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
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|---|---|---|---|---|---|---|---|---|---|---|
fract_eq_fract {a b : α} : fract a = fract b ↔ ∃ z : ℤ, a - b = z | ⟨λ h, ⟨⌊a⌋ - ⌊b⌋, begin
unfold fract at h, rw [int.cast_sub, sub_eq_sub_iff_sub_eq_sub.1 h],
end⟩, begin
rintro ⟨z, hz⟩,
refine fract_eq_iff.2 ⟨fract_nonneg _, fract_lt_one _, z + ⌊b⌋, _⟩,
rw [eq_add_of_sub_eq hz, add_comm, int.cast_add],
exact add_sub_sub_cancel _ _ _,
end⟩ | lemma | int.fract_eq_fract | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [
"int.cast_add",
"int.cast_sub"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fract_eq_self {a : α} : fract a = a ↔ 0 ≤ a ∧ a < 1 | fract_eq_iff.trans $ and.assoc.symm.trans $ and_iff_left ⟨0, by simp⟩ | lemma | int.fract_eq_self | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fract_fract (a : α) : fract (fract a) = fract a | fract_eq_self.2 ⟨fract_nonneg _, fract_lt_one _⟩ | lemma | int.fract_fract | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fract_add (a b : α) : ∃ z : ℤ, fract (a + b) - fract a - fract b = z | ⟨⌊a⌋ + ⌊b⌋ - ⌊a + b⌋, by { unfold fract, simp [sub_eq_add_neg], abel }⟩ | lemma | int.fract_add | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fract_neg {x : α} (hx : fract x ≠ 0) :
fract (-x) = 1 - fract x | begin
rw fract_eq_iff,
split,
{ rw [le_sub_iff_add_le, zero_add],
exact (fract_lt_one x).le, },
refine ⟨sub_lt_self _ (lt_of_le_of_ne' (fract_nonneg x) hx), -⌊x⌋ - 1, _⟩,
simp only [sub_sub_eq_add_sub, cast_sub, cast_neg, cast_one, sub_left_inj],
conv in (-x) {rw ← floor_add_fract x},
simp [-floor_add... | lemma | int.fract_neg | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [
"lt_of_le_of_ne'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fract_neg_eq_zero {x : α} : fract (-x) = 0 ↔ fract x = 0 | begin
simp only [fract_eq_iff, le_refl, zero_lt_one, tsub_zero, true_and],
split; rintros ⟨z, hz⟩; use [-z]; simp [← hz],
end | lemma | int.fract_neg_eq_zero | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [
"tsub_zero",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fract_mul_nat (a : α) (b : ℕ) : ∃ z : ℤ, fract a * b - fract (a * b) = z | begin
induction b with c hc,
use 0, simp,
rcases hc with ⟨z, hz⟩,
rw [nat.succ_eq_add_one, nat.cast_add, mul_add, mul_add, nat.cast_one, mul_one, mul_one],
rcases fract_add (a * c) a with ⟨y, hy⟩,
use z - y,
rw [int.cast_sub, ←hz, ←hy],
abel
end | lemma | int.fract_mul_nat | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [
"int.cast_sub",
"mul_one",
"nat.cast_add",
"nat.cast_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preimage_fract (s : set α) : fract ⁻¹' s = ⋃ m : ℤ, (λ x, x - m) ⁻¹' (s ∩ Ico (0 : α) 1) | begin
ext x,
simp only [mem_preimage, mem_Union, mem_inter_iff],
refine ⟨λ h, ⟨⌊x⌋, h, fract_nonneg x, fract_lt_one x⟩, _⟩,
rintro ⟨m, hms, hm0, hm1⟩,
obtain rfl : ⌊x⌋ = m, from floor_eq_iff.2 ⟨sub_nonneg.1 hm0, sub_lt_iff_lt_add'.1 hm1⟩,
exact hms
end | lemma | int.preimage_fract | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image_fract (s : set α) : fract '' s = ⋃ m : ℤ, (λ x, x - m) '' s ∩ Ico 0 1 | begin
ext x,
simp only [mem_image, mem_inter_iff, mem_Union], split,
{ rintro ⟨y, hy, rfl⟩,
exact ⟨⌊y⌋, ⟨y, hy, rfl⟩, fract_nonneg y, fract_lt_one y⟩ },
{ rintro ⟨m, ⟨y, hys, rfl⟩, h0, h1⟩,
obtain rfl : ⌊y⌋ = m, from floor_eq_iff.2 ⟨sub_nonneg.1 h0, sub_lt_iff_lt_add'.1 h1⟩,
exact ⟨y, hys, rfl⟩ }
en... | lemma | int.image_fract | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fract_div_mul_self_mem_Ico (a b : k) (ha : 0 < a) : fract (b/a) * a ∈ Ico 0 a | ⟨(zero_le_mul_right ha).2 (fract_nonneg (b/a)), (mul_lt_iff_lt_one_left ha).2 (fract_lt_one (b/a))⟩ | lemma | int.fract_div_mul_self_mem_Ico | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [
