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 |
|---|---|---|---|---|---|---|---|---|---|---|
mk_eq_zero {a : α} : associates.mk a = 0 ↔ a = 0 | ⟨assume h, (associated_zero_iff_eq_zero a).1 $ quotient.exact h, assume h, h.symm ▸ rfl⟩ | theorem | associates.mk_eq_zero | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associated_zero_iff_eq_zero",
"associates.mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_ne_zero {a : α} : associates.mk a ≠ 0 ↔ a ≠ 0 | not_congr mk_eq_zero | theorem | associates.mk_ne_zero | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associates.mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_non_zero_rep {a : associates α} : a ≠ 0 → ∃ a0 : α, a0 ≠ 0 ∧ associates.mk a0 = a | quotient.induction_on a (λ b nz, ⟨b, mt (congr_arg quotient.mk) nz, rfl⟩) | lemma | associates.exists_non_zero_rep | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associates",
"associates.mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prime.le_or_le {p : associates α} (hp : prime p) {a b : associates α} (h : p ≤ a * b) :
p ≤ a ∨ p ≤ b | hp.2.2 a b h | lemma | associates.prime.le_or_le | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associates",
"prime"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prime_mk (p : α) : prime (associates.mk p) ↔ _root_.prime p | begin
rw [prime, _root_.prime, forall_associated],
transitivity,
{ apply and_congr, refl,
apply and_congr, refl,
apply forall_congr, assume a,
exact forall_associated },
apply and_congr mk_ne_zero,
apply and_congr,
{ rw [is_unit_mk], },
refine forall₂_congr (λ a b, _),
rw [mk_mul_mk, mk_dvd_... | lemma | associates.prime_mk | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associates.mk",
"forall₂_congr",
"prime"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
irreducible_mk (a : α) : irreducible (associates.mk a) ↔ irreducible a | begin
simp only [irreducible_iff, is_unit_mk],
apply and_congr iff.rfl,
split,
{ rintro h x y rfl,
simpa [is_unit_mk] using h (associates.mk x) (associates.mk y) rfl },
{ intros h x y,
refine quotient.induction_on₂ x y (assume x y a_eq, _),
rcases quotient.exact a_eq.symm with ⟨u, a_eq⟩,
rw mu... | theorem | associates.irreducible_mk | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associates.mk",
"irreducible",
"irreducible_iff",
"is_unit",
"mul_assoc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_dvd_not_unit_mk_iff {a b : α} :
dvd_not_unit (associates.mk a) (associates.mk b) ↔
dvd_not_unit a b | begin
rw [dvd_not_unit, dvd_not_unit, mk_ne_zero],
apply and_congr_right, intro ane0,
split,
{ contrapose!, rw forall_associated,
intros h x hx hbax,
rw [mk_mul_mk, mk_eq_mk_iff_associated] at hbax,
cases hbax with u hu,
apply h (x * ↑u⁻¹),
{ rw is_unit_mk at hx,
rw associated.is_unit_... | theorem | associates.mk_dvd_not_unit_mk_iff | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associated.is_unit_iff",
"associates.mk",
"dvd_not_unit",
"mul_assoc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dvd_not_unit_of_lt {a b : associates α} (hlt : a < b) :
dvd_not_unit a b | begin
split, { rintro rfl, apply not_lt_of_le _ hlt, apply dvd_zero },
rcases hlt with ⟨⟨x, rfl⟩, ndvd⟩,
refine ⟨x, _, rfl⟩,
contrapose! ndvd,
rcases ndvd with ⟨u, rfl⟩,
simp,
end | theorem | associates.dvd_not_unit_of_lt | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associates",
"dvd_not_unit",
"dvd_zero",
"not_lt_of_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
irreducible_iff_prime_iff :
(∀ a : α, irreducible a ↔ prime a) ↔ (∀ a : (associates α), irreducible a ↔ prime a) | by simp_rw [forall_associated, irreducible_mk, prime_mk] | theorem | associates.irreducible_iff_prime_iff | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associates",
"irreducible",
"prime"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_of_mul_le_mul_left (a b c : associates α) (ha : a ≠ 0) :
a * b ≤ a * c → b ≤ c | | ⟨d, hd⟩ := ⟨d, mul_left_cancel₀ ha $ by rwa ← mul_assoc⟩ | lemma | associates.le_of_mul_le_mul_left | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associates",
"le_of_mul_le_mul_left",
"mul_left_cancel₀"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_or_eq_of_le_of_prime :
∀(p m : associates α), prime p → m ≤ p → (m = 1 ∨ m = p) | | _ m ⟨hp0, hp1, h⟩ ⟨d, rfl⟩ :=
match h m d dvd_rfl with
| or.inl h := classical.by_cases (assume : m = 0, by simp [this]) $
assume : m ≠ 0,
have m * d ≤ m * 1, by simpa using h,
have d ≤ 1, from associates.le_of_mul_le_mul_left m d 1 ‹m ≠ 0› this,
have d = 1, from bot_unique this,
by simp [this]
| or.inr h :... | lemma | associates.one_or_eq_of_le_of_prime | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associates",
"associates.le_of_mul_le_mul_left",
"bot_unique",
"dvd_rfl",
"mul_comm",
"prime"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dvd_not_unit_iff_lt {a b : associates α} :
dvd_not_unit a b ↔ a < b | dvd_and_not_dvd_iff.symm | theorem | associates.dvd_not_unit_iff_lt | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associates",
"dvd_not_unit"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_one_iff {p : associates α} : p ≤ 1 ↔ p = 1 | by rw [← associates.bot_eq_one, le_bot_iff] | lemma | associates.le_one_iff | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associates",
