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
value |
|---|---|---|---|---|---|---|---|---|---|---|
zero_at_filter_add_submonoid [topological_space β]
[add_zero_class β] [has_continuous_add β] (l : filter α) : add_submonoid (α → β) | { carrier := zero_at_filter l,
add_mem' := λ a b ha hb, ha.add hb,
zero_mem' := zero_zero_at_filter l, } | def | filter.zero_at_filter_add_submonoid | order.filter | src/order/filter/zero_and_bounded_at_filter.lean | [
"algebra.module.submodule.basic",
"topology.algebra.monoid",
"analysis.asymptotics.asymptotics"
] | [
"add_submonoid",
"add_zero_class",
"filter",
"has_continuous_add",
"topological_space"
] | `zero_at_filter_add_submonoid l` is the additive submonoid of `f : α → β`
which tend to zero along `l`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bounded_at_filter [has_norm β] (l : filter α) (f : α → β) : Prop | asymptotics.is_O l f (1 : α → ℝ) | def | filter.bounded_at_filter | order.filter | src/order/filter/zero_and_bounded_at_filter.lean | [
"algebra.module.submodule.basic",
"topology.algebra.monoid",
"analysis.asymptotics.asymptotics"
] | [
"asymptotics.is_O",
"filter",
"has_norm"
] | If `l` is a filter on `α`, then a function `f: α → β` is `bounded_at_filter l`
if `f =O[l] 1`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
zero_at_filter.bounded_at_filter [normed_add_comm_group β] {l : filter α} {f : α → β}
(hf : zero_at_filter l f) : bounded_at_filter l f | begin
rw [zero_at_filter, ← asymptotics.is_o_const_iff (one_ne_zero' ℝ)] at hf,
exact hf.is_O,
end | lemma | filter.zero_at_filter.bounded_at_filter | order.filter | src/order/filter/zero_and_bounded_at_filter.lean | [
"algebra.module.submodule.basic",
"topology.algebra.monoid",
"analysis.asymptotics.asymptotics"
] | [
"asymptotics.is_o_const_iff",
"filter",
"normed_add_comm_group",
"one_ne_zero'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
const_bounded_at_filter [normed_field β] (l : filter α) (c : β) :
bounded_at_filter l (function.const α c : α → β) | asymptotics.is_O_const_const c one_ne_zero l | lemma | filter.const_bounded_at_filter | order.filter | src/order/filter/zero_and_bounded_at_filter.lean | [
"algebra.module.submodule.basic",
"topology.algebra.monoid",
"analysis.asymptotics.asymptotics"
] | [
"asymptotics.is_O_const_const",
"filter",
"normed_field",
"one_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bounded_at_filter.add [normed_add_comm_group β] {l : filter α} {f g : α → β}
(hf : bounded_at_filter l f) (hg : bounded_at_filter l g) :
bounded_at_filter l (f + g) | by simpa using hf.add hg | lemma | filter.bounded_at_filter.add | order.filter | src/order/filter/zero_and_bounded_at_filter.lean | [
"algebra.module.submodule.basic",
"topology.algebra.monoid",
"analysis.asymptotics.asymptotics"
] | [
"filter",
"normed_add_comm_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bounded_at_filter.neg [normed_add_comm_group β] {l : filter α} {f : α → β}
(hf : bounded_at_filter l f) :
bounded_at_filter l (-f) | hf.neg_left | lemma | filter.bounded_at_filter.neg | order.filter | src/order/filter/zero_and_bounded_at_filter.lean | [
"algebra.module.submodule.basic",
"topology.algebra.monoid",
"analysis.asymptotics.asymptotics"
] | [
"filter",
"normed_add_comm_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bounded_at_filter.smul {𝕜 : Type*} [normed_field 𝕜] [normed_add_comm_group β]
[normed_space 𝕜 β] {l : filter α} {f : α → β} (c : 𝕜) (hf : bounded_at_filter l f) :
bounded_at_filter l (c • f) | hf.const_smul_left c | lemma | filter.bounded_at_filter.smul | order.filter | src/order/filter/zero_and_bounded_at_filter.lean | [
"algebra.module.submodule.basic",
"topology.algebra.monoid",
"analysis.asymptotics.asymptotics"
] | [
"filter",
"normed_add_comm_group",
"normed_field",
"normed_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bounded_at_filter.mul [normed_field β] {l : filter α} {f g : α → β}
(hf : bounded_at_filter l f) (hg : bounded_at_filter l g) :
