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 |
|---|---|---|---|---|---|---|---|---|---|---|
canonically_ordered_add_monoid.to_add_cancel_comm_monoid : add_cancel_comm_monoid α | { add_left_cancel := λ a b c h, by simpa only [add_tsub_cancel_left] using congr_arg (λ x, x - a) h,
..(by apply_instance : add_comm_monoid α) } | def | canonically_ordered_add_monoid.to_add_cancel_comm_monoid | algebra.order.sub | src/algebra/order/sub/canonical.lean | [
"algebra.order.monoid.canonical.defs",
"algebra.order.sub.defs"
] | [
"add_cancel_comm_monoid",
"add_comm_monoid",
"add_tsub_cancel_left"
] | A `canonically_ordered_add_monoid` with ordered subtraction and order-reflecting addition is
cancellative. This is not an instance at it would form a typeclass loop.
See note [reducible non-instances]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tsub_pos_iff_lt : 0 < a - b ↔ b < a | by rw [tsub_pos_iff_not_le, not_le] | lemma | tsub_pos_iff_lt | algebra.order.sub | src/algebra/order/sub/canonical.lean | [
"algebra.order.monoid.canonical.defs",
"algebra.order.sub.defs"
] | [
"tsub_pos_iff_not_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsub_eq_tsub_min (a b : α) : a - b = a - min a b | begin
cases le_total a b with h h,
{ rw [min_eq_left h, tsub_self, tsub_eq_zero_of_le h] },
{ rw [min_eq_right h] },
end | lemma | tsub_eq_tsub_min | algebra.order.sub | src/algebra/order/sub/canonical.lean | [
"algebra.order.monoid.canonical.defs",
"algebra.order.sub.defs"
] | [
"tsub_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_tsub_iff_right (hc : add_le_cancellable c) : a < b - c ↔ a + c < b | ⟨lt_imp_lt_of_le_imp_le tsub_le_iff_right.mpr, hc.lt_tsub_of_add_lt_right⟩ | lemma | add_le_cancellable.lt_tsub_iff_right | algebra.order.sub | src/algebra/order/sub/canonical.lean | [
"algebra.order.monoid.canonical.defs",
"algebra.order.sub.defs"
] | [
"lt_tsub_iff_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_tsub_iff_left (hc : add_le_cancellable c) : a < b - c ↔ c + a < b | ⟨lt_imp_lt_of_le_imp_le tsub_le_iff_left.mpr, hc.lt_tsub_of_add_lt_left⟩ | lemma | add_le_cancellable.lt_tsub_iff_left | algebra.order.sub | src/algebra/order/sub/canonical.lean | [
"algebra.order.monoid.canonical.defs",
"algebra.order.sub.defs"
] | [
"lt_tsub_iff_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsub_lt_tsub_iff_right (hc : add_le_cancellable c) (h : c ≤ a) :
a - c < b - c ↔ a < b | by rw [hc.lt_tsub_iff_left, add_tsub_cancel_of_le h] | lemma | add_le_cancellable.tsub_lt_tsub_iff_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_lt_tsub_iff_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsub_lt_self (ha : add_le_cancellable a) (h₁ : 0 < a) (h₂ : 0 < b) : a - b < a | begin
refine tsub_le_self.lt_of_ne (λ h, _),
rw [← h, tsub_pos_iff_lt] at h₁,
exact h₂.not_le (ha.add_le_iff_nonpos_left.1 $ add_le_of_le_tsub_left_of_le h₁.le h.ge),
end | lemma | add_le_cancellable.tsub_lt_self | algebra.order.sub | src/algebra/order/sub/canonical.lean | [
"algebra.order.monoid.canonical.defs",
"algebra.order.sub.defs"
] | [
"add_le_of_le_tsub_left_of_le",
"tsub_lt_self",
"tsub_pos_iff_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsub_lt_self_iff (ha : add_le_cancellable a) : a - b < a ↔ 0 < a ∧ 0 < b | begin
refine ⟨λ h, ⟨(zero_le _).trans_lt h, (zero_le b).lt_of_ne _⟩, λ h, ha.tsub_lt_self h.1 h.2⟩,
rintro rfl,
rw [tsub_zero] at h,
exact h.false
end | lemma | add_le_cancellable.tsub_lt_self_iff | algebra.order.sub | src/algebra/order/sub/canonical.lean | [
"algebra.order.monoid.canonical.defs",
"algebra.order.sub.defs"
] | [
"tsub_lt_self_iff",
"tsub_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsub_lt_tsub_iff_left_of_le (ha : add_le_cancellable a) (hb : add_le_cancellable b)
(h : b ≤ a) : a - b < a - c ↔ c < b | lt_iff_lt_of_le_iff_le $ ha.tsub_le_tsub_iff_left hb h | lemma | add_le_cancellable.tsub_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_iff_lt_of_le_iff_le",
"tsub_lt_tsub_iff_left_of_le"
] | See `lt_tsub_iff_left_of_le_of_le` for a weaker statement in a partial order. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tsub_lt_tsub_iff_right (h : c ≤ a) : a - c < b - c ↔ a < b | contravariant.add_le_cancellable.tsub_lt_tsub_iff_right h | lemma | tsub_lt_tsub_iff_right | algebra.order.sub | src/algebra/order/sub/canonical.lean | [
"algebra.order.monoid.canonical.defs",
"algebra.order.sub.defs"
] | [] | This lemma also holds for `ennreal`, but we need a different proof for that. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tsub_lt_self : 0 < a → 0 < b → a - b < a | contravariant.add_le_cancellable.tsub_lt_self | lemma | tsub_lt_self | 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_self_iff : a - b < a ↔ 0 < a ∧ 0 < b | contravariant.add_le_cancellable.tsub_lt_self_iff | lemma | tsub_lt_self_iff | 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 (h : b ≤ a) : a - b < a - c ↔ c < b | contravariant.add_le_cancellable.tsub_lt_tsub_iff_left_of_le contravariant.add_le_cancellable h | lemma | tsub_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_of_le_of_le` for a weaker statement in a partial order. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tsub_add_eq_max : a - b + b = max a b | begin
cases le_total a b with h h,
{ rw [max_eq_right h, tsub_eq_zero_of_le h, zero_add] },