"associates.bot_eq_one",
"le_bot_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dvd_not_unit.is_unit_of_irreducible_right [comm_monoid_with_zero α] {p q : α}
(h : dvd_not_unit p q) (hq : irreducible q) : is_unit p | begin
obtain ⟨hp', x, hx, hx'⟩ := h,
exact or.resolve_right ((irreducible_iff.1 hq).right p x hx') hx
end | lemma | dvd_not_unit.is_unit_of_irreducible_right | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"comm_monoid_with_zero",
"dvd_not_unit",
"irreducible",
"is_unit"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
not_irreducible_of_not_unit_dvd_not_unit [comm_monoid_with_zero α] {p q : α}
(hp : ¬is_unit p) (h : dvd_not_unit p q) : ¬ irreducible q | mt h.is_unit_of_irreducible_right hp | lemma | not_irreducible_of_not_unit_dvd_not_unit | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"comm_monoid_with_zero",
"dvd_not_unit",
"irreducible",
"is_unit"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dvd_not_unit.not_unit [comm_monoid_with_zero α] {p q : α}
(hp : dvd_not_unit p q) : ¬ is_unit q | begin
obtain ⟨-, x, hx, rfl⟩ := hp,
exact λ hc, hx (is_unit_iff_dvd_one.mpr (dvd_of_mul_left_dvd (is_unit_iff_dvd_one.mp hc))),
end | lemma | dvd_not_unit.not_unit | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"comm_monoid_with_zero",
"dvd_not_unit",
"dvd_of_mul_left_dvd",
"is_unit"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dvd_not_unit_of_dvd_not_unit_associated [comm_monoid_with_zero α]
[nontrivial α] {p q r : α} (h : dvd_not_unit p q) (h' : associated q r) : dvd_not_unit p r | begin
obtain ⟨u, rfl⟩ := associated.symm h',
obtain ⟨hp, x, hx⟩ := h,
refine ⟨hp, x * ↑(u⁻¹), dvd_not_unit.not_unit ⟨u⁻¹.ne_zero, x, hx.left, mul_comm _ _⟩, _⟩,
rw [← mul_assoc, ← hx.right, mul_assoc, units.mul_inv, mul_one]
end | lemma | dvd_not_unit_of_dvd_not_unit_associated | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associated",
"associated.symm",
"comm_monoid_with_zero",
"dvd_not_unit",
"dvd_not_unit.not_unit",
"mul_assoc",
"mul_comm",
"mul_one",
"ne_zero",
"nontrivial",
"units.mul_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_unit_of_associated_mul [cancel_comm_monoid_with_zero α]
{p b : α} (h : associated (p * b) p) (hp : p ≠ 0) : is_unit b | begin
cases h with a ha,
refine is_unit_of_mul_eq_one b a ((mul_right_inj' hp).mp _),
rwa [← mul_assoc, mul_one],
end | lemma | is_unit_of_associated_mul | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associated",
"cancel_comm_monoid_with_zero",
"is_unit",
"is_unit_of_mul_eq_one",
"mul_assoc",
"mul_one",
"mul_right_inj'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dvd_not_unit.not_associated [cancel_comm_monoid_with_zero α] {p q : α}
(h : dvd_not_unit p q) : ¬ associated p q | begin
rintro ⟨a, rfl⟩,
obtain ⟨hp, x, hx, hx'⟩ := h,
rcases (mul_right_inj' hp).mp hx' with rfl,
exact hx a.is_unit,
end | lemma | dvd_not_unit.not_associated | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associated",
"cancel_comm_monoid_with_zero",
"dvd_not_unit",
"mul_right_inj'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dvd_not_unit.ne [cancel_comm_monoid_with_zero α] {p q : α}
(h : dvd_not_unit p q) : p ≠ q | begin
by_contra hcontra,
obtain ⟨hp, x, hx', hx''⟩ := h,
conv_lhs at hx'' {rw [← hcontra, ← mul_one p]},
rw (mul_left_cancel₀ hp hx'').symm at hx',
exact hx' is_unit_one,
end | lemma | dvd_not_unit.ne | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"by_contra",
"cancel_comm_monoid_with_zero",
"dvd_not_unit",
"is_unit_one",
"mul_left_cancel₀",
"mul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_injective_of_not_unit [cancel_comm_monoid_with_zero α] {q : α}
(hq : ¬ is_unit q) (hq' : q ≠ 0): function.injective (λ (n : ℕ), q^n) | begin
refine injective_of_lt_imp_ne (λ n m h, dvd_not_unit.ne ⟨pow_ne_zero n hq', q^(m - n), _, _⟩),
{ exact not_is_unit_of_not_is_unit_dvd hq (dvd_pow (dvd_refl _) (nat.sub_pos_of_lt h).ne') },
{ exact (pow_mul_pow_sub q h.le).symm }
end | lemma | pow_injective_of_not_unit | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"cancel_comm_monoid_with_zero",
"dvd_not_unit.ne",
"dvd_pow",
"dvd_refl",
"injective_of_lt_imp_ne",
"is_unit",
"not_is_unit_of_not_is_unit_dvd",
"pow_mul_pow_sub"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dvd_prime_pow [cancel_comm_monoid_with_zero α] {p q : α} (hp : prime p) (n : ℕ) :
q ∣ p^n ↔ ∃ i ≤ n, associated q (p ^ i) | begin
induction n with n ih generalizing q,
{ simp [← is_unit_iff_dvd_one, associated_one_iff_is_unit] },
refine ⟨λ h, _, λ ⟨i, hi, hq⟩, hq.dvd.trans (pow_dvd_pow p hi)⟩,
rw pow_succ at h,
rcases hp.left_dvd_or_dvd_right_of_dvd_mul h with (⟨q, rfl⟩ | hno),
{ rw [mul_dvd_mul_iff_left hp.ne_zero, ih] at h,
... | lemma | dvd_prime_pow | algebra | src/algebra/associated.lean | [
"algebra.divisibility.basic",
"algebra.group_power.lemmas",
"algebra.parity"
] | [
"associated",
"associated_one_iff_is_unit",
"cancel_comm_monoid_with_zero",
"ih",
"is_unit_iff_dvd_one",
"mul_dvd_mul_iff_left",
"pow_dvd_pow",
"pow_succ",