"mul_lt_iff_lt_one_left",
"zero_le_mul_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fract_div_mul_self_add_zsmul_eq (a b : k) (ha : a ≠ 0) :
fract (b/a) * a + ⌊b/a⌋ • a = b | by rw [zsmul_eq_mul, ← add_mul, fract_add_floor, div_mul_cancel b ha] | lemma | int.fract_div_mul_self_add_zsmul_eq | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [
"div_mul_cancel",
"zsmul_eq_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_floor_div_mul_nonneg (a : k) (hb : 0 < b) : 0 ≤ a - ⌊a / b⌋ * b | sub_nonneg_of_le $ (le_div_iff hb).1 $ floor_le _ | lemma | int.sub_floor_div_mul_nonneg | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [
"le_div_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_floor_div_mul_lt (a : k) (hb : 0 < b) : a - ⌊a / b⌋ * b < b | sub_lt_iff_lt_add.2 $ by { rw [←one_add_mul, ←div_lt_iff hb, add_comm], exact lt_floor_add_one _ } | lemma | int.sub_floor_div_mul_lt | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fract_div_nat_cast_eq_div_nat_cast_mod {m n : ℕ} :
fract ((m : k) / n) = ↑(m % n) / n | begin
rcases n.eq_zero_or_pos with rfl | hn, { simp, },
have hn' : 0 < (n : k), { norm_cast, assumption, },
refine fract_eq_iff.mpr ⟨by positivity, _, m / n, _⟩,
{ simpa only [div_lt_one hn', nat.cast_lt] using m.mod_lt hn, },
{ rw [sub_eq_iff_eq_add', ← mul_right_inj' hn'.ne.symm, mul_div_cancel' _ hn'.ne.sy... | lemma | int.fract_div_nat_cast_eq_div_nat_cast_mod | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [
"div_lt_one",
"mul_div_cancel'",
"mul_right_inj'",
"nat.cast_add",
"nat.cast_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fract_div_int_cast_eq_div_int_cast_mod {m : ℤ} {n : ℕ} :
fract ((m : k) / n) = ↑(m % n) / n | begin
rcases n.eq_zero_or_pos with rfl | hn, { simp, },
replace hn : 0 < (n : k), { norm_cast, assumption, },
have : ∀ {l : ℤ} (hl : 0 ≤ l), fract ((l : k) / n) = ↑(l % n) / n,
{ intros,
obtain ⟨l₀, rfl | rfl⟩ := l.eq_coe_or_neg,
{ rw [cast_coe_nat, ← coe_nat_mod, cast_coe_nat, fract_div_nat_cast_eq_div... | lemma | int.fract_div_int_cast_eq_div_int_cast_mod | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [
"add_div",
"mul_div_cancel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gc_ceil_coe : galois_connection ceil (coe : ℤ → α) | floor_ring.gc_ceil_coe | lemma | int.gc_ceil_coe | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [
"galois_connection"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ceil_le : ⌈a⌉ ≤ z ↔ a ≤ z | gc_ceil_coe a z | lemma | int.ceil_le | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
floor_neg : ⌊-a⌋ = -⌈a⌉ | eq_of_forall_le_iff (λ z, by rw [le_neg, ceil_le, le_floor, int.cast_neg, le_neg]) | lemma | int.floor_neg | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [
"eq_of_forall_le_iff",
"int.cast_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ceil_neg : ⌈-a⌉ = -⌊a⌋ | eq_of_forall_ge_iff (λ z, by rw [neg_le, ceil_le, le_floor, int.cast_neg, neg_le]) | lemma | int.ceil_neg | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [
"eq_of_forall_ge_iff",
"int.cast_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_ceil : z < ⌈a⌉ ↔ (z : α) < a | lt_iff_lt_of_le_iff_le ceil_le | lemma | int.lt_ceil | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [
"lt_iff_lt_of_le_iff_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_one_le_ceil_iff : z + 1 ≤ ⌈a⌉ ↔ (z : α) < a | by rw [← lt_ceil, add_one_le_iff] | lemma | int.add_one_le_ceil_iff | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_le_ceil_iff : 1 ≤ ⌈a⌉ ↔ 0 < a | by rw [← zero_add (1 : ℤ), add_one_le_ceil_iff, cast_zero] | lemma | int.one_le_ceil_iff | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ceil_le_floor_add_one (a : α) : ⌈a⌉ ≤ ⌊a⌋ + 1 | by { rw [ceil_le, int.cast_add, int.cast_one], exact (lt_floor_add_one a).le } | lemma | int.ceil_le_floor_add_one | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [
"int.cast_add",
"int.cast_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_ceil (a : α) : a ≤ ⌈a⌉ | gc_ceil_coe.le_u_l a | lemma | int.le_ceil | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ceil_int_cast (z : ℤ) : ⌈(z : α)⌉ = z | eq_of_forall_ge_iff $ λ a, by rw [ceil_le, int.cast_le] | lemma | int.ceil_int_cast | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [
"eq_of_forall_ge_iff",
"int.cast_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ceil_nat_cast (n : ℕ) : ⌈(n : α)⌉ = n | eq_of_forall_ge_iff $ λ a, by rw [ceil_le, ← cast_coe_nat, cast_le] | lemma | int.ceil_nat_cast | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [
"eq_of_forall_ge_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ceil_mono : monotone (ceil : α → ℤ) | gc_ceil_coe.monotone_l | lemma | int.ceil_mono | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ceil_add_int (a : α) (z : ℤ) : ⌈a + z⌉ = ⌈a⌉ + z | by rw [←neg_inj, neg_add', ←floor_neg, ←floor_neg, neg_add', floor_sub_int] | lemma | int.ceil_add_int | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ceil_add_nat (a : α) (n : ℕ) : ⌈a + n⌉ = ⌈a⌉ + n | by rw [← int.cast_coe_nat, ceil_add_int] | lemma | int.ceil_add_nat | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [
"int.cast_coe_nat"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ceil_add_one (a : α) : ⌈a + 1⌉ = ⌈a⌉ + 1 | by { convert ceil_add_int a (1 : ℤ), exact cast_one.symm } | lemma | int.ceil_add_one | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ceil_sub_int (a : α) (z : ℤ) : ⌈a - z⌉ = ⌈a⌉ - z | eq.trans (by rw [int.cast_neg, sub_eq_add_neg]) (ceil_add_int _ _) | lemma | int.ceil_sub_int | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [
"int.cast_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ceil_sub_nat (a : α) (n : ℕ) : ⌈a - n⌉ = ⌈a⌉ - n | by convert ceil_sub_int a n using 1; simp | lemma | int.ceil_sub_nat | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ceil_sub_one (a : α) : ⌈a - 1⌉ = ⌈a⌉ - 1 | by rw [eq_sub_iff_add_eq, ← ceil_add_one, sub_add_cancel] | lemma | int.ceil_sub_one | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ceil_lt_add_one (a : α) : (⌈a⌉ : α) < a + 1 | by { rw [← lt_ceil, ← int.cast_one, ceil_add_int], apply lt_add_one } | lemma | int.ceil_lt_add_one | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [
"int.cast_one",
"lt_add_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ceil_add_le (a b : α) : ⌈a + b⌉ ≤ ⌈a⌉ + ⌈b⌉ | begin
rw [ceil_le, int.cast_add],
exact add_le_add (le_ceil _) (le_ceil _),
end | lemma | int.ceil_add_le | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [
"int.cast_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ceil_add_ceil_le (a b : α) : ⌈a⌉ + ⌈b⌉ ≤ ⌈a + b⌉ + 1 | begin
rw [←le_sub_iff_add_le, ceil_le, int.cast_sub, int.cast_add, int.cast_one, le_sub_comm],
refine (ceil_lt_add_one _).le.trans _,
rw [le_sub_iff_add_le', ←add_assoc, add_le_add_iff_right],
exact le_ceil _,
end | lemma | int.ceil_add_ceil_le | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [
"int.cast_add",
"int.cast_one",
