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 |
|---|---|---|---|---|---|---|---|---|---|---|
pos_mul_reflect_lt_iff_mul_pos_reflect_lt :
pos_mul_reflect_lt α ↔ mul_pos_reflect_lt α | by simp ! only [mul_comm] | lemma | pos_mul_reflect_lt_iff_mul_pos_reflect_lt | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"mul_comm",
"mul_pos_reflect_lt",
"pos_mul_reflect_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pos_mul_mono_iff_mul_pos_mono :
pos_mul_mono α ↔ mul_pos_mono α | by simp ! only [mul_comm] | lemma | pos_mul_mono_iff_mul_pos_mono | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"mul_comm",
"mul_pos_mono",
"pos_mul_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pos_mul_mono_rev_iff_mul_pos_mono_rev :
pos_mul_mono_rev α ↔ mul_pos_mono_rev α | by simp ! only [mul_comm] | lemma | pos_mul_mono_rev_iff_mul_pos_mono_rev | algebra.order.ring | src/algebra/order/ring/lemmas.lean | [
"algebra.covariant_and_contravariant",
"algebra.group_with_zero.defs"
] | [
"mul_comm",
"mul_pos_mono_rev",
"pos_mul_mono_rev"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_def {a b : with_top α} : a * b = if a = 0 ∨ b = 0 then 0 else option.map₂ (*) a b | rfl | lemma | with_top.mul_def | algebra.order.ring | src/algebra/order/ring/with_top.lean | [
"algebra.hom.ring",
"algebra.order.monoid.with_top",
"algebra.order.ring.canonical"
] | [
"option.map₂",
"with_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_top' {a : with_top α} : a * ⊤ = if a = 0 then 0 else ⊤ | by induction a using with_top.rec_top_coe; simp [mul_def]; refl | lemma | with_top.mul_top' | algebra.order.ring | src/algebra/order/ring/with_top.lean | [
"algebra.hom.ring",
"algebra.order.monoid.with_top",
"algebra.order.ring.canonical"
] | [
"with_top",
"with_top.rec_top_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_top {a : with_top α} (h : a ≠ 0) : a * ⊤ = ⊤ | by rw [mul_top', if_neg h] | lemma | with_top.mul_top | algebra.order.ring | src/algebra/order/ring/with_top.lean | [
"algebra.hom.ring",
"algebra.order.monoid.with_top",
"algebra.order.ring.canonical"
] | [
"with_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
top_mul' {a : with_top α} : ⊤ * a = if a = 0 then 0 else ⊤ | by induction a using with_top.rec_top_coe; simp [mul_def]; refl | lemma | with_top.top_mul' | algebra.order.ring | src/algebra/order/ring/with_top.lean | [
"algebra.hom.ring",
"algebra.order.monoid.with_top",
"algebra.order.ring.canonical"
] | [
"with_top",
"with_top.rec_top_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
top_mul {a : with_top α} (h : a ≠ 0) : ⊤ * a = ⊤ | by rw [top_mul', if_neg h] | lemma | with_top.top_mul | algebra.order.ring | src/algebra/order/ring/with_top.lean | [
"algebra.hom.ring",
"algebra.order.monoid.with_top",
"algebra.order.ring.canonical"
] | [
"with_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
top_mul_top : (⊤ * ⊤ : with_top α) = ⊤ | top_mul top_ne_zero | lemma | with_top.top_mul_top | algebra.order.ring | src/algebra/order/ring/with_top.lean | [
"algebra.hom.ring",
"algebra.order.monoid.with_top",
"algebra.order.ring.canonical"
] | [
"with_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_eq_top_iff {a b : with_top α} : a * b = ⊤ ↔ a ≠ 0 ∧ b = ⊤ ∨ a = ⊤ ∧ b ≠ 0 | begin
rw [mul_def, ite_eq_iff, ← none_eq_top, option.map₂_eq_none_iff],
have ha : a = 0 → a ≠ none := λ h, h.symm ▸ zero_ne_top,
have hb : b = 0 → b ≠ none := λ h, h.symm ▸ zero_ne_top,
tauto
end | theorem | with_top.mul_eq_top_iff | algebra.order.ring | src/algebra/order/ring/with_top.lean | [
"algebra.hom.ring",
"algebra.order.monoid.with_top",
"algebra.order.ring.canonical"
] | [
"ite_eq_iff",
"option.map₂_eq_none_iff",
"with_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_lt_top' [has_lt α] {a b : with_top α} (ha : a < ⊤) (hb : b < ⊤) : a * b < ⊤ | begin
rw [with_top.lt_top_iff_ne_top] at *,
simp only [ne.def, mul_eq_top_iff, *, and_false, false_and, false_or, not_false_iff]
end | theorem | with_top.mul_lt_top' | algebra.order.ring | src/algebra/order/ring/with_top.lean | [
"algebra.hom.ring",
"algebra.order.monoid.with_top",
"algebra.order.ring.canonical"
] | [
"with_top",
"with_top.lt_top_iff_ne_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_lt_top [has_lt α] {a b : with_top α} (ha : a ≠ ⊤) (hb : b ≠ ⊤) : a * b < ⊤ | mul_lt_top' (with_top.lt_top_iff_ne_top.2 ha) (with_top.lt_top_iff_ne_top.2 hb) | theorem | with_top.mul_lt_top | algebra.order.ring | src/algebra/order/ring/with_top.lean | [
"algebra.hom.ring",
"algebra.order.monoid.with_top",
"algebra.order.ring.canonical"
] | [
"with_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mul {a b : α} : (↑(a * b) : with_top α) = a * b | decidable.by_cases (assume : a = 0, by simp [this]) $ assume ha,
decidable.by_cases (assume : b = 0, by simp [this]) $ assume hb,
by { simp [*, mul_def] } | lemma | with_top.coe_mul | algebra.order.ring | src/algebra/order/ring/with_top.lean | [
"algebra.hom.ring",
"algebra.order.monoid.with_top",
"algebra.order.ring.canonical"
] | [
"with_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_coe {b : α} (hb : b ≠ 0) : ∀{a : with_top α}, a * b = a.bind (λa:α, ↑(a * b)) | | none := show (if (⊤:with_top α) = 0 ∨ (b:with_top α) = 0 then 0 else ⊤ : with_top α) = ⊤,
