pred | src/order/succ_pred/linear_locally_finite.lean | [
"order.locally_finite",
"order.succ_pred.basic",
"order.hom.basic",
"data.countable.basic",
"logic.encodable.basic"
] | [
"bot_le",
"equiv.coe_fn_mk",
"finset.range",
"int.to_nat_coe_nat",
"int.to_nat_le",
"int.to_nat_le_to_nat",
"int.to_nat_of_nonneg",
"inv_fun",
"is_max",
"is_top_iff_eq_top",
"is_top_iff_is_max",
"iterate_succ_to_Z",
"le_top",
"order_bot",
"order_top",
"subtype.coe_mk",
"subtype.mk_le... | If the order has both a bot and a top, `to_Z` gives an `order_iso` between `ι` and
`finset.range n` for some `n`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
refl_trans_gen_of_succ_of_le (r : α → α → Prop) {n m : α}
(h : ∀ i ∈ Ico n m, r i (succ i)) (hnm : n ≤ m) : refl_trans_gen r n m | begin
revert h, refine succ.rec _ _ hnm,
{ intros h, exact refl_trans_gen.refl },
{ intros m hnm ih h,
have : refl_trans_gen r n m := ih (λ i hi, h i ⟨hi.1, hi.2.trans_le $ le_succ m⟩),
cases (le_succ m).eq_or_lt with hm hm, { rwa [← hm] },
exact this.tail (h m ⟨hnm, hm⟩) }
end | lemma | refl_trans_gen_of_succ_of_le | order.succ_pred | src/order/succ_pred/relation.lean | [
"order.succ_pred.basic"
] | [
"ih",
"succ.rec"
] | For `n ≤ m`, `(n, m)` is in the reflexive-transitive closure of `~` if `i ~ succ i`
for all `i` between `n` and `m`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
refl_trans_gen_of_succ_of_ge (r : α → α → Prop) {n m : α}
(h : ∀ i ∈ Ico m n, r (succ i) i) (hmn : m ≤ n) : refl_trans_gen r n m | by { rw [← refl_trans_gen_swap], exact refl_trans_gen_of_succ_of_le (swap r) h hmn } | lemma | refl_trans_gen_of_succ_of_ge | order.succ_pred | src/order/succ_pred/relation.lean | [
"order.succ_pred.basic"
] | [
"refl_trans_gen_of_succ_of_le"
] | For `m ≤ n`, `(n, m)` is in the reflexive-transitive closure of `~` if `succ i ~ i`
for all `i` between `n` and `m`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
trans_gen_of_succ_of_lt (r : α → α → Prop) {n m : α}
(h : ∀ i ∈ Ico n m, r i (succ i)) (hnm : n < m) : trans_gen r n m | (refl_trans_gen_iff_eq_or_trans_gen.mp $ refl_trans_gen_of_succ_of_le r h hnm.le).resolve_left
hnm.ne' | lemma | trans_gen_of_succ_of_lt | order.succ_pred | src/order/succ_pred/relation.lean | [
"order.succ_pred.basic"
] | [
"refl_trans_gen_of_succ_of_le"
] | For `n < m`, `(n, m)` is in the transitive closure of a relation `~` if `i ~ succ i`
for all `i` between `n` and `m`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
trans_gen_of_succ_of_gt (r : α → α → Prop) {n m : α}
(h : ∀ i ∈ Ico m n, r (succ i) i) (hmn : m < n) : trans_gen r n m | (refl_trans_gen_iff_eq_or_trans_gen.mp $ refl_trans_gen_of_succ_of_ge r h hmn.le).resolve_left
hmn.ne | lemma | trans_gen_of_succ_of_gt | order.succ_pred | src/order/succ_pred/relation.lean | [
"order.succ_pred.basic"
] | [
"refl_trans_gen_of_succ_of_ge"
] | For `m < n`, `(n, m)` is in the transitive closure of a relation `~` if `succ i ~ i`
for all `i` between `n` and `m`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
refl_trans_gen_of_succ (r : α → α → Prop) {n m : α}
(h1 : ∀ i ∈ Ico n m, r i (succ i)) (h2 : ∀ i ∈ Ico m n, r (succ i) i) : refl_trans_gen r n m | (le_total n m).elim (refl_trans_gen_of_succ_of_le r h1) $ refl_trans_gen_of_succ_of_ge r h2 | lemma | refl_trans_gen_of_succ | order.succ_pred | src/order/succ_pred/relation.lean | [
"order.succ_pred.basic"
] | [
"refl_trans_gen_of_succ_of_ge",
"refl_trans_gen_of_succ_of_le"
] | `(n, m)` is in the reflexive-transitive closure of `~` if `i ~ succ i` and `succ i ~ i`
for all `i` between `n` and `m`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
