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 |
|---|---|---|---|---|---|---|---|---|---|---|
ceil_nat_cast (n : ℕ) : ⌈(n : α)⌉₊ = n | eq_of_forall_ge_iff $ λ a, by rw [ceil_le, cast_le] | lemma | nat.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 → ℕ) | gc_ceil_coe.monotone_l | lemma | nat.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_zero : ⌈(0 : α)⌉₊ = 0 | by rw [← nat.cast_zero, ceil_nat_cast] | lemma | nat.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"
] | [
"nat.cast_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ceil_one : ⌈(1 : α)⌉₊ = 1 | by rw [←nat.cast_one, ceil_nat_cast] | lemma | nat.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_eq_zero : ⌈a⌉₊ = 0 ↔ a ≤ 0 | by rw [← le_zero_iff, ceil_le, nat.cast_zero] | lemma | nat.ceil_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"
] | [
"le_zero_iff",
"nat.cast_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ceil_pos : 0 < ⌈a⌉₊ ↔ 0 < a | by rw [lt_ceil, cast_zero] | lemma | nat.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 | |
lt_of_ceil_lt (h : ⌈a⌉₊ < n) : a < n | (le_ceil a).trans_lt (nat.cast_lt.2 h) | lemma | nat.lt_of_ceil_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 | |
le_of_ceil_le (h : ⌈a⌉₊ ≤ n) : a ≤ n | (le_ceil a).trans (nat.cast_le.2 h) | lemma | nat.le_of_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_le_ceil (a : α) : ⌊a⌋₊ ≤ ⌈a⌉₊ | begin
obtain ha | ha := le_total a 0,
{ rw floor_of_nonpos ha,
exact nat.zero_le _ },
{ exact cast_le.1 ((floor_le ha).trans $ le_ceil _) }
end | lemma | nat.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_of_pos {a b : α} (h : a < b) (h' : 0 < b) : ⌊a⌋₊ < ⌈b⌉₊ | begin
rcases le_or_lt 0 a with ha|ha,
{ rw floor_lt ha, exact h.trans_le (le_ceil _) },
{ rwa [floor_of_nonpos ha.le, lt_ceil, nat.cast_zero] }
end | lemma | nat.floor_lt_ceil_of_lt_of_pos | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [
"nat.cast_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ceil_eq_iff (hn : n ≠ 0) : ⌈a⌉₊ = n ↔ ↑(n - 1) < a ∧ a ≤ n | by rw [← ceil_le, ← not_le, ← ceil_le, not_le,
tsub_lt_iff_right (nat.add_one_le_iff.2 (pos_iff_ne_zero.2 hn)), nat.lt_add_one_iff,
le_antisymm_iff, and.comm] | lemma | nat.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"
] | [
"nat.lt_add_one_iff",
"tsub_lt_iff_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preimage_ceil_zero : (nat.ceil : α → ℕ) ⁻¹' {0} = Iic 0 | ext $ λ x, ceil_eq_zero | lemma | nat.preimage_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"
] | [
"nat.ceil"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preimage_ceil_of_ne_zero (hn : n ≠ 0) : (nat.ceil : α → ℕ) ⁻¹' {n} = Ioc ↑(n - 1) n | ext $ λ x, ceil_eq_iff hn | lemma | nat.preimage_ceil_of_ne_zero | 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 | |
preimage_Ioo {a b : α} (ha : 0 ≤ a) :
((coe : ℕ → α) ⁻¹' (set.Ioo a b)) = set.Ioo ⌊a⌋₊ ⌈b⌉₊ | by { ext, simp [floor_lt, lt_ceil, ha] } | lemma | nat.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 : ℕ → α) ⁻¹' (set.Ico a b)) = set.Ico ⌈a⌉₊ ⌈b⌉₊ | by { ext, simp [ceil_le, lt_ceil] } | lemma | nat.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} (ha : 0 ≤ a) (hb : 0 ≤ b) :
((coe : ℕ → α) ⁻¹' (set.Ioc a b)) = set.Ioc ⌊a⌋₊ ⌊b⌋₊ | by { ext, simp [floor_lt, le_floor_iff, hb, ha] } | lemma | nat.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} (hb : 0 ≤ b) :
((coe : ℕ → α) ⁻¹' (set.Icc a b)) = set.Icc ⌈a⌉₊ ⌊b⌋₊ | by { ext, simp [ceil_le, hb, le_floor_iff] } | lemma | nat.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 {a : α} (ha : 0 ≤ a) : ((coe : ℕ → α) ⁻¹' (set.Ioi a)) = set.Ioi ⌊a⌋₊ | by { ext, simp [floor_lt, ha] } | lemma | nat.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 {a : α} : ((coe : ℕ → α) ⁻¹' (set.Ici a)) = set.Ici ⌈a⌉₊ | by { ext, simp [ceil_le] } | lemma | nat.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 {a : α} : ((coe : ℕ → α) ⁻¹' (set.Iio a)) = set.Iio ⌈a⌉₊ | by { ext, simp [lt_ceil] } | lemma | nat.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 {a : α} (ha : 0 ≤ a) : ((coe : ℕ → α) ⁻¹' (set.Iic a)) = set.Iic ⌊a⌋₊ | by { ext, simp [le_floor_iff, ha] } | lemma | nat.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 | |