by simp [hb]
| (some a) := show ↑a * ↑b = ↑(a * b), from coe_mul.symm | lemma | with_top.mul_coe | algebra.order.ring | src/algebra/order/ring/with_top.lean | [
"algebra.hom.ring",
"algebra.order.monoid.with_top",
"algebra.order.ring.canonical"
] | [
"with_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
untop'_zero_mul (a b : with_top α) : (a * b).untop' 0 = a.untop' 0 * b.untop' 0 | begin
by_cases ha : a = 0, { rw [ha, zero_mul, ← coe_zero, untop'_coe, zero_mul] },
by_cases hb : b = 0, { rw [hb, mul_zero, ← coe_zero, untop'_coe, mul_zero] },
induction a using with_top.rec_top_coe, { rw [top_mul hb, untop'_top, zero_mul] },
induction b using with_top.rec_top_coe, { rw [mul_top ha, untop'_to... | lemma | with_top.untop'_zero_mul | algebra.order.ring | src/algebra/order/ring/with_top.lean | [
"algebra.hom.ring",
"algebra.order.monoid.with_top",
"algebra.order.ring.canonical"
] | [
"mul_zero",
"with_top",
"with_top.rec_top_coe",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.monoid_with_zero_hom.with_top_map
{R S : Type*} [mul_zero_one_class R] [decidable_eq R] [nontrivial R]
[mul_zero_one_class S] [decidable_eq S] [nontrivial S] (f : R →*₀ S) (hf : function.injective f) :
with_top R →*₀ with_top S | { to_fun := with_top.map f,
map_mul' := λ x y,
begin
have : ∀ z, map f z = 0 ↔ z = 0,
from λ z, (option.map_injective hf).eq_iff' f.to_zero_hom.with_top_map.map_zero,
rcases decidable.eq_or_ne x 0 with rfl|hx, { simp },
rcases decidable.eq_or_ne y 0 with rfl|hy, { simp },
induction... | def | monoid_with_zero_hom.with_top_map | algebra.order.ring | src/algebra/order/ring/with_top.lean | [
"algebra.hom.ring",
"algebra.order.monoid.with_top",
"algebra.order.ring.canonical"
] | [
"decidable.eq_or_ne",
"map_mul",
"mul_zero_one_class",
"nontrivial",
"option.map_injective",
"with_top",
"with_top.map",
"with_top.rec_top_coe"
] | A version of `with_top.map` for `monoid_with_zero_hom`s. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
distrib' (a b c : with_top α) : (a + b) * c = a * c + b * c | begin
induction c using with_top.rec_top_coe,
{ by_cases ha : a = 0; simp [ha] },
{ by_cases hc : c = 0, { simp [hc] },
simp only [mul_coe hc], cases a; cases b,
repeat { refl <|> exact congr_arg some (add_mul _ _ _) } }
end | lemma | with_top.distrib' | algebra.order.ring | src/algebra/order/ring/with_top.lean | [
"algebra.hom.ring",
"algebra.order.monoid.with_top",
"algebra.order.ring.canonical"
] | [
"with_top",
"with_top.rec_top_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.ring_hom.with_top_map
{R S : Type*} [canonically_ordered_comm_semiring R] [decidable_eq R] [nontrivial R]
[canonically_ordered_comm_semiring S] [decidable_eq S] [nontrivial S]
(f : R →+* S) (hf : function.injective f) :
with_top R →+* with_top S | { to_fun := with_top.map f,
.. f.to_monoid_with_zero_hom.with_top_map hf, .. f.to_add_monoid_hom.with_top_map } | def | ring_hom.with_top_map | algebra.order.ring | src/algebra/order/ring/with_top.lean | [
"algebra.hom.ring",
"algebra.order.monoid.with_top",
"algebra.order.ring.canonical"
] | [
"canonically_ordered_comm_semiring",
"nontrivial",
"with_top",
"with_top.map"
] | A version of `with_top.map` for `ring_hom`s. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_def {a b : with_bot α} :
a * b = if a = 0 ∨ b = 0 then 0 else option.map₂ (*) a b | rfl | lemma | with_bot.mul_def | algebra.order.ring | src/algebra/order/ring/with_top.lean | [
"algebra.hom.ring",
"algebra.order.monoid.with_top",
"algebra.order.ring.canonical"
] | [
"option.map₂",
"with_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_bot {a : with_bot α} (h : a ≠ 0) : a * ⊥ = ⊥ | with_top.mul_top h | lemma | with_bot.mul_bot | algebra.order.ring | src/algebra/order/ring/with_top.lean | [
"algebra.hom.ring",
"algebra.order.monoid.with_top",
"algebra.order.ring.canonical"
] | [
"with_bot",
"with_top.mul_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bot_mul {a : with_bot α} (h : a ≠ 0) : ⊥ * a = ⊥ | with_top.top_mul h | lemma | with_bot.bot_mul | algebra.order.ring | src/algebra/order/ring/with_top.lean | [
"algebra.hom.ring",
"algebra.order.monoid.with_top",
"algebra.order.ring.canonical"
] | [
"with_bot",
"with_top.top_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bot_mul_bot : (⊥ * ⊥ : with_bot α) = ⊥ | with_top.top_mul_top | lemma | with_bot.bot_mul_bot | algebra.order.ring | src/algebra/order/ring/with_top.lean | [
"algebra.hom.ring",
"algebra.order.monoid.with_top",
"algebra.order.ring.canonical"
] | [
"with_bot",
"with_top.top_mul_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_eq_bot_iff {a b : with_bot α} : a * b = ⊥ ↔ a ≠ 0 ∧ b = ⊥ ∨ a = ⊥ ∧ b ≠ 0 | with_top.mul_eq_top_iff | theorem | with_bot.mul_eq_bot_iff | algebra.order.ring | src/algebra/order/ring/with_top.lean | [
"algebra.hom.ring",
"algebra.order.monoid.with_top",
"algebra.order.ring.canonical"
] | [
"with_bot",
"with_top.mul_eq_top_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bot_lt_mul' [has_lt α] {a b : with_bot α} (ha : ⊥ < a) (hb : ⊥ < b) : ⊥ < a * b | @with_top.mul_lt_top' αᵒᵈ _ _ _ _ _ _ ha hb | theorem | with_bot.bot_lt_mul' | algebra.order.ring | src/algebra/order/ring/with_top.lean | [