"int.cast_sub"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ceil_pos : 0 < ⌈a⌉ ↔ 0 < a | by rw [lt_ceil, cast_zero] | lemma | int.ceil_pos | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ceil_zero : ⌈(0 : α)⌉ = 0 | by rw [← cast_zero, ceil_int_cast] | lemma | int.ceil_zero | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ceil_one : ⌈(1 : α)⌉ = 1 | by rw [← cast_one, ceil_int_cast] | lemma | int.ceil_one | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ceil_nonneg (ha : 0 ≤ a) : 0 ≤ ⌈a⌉ | by exact_mod_cast ha.trans (le_ceil a) | lemma | int.ceil_nonneg | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ceil_eq_iff : ⌈a⌉ = z ↔ ↑z - 1 < a ∧ a ≤ z | by rw [←ceil_le, ←int.cast_one, ←int.cast_sub, ←lt_ceil, int.sub_one_lt_iff, le_antisymm_iff,
and.comm] | lemma | int.ceil_eq_iff | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [
"int.sub_one_lt_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ceil_eq_zero_iff : ⌈a⌉ = 0 ↔ a ∈ Ioc (-1 : α) 0 | by simp [ceil_eq_iff] | lemma | int.ceil_eq_zero_iff | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ceil_eq_on_Ioc (z : ℤ) : ∀ a ∈ set.Ioc (z - 1 : α) z, ⌈a⌉ = z | λ a ⟨h₀, h₁⟩, ceil_eq_iff.mpr ⟨h₀, h₁⟩ | lemma | int.ceil_eq_on_Ioc | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [
"set.Ioc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ceil_eq_on_Ioc' (z : ℤ) : ∀ a ∈ set.Ioc (z - 1 : α) z, (⌈a⌉ : α) = z | λ a ha, by exact_mod_cast ceil_eq_on_Ioc z a ha | lemma | int.ceil_eq_on_Ioc' | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [
"set.Ioc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
floor_le_ceil (a : α) : ⌊a⌋ ≤ ⌈a⌉ | cast_le.1 $ (floor_le _).trans $ le_ceil _ | lemma | int.floor_le_ceil | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
floor_lt_ceil_of_lt {a b : α} (h : a < b) : ⌊a⌋ < ⌈b⌉ | cast_lt.1 $ (floor_le a).trans_lt $ h.trans_le $ le_ceil b | lemma | int.floor_lt_ceil_of_lt | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preimage_ceil_singleton (m : ℤ) : (ceil : α → ℤ) ⁻¹' {m} = Ioc (m - 1) m | ext $ λ x, ceil_eq_iff | lemma | int.preimage_ceil_singleton | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fract_eq_zero_or_add_one_sub_ceil (a : α) : fract a = 0 ∨ fract a = a + 1 - (⌈a⌉ : α) | begin
cases eq_or_ne (fract a) 0 with ha ha, { exact or.inl ha, }, right,
suffices : (⌈a⌉ : α) = ⌊a⌋ + 1, { rw [this, ← self_sub_fract], abel, },
norm_cast,
rw ceil_eq_iff,
refine ⟨_, _root_.le_of_lt $ by simp⟩,
rw [cast_add, cast_one, add_tsub_cancel_right, ← self_sub_fract a, sub_lt_self_iff],
exact ha.... | lemma | int.fract_eq_zero_or_add_one_sub_ceil | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [
"add_tsub_cancel_right",
"eq_or_ne"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ceil_eq_add_one_sub_fract (ha : fract a ≠ 0) : (⌈a⌉ : α) = a + 1 - fract a | by { rw (or_iff_right ha).mp (fract_eq_zero_or_add_one_sub_ceil a), abel, } | lemma | int.ceil_eq_add_one_sub_fract | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [
"or_iff_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ceil_sub_self_eq (ha : fract a ≠ 0) : (⌈a⌉ : α) - a = 1 - fract a | by { rw (or_iff_right ha).mp (fract_eq_zero_or_add_one_sub_ceil a), abel, } | lemma | int.ceil_sub_self_eq | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [
"or_iff_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preimage_Ioo {a b : α} : ((coe : ℤ → α) ⁻¹' (set.Ioo a b)) = set.Ioo ⌊a⌋ ⌈b⌉ | by { ext, simp [floor_lt, lt_ceil] } | lemma | int.preimage_Ioo | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [
"set.Ioo"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preimage_Ico {a b : α} : ((coe : ℤ → α) ⁻¹' (set.Ico a b)) = set.Ico ⌈a⌉ ⌈b⌉ | by { ext, simp [ceil_le, lt_ceil] } | lemma | int.preimage_Ico | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [
"set.Ico"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preimage_Ioc {a b : α} : ((coe : ℤ → α) ⁻¹' (set.Ioc a b)) = set.Ioc ⌊a⌋ ⌊b⌋ | by { ext, simp [floor_lt, le_floor] } | lemma | int.preimage_Ioc | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [
"set.Ioc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preimage_Icc {a b : α} : ((coe : ℤ → α) ⁻¹' (set.Icc a b)) = set.Icc ⌈a⌉ ⌊b⌋ | by { ext, simp [ceil_le, le_floor] } | lemma | int.preimage_Icc | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [
"set.Icc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preimage_Ioi : ((coe : ℤ → α) ⁻¹' (set.Ioi a)) = set.Ioi ⌊a⌋ | by { ext, simp [floor_lt] } | lemma | int.preimage_Ioi | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [
"set.Ioi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preimage_Ici : ((coe : ℤ → α) ⁻¹' (set.Ici a)) = set.Ici ⌈a⌉ | by { ext, simp [ceil_le] } | lemma | int.preimage_Ici | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [
"set.Ici"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preimage_Iio : ((coe : ℤ → α) ⁻¹' (set.Iio a)) = set.Iio ⌈a⌉ | by { ext, simp [lt_ceil] } | lemma | int.preimage_Iio | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [
"set.Iio"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preimage_Iic : ((coe : ℤ → α) ⁻¹' (set.Iic a)) = set.Iic ⌊a⌋ | by { ext, simp [le_floor] } | lemma | int.preimage_Iic | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [
"set.Iic"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
round (x : α) : ℤ | if 2 * fract x < 1 then ⌊x⌋ else ⌈x⌉ | def | round | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [] | `round` rounds a number to the nearest integer. `round (1 / 2) = 1` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
round_zero : round (0 : α) = 0 | by simp [round] | lemma | round_zero | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [
"round"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
round_one : round (1 : α) = 1 | by simp [round] | lemma | round_one | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [
"round"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
round_nat_cast (n : ℕ) : round (n : α) = n | by simp [round] | lemma | round_nat_cast | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [
"round"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
round_int_cast (n : ℤ) : round (n : α) = n | by simp [round] | lemma | round_int_cast | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [
"round"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
round_add_int (x : α) (y : ℤ) : round (x + y) = round x + y | by rw [round, round, int.fract_add_int, int.floor_add_int, int.ceil_add_int, ← apply_ite2, if_t_t] | lemma | round_add_int | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [
"apply_ite2",
"int.ceil_add_int",
"int.floor_add_int",
"int.fract_add_int",
"round"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
round_add_one (a : α) : round (a + 1) = round a + 1 | by { convert round_add_int a 1, exact int.cast_one.symm } | lemma | round_add_one | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [
"round",
"round_add_int"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
round_sub_int (x : α) (y : ℤ) : round (x - y) = round x - y | by { rw [sub_eq_add_neg], norm_cast, rw [round_add_int, sub_eq_add_neg] } | lemma | round_sub_int | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [
"round",
"round_add_int"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
round_sub_one (a : α) : round (a - 1) = round a - 1 | by { convert round_sub_int a 1, exact int.cast_one.symm } | lemma | round_sub_one | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [
"round",