{ rw [max_eq_left h, tsub_add_cancel_of_le h] }
end | lemma | tsub_add_eq_max | 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_tsub_eq_max : a + (b - a) = max a b | by rw [add_comm, max_comm, tsub_add_eq_max] | lemma | add_tsub_eq_max | algebra.order.sub | src/algebra/order/sub/canonical.lean | [
"algebra.order.monoid.canonical.defs",
"algebra.order.sub.defs"
] | [
"tsub_add_eq_max"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsub_min : a - min a b = a - b | begin
cases le_total a b with h h,
{ rw [min_eq_left h, tsub_self, tsub_eq_zero_of_le h] },
{ rw [min_eq_right h] }
end | lemma | tsub_min | algebra.order.sub | src/algebra/order/sub/canonical.lean | [
"algebra.order.monoid.canonical.defs",
"algebra.order.sub.defs"
] | [
"tsub_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsub_add_min : a - b + min a b = a | by { rw [← tsub_min, tsub_add_cancel_of_le], apply min_le_left } | lemma | tsub_add_min | algebra.order.sub | src/algebra/order/sub/canonical.lean | [
"algebra.order.monoid.canonical.defs",
"algebra.order.sub.defs"
] | [
"tsub_add_cancel_of_le",
"tsub_min"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_ordered_sub (α : Type*) [has_le α] [has_add α] [has_sub α] : Prop | (tsub_le_iff_right : ∀ a b c : α, a - b ≤ c ↔ a ≤ c + b) | class | has_ordered_sub | algebra.order.sub | src/algebra/order/sub/defs.lean | [
"algebra.covariant_and_contravariant",
"algebra.group.basic",
"algebra.order.monoid.lemmas",
"order.lattice"
] | [
"tsub_le_iff_right"
] | `has_ordered_sub α` means that `α` has a subtraction characterized by `a - b ≤ c ↔ a ≤ c + b`.
In other words, `a - b` is the least `c` such that `a ≤ b + c`.
This is satisfied both by the subtraction in additive ordered groups and by truncated subtraction
in canonically ordered monoids on many specific types. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tsub_le_iff_right : a - b ≤ c ↔ a ≤ c + b | has_ordered_sub.tsub_le_iff_right a b c | lemma | tsub_le_iff_right | algebra.order.sub | src/algebra/order/sub/defs.lean | [
"algebra.covariant_and_contravariant",
"algebra.group.basic",
"algebra.order.monoid.lemmas",
"order.lattice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_tsub_le_right : a + b - b ≤ a | tsub_le_iff_right.mpr le_rfl | lemma | add_tsub_le_right | algebra.order.sub | src/algebra/order/sub/defs.lean | [
"algebra.covariant_and_contravariant",
"algebra.group.basic",
"algebra.order.monoid.lemmas",
"order.lattice"
] | [
"le_rfl"
] | See `add_tsub_cancel_right` for the equality if `contravariant_class α α (+) (≤)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
le_tsub_add : b ≤ (b - a) + a | tsub_le_iff_right.mp le_rfl | lemma | le_tsub_add | algebra.order.sub | src/algebra/order/sub/defs.lean | [
"algebra.covariant_and_contravariant",
"algebra.group.basic",
"algebra.order.monoid.lemmas",
"order.lattice"
] | [
"le_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsub_le_iff_left : a - b ≤ c ↔ a ≤ b + c | by rw [tsub_le_iff_right, add_comm] | lemma | tsub_le_iff_left | algebra.order.sub | src/algebra/order/sub/defs.lean | [
"algebra.covariant_and_contravariant",
"algebra.group.basic",
"algebra.order.monoid.lemmas",
"order.lattice"
] | [
"tsub_le_iff_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_add_tsub : a ≤ b + (a - b) | tsub_le_iff_left.mp le_rfl | lemma | le_add_tsub | algebra.order.sub | src/algebra/order/sub/defs.lean | [
"algebra.covariant_and_contravariant",
"algebra.group.basic",
"algebra.order.monoid.lemmas",
"order.lattice"
] | [
"le_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_tsub_le_left : a + b - a ≤ b | tsub_le_iff_left.mpr le_rfl | lemma | add_tsub_le_left | algebra.order.sub | src/algebra/order/sub/defs.lean | [
"algebra.covariant_and_contravariant",
"algebra.group.basic",
"algebra.order.monoid.lemmas",
"order.lattice"
] | [
"le_rfl"
] | See `add_tsub_cancel_left` for the equality if `contravariant_class α α (+) (≤)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tsub_le_tsub_right (h : a ≤ b) (c : α) : a - c ≤ b - c | tsub_le_iff_left.mpr $ h.trans le_add_tsub | lemma | tsub_le_tsub_right | algebra.order.sub | src/algebra/order/sub/defs.lean | [
"algebra.covariant_and_contravariant",
"algebra.group.basic",
"algebra.order.monoid.lemmas",
"order.lattice"
] | [
"le_add_tsub"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsub_le_iff_tsub_le : a - b ≤ c ↔ a - c ≤ b | by rw [tsub_le_iff_left, tsub_le_iff_right] | lemma | tsub_le_iff_tsub_le | algebra.order.sub | src/algebra/order/sub/defs.lean | [
"algebra.covariant_and_contravariant",
"algebra.group.basic",
"algebra.order.monoid.lemmas",
"order.lattice"
] | [
"tsub_le_iff_left",
"tsub_le_iff_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsub_tsub_le : b - (b - a) ≤ a | tsub_le_iff_right.mpr le_add_tsub | lemma | tsub_tsub_le | algebra.order.sub | src/algebra/order/sub/defs.lean | [
"algebra.covariant_and_contravariant",
"algebra.group.basic",
"algebra.order.monoid.lemmas",
"order.lattice"
] | [
"le_add_tsub"
] | See `tsub_tsub_cancel_of_le` for the equality. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tsub_le_tsub_left (h : a ≤ b) (c : α) : c - b ≤ c - a | tsub_le_iff_left.mpr $ le_add_tsub.trans $ add_le_add_right h _ | lemma | tsub_le_tsub_left | algebra.order.sub | src/algebra/order/sub/defs.lean | [
"algebra.covariant_and_contravariant",