"algebra.hom.ring",
"algebra.order.monoid.with_top",
"algebra.order.ring.canonical"
] | [
"with_bot",
"with_top.mul_lt_top'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bot_lt_mul [has_lt α] {a b : with_bot α} (ha : a ≠ ⊥) (hb : b ≠ ⊥) : ⊥ < a * b | @with_top.mul_lt_top αᵒᵈ _ _ _ _ _ _ ha hb | theorem | with_bot.bot_lt_mul | algebra.order.ring | src/algebra/order/ring/with_top.lean | [
"algebra.hom.ring",
"algebra.order.monoid.with_top",
"algebra.order.ring.canonical"
] | [
"with_bot",
"with_top.mul_lt_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mul {a b : α} : (↑(a * b) : with_bot α) = a * b | with_top.coe_mul | lemma | with_bot.coe_mul | algebra.order.ring | src/algebra/order/ring/with_top.lean | [
"algebra.hom.ring",
"algebra.order.monoid.with_top",
"algebra.order.ring.canonical"
] | [
"with_bot",
"with_top.coe_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_coe {b : α} (hb : b ≠ 0) {a : with_bot α} : a * b = a.bind (λa:α, ↑(a * b)) | with_top.mul_coe hb | lemma | with_bot.mul_coe | algebra.order.ring | src/algebra/order/ring/with_top.lean | [
"algebra.hom.ring",
"algebra.order.monoid.with_top",
"algebra.order.ring.canonical"
] | [
"with_bot",
"with_top.mul_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_hom.le_map_tsub [preorder β] [has_add β] [has_sub β] [has_ordered_sub β]
(f : add_hom α β) (hf : monotone f) (a b : α) :
f a - f b ≤ f (a - b) | by { rw [tsub_le_iff_right, ← f.map_add], exact hf le_tsub_add } | lemma | add_hom.le_map_tsub | algebra.order.sub | src/algebra/order/sub/basic.lean | [
"order.hom.basic",
"algebra.hom.equiv.basic",
"algebra.ring.basic",
"algebra.order.sub.defs"
] | [
"add_hom",
"has_ordered_sub",
"le_tsub_add",
"monotone",
"tsub_le_iff_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_mul_tsub {R : Type*} [distrib R] [preorder R] [has_sub R] [has_ordered_sub R]
[covariant_class R R (*) (≤)] {a b c : R} :
a * b - a * c ≤ a * (b - c) | (add_hom.mul_left a).le_map_tsub (monotone_id.const_mul' a) _ _ | lemma | le_mul_tsub | algebra.order.sub | src/algebra/order/sub/basic.lean | [
"order.hom.basic",
"algebra.hom.equiv.basic",
"algebra.ring.basic",
"algebra.order.sub.defs"
] | [
"add_hom.mul_left",
"covariant_class",
"distrib",
"has_ordered_sub"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_tsub_mul {R : Type*} [comm_semiring R] [preorder R] [has_sub R] [has_ordered_sub R]
[covariant_class R R (*) (≤)] {a b c : R} :
a * c - b * c ≤ (a - b) * c | by simpa only [mul_comm _ c] using le_mul_tsub | lemma | le_tsub_mul | algebra.order.sub | src/algebra/order/sub/basic.lean | [
"order.hom.basic",
"algebra.hom.equiv.basic",
"algebra.ring.basic",
"algebra.order.sub.defs"
] | [
"comm_semiring",
"covariant_class",
"has_ordered_sub",
"le_mul_tsub",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
order_iso.map_tsub {M N : Type*} [preorder M] [has_add M] [has_sub M] [has_ordered_sub M]
[partial_order N] [has_add N] [has_sub N] [has_ordered_sub N] (e : M ≃o N)
(h_add : ∀ a b, e (a + b) = e a + e b) (a b : M) :
e (a - b) = e a - e b | begin
set e_add : M ≃+ N := { map_add' := h_add, .. e },
refine le_antisymm _ (e_add.to_add_hom.le_map_tsub e.monotone a b),
suffices : e (e.symm (e a) - e.symm (e b)) ≤ e (e.symm (e a - e b)), by simpa,
exact e.monotone (e_add.symm.to_add_hom.le_map_tsub e.symm.monotone _ _)
end | lemma | order_iso.map_tsub | algebra.order.sub | src/algebra/order/sub/basic.lean | [
"order.hom.basic",
"algebra.hom.equiv.basic",
"algebra.ring.basic",
"algebra.order.sub.defs"
] | [
"has_ordered_sub"
] | An order isomorphism between types with ordered subtraction preserves subtraction provided that
it preserves addition. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add_monoid_hom.le_map_tsub [preorder β] [add_comm_monoid β] [has_sub β]
[has_ordered_sub β] (f : α →+ β) (hf : monotone f) (a b : α) :
f a - f b ≤ f (a - b) | f.to_add_hom.le_map_tsub hf a b | lemma | add_monoid_hom.le_map_tsub | algebra.order.sub | src/algebra/order/sub/basic.lean | [
"order.hom.basic",
"algebra.hom.equiv.basic",
"algebra.ring.basic",
"algebra.order.sub.defs"
] | [
"add_comm_monoid",
"has_ordered_sub",
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_tsub_cancel_of_le (h : a ≤ b) : a + (b - a) = b | begin
refine le_antisymm _ le_add_tsub,
obtain ⟨c, rfl⟩ := exists_add_of_le h,
exact add_le_add_left add_tsub_le_left a,
end | lemma | add_tsub_cancel_of_le | algebra.order.sub | src/algebra/order/sub/canonical.lean | [
"algebra.order.monoid.canonical.defs",
"algebra.order.sub.defs"
] | [
"add_tsub_le_left",
"le_add_tsub"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsub_add_cancel_of_le (h : a ≤ b) : b - a + a = b | by { rw [add_comm], exact add_tsub_cancel_of_le h } | lemma | tsub_add_cancel_of_le | algebra.order.sub | src/algebra/order/sub/canonical.lean | [
"algebra.order.monoid.canonical.defs",
"algebra.order.sub.defs"
] | [
"add_tsub_cancel_of_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_le_of_le_tsub_right_of_le (h : b ≤ c) (h2 : a ≤ c - b) : a + b ≤ c | (add_le_add_right h2 b).trans_eq $ tsub_add_cancel_of_le h | lemma | add_le_of_le_tsub_right_of_le | algebra.order.sub | src/algebra/order/sub/canonical.lean | [