bounded_at_filter l (f * g) | begin
refine (hf.mul hg).trans _,
convert asymptotics.is_O_refl _ l,
ext x,
simp,
end | lemma | filter.bounded_at_filter.mul | order.filter | src/order/filter/zero_and_bounded_at_filter.lean | [
"algebra.module.submodule.basic",
"topology.algebra.monoid",
"analysis.asymptotics.asymptotics"
] | [
"asymptotics.is_O_refl",
"filter",
"normed_field"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bounded_filter_submodule [normed_field β] (l : filter α) : submodule β (α → β) | { carrier := bounded_at_filter l,
zero_mem' := const_bounded_at_filter l 0,
add_mem' := λ f g hf hg, hf.add hg,
smul_mem' := λ c f hf, hf.smul c } | def | filter.bounded_filter_submodule | order.filter | src/order/filter/zero_and_bounded_at_filter.lean | [
"algebra.module.submodule.basic",
"topology.algebra.monoid",
"analysis.asymptotics.asymptotics"
] | [
"filter",
"normed_field",
"submodule"
] | The submodule of functions that are bounded along a filter `l`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bounded_filter_subalgebra [normed_field β] (l : filter α) :
subalgebra β (α → β) | begin
refine submodule.to_subalgebra (bounded_filter_submodule l) _ (λ f g hf hg, _),
{ exact const_bounded_at_filter l (1:β) },
{ simpa only [pi.one_apply, mul_one, norm_mul] using hf.mul hg, },
end | def | filter.bounded_filter_subalgebra | order.filter | src/order/filter/zero_and_bounded_at_filter.lean | [
"algebra.module.submodule.basic",
"topology.algebra.monoid",
"analysis.asymptotics.asymptotics"
] | [
"filter",
"mul_one",
"norm_mul",
"normed_field",
"pi.one_apply",
"subalgebra",
"submodule.to_subalgebra"
] | The subalgebra of functions that are bounded along a filter `l`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_himp (α : Type*) | (himp : α → α → α) | class | has_himp | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [] | Syntax typeclass for Heyting implication `⇨`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_hnot (α : Type*) | (hnot : α → α) | class | has_hnot | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [] | Syntax typeclass for Heyting negation `￢`.
The difference between `has_compl` and `has_hnot` is that the former belongs to Heyting algebras,
while the latter belongs to co-Heyting algebras. They are both pseudo-complements, but `compl`
underestimates while `hnot` overestimates. In boolean algebras, they are equal. See... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fst_himp [has_himp α] [has_himp β] (a b : α × β) : (a ⇨ b).1 = a.1 ⇨ b.1 | rfl | lemma | fst_himp | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"has_himp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
snd_himp [has_himp α] [has_himp β] (a b : α × β) : (a ⇨ b).2 = a.2 ⇨ b.2 | rfl | lemma | snd_himp | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"has_himp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fst_hnot [has_hnot α] [has_hnot β] (a : α × β) : (￢a).1 = ￢a.1 | rfl | lemma | fst_hnot | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"has_hnot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
snd_hnot [has_hnot α] [has_hnot β] (a : α × β) : (￢a).2 = ￢a.2 | rfl | lemma | snd_hnot | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"has_hnot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fst_sdiff [has_sdiff α] [has_sdiff β] (a b : α × β) : (a \ b).1 = a.1 \ b.1 | rfl | lemma | fst_sdiff | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
snd_sdiff [has_sdiff α] [has_sdiff β] (a b : α × β) : (a \ b).2 = a.2 \ b.2 | rfl | lemma | snd_sdiff | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fst_compl [has_compl α] [has_compl β] (a : α × β) : aᶜ.1 = a.1ᶜ | rfl | lemma | fst_compl | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"has_compl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
snd_compl [has_compl α] [has_compl β] (a : α × β) : aᶜ.2 = a.2ᶜ | rfl | lemma | snd_compl | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"has_compl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