"algebra.group.basic",
"algebra.order.monoid.lemmas",
"order.lattice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsub_le_tsub (hab : a ≤ b) (hcd : c ≤ d) : a - d ≤ b - c | (tsub_le_tsub_right hab _).trans $ tsub_le_tsub_left hcd _ | lemma | tsub_le_tsub | algebra.order.sub | src/algebra/order/sub/defs.lean | [
"algebra.covariant_and_contravariant",
"algebra.group.basic",
"algebra.order.monoid.lemmas",
"order.lattice"
] | [
"tsub_le_tsub_left",
"tsub_le_tsub_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antitone_const_tsub : antitone (λ x, c - x) | λ x y hxy, tsub_le_tsub rfl.le hxy | lemma | antitone_const_tsub | algebra.order.sub | src/algebra/order/sub/defs.lean | [
"algebra.covariant_and_contravariant",
"algebra.group.basic",
"algebra.order.monoid.lemmas",
"order.lattice"
] | [
"antitone",
"tsub_le_tsub"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_tsub_le_assoc : a + b - c ≤ a + (b - c) | by { rw [tsub_le_iff_left, add_left_comm], exact add_le_add_left le_add_tsub a } | lemma | add_tsub_le_assoc | algebra.order.sub | src/algebra/order/sub/defs.lean | [
"algebra.covariant_and_contravariant",
"algebra.group.basic",
"algebra.order.monoid.lemmas",
"order.lattice"
] | [
"le_add_tsub",
"tsub_le_iff_left"
] | See `add_tsub_assoc_of_le` for the equality. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add_tsub_le_tsub_add : a + b - c ≤ a - c + b | by { rw [add_comm, add_comm _ b], exact add_tsub_le_assoc } | lemma | add_tsub_le_tsub_add | algebra.order.sub | src/algebra/order/sub/defs.lean | [
"algebra.covariant_and_contravariant",
"algebra.group.basic",
"algebra.order.monoid.lemmas",
"order.lattice"
] | [
"add_tsub_le_assoc"
] | See `tsub_add_eq_add_tsub` for the equality. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add_le_add_add_tsub : a + b ≤ (a + c) + (b - c) | by { rw [add_assoc], exact add_le_add_left le_add_tsub a } | lemma | add_le_add_add_tsub | algebra.order.sub | src/algebra/order/sub/defs.lean | [
"algebra.covariant_and_contravariant",
"algebra.group.basic",
"algebra.order.monoid.lemmas",
"order.lattice"
] | [
"le_add_tsub"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_tsub_add_add : a + b ≤ (a - c) + (b + c) | by { rw [add_comm a, add_comm (a - c)], exact add_le_add_add_tsub } | lemma | le_tsub_add_add | algebra.order.sub | src/algebra/order/sub/defs.lean | [
"algebra.covariant_and_contravariant",
"algebra.group.basic",
"algebra.order.monoid.lemmas",
"order.lattice"
] | [
"add_le_add_add_tsub"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsub_le_tsub_add_tsub : a - c ≤ (a - b) + (b - c) | begin
rw [tsub_le_iff_left, ← add_assoc, add_right_comm],
exact le_add_tsub.trans (add_le_add_right le_add_tsub _),
end | lemma | tsub_le_tsub_add_tsub | algebra.order.sub | src/algebra/order/sub/defs.lean | [
"algebra.covariant_and_contravariant",
"algebra.group.basic",
"algebra.order.monoid.lemmas",
"order.lattice"
] | [
"le_add_tsub",
"tsub_le_iff_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsub_tsub_tsub_le_tsub : (c - a) - (c - b) ≤ b - a | begin
rw [tsub_le_iff_left, tsub_le_iff_left, add_left_comm],
exact le_tsub_add.trans (add_le_add_left le_add_tsub _),
end | lemma | tsub_tsub_tsub_le_tsub | algebra.order.sub | src/algebra/order/sub/defs.lean | [
"algebra.covariant_and_contravariant",
"algebra.group.basic",
"algebra.order.monoid.lemmas",
"order.lattice"
] | [
"le_add_tsub",
"tsub_le_iff_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsub_tsub_le_tsub_add {a b c : α} : a - (b - c) ≤ a - b + c | tsub_le_iff_right.2 $ calc
a ≤ a - b + b : le_tsub_add
... ≤ a - b + (c + (b - c)) : add_le_add_left le_add_tsub _
... = a - b + c + (b - c) : (add_assoc _ _ _).symm | lemma | tsub_tsub_le_tsub_add | algebra.order.sub | src/algebra/order/sub/defs.lean | [
"algebra.covariant_and_contravariant",
"algebra.group.basic",
"algebra.order.monoid.lemmas",
"order.lattice"
] | [
"le_add_tsub",
"le_tsub_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_tsub_add_le_tsub_add_tsub : a + b - (c + d) ≤ a - c + (b - d) | begin
rw [add_comm c, tsub_le_iff_left, add_assoc, ←tsub_le_iff_left, ←tsub_le_iff_left],
refine (tsub_le_tsub_right add_tsub_le_assoc c).trans _,
rw [add_comm a, add_comm (a - c)],
exact add_tsub_le_assoc,
end | lemma | add_tsub_add_le_tsub_add_tsub | algebra.order.sub | src/algebra/order/sub/defs.lean | [
"algebra.covariant_and_contravariant",
"algebra.group.basic",
"algebra.order.monoid.lemmas",
"order.lattice"
] | [
"add_tsub_le_assoc",
"tsub_le_iff_left",
"tsub_le_tsub_right"
] | See `tsub_add_tsub_comm` for the equality. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add_tsub_add_le_tsub_left : a + b - (a + c) ≤ b - c | by { rw [tsub_le_iff_left, add_assoc], exact add_le_add_left le_add_tsub _ } | lemma | add_tsub_add_le_tsub_left | algebra.order.sub | src/algebra/order/sub/defs.lean | [
"algebra.covariant_and_contravariant",
"algebra.group.basic",
"algebra.order.monoid.lemmas",
"order.lattice"
] | [
"le_add_tsub",
"tsub_le_iff_left"
] | See `add_tsub_add_eq_tsub_left` for the equality. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add_tsub_add_le_tsub_right : a + c - (b + c) ≤ a - b | by { rw [tsub_le_iff_left, add_right_comm], exact add_le_add_right le_add_tsub c } | lemma | add_tsub_add_le_tsub_right | algebra.order.sub | src/algebra/order/sub/defs.lean | [