"algebra.order.monoid.canonical.defs",
"algebra.order.sub.defs"
] | [
"tsub_add_cancel_of_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_le_of_le_tsub_left_of_le (h : a ≤ c) (h2 : b ≤ c - a) : a + b ≤ c | (add_le_add_left h2 a).trans_eq $ add_tsub_cancel_of_le h | lemma | add_le_of_le_tsub_left_of_le | algebra.order.sub | src/algebra/order/sub/canonical.lean | [
"algebra.order.monoid.canonical.defs",
"algebra.order.sub.defs"
] | [
"add_tsub_cancel_of_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsub_le_tsub_iff_right (h : c ≤ b) : a - c ≤ b - c ↔ a ≤ b | by rw [tsub_le_iff_right, tsub_add_cancel_of_le h] | lemma | tsub_le_tsub_iff_right | algebra.order.sub | src/algebra/order/sub/canonical.lean | [
"algebra.order.monoid.canonical.defs",
"algebra.order.sub.defs"
] | [
"tsub_add_cancel_of_le",
"tsub_le_iff_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsub_left_inj (h1 : c ≤ a) (h2 : c ≤ b) : a - c = b - c ↔ a = b | by simp_rw [le_antisymm_iff, tsub_le_tsub_iff_right h1, tsub_le_tsub_iff_right h2] | lemma | tsub_left_inj | algebra.order.sub | src/algebra/order/sub/canonical.lean | [
"algebra.order.monoid.canonical.defs",
"algebra.order.sub.defs"
] | [
"tsub_le_tsub_iff_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsub_inj_left (h₁ : a ≤ b) (h₂ : a ≤ c) : b - a = c - a → b = c | (tsub_left_inj h₁ h₂).1 | lemma | tsub_inj_left | algebra.order.sub | src/algebra/order/sub/canonical.lean | [
"algebra.order.monoid.canonical.defs",
"algebra.order.sub.defs"
] | [
"tsub_left_inj"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_of_tsub_lt_tsub_right_of_le (h : c ≤ b) (h2 : a - c < b - c) : a < b | by { refine ((tsub_le_tsub_iff_right h).mp h2.le).lt_of_ne _, rintro rfl, exact h2.false } | lemma | lt_of_tsub_lt_tsub_right_of_le | algebra.order.sub | src/algebra/order/sub/canonical.lean | [
"algebra.order.monoid.canonical.defs",
"algebra.order.sub.defs"
] | [
"tsub_le_tsub_iff_right"
] | See `lt_of_tsub_lt_tsub_right` for a stronger statement in a linear order. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tsub_add_tsub_cancel (hab : b ≤ a) (hcb : c ≤ b) : (a - b) + (b - c) = a - c | begin
convert tsub_add_cancel_of_le (tsub_le_tsub_right hab c) using 2,
rw [tsub_tsub, add_tsub_cancel_of_le hcb],
end | lemma | tsub_add_tsub_cancel | algebra.order.sub | src/algebra/order/sub/canonical.lean | [
"algebra.order.monoid.canonical.defs",
"algebra.order.sub.defs"
] | [
"add_tsub_cancel_of_le",
"tsub_add_cancel_of_le",
"tsub_le_tsub_right",
"tsub_tsub"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsub_tsub_tsub_cancel_right (h : c ≤ b) : (a - c) - (b - c) = a - b | by rw [tsub_tsub, add_tsub_cancel_of_le h] | lemma | tsub_tsub_tsub_cancel_right | algebra.order.sub | src/algebra/order/sub/canonical.lean | [
"algebra.order.monoid.canonical.defs",
"algebra.order.sub.defs"
] | [
"add_tsub_cancel_of_le",
"tsub_tsub"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_tsub_iff_add_eq_of_le (hc : add_le_cancellable c) (h : c ≤ b) :
a = b - c ↔ a + c = b | ⟨by { rintro rfl, exact tsub_add_cancel_of_le h }, hc.eq_tsub_of_add_eq⟩ | lemma | add_le_cancellable.eq_tsub_iff_add_eq_of_le | algebra.order.sub | src/algebra/order/sub/canonical.lean | [
"algebra.order.monoid.canonical.defs",
"algebra.order.sub.defs"
] | [
"eq_tsub_iff_add_eq_of_le",
"tsub_add_cancel_of_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsub_eq_iff_eq_add_of_le (hb : add_le_cancellable b) (h : b ≤ a) :
a - b = c ↔ a = c + b | by rw [eq_comm, hb.eq_tsub_iff_add_eq_of_le h, eq_comm] | lemma | add_le_cancellable.tsub_eq_iff_eq_add_of_le | algebra.order.sub | src/algebra/order/sub/canonical.lean | [
"algebra.order.monoid.canonical.defs",
"algebra.order.sub.defs"
] | [
"tsub_eq_iff_eq_add_of_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_tsub_assoc_of_le (hc : add_le_cancellable c) (h : c ≤ b) (a : α) :
a + b - c = a + (b - c) | by conv_lhs { rw [← add_tsub_cancel_of_le h, add_comm c, ← add_assoc,
hc.add_tsub_cancel_right] } | lemma | add_le_cancellable.add_tsub_assoc_of_le | algebra.order.sub | src/algebra/order/sub/canonical.lean | [
"algebra.order.monoid.canonical.defs",
"algebra.order.sub.defs"
] | [
"add_tsub_assoc_of_le",
"add_tsub_cancel_of_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsub_add_eq_add_tsub (hb : add_le_cancellable b) (h : b ≤ a) :
a - b + c = a + c - b | by rw [add_comm a, hb.add_tsub_assoc_of_le h, add_comm] | lemma | add_le_cancellable.tsub_add_eq_add_tsub | algebra.order.sub | src/algebra/order/sub/canonical.lean | [
"algebra.order.monoid.canonical.defs",
"algebra.order.sub.defs"
] | [
"tsub_add_eq_add_tsub"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsub_tsub_assoc (hbc : add_le_cancellable (b - c)) (h₁ : b ≤ a) (h₂ : c ≤ b) :
a - (b - c) = a - b + c | hbc.tsub_eq_of_eq_add $ by rw [add_assoc, add_tsub_cancel_of_le h₂, tsub_add_cancel_of_le h₁] | lemma | add_le_cancellable.tsub_tsub_assoc | algebra.order.sub | src/algebra/order/sub/canonical.lean | [
"algebra.order.monoid.canonical.defs",
"algebra.order.sub.defs"
] | [
"add_tsub_cancel_of_le",
