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 |
|---|---|---|---|---|---|---|---|---|---|---|
lt_of_smul_lt_smul_of_nonneg (h : c • a < c • b) (hc : 0 ≤ c) : a < b | hc.eq_or_lt.elim (λ hc, false.elim $ lt_irrefl (0:M) $ by rwa [← hc, zero_smul, zero_smul] at h)
(ordered_smul.lt_of_smul_lt_smul_of_pos h) | lemma | lt_of_smul_lt_smul_of_nonneg | algebra.order | src/algebra/order/smul.lean | [
"algebra.module.pi",
"algebra.module.prod",
"algebra.order.monoid.prod",
"algebra.order.pi",
"data.set.pointwise.smul",
"tactic.positivity"
] | [
"zero_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_lt_smul_iff_of_pos (hc : 0 < c) : c • a < c • b ↔ a < b | ⟨λ h, lt_of_smul_lt_smul_of_nonneg h hc.le, λ h, smul_lt_smul_of_pos h hc⟩ | lemma | smul_lt_smul_iff_of_pos | algebra.order | src/algebra/order/smul.lean | [
"algebra.module.pi",
"algebra.module.prod",
"algebra.order.monoid.prod",
"algebra.order.pi",
"data.set.pointwise.smul",
"tactic.positivity"
] | [
"lt_of_smul_lt_smul_of_nonneg",
"smul_lt_smul_of_pos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_pos_iff_of_pos (hc : 0 < c) : 0 < c • a ↔ 0 < a | calc 0 < c • a ↔ c • 0 < c • a : by rw smul_zero
... ↔ 0 < a : smul_lt_smul_iff_of_pos hc | lemma | smul_pos_iff_of_pos | algebra.order | src/algebra/order/smul.lean | [
"algebra.module.pi",
"algebra.module.prod",
"algebra.order.monoid.prod",
"algebra.order.pi",
"data.set.pointwise.smul",
"tactic.positivity"
] | [
"smul_lt_smul_iff_of_pos",
"smul_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monotone_smul_left (hc : 0 ≤ c) : monotone (has_smul.smul c : M → M) | λ a b h, smul_le_smul_of_nonneg h hc | lemma | monotone_smul_left | algebra.order | src/algebra/order/smul.lean | [
"algebra.module.pi",
"algebra.module.prod",
"algebra.order.monoid.prod",
"algebra.order.pi",
"data.set.pointwise.smul",
"tactic.positivity"
] | [
"monotone",
"smul_le_smul_of_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_mono_smul_left (hc : 0 < c) : strict_mono (has_smul.smul c : M → M) | λ a b h, smul_lt_smul_of_pos h hc | lemma | strict_mono_smul_left | algebra.order | src/algebra/order/smul.lean | [
"algebra.module.pi",
"algebra.module.prod",
"algebra.order.monoid.prod",
"algebra.order.pi",
"data.set.pointwise.smul",
"tactic.positivity"
] | [
"smul_lt_smul_of_pos",
"strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_lower_bounds_subset_lower_bounds_smul (hc : 0 ≤ c) :
c • lower_bounds s ⊆ lower_bounds (c • s) | (monotone_smul_left hc).image_lower_bounds_subset_lower_bounds_image | lemma | smul_lower_bounds_subset_lower_bounds_smul | algebra.order | src/algebra/order/smul.lean | [
"algebra.module.pi",
"algebra.module.prod",
"algebra.order.monoid.prod",
"algebra.order.pi",
"data.set.pointwise.smul",
"tactic.positivity"
] | [
"lower_bounds",
"monotone_smul_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_upper_bounds_subset_upper_bounds_smul (hc : 0 ≤ c) :
c • upper_bounds s ⊆ upper_bounds (c • s) | (monotone_smul_left hc).image_upper_bounds_subset_upper_bounds_image | lemma | smul_upper_bounds_subset_upper_bounds_smul | algebra.order | src/algebra/order/smul.lean | [
"algebra.module.pi",
"algebra.module.prod",
"algebra.order.monoid.prod",
"algebra.order.pi",
"data.set.pointwise.smul",
"tactic.positivity"
] | [
"monotone_smul_left",
"upper_bounds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bdd_below.smul_of_nonneg (hs : bdd_below s) (hc : 0 ≤ c) : bdd_below (c • s) | (monotone_smul_left hc).map_bdd_below hs | lemma | bdd_below.smul_of_nonneg | algebra.order | src/algebra/order/smul.lean | [
"algebra.module.pi",
"algebra.module.prod",
"algebra.order.monoid.prod",
"algebra.order.pi",
"data.set.pointwise.smul",
"tactic.positivity"
] | [
"bdd_below",
"monotone_smul_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bdd_above.smul_of_nonneg (hs : bdd_above s) (hc : 0 ≤ c) : bdd_above (c • s) | (monotone_smul_left hc).map_bdd_above hs | lemma | bdd_above.smul_of_nonneg | algebra.order | src/algebra/order/smul.lean | [
"algebra.module.pi",
"algebra.module.prod",
"algebra.order.monoid.prod",
"algebra.order.pi",
"data.set.pointwise.smul",
"tactic.positivity"
] | [
"bdd_above",
"monotone_smul_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ordered_smul.mk'' [ordered_semiring 𝕜] [linear_ordered_add_comm_monoid M] [smul_with_zero 𝕜 M]
(h : ∀ ⦃c : 𝕜⦄, 0 < c → strict_mono (λ a : M, c • a)) :
ordered_smul 𝕜 M | { smul_lt_smul_of_pos := λ a b c hab hc, h hc hab,
lt_of_smul_lt_smul_of_pos := λ a b c hab hc, (h hc).lt_iff_lt.1 hab } | lemma | ordered_smul.mk'' | algebra.order | src/algebra/order/smul.lean | [
"algebra.module.pi",
"algebra.module.prod",
"algebra.order.monoid.prod",
"algebra.order.pi",
"data.set.pointwise.smul",
"tactic.positivity"
] | [
"linear_ordered_add_comm_monoid",
"ordered_semiring",
"ordered_smul",
"smul_lt_smul_of_pos",
"smul_with_zero",
"strict_mono"
] | To prove that a linear ordered monoid is an ordered module, it suffices to verify only the first