trans_gen_of_succ_of_ne (r : α → α → Prop) {n m : α}
(h1 : ∀ i ∈ Ico n m, r i (succ i)) (h2 : ∀ i ∈ Ico m n, r (succ i) i)
(hnm : n ≠ m) : trans_gen r n m | (refl_trans_gen_iff_eq_or_trans_gen.mp (refl_trans_gen_of_succ r h1 h2)).resolve_left hnm.symm | lemma | trans_gen_of_succ_of_ne | order.succ_pred | src/order/succ_pred/relation.lean | [
"order.succ_pred.basic"
] | [
"refl_trans_gen_of_succ"
] | For `n ≠ m`,`(n, m)` is in the transitive closure of a relation `~` if `i ~ succ i` and
`succ i ~ i` for all `i` between `n` and `m`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
trans_gen_of_succ_of_reflexive (r : α → α → Prop) {n m : α} (hr : reflexive r)
(h1 : ∀ i ∈ Ico n m, r i (succ i)) (h2 : ∀ i ∈ Ico m n, r (succ i) i) : trans_gen r n m | begin
rcases eq_or_ne m n with rfl|hmn, { exact trans_gen.single (hr m) },
exact trans_gen_of_succ_of_ne r h1 h2 hmn.symm
end | lemma | trans_gen_of_succ_of_reflexive | order.succ_pred | src/order/succ_pred/relation.lean | [
"order.succ_pred.basic"
] | [
"eq_or_ne",
"trans_gen_of_succ_of_ne"
] | `(n, m)` is in the transitive closure of a reflexive relation `~` if `i ~ succ i` and
`succ i ~ i` for all `i` between `n` and `m`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
refl_trans_gen_of_pred_of_ge (r : α → α → Prop) {n m : α}
(h : ∀ i ∈ Ioc m n, r i (pred i)) (hnm : m ≤ n) : refl_trans_gen r n m | @refl_trans_gen_of_succ_of_le αᵒᵈ _ _ _ r n m (λ x hx, h x ⟨hx.2, hx.1⟩) hnm | lemma | refl_trans_gen_of_pred_of_ge | order.succ_pred | src/order/succ_pred/relation.lean | [
"order.succ_pred.basic"
] | [
"refl_trans_gen_of_succ_of_le"
] | For `m ≤ n`, `(n, m)` is in the reflexive-transitive closure of `~` if `i ~ pred i`
for all `i` between `n` and `m`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
refl_trans_gen_of_pred_of_le (r : α → α → Prop) {n m : α}
(h : ∀ i ∈ Ioc n m, r (pred i) i) (hmn : n ≤ m) : refl_trans_gen r n m | @refl_trans_gen_of_succ_of_ge αᵒᵈ _ _ _ r n m (λ x hx, h x ⟨hx.2, hx.1⟩) hmn | lemma | refl_trans_gen_of_pred_of_le | order.succ_pred | src/order/succ_pred/relation.lean | [
"order.succ_pred.basic"
] | [
"refl_trans_gen_of_succ_of_ge"
] | For `n ≤ m`, `(n, m)` is in the reflexive-transitive closure of `~` if `pred i ~ i`
for all `i` between `n` and `m`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
trans_gen_of_pred_of_gt (r : α → α → Prop) {n m : α}
(h : ∀ i ∈ Ioc m n, r i (pred i)) (hnm : m < n) : trans_gen r n m | @trans_gen_of_succ_of_lt αᵒᵈ _ _ _ r _ _ (λ x hx, h x ⟨hx.2, hx.1⟩) hnm | lemma | trans_gen_of_pred_of_gt | order.succ_pred | src/order/succ_pred/relation.lean | [
"order.succ_pred.basic"
] | [
"trans_gen_of_succ_of_lt"
] | For `m < n`, `(n, m)` is in the transitive closure of a relation `~` for `n ≠ m` if `i ~ pred i`
for all `i` between `n` and `m`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
trans_gen_of_pred_of_lt (r : α → α → Prop) {n m : α}
(h : ∀ i ∈ Ioc n m, r (pred i) i) (hmn : n < m) : trans_gen r n m | @trans_gen_of_succ_of_gt αᵒᵈ _ _ _ r _ _ (λ x hx, h x ⟨hx.2, hx.1⟩) hmn | lemma | trans_gen_of_pred_of_lt | order.succ_pred | src/order/succ_pred/relation.lean | [
"order.succ_pred.basic"
] | [
"trans_gen_of_succ_of_gt"
] | For `n < m`, `(n, m)` is in the transitive closure of a relation `~` for `n ≠ m` if `pred i ~ i`
for all `i` between `n` and `m`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
refl_trans_gen_of_pred (r : α → α → Prop) {n m : α}
(h1 : ∀ i ∈ Ioc m n, r i (pred i)) (h2 : ∀ i ∈ Ioc n m, r (pred i) i) : refl_trans_gen r n m | @refl_trans_gen_of_succ αᵒᵈ _ _ _ r n m (λ x hx, h1 x ⟨hx.2, hx.1⟩)
(λ x hx, h2 x ⟨hx.2, hx.1⟩) | lemma | refl_trans_gen_of_pred | order.succ_pred | src/order/succ_pred/relation.lean | [