himp_def [Π i, has_himp (π i)] (a b : Π i, π i) : (a ⇨ b) = λ i, a i ⇨ b i | rfl | lemma | pi.himp_def | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"has_himp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hnot_def [Π i, has_hnot (π i)] (a : Π i, π i) : ￢a = λ i, ￢a i | rfl | lemma | pi.hnot_def | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"has_hnot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
himp_apply [Π i, has_himp (π i)] (a b : Π i, π i) (i : ι) : (a ⇨ b) i = a i ⇨ b i | rfl | lemma | pi.himp_apply | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"has_himp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hnot_apply [Π i, has_hnot (π i)] (a : Π i, π i) (i : ι) : (￢a) i = ￢a i | rfl | lemma | pi.hnot_apply | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"has_hnot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
generalized_heyting_algebra (α : Type*) extends lattice α, has_top α, has_himp α | (le_top : ∀ a : α, a ≤ ⊤)
(le_himp_iff (a b c : α) : a ≤ b ⇨ c ↔ a ⊓ b ≤ c) | class | generalized_heyting_algebra | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"has_himp",
"has_top",
"lattice",
"le_himp_iff",
"le_top"
] | A generalized Heyting algebra is a lattice with an additional binary operation `⇨` called
Heyting implication such that `a ⇨` is right adjoint to `a ⊓`.
This generalizes `heyting_algebra` by not requiring a bottom element. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
generalized_coheyting_algebra (α : Type*) extends lattice α, has_bot α, has_sdiff α | (bot_le : ∀ a : α, ⊥ ≤ a)
(sdiff_le_iff (a b c : α) : a \ b ≤ c ↔ a ≤ b ⊔ c) | class | generalized_coheyting_algebra | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"bot_le",
"has_bot",
"lattice",
"sdiff_le_iff"
] | A generalized co-Heyting algebra is a lattice with an additional binary difference operation `\`
such that `\ a` is right adjoint to `⊔ a`.
This generalizes `coheyting_algebra` by not requiring a top element. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
heyting_algebra (α : Type*) extends generalized_heyting_algebra α, has_bot α, has_compl α | (bot_le : ∀ a : α, ⊥ ≤ a)
(himp_bot (a : α) : a ⇨ ⊥ = aᶜ) | class | heyting_algebra | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"bot_le",
"generalized_heyting_algebra",
"has_bot",
"has_compl",
"himp_bot"
] | A Heyting algebra is a bounded lattice with an additional binary operation `⇨` called Heyting
implication such that `a ⇨` is right adjoint to `a ⊓`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coheyting_algebra (α : Type*)
extends generalized_coheyting_algebra α, has_top α, has_hnot α | (le_top : ∀ a : α, a ≤ ⊤)
(top_sdiff (a : α) : ⊤ \ a = ￢a) | class | coheyting_algebra | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"generalized_coheyting_algebra",
"has_hnot",
"has_top",
"le_top",
"top_sdiff"
] | A co-Heyting algebra is a bounded lattice with an additional binary difference operation `\`
such that `\ a` is right adjoint to `⊔ a`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
biheyting_algebra (α : Type*) extends heyting_algebra α, has_sdiff α, has_hnot α | (sdiff_le_iff (a b c : α) : a \ b ≤ c ↔ a ≤ b ⊔ c)
(top_sdiff (a : α) : ⊤ \ a = ￢a) | class | biheyting_algebra | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"has_hnot",
"heyting_algebra",
"sdiff_le_iff",
"top_sdiff"
] | A bi-Heyting algebra is a Heyting algebra that is also a co-Heyting algebra. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
generalized_heyting_algebra.to_order_top [generalized_heyting_algebra α] : order_top α | { ..‹generalized_heyting_algebra α› } | instance | generalized_heyting_algebra.to_order_top | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"generalized_heyting_algebra",
"order_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
generalized_coheyting_algebra.to_order_bot [generalized_coheyting_algebra α] :