"prime"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bdd_above_inv : bdd_above s⁻¹ ↔ bdd_below s | (order_iso.inv G).bdd_above_preimage | lemma | bdd_above_inv | algebra | src/algebra/bounds.lean | [
"algebra.order.group.order_iso",
"algebra.order.monoid.order_dual",
"data.set.pointwise.basic",
"order.bounds.order_iso",
"order.conditionally_complete_lattice.basic"
] | [
"bdd_above",
"bdd_below",
"order_iso.inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bdd_below_inv : bdd_below s⁻¹ ↔ bdd_above s | (order_iso.inv G).bdd_below_preimage | lemma | bdd_below_inv | algebra | src/algebra/bounds.lean | [
"algebra.order.group.order_iso",
"algebra.order.monoid.order_dual",
"data.set.pointwise.basic",
"order.bounds.order_iso",
"order.conditionally_complete_lattice.basic"
] | [
"bdd_above",
"bdd_below",
"order_iso.inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bdd_above.inv (h : bdd_above s) : bdd_below s⁻¹ | bdd_below_inv.2 h | lemma | bdd_above.inv | algebra | src/algebra/bounds.lean | [
"algebra.order.group.order_iso",
"algebra.order.monoid.order_dual",
"data.set.pointwise.basic",
"order.bounds.order_iso",
"order.conditionally_complete_lattice.basic"
] | [
"bdd_above",
"bdd_below"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bdd_below.inv (h : bdd_below s) : bdd_above s⁻¹ | bdd_above_inv.2 h | lemma | bdd_below.inv | algebra | src/algebra/bounds.lean | [
"algebra.order.group.order_iso",
"algebra.order.monoid.order_dual",
"data.set.pointwise.basic",
"order.bounds.order_iso",
"order.conditionally_complete_lattice.basic"
] | [
"bdd_above",
"bdd_below"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_lub_inv : is_lub s⁻¹ a ↔ is_glb s a⁻¹ | (order_iso.inv G).is_lub_preimage | lemma | is_lub_inv | algebra | src/algebra/bounds.lean | [
"algebra.order.group.order_iso",
"algebra.order.monoid.order_dual",
"data.set.pointwise.basic",
"order.bounds.order_iso",
"order.conditionally_complete_lattice.basic"
] | [
"is_glb",
"is_lub",
"order_iso.inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_lub_inv' : is_lub s⁻¹ a⁻¹ ↔ is_glb s a | (order_iso.inv G).is_lub_preimage' | lemma | is_lub_inv' | algebra | src/algebra/bounds.lean | [
"algebra.order.group.order_iso",
"algebra.order.monoid.order_dual",
"data.set.pointwise.basic",
"order.bounds.order_iso",
"order.conditionally_complete_lattice.basic"
] | [
"is_glb",
"is_lub",
"order_iso.inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_glb.inv (h : is_glb s a) : is_lub s⁻¹ a⁻¹ | is_lub_inv'.2 h | lemma | is_glb.inv | algebra | src/algebra/bounds.lean | [
"algebra.order.group.order_iso",
"algebra.order.monoid.order_dual",
"data.set.pointwise.basic",
"order.bounds.order_iso",
"order.conditionally_complete_lattice.basic"
] | [
"is_glb",
"is_lub"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_glb_inv : is_glb s⁻¹ a ↔ is_lub s a⁻¹ | (order_iso.inv G).is_glb_preimage | lemma | is_glb_inv | algebra | src/algebra/bounds.lean | [
"algebra.order.group.order_iso",
"algebra.order.monoid.order_dual",
"data.set.pointwise.basic",
"order.bounds.order_iso",
"order.conditionally_complete_lattice.basic"
] | [
"is_glb",
"is_lub",
"order_iso.inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_glb_inv' : is_glb s⁻¹ a⁻¹ ↔ is_lub s a | (order_iso.inv G).is_glb_preimage' | lemma | is_glb_inv' | algebra | src/algebra/bounds.lean | [
"algebra.order.group.order_iso",
"algebra.order.monoid.order_dual",
"data.set.pointwise.basic",
"order.bounds.order_iso",
"order.conditionally_complete_lattice.basic"
] | [
"is_glb",
"is_lub",
"order_iso.inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_lub.inv (h : is_lub s a) : is_glb s⁻¹ a⁻¹ | is_glb_inv'.2 h | lemma | is_lub.inv | algebra | src/algebra/bounds.lean | [
"algebra.order.group.order_iso",
"algebra.order.monoid.order_dual",
"data.set.pointwise.basic",
"order.bounds.order_iso",
"order.conditionally_complete_lattice.basic"
] | [
"is_glb",
"is_lub"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_mem_upper_bounds_mul {s t : set M} {a b : M} (ha : a ∈ upper_bounds s)
(hb : b ∈ upper_bounds t) :
a * b ∈ upper_bounds (s * t) | forall_image2_iff.2 $ λ x hx y hy, mul_le_mul' (ha hx) (hb hy) | lemma | mul_mem_upper_bounds_mul | algebra | src/algebra/bounds.lean | [
"algebra.order.group.order_iso",
"algebra.order.monoid.order_dual",
"data.set.pointwise.basic",
"order.bounds.order_iso",
"order.conditionally_complete_lattice.basic"
] | [
"mul_le_mul'",
"upper_bounds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subset_upper_bounds_mul (s t : set M) :
upper_bounds s * upper_bounds t ⊆ upper_bounds (s * t) | image2_subset_iff.2 $ λ x hx y hy, mul_mem_upper_bounds_mul hx hy | lemma | subset_upper_bounds_mul | algebra | src/algebra/bounds.lean | [
"algebra.order.group.order_iso",
"algebra.order.monoid.order_dual",
"data.set.pointwise.basic",
"order.bounds.order_iso",
"order.conditionally_complete_lattice.basic"
] | [
"mul_mem_upper_bounds_mul",
"upper_bounds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_mem_lower_bounds_mul {s t : set M} {a b : M} (ha : a ∈ lower_bounds s)
(hb : b ∈ lower_bounds t) : a * b ∈ lower_bounds (s * t) | @mul_mem_upper_bounds_mul Mᵒᵈ _ _ _ _ _ _ _ _ ha hb | lemma | mul_mem_lower_bounds_mul | algebra | src/algebra/bounds.lean | [