"order.succ_pred.basic"
] | [
"refl_trans_gen_of_succ"
] | `(n, m)` is in the reflexive-transitive closure of `~` if `i ~ pred i` and `pred i ~ i`
for all `i` between `n` and `m`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
trans_gen_of_pred_of_ne (r : α → α → Prop) {n m : α}
(h1 : ∀ i ∈ Ioc m n, r i (pred i)) (h2 : ∀ i ∈ Ioc n m, r (pred i) i)
(hnm : n ≠ m) : trans_gen r n m | @trans_gen_of_succ_of_ne αᵒᵈ _ _ _ r n m (λ x hx, h1 x ⟨hx.2, hx.1⟩)
(λ x hx, h2 x ⟨hx.2, hx.1⟩) hnm | lemma | trans_gen_of_pred_of_ne | order.succ_pred | src/order/succ_pred/relation.lean | [
"order.succ_pred.basic"
] | [
"trans_gen_of_succ_of_ne"
] | For `n ≠ m`, `(n, m)` is in the transitive closure of a relation `~` if `i ~ pred i` and
`pred i ~ i` for all `i` between `n` and `m`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
trans_gen_of_pred_of_reflexive (r : α → α → Prop) {n m : α} (hr : reflexive r)
(h1 : ∀ i ∈ Ioc m n, r i (pred i)) (h2 : ∀ i ∈ Ioc n m, r (pred i) i) : trans_gen r n m | @trans_gen_of_succ_of_reflexive αᵒᵈ _ _ _ r n m hr (λ x hx, h1 x ⟨hx.2, hx.1⟩)
(λ x hx, h2 x ⟨hx.2, hx.1⟩) | lemma | trans_gen_of_pred_of_reflexive | order.succ_pred | src/order/succ_pred/relation.lean | [
"order.succ_pred.basic"
] | [
"trans_gen_of_succ_of_reflexive"
] | `(n, m)` is in the transitive closure of a reflexive relation `~` if `i ~ pred i` and
`pred i ~ i` for all `i` between `n` and `m`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_upper_set (s : set α) : Prop | ∀ ⦃a b : α⦄, a ≤ b → a ∈ s → b ∈ s | def | is_upper_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"
] | [] | An upper set in an order `α` is a set such that any element greater than one of its members is
also a member. Also called up-set, upward-closed set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_lower_set (s : set α) : Prop | ∀ ⦃a b : α⦄, b ≤ a → a ∈ s → b ∈ s | def | is_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"
] | [] | A lower set in an order `α` is a set such that any element less than one of its members is also
a member. Also called down-set, downward-closed set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_upper_set_empty : is_upper_set (∅ : set α) | λ _ _ _, id | lemma | is_upper_set_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"
] | [
"is_upper_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_lower_set_empty : is_lower_set (∅ : set α) | λ _ _ _, id | lemma | is_lower_set_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"
] | [
"is_lower_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_upper_set_univ : is_upper_set (univ : set α) | λ _ _ _, id | lemma | is_upper_set_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"
] | [
"is_upper_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_lower_set_univ : is_lower_set (univ : set α) | λ _ _ _, id | lemma | is_lower_set_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"
] | [
"is_lower_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_upper_set.compl (hs : is_upper_set s) : is_lower_set sᶜ | λ a b h hb ha, hb $ hs h ha | lemma | is_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"
] | [
"is_lower_set",
"is_upper_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_lower_set.compl (hs : is_lower_set s) : is_upper_set sᶜ | λ a b h hb ha, hb $ hs h ha | lemma | is_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"
] | [
"is_lower_set",
"is_upper_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_upper_set_compl : is_upper_set sᶜ ↔ is_lower_set s | ⟨λ h, by { convert h.compl, rw compl_compl }, is_lower_set.compl⟩ | lemma | is_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"