"tsub_add_cancel_of_le",
"tsub_tsub_assoc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsub_add_tsub_comm (hb : add_le_cancellable b) (hd : add_le_cancellable d)
(hba : b ≤ a) (hdc : d ≤ c) : a - b + (c - d) = a + c - (b + d) | by rw [hb.tsub_add_eq_add_tsub hba, ←hd.add_tsub_assoc_of_le hdc, tsub_tsub, add_comm d] | lemma | add_le_cancellable.tsub_add_tsub_comm | algebra.order.sub | src/algebra/order/sub/canonical.lean | [
"algebra.order.monoid.canonical.defs",
"algebra.order.sub.defs"
] | [
"tsub_add_tsub_comm",
"tsub_tsub"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_tsub_iff_left (ha : add_le_cancellable a) (h : a ≤ c) : b ≤ c - a ↔ a + b ≤ c | ⟨add_le_of_le_tsub_left_of_le h, ha.le_tsub_of_add_le_left⟩ | lemma | add_le_cancellable.le_tsub_iff_left | algebra.order.sub | src/algebra/order/sub/canonical.lean | [
"algebra.order.monoid.canonical.defs",
"algebra.order.sub.defs"
] | [
"le_tsub_iff_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_tsub_iff_right (ha : add_le_cancellable a) (h : a ≤ c) : b ≤ c - a ↔ b + a ≤ c | by { rw [add_comm], exact ha.le_tsub_iff_left h } | lemma | add_le_cancellable.le_tsub_iff_right | algebra.order.sub | src/algebra/order/sub/canonical.lean | [
"algebra.order.monoid.canonical.defs",
"algebra.order.sub.defs"
] | [
"le_tsub_iff_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsub_lt_iff_left (hb : add_le_cancellable b) (hba : b ≤ a) :
a - b < c ↔ a < b + c | begin
refine ⟨hb.lt_add_of_tsub_lt_left, _⟩,
intro h, refine (tsub_le_iff_left.mpr h.le).lt_of_ne _,
rintro rfl, exact h.ne' (add_tsub_cancel_of_le hba)
end | lemma | add_le_cancellable.tsub_lt_iff_left | algebra.order.sub | src/algebra/order/sub/canonical.lean | [
"algebra.order.monoid.canonical.defs",
"algebra.order.sub.defs"
] | [
"add_tsub_cancel_of_le",
"tsub_lt_iff_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsub_lt_iff_right (hb : add_le_cancellable b) (hba : b ≤ a) :
a - b < c ↔ a < c + b | by { rw [add_comm], exact hb.tsub_lt_iff_left hba } | lemma | add_le_cancellable.tsub_lt_iff_right | algebra.order.sub | src/algebra/order/sub/canonical.lean | [
"algebra.order.monoid.canonical.defs",
"algebra.order.sub.defs"
] | [
"tsub_lt_iff_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsub_lt_iff_tsub_lt (hb : add_le_cancellable b) (hc : add_le_cancellable c)
(h₁ : b ≤ a) (h₂ : c ≤ a) : a - b < c ↔ a - c < b | by rw [hb.tsub_lt_iff_left h₁, hc.tsub_lt_iff_right h₂] | lemma | add_le_cancellable.tsub_lt_iff_tsub_lt | algebra.order.sub | src/algebra/order/sub/canonical.lean | [
"algebra.order.monoid.canonical.defs",
"algebra.order.sub.defs"
] | [
"tsub_lt_iff_tsub_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_tsub_iff_le_tsub (ha : add_le_cancellable a) (hc : add_le_cancellable c)
(h₁ : a ≤ b) (h₂ : c ≤ b) : a ≤ b - c ↔ c ≤ b - a | by rw [ha.le_tsub_iff_left h₁, hc.le_tsub_iff_right h₂] | lemma | add_le_cancellable.le_tsub_iff_le_tsub | algebra.order.sub | src/algebra/order/sub/canonical.lean | [
"algebra.order.monoid.canonical.defs",
"algebra.order.sub.defs"
] | [
"le_tsub_iff_le_tsub"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_tsub_iff_right_of_le (hc : add_le_cancellable c) (h : c ≤ b) :
a < b - c ↔ a + c < b | begin
refine ⟨λ h', (add_le_of_le_tsub_right_of_le h h'.le).lt_of_ne _, hc.lt_tsub_of_add_lt_right⟩,
rintro rfl,
exact h'.ne' hc.add_tsub_cancel_right,
end | lemma | add_le_cancellable.lt_tsub_iff_right_of_le | algebra.order.sub | src/algebra/order/sub/canonical.lean | [
"algebra.order.monoid.canonical.defs",
"algebra.order.sub.defs"
] | [
"add_le_of_le_tsub_right_of_le",
"lt_tsub_iff_right_of_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_tsub_iff_left_of_le (hc : add_le_cancellable c) (h : c ≤ b) :
a < b - c ↔ c + a < b | by { rw [add_comm], exact hc.lt_tsub_iff_right_of_le h } | lemma | add_le_cancellable.lt_tsub_iff_left_of_le | algebra.order.sub | src/algebra/order/sub/canonical.lean | [
"algebra.order.monoid.canonical.defs",
"algebra.order.sub.defs"
] | [
"lt_tsub_iff_left_of_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsub_inj_right (hab : add_le_cancellable (a - b)) (h₁ : b ≤ a) (h₂ : c ≤ a)
(h₃ : a - b = a - c) : b = c | by { rw ← hab.inj, rw [tsub_add_cancel_of_le h₁, h₃, tsub_add_cancel_of_le h₂] } | lemma | add_le_cancellable.tsub_inj_right | algebra.order.sub | src/algebra/order/sub/canonical.lean | [
"algebra.order.monoid.canonical.defs",
"algebra.order.sub.defs"
] | [
"tsub_add_cancel_of_le",
"tsub_inj_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_of_tsub_lt_tsub_left_of_le [contravariant_class α α (+) (<)]
(hb : add_le_cancellable b) (hca : c ≤ a) (h : a - b < a - c) : c < b | begin
conv_lhs at h { rw [← tsub_add_cancel_of_le hca] },
exact lt_of_add_lt_add_left (hb.lt_add_of_tsub_lt_right h),
end | lemma | add_le_cancellable.lt_of_tsub_lt_tsub_left_of_le | algebra.order.sub | src/algebra/order/sub/canonical.lean | [
"algebra.order.monoid.canonical.defs",
"algebra.order.sub.defs"
] | [
"contravariant_class",
"lt_of_tsub_lt_tsub_left_of_le",
"tsub_add_cancel_of_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsub_lt_tsub_left_of_le (hab : add_le_cancellable (a - b)) (h₁ : b ≤ a)