"algebra.order.group.order_iso",
"algebra.order.monoid.order_dual",
"data.set.pointwise.basic",
"order.bounds.order_iso",
"order.conditionally_complete_lattice.basic"
] | [
"lower_bounds",
"mul_mem_upper_bounds_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subset_lower_bounds_mul (s t : set M) :
lower_bounds s * lower_bounds t ⊆ lower_bounds (s * t) | @subset_upper_bounds_mul Mᵒᵈ _ _ _ _ _ _ | lemma | subset_lower_bounds_mul | algebra | src/algebra/bounds.lean | [
"algebra.order.group.order_iso",
"algebra.order.monoid.order_dual",
"data.set.pointwise.basic",
"order.bounds.order_iso",
"order.conditionally_complete_lattice.basic"
] | [
"lower_bounds",
"subset_upper_bounds_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bdd_above.mul {s t : set M} (hs : bdd_above s) (ht : bdd_above t) :
bdd_above (s * t) | (hs.mul ht).mono (subset_upper_bounds_mul s t) | lemma | bdd_above.mul | algebra | src/algebra/bounds.lean | [
"algebra.order.group.order_iso",
"algebra.order.monoid.order_dual",
"data.set.pointwise.basic",
"order.bounds.order_iso",
"order.conditionally_complete_lattice.basic"
] | [
"bdd_above",
"subset_upper_bounds_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bdd_below.mul {s t : set M} (hs : bdd_below s) (ht : bdd_below t) :
bdd_below (s * t) | (hs.mul ht).mono (subset_lower_bounds_mul s t) | lemma | bdd_below.mul | algebra | src/algebra/bounds.lean | [
"algebra.order.group.order_iso",
"algebra.order.monoid.order_dual",
"data.set.pointwise.basic",
"order.bounds.order_iso",
"order.conditionally_complete_lattice.basic"
] | [
"bdd_below",
"subset_lower_bounds_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
csupr_mul (hf : bdd_above (set.range f)) (a : G) :
(⨆ i, f i) * a = ⨆ i, f i * a | (order_iso.mul_right a).map_csupr hf | lemma | csupr_mul | algebra | src/algebra/bounds.lean | [
"algebra.order.group.order_iso",
"algebra.order.monoid.order_dual",
"data.set.pointwise.basic",
"order.bounds.order_iso",
"order.conditionally_complete_lattice.basic"
] | [
"bdd_above",
"order_iso.mul_right",
"set.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
csupr_div (hf : bdd_above (set.range f)) (a : G) :
(⨆ i, f i) / a = ⨆ i, f i / a | by simp only [div_eq_mul_inv, csupr_mul hf] | lemma | csupr_div | algebra | src/algebra/bounds.lean | [
"algebra.order.group.order_iso",
"algebra.order.monoid.order_dual",
"data.set.pointwise.basic",
"order.bounds.order_iso",
"order.conditionally_complete_lattice.basic"
] | [
"bdd_above",
"csupr_mul",
"div_eq_mul_inv",
"set.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_csupr (hf : bdd_above (set.range f)) (a : G) :
a * (⨆ i, f i) = ⨆ i, a * f i | (order_iso.mul_left a).map_csupr hf | lemma | mul_csupr | algebra | src/algebra/bounds.lean | [
"algebra.order.group.order_iso",
"algebra.order.monoid.order_dual",
"data.set.pointwise.basic",
"order.bounds.order_iso",
"order.conditionally_complete_lattice.basic"
] | [
"bdd_above",
"order_iso.mul_left",
"set.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
covariant : Prop | ∀ (m) {n₁ n₂}, r n₁ n₂ → r (μ m n₁) (μ m n₂) | def | covariant | algebra | src/algebra/covariant_and_contravariant.lean | [
"algebra.group.defs",
"order.basic",
"order.monotone.basic"
] | [] | `covariant` is useful to formulate succintly statements about the interactions between an
action of a Type on another one and a relation on the acted-upon Type.
See the `covariant_class` doc-string for its meaning. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
contravariant : Prop | ∀ (m) {n₁ n₂}, r (μ m n₁) (μ m n₂) → r n₁ n₂ | def | contravariant | algebra | src/algebra/covariant_and_contravariant.lean | [
"algebra.group.defs",
"order.basic",
"order.monotone.basic"
] | [] | `contravariant` is useful to formulate succintly statements about the interactions between an
action of a Type on another one and a relation on the acted-upon Type.
See the `contravariant_class` doc-string for its meaning. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
covariant_class : Prop | (elim : covariant M N μ r) | class | covariant_class | algebra | src/algebra/covariant_and_contravariant.lean | [
"algebra.group.defs",
"order.basic",
"order.monotone.basic"
] | [
"covariant"
] | Given an action `μ` of a Type `M` on a Type `N` and a relation `r` on `N`, informally, the
`covariant_class` says that "the action `μ` preserves the relation `r`."
More precisely, the `covariant_class` is a class taking two Types `M N`, together with an "action"
`μ : M → N → N` and a relation `r : N → N → Prop`. Its ... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
contravariant_class : Prop | (elim : contravariant M N μ r) | class | contravariant_class | algebra | src/algebra/covariant_and_contravariant.lean | [
"algebra.group.defs",
"order.basic",
"order.monotone.basic"
] | [
"contravariant"
] | Given an action `μ` of a Type `M` on a Type `N` and a relation `r` on `N`, informally, the
`contravariant_class` says that "if the result of the action `μ` on a pair satisfies the
relation `r`, then the initial pair satisfied the relation `r`."