"algebra.covariant_and_contravariant",
"algebra.group.basic",
"algebra.order.monoid.lemmas",
"order.lattice"
] | [
"le_add_tsub",
"tsub_le_iff_left"
] | See `add_tsub_add_eq_tsub_right` for the equality. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
le_add_tsub_swap (hb : add_le_cancellable b) : a ≤ b + a - b | hb le_add_tsub | lemma | add_le_cancellable.le_add_tsub_swap | algebra.order.sub | src/algebra/order/sub/defs.lean | [
"algebra.covariant_and_contravariant",
"algebra.group.basic",
"algebra.order.monoid.lemmas",
"order.lattice"
] | [
"le_add_tsub",
"le_add_tsub_swap"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_add_tsub (hb : add_le_cancellable b) : a ≤ a + b - b | by { rw add_comm, exact hb.le_add_tsub_swap } | lemma | add_le_cancellable.le_add_tsub | algebra.order.sub | src/algebra/order/sub/defs.lean | [
"algebra.covariant_and_contravariant",
"algebra.group.basic",
"algebra.order.monoid.lemmas",
"order.lattice"
] | [
"le_add_tsub"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_tsub_of_add_le_left (ha : add_le_cancellable a) (h : a + b ≤ c) : b ≤ c - a | ha $ h.trans le_add_tsub | lemma | add_le_cancellable.le_tsub_of_add_le_left | algebra.order.sub | src/algebra/order/sub/defs.lean | [
"algebra.covariant_and_contravariant",
"algebra.group.basic",
"algebra.order.monoid.lemmas",
"order.lattice"
] | [
"le_add_tsub",
"le_tsub_of_add_le_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_tsub_of_add_le_right (hb : add_le_cancellable b) (h : a + b ≤ c) : a ≤ c - b | hb.le_tsub_of_add_le_left $ by rwa add_comm | lemma | add_le_cancellable.le_tsub_of_add_le_right | algebra.order.sub | src/algebra/order/sub/defs.lean | [
"algebra.covariant_and_contravariant",
"algebra.group.basic",
"algebra.order.monoid.lemmas",
"order.lattice"
] | [
"le_tsub_of_add_le_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_add_tsub_swap : a ≤ b + a - b | contravariant.add_le_cancellable.le_add_tsub_swap | lemma | le_add_tsub_swap | algebra.order.sub | src/algebra/order/sub/defs.lean | [
"algebra.covariant_and_contravariant",
"algebra.group.basic",
"algebra.order.monoid.lemmas",
"order.lattice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_add_tsub' : a ≤ a + b - b | contravariant.add_le_cancellable.le_add_tsub | lemma | le_add_tsub' | algebra.order.sub | src/algebra/order/sub/defs.lean | [
"algebra.covariant_and_contravariant",
"algebra.group.basic",
"algebra.order.monoid.lemmas",
"order.lattice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_tsub_of_add_le_left (h : a + b ≤ c) : b ≤ c - a | contravariant.add_le_cancellable.le_tsub_of_add_le_left h | lemma | le_tsub_of_add_le_left | algebra.order.sub | src/algebra/order/sub/defs.lean | [
"algebra.covariant_and_contravariant",
"algebra.group.basic",
"algebra.order.monoid.lemmas",
"order.lattice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_tsub_of_add_le_right (h : a + b ≤ c) : a ≤ c - b | contravariant.add_le_cancellable.le_tsub_of_add_le_right h | lemma | le_tsub_of_add_le_right | algebra.order.sub | src/algebra/order/sub/defs.lean | [
"algebra.covariant_and_contravariant",
"algebra.group.basic",
"algebra.order.monoid.lemmas",
"order.lattice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsub_nonpos : a - b ≤ 0 ↔ a ≤ b | by rw [tsub_le_iff_left, add_zero] | lemma | tsub_nonpos | algebra.order.sub | src/algebra/order/sub/defs.lean | [
"algebra.covariant_and_contravariant",
"algebra.group.basic",
"algebra.order.monoid.lemmas",
"order.lattice"
] | [
"tsub_le_iff_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsub_tsub (b a c : α) : b - a - c = b - (a + c) | begin
apply le_antisymm,
{ rw [tsub_le_iff_left, tsub_le_iff_left, ← add_assoc, ← tsub_le_iff_left] },
{ rw [tsub_le_iff_left, add_assoc, ← tsub_le_iff_left, ← tsub_le_iff_left] }
end | lemma | tsub_tsub | algebra.order.sub | src/algebra/order/sub/defs.lean | [
"algebra.covariant_and_contravariant",
"algebra.group.basic",
"algebra.order.monoid.lemmas",
"order.lattice"
] | [
"tsub_le_iff_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsub_add_eq_tsub_tsub (a b c : α) : a - (b + c) = a - b - c | (tsub_tsub _ _ _).symm | lemma | tsub_add_eq_tsub_tsub | algebra.order.sub | src/algebra/order/sub/defs.lean | [
"algebra.covariant_and_contravariant",
"algebra.group.basic",
"algebra.order.monoid.lemmas",
"order.lattice"
] | [
"tsub_tsub"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsub_add_eq_tsub_tsub_swap (a b c : α) : a - (b + c) = a - c - b | by { rw [add_comm], apply tsub_add_eq_tsub_tsub } | lemma | tsub_add_eq_tsub_tsub_swap | algebra.order.sub | src/algebra/order/sub/defs.lean | [
"algebra.covariant_and_contravariant",
"algebra.group.basic",
"algebra.order.monoid.lemmas",
"order.lattice"
] | [
"tsub_add_eq_tsub_tsub"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsub_right_comm : a - b - c = a - c - b | by simp_rw [← tsub_add_eq_tsub_tsub, add_comm] | lemma | tsub_right_comm | algebra.order.sub | src/algebra/order/sub/defs.lean | [
"algebra.covariant_and_contravariant",
"algebra.group.basic",
"algebra.order.monoid.lemmas",
"order.lattice"
] | [
"tsub_add_eq_tsub_tsub"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsub_eq_of_eq_add (hb : add_le_cancellable b) (h : a = c + b) : a - b = c | le_antisymm (tsub_le_iff_right.mpr h.le) $