order_bot α | { ..‹generalized_coheyting_algebra α› } | instance | generalized_coheyting_algebra.to_order_bot | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"generalized_coheyting_algebra",
"order_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
heyting_algebra.to_bounded_order [heyting_algebra α] : bounded_order α | { ..‹heyting_algebra α› } | instance | heyting_algebra.to_bounded_order | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"bounded_order",
"heyting_algebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coheyting_algebra.to_bounded_order [coheyting_algebra α] : bounded_order α | { ..‹coheyting_algebra α› } | instance | coheyting_algebra.to_bounded_order | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"bounded_order",
"coheyting_algebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
biheyting_algebra.to_coheyting_algebra [biheyting_algebra α] : coheyting_algebra α | { ..‹biheyting_algebra α› } | instance | biheyting_algebra.to_coheyting_algebra | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"biheyting_algebra",
"coheyting_algebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
heyting_algebra.of_himp [distrib_lattice α] [bounded_order α] (himp : α → α → α)
(le_himp_iff : ∀ a b c, a ≤ himp b c ↔ a ⊓ b ≤ c) : heyting_algebra α | { himp := himp,
compl := λ a, himp a ⊥,
le_himp_iff := le_himp_iff,
himp_bot := λ a, rfl,
..‹distrib_lattice α›, ..‹bounded_order α› } | def | heyting_algebra.of_himp | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"bounded_order",
"distrib_lattice",
"heyting_algebra",
"himp_bot",
"le_himp_iff"
] | Construct a Heyting algebra from the lattice structure and Heyting implication alone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
heyting_algebra.of_compl [distrib_lattice α] [bounded_order α] (compl : α → α)
(le_himp_iff : ∀ a b c, a ≤ compl b ⊔ c ↔ a ⊓ b ≤ c) : heyting_algebra α | { himp := λ a, (⊔) (compl a),
compl := compl,
le_himp_iff := le_himp_iff,
himp_bot := λ a, sup_bot_eq,
..‹distrib_lattice α›, ..‹bounded_order α› } | def | heyting_algebra.of_compl | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"bounded_order",
"distrib_lattice",
"heyting_algebra",
"himp_bot",
"le_himp_iff",
"sup_bot_eq"
] | Construct a Heyting algebra from the lattice structure and complement operator alone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coheyting_algebra.of_sdiff [distrib_lattice α] [bounded_order α] (sdiff : α → α → α)
(sdiff_le_iff : ∀ a b c, sdiff a b ≤ c ↔ a ≤ b ⊔ c) : coheyting_algebra α | { sdiff := sdiff,
hnot := λ a, sdiff ⊤ a,
sdiff_le_iff := sdiff_le_iff,
top_sdiff := λ a, rfl,
..‹distrib_lattice α›, ..‹bounded_order α› } | def | coheyting_algebra.of_sdiff | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"bounded_order",
"coheyting_algebra",
"distrib_lattice",
"sdiff_le_iff",
"top_sdiff"
] | Construct a co-Heyting algebra from the lattice structure and the difference alone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coheyting_algebra.of_hnot [distrib_lattice α] [bounded_order α] (hnot : α → α)
(sdiff_le_iff : ∀ a b c, a ⊓ hnot b ≤ c ↔ a ≤ b ⊔ c) : coheyting_algebra α | { sdiff := λ a b, (a ⊓ hnot b),
hnot := hnot,
sdiff_le_iff := sdiff_le_iff,
top_sdiff := λ a, top_inf_eq,
..‹distrib_lattice α›, ..‹bounded_order α› } | def | coheyting_algebra.of_hnot | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"bounded_order",
"coheyting_algebra",
"distrib_lattice",
"sdiff_le_iff",
"top_inf_eq",
"top_sdiff"
] | Construct a co-Heyting algebra from the difference and Heyting negation alone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
le_himp_iff : a ≤ b ⇨ c ↔ a ⊓ b ≤ c | generalized_heyting_algebra.le_himp_iff _ _ _ | lemma | le_himp_iff | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_himp_iff' : a ≤ b ⇨ c ↔ b ⊓ a ≤ c | by rw [le_himp_iff, inf_comm] | lemma | le_himp_iff' | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"inf_comm",
"le_himp_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_himp_comm : a ≤ b ⇨ c ↔ b ≤ a ⇨ c | by rw [le_himp_iff, le_himp_iff'] | lemma | le_himp_comm | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"le_himp_iff",