More precisely, the `contravariant_class` is a class taking two Types `M N... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
rel_iff_cov [covariant_class M N μ r] [contravariant_class M N μ r] (m : M) {a b : N} :
r (μ m a) (μ m b) ↔ r a b | ⟨contravariant_class.elim _, covariant_class.elim _⟩ | lemma | rel_iff_cov | algebra | src/algebra/covariant_and_contravariant.lean | [
"algebra.group.defs",
"order.basic",
"order.monotone.basic"
] | [
"contravariant_class",
"covariant_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
covariant.flip (h : covariant M N μ r) : covariant M N μ (flip r) | λ a b c hbc, h a hbc | lemma | covariant.flip | algebra | src/algebra/covariant_and_contravariant.lean | [
"algebra.group.defs",
"order.basic",
"order.monotone.basic"
] | [
"covariant"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
contravariant.flip (h : contravariant M N μ r) : contravariant M N μ (flip r) | λ a b c hbc, h a hbc | lemma | contravariant.flip | algebra | src/algebra/covariant_and_contravariant.lean | [
"algebra.group.defs",
"order.basic",
"order.monotone.basic"
] | [
"contravariant"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
act_rel_act_of_rel (m : M) {a b : N} (ab : r a b) :
r (μ m a) (μ m b) | covariant_class.elim _ ab | lemma | act_rel_act_of_rel | algebra | src/algebra/covariant_and_contravariant.lean | [
"algebra.group.defs",
"order.basic",
"order.monotone.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
group.covariant_iff_contravariant [group N] :
covariant N N (*) r ↔ contravariant N N (*) r | begin
refine ⟨λ h a b c bc, _, λ h a b c bc, _⟩,
{ rw [← inv_mul_cancel_left a b, ← inv_mul_cancel_left a c],
exact h a⁻¹ bc },
{ rw [← inv_mul_cancel_left a b, ← inv_mul_cancel_left a c] at bc,
exact h a⁻¹ bc }
end | lemma | group.covariant_iff_contravariant | algebra | src/algebra/covariant_and_contravariant.lean | [
"algebra.group.defs",
"order.basic",
"order.monotone.basic"
] | [
"contravariant",
"covariant",
"group",
"inv_mul_cancel_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
group.covconv [group N] [covariant_class N N (*) r] :
contravariant_class N N (*) r | ⟨group.covariant_iff_contravariant.mp covariant_class.elim⟩ | instance | group.covconv | algebra | src/algebra/covariant_and_contravariant.lean | [
"algebra.group.defs",
"order.basic",
"order.monotone.basic"
] | [
"contravariant_class",
"covariant_class",
"group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
group.covariant_swap_iff_contravariant_swap [group N] :
covariant N N (swap (*)) r ↔ contravariant N N (swap (*)) r | begin
refine ⟨λ h a b c bc, _, λ h a b c bc, _⟩,
{ rw [← mul_inv_cancel_right b a, ← mul_inv_cancel_right c a],
exact h a⁻¹ bc },
{ rw [← mul_inv_cancel_right b a, ← mul_inv_cancel_right c a] at bc,
exact h a⁻¹ bc }
end | lemma | group.covariant_swap_iff_contravariant_swap | algebra | src/algebra/covariant_and_contravariant.lean | [
"algebra.group.defs",
"order.basic",
"order.monotone.basic"
] | [
"contravariant",
"covariant",
"group",
"mul_inv_cancel_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
group.covconv_swap [group N] [covariant_class N N (swap (*)) r] :
contravariant_class N N (swap (*)) r | ⟨group.covariant_swap_iff_contravariant_swap.mp covariant_class.elim⟩ | instance | group.covconv_swap | algebra | src/algebra/covariant_and_contravariant.lean | [
"algebra.group.defs",
"order.basic",
"order.monotone.basic"
] | [
"contravariant_class",
"covariant_class",
"group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
act_rel_of_rel_of_act_rel (ab : r a b) (rl : r (μ m b) c) :
r (μ m a) c | trans (act_rel_act_of_rel m ab) rl | lemma | act_rel_of_rel_of_act_rel | algebra | src/algebra/covariant_and_contravariant.lean | [
"algebra.group.defs",
"order.basic",
"order.monotone.basic"
] | [
"act_rel_act_of_rel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rel_act_of_rel_of_rel_act (ab : r a b) (rr : r c (μ m a)) :
r c (μ m b) | trans rr (act_rel_act_of_rel _ ab) | lemma | rel_act_of_rel_of_rel_act | algebra | src/algebra/covariant_and_contravariant.lean | [
"algebra.group.defs",
"order.basic",
"order.monotone.basic"
] | [
"act_rel_act_of_rel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
act_rel_act_of_rel_of_rel (ab : r a b) (cd : r c d) :
r (mu a c) (mu b d) | trans (act_rel_act_of_rel c ab : _) (act_rel_act_of_rel b cd) | lemma | act_rel_act_of_rel_of_rel | algebra | src/algebra/covariant_and_contravariant.lean | [
"algebra.group.defs",
"order.basic",
"order.monotone.basic"
] | [
"act_rel_act_of_rel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rel_of_act_rel_act (m : M) {a b : N} (ab : r (μ m a) (μ m b)) :
r a b | contravariant_class.elim _ ab | lemma | rel_of_act_rel_act | algebra | src/algebra/covariant_and_contravariant.lean | [
"algebra.group.defs",
"order.basic",
"order.monotone.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
act_rel_of_act_rel_of_rel_act_rel (ab : r (μ m a) b) (rl : r (μ m b) (μ m c)) :
r (μ m a) c | trans ab (rel_of_act_rel_act m rl) | lemma | act_rel_of_act_rel_of_rel_act_rel | algebra | src/algebra/covariant_and_contravariant.lean | [
"algebra.group.defs",
"order.basic",
"order.monotone.basic"
] | [
"rel_of_act_rel_act"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rel_act_of_act_rel_act_of_rel_act (ab : r (μ m a) (μ m b)) (rr : r b (μ m c)) :
r a (μ m c) | trans (rel_of_act_rel_act m ab) rr | lemma | rel_act_of_act_rel_act_of_rel_act | algebra | src/algebra/covariant_and_contravariant.lean | [
"algebra.group.defs",
"order.basic",
"order.monotone.basic"
] | [