] | [
"compl_compl",
"is_lower_set",
"is_upper_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_lower_set_compl : is_lower_set sᶜ ↔ is_upper_set s | ⟨λ h, by { convert h.compl, rw compl_compl }, is_upper_set.compl⟩ | lemma | is_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"
] | [
"compl_compl",
"is_lower_set",
"is_upper_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_upper_set.union (hs : is_upper_set s) (ht : is_upper_set t) : is_upper_set (s ∪ t) | λ a b h, or.imp (hs h) (ht h) | lemma | is_upper_set.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"
] | [
"is_upper_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_lower_set.union (hs : is_lower_set s) (ht : is_lower_set t) : is_lower_set (s ∪ t) | λ a b h, or.imp (hs h) (ht h) | lemma | is_lower_set.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"
] | [
"is_lower_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_upper_set.inter (hs : is_upper_set s) (ht : is_upper_set t) : is_upper_set (s ∩ t) | λ a b h, and.imp (hs h) (ht h) | lemma | is_upper_set.inter | 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.inter (hs : is_lower_set s) (ht : is_lower_set t) : is_lower_set (s ∩ t) | λ a b h, and.imp (hs h) (ht h) | lemma | is_lower_set.inter | 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 | |
is_upper_set_Union {f : ι → set α} (hf : ∀ i, is_upper_set (f i)) : is_upper_set (⋃ i, f i) | λ a b h, Exists₂.imp $ forall_range_iff.2 $ λ i, hf i h | lemma | is_upper_set_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"
] | [
"Exists₂.imp",
"is_upper_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_lower_set_Union {f : ι → set α} (hf : ∀ i, is_lower_set (f i)) : is_lower_set (⋃ i, f i) | λ a b h, Exists₂.imp $ forall_range_iff.2 $ λ i, hf i h | lemma | is_lower_set_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"
] | [
"Exists₂.imp",
"is_lower_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_upper_set_Union₂ {f : Π i, κ i → set α} (hf : ∀ i j, is_upper_set (f i j)) :
is_upper_set (⋃ i j, f i j) | is_upper_set_Union $ λ i, is_upper_set_Union $ hf i | lemma | is_upper_set_Union₂ | 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",
"is_upper_set_Union"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_lower_set_Union₂ {f : Π i, κ i → set α} (hf : ∀ i j, is_lower_set (f i j)) :
is_lower_set (⋃ i j, f i j) | is_lower_set_Union $ λ i, is_lower_set_Union $ hf i | lemma | is_lower_set_Union₂ | 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",
"is_lower_set_Union"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_upper_set_sUnion {S : set (set α)} (hf : ∀ s ∈ S, is_upper_set s) : is_upper_set (⋃₀ S) | λ a b h, Exists₂.imp $ λ s hs, hf s hs h | lemma | is_upper_set_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"
] | [
"Exists₂.imp",
"is_upper_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_lower_set_sUnion {S : set (set α)} (hf : ∀ s ∈ S, is_lower_set s) : is_lower_set (⋃₀ S) | λ a b h, Exists₂.imp $ λ s hs, hf s hs h | lemma | is_lower_set_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"
] | [
"Exists₂.imp",
"is_lower_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_upper_set_Inter {f : ι → set α} (hf : ∀ i, is_upper_set (f i)) : is_upper_set (⋂ i, f i) | λ a b h, forall₂_imp $ forall_range_iff.2 $ λ i, hf i h | lemma | is_upper_set_Inter | 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"
] | [
"forall₂_imp",
"is_upper_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_lower_set_Inter {f : ι → set α} (hf : ∀ i, is_lower_set (f i)) : is_lower_set (⋂ i, f i) | λ a b h, forall₂_imp $ forall_range_iff.2 $ λ i, hf i h | lemma | is_lower_set_Inter | 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"