by { rw h, exact hb.le_add_tsub } | lemma | add_le_cancellable.tsub_eq_of_eq_add | algebra.order.sub | src/algebra/order/sub/defs.lean | [
"algebra.covariant_and_contravariant",
"algebra.group.basic",
"algebra.order.monoid.lemmas",
"order.lattice"
] | [
"tsub_eq_of_eq_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_tsub_of_add_eq (hc : add_le_cancellable c) (h : a + c = b) : a = b - c | (hc.tsub_eq_of_eq_add h.symm).symm | lemma | add_le_cancellable.eq_tsub_of_add_eq | algebra.order.sub | src/algebra/order/sub/defs.lean | [
"algebra.covariant_and_contravariant",
"algebra.group.basic",
"algebra.order.monoid.lemmas",
"order.lattice"
] | [
"eq_tsub_of_add_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsub_eq_of_eq_add_rev (hb : add_le_cancellable b) (h : a = b + c) : a - b = c | hb.tsub_eq_of_eq_add $ by rw [add_comm, h] | theorem | add_le_cancellable.tsub_eq_of_eq_add_rev | algebra.order.sub | src/algebra/order/sub/defs.lean | [
"algebra.covariant_and_contravariant",
"algebra.group.basic",
"algebra.order.monoid.lemmas",
"order.lattice"
] | [
"tsub_eq_of_eq_add_rev"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_tsub_cancel_right (hb : add_le_cancellable b) : a + b - b = a | hb.tsub_eq_of_eq_add $ by rw [add_comm] | lemma | add_le_cancellable.add_tsub_cancel_right | algebra.order.sub | src/algebra/order/sub/defs.lean | [
"algebra.covariant_and_contravariant",
"algebra.group.basic",
"algebra.order.monoid.lemmas",
"order.lattice"
] | [
"add_tsub_cancel_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_tsub_cancel_left (ha : add_le_cancellable a) : a + b - a = b | ha.tsub_eq_of_eq_add $ add_comm a b | lemma | add_le_cancellable.add_tsub_cancel_left | algebra.order.sub | src/algebra/order/sub/defs.lean | [
"algebra.covariant_and_contravariant",
"algebra.group.basic",
"algebra.order.monoid.lemmas",
"order.lattice"
] | [
"add_tsub_cancel_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_add_of_tsub_lt_left (hb : add_le_cancellable b) (h : a - b < c) : a < b + c | begin
rw [lt_iff_le_and_ne, ← tsub_le_iff_left],
refine ⟨h.le, _⟩,
rintro rfl,
simpa [hb] using h,
end | lemma | add_le_cancellable.lt_add_of_tsub_lt_left | algebra.order.sub | src/algebra/order/sub/defs.lean | [
"algebra.covariant_and_contravariant",
"algebra.group.basic",
"algebra.order.monoid.lemmas",
"order.lattice"
] | [
"lt_add_of_tsub_lt_left",
"lt_iff_le_and_ne",
"tsub_le_iff_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_add_of_tsub_lt_right (hc : add_le_cancellable c) (h : a - c < b) : a < b + c | begin
rw [lt_iff_le_and_ne, ← tsub_le_iff_right],
refine ⟨h.le, _⟩,
rintro rfl,
simpa [hc] using h,
end | lemma | add_le_cancellable.lt_add_of_tsub_lt_right | algebra.order.sub | src/algebra/order/sub/defs.lean | [
"algebra.covariant_and_contravariant",
"algebra.group.basic",
"algebra.order.monoid.lemmas",
"order.lattice"
] | [
"lt_add_of_tsub_lt_right",
"lt_iff_le_and_ne",
"tsub_le_iff_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_tsub_of_add_lt_right (hc : add_le_cancellable c) (h : a + c < b) : a < b - c | (hc.le_tsub_of_add_le_right h.le).lt_of_ne $ by { rintro rfl, exact h.not_le le_tsub_add } | lemma | add_le_cancellable.lt_tsub_of_add_lt_right | algebra.order.sub | src/algebra/order/sub/defs.lean | [
"algebra.covariant_and_contravariant",
"algebra.group.basic",
"algebra.order.monoid.lemmas",
"order.lattice"
] | [
"le_tsub_add",
"lt_tsub_of_add_lt_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_tsub_of_add_lt_left (ha : add_le_cancellable a) (h : a + c < b) : c < b - a | ha.lt_tsub_of_add_lt_right $ by rwa add_comm | lemma | add_le_cancellable.lt_tsub_of_add_lt_left | algebra.order.sub | src/algebra/order/sub/defs.lean | [
"algebra.covariant_and_contravariant",
"algebra.group.basic",
"algebra.order.monoid.lemmas",
"order.lattice"
] | [
"lt_tsub_of_add_lt_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsub_eq_of_eq_add (h : a = c + b) : a - b = c | contravariant.add_le_cancellable.tsub_eq_of_eq_add h | lemma | tsub_eq_of_eq_add | algebra.order.sub | src/algebra/order/sub/defs.lean | [
"algebra.covariant_and_contravariant",
"algebra.group.basic",
"algebra.order.monoid.lemmas",
"order.lattice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_tsub_of_add_eq (h : a + c = b) : a = b - c | contravariant.add_le_cancellable.eq_tsub_of_add_eq h | lemma | eq_tsub_of_add_eq | algebra.order.sub | src/algebra/order/sub/defs.lean | [
"algebra.covariant_and_contravariant",
"algebra.group.basic",
"algebra.order.monoid.lemmas",
"order.lattice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsub_eq_of_eq_add_rev (h : a = b + c) : a - b = c | contravariant.add_le_cancellable.tsub_eq_of_eq_add_rev h | lemma | tsub_eq_of_eq_add_rev | algebra.order.sub | src/algebra/order/sub/defs.lean | [
"algebra.covariant_and_contravariant",
"algebra.group.basic",
"algebra.order.monoid.lemmas",
"order.lattice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_tsub_cancel_right (a b : α) : a + b - b = a | contravariant.add_le_cancellable.add_tsub_cancel_right | lemma | add_tsub_cancel_right | algebra.order.sub | src/algebra/order/sub/defs.lean | [
"algebra.covariant_and_contravariant",
"algebra.group.basic",
"algebra.order.monoid.lemmas",
"order.lattice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_tsub_cancel_left (a b : α) : a + b - a = b | contravariant.add_le_cancellable.add_tsub_cancel_left | lemma | add_tsub_cancel_left | algebra.order.sub | src/algebra/order/sub/defs.lean | [