"rel_of_act_rel_act"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
covariant.monotone_of_const [covariant_class M N μ (≤)] (m : M) : monotone (μ m) | λ a b ha, covariant_class.elim m ha | lemma | covariant.monotone_of_const | algebra | src/algebra/covariant_and_contravariant.lean | [
"algebra.group.defs",
"order.basic",
"order.monotone.basic"
] | [
"covariant_class",
"monotone"
] | The partial application of a constant to a covariant operator is monotone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monotone.covariant_of_const [covariant_class M N μ (≤)] (hf : monotone f) (m : M) :
monotone (λ n, f (μ m n)) | hf.comp $ covariant.monotone_of_const m | lemma | monotone.covariant_of_const | algebra | src/algebra/covariant_and_contravariant.lean | [
"algebra.group.defs",
"order.basic",
"order.monotone.basic"
] | [
"covariant.monotone_of_const",
"covariant_class",
"monotone"
] | A monotone function remains monotone when composed with the partial application
of a covariant operator. E.g., `∀ (m : ℕ), monotone f → monotone (λ n, f (m + n))`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monotone.covariant_of_const' {μ : N → N → N} [covariant_class N N (swap μ) (≤)]
(hf : monotone f) (m : N) :
monotone (λ n, f (μ n m)) | hf.comp $ covariant.monotone_of_const m | lemma | monotone.covariant_of_const' | algebra | src/algebra/covariant_and_contravariant.lean | [
"algebra.group.defs",
"order.basic",
"order.monotone.basic"
] | [
"covariant.monotone_of_const",
"covariant_class",
"monotone"
] | Same as `monotone.covariant_of_const`, but with the constant on the other side of
the operator. E.g., `∀ (m : ℕ), monotone f → monotone (λ n, f (n + m))`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
antitone.covariant_of_const [covariant_class M N μ (≤)] (hf : antitone f) (m : M) :
antitone (λ n, f (μ m n)) | hf.comp_monotone $ covariant.monotone_of_const m | lemma | antitone.covariant_of_const | algebra | src/algebra/covariant_and_contravariant.lean | [
"algebra.group.defs",
"order.basic",
"order.monotone.basic"
] | [
"antitone",
"covariant.monotone_of_const",
"covariant_class"
] | Dual of `monotone.covariant_of_const` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
antitone.covariant_of_const' {μ : N → N → N} [covariant_class N N (swap μ) (≤)]
(hf : antitone f) (m : N) :
antitone (λ n, f (μ n m)) | hf.comp_monotone $ covariant.monotone_of_const m | lemma | antitone.covariant_of_const' | algebra | src/algebra/covariant_and_contravariant.lean | [
"algebra.group.defs",
"order.basic",
"order.monotone.basic"
] | [
"antitone",
"covariant.monotone_of_const",
"covariant_class"
] | Dual of `monotone.covariant_of_const'` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
covariant_le_of_covariant_lt [partial_order N] :
covariant M N μ (<) → covariant M N μ (≤) | begin
refine λ h a b c bc, _,
rcases le_iff_eq_or_lt.mp bc with rfl | bc,
{ exact rfl.le },
{ exact (h _ bc).le }
end | lemma | covariant_le_of_covariant_lt | algebra | src/algebra/covariant_and_contravariant.lean | [
"algebra.group.defs",
"order.basic",
"order.monotone.basic"
] | [
"covariant"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
contravariant_lt_of_contravariant_le [partial_order N] :
contravariant M N μ (≤) → contravariant M N μ (<) | begin
refine λ h a b c bc, lt_iff_le_and_ne.mpr ⟨h a bc.le, _⟩,
rintro rfl,
exact lt_irrefl _ bc,
end | lemma | contravariant_lt_of_contravariant_le | algebra | src/algebra/covariant_and_contravariant.lean | [
"algebra.group.defs",
"order.basic",
"order.monotone.basic"
] | [
"contravariant"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
covariant_le_iff_contravariant_lt [linear_order N] :
covariant M N μ (≤) ↔ contravariant M N μ (<) | ⟨ λ h a b c bc, not_le.mp (λ k, not_le.mpr bc (h _ k)),
λ h a b c bc, not_lt.mp (λ k, not_lt.mpr bc (h _ k))⟩ | lemma | covariant_le_iff_contravariant_lt | algebra | src/algebra/covariant_and_contravariant.lean | [
"algebra.group.defs",
"order.basic",
"order.monotone.basic"
] | [
"contravariant",
"covariant"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
covariant_lt_iff_contravariant_le [linear_order N] :
covariant M N μ (<) ↔ contravariant M N μ (≤) | ⟨ λ h a b c bc, not_lt.mp (λ k, not_lt.mpr bc (h _ k)),
λ h a b c bc, not_le.mp (λ k, not_le.mpr bc (h _ k))⟩ | lemma | covariant_lt_iff_contravariant_le | algebra | src/algebra/covariant_and_contravariant.lean | [
"algebra.group.defs",
"order.basic",
"order.monotone.basic"
] | [
"contravariant",
"covariant"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
covariant_flip_mul_iff [comm_semigroup N] :
covariant N N (flip (*)) (r) ↔ covariant N N (*) (r) | by rw is_symm_op.flip_eq | lemma | covariant_flip_mul_iff | algebra | src/algebra/covariant_and_contravariant.lean | [
"algebra.group.defs",
"order.basic",
"order.monotone.basic"
] | [
"comm_semigroup",
"covariant",
"is_symm_op.flip_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
contravariant_flip_mul_iff [comm_semigroup N] :
contravariant N N (flip (*)) (r) ↔ contravariant N N (*) (r) | by rw is_symm_op.flip_eq | lemma | contravariant_flip_mul_iff | algebra | src/algebra/covariant_and_contravariant.lean | [
"algebra.group.defs",
"order.basic",
"order.monotone.basic"
] | [
"comm_semigroup",
"contravariant",
"is_symm_op.flip_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
contravariant_mul_lt_of_covariant_mul_le [has_mul N] [linear_order N]
[covariant_class N N (*) (≤)] : contravariant_class N N (*) (<) | { elim := (covariant_le_iff_contravariant_lt N N (*)).mp covariant_class.elim } | instance | contravariant_mul_lt_of_covariant_mul_le | algebra | src/algebra/covariant_and_contravariant.lean | [
