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
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|---|---|---|---|---|---|---|---|---|---|---|
exists_nat_one_div_lt {ε : α} (hε : 0 < ε) : ∃ n : ℕ, 1 / (n + 1: α) < ε | begin
cases exists_nat_gt (1/ε) with n hn,
use n,
rw [div_lt_iff, ← div_lt_iff' hε],
{ apply hn.trans,
simp [zero_lt_one] },
{ exact n.cast_add_one_pos }
end | theorem | exists_nat_one_div_lt | algebra.order | src/algebra/order/archimedean.lean | [
"data.int.least_greatest",
"data.rat.floor"
] | [
"div_lt_iff",
"div_lt_iff'",
"exists_nat_gt",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_pos_rat_lt {x : α} (x0 : 0 < x) : ∃ q : ℚ, 0 < q ∧ (q : α) < x | by simpa only [rat.cast_pos] using exists_rat_btwn x0 | theorem | exists_pos_rat_lt | algebra.order | src/algebra/order/archimedean.lean | [
"data.int.least_greatest",
"data.rat.floor"
] | [
"exists_rat_btwn",
"rat.cast_pos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_rat_near (x : α) (ε0 : 0 < ε) : ∃ q : ℚ, |x - q| < ε | let ⟨q, h₁, h₂⟩ := exists_rat_btwn $ ((sub_lt_self_iff x).2 ε0).trans ((lt_add_iff_pos_left x).2 ε0)
in ⟨q, abs_sub_lt_iff.2 ⟨sub_lt_comm.1 h₁, sub_lt_iff_lt_add.2 h₂⟩⟩ | lemma | exists_rat_near | algebra.order | src/algebra/order/archimedean.lean | [
"data.int.least_greatest",
"data.rat.floor"
] | [
"exists_rat_btwn"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
archimedean_iff_nat_lt : archimedean α ↔ ∀ x : α, ∃ n : ℕ, x < n | ⟨@exists_nat_gt α _, λ H, ⟨λ x y y0,
(H (x / y)).imp $ λ n h, le_of_lt $
by rwa [div_lt_iff y0, ← nsmul_eq_mul] at h⟩⟩ | lemma | archimedean_iff_nat_lt | algebra.order | src/algebra/order/archimedean.lean | [
"data.int.least_greatest",
"data.rat.floor"
] | [
"archimedean",
"div_lt_iff",
"exists_nat_gt",
"nsmul_eq_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
archimedean_iff_nat_le : archimedean α ↔ ∀ x : α, ∃ n : ℕ, x ≤ n | archimedean_iff_nat_lt.trans
⟨λ H x, (H x).imp $ λ _, le_of_lt,
λ H x, let ⟨n, h⟩ := H x in ⟨n+1,
lt_of_le_of_lt h (nat.cast_lt.2 (lt_add_one _))⟩⟩ | lemma | archimedean_iff_nat_le | algebra.order | src/algebra/order/archimedean.lean | [
"data.int.least_greatest",
"data.rat.floor"
] | [
"archimedean",
"lt_add_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
archimedean_iff_int_lt : archimedean α ↔ ∀ x : α, ∃ n : ℤ, x < n | ⟨@exists_int_gt α _,
begin
rw archimedean_iff_nat_lt,
intros h x,
obtain ⟨n, h⟩ := h x,
refine ⟨n.to_nat, h.trans_le _⟩,
exact_mod_cast int.le_to_nat _,
end⟩ | lemma | archimedean_iff_int_lt | algebra.order | src/algebra/order/archimedean.lean | [
"data.int.least_greatest",
"data.rat.floor"
] | [
"archimedean",
"archimedean_iff_nat_lt",
"exists_int_gt",
"int.le_to_nat"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
archimedean_iff_int_le : archimedean α ↔ ∀ x : α, ∃ n : ℤ, x ≤ n | archimedean_iff_int_lt.trans
⟨λ H x, (H x).imp $ λ _, le_of_lt,
λ H x, let ⟨n, h⟩ := H x in ⟨n+1,
lt_of_le_of_lt h (int.cast_lt.2 (lt_add_one _))⟩⟩ | lemma | archimedean_iff_int_le | algebra.order | src/algebra/order/archimedean.lean | [
"data.int.least_greatest",
"data.rat.floor"
] | [
"archimedean",
"lt_add_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
archimedean_iff_rat_lt : archimedean α ↔ ∀ x : α, ∃ q : ℚ, x < q | ⟨@exists_rat_gt α _,