floor_add_nat (ha : 0 ≤ a) (n : ℕ) : ⌊a + n⌋₊ = ⌊a⌋₊ + n | eq_of_forall_le_iff $ λ b, begin
rw [le_floor_iff (add_nonneg ha n.cast_nonneg)],
obtain hb | hb := le_total n b,
{ obtain ⟨d, rfl⟩ := exists_add_of_le hb,
rw [nat.cast_add, add_comm n, add_comm (n : α), add_le_add_iff_right, add_le_add_iff_right,
le_floor_iff ha] },
{ obtain ⟨d, rfl⟩ := exists_add_of... | lemma | nat.floor_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"
] | [
"eq_of_forall_le_iff",
"iff_of_true",
"nat.cast_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
floor_add_one (ha : 0 ≤ a) : ⌊a + 1⌋₊ = ⌊a⌋₊ + 1 | by { convert floor_add_nat ha 1, exact cast_one.symm } | lemma | nat.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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
floor_sub_nat [has_sub α] [has_ordered_sub α] [has_exists_add_of_le α] (a : α) (n : ℕ) :
⌊a - n⌋₊ = ⌊a⌋₊ - n | begin
obtain ha | ha := le_total a 0,
{ rw [floor_of_nonpos ha, floor_of_nonpos (tsub_nonpos_of_le (ha.trans n.cast_nonneg)),
zero_tsub] },
cases le_total a n,
{ rw [floor_of_nonpos (tsub_nonpos_of_le h), eq_comm, tsub_eq_zero_iff_le],
exact nat.cast_le.1 ((nat.floor_le ha).trans h) },
{ rw [eq_tsub... | lemma | nat.floor_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"
] | [
"eq_tsub_iff_add_eq_of_le",
"has_exists_add_of_le",
"has_ordered_sub",
"le_tsub_of_add_le_left",
"nat.floor_le",
"tsub_add_cancel_of_le",
"tsub_eq_zero_iff_le",
"zero_tsub"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ceil_add_nat (ha : 0 ≤ a) (n : ℕ) : ⌈a + n⌉₊ = ⌈a⌉₊ + n | eq_of_forall_ge_iff $ λ b, begin
rw [←not_lt, ←not_lt, not_iff_not],
rw [lt_ceil],
obtain hb | hb := le_or_lt n b,
{ obtain ⟨d, rfl⟩ := exists_add_of_le hb,
rw [nat.cast_add, add_comm n, add_comm (n : α), add_lt_add_iff_right, add_lt_add_iff_right,
lt_ceil] },
{ exact iff_of_true (lt_add_of_nonneg_o... | lemma | nat.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"
] | [
"eq_of_forall_ge_iff",
"iff_of_true",
"nat.cast_add",
"not_iff_not"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ceil_add_one (ha : 0 ≤ a) : ⌈a + 1⌉₊ = ⌈a⌉₊ + 1 | by { convert ceil_add_nat ha 1, exact cast_one.symm } | lemma | nat.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_lt_add_one (ha : 0 ≤ a) : (⌈a⌉₊ : α) < a + 1 | lt_ceil.1 $ (nat.lt_succ_self _).trans_le (ceil_add_one ha).ge | lemma | nat.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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ceil_add_le (a b : α) : ⌈a + b⌉₊ ≤ ⌈a⌉₊ + ⌈b⌉₊ | begin
rw [ceil_le, nat.cast_add],
exact add_le_add (le_ceil _) (le_ceil _),
end | lemma | nat.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"
] | [
"nat.cast_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_one_lt_floor (a : α) : a - 1 < ⌊a⌋₊ | sub_lt_iff_lt_add.2 $ lt_floor_add_one a | lemma | nat.sub_one_lt_floor | 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_div_nat (a : α) (n : ℕ) : ⌊a / n⌋₊ = ⌊a⌋₊ / n | begin
cases le_total a 0 with ha ha,
{ rw [floor_of_nonpos, floor_of_nonpos ha],
{ simp },
apply div_nonpos_of_nonpos_of_nonneg ha n.cast_nonneg },
obtain rfl | hn := n.eq_zero_or_pos,
{ rw [cast_zero, div_zero, nat.div_zero, floor_zero] },
refine (floor_eq_iff _).2 _,
{ exact div_nonneg ha n.cast_n... | lemma | nat.floor_div_nat | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [
"div_le_div_of_le_of_nonneg",
"div_lt_iff",
"div_nonneg",
"div_nonpos_of_nonpos_of_nonneg",
"div_zero",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