"le_himp_iff'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_himp : a ≤ b ⇨ a | le_himp_iff.2 inf_le_left | lemma | le_himp | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"inf_le_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_himp_iff_left : a ≤ a ⇨ b ↔ a ≤ b | by rw [le_himp_iff, inf_idem] | lemma | le_himp_iff_left | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"inf_idem",
"le_himp_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
himp_self : a ⇨ a = ⊤ | top_le_iff.1 $ le_himp_iff.2 inf_le_right | lemma | himp_self | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"inf_le_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
himp_inf_le : (a ⇨ b) ⊓ a ≤ b | le_himp_iff.1 le_rfl | lemma | himp_inf_le | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"le_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inf_himp_le : a ⊓ (a ⇨ b) ≤ b | by rw [inf_comm, ←le_himp_iff] | lemma | inf_himp_le | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"inf_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inf_himp (a b : α) : a ⊓ (a ⇨ b) = a ⊓ b | le_antisymm (le_inf inf_le_left $ by rw [inf_comm, ←le_himp_iff]) $ inf_le_inf_left _ le_himp | lemma | inf_himp | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"inf_comm",
"inf_le_inf_left",
"inf_le_left",
"le_himp",
"le_inf"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
himp_inf_self (a b : α) : (a ⇨ b) ⊓ a = b ⊓ a | by rw [inf_comm, inf_himp, inf_comm] | lemma | himp_inf_self | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"inf_comm",
"inf_himp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
himp_eq_top_iff : a ⇨ b = ⊤ ↔ a ≤ b | by rw [←top_le_iff, le_himp_iff, top_inf_eq] | lemma | himp_eq_top_iff | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"le_himp_iff",
"top_inf_eq"
] | The **deduction theorem** in the Heyting algebra model of intuitionistic logic:
an implication holds iff the conclusion follows from the hypothesis. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
himp_top : a ⇨ ⊤ = ⊤ | himp_eq_top_iff.2 le_top | lemma | himp_top | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"le_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
top_himp : ⊤ ⇨ a = a | eq_of_forall_le_iff $ λ b, by rw [le_himp_iff, inf_top_eq] | lemma | top_himp | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"eq_of_forall_le_iff",
"inf_top_eq",
"le_himp_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
himp_himp (a b c : α) : a ⇨ b ⇨ c = a ⊓ b ⇨ c | eq_of_forall_le_iff $ λ d, by simp_rw [le_himp_iff, inf_assoc] | lemma | himp_himp | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"eq_of_forall_le_iff",
"inf_assoc",
"le_himp_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
himp_le_himp_himp_himp : b ⇨ c ≤ (a ⇨ b) ⇨ a ⇨ c | begin
rw [le_himp_iff, le_himp_iff, inf_assoc, himp_inf_self, ←inf_assoc, himp_inf_self, inf_assoc],
exact inf_le_left,
end | lemma | himp_le_himp_himp_himp | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"himp_inf_self",
"inf_assoc",
"inf_le_left",
"le_himp_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
himp_left_comm (a b c : α) : a ⇨ b ⇨ c = b ⇨ a ⇨ c | by simp_rw [himp_himp, inf_comm] | lemma | himp_left_comm | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"himp_himp",
"inf_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
himp_idem : b ⇨ b ⇨ a = b ⇨ a | by rw [himp_himp, inf_idem] | lemma | himp_idem | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"himp_himp",
"inf_idem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
himp_inf_distrib (a b c : α) : a ⇨ b ⊓ c = (a ⇨ b) ⊓ (a ⇨ c) | eq_of_forall_le_iff $ λ d, by simp_rw [le_himp_iff, le_inf_iff, le_himp_iff] | lemma | himp_inf_distrib | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"eq_of_forall_le_iff",
"le_himp_iff",