axiom of `ordered_smul`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nat.ordered_smul [linear_ordered_cancel_add_comm_monoid M] : ordered_smul ℕ M | ordered_smul.mk'' $ λ n hn a b hab, begin
cases n,
{ cases hn },
induction n with n ih,
{ simp only [one_nsmul, hab], },
{ simp only [succ_nsmul _ n.succ, add_lt_add hab (ih n.succ_pos)] }
end | instance | nat.ordered_smul | algebra.order | src/algebra/order/smul.lean | [
"algebra.module.pi",
"algebra.module.prod",
"algebra.order.monoid.prod",
"algebra.order.pi",
"data.set.pointwise.smul",
"tactic.positivity"
] | [
"ih",
"linear_ordered_cancel_add_comm_monoid",
"ordered_smul",
"ordered_smul.mk''"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
int.ordered_smul [linear_ordered_add_comm_group M] : ordered_smul ℤ M | ordered_smul.mk'' $ λ n hn, begin
cases n,
{ simp only [int.of_nat_eq_coe, int.coe_nat_pos, coe_nat_zsmul] at ⊢ hn,
exact strict_mono_smul_left hn },
{ cases (int.neg_succ_not_pos _).1 hn }
end | instance | int.ordered_smul | algebra.order | src/algebra/order/smul.lean | [
"algebra.module.pi",
"algebra.module.prod",
"algebra.order.monoid.prod",
"algebra.order.pi",
"data.set.pointwise.smul",
"tactic.positivity"
] | [
"int.coe_nat_pos",
"int.neg_succ_not_pos",
"linear_ordered_add_comm_group",
"ordered_smul",
"ordered_smul.mk''",
"strict_mono_smul_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_ordered_semiring.to_ordered_smul : ordered_smul R R | ordered_smul.mk'' $ λ c, strict_mono_mul_left_of_pos | instance | linear_ordered_semiring.to_ordered_smul | algebra.order | src/algebra/order/smul.lean | [
"algebra.module.pi",
"algebra.module.prod",
"algebra.order.monoid.prod",
"algebra.order.pi",
"data.set.pointwise.smul",
"tactic.positivity"
] | [
"ordered_smul",
"ordered_smul.mk''",
"strict_mono_mul_left_of_pos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_max (ha : 0 ≤ a) (b₁ b₂ : M) : a • max b₁ b₂ = max (a • b₁) (a • b₂) | (monotone_smul_left ha : monotone (_ : M → M)).map_max | lemma | smul_max | algebra.order | src/algebra/order/smul.lean | [
"algebra.module.pi",
"algebra.module.prod",
"algebra.order.monoid.prod",
"algebra.order.pi",
"data.set.pointwise.smul",
"tactic.positivity"
] | [
"monotone",
"monotone_smul_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_min (ha : 0 ≤ a) (b₁ b₂ : M) : a • min b₁ b₂ = min (a • b₁) (a • b₂) | (monotone_smul_left ha : monotone (_ : M → M)).map_min | lemma | smul_min | algebra.order | src/algebra/order/smul.lean | [
"algebra.module.pi",
"algebra.module.prod",
"algebra.order.monoid.prod",
"algebra.order.pi",
"data.set.pointwise.smul",
"tactic.positivity"
] | [
"monotone",
"monotone_smul_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ordered_smul.mk' (h : ∀ ⦃a b : M⦄ ⦃c : 𝕜⦄, a < b → 0 < c → c • a ≤ c • b) :
ordered_smul 𝕜 M | begin
have hlt' : ∀ ⦃a b : M⦄ ⦃c : 𝕜⦄, a < b → 0 < c → c • a < c • b,
{ refine λ a b c hab hc, (h hab hc).lt_of_ne _,
rw [ne.def, hc.ne'.is_unit.smul_left_cancel],
exact hab.ne },
refine { smul_lt_smul_of_pos := hlt', .. },
intros a b c hab hc,
obtain ⟨c, rfl⟩ := hc.ne'.is_unit,
rw [← inv_smul_smul... | lemma | ordered_smul.mk' | algebra.order | src/algebra/order/smul.lean | [
"algebra.module.pi",
"algebra.module.prod",
"algebra.order.monoid.prod",
"algebra.order.pi",
"data.set.pointwise.smul",
"tactic.positivity"
] | [
"inv_smul_smul",
"ordered_smul",
"pos_of_mul_pos_right",
"smul_lt_smul_of_pos",
"zero_lt_one"
] | To prove that a vector space over a linear ordered field is ordered, it suffices to verify only
the first axiom of `ordered_smul`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pi.ordered_smul {M : ι → Type*} [Π i, ordered_add_comm_monoid (M i)]
[Π i, mul_action_with_zero 𝕜 (M i)] [∀ i, ordered_smul 𝕜 (M i)] :
ordered_smul 𝕜 (Π i, M i) | ordered_smul.mk' $ λ v u c h hc i, smul_le_smul_of_nonneg (h.le i) hc.le | instance | pi.ordered_smul | algebra.order | src/algebra/order/smul.lean | [
"algebra.module.pi",
"algebra.module.prod",
"algebra.order.monoid.prod",
"algebra.order.pi",
"data.set.pointwise.smul",
"tactic.positivity"
] | [
"mul_action_with_zero",
"ordered_add_comm_monoid",
"ordered_smul",
"ordered_smul.mk'",
"smul_le_smul_of_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi.ordered_smul' [ordered_smul 𝕜 M] : ordered_smul 𝕜 (ι → M) | pi.ordered_smul | instance | pi.ordered_smul' | algebra.order | src/algebra/order/smul.lean | [
"algebra.module.pi",
"algebra.module.prod",
"algebra.order.monoid.prod",
"algebra.order.pi",
"data.set.pointwise.smul",
"tactic.positivity"
] | [
"ordered_smul",
"pi.ordered_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi.ordered_smul'' : ordered_smul 𝕜 (ι → 𝕜) | @pi.ordered_smul' ι 𝕜 𝕜 _ _ _ _ | instance | pi.ordered_smul'' | algebra.order | src/algebra/order/smul.lean | [
"algebra.module.pi",
"algebra.module.prod",
"algebra.order.monoid.prod",
"algebra.order.pi",
"data.set.pointwise.smul",