] | [
"forall₂_imp",
"is_lower_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_upper_set_Inter₂ {f : Π i, κ i → set α} (hf : ∀ i j, is_upper_set (f i j)) :
is_upper_set (⋂ i j, f i j) | is_upper_set_Inter $ λ i, is_upper_set_Inter $ hf i | lemma | is_upper_set_Inter₂ | 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",
"is_upper_set_Inter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_lower_set_Inter₂ {f : Π i, κ i → set α} (hf : ∀ i j, is_lower_set (f i j)) :
is_lower_set (⋂ i j, f i j) | is_lower_set_Inter $ λ i, is_lower_set_Inter $ hf i | lemma | is_lower_set_Inter₂ | 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",
"is_lower_set_Inter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_upper_set_sInter {S : set (set α)} (hf : ∀ s ∈ S, is_upper_set s) : is_upper_set (⋂₀ S) | λ a b h, forall₂_imp $ λ s hs, hf s hs h | lemma | is_upper_set_sInter | 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"
] | [
"forall₂_imp",
"is_upper_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_lower_set_sInter {S : set (set α)} (hf : ∀ s ∈ S, is_lower_set s) : is_lower_set (⋂₀ S) | λ a b h, forall₂_imp $ λ s hs, hf s hs h | lemma | is_lower_set_sInter | 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"
] | [
"forall₂_imp",
"is_lower_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_lower_set_preimage_of_dual_iff : is_lower_set (of_dual ⁻¹' s) ↔ is_upper_set s | iff.rfl | lemma | is_lower_set_preimage_of_dual_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"
] | [
"is_lower_set",
"is_upper_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_upper_set_preimage_of_dual_iff : is_upper_set (of_dual ⁻¹' s) ↔ is_lower_set s | iff.rfl | lemma | is_upper_set_preimage_of_dual_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"
] | [
"is_lower_set",
"is_upper_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_lower_set_preimage_to_dual_iff {s : set αᵒᵈ} :
is_lower_set (to_dual ⁻¹' s) ↔ is_upper_set s | iff.rfl | lemma | is_lower_set_preimage_to_dual_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"
] | [
"is_lower_set",
"is_upper_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_upper_set_preimage_to_dual_iff {s : set αᵒᵈ} :
is_upper_set (to_dual ⁻¹' s) ↔ is_lower_set s | iff.rfl | lemma | is_upper_set_preimage_to_dual_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"
] | [
"is_lower_set",
"is_upper_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_upper_set_Ici : is_upper_set (Ici a) | λ _ _, ge_trans | lemma | is_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"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_lower_set_Iic : is_lower_set (Iic a) | λ _ _, le_trans | lemma | is_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"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_upper_set_Ioi : is_upper_set (Ioi a) | λ _ _, flip lt_of_lt_of_le | lemma | is_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"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_lower_set_Iio : is_lower_set (Iio a) | λ _ _, lt_of_le_of_lt | lemma | is_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"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_upper_set_iff_Ici_subset : is_upper_set s ↔ ∀ ⦃a⦄, a ∈ s → Ici a ⊆ s | by simp [is_upper_set, subset_def, @forall_swap (_ ∈ s)] | lemma | is_upper_set_iff_Ici_subset | 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"
] | [
"forall_swap",