"le_inf_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sup_himp_distrib (a b c : α) : a ⊔ b ⇨ c = (a ⇨ c) ⊓ (b ⇨ c) | eq_of_forall_le_iff $ λ d, by { rw [le_inf_iff, le_himp_comm, sup_le_iff], simp_rw le_himp_comm } | lemma | sup_himp_distrib | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"eq_of_forall_le_iff",
"le_himp_comm",
"le_inf_iff",
"sup_le_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
himp_le_himp_left (h : a ≤ b) : c ⇨ a ≤ c ⇨ b | le_himp_iff.2 $ himp_inf_le.trans h | lemma | himp_le_himp_left | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
himp_le_himp_right (h : a ≤ b) : b ⇨ c ≤ a ⇨ c | le_himp_iff.2 $ (inf_le_inf_left _ h).trans himp_inf_le | lemma | himp_le_himp_right | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"himp_inf_le",
"inf_le_inf_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
himp_le_himp (hab : a ≤ b) (hcd : c ≤ d) : b ⇨ c ≤ a ⇨ d | (himp_le_himp_right hab).trans $ himp_le_himp_left hcd | lemma | himp_le_himp | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"himp_le_himp_left",
"himp_le_himp_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sup_himp_self_left (a b : α) : (a ⊔ b) ⇨ a = b ⇨ a | by rw [sup_himp_distrib, himp_self, top_inf_eq] | lemma | sup_himp_self_left | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"himp_self",
"sup_himp_distrib",
"top_inf_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sup_himp_self_right (a b : α) : (a ⊔ b) ⇨ b = a ⇨ b | by rw [sup_himp_distrib, himp_self, inf_top_eq] | lemma | sup_himp_self_right | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"himp_self",
"inf_top_eq",
"sup_himp_distrib"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
codisjoint.himp_eq_right (h : codisjoint a b) : b ⇨ a = a | by { conv_rhs { rw ←@top_himp _ _ a }, rw [←h.eq_top, sup_himp_self_left] } | lemma | codisjoint.himp_eq_right | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"codisjoint",
"sup_himp_self_left",
"top_himp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
codisjoint.himp_eq_left (h : codisjoint a b) : a ⇨ b = b | h.symm.himp_eq_right | lemma | codisjoint.himp_eq_left | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"codisjoint"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
codisjoint.himp_inf_cancel_right (h : codisjoint a b) : a ⇨ (a ⊓ b) = b | by rw [himp_inf_distrib, himp_self, top_inf_eq, h.himp_eq_left] | lemma | codisjoint.himp_inf_cancel_right | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"codisjoint",
"himp_inf_distrib",
"himp_self",
"top_inf_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
codisjoint.himp_inf_cancel_left (h : codisjoint a b) : b ⇨ (a ⊓ b) = a | by rw [himp_inf_distrib, himp_self, inf_top_eq, h.himp_eq_right] | lemma | codisjoint.himp_inf_cancel_left | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"codisjoint",
"himp_inf_distrib",
"himp_self",
"inf_top_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
codisjoint.himp_le_of_right_le (hac : codisjoint a c) (hba : b ≤ a) : c ⇨ b ≤ a | (himp_le_himp_left hba).trans_eq hac.himp_eq_right | lemma | codisjoint.himp_le_of_right_le | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"codisjoint",
"himp_le_himp_left"
] | See `himp_le` for a stronger version in Boolean algebras. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
le_himp_himp : a ≤ (a ⇨ b) ⇨ b | le_himp_iff.2 inf_himp_le | lemma | le_himp_himp | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"inf_himp_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
himp_triangle (a b c : α) : (a ⇨ b) ⊓ (b ⇨ c) ≤ a ⇨ c | by { rw [le_himp_iff, inf_right_comm, ←le_himp_iff], exact himp_inf_le.trans le_himp_himp } | lemma | himp_triangle | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"inf_right_comm",
"le_himp_himp",
"le_himp_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
himp_inf_himp_cancel (hba : b ≤ a) (hcb : c ≤ b) : (a ⇨ b) ⊓ (b ⇨ c) = a ⇨ c | (himp_triangle _ _ _).antisymm $ le_inf (himp_le_himp_left hcb) (himp_le_himp_right hba) | lemma | himp_inf_himp_cancel | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"himp_le_himp_left",