"tactic.positivity"
] | [
"ordered_smul",
"pi.ordered_smul'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_le_smul_iff_of_pos (hc : 0 < c) : c • a ≤ c • b ↔ a ≤ b | ⟨λ h, inv_smul_smul₀ hc.ne' a ▸ inv_smul_smul₀ hc.ne' b ▸
smul_le_smul_of_nonneg h (inv_nonneg.2 hc.le),
λ h, smul_le_smul_of_nonneg h hc.le⟩ | lemma | smul_le_smul_iff_of_pos | algebra.order | src/algebra/order/smul.lean | [
"algebra.module.pi",
"algebra.module.prod",
"algebra.order.monoid.prod",
"algebra.order.pi",
"data.set.pointwise.smul",
"tactic.positivity"
] | [
"inv_smul_smul₀",
"smul_le_smul_of_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_smul_le_iff (h : 0 < c) : c⁻¹ • a ≤ b ↔ a ≤ c • b | by { rw [←smul_le_smul_iff_of_pos h, smul_inv_smul₀ h.ne'], apply_instance } | lemma | inv_smul_le_iff | algebra.order | src/algebra/order/smul.lean | [
"algebra.module.pi",
"algebra.module.prod",
"algebra.order.monoid.prod",
"algebra.order.pi",
"data.set.pointwise.smul",
"tactic.positivity"
] | [
"smul_inv_smul₀"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_smul_lt_iff (h : 0 < c) : c⁻¹ • a < b ↔ a < c • b | by { rw [←smul_lt_smul_iff_of_pos h, smul_inv_smul₀ h.ne'], apply_instance } | lemma | inv_smul_lt_iff | algebra.order | src/algebra/order/smul.lean | [
"algebra.module.pi",
"algebra.module.prod",
"algebra.order.monoid.prod",
"algebra.order.pi",
"data.set.pointwise.smul",
"tactic.positivity"
] | [
"smul_inv_smul₀"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_inv_smul_iff (h : 0 < c) : a ≤ c⁻¹ • b ↔ c • a ≤ b | by { rw [←smul_le_smul_iff_of_pos h, smul_inv_smul₀ h.ne'], apply_instance } | lemma | le_inv_smul_iff | algebra.order | src/algebra/order/smul.lean | [
"algebra.module.pi",
"algebra.module.prod",
"algebra.order.monoid.prod",
"algebra.order.pi",
"data.set.pointwise.smul",
"tactic.positivity"
] | [
"smul_inv_smul₀"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_inv_smul_iff (h : 0 < c) : a < c⁻¹ • b ↔ c • a < b | by { rw [←smul_lt_smul_iff_of_pos h, smul_inv_smul₀ h.ne'], apply_instance } | lemma | lt_inv_smul_iff | algebra.order | src/algebra/order/smul.lean | [
"algebra.module.pi",
"algebra.module.prod",
"algebra.order.monoid.prod",
"algebra.order.pi",
"data.set.pointwise.smul",
"tactic.positivity"
] | [
"smul_inv_smul₀"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
order_iso.smul_left (hc : 0 < c) : M ≃o M | { to_fun := λ b, c • b,
inv_fun := λ b, c⁻¹ • b,
left_inv := inv_smul_smul₀ hc.ne',
right_inv := smul_inv_smul₀ hc.ne',
map_rel_iff' := λ b₁ b₂, smul_le_smul_iff_of_pos hc } | def | order_iso.smul_left | algebra.order | src/algebra/order/smul.lean | [
"algebra.module.pi",
"algebra.module.prod",
"algebra.order.monoid.prod",
"algebra.order.pi",
"data.set.pointwise.smul",
"tactic.positivity"
] | [
"inv_fun",
"inv_smul_smul₀",
"smul_inv_smul₀",
"smul_le_smul_iff_of_pos"
] | Left scalar multiplication as an order isomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lower_bounds_smul_of_pos (hc : 0 < c) : lower_bounds (c • s) = c • lower_bounds s | (order_iso.smul_left _ hc).lower_bounds_image | lemma | lower_bounds_smul_of_pos | algebra.order | src/algebra/order/smul.lean | [
"algebra.module.pi",
"algebra.module.prod",
"algebra.order.monoid.prod",
"algebra.order.pi",
"data.set.pointwise.smul",
"tactic.positivity"
] | [
"lower_bounds",
"order_iso.smul_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
upper_bounds_smul_of_pos (hc : 0 < c) : upper_bounds (c • s) = c • upper_bounds s | (order_iso.smul_left _ hc).upper_bounds_image | lemma | upper_bounds_smul_of_pos | algebra.order | src/algebra/order/smul.lean | [
"algebra.module.pi",
"algebra.module.prod",
"algebra.order.monoid.prod",
"algebra.order.pi",
"data.set.pointwise.smul",
"tactic.positivity"
] | [
"order_iso.smul_left",
"upper_bounds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bdd_below_smul_iff_of_pos (hc : 0 < c) : bdd_below (c • s) ↔ bdd_below s | (order_iso.smul_left _ hc).bdd_below_image | lemma | bdd_below_smul_iff_of_pos | algebra.order | src/algebra/order/smul.lean | [
"algebra.module.pi",
"algebra.module.prod",
"algebra.order.monoid.prod",
"algebra.order.pi",
"data.set.pointwise.smul",
"tactic.positivity"
] | [
"bdd_below",
"order_iso.smul_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bdd_above_smul_iff_of_pos (hc : 0 < c) : bdd_above (c • s) ↔ bdd_above s | (order_iso.smul_left _ hc).bdd_above_image | lemma | bdd_above_smul_iff_of_pos | algebra.order | src/algebra/order/smul.lean | [
"algebra.module.pi",
"algebra.module.prod",
"algebra.order.monoid.prod",
"algebra.order.pi",
"data.set.pointwise.smul",
"tactic.positivity"
] | [
"bdd_above",
"order_iso.smul_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_nonneg_of_pos_of_nonneg (ha : 0 < a) (hb : 0 ≤ b) : 0 ≤ a • b | smul_nonneg ha.le hb | lemma | tactic.smul_nonneg_of_pos_of_nonneg | algebra.order | src/algebra/order/smul.lean | [
"algebra.module.pi",
"algebra.module.prod",
"algebra.order.monoid.prod",