floor_div_eq_div (m n : ℕ) : ⌊(m : α) / n⌋₊ = m / n | by { convert floor_div_nat (m : α) n, rw m.floor_coe } | lemma | nat.floor_div_eq_div | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [] | Natural division is the floor of field division. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
subsingleton_floor_semiring {α} [linear_ordered_semiring α] :
subsingleton (floor_semiring α) | begin
refine ⟨λ H₁ H₂, _⟩,
have : H₁.ceil = H₂.ceil,
from funext (λ a, H₁.gc_ceil.l_unique H₂.gc_ceil $ λ n, rfl),
have : H₁.floor = H₂.floor,
{ ext a,
cases lt_or_le a 0,
{ rw [H₁.floor_of_neg, H₂.floor_of_neg]; exact h },
{ refine eq_of_forall_le_iff (λ n, _),
rw [H₁.gc_floor, H₂.gc_floo... | lemma | subsingleton_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"
] | [
"eq_of_forall_le_iff",
"floor_semiring",
"linear_ordered_semiring"
] | There exists at most one `floor_semiring` structure on a linear ordered semiring. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
floor_ring (α) [linear_ordered_ring α] | (floor : α → ℤ)
(ceil : α → ℤ)
(gc_coe_floor : galois_connection coe floor)
(gc_ceil_coe : galois_connection ceil coe) | class | 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"
] | [
"galois_connection",
"linear_ordered_ring"
] | A `floor_ring` is a linear ordered ring over `α` with a function
`floor : α → ℤ` satisfying `∀ (z : ℤ) (a : α), z ≤ floor a ↔ (z : α) ≤ a)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
floor_ring.of_floor (α) [linear_ordered_ring α] (floor : α → ℤ)
(gc_coe_floor : galois_connection coe floor) : floor_ring α | { floor := floor,
ceil := λ a, -floor (-a),
gc_coe_floor := gc_coe_floor,
gc_ceil_coe := λ a z, by rw [neg_le, ←gc_coe_floor, int.cast_neg, neg_le_neg_iff] } | def | floor_ring.of_floor | 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",
"galois_connection",
"int.cast_neg",
"linear_ordered_ring"
] | A `floor_ring` constructor from the `floor` function alone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
floor_ring.of_ceil (α) [linear_ordered_ring α] (ceil : α → ℤ)
(gc_ceil_coe : galois_connection ceil coe) : floor_ring α | { floor := λ a, -ceil (-a),
ceil := ceil,
gc_coe_floor := λ a z, by rw [le_neg, gc_ceil_coe, int.cast_neg, neg_le_neg_iff],
gc_ceil_coe := gc_ceil_coe } | def | floor_ring.of_ceil | 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",
"galois_connection",
"int.cast_neg",
"linear_ordered_ring"
] | A `floor_ring` constructor from the `ceil` function alone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
floor : α → ℤ | floor_ring.floor | def | 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.floor a` is the greatest integer `z` such that `z ≤ a`. It is denoted with `⌊a⌋`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ceil : α → ℤ | floor_ring.ceil | def | 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 a` is the smallest integer `z` such that `a ≤ z`. It is denoted with `⌈a⌉`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fract (a : α) : α | a - floor a | def | int.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.fract a`, the fractional part of `a`, is `a` minus its floor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
floor_int : (int.floor : ℤ → ℤ) = id | rfl | lemma | int.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"
] | [
"int.floor"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ceil_int : (int.ceil : ℤ → ℤ) = id | rfl | lemma | int.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"
] | [
"int.ceil"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fract_int : (int.fract : ℤ → ℤ) = 0 | funext $ λ x, by simp [fract] | lemma | int.fract_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.fract"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
floor_ring_floor_eq : @floor_ring.floor = @int.floor | rfl | lemma | int.floor_ring_floor_eq | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [
"int.floor"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