"algebra.covariant_and_contravariant",
"algebra.group.basic",
"algebra.order.monoid.lemmas",
"order.lattice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_add_of_tsub_lt_left (h : a - b < c) : a < b + c | contravariant.add_le_cancellable.lt_add_of_tsub_lt_left h | lemma | lt_add_of_tsub_lt_left | algebra.order.sub | src/algebra/order/sub/defs.lean | [
"algebra.covariant_and_contravariant",
"algebra.group.basic",
"algebra.order.monoid.lemmas",
"order.lattice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_add_of_tsub_lt_right (h : a - c < b) : a < b + c | contravariant.add_le_cancellable.lt_add_of_tsub_lt_right h | lemma | lt_add_of_tsub_lt_right | algebra.order.sub | src/algebra/order/sub/defs.lean | [
"algebra.covariant_and_contravariant",
"algebra.group.basic",
"algebra.order.monoid.lemmas",
"order.lattice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_tsub_of_add_lt_left : a + c < b → c < b - a | contravariant.add_le_cancellable.lt_tsub_of_add_lt_left | lemma | lt_tsub_of_add_lt_left | algebra.order.sub | src/algebra/order/sub/defs.lean | [
"algebra.covariant_and_contravariant",
"algebra.group.basic",
"algebra.order.monoid.lemmas",
"order.lattice"
] | [] | This lemma (and some of its corollaries) also holds for `ennreal`, but this proof doesn't work
for it. Maybe we should add this lemma as field to `has_ordered_sub`? | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lt_tsub_of_add_lt_right : a + c < b → a < b - c | contravariant.add_le_cancellable.lt_tsub_of_add_lt_right | lemma | lt_tsub_of_add_lt_right | algebra.order.sub | src/algebra/order/sub/defs.lean | [
"algebra.covariant_and_contravariant",
"algebra.group.basic",
"algebra.order.monoid.lemmas",
"order.lattice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_tsub_add_eq_tsub_right (a c b : α) : (a + c) - (b + c) = a - b | begin
refine add_tsub_add_le_tsub_right.antisymm (tsub_le_iff_right.2 $ le_of_add_le_add_right _), swap,
rw add_assoc,
exact le_tsub_add,
end | lemma | add_tsub_add_eq_tsub_right | algebra.order.sub | src/algebra/order/sub/defs.lean | [
"algebra.covariant_and_contravariant",
"algebra.group.basic",
"algebra.order.monoid.lemmas",
"order.lattice"
] | [
"le_tsub_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_tsub_add_eq_tsub_left (a b c : α) : (a + b) - (a + c) = b - c | by rw [add_comm a b, add_comm a c, add_tsub_add_eq_tsub_right] | lemma | add_tsub_add_eq_tsub_left | algebra.order.sub | src/algebra/order/sub/defs.lean | [
"algebra.covariant_and_contravariant",
"algebra.group.basic",
"algebra.order.monoid.lemmas",
"order.lattice"
] | [
"add_tsub_add_eq_tsub_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_of_tsub_lt_tsub_right (h : a - c < b - c) : a < b | lt_imp_lt_of_le_imp_le (λ h, tsub_le_tsub_right h c) h | lemma | lt_of_tsub_lt_tsub_right | algebra.order.sub | src/algebra/order/sub/defs.lean | [
"algebra.covariant_and_contravariant",
"algebra.group.basic",
"algebra.order.monoid.lemmas",
"order.lattice"
] | [
"lt_imp_lt_of_le_imp_le",
"tsub_le_tsub_right"
] | See `lt_of_tsub_lt_tsub_right_of_le` for a weaker statement in a partial order. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lt_tsub_iff_right : a < b - c ↔ a + c < b | lt_iff_lt_of_le_iff_le tsub_le_iff_right | lemma | lt_tsub_iff_right | algebra.order.sub | src/algebra/order/sub/defs.lean | [
"algebra.covariant_and_contravariant",
"algebra.group.basic",
"algebra.order.monoid.lemmas",
"order.lattice"
] | [
"lt_iff_lt_of_le_iff_le",
"tsub_le_iff_right"
] | See `lt_tsub_iff_right_of_le` for a weaker statement in a partial order. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lt_tsub_iff_left : a < b - c ↔ c + a < b | lt_iff_lt_of_le_iff_le tsub_le_iff_left | lemma | lt_tsub_iff_left | algebra.order.sub | src/algebra/order/sub/defs.lean | [
"algebra.covariant_and_contravariant",
"algebra.group.basic",
"algebra.order.monoid.lemmas",
"order.lattice"
] | [
"lt_iff_lt_of_le_iff_le",
"tsub_le_iff_left"
] | See `lt_tsub_iff_left_of_le` for a weaker statement in a partial order. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lt_tsub_comm : a < b - c ↔ c < b - a | lt_tsub_iff_left.trans lt_tsub_iff_right.symm | lemma | lt_tsub_comm | algebra.order.sub | src/algebra/order/sub/defs.lean | [
"algebra.covariant_and_contravariant",
"algebra.group.basic",
"algebra.order.monoid.lemmas",
"order.lattice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_of_tsub_lt_tsub_left (h : a - b < a - c) : c < b | lt_imp_lt_of_le_imp_le (λ h, tsub_le_tsub_left h a) h | lemma | lt_of_tsub_lt_tsub_left | algebra.order.sub | src/algebra/order/sub/defs.lean | [
"algebra.covariant_and_contravariant",
"algebra.group.basic",
"algebra.order.monoid.lemmas",
"order.lattice"
] | [
"lt_imp_lt_of_le_imp_le",
"tsub_le_tsub_left"
] | See `lt_of_tsub_lt_tsub_left_of_le` for a weaker statement in a partial order. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tsub_zero (a : α) : a - 0 = a | add_le_cancellable.tsub_eq_of_eq_add add_le_cancellable_zero (add_zero _).symm | lemma | tsub_zero | algebra.order.sub | src/algebra/order/sub/defs.lean | [
"algebra.covariant_and_contravariant",
"algebra.group.basic",
"algebra.order.monoid.lemmas",
"order.lattice"
] | [