"himp_le_himp_right",
"himp_triangle",
"le_inf"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
generalized_heyting_algebra.to_distrib_lattice : distrib_lattice α | distrib_lattice.of_inf_sup_le $ λ a b c,
by simp_rw [@inf_comm _ _ a, ←le_himp_iff, sup_le_iff, le_himp_iff, ←sup_le_iff] | instance | generalized_heyting_algebra.to_distrib_lattice | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"distrib_lattice",
"distrib_lattice.of_inf_sup_le",
"inf_comm",
"le_himp_iff",
"sup_le_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod.generalized_heyting_algebra [generalized_heyting_algebra β] :
generalized_heyting_algebra (α × β) | { le_himp_iff := λ a b c, and_congr le_himp_iff le_himp_iff,
..prod.lattice α β, ..prod.order_top α β, ..prod.has_himp, ..prod.has_compl } | instance | prod.generalized_heyting_algebra | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"generalized_heyting_algebra",
"le_himp_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi.generalized_heyting_algebra {α : ι → Type*} [Π i, generalized_heyting_algebra (α i)] :
generalized_heyting_algebra (Π i, α i) | by { pi_instance, exact λ a b c, forall_congr (λ i, le_himp_iff) } | instance | pi.generalized_heyting_algebra | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"generalized_heyting_algebra",
"le_himp_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sdiff_le_iff : a \ b ≤ c ↔ a ≤ b ⊔ c | generalized_coheyting_algebra.sdiff_le_iff _ _ _ | lemma | sdiff_le_iff | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sdiff_le_iff' : a \ b ≤ c ↔ a ≤ c ⊔ b | by rw [sdiff_le_iff, sup_comm] | lemma | sdiff_le_iff' | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"sdiff_le_iff",
"sup_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sdiff_le_comm : a \ b ≤ c ↔ a \ c ≤ b | by rw [sdiff_le_iff, sdiff_le_iff'] | lemma | sdiff_le_comm | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"sdiff_le_iff",
"sdiff_le_iff'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sdiff_le : a \ b ≤ a | sdiff_le_iff.2 le_sup_right | lemma | sdiff_le | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"le_sup_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
disjoint.disjoint_sdiff_left (h : disjoint a b) : disjoint (a \ c) b | h.mono_left sdiff_le | lemma | disjoint.disjoint_sdiff_left | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"disjoint",
"sdiff_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
disjoint.disjoint_sdiff_right (h : disjoint a b) : disjoint a (b \ c) | h.mono_right sdiff_le | lemma | disjoint.disjoint_sdiff_right | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"disjoint",
"sdiff_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sdiff_le_iff_left : a \ b ≤ b ↔ a ≤ b | by rw [sdiff_le_iff, sup_idem] | lemma | sdiff_le_iff_left | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"sdiff_le_iff",
"sup_idem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sdiff_self : a \ a = ⊥ | le_bot_iff.1 $ sdiff_le_iff.2 le_sup_left | lemma | sdiff_self | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"le_sup_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_sup_sdiff : a ≤ b ⊔ a \ b | sdiff_le_iff.1 le_rfl | lemma | le_sup_sdiff | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"le_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_sdiff_sup : a ≤ a \ b ⊔ b | by rw [sup_comm, ←sdiff_le_iff] | lemma | le_sdiff_sup | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"sup_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sup_sdiff_left : a ⊔ a \ b = a | sup_of_le_left sdiff_le | lemma | sup_sdiff_left | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"sdiff_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sup_sdiff_right : a \ b ⊔ a = a | sup_of_le_right sdiff_le | lemma | sup_sdiff_right | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"sdiff_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inf_sdiff_left : a \ b ⊓ a = a \ b | inf_of_le_left sdiff_le | lemma | inf_sdiff_left | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"sdiff_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inf_sdiff_right : a ⊓ a \ b = a \ b | inf_of_le_right sdiff_le | lemma | inf_sdiff_right | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"sdiff_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sup_sdiff_self (a b : α) : a ⊔ b \ a = a ⊔ b | le_antisymm (sup_le_sup_left sdiff_le _) (sup_le le_sup_left le_sup_sdiff) | lemma | sup_sdiff_self | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"le_sup_left",
"le_sup_sdiff",
"sdiff_le",
"sup_le",
"sup_le_sup_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sdiff_sup_self (a b : α) : b \ a ⊔ a = b ⊔ a | by rw [sup_comm, sup_sdiff_self, sup_comm] | lemma | sdiff_sup_self | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"sup_comm",
"sup_sdiff_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sup_sdiff_eq_sup (h : c ≤ a) : a ⊔ b \ c = a ⊔ b | sup_congr_left (sdiff_le.trans le_sup_right) $ le_sup_sdiff.trans $ sup_le_sup_right h _ | lemma | sup_sdiff_eq_sup | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"le_sup_right",
"sup_congr_left",
"sup_le_sup_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sup_sdiff_cancel' (hab : a ≤ b) (hbc : b ≤ c) : b ⊔ c \ a = c | by rw [sup_sdiff_eq_sup hab, sup_of_le_right hbc] | lemma | sup_sdiff_cancel' | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"sup_sdiff_eq_sup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sup_sdiff_cancel_right (h : a ≤ b) : a ⊔ b \ a = b | sup_sdiff_cancel' le_rfl h | lemma | sup_sdiff_cancel_right | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"le_rfl",
"sup_sdiff_cancel'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sdiff_sup_cancel (h : b ≤ a) : a \ b ⊔ b = a | by rw [sup_comm, sup_sdiff_cancel_right h] | lemma | sdiff_sup_cancel | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"sup_comm",
"sup_sdiff_cancel_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sup_le_of_le_sdiff_left (h : b ≤ c \ a) (hac : a ≤ c) : a ⊔ b ≤ c | sup_le hac $ h.trans sdiff_le | lemma | sup_le_of_le_sdiff_left | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"sdiff_le",
"sup_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sup_le_of_le_sdiff_right (h : a ≤ c \ b) (hbc : b ≤ c) : a ⊔ b ≤ c | sup_le (h.trans sdiff_le) hbc | lemma | sup_le_of_le_sdiff_right | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"sdiff_le",
"sup_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sdiff_eq_bot_iff : a \ b = ⊥ ↔ a ≤ b | by rw [←le_bot_iff, sdiff_le_iff, sup_bot_eq] | lemma | sdiff_eq_bot_iff | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"sdiff_le_iff",
"sup_bot_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sdiff_bot : a \ ⊥ = a | eq_of_forall_ge_iff $ λ b, by rw [sdiff_le_iff, bot_sup_eq] | lemma | sdiff_bot | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"bot_sup_eq",
"eq_of_forall_ge_iff",
"sdiff_le_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bot_sdiff : ⊥ \ a = ⊥ | sdiff_eq_bot_iff.2 bot_le | lemma | bot_sdiff | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"bot_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sdiff_sdiff_sdiff_le_sdiff : a \ b \ (a \ c) ≤ c \ b | begin
rw [sdiff_le_iff, sdiff_le_iff, sup_left_comm, sup_sdiff_self, sup_left_comm, sdiff_sup_self,
sup_left_comm],
exact le_sup_left,
end | lemma | sdiff_sdiff_sdiff_le_sdiff | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"le_sup_left",
"sdiff_le_iff",
"sdiff_sup_self",
"sup_left_comm",
"sup_sdiff_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sdiff_sdiff (a b c : α) : a \ b \ c = a \ (b ⊔ c) | eq_of_forall_ge_iff $ λ d, by simp_rw [sdiff_le_iff, sup_assoc] | lemma | sdiff_sdiff | order.heyting | src/order/heyting/basic.lean | [
"order.prop_instances"
] | [
"eq_of_forall_ge_iff",
"sdiff_le_iff",
"sup_assoc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.