floor_ring_ceil_eq : @floor_ring.ceil = @int.ceil | rfl | lemma | int.floor_ring_ceil_eq | 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"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gc_coe_floor : galois_connection (coe : ℤ → α) floor | floor_ring.gc_coe_floor | lemma | int.gc_coe_floor | 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 | |
le_floor : z ≤ ⌊a⌋ ↔ (z : α) ≤ a | (gc_coe_floor z a).symm | lemma | int.le_floor | 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 : ⌊a⌋ < z ↔ a < z | lt_iff_lt_of_le_iff_le le_floor | lemma | int.floor_lt | 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 | |
floor_le (a : α) : (⌊a⌋ : α) ≤ a | gc_coe_floor.l_u_le a | lemma | int.floor_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_nonneg : 0 ≤ ⌊a⌋ ↔ 0 ≤ a | by rw [le_floor, int.cast_zero] | lemma | 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"
] | [
"int.cast_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
floor_le_sub_one_iff : ⌊a⌋ ≤ z - 1 ↔ a < z | by rw [← floor_lt, le_sub_one_iff] | lemma | int.floor_le_sub_one_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 | |
floor_le_neg_one_iff : ⌊a⌋ ≤ -1 ↔ a < 0 | by rw [← zero_sub (1 : ℤ), floor_le_sub_one_iff, cast_zero] | lemma | int.floor_le_neg_one_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 | |
floor_nonpos (ha : a ≤ 0) : ⌊a⌋ ≤ 0 | begin
rw [← @cast_le α, int.cast_zero],
exact (floor_le a).trans ha,
end | lemma | int.floor_nonpos | 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_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_succ_floor (a : α) : a < ⌊a⌋.succ | floor_lt.1 $ int.lt_succ_self _ | lemma | int.lt_succ_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.lt_succ_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_floor_add_one (a : α) : a < ⌊a⌋ + 1 | by simpa only [int.succ, int.cast_add, int.cast_one] using lt_succ_floor a | lemma | int.lt_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",
"int.succ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_one_lt_floor (a : α) : a - 1 < ⌊a⌋ | sub_lt_iff_lt_add.2 (lt_floor_add_one a) | lemma | int.sub_one_lt_floor | 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_int_cast (z : ℤ) : ⌊(z : α)⌋ = z | eq_of_forall_le_iff $ λ a, by rw [le_floor, int.cast_le] | lemma | int.floor_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_le_iff",
"int.cast_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
floor_nat_cast (n : ℕ) : ⌊(n : α)⌋ = n | eq_of_forall_le_iff $ λ a, by rw [le_floor, ← cast_coe_nat, cast_le] | lemma | int.floor_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_le_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
floor_zero : ⌊(0 : α)⌋ = 0 | by rw [← cast_zero, floor_int_cast] | lemma | int.floor_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 | |
floor_one : ⌊(1 : α)⌋ = 1 | by rw [← cast_one, floor_int_cast] | lemma | int.floor_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 | |
floor_mono : monotone (floor : α → ℤ) | gc_coe_floor.monotone_u | lemma | int.floor_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 | |
floor_pos : 0 < ⌊a⌋ ↔ 1 ≤ a | by { convert le_floor, exact cast_one.symm } | lemma | int.floor_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 | |
floor_add_int (a : α) (z : ℤ) : ⌊a + z⌋ = ⌊a⌋ + z | eq_of_forall_le_iff $ λ a, by rw [le_floor,
← sub_le_iff_le_add, ← sub_le_iff_le_add, le_floor, int.cast_sub] | lemma | int.floor_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"
] | [
"eq_of_forall_le_iff",
"int.cast_sub"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
floor_add_one (a : α) : ⌊a + 1⌋ = ⌊a⌋ + 1 | by { convert floor_add_int a 1, exact cast_one.symm } | lemma | int.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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_floor_add (a b : α) : ⌊a⌋ + ⌊b⌋ ≤ ⌊a + b⌋ | begin
rw [le_floor, int.cast_add],
exact add_le_add (floor_le _) (floor_le _),
end | lemma | int.le_floor_add | 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 | |