"round_sub_int"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
round_add_nat (x : α) (y : ℕ) : round (x + y) = round x + y | by rw [round, round, fract_add_nat, int.floor_add_nat, int.ceil_add_nat, ← apply_ite2, if_t_t] | lemma | round_add_nat | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [
"apply_ite2",
"int.ceil_add_nat",
"int.floor_add_nat",
"round"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
round_sub_nat (x : α) (y : ℕ) : round (x - y) = round x - y | by { rw [sub_eq_add_neg, ← int.cast_coe_nat], norm_cast, rw [round_add_int, sub_eq_add_neg] } | lemma | round_sub_nat | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [
"int.cast_coe_nat",
"round",
"round_add_int"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
round_int_add (x : α) (y : ℤ) : round ((y : α) + x) = y + round x | by { rw [add_comm, round_add_int, add_comm] } | lemma | round_int_add | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [
"round",
"round_add_int"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
round_nat_add (x : α) (y : ℕ) : round ((y : α) + x) = y + round x | by { rw [add_comm, round_add_nat, add_comm] } | lemma | round_nat_add | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [
"round",
"round_add_nat"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_sub_round_eq_min (x : α) : |x - round x| = min (fract x) (1 - fract x) | begin
simp_rw [round, min_def_lt, two_mul, ← lt_tsub_iff_left],
cases lt_or_ge (fract x) (1 - fract x) with hx hx,
{ rw [if_pos hx, if_pos hx, self_sub_floor, abs_fract], },
{ have : 0 < fract x,
{ replace hx : 0 < fract x + fract x := lt_of_lt_of_le zero_lt_one (tsub_le_iff_left.mp hx),
simpa only [←... | lemma | abs_sub_round_eq_min | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [
"abs_sub_comm",
"lt_tsub_iff_left",
"min_def_lt",
"round",
"two_mul",
"zero_lt_mul_left",
"zero_lt_one",
"zero_lt_two"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
round_le (x : α) (z : ℤ) : |x - round x| ≤ |x - z| | begin
rw [abs_sub_round_eq_min, min_le_iff],
rcases le_or_lt (z : α) x with hx | hx; [left, right],
{ conv_rhs { rw [abs_eq_self.mpr (sub_nonneg.mpr hx), ← fract_add_floor x, add_sub_assoc], },
simpa only [le_add_iff_nonneg_right, sub_nonneg, cast_le] using le_floor.mpr hx, },
{ rw abs_eq_neg_self.mpr (sub_... | lemma | round_le | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [
"abs_sub_round_eq_min",
"min_le_iff",
"round"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
round_eq (x : α) : round x = ⌊x + 1 / 2⌋ | begin
simp_rw [round, (by simp only [lt_div_iff', two_pos] : 2 * fract x < 1 ↔ fract x < 1 / 2)],
cases lt_or_ge (fract x) (1 / 2) with hx hx,
{ conv_rhs { rw [← fract_add_floor x, add_assoc, add_left_comm, floor_int_add], },
rw [if_pos hx, self_eq_add_right, floor_eq_iff, cast_zero, zero_add],
split; lin... | lemma | round_eq | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [
"lt_div_iff'",
"round"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
round_two_inv : round (2⁻¹ : α) = 1 | by simp only [round_eq, ← one_div, add_halves', floor_one] | lemma | round_two_inv | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [
"add_halves'",
"one_div",
"round",
"round_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
round_neg_two_inv : round (-2⁻¹ : α) = 0 | by simp only [round_eq, ← one_div, add_left_neg, floor_zero] | lemma | round_neg_two_inv | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [
"one_div",
"round",
"round_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
round_eq_zero_iff {x : α} : round x = 0 ↔ x ∈ Ico (-(1 / 2)) ((1 : α)/2) | begin
rw [round_eq, floor_eq_zero_iff, add_mem_Ico_iff_left],