"add_le_cancellable.tsub_eq_of_eq_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub : Π (a b : with_top α), with_top α | | _ ⊤ := 0
| ⊤ (x : α) := ⊤
| (x : α) (y : α) := (x - y : α) | def | with_top.sub | algebra.order.sub | src/algebra/order/sub/with_top.lean | [
"algebra.order.sub.defs",
"algebra.order.monoid.with_top"
] | [
"with_top"
] | If `α` has subtraction and `0`, we can extend the subtraction to `with_top α`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_sub {a b : α} : (↑(a - b) : with_top α) = ↑a - ↑b | rfl | lemma | with_top.coe_sub | algebra.order.sub | src/algebra/order/sub/with_top.lean | [
"algebra.order.sub.defs",
"algebra.order.monoid.with_top"
] | [
"with_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
top_sub_coe {a : α} : (⊤ : with_top α) - a = ⊤ | rfl | lemma | with_top.top_sub_coe | algebra.order.sub | src/algebra/order/sub/with_top.lean | [
"algebra.order.sub.defs",
"algebra.order.monoid.with_top"
] | [
"with_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_top {a : with_top α} : a - ⊤ = 0 | by { cases a; refl } | lemma | with_top.sub_top | algebra.order.sub | src/algebra/order/sub/with_top.lean | [
"algebra.order.sub.defs",
"algebra.order.monoid.with_top"
] | [
"with_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_eq_top_iff : ∀ {a b : with_top α}, a - b = ⊤ ↔ a = ⊤ ∧ b ≠ ⊤ | | _ ⊤ := by simp
| ⊤ (b : α) := by simp
| (a : α) (b : α) := by simp only [← coe_sub, coe_ne_top, false_and] | theorem | with_top.sub_eq_top_iff | algebra.order.sub | src/algebra/order/sub/with_top.lean | [
"algebra.order.sub.defs",
"algebra.order.monoid.with_top"
] | [
"with_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_sub [has_sub β] [has_zero β] {f : α → β} (h : ∀ x y, f (x - y) = f x - f y)
(h₀ : f 0 = 0) :
∀ x y : with_top α, (x - y).map f = x.map f - y.map f | | _ ⊤ := by simp only [h₀, sub_top, with_top.map_zero, coe_zero, map_top]
| ⊤ (x : α) := rfl
| (x : α) (y : α) := by simp only [← coe_sub, map_coe, h] | lemma | with_top.map_sub | algebra.order.sub | src/algebra/order/sub/with_top.lean | [
"algebra.order.sub.defs",
"algebra.order.monoid.with_top"
] | [
"with_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_degree_list_sum_le (l : list S[X]) :
nat_degree l.sum ≤ (l.map nat_degree).foldr max 0 | list.sum_le_foldr_max nat_degree (by simp) nat_degree_add_le _ | lemma | polynomial.nat_degree_list_sum_le | algebra.polynomial | src/algebra/polynomial/big_operators.lean | [
"algebra.order.with_zero",
"data.polynomial.monic"
] | [
"list.sum_le_foldr_max"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_degree_multiset_sum_le (l : multiset S[X]) :
nat_degree l.sum ≤ (l.map nat_degree).foldr max max_left_comm 0 | quotient.induction_on l (by simpa using nat_degree_list_sum_le) | lemma | polynomial.nat_degree_multiset_sum_le | algebra.polynomial | src/algebra/polynomial/big_operators.lean | [
"algebra.order.with_zero",
"data.polynomial.monic"
] | [
"multiset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_degree_sum_le (f : ι → S[X]) :
nat_degree (∑ i in s, f i) ≤ s.fold max 0 (nat_degree ∘ f) | by simpa using nat_degree_multiset_sum_le (s.val.map f) | lemma | polynomial.nat_degree_sum_le | algebra.polynomial | src/algebra/polynomial/big_operators.lean | [
"algebra.order.with_zero",
"data.polynomial.monic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
degree_list_sum_le (l : list S[X]) :
degree l.sum ≤ (l.map nat_degree).maximum | begin
by_cases h : l.sum = 0,
{ simp [h] },
{ rw degree_eq_nat_degree h,
suffices : (l.map nat_degree).maximum = ((l.map nat_degree).foldr max 0 : ℕ),
{ rw this,
simpa [this] using nat_degree_list_sum_le l },
rw ← list.foldr_max_of_ne_nil,
{ congr },
contrapose! h,
rw [list.map_eq_ni... | lemma | polynomial.degree_list_sum_le | algebra.polynomial | src/algebra/polynomial/big_operators.lean | [
"algebra.order.with_zero",
"data.polynomial.monic"
] | [
"list.foldr_max_of_ne_nil",
"list.map_eq_nil"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_degree_list_prod_le (l : list S[X]) :
nat_degree l.prod ≤ (l.map nat_degree).sum | begin
induction l with hd tl IH,
{ simp },
{ simpa using nat_degree_mul_le.trans (add_le_add_left IH _) }
end | lemma | polynomial.nat_degree_list_prod_le | algebra.polynomial | src/algebra/polynomial/big_operators.lean | [
"algebra.order.with_zero",
"data.polynomial.monic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
degree_list_prod_le (l : list S[X]) :
degree l.prod ≤ (l.map degree).sum | begin
induction l with hd tl IH,
{ simp },
{ simpa using (degree_mul_le _ _).trans (add_le_add_left IH _) }
end | lemma | polynomial.degree_list_prod_le | algebra.polynomial | src/algebra/polynomial/big_operators.lean | [
"algebra.order.with_zero",
"data.polynomial.monic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_list_prod_of_nat_degree_le (l : list S[X]) (n : ℕ)
(hl : ∀ p ∈ l, nat_degree p ≤ n) :
coeff (list.prod l) (l.length * n) = (l.map (λ p, coeff p n)).prod | begin
induction l with hd tl IH,
{ simp },
{ have hl' : ∀ (p ∈ tl), nat_degree p ≤ n := λ p hp, hl p (list.mem_cons_of_mem _ hp),
simp only [list.prod_cons, list.map, list.length],
rw [add_mul, one_mul, add_comm, ←IH hl', mul_comm tl.length],
have h : nat_degree tl.prod ≤ n * tl.length,
{ refine (... | lemma | polynomial.coeff_list_prod_of_nat_degree_le | algebra.polynomial | src/algebra/polynomial/big_operators.lean | [