"is_upper_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_lower_set_iff_Iic_subset : is_lower_set s ↔ ∀ ⦃a⦄, a ∈ s → Iic a ⊆ s | by simp [is_lower_set, subset_def, @forall_swap (_ ∈ s)] | lemma | is_lower_set_iff_Iic_subset | 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"
] | [
"forall_swap",
"is_lower_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_upper_set.ord_connected (h : is_upper_set s) : s.ord_connected | ⟨λ a ha b _, Icc_subset_Ici_self.trans $ h.Ici_subset ha⟩ | lemma | is_upper_set.ord_connected | 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.ord_connected (h : is_lower_set s) : s.ord_connected | ⟨λ a _ b hb, Icc_subset_Iic_self.trans $ h.Iic_subset hb⟩ | lemma | is_lower_set.ord_connected | 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 | |
is_upper_set.preimage (hs : is_upper_set s) {f : β → α} (hf : monotone f) :
is_upper_set (f ⁻¹' s : set β) | λ x y hxy, hs $ hf hxy | lemma | is_upper_set.preimage | 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",
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_lower_set.preimage (hs : is_lower_set s) {f : β → α} (hf : monotone f) :
is_lower_set (f ⁻¹' s : set β) | λ x y hxy, hs $ hf hxy | lemma | is_lower_set.preimage | 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",
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_upper_set.image (hs : is_upper_set s) (f : α ≃o β) : is_upper_set (f '' s : set β) | by { change is_upper_set ((f : α ≃ β) '' s), rw set.image_equiv_eq_preimage_symm,
exact hs.preimage f.symm.monotone } | lemma | is_upper_set.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"
] | [
"is_upper_set",
"set.image_equiv_eq_preimage_symm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_lower_set.image (hs : is_lower_set s) (f : α ≃o β) : is_lower_set (f '' s : set β) | by { change is_lower_set ((f : α ≃ β) '' s), rw set.image_equiv_eq_preimage_symm,
exact hs.preimage f.symm.monotone } | lemma | is_lower_set.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"
] | [
"is_lower_set",
"set.image_equiv_eq_preimage_symm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set.monotone_mem : monotone (∈ s) ↔ is_upper_set s | iff.rfl | lemma | set.monotone_mem | 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",
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set.antitone_mem : antitone (∈ s) ↔ is_lower_set s | forall_swap | lemma | set.antitone_mem | 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",
"forall_swap",
"is_lower_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_upper_set_set_of : is_upper_set {a | p a} ↔ monotone p | iff.rfl | lemma | is_upper_set_set_of | 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",
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_lower_set_set_of : is_lower_set {a | p a} ↔ antitone p | forall_swap | lemma | is_lower_set_set_of | 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",
"forall_swap",
"is_lower_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_lower_set.top_mem (hs : is_lower_set s) : ⊤ ∈ s ↔ s = univ | ⟨λ h, eq_univ_of_forall $ λ a, hs le_top h, λ h, h.symm ▸ mem_univ _⟩ | lemma | is_lower_set.top_mem | 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",
"le_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_upper_set.top_mem (hs : is_upper_set s) : ⊤ ∈ s ↔ s.nonempty | ⟨λ h, ⟨_, h⟩, λ ⟨a, ha⟩, hs le_top ha⟩ | lemma | is_upper_set.top_mem | 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",
"le_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_upper_set.not_top_mem (hs : is_upper_set s) : ⊤ ∉ s ↔ s = ∅ | hs.top_mem.not.trans not_nonempty_iff_eq_empty | lemma | is_upper_set.not_top_mem | 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_upper_set.bot_mem (hs : is_upper_set s) : ⊥ ∈ s ↔ s = univ | ⟨λ h, eq_univ_of_forall $ λ a, hs bot_le h, λ h, h.symm ▸ mem_univ _⟩ | lemma | is_upper_set.bot_mem | 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"