norm_num,
end | lemma | round_eq_zero_iff | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [
"round",
"round_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_sub_round (x : α) : |x - round x| ≤ 1 / 2 | begin
rw [round_eq, abs_sub_le_iff],
have := floor_le (x + 1 / 2),
have := lt_floor_add_one (x + 1 / 2),
split; linarith
end | lemma | abs_sub_round | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [
"abs_sub_le_iff",
"round",
"round_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_sub_round_div_nat_cast_eq {m n : ℕ} : | |(m : α) / n - round ((m : α) / n)| = ↑(min (m % n) (n - m % n)) / n :=
begin
rcases n.eq_zero_or_pos with rfl | hn, { simp, },
have hn' : 0 < (n : α), { norm_cast, assumption, },
rw [abs_sub_round_eq_min, nat.cast_min, ← min_div_div_right hn'.le,
fract_div_nat_cast_eq_div_nat_cast_mod, nat.cast_sub (m.mod_lt... | lemma | abs_sub_round_div_nat_cast_eq | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [
"abs_sub_round_eq_min",
"div_self",
"min_div_div_right",
"nat.cast_min",
"nat.cast_sub",
"round",
"sub_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
floor_congr (h : ∀ n : ℕ, (n : α) ≤ a ↔ (n : β) ≤ b) : ⌊a⌋₊ = ⌊b⌋₊ | begin
have h₀ : 0 ≤ a ↔ 0 ≤ b := by simpa only [cast_zero] using h 0,
obtain ha | ha := lt_or_le a 0,
{ rw [floor_of_nonpos ha.le, floor_of_nonpos (le_of_not_le $ h₀.not.mp ha.not_le)] },
exact (le_floor $ (h _).1 $ floor_le ha).antisymm (le_floor $ (h _).2 $ floor_le $ h₀.1 ha),
end | lemma | nat.floor_congr | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ceil_congr (h : ∀ n : ℕ, a ≤ n ↔ b ≤ n) : ⌈a⌉₊ = ⌈b⌉₊ | (ceil_le.2 $ (h _).2 $ le_ceil _).antisymm $ ceil_le.2 $ (h _).1 $ le_ceil _ | lemma | nat.ceil_congr | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_floor (f : F) (hf : strict_mono f) (a : α) : ⌊f a⌋₊ = ⌊a⌋₊ | floor_congr $ λ n, by rw [←map_nat_cast f, hf.le_iff_le] | lemma | nat.map_floor | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [
"strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_ceil (f : F) (hf : strict_mono f) (a : α) : ⌈f a⌉₊ = ⌈a⌉₊ | ceil_congr $ λ n, by rw [←map_nat_cast f, hf.le_iff_le] | lemma | nat.map_ceil | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [
"strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
floor_congr (h : ∀ n : ℤ, (n : α) ≤ a ↔ (n : β) ≤ b) : ⌊a⌋ = ⌊b⌋ | (le_floor.2 $ (h _).1 $ floor_le _).antisymm $ le_floor.2 $ (h _).2 $ floor_le _ | lemma | int.floor_congr | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ceil_congr (h : ∀ n : ℤ, a ≤ n ↔ b ≤ n) : ⌈a⌉ = ⌈b⌉ | (ceil_le.2 $ (h _).2 $ le_ceil _).antisymm $ ceil_le.2 $ (h _).1 $ le_ceil _ | lemma | int.ceil_congr | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_floor (f : F) (hf : strict_mono f) (a : α) : ⌊f a⌋ = ⌊a⌋ | floor_congr $ λ n, by rw [←map_int_cast f, hf.le_iff_le] | lemma | int.map_floor | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [
"strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_ceil (f : F) (hf : strict_mono f) (a : α) : ⌈f a⌉ = ⌈a⌉ | ceil_congr $ λ n, by rw [←map_int_cast f, hf.le_iff_le] | lemma | int.map_ceil | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [
"strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_fract (f : F) (hf : strict_mono f) (a : α) : fract (f a) = f (fract a) | by simp_rw [fract, map_sub, map_int_cast, map_floor _ hf] | lemma | int.map_fract | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [
"map_int_cast",
"strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_round (f : F) (hf : strict_mono f) (a : α) : round (f a) = round a | by simp_rw [round_eq, ←map_floor _ hf, map_add, one_div, map_inv₀, map_bit0, map_one] | lemma | int.map_round | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [
"map_bit0",
"map_inv₀",
"map_one",
"one_div",
"round",
"round_eq",
"strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.floor_ring.to_floor_semiring : floor_semiring α | { floor := λ a, ⌊a⌋.to_nat,
ceil := λ a, ⌈a⌉.to_nat,
floor_of_neg := λ a ha, int.to_nat_of_nonpos (int.floor_nonpos ha.le),
gc_floor := λ a n ha,
by rw [int.le_to_nat_iff (int.floor_nonneg.2 ha), int.le_floor, int.cast_coe_nat],
gc_ceil := λ a n, by rw [int.to_nat_le, int.ceil_le, int.cast_coe_nat] } | instance | floor_ring.to_floor_semiring | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [
"floor_semiring",
"int.cast_coe_nat",
"int.ceil_le",
"int.floor_nonpos",
"int.le_floor",
"int.le_to_nat_iff",
"int.to_nat_le",
"int.to_nat_of_nonpos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
int.floor_to_nat (a : α) : ⌊a⌋.to_nat = ⌊a⌋₊ | rfl | lemma | int.floor_to_nat | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
int.ceil_to_nat (a : α) : ⌈a⌉.to_nat = ⌈a⌉₊ | rfl | lemma | int.ceil_to_nat | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat.floor_int : (nat.floor : ℤ → ℕ) = int.to_nat | rfl | lemma | nat.floor_int | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [
"nat.floor"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat.ceil_int : (nat.ceil : ℤ → ℕ) = int.to_nat | rfl | lemma | nat.ceil_int | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [
"nat.ceil"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat.cast_floor_eq_int_floor (ha : 0 ≤ a) : (⌊a⌋₊ : ℤ) = ⌊a⌋ | by rw [←int.floor_to_nat, int.to_nat_of_nonneg (int.floor_nonneg.2 ha)] | lemma | nat.cast_floor_eq_int_floor | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [
"int.to_nat_of_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat.cast_floor_eq_cast_int_floor (ha : 0 ≤ a) : (⌊a⌋₊ : α) = ⌊a⌋ | by rw [←nat.cast_floor_eq_int_floor ha, int.cast_coe_nat] | lemma | nat.cast_floor_eq_cast_int_floor | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [
"int.cast_coe_nat"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat.cast_ceil_eq_int_ceil (ha : 0 ≤ a) : (⌈a⌉₊ : ℤ) = ⌈a⌉ | by { rw [←int.ceil_to_nat, int.to_nat_of_nonneg (int.ceil_nonneg ha)] } | lemma | nat.cast_ceil_eq_int_ceil | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [
"int.ceil_nonneg",
"int.to_nat_of_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat.cast_ceil_eq_cast_int_ceil (ha : 0 ≤ a) : (⌈a⌉₊ : α) = ⌈a⌉ | by rw [←nat.cast_ceil_eq_int_ceil ha, int.cast_coe_nat] | lemma | nat.cast_ceil_eq_cast_int_ceil | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [
"int.cast_coe_nat"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subsingleton_floor_ring {α} [linear_ordered_ring α] :
subsingleton (floor_ring α) | begin
refine ⟨λ H₁ H₂, _⟩,
have : H₁.floor = H₂.floor := funext (λ a, H₁.gc_coe_floor.u_unique H₂.gc_coe_floor $ λ _, rfl),
have : H₁.ceil = H₂.ceil := funext (λ a, H₁.gc_ceil_coe.l_unique H₂.gc_ceil_coe $ λ _, rfl),
cases H₁, cases H₂, congr; assumption
end | lemma | subsingleton_floor_ring | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [
"floor_ring",
"linear_ordered_ring"
] | There exists at most one `floor_ring` structure on a given linear ordered ring. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
int_floor_nonneg [linear_ordered_ring α] [floor_ring α] {a : α} (ha : 0 ≤ a) :
0 ≤ ⌊a⌋ | int.floor_nonneg.2 ha | lemma | tactic.int_floor_nonneg | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [
"floor_ring",
"linear_ordered_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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