(h : c < b) : a - b < a - c | (tsub_le_tsub_left h.le _).lt_of_ne $ λ h', h.ne' $ hab.tsub_inj_right h₁ (h.le.trans h₁) h' | lemma | add_le_cancellable.tsub_lt_tsub_left_of_le | algebra.order.sub | src/algebra/order/sub/canonical.lean | [
"algebra.order.monoid.canonical.defs",
"algebra.order.sub.defs"
] | [
"tsub_le_tsub_left",
"tsub_lt_tsub_left_of_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsub_lt_tsub_right_of_le (hc : add_le_cancellable c) (h : c ≤ a) (h2 : a < b) :
a - c < b - c | by { apply hc.lt_tsub_of_add_lt_left, rwa [add_tsub_cancel_of_le h] } | lemma | add_le_cancellable.tsub_lt_tsub_right_of_le | algebra.order.sub | src/algebra/order/sub/canonical.lean | [
"algebra.order.monoid.canonical.defs",
"algebra.order.sub.defs"
] | [
"add_tsub_cancel_of_le",
"tsub_lt_tsub_right_of_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsub_lt_tsub_iff_left_of_le_of_le [contravariant_class α α (+) (<)]
(hb : add_le_cancellable b) (hab : add_le_cancellable (a - b)) (h₁ : b ≤ a) (h₂ : c ≤ a) :
a - b < a - c ↔ c < b | ⟨hb.lt_of_tsub_lt_tsub_left_of_le h₂, hab.tsub_lt_tsub_left_of_le h₁⟩ | lemma | add_le_cancellable.tsub_lt_tsub_iff_left_of_le_of_le | algebra.order.sub | src/algebra/order/sub/canonical.lean | [
"algebra.order.monoid.canonical.defs",
"algebra.order.sub.defs"
] | [
"contravariant_class",
"tsub_lt_tsub_iff_left_of_le_of_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_tsub_tsub_cancel (hac : add_le_cancellable (a - c)) (h : c ≤ a) :
(a + b) - (a - c) = b + c | hac.tsub_eq_of_eq_add $ by rw [add_assoc, add_tsub_cancel_of_le h, add_comm] | lemma | add_le_cancellable.add_tsub_tsub_cancel | algebra.order.sub | src/algebra/order/sub/canonical.lean | [
"algebra.order.monoid.canonical.defs",
"algebra.order.sub.defs"
] | [
"add_tsub_cancel_of_le",
"add_tsub_tsub_cancel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsub_tsub_cancel_of_le (hba : add_le_cancellable (b - a)) (h : a ≤ b) :
b - (b - a) = a | hba.tsub_eq_of_eq_add (add_tsub_cancel_of_le h).symm | lemma | add_le_cancellable.tsub_tsub_cancel_of_le | algebra.order.sub | src/algebra/order/sub/canonical.lean | [
"algebra.order.monoid.canonical.defs",
"algebra.order.sub.defs"
] | [
"add_tsub_cancel_of_le",
"tsub_tsub_cancel_of_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsub_tsub_tsub_cancel_left (hab : add_le_cancellable (a - b)) (h : b ≤ a) :
a - c - (a - b) = b - c | by rw [tsub_right_comm, hab.tsub_tsub_cancel_of_le h] | lemma | add_le_cancellable.tsub_tsub_tsub_cancel_left | algebra.order.sub | src/algebra/order/sub/canonical.lean | [
"algebra.order.monoid.canonical.defs",
"algebra.order.sub.defs"
] | [
"tsub_right_comm",
"tsub_tsub_tsub_cancel_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_tsub_iff_add_eq_of_le (h : c ≤ b) : a = b - c ↔ a + c = b | contravariant.add_le_cancellable.eq_tsub_iff_add_eq_of_le h | lemma | eq_tsub_iff_add_eq_of_le | algebra.order.sub | src/algebra/order/sub/canonical.lean | [
"algebra.order.monoid.canonical.defs",
"algebra.order.sub.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsub_eq_iff_eq_add_of_le (h : b ≤ a) : a - b = c ↔ a = c + b | contravariant.add_le_cancellable.tsub_eq_iff_eq_add_of_le h | lemma | tsub_eq_iff_eq_add_of_le | algebra.order.sub | src/algebra/order/sub/canonical.lean | [
"algebra.order.monoid.canonical.defs",
"algebra.order.sub.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_tsub_assoc_of_le (h : c ≤ b) (a : α) : a + b - c = a + (b - c) | contravariant.add_le_cancellable.add_tsub_assoc_of_le h a | lemma | add_tsub_assoc_of_le | algebra.order.sub | src/algebra/order/sub/canonical.lean | [
"algebra.order.monoid.canonical.defs",
"algebra.order.sub.defs"
] | [] | See `add_tsub_le_assoc` for an inequality. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tsub_add_eq_add_tsub (h : b ≤ a) : a - b + c = a + c - b | contravariant.add_le_cancellable.tsub_add_eq_add_tsub h | lemma | tsub_add_eq_add_tsub | algebra.order.sub | src/algebra/order/sub/canonical.lean | [
"algebra.order.monoid.canonical.defs",
"algebra.order.sub.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsub_tsub_assoc (h₁ : b ≤ a) (h₂ : c ≤ b) : a - (b - c) = a - b + c | contravariant.add_le_cancellable.tsub_tsub_assoc h₁ h₂ | lemma | tsub_tsub_assoc | algebra.order.sub | src/algebra/order/sub/canonical.lean | [
"algebra.order.monoid.canonical.defs",
"algebra.order.sub.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsub_add_tsub_comm (hba : b ≤ a) (hdc : d ≤ c) : a - b + (c - d) = a + c - (b + d) | contravariant.add_le_cancellable.tsub_add_tsub_comm contravariant.add_le_cancellable hba hdc | lemma | tsub_add_tsub_comm | algebra.order.sub | src/algebra/order/sub/canonical.lean | [
"algebra.order.monoid.canonical.defs",
"algebra.order.sub.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_tsub_iff_left (h : a ≤ c) : b ≤ c - a ↔ a + b ≤ c | contravariant.add_le_cancellable.le_tsub_iff_left h | lemma | le_tsub_iff_left | algebra.order.sub | src/algebra/order/sub/canonical.lean | [
"algebra.order.monoid.canonical.defs",
"algebra.order.sub.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_tsub_iff_right (h : a ≤ c) : b ≤ c - a ↔ b + a ≤ c | contravariant.add_le_cancellable.le_tsub_iff_right h | lemma | le_tsub_iff_right | algebra.order.sub | src/algebra/order/sub/canonical.lean | [