"algebra.group.defs",
"order.basic",
"order.monotone.basic"
] | [
"contravariant_class",
"covariant_class",
"covariant_le_iff_contravariant_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
covariant_mul_lt_of_contravariant_mul_le [has_mul N] [linear_order N]
[contravariant_class N N (*) (≤)] : covariant_class N N (*) (<) | { elim := (covariant_lt_iff_contravariant_le N N (*)).mpr contravariant_class.elim } | instance | covariant_mul_lt_of_contravariant_mul_le | algebra | src/algebra/covariant_and_contravariant.lean | [
"algebra.group.defs",
"order.basic",
"order.monotone.basic"
] | [
"contravariant_class",
"covariant_class",
"covariant_lt_iff_contravariant_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
covariant_swap_mul_le_of_covariant_mul_le [comm_semigroup N] [has_le N]
[covariant_class N N (*) (≤)] : covariant_class N N (swap (*)) (≤) | { elim := (covariant_flip_mul_iff N (≤)).mpr covariant_class.elim } | instance | covariant_swap_mul_le_of_covariant_mul_le | algebra | src/algebra/covariant_and_contravariant.lean | [
"algebra.group.defs",
"order.basic",
"order.monotone.basic"
] | [
"comm_semigroup",
"covariant_class",
"covariant_flip_mul_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
contravariant_swap_mul_le_of_contravariant_mul_le [comm_semigroup N] [has_le N]
[contravariant_class N N (*) (≤)] : contravariant_class N N (swap (*)) (≤) | { elim := (contravariant_flip_mul_iff N (≤)).mpr contravariant_class.elim } | instance | contravariant_swap_mul_le_of_contravariant_mul_le | algebra | src/algebra/covariant_and_contravariant.lean | [
"algebra.group.defs",
"order.basic",
"order.monotone.basic"
] | [
"comm_semigroup",
"contravariant_class",
"contravariant_flip_mul_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
contravariant_swap_mul_lt_of_contravariant_mul_lt [comm_semigroup N] [has_lt N]
[contravariant_class N N (*) (<)] : contravariant_class N N (swap (*)) (<) | { elim := (contravariant_flip_mul_iff N (<)).mpr contravariant_class.elim } | instance | contravariant_swap_mul_lt_of_contravariant_mul_lt | algebra | src/algebra/covariant_and_contravariant.lean | [
"algebra.group.defs",
"order.basic",
"order.monotone.basic"
] | [
"comm_semigroup",
"contravariant_class",
"contravariant_flip_mul_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
covariant_swap_mul_lt_of_covariant_mul_lt [comm_semigroup N] [has_lt N]
[covariant_class N N (*) (<)] : covariant_class N N (swap (*)) (<) | { elim := (covariant_flip_mul_iff N (<)).mpr covariant_class.elim } | instance | covariant_swap_mul_lt_of_covariant_mul_lt | algebra | src/algebra/covariant_and_contravariant.lean | [
"algebra.group.defs",
"order.basic",
"order.monotone.basic"
] | [
"comm_semigroup",
"covariant_class",
"covariant_flip_mul_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left_cancel_semigroup.covariant_mul_lt_of_covariant_mul_le
[left_cancel_semigroup N] [partial_order N] [covariant_class N N (*) (≤)] :
covariant_class N N (*) (<) | { elim := λ a b c bc, by { cases lt_iff_le_and_ne.mp bc with bc cb,
exact lt_iff_le_and_ne.mpr ⟨covariant_class.elim a bc, (mul_ne_mul_right a).mpr cb⟩ } } | instance | left_cancel_semigroup.covariant_mul_lt_of_covariant_mul_le | algebra | src/algebra/covariant_and_contravariant.lean | [
"algebra.group.defs",
"order.basic",
"order.monotone.basic"
] | [
"covariant_class",
"left_cancel_semigroup",
"mul_ne_mul_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right_cancel_semigroup.covariant_swap_mul_lt_of_covariant_swap_mul_le
[right_cancel_semigroup N] [partial_order N] [covariant_class N N (swap (*)) (≤)] :
covariant_class N N (swap (*)) (<) | { elim := λ a b c bc, by { cases lt_iff_le_and_ne.mp bc with bc cb,
exact lt_iff_le_and_ne.mpr ⟨covariant_class.elim a bc, (mul_ne_mul_left a).mpr cb⟩ } } | instance | right_cancel_semigroup.covariant_swap_mul_lt_of_covariant_swap_mul_le | algebra | src/algebra/covariant_and_contravariant.lean | [
"algebra.group.defs",
"order.basic",
"order.monotone.basic"
] | [
"covariant_class",
"mul_ne_mul_left",
"right_cancel_semigroup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left_cancel_semigroup.contravariant_mul_le_of_contravariant_mul_lt
[left_cancel_semigroup N] [partial_order N] [contravariant_class N N (*) (<)] :
contravariant_class N N (*) (≤) | { elim := λ a b c bc, by { cases le_iff_eq_or_lt.mp bc with h h,
{ exact ((mul_right_inj a).mp h).le },
{ exact (contravariant_class.elim _ h).le } } } | instance | left_cancel_semigroup.contravariant_mul_le_of_contravariant_mul_lt | algebra | src/algebra/covariant_and_contravariant.lean | [
"algebra.group.defs",
"order.basic",
"order.monotone.basic"
] | [
"contravariant_class",
"left_cancel_semigroup",
"mul_right_inj"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right_cancel_semigroup.contravariant_swap_mul_le_of_contravariant_swap_mul_lt
[right_cancel_semigroup N] [partial_order N] [contravariant_class N N (swap (*)) (<)] :
contravariant_class N N (swap (*)) (≤) | { elim := λ a b c bc, by { cases le_iff_eq_or_lt.mp bc with h h,
{ exact ((mul_left_inj a).mp h).le },
{ exact (contravariant_class.elim _ h).le } } } | instance | right_cancel_semigroup.contravariant_swap_mul_le_of_contravariant_swap_mul_lt | algebra | src/algebra/covariant_and_contravariant.lean | [
"algebra.group.defs",
"order.basic",
"order.monotone.basic"
] | [
"contravariant_class",
"mul_left_inj",