"algebra.order.pi",
"data.set.pointwise.smul",
"tactic.positivity"
] | [
"smul_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_nonneg_of_nonneg_of_pos (ha : 0 ≤ a) (hb : 0 < b) : 0 ≤ a • b | smul_nonneg ha hb.le | lemma | tactic.smul_nonneg_of_nonneg_of_pos | algebra.order | src/algebra/order/smul.lean | [
"algebra.module.pi",
"algebra.module.prod",
"algebra.order.monoid.prod",
"algebra.order.pi",
"data.set.pointwise.smul",
"tactic.positivity"
] | [
"smul_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_ne_zero_of_pos_of_ne_zero [preorder R] (ha : 0 < a) (hb : b ≠ 0) : a • b ≠ 0 | smul_ne_zero ha.ne' hb | lemma | tactic.smul_ne_zero_of_pos_of_ne_zero | algebra.order | src/algebra/order/smul.lean | [
"algebra.module.pi",
"algebra.module.prod",
"algebra.order.monoid.prod",
"algebra.order.pi",
"data.set.pointwise.smul",
"tactic.positivity"
] | [
"smul_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_ne_zero_of_ne_zero_of_pos [preorder M] (ha : a ≠ 0) (hb : 0 < b) : a • b ≠ 0 | smul_ne_zero ha hb.ne' | lemma | tactic.smul_ne_zero_of_ne_zero_of_pos | algebra.order | src/algebra/order/smul.lean | [
"algebra.module.pi",
"algebra.module.prod",
"algebra.order.monoid.prod",
"algebra.order.pi",
"data.set.pointwise.smul",
"tactic.positivity"
] | [
"smul_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
positivity_smul : expr → tactic strictness | | e@`(%%a • %%b) := do
strictness_a ← core a,
strictness_b ← core b,
match strictness_a, strictness_b with
| positive pa, positive pb := positive <$> mk_app ``smul_pos [pa, pb]
| positive pa, nonnegative pb := nonnegative <$> mk_app ``smul_nonneg_of_pos_of_nonneg [pa, pb]
| nonnegative pa, positive pb := no... | def | tactic.positivity_smul | algebra.order | src/algebra/order/smul.lean | [
"algebra.module.pi",
"algebra.module.prod",
"algebra.order.monoid.prod",
"algebra.order.pi",
"data.set.pointwise.smul",
"tactic.positivity"
] | [
"smul_nonneg"
] | Extension for the `positivity` tactic: scalar multiplication is nonnegative/positive/nonzero if
both sides are. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_Ico_div (a b : α) : ℤ | (exists_unique_sub_zsmul_mem_Ico hp b a).some | def | to_Ico_div | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"exists_unique_sub_zsmul_mem_Ico"
] | The unique integer such that this multiple of `p`, subtracted from `b`, is in `Ico a (a + p)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sub_to_Ico_div_zsmul_mem_Ico (a b : α) : b - to_Ico_div hp a b • p ∈ set.Ico a (a + p) | (exists_unique_sub_zsmul_mem_Ico hp b a).some_spec.1 | lemma | sub_to_Ico_div_zsmul_mem_Ico | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"exists_unique_sub_zsmul_mem_Ico",
"set.Ico",
"to_Ico_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ico_div_eq_of_sub_zsmul_mem_Ico (h : b - n • p ∈ set.Ico a (a + p)) :
to_Ico_div hp a b = n | ((exists_unique_sub_zsmul_mem_Ico hp b a).some_spec.2 _ h).symm | lemma | to_Ico_div_eq_of_sub_zsmul_mem_Ico | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"exists_unique_sub_zsmul_mem_Ico",
"set.Ico",
"to_Ico_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ioc_div (a b : α) : ℤ | (exists_unique_sub_zsmul_mem_Ioc hp b a).some | def | to_Ioc_div | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"exists_unique_sub_zsmul_mem_Ioc"
] | The unique integer such that this multiple of `p`, subtracted from `b`, is in `Ioc a (a + p)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sub_to_Ioc_div_zsmul_mem_Ioc (a b : α) : b - to_Ioc_div hp a b • p ∈ set.Ioc a (a + p) | (exists_unique_sub_zsmul_mem_Ioc hp b a).some_spec.1 | lemma | sub_to_Ioc_div_zsmul_mem_Ioc | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"exists_unique_sub_zsmul_mem_Ioc",
"set.Ioc",
"to_Ioc_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ioc_div_eq_of_sub_zsmul_mem_Ioc (h : b - n • p ∈ set.Ioc a (a + p)) :
to_Ioc_div hp a b = n | ((exists_unique_sub_zsmul_mem_Ioc hp b a).some_spec.2 _ h).symm | lemma | to_Ioc_div_eq_of_sub_zsmul_mem_Ioc | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"exists_unique_sub_zsmul_mem_Ioc",
"set.Ioc",
"to_Ioc_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ico_mod (a b : α) : α | b - to_Ico_div hp a b • p | def | to_Ico_mod | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"to_Ico_div"
] | Reduce `b` to the interval `Ico a (a + p)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_Ioc_mod (a b : α) : α | b - to_Ioc_div hp a b • p | def | to_Ioc_mod | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"to_Ioc_div"
] | Reduce `b` to the interval `Ioc a (a + p)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_Ico_mod_mem_Ico (a b : α) : to_Ico_mod hp a b ∈ set.Ico a (a + p) | sub_to_Ico_div_zsmul_mem_Ico hp a b | lemma | to_Ico_mod_mem_Ico | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"set.Ico",
"sub_to_Ico_div_zsmul_mem_Ico",