le_floor_add_floor (a b : α) : ⌊a + b⌋ - 1 ≤ ⌊a⌋ + ⌊b⌋ | begin
rw [←sub_le_iff_le_add, le_floor, int.cast_sub, sub_le_comm, int.cast_sub, int.cast_one],
refine le_trans _ (sub_one_lt_floor _).le,
rw [sub_le_iff_le_add', ←add_sub_assoc, sub_le_sub_iff_right],
exact floor_le _,
end | lemma | int.le_floor_add_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_one",
"int.cast_sub"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
floor_int_add (z : ℤ) (a : α) : ⌊↑z + a⌋ = z + ⌊a⌋ | by simpa only [add_comm] using floor_add_int a z | lemma | int.floor_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
floor_add_nat (a : α) (n : ℕ) : ⌊a + n⌋ = ⌊a⌋ + n | by rw [← int.cast_coe_nat, floor_add_int] | lemma | int.floor_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 | |
floor_nat_add (n : ℕ) (a : α) : ⌊↑n + a⌋ = n + ⌊a⌋ | by rw [← int.cast_coe_nat, floor_int_add] | lemma | int.floor_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"
] | [
"int.cast_coe_nat"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
floor_sub_int (a : α) (z : ℤ) : ⌊a - z⌋ = ⌊a⌋ - z | eq.trans (by rw [int.cast_neg, sub_eq_add_neg]) (floor_add_int _ _) | lemma | int.floor_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 | |
floor_sub_nat (a : α) (n : ℕ) : ⌊a - n⌋ = ⌊a⌋ - n | by rw [← int.cast_coe_nat, floor_sub_int] | lemma | int.floor_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"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_sub_lt_one_of_floor_eq_floor {α : Type*} [linear_ordered_comm_ring α] [floor_ring α]
{a b : α} (h : ⌊a⌋ = ⌊b⌋) : |a - b| < 1 | begin
have : a < ⌊a⌋ + 1 := lt_floor_add_one a,
have : b < ⌊b⌋ + 1 := lt_floor_add_one b,
have : (⌊a⌋ : α) = ⌊b⌋ := int.cast_inj.2 h,
have : (⌊a⌋ : α) ≤ a := floor_le a,
have : (⌊b⌋ : α) ≤ b := floor_le b,
exact abs_sub_lt_iff.2 ⟨by linarith, by linarith⟩
end | lemma | int.abs_sub_lt_one_of_floor_eq_floor | 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_comm_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
floor_eq_iff : ⌊a⌋ = z ↔ ↑z ≤ a ∧ a < z + 1 | by rw [le_antisymm_iff, le_floor, ←int.lt_add_one_iff, floor_lt, int.cast_add, int.cast_one,
and.comm] | lemma | int.floor_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.cast_add",
"int.cast_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
floor_eq_zero_iff : ⌊a⌋ = 0 ↔ a ∈ Ico (0 : α) 1 | by simp [floor_eq_iff] | lemma | int.floor_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 | |
floor_eq_on_Ico (n : ℤ) : ∀ a ∈ set.Ico (n : α) (n + 1), ⌊a⌋ = n | λ a ⟨h₀, h₁⟩, floor_eq_iff.mpr ⟨h₀, h₁⟩ | lemma | int.floor_eq_on_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 | |
floor_eq_on_Ico' (n : ℤ) : ∀ a ∈ set.Ico (n : α) (n + 1), (⌊a⌋ : α) = n | λ a ha, congr_arg _ $ floor_eq_on_Ico n a ha | lemma | int.floor_eq_on_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_floor_singleton (m : ℤ) : (floor : α → ℤ) ⁻¹' {m} = Ico m (m + 1) | ext $ λ x, floor_eq_iff | lemma | int.preimage_floor_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 | |
self_sub_floor (a : α) : a - ⌊a⌋ = fract a | rfl | lemma | int.self_sub_floor | 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_add_fract (a : α) : (⌊a⌋ : α) + fract a = a | add_sub_cancel'_right _ _ | lemma | int.floor_add_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_floor (a : α) : fract a + ⌊a⌋ = a | sub_add_cancel _ _ | lemma | int.fract_add_floor | 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_int (a : α) (m : ℤ) : fract (a + m) = fract a | by { rw fract, simp } | lemma | int.fract_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 | |