] | [
"bot_le",
"is_upper_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_lower_set.bot_mem (hs : is_lower_set s) : ⊥ ∈ s ↔ s.nonempty | ⟨λ h, ⟨_, h⟩, λ ⟨a, ha⟩, hs bot_le ha⟩ | lemma | is_lower_set.bot_mem | 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"
] | [
"bot_le",
"is_lower_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_lower_set.not_bot_mem (hs : is_lower_set s) : ⊥ ∉ s ↔ s = ∅ | hs.bot_mem.not.trans not_nonempty_iff_eq_empty | lemma | is_lower_set.not_bot_mem | 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 | |
is_upper_set.not_bdd_above (hs : is_upper_set s) : s.nonempty → ¬ bdd_above s | begin
rintro ⟨a, ha⟩ ⟨b, hb⟩,
obtain ⟨c, hc⟩ := exists_gt b,
exact hc.not_le (hb $ hs ((hb ha).trans hc.le) ha),
end | lemma | is_upper_set.not_bdd_above | 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",
"is_upper_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
not_bdd_above_Ici : ¬ bdd_above (Ici a) | (is_upper_set_Ici _).not_bdd_above nonempty_Ici | lemma | not_bdd_above_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"
] | [
"bdd_above",
"is_upper_set_Ici"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
not_bdd_above_Ioi : ¬ bdd_above (Ioi a) | (is_upper_set_Ioi _).not_bdd_above nonempty_Ioi | lemma | not_bdd_above_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"
] | [
"bdd_above",
"is_upper_set_Ioi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_lower_set.not_bdd_below (hs : is_lower_set s) : s.nonempty → ¬ bdd_below s | begin
rintro ⟨a, ha⟩ ⟨b, hb⟩,
obtain ⟨c, hc⟩ := exists_lt b,
exact hc.not_le (hb $ hs (hc.le.trans $ hb ha) ha),
end | lemma | is_lower_set.not_bdd_below | 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",
"is_lower_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
not_bdd_below_Iic : ¬ bdd_below (Iic a) | (is_lower_set_Iic _).not_bdd_below nonempty_Iic | lemma | not_bdd_below_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"
] | [
"bdd_below",
"is_lower_set_Iic"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
not_bdd_below_Iio : ¬ bdd_below (Iio a) | (is_lower_set_Iio _).not_bdd_below nonempty_Iio | lemma | not_bdd_below_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"
] | [
"bdd_below",
"is_lower_set_Iio"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_upper_set_iff_forall_lt : is_upper_set s ↔ ∀ ⦃a b : α⦄, a < b → a ∈ s → b ∈ s | forall_congr $ λ a, by simp [le_iff_eq_or_lt, or_imp_distrib, forall_and_distrib] | lemma | is_upper_set_iff_forall_lt | 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"
] | [
"forall_and_distrib",
"is_upper_set",
"le_iff_eq_or_lt",
"or_imp_distrib"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_lower_set_iff_forall_lt : is_lower_set s ↔ ∀ ⦃a b : α⦄, b < a → a ∈ s → b ∈ s | forall_congr $ λ a, by simp [le_iff_eq_or_lt, or_imp_distrib, forall_and_distrib] | lemma | is_lower_set_iff_forall_lt | 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"
] | [
"forall_and_distrib",
"is_lower_set",
"le_iff_eq_or_lt",
"or_imp_distrib"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_upper_set_iff_Ioi_subset : is_upper_set s ↔ ∀ ⦃a⦄, a ∈ s → Ioi a ⊆ s | by simp [is_upper_set_iff_forall_lt, subset_def, @forall_swap (_ ∈ s)] | lemma | is_upper_set_iff_Ioi_subset | 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"
] | [
"forall_swap",
"is_upper_set",