"to_Ico_mod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ico_mod_mem_Ico' (b : α) : to_Ico_mod hp 0 b ∈ set.Ico 0 p | by { convert to_Ico_mod_mem_Ico hp 0 b, exact (zero_add p).symm, } | lemma | to_Ico_mod_mem_Ico' | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"set.Ico",
"to_Ico_mod",
"to_Ico_mod_mem_Ico"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ioc_mod_mem_Ioc (a b : α) : to_Ioc_mod hp a b ∈ set.Ioc a (a + p) | sub_to_Ioc_div_zsmul_mem_Ioc hp a b | lemma | to_Ioc_mod_mem_Ioc | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"set.Ioc",
"sub_to_Ioc_div_zsmul_mem_Ioc",
"to_Ioc_mod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left_le_to_Ico_mod (a b : α) : a ≤ to_Ico_mod hp a b | (set.mem_Ico.1 (to_Ico_mod_mem_Ico hp a b)).1 | lemma | left_le_to_Ico_mod | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"to_Ico_mod",
"to_Ico_mod_mem_Ico"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left_lt_to_Ioc_mod (a b : α) : a < to_Ioc_mod hp a b | (set.mem_Ioc.1 (to_Ioc_mod_mem_Ioc hp a b)).1 | lemma | left_lt_to_Ioc_mod | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"to_Ioc_mod",
"to_Ioc_mod_mem_Ioc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ico_mod_lt_right (a b : α) : to_Ico_mod hp a b < a + p | (set.mem_Ico.1 (to_Ico_mod_mem_Ico hp a b)).2 | lemma | to_Ico_mod_lt_right | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"to_Ico_mod",
"to_Ico_mod_mem_Ico"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ioc_mod_le_right (a b : α) : to_Ioc_mod hp a b ≤ a + p | (set.mem_Ioc.1 (to_Ioc_mod_mem_Ioc hp a b)).2 | lemma | to_Ioc_mod_le_right | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"to_Ioc_mod",
"to_Ioc_mod_mem_Ioc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
self_sub_to_Ico_div_zsmul (a b : α) : b - to_Ico_div hp a b • p = to_Ico_mod hp a b | rfl | lemma | self_sub_to_Ico_div_zsmul | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"to_Ico_div",
"to_Ico_mod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
self_sub_to_Ioc_div_zsmul (a b : α) : b - to_Ioc_div hp a b • p = to_Ioc_mod hp a b | rfl | lemma | self_sub_to_Ioc_div_zsmul | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"to_Ioc_div",
"to_Ioc_mod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ico_div_zsmul_sub_self (a b : α) :
to_Ico_div hp a b • p - b = -to_Ico_mod hp a b | by rw [to_Ico_mod, neg_sub] | lemma | to_Ico_div_zsmul_sub_self | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"to_Ico_div",
"to_Ico_mod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ioc_div_zsmul_sub_self (a b : α) :
to_Ioc_div hp a b • p - b = -to_Ioc_mod hp a b | by rw [to_Ioc_mod, neg_sub] | lemma | to_Ioc_div_zsmul_sub_self | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"to_Ioc_div",
"to_Ioc_mod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ico_mod_sub_self (a b : α) : to_Ico_mod hp a b - b = -to_Ico_div hp a b • p | by rw [to_Ico_mod, sub_sub_cancel_left, neg_smul] | lemma | to_Ico_mod_sub_self | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"neg_smul",
"to_Ico_div",
"to_Ico_mod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ioc_mod_sub_self (a b : α) : to_Ioc_mod hp a b - b = -to_Ioc_div hp a b • p | by rw [to_Ioc_mod, sub_sub_cancel_left, neg_smul] | lemma | to_Ioc_mod_sub_self | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"neg_smul",
"to_Ioc_div",
"to_Ioc_mod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
self_sub_to_Ico_mod (a b : α) : b - to_Ico_mod hp a b = to_Ico_div hp a b • p | by rw [to_Ico_mod, sub_sub_cancel] | lemma | self_sub_to_Ico_mod | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"to_Ico_div",
"to_Ico_mod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
self_sub_to_Ioc_mod (a b : α) : b - to_Ioc_mod hp a b = to_Ioc_div hp a b • p | by rw [to_Ioc_mod, sub_sub_cancel] | lemma | self_sub_to_Ioc_mod | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"to_Ioc_div",
"to_Ioc_mod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ico_mod_add_to_Ico_div_zsmul (a b : α) :
to_Ico_mod hp a b + to_Ico_div hp a b • p = b | by rw [to_Ico_mod, sub_add_cancel] | lemma | to_Ico_mod_add_to_Ico_div_zsmul | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"to_Ico_div",
"to_Ico_mod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ioc_mod_add_to_Ioc_div_zsmul (a b : α) :
to_Ioc_mod hp a b + to_Ioc_div hp a b • p = b | by rw [to_Ioc_mod, sub_add_cancel] | lemma | to_Ioc_mod_add_to_Ioc_div_zsmul | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"to_Ioc_div",
"to_Ioc_mod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ico_div_zsmul_sub_to_Ico_mod (a b : α) :
to_Ico_div hp a b • p + to_Ico_mod hp a b = b | by rw [add_comm, to_Ico_mod_add_to_Ico_div_zsmul] | lemma | to_Ico_div_zsmul_sub_to_Ico_mod | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"to_Ico_div",
"to_Ico_mod",
"to_Ico_mod_add_to_Ico_div_zsmul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ioc_div_zsmul_sub_to_Ioc_mod (a b : α) :