"algebra.order.monoid.canonical.defs",
"algebra.order.sub.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsub_lt_iff_left (hbc : b ≤ a) : a - b < c ↔ a < b + c | contravariant.add_le_cancellable.tsub_lt_iff_left hbc | lemma | tsub_lt_iff_left | algebra.order.sub | src/algebra/order/sub/canonical.lean | [
"algebra.order.monoid.canonical.defs",
"algebra.order.sub.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsub_lt_iff_right (hbc : b ≤ a) : a - b < c ↔ a < c + b | contravariant.add_le_cancellable.tsub_lt_iff_right hbc | lemma | tsub_lt_iff_right | algebra.order.sub | src/algebra/order/sub/canonical.lean | [
"algebra.order.monoid.canonical.defs",
"algebra.order.sub.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsub_lt_iff_tsub_lt (h₁ : b ≤ a) (h₂ : c ≤ a) : a - b < c ↔ a - c < b | contravariant.add_le_cancellable.tsub_lt_iff_tsub_lt contravariant.add_le_cancellable h₁ h₂ | lemma | tsub_lt_iff_tsub_lt | algebra.order.sub | src/algebra/order/sub/canonical.lean | [
"algebra.order.monoid.canonical.defs",
"algebra.order.sub.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_tsub_iff_le_tsub (h₁ : a ≤ b) (h₂ : c ≤ b) : a ≤ b - c ↔ c ≤ b - a | contravariant.add_le_cancellable.le_tsub_iff_le_tsub contravariant.add_le_cancellable h₁ h₂ | lemma | le_tsub_iff_le_tsub | algebra.order.sub | src/algebra/order/sub/canonical.lean | [
"algebra.order.monoid.canonical.defs",
"algebra.order.sub.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_tsub_iff_right_of_le (h : c ≤ b) : a < b - c ↔ a + c < b | contravariant.add_le_cancellable.lt_tsub_iff_right_of_le h | lemma | lt_tsub_iff_right_of_le | algebra.order.sub | src/algebra/order/sub/canonical.lean | [
"algebra.order.monoid.canonical.defs",
"algebra.order.sub.defs"
] | [] | See `lt_tsub_iff_right` for a stronger statement in a linear order. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lt_tsub_iff_left_of_le (h : c ≤ b) : a < b - c ↔ c + a < b | contravariant.add_le_cancellable.lt_tsub_iff_left_of_le h | lemma | lt_tsub_iff_left_of_le | algebra.order.sub | src/algebra/order/sub/canonical.lean | [
"algebra.order.monoid.canonical.defs",
"algebra.order.sub.defs"
] | [] | See `lt_tsub_iff_left` for a stronger statement in a linear order. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lt_of_tsub_lt_tsub_left_of_le [contravariant_class α α (+) (<)]
(hca : c ≤ a) (h : a - b < a - c) : c < b | contravariant.add_le_cancellable.lt_of_tsub_lt_tsub_left_of_le hca h | lemma | lt_of_tsub_lt_tsub_left_of_le | algebra.order.sub | src/algebra/order/sub/canonical.lean | [
"algebra.order.monoid.canonical.defs",
"algebra.order.sub.defs"
] | [
"contravariant_class"
] | See `lt_of_tsub_lt_tsub_left` for a stronger statement in a linear order. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tsub_lt_tsub_left_of_le : b ≤ a → c < b → a - b < a - c | contravariant.add_le_cancellable.tsub_lt_tsub_left_of_le | lemma | tsub_lt_tsub_left_of_le | algebra.order.sub | src/algebra/order/sub/canonical.lean | [
"algebra.order.monoid.canonical.defs",
"algebra.order.sub.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsub_lt_tsub_right_of_le (h : c ≤ a) (h2 : a < b) : a - c < b - c | contravariant.add_le_cancellable.tsub_lt_tsub_right_of_le h h2 | lemma | tsub_lt_tsub_right_of_le | algebra.order.sub | src/algebra/order/sub/canonical.lean | [
"algebra.order.monoid.canonical.defs",
"algebra.order.sub.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsub_inj_right (h₁ : b ≤ a) (h₂ : c ≤ a) (h₃ : a - b = a - c) : b = c | contravariant.add_le_cancellable.tsub_inj_right h₁ h₂ h₃ | lemma | tsub_inj_right | algebra.order.sub | src/algebra/order/sub/canonical.lean | [
"algebra.order.monoid.canonical.defs",
"algebra.order.sub.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsub_lt_tsub_iff_left_of_le_of_le [contravariant_class α α (+) (<)]
(h₁ : b ≤ a) (h₂ : c ≤ a) : a - b < a - c ↔ c < b | contravariant.add_le_cancellable.tsub_lt_tsub_iff_left_of_le_of_le
contravariant.add_le_cancellable h₁ h₂ | lemma | tsub_lt_tsub_iff_left_of_le_of_le | algebra.order.sub | src/algebra/order/sub/canonical.lean | [
"algebra.order.monoid.canonical.defs",
"algebra.order.sub.defs"
] | [
"contravariant_class"
] | See `tsub_lt_tsub_iff_left_of_le` for a stronger statement in a linear order. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add_tsub_tsub_cancel (h : c ≤ a) : (a + b) - (a - c) = b + c | contravariant.add_le_cancellable.add_tsub_tsub_cancel h | lemma | add_tsub_tsub_cancel | algebra.order.sub | src/algebra/order/sub/canonical.lean | [
"algebra.order.monoid.canonical.defs",
"algebra.order.sub.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsub_tsub_cancel_of_le (h : a ≤ b) : b - (b - a) = a | contravariant.add_le_cancellable.tsub_tsub_cancel_of_le h | lemma | tsub_tsub_cancel_of_le | algebra.order.sub | src/algebra/order/sub/canonical.lean | [
"algebra.order.monoid.canonical.defs",
"algebra.order.sub.defs"