"is_upper_set_iff_forall_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_lower_set_iff_Iio_subset : is_lower_set s ↔ ∀ ⦃a⦄, a ∈ s → Iio a ⊆ s | by simp [is_lower_set_iff_forall_lt, subset_def, @forall_swap (_ ∈ s)] | lemma | is_lower_set_iff_Iio_subset | 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"
] | [
"forall_swap",
"is_lower_set",
"is_lower_set_iff_forall_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_upper_set.total (hs : is_upper_set s) (ht : is_upper_set t) : s ⊆ t ∨ t ⊆ s | begin
by_contra' h,
simp_rw set.not_subset at h,
obtain ⟨⟨a, has, hat⟩, b, hbt, hbs⟩ := h,
obtain hab | hba := le_total a b,
{ exact hbs (hs hab has) },
{ exact hat (ht hba hbt) }
end | lemma | is_upper_set.total | 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",
"set.not_subset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_lower_set.total (hs : is_lower_set s) (ht : is_lower_set t) : s ⊆ t ∨ t ⊆ s | hs.to_dual.total ht.to_dual | lemma | is_lower_set.total | 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 | |
upper_set (α : Type*) [has_le α] | (carrier : set α)
(upper' : is_upper_set carrier) | structure | upper_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"
] | [
"is_upper_set"
] | The type of upper sets of an order. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lower_set (α : Type*) [has_le α] | (carrier : set α)
(lower' : is_lower_set carrier) | structure | 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"
] | [
"is_lower_set"
] | The type of lower sets of an order. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ext {s t : upper_set α} : (s : set α) = t → s = t | set_like.ext' | lemma | upper_set.ext | 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'",
"upper_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
carrier_eq_coe (s : upper_set α) : s.carrier = s | rfl | lemma | upper_set.carrier_eq_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"
] | [
"upper_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
upper (s : upper_set α) : is_upper_set (s : set α) | s.upper' | lemma | upper_set.upper | 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_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_mk (carrier : set α) (upper') {a : α} : a ∈ mk carrier upper' ↔ a ∈ carrier | iff.rfl | lemma | upper_set.mem_mk | 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 | |
ext {s t : lower_set α} : (s : set α) = t → s = t | set_like.ext' | lemma | lower_set.ext | 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_like.ext'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
carrier_eq_coe (s : lower_set α) : s.carrier = s | rfl | lemma | lower_set.carrier_eq_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"
] | [
"lower_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lower (s : lower_set α) : is_lower_set (s : set α) | s.lower' | lemma | lower_set.lower | 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_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_mk (carrier : set α) (lower') {a : α} : a ∈ mk carrier lower' ↔ a ∈ carrier | iff.rfl | lemma | lower_set.mem_mk | 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_subset_coe : (s : set α) ⊆ t ↔ t ≤ s | iff.rfl | lemma | upper_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 : ((⊤ : upper_set α) : set α) = ∅ | rfl | lemma | upper_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"
] | [
"upper_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_bot : ((⊥ : upper_set α) : set α) = univ | rfl | lemma | upper_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"
] | [
"upper_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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