to_Ioc_div hp a b • p + to_Ioc_mod hp a b = b | by rw [add_comm, to_Ioc_mod_add_to_Ioc_div_zsmul] | lemma | to_Ioc_div_zsmul_sub_to_Ioc_mod | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"to_Ioc_div",
"to_Ioc_mod",
"to_Ioc_mod_add_to_Ioc_div_zsmul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ico_mod_eq_iff : to_Ico_mod hp a b = c ↔ c ∈ set.Ico a (a + p) ∧ ∃ z : ℤ, b = c + z • p | begin
refine ⟨λ h, ⟨h ▸ to_Ico_mod_mem_Ico hp a b, to_Ico_div hp a b,
h ▸ (to_Ico_mod_add_to_Ico_div_zsmul _ _ _).symm⟩, _⟩,
simp_rw ←@sub_eq_iff_eq_add,
rintro ⟨hc, n, rfl⟩,
rw [←to_Ico_div_eq_of_sub_zsmul_mem_Ico hp hc, to_Ico_mod],
end | lemma | to_Ico_mod_eq_iff | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"set.Ico",
"to_Ico_div",
"to_Ico_mod",
"to_Ico_mod_add_to_Ico_div_zsmul",
"to_Ico_mod_mem_Ico"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ioc_mod_eq_iff : to_Ioc_mod hp a b = c ↔ c ∈ set.Ioc a (a + p) ∧ ∃ z : ℤ, b = c + z • p | begin
refine ⟨λ h, ⟨h ▸ to_Ioc_mod_mem_Ioc hp a b, to_Ioc_div hp a b,
h ▸ (to_Ioc_mod_add_to_Ioc_div_zsmul hp _ _).symm⟩, _⟩,
simp_rw ←@sub_eq_iff_eq_add,
rintro ⟨hc, n, rfl⟩,
rw [←to_Ioc_div_eq_of_sub_zsmul_mem_Ioc hp hc, to_Ioc_mod],
end | lemma | to_Ioc_mod_eq_iff | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"set.Ioc",
"to_Ioc_div",
"to_Ioc_mod",
"to_Ioc_mod_add_to_Ioc_div_zsmul",
"to_Ioc_mod_mem_Ioc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ico_div_apply_left (a : α) : to_Ico_div hp a a = 0 | to_Ico_div_eq_of_sub_zsmul_mem_Ico hp $ by simp [hp] | lemma | to_Ico_div_apply_left | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"to_Ico_div",
"to_Ico_div_eq_of_sub_zsmul_mem_Ico"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ioc_div_apply_left (a : α) : to_Ioc_div hp a a = -1 | to_Ioc_div_eq_of_sub_zsmul_mem_Ioc hp $ by simp [hp] | lemma | to_Ioc_div_apply_left | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"to_Ioc_div",
"to_Ioc_div_eq_of_sub_zsmul_mem_Ioc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ico_mod_apply_left (a : α) : to_Ico_mod hp a a = a | by { rw [to_Ico_mod_eq_iff hp, set.left_mem_Ico], exact ⟨lt_add_of_pos_right _ hp, 0, by simp⟩ } | lemma | to_Ico_mod_apply_left | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"set.left_mem_Ico",
"to_Ico_mod",
"to_Ico_mod_eq_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ioc_mod_apply_left (a : α) : to_Ioc_mod hp a a = a + p | by { rw [to_Ioc_mod_eq_iff hp, set.right_mem_Ioc], exact ⟨lt_add_of_pos_right _ hp, -1, by simp⟩ } | lemma | to_Ioc_mod_apply_left | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"set.right_mem_Ioc",
"to_Ioc_mod",
"to_Ioc_mod_eq_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ico_div_apply_right (a : α) : to_Ico_div hp a (a + p) = 1 | to_Ico_div_eq_of_sub_zsmul_mem_Ico hp $ by simp [hp] | lemma | to_Ico_div_apply_right | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"to_Ico_div",
"to_Ico_div_eq_of_sub_zsmul_mem_Ico"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ioc_div_apply_right (a : α) : to_Ioc_div hp a (a + p) = 0 | to_Ioc_div_eq_of_sub_zsmul_mem_Ioc hp $ by simp [hp] | lemma | to_Ioc_div_apply_right | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"to_Ioc_div",
"to_Ioc_div_eq_of_sub_zsmul_mem_Ioc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ico_mod_apply_right (a : α) : to_Ico_mod hp a (a + p) = a | by { rw [to_Ico_mod_eq_iff hp, set.left_mem_Ico], exact ⟨lt_add_of_pos_right _ hp, 1, by simp⟩ } | lemma | to_Ico_mod_apply_right | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"set.left_mem_Ico",
"to_Ico_mod",
"to_Ico_mod_eq_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ioc_mod_apply_right (a : α) : to_Ioc_mod hp a (a + p) = a + p | by { rw [to_Ioc_mod_eq_iff hp, set.right_mem_Ioc], exact ⟨lt_add_of_pos_right _ hp, 0, by simp⟩ } | lemma | to_Ioc_mod_apply_right | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"set.right_mem_Ioc",
"to_Ioc_mod",
"to_Ioc_mod_eq_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ico_div_add_zsmul (a b : α) (m : ℤ) :
to_Ico_div hp a (b + m • p) = to_Ico_div hp a b + m | to_Ico_div_eq_of_sub_zsmul_mem_Ico hp $
by simpa only [add_smul, add_sub_add_right_eq_sub] using sub_to_Ico_div_zsmul_mem_Ico hp a b | lemma | to_Ico_div_add_zsmul | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"add_smul",
"sub_to_Ico_div_zsmul_mem_Ico",
"to_Ico_div",
"to_Ico_div_eq_of_sub_zsmul_mem_Ico"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ico_div_add_zsmul' (a b: α) (m : ℤ) :
to_Ico_div hp (a + m • p) b = to_Ico_div hp a b - m | begin
refine to_Ico_div_eq_of_sub_zsmul_mem_Ico _ _,
rw [sub_smul, ←sub_add, add_right_comm],
simpa using sub_to_Ico_div_zsmul_mem_Ico hp a b,
end | lemma | to_Ico_div_add_zsmul' | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"sub_smul",
"sub_to_Ico_div_zsmul_mem_Ico",
"to_Ico_div",
"to_Ico_div_eq_of_sub_zsmul_mem_Ico"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ioc_div_add_zsmul (a b : α) (m : ℤ) :