"algebra.order.with_zero",
"data.polynomial.monic"
] | [
"list.prod",
"list.prod_cons",
"mul_comm",
"mul_zero",
"one_mul",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_degree_multiset_prod_le :
t.prod.nat_degree ≤ (t.map nat_degree).sum | quotient.induction_on t (by simpa using nat_degree_list_prod_le) | lemma | polynomial.nat_degree_multiset_prod_le | algebra.polynomial | src/algebra/polynomial/big_operators.lean | [
"algebra.order.with_zero",
"data.polynomial.monic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_degree_prod_le : (∏ i in s, f i).nat_degree ≤ ∑ i in s, (f i).nat_degree | by simpa using nat_degree_multiset_prod_le (s.1.map f) | lemma | polynomial.nat_degree_prod_le | algebra.polynomial | src/algebra/polynomial/big_operators.lean | [
"algebra.order.with_zero",
"data.polynomial.monic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
degree_multiset_prod_le :
t.prod.degree ≤ (t.map polynomial.degree).sum | quotient.induction_on t (by simpa using degree_list_prod_le) | lemma | polynomial.degree_multiset_prod_le | algebra.polynomial | src/algebra/polynomial/big_operators.lean | [
"algebra.order.with_zero",
"data.polynomial.monic"
] | [
"polynomial.degree"
] | The degree of a product of polynomials is at most the sum of the degrees,
where the degree of the zero polynomial is ⊥. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
degree_prod_le : (∏ i in s, f i).degree ≤ ∑ i in s, (f i).degree | by simpa only [multiset.map_map] using degree_multiset_prod_le (s.1.map f) | lemma | polynomial.degree_prod_le | algebra.polynomial | src/algebra/polynomial/big_operators.lean | [
"algebra.order.with_zero",
"data.polynomial.monic"
] | [
"multiset.map_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
leading_coeff_multiset_prod' (h : (t.map leading_coeff).prod ≠ 0) :
t.prod.leading_coeff = (t.map leading_coeff).prod | begin
induction t using multiset.induction_on with a t ih, { simp },
simp only [multiset.map_cons, multiset.prod_cons] at h ⊢,
rw polynomial.leading_coeff_mul'; { rwa ih, apply right_ne_zero_of_mul h }
end | lemma | polynomial.leading_coeff_multiset_prod' | algebra.polynomial | src/algebra/polynomial/big_operators.lean | [
"algebra.order.with_zero",
"data.polynomial.monic"
] | [
"ih",
"multiset.induction_on",
"multiset.map_cons",
"multiset.prod_cons",
"polynomial.leading_coeff_mul'",
"right_ne_zero_of_mul"
] | The leading coefficient of a product of polynomials is equal to
the product of the leading coefficients, provided that this product is nonzero.
See `polynomial.leading_coeff_multiset_prod` (without the `'`) for a version for integral domains,
where this condition is automatically satisfied. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
leading_coeff_prod' (h : ∏ i in s, (f i).leading_coeff ≠ 0) :
(∏ i in s, f i).leading_coeff = ∏ i in s, (f i).leading_coeff | by simpa using leading_coeff_multiset_prod' (s.1.map f) (by simpa using h) | lemma | polynomial.leading_coeff_prod' | algebra.polynomial | src/algebra/polynomial/big_operators.lean | [
"algebra.order.with_zero",
"data.polynomial.monic"
] | [] | The leading coefficient of a product of polynomials is equal to
the product of the leading coefficients, provided that this product is nonzero.
See `polynomial.leading_coeff_prod` (without the `'`) for a version for integral domains,
where this condition is automatically satisfied. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nat_degree_multiset_prod' (h : (t.map (λ f, leading_coeff f)).prod ≠ 0) :
t.prod.nat_degree = (t.map (λ f, nat_degree f)).sum | begin
revert h,
refine multiset.induction_on t _ (λ a t ih ht, _), { simp },
rw [multiset.map_cons, multiset.prod_cons] at ht ⊢,
rw [multiset.sum_cons, polynomial.nat_degree_mul', ih],
{ apply right_ne_zero_of_mul ht },
{ rwa polynomial.leading_coeff_multiset_prod', apply right_ne_zero_of_mul ht },
end | lemma | polynomial.nat_degree_multiset_prod' | algebra.polynomial | src/algebra/polynomial/big_operators.lean | [
"algebra.order.with_zero",
"data.polynomial.monic"
] | [
"ih",
"multiset.induction_on",
"multiset.map_cons",
"multiset.prod_cons",
"polynomial.leading_coeff_multiset_prod'",
"polynomial.nat_degree_mul'",
"right_ne_zero_of_mul"
] | The degree of a product of polynomials is equal to
the sum of the degrees, provided that the product of leading coefficients is nonzero.
See `polynomial.nat_degree_multiset_prod` (without the `'`) for a version for integral domains,
where this condition is automatically satisfied. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nat_degree_prod' (h : ∏ i in s, (f i).leading_coeff ≠ 0) :
(∏ i in s, f i).nat_degree = ∑ i in s, (f i).nat_degree | by simpa using nat_degree_multiset_prod' (s.1.map f) (by simpa using h) | lemma | polynomial.nat_degree_prod' | algebra.polynomial | src/algebra/polynomial/big_operators.lean | [
"algebra.order.with_zero",
"data.polynomial.monic"
] | [] | The degree of a product of polynomials is equal to
the sum of the degrees, provided that the product of leading coefficients is nonzero.
See `polynomial.nat_degree_prod` (without the `'`) for a version for integral domains,
where this condition is automatically satisfied. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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