] | [] | See `tsub_tsub_le` for an inequality. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tsub_tsub_tsub_cancel_left (h : b ≤ a) : a - c - (a - b) = b - c | contravariant.add_le_cancellable.tsub_tsub_tsub_cancel_left h | lemma | tsub_tsub_tsub_cancel_left | algebra.order.sub | src/algebra/order/sub/canonical.lean | [
"algebra.order.monoid.canonical.defs",
"algebra.order.sub.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_tsub_cancel_iff_le : a + (b - a) = b ↔ a ≤ b | ⟨λ h, le_iff_exists_add.mpr ⟨b - a, h.symm⟩, add_tsub_cancel_of_le⟩ | lemma | add_tsub_cancel_iff_le | algebra.order.sub | src/algebra/order/sub/canonical.lean | [
"algebra.order.monoid.canonical.defs",
"algebra.order.sub.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsub_add_cancel_iff_le : b - a + a = b ↔ a ≤ b | by { rw [add_comm], exact add_tsub_cancel_iff_le } | lemma | tsub_add_cancel_iff_le | algebra.order.sub | src/algebra/order/sub/canonical.lean | [
"algebra.order.monoid.canonical.defs",
"algebra.order.sub.defs"
] | [
"add_tsub_cancel_iff_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsub_eq_zero_iff_le : a - b = 0 ↔ a ≤ b | by rw [← nonpos_iff_eq_zero, tsub_le_iff_left, add_zero] | lemma | tsub_eq_zero_iff_le | algebra.order.sub | src/algebra/order/sub/canonical.lean | [
"algebra.order.monoid.canonical.defs",
"algebra.order.sub.defs"
] | [
"tsub_le_iff_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsub_self (a : α) : a - a = 0 | tsub_eq_zero_of_le le_rfl | lemma | tsub_self | algebra.order.sub | src/algebra/order/sub/canonical.lean | [
"algebra.order.monoid.canonical.defs",
"algebra.order.sub.defs"
] | [
"le_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsub_le_self : a - b ≤ a | tsub_le_iff_left.mpr $ le_add_left le_rfl | lemma | tsub_le_self | algebra.order.sub | src/algebra/order/sub/canonical.lean | [
"algebra.order.monoid.canonical.defs",
"algebra.order.sub.defs"
] | [
"le_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_tsub (a : α) : 0 - a = 0 | tsub_eq_zero_of_le $ zero_le a | lemma | zero_tsub | algebra.order.sub | src/algebra/order/sub/canonical.lean | [
"algebra.order.monoid.canonical.defs",
"algebra.order.sub.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsub_self_add (a b : α) : a - (a + b) = 0 | tsub_eq_zero_of_le $ self_le_add_right _ _ | lemma | tsub_self_add | algebra.order.sub | src/algebra/order/sub/canonical.lean | [
"algebra.order.monoid.canonical.defs",
"algebra.order.sub.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsub_pos_iff_not_le : 0 < a - b ↔ ¬ a ≤ b | by rw [pos_iff_ne_zero, ne.def, tsub_eq_zero_iff_le] | lemma | tsub_pos_iff_not_le | algebra.order.sub | src/algebra/order/sub/canonical.lean | [
"algebra.order.monoid.canonical.defs",
"algebra.order.sub.defs"
] | [
"tsub_eq_zero_iff_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsub_pos_of_lt (h : a < b) : 0 < b - a | tsub_pos_iff_not_le.mpr h.not_le | lemma | tsub_pos_of_lt | algebra.order.sub | src/algebra/order/sub/canonical.lean | [
"algebra.order.monoid.canonical.defs",
"algebra.order.sub.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsub_lt_of_lt (h : a < b) : a - c < b | lt_of_le_of_lt tsub_le_self h | lemma | tsub_lt_of_lt | algebra.order.sub | src/algebra/order/sub/canonical.lean | [
"algebra.order.monoid.canonical.defs",
"algebra.order.sub.defs"
] | [
"tsub_le_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsub_le_tsub_iff_left (ha : add_le_cancellable a) (hc : add_le_cancellable c)
(h : c ≤ a) : a - b ≤ a - c ↔ c ≤ b | begin
refine ⟨_, λ h, tsub_le_tsub_left h a⟩,
rw [tsub_le_iff_left, ← hc.add_tsub_assoc_of_le h, hc.le_tsub_iff_right (h.trans le_add_self),
add_comm b],
apply ha,
end | lemma | add_le_cancellable.tsub_le_tsub_iff_left | algebra.order.sub | src/algebra/order/sub/canonical.lean | [
"algebra.order.monoid.canonical.defs",
"algebra.order.sub.defs"
] | [
"tsub_le_iff_left",
"tsub_le_tsub_iff_left",
"tsub_le_tsub_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsub_right_inj (ha : add_le_cancellable a) (hb : add_le_cancellable b)
(hc : add_le_cancellable c) (hba : b ≤ a) (hca : c ≤ a) : a - b = a - c ↔ b = c | by simp_rw [le_antisymm_iff, ha.tsub_le_tsub_iff_left hb hba, ha.tsub_le_tsub_iff_left hc hca,
and_comm] | lemma | add_le_cancellable.tsub_right_inj | algebra.order.sub | src/algebra/order/sub/canonical.lean | [
"algebra.order.monoid.canonical.defs",
"algebra.order.sub.defs"
] | [
"tsub_right_inj"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsub_le_tsub_iff_left (h : c ≤ a) : a - b ≤ a - c ↔ c ≤ b | contravariant.add_le_cancellable.tsub_le_tsub_iff_left contravariant.add_le_cancellable h | lemma | tsub_le_tsub_iff_left | algebra.order.sub | src/algebra/order/sub/canonical.lean | [
"algebra.order.monoid.canonical.defs",
"algebra.order.sub.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsub_right_inj (hba : b ≤ a) (hca : c ≤ a) : a - b = a - c ↔ b = c | contravariant.add_le_cancellable.tsub_right_inj contravariant.add_le_cancellable
contravariant.add_le_cancellable hba hca | lemma | tsub_right_inj | algebra.order.sub | src/algebra/order/sub/canonical.lean | [
"algebra.order.monoid.canonical.defs",
"algebra.order.sub.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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