to_Ioc_div hp a (b + m • p) = to_Ioc_div hp a b + m | to_Ioc_div_eq_of_sub_zsmul_mem_Ioc hp $
by simpa only [add_smul, add_sub_add_right_eq_sub] using sub_to_Ioc_div_zsmul_mem_Ioc hp a b | lemma | to_Ioc_div_add_zsmul | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"add_smul",
"sub_to_Ioc_div_zsmul_mem_Ioc",
"to_Ioc_div",
"to_Ioc_div_eq_of_sub_zsmul_mem_Ioc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ioc_div_add_zsmul' (a b : α) (m : ℤ) :
to_Ioc_div hp (a + m • p) b = to_Ioc_div hp a b - m | begin
refine to_Ioc_div_eq_of_sub_zsmul_mem_Ioc _ _,
rw [sub_smul, ←sub_add, add_right_comm],
simpa using sub_to_Ioc_div_zsmul_mem_Ioc hp a b,
end | lemma | to_Ioc_div_add_zsmul' | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"sub_smul",
"sub_to_Ioc_div_zsmul_mem_Ioc",
"to_Ioc_div",
"to_Ioc_div_eq_of_sub_zsmul_mem_Ioc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ico_div_zsmul_add (a b : α) (m : ℤ) :
to_Ico_div hp a (m • p + b) = m + to_Ico_div hp a b | by rw [add_comm, to_Ico_div_add_zsmul, add_comm] | lemma | to_Ico_div_zsmul_add | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"to_Ico_div",
"to_Ico_div_add_zsmul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ioc_div_zsmul_add (a b : α) (m : ℤ) :
to_Ioc_div hp a (m • p + b) = m + to_Ioc_div hp a b | by rw [add_comm, to_Ioc_div_add_zsmul, add_comm] | lemma | to_Ioc_div_zsmul_add | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"to_Ioc_div",
"to_Ioc_div_add_zsmul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ico_div_sub_zsmul (a b : α) (m : ℤ) :
to_Ico_div hp a (b - m • p) = to_Ico_div hp a b - m | by rw [sub_eq_add_neg, ←neg_smul, to_Ico_div_add_zsmul, sub_eq_add_neg] | lemma | to_Ico_div_sub_zsmul | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"to_Ico_div",
"to_Ico_div_add_zsmul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ico_div_sub_zsmul' (a b : α) (m : ℤ) :
to_Ico_div hp (a - m • p) b = to_Ico_div hp a b + m | by rw [sub_eq_add_neg, ←neg_smul, to_Ico_div_add_zsmul', sub_neg_eq_add] | lemma | to_Ico_div_sub_zsmul' | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"to_Ico_div",
"to_Ico_div_add_zsmul'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ioc_div_sub_zsmul (a b : α) (m : ℤ) :
to_Ioc_div hp a (b - m • p) = to_Ioc_div hp a b - m | by rw [sub_eq_add_neg, ←neg_smul, to_Ioc_div_add_zsmul, sub_eq_add_neg] | lemma | to_Ioc_div_sub_zsmul | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"to_Ioc_div",
"to_Ioc_div_add_zsmul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ioc_div_sub_zsmul' (a b : α) (m : ℤ) :
to_Ioc_div hp (a - m • p) b = to_Ioc_div hp a b + m | by rw [sub_eq_add_neg, ←neg_smul, to_Ioc_div_add_zsmul', sub_neg_eq_add] | lemma | to_Ioc_div_sub_zsmul' | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"to_Ioc_div",
"to_Ioc_div_add_zsmul'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ico_div_add_right (a b : α) : to_Ico_div hp a (b + p) = to_Ico_div hp a b + 1 | by simpa only [one_zsmul] using to_Ico_div_add_zsmul hp a b 1 | lemma | to_Ico_div_add_right | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"to_Ico_div",
"to_Ico_div_add_zsmul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ico_div_add_right' (a b : α) : to_Ico_div hp (a + p) b = to_Ico_div hp a b - 1 | by simpa only [one_zsmul] using to_Ico_div_add_zsmul' hp a b 1 | lemma | to_Ico_div_add_right' | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"to_Ico_div",
"to_Ico_div_add_zsmul'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ioc_div_add_right (a b : α) : to_Ioc_div hp a (b + p) = to_Ioc_div hp a b + 1 | by simpa only [one_zsmul] using to_Ioc_div_add_zsmul hp a b 1 | lemma | to_Ioc_div_add_right | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"to_Ioc_div",
"to_Ioc_div_add_zsmul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ioc_div_add_right' (a b : α) : to_Ioc_div hp (a + p) b = to_Ioc_div hp a b - 1 | by simpa only [one_zsmul] using to_Ioc_div_add_zsmul' hp a b 1 | lemma | to_Ioc_div_add_right' | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"to_Ioc_div",
"to_Ioc_div_add_zsmul'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ico_div_add_left (a b : α) : to_Ico_div hp a (p + b) = to_Ico_div hp a b + 1 | by rw [add_comm, to_Ico_div_add_right] | lemma | to_Ico_div_add_left | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"to_Ico_div",
"to_Ico_div_add_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ico_div_add_left' (a b : α) : to_Ico_div hp (p + a) b = to_Ico_div hp a b - 1 | by rw [add_comm, to_Ico_div_add_right'] | lemma | to_Ico_div_add_left' | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"to_Ico_div",
"to_Ico_div_add_right'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ioc_div_add_left (a b : α) : to_Ioc_div hp a (p + b) = to_Ioc_div hp a b + 1 | by rw [add_comm, to_Ioc_div_add_right] | lemma | to_Ioc_div_add_left | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"to_Ioc_div",