fract_add_nat (a : α) (m : ℕ) : fract (a + m) = fract a | by { rw fract, simp } | lemma | int.fract_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fract_sub_int (a : α) (m : ℤ) : fract (a - m) = fract a | by { rw fract, simp } | lemma | int.fract_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fract_int_add (m : ℤ) (a : α) : fract (↑m + a) = fract a | by rw [add_comm, fract_add_int] | lemma | int.fract_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fract_sub_nat (a : α) (n : ℕ) : fract (a - n) = fract a | by { rw fract, simp } | lemma | int.fract_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 | |
fract_int_nat (n : ℕ) (a : α) : fract (↑n + a) = fract a | by rw [add_comm, fract_add_nat] | lemma | int.fract_int_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 | |
fract_add_le (a b : α) : fract (a + b) ≤ fract a + fract b | begin
rw [fract, fract, fract, sub_add_sub_comm, sub_le_sub_iff_left, ←int.cast_add, int.cast_le],
exact le_floor_add _ _,
end | lemma | int.fract_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_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fract_add_fract_le (a b : α) : fract a + fract b ≤ fract (a + b) + 1 | begin
rw [fract, fract, fract, sub_add_sub_comm, sub_add, sub_le_sub_iff_left],
exact_mod_cast le_floor_add_floor a b,
end | lemma | int.fract_add_fract_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 | |
self_sub_fract (a : α) : a - fract a = ⌊a⌋ | sub_sub_cancel _ _ | lemma | int.self_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fract_sub_self (a : α) : fract a - a = -⌊a⌋ | sub_sub_cancel_left _ _ | lemma | int.fract_sub_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_nonneg (a : α) : 0 ≤ fract a | sub_nonneg.2 $ floor_le _ | lemma | int.fract_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 | |
fract_pos : 0 < fract a ↔ a ≠ ⌊a⌋ | (fract_nonneg a).lt_iff_ne.trans $ ne_comm.trans sub_ne_zero | lemma | int.fract_pos | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [] | The fractional part of `a` is positive if and only if `a ≠ ⌊a⌋`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fract_lt_one (a : α) : fract a < 1 | sub_lt_comm.1 $ sub_one_lt_floor _ | lemma | int.fract_lt_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 | |
fract_zero : fract (0 : α) = 0 | by rw [fract, floor_zero, cast_zero, sub_self] | lemma | int.fract_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 | |
fract_one : fract (1 : α) = 0 | by simp [fract] | lemma | int.fract_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 | |
abs_fract : |int.fract a| = int.fract a | abs_eq_self.mpr $ fract_nonneg a | lemma | int.abs_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.fract"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_one_sub_fract : |1 - fract a| = 1 - fract a | abs_eq_self.mpr $ sub_nonneg.mpr (fract_lt_one a).le | lemma | int.abs_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fract_int_cast (z : ℤ) : fract (z : α) = 0 | by { unfold fract, rw floor_int_cast, exact sub_self _ } | lemma | int.fract_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fract_nat_cast (n : ℕ) : fract (n : α) = 0 | by simp [fract] | lemma | int.fract_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fract_floor (a : α) : fract (⌊a⌋ : α) = 0 | fract_int_cast _ | lemma | int.fract_floor | 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_fract (a : α) : ⌊fract a⌋ = 0 | by rw [floor_eq_iff, int.cast_zero, zero_add]; exact ⟨fract_nonneg _, fract_lt_one _⟩ | lemma | int.floor_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_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fract_eq_iff {a b : α} : fract a = b ↔ 0 ≤ b ∧ b < 1 ∧ ∃ z : ℤ, a - b = z | ⟨λ h, by { rw ←h, exact ⟨fract_nonneg _, fract_lt_one _, ⟨⌊a⌋, sub_sub_cancel _ _⟩⟩},
begin
rintro ⟨h₀, h₁, z, hz⟩,
show a - ⌊a⌋ = b, apply eq.symm,
rw [eq_sub_iff_add_eq, add_comm, ←eq_sub_iff_add_eq],
rw [hz, int.cast_inj, floor_eq_iff, ←hz],
clear hz, split; simpa [sub_eq_add_neg, add_assoc]
... | lemma | int.fract_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.cast_inj"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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