"right_cancel_semigroup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cubic (R : Type*) | (a b c d : R) | structure | cubic | algebra | src/algebra/cubic_discriminant.lean | [
"data.polynomial.splits"
] | [] | The structure representing a cubic polynomial. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_poly (P : cubic R) : R[X] | C P.a * X ^ 3 + C P.b * X ^ 2 + C P.c * X + C P.d | def | cubic.to_poly | algebra | src/algebra/cubic_discriminant.lean | [
"data.polynomial.splits"
] | [
"cubic"
] | Convert a cubic polynomial to a polynomial. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
C_mul_prod_X_sub_C_eq [comm_ring S] {w x y z : S} :
C w * (X - C x) * (X - C y) * (X - C z)
= to_poly ⟨w, w * -(x + y + z), w * (x * y + x * z + y * z), w * -(x * y * z)⟩ | by { simp only [to_poly, C_neg, C_add, C_mul], ring1 } | theorem | cubic.C_mul_prod_X_sub_C_eq | algebra | src/algebra/cubic_discriminant.lean | [
"data.polynomial.splits"
] | [
"comm_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_X_sub_C_eq [comm_ring S] {x y z : S} :
(X - C x) * (X - C y) * (X - C z)
= to_poly ⟨1, -(x + y + z), (x * y + x * z + y * z), -(x * y * z)⟩ | by rw [← one_mul $ X - C x, ← C_1, C_mul_prod_X_sub_C_eq, one_mul, one_mul, one_mul] | theorem | cubic.prod_X_sub_C_eq | algebra | src/algebra/cubic_discriminant.lean | [
"data.polynomial.splits"
] | [
"comm_ring",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeffs :
(∀ n > 3, P.to_poly.coeff n = 0) ∧ P.to_poly.coeff 3 = P.a ∧ P.to_poly.coeff 2 = P.b
∧ P.to_poly.coeff 1 = P.c ∧ P.to_poly.coeff 0 = P.d | begin
simp only [to_poly, coeff_add, coeff_C, coeff_C_mul_X, coeff_C_mul_X_pow],
norm_num,
intros n hn,
repeat { rw [if_neg] },
any_goals { linarith only [hn] },
repeat { rw [zero_add] }
end | lemma | cubic.coeffs | algebra | src/algebra/cubic_discriminant.lean | [
"data.polynomial.splits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_eq_zero {n : ℕ} (hn : 3 < n) : P.to_poly.coeff n = 0 | coeffs.1 n hn | lemma | cubic.coeff_eq_zero | algebra | src/algebra/cubic_discriminant.lean | [
"data.polynomial.splits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_eq_a : P.to_poly.coeff 3 = P.a | coeffs.2.1 | lemma | cubic.coeff_eq_a | algebra | src/algebra/cubic_discriminant.lean | [
"data.polynomial.splits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_eq_b : P.to_poly.coeff 2 = P.b | coeffs.2.2.1 | lemma | cubic.coeff_eq_b | algebra | src/algebra/cubic_discriminant.lean | [
"data.polynomial.splits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_eq_c : P.to_poly.coeff 1 = P.c | coeffs.2.2.2.1 | lemma | cubic.coeff_eq_c | algebra | src/algebra/cubic_discriminant.lean | [
"data.polynomial.splits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_eq_d : P.to_poly.coeff 0 = P.d | coeffs.2.2.2.2 | lemma | cubic.coeff_eq_d | algebra | src/algebra/cubic_discriminant.lean | [
"data.polynomial.splits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
a_of_eq (h : P.to_poly = Q.to_poly) : P.a = Q.a | by rw [← coeff_eq_a, h, coeff_eq_a] | lemma | cubic.a_of_eq | algebra | src/algebra/cubic_discriminant.lean | [
"data.polynomial.splits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
b_of_eq (h : P.to_poly = Q.to_poly) : P.b = Q.b | by rw [← coeff_eq_b, h, coeff_eq_b] | lemma | cubic.b_of_eq | algebra | src/algebra/cubic_discriminant.lean | [
"data.polynomial.splits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
c_of_eq (h : P.to_poly = Q.to_poly) : P.c = Q.c | by rw [← coeff_eq_c, h, coeff_eq_c] | lemma | cubic.c_of_eq | algebra | src/algebra/cubic_discriminant.lean | [
"data.polynomial.splits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
d_of_eq (h : P.to_poly = Q.to_poly) : P.d = Q.d | by rw [← coeff_eq_d, h, coeff_eq_d] | lemma | cubic.d_of_eq | algebra | src/algebra/cubic_discriminant.lean | [
"data.polynomial.splits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_poly_injective (P Q : cubic R) : P.to_poly = Q.to_poly ↔ P = Q | ⟨λ h, ext P Q (a_of_eq h) (b_of_eq h) (c_of_eq h) (d_of_eq h), congr_arg to_poly⟩ | lemma | cubic.to_poly_injective | algebra | src/algebra/cubic_discriminant.lean | [
"data.polynomial.splits"
] | [
"cubic"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_a_eq_zero (ha : P.a = 0) : P.to_poly = C P.b * X ^ 2 + C P.c * X + C P.d | by rw [to_poly, ha, C_0, zero_mul, zero_add] | lemma | cubic.of_a_eq_zero | algebra | src/algebra/cubic_discriminant.lean | [
"data.polynomial.splits"
] | [
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_a_eq_zero' : to_poly ⟨0, b, c, d⟩ = C b * X ^ 2 + C c * X + C d | of_a_eq_zero rfl | lemma | cubic.of_a_eq_zero' | algebra | src/algebra/cubic_discriminant.lean | [
"data.polynomial.splits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_b_eq_zero (ha : P.a = 0) (hb : P.b = 0) : P.to_poly = C P.c * X + C P.d | by rw [of_a_eq_zero ha, hb, C_0, zero_mul, zero_add] | lemma | cubic.of_b_eq_zero | algebra | src/algebra/cubic_discriminant.lean | [
"data.polynomial.splits"
] | [
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_b_eq_zero' : to_poly ⟨0, 0, c, d⟩ = C c * X + C d | of_b_eq_zero rfl rfl | lemma | cubic.of_b_eq_zero' | algebra | src/algebra/cubic_discriminant.lean | [
"data.polynomial.splits"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_c_eq_zero (ha : P.a = 0) (hb : P.b = 0) (hc : P.c = 0) : P.to_poly = C P.d | by rw [of_b_eq_zero ha hb, hc, C_0, zero_mul, zero_add] | lemma | cubic.of_c_eq_zero | algebra | src/algebra/cubic_discriminant.lean | [
"data.polynomial.splits"
] | [
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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