"to_Ioc_div_add_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ioc_div_add_left' (a b : α) : to_Ioc_div hp (p + a) b = to_Ioc_div hp a b - 1 | by rw [add_comm, to_Ioc_div_add_right'] | lemma | to_Ioc_div_add_left' | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"to_Ioc_div",
"to_Ioc_div_add_right'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ico_div_sub (a b : α) : to_Ico_div hp a (b - p) = to_Ico_div hp a b - 1 | by simpa only [one_zsmul] using to_Ico_div_sub_zsmul hp a b 1 | lemma | to_Ico_div_sub | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"to_Ico_div",
"to_Ico_div_sub_zsmul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ico_div_sub' (a b : α) : to_Ico_div hp (a - p) b = to_Ico_div hp a b + 1 | by simpa only [one_zsmul] using to_Ico_div_sub_zsmul' hp a b 1 | lemma | to_Ico_div_sub' | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"to_Ico_div",
"to_Ico_div_sub_zsmul'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ioc_div_sub (a b : α) : to_Ioc_div hp a (b - p) = to_Ioc_div hp a b - 1 | by simpa only [one_zsmul] using to_Ioc_div_sub_zsmul hp a b 1 | lemma | to_Ioc_div_sub | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"to_Ioc_div",
"to_Ioc_div_sub_zsmul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ioc_div_sub' (a b : α) : to_Ioc_div hp (a - p) b = to_Ioc_div hp a b + 1 | by simpa only [one_zsmul] using to_Ioc_div_sub_zsmul' hp a b 1 | lemma | to_Ioc_div_sub' | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"to_Ioc_div",
"to_Ioc_div_sub_zsmul'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ico_div_sub_eq_to_Ico_div_add (a b c : α) :
to_Ico_div hp a (b - c) = to_Ico_div hp (a + c) b | begin
apply to_Ico_div_eq_of_sub_zsmul_mem_Ico,
rw [←sub_right_comm, set.sub_mem_Ico_iff_left, add_right_comm],
exact sub_to_Ico_div_zsmul_mem_Ico hp (a + c) b,
end | lemma | to_Ico_div_sub_eq_to_Ico_div_add | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"set.sub_mem_Ico_iff_left",
"sub_to_Ico_div_zsmul_mem_Ico",
"to_Ico_div",
"to_Ico_div_eq_of_sub_zsmul_mem_Ico"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ioc_div_sub_eq_to_Ioc_div_add (a b c : α) :
to_Ioc_div hp a (b - c) = to_Ioc_div hp (a + c) b | begin
apply to_Ioc_div_eq_of_sub_zsmul_mem_Ioc,
rw [←sub_right_comm, set.sub_mem_Ioc_iff_left, add_right_comm],
exact sub_to_Ioc_div_zsmul_mem_Ioc hp (a + c) b,
end | lemma | to_Ioc_div_sub_eq_to_Ioc_div_add | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"set.sub_mem_Ioc_iff_left",
"sub_to_Ioc_div_zsmul_mem_Ioc",
"to_Ioc_div",
"to_Ioc_div_eq_of_sub_zsmul_mem_Ioc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ico_div_sub_eq_to_Ico_div_add' (a b c : α) :
to_Ico_div hp (a - c) b = to_Ico_div hp a (b + c) | by rw [←sub_neg_eq_add, to_Ico_div_sub_eq_to_Ico_div_add, sub_eq_add_neg] | lemma | to_Ico_div_sub_eq_to_Ico_div_add' | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"to_Ico_div",
"to_Ico_div_sub_eq_to_Ico_div_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ioc_div_sub_eq_to_Ioc_div_add' (a b c : α) :
to_Ioc_div hp (a - c) b = to_Ioc_div hp a (b + c) | by rw [←sub_neg_eq_add, to_Ioc_div_sub_eq_to_Ioc_div_add, sub_eq_add_neg] | lemma | to_Ioc_div_sub_eq_to_Ioc_div_add' | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"to_Ioc_div",
"to_Ioc_div_sub_eq_to_Ioc_div_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ico_div_neg (a b : α) : to_Ico_div hp a (-b) = -(to_Ioc_div hp (-a) b + 1) | begin
suffices : to_Ico_div hp a (-b) = -(to_Ioc_div hp (-(a + p)) b),
{ rwa [neg_add, ←sub_eq_add_neg, to_Ioc_div_sub_eq_to_Ioc_div_add',
to_Ioc_div_add_right] at this },
rw [← neg_eq_iff_eq_neg, eq_comm],
apply to_Ioc_div_eq_of_sub_zsmul_mem_Ioc,
obtain ⟨hc, ho⟩ := sub_to_Ico_div_zsmul_mem_Ico hp a (-... | lemma | to_Ico_div_neg | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"sub_to_Ico_div_zsmul_mem_Ico",
"to_Ico_div",
"to_Ioc_div",
"to_Ioc_div_add_right",
"to_Ioc_div_eq_of_sub_zsmul_mem_Ioc",
"to_Ioc_div_sub_eq_to_Ioc_div_add'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ico_div_neg' (a b : α) : to_Ico_div hp (-a) b = -(to_Ioc_div hp a (-b) + 1) | by simpa only [neg_neg] using to_Ico_div_neg hp (-a) (-b) | lemma | to_Ico_div_neg' | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"to_Ico_div",
"to_Ico_div_neg",
"to_Ioc_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ioc_div_neg (a b : α) : to_Ioc_div hp a (-b) = -(to_Ico_div hp (-a) b + 1) | by rw [←neg_neg b, to_Ico_div_neg, neg_neg, neg_neg, neg_add', neg_neg, add_sub_cancel] | lemma | to_Ioc_div_neg | algebra.order | src/algebra/order/to_interval_mod.lean | [
"algebra.modeq",
"algebra.module.basic",
"algebra.order.archimedean",
"algebra.periodic",
"data.int.succ_pred",
"group_theory.quotient_group",
"order.circular"
] | [
"to_Ico_div",
"to_Ico_div_neg",
"to_Ioc_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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