λ H, archimedean_iff_nat_lt.2 $ λ x,
let ⟨q, h⟩ := H x in
⟨⌈q⌉₊, lt_of_lt_of_le h $
by simpa only [rat.cast_coe_nat] using (@rat.cast_le α _ _ _).2 (nat.le_ceil _)⟩⟩ | lemma | archimedean_iff_rat_lt | algebra.order | src/algebra/order/archimedean.lean | [
"data.int.least_greatest",
"data.rat.floor"
] | [
"archimedean",
"exists_rat_gt",
"nat.le_ceil",
"rat.cast_coe_nat",
"rat.cast_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
archimedean_iff_rat_le : archimedean α ↔ ∀ x : α, ∃ q : ℚ, x ≤ q | archimedean_iff_rat_lt.trans
⟨λ H x, (H x).imp $ λ _, le_of_lt,
λ H x, let ⟨n, h⟩ := H x in ⟨n+1,
lt_of_le_of_lt h (rat.cast_lt.2 (lt_add_one _))⟩⟩ | lemma | archimedean_iff_rat_le | algebra.order | src/algebra/order/archimedean.lean | [
"data.int.least_greatest",
"data.rat.floor"
] | [
"archimedean",
"lt_add_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
archimedean.floor_ring (α) [linear_ordered_ring α] [archimedean α] :
floor_ring α | floor_ring.of_floor α (λ a, classical.some (exists_floor a))
(λ z a, (classical.some_spec (exists_floor a) z).symm) | def | archimedean.floor_ring | algebra.order | src/algebra/order/archimedean.lean | [
"data.int.least_greatest",
"data.rat.floor"
] | [
"archimedean",
"exists_floor",
"floor_ring",
"floor_ring.of_floor",
"linear_ordered_ring"
] | A linear ordered archimedean ring is a floor ring. This is not an `instance` because in some
cases we have a computable `floor` function. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
floor_ring.archimedean (α) [linear_ordered_field α] [floor_ring α] : archimedean α | begin
rw archimedean_iff_int_le,
exact λ x, ⟨⌈x⌉, int.le_ceil x⟩
end | instance | floor_ring.archimedean | algebra.order | src/algebra/order/archimedean.lean | [
"data.int.least_greatest",
"data.rat.floor"
] | [
"archimedean",
"archimedean_iff_int_le",
"floor_ring",
"int.le_ceil",
"linear_ordered_field"
] | A linear ordered field that is a floor ring is archimedean. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monovary_on.sum_smul_sum_le_card_smul_sum (hfg : monovary_on f g s) :
(∑ i in s, f i) • ∑ i in s, g i ≤ s.card • ∑ i in s, f i • g i | begin
classical,
obtain ⟨σ, hσ, hs⟩ := s.countable_to_set.exists_cycle_on,
rw [←card_range s.card, sum_smul_sum_eq_sum_perm hσ],
exact sum_le_card_nsmul _ _ _ (λ n _, hfg.sum_smul_comp_perm_le_sum_smul $ λ x hx, hs $ λ h, hx $
is_fixed_pt.perm_pow h _),
end | lemma | monovary_on.sum_smul_sum_le_card_smul_sum | algebra.order | src/algebra/order/chebyshev.lean | [
"algebra.big_operators.order",
"algebra.order.rearrangement",
"group_theory.perm.cycle.basic"
] | [
"monovary_on"
] | **Chebyshev's Sum Inequality**: When `f` and `g` monovary together (eg they are both
monotone/antitone), the scalar product of their sum is less than the size of the set times their
scalar product. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
antivary_on.card_smul_sum_le_sum_smul_sum (hfg : antivary_on f g s) :
s.card • ∑ i in s, f i • g i ≤ (∑ i in s, f i) • ∑ i in s, g i | by convert hfg.dual_right.sum_smul_sum_le_card_smul_sum | lemma | antivary_on.card_smul_sum_le_sum_smul_sum | algebra.order | src/algebra/order/chebyshev.lean | [
"algebra.big_operators.order",
"algebra.order.rearrangement",
"group_theory.perm.cycle.basic"
] | [
"antivary_on"
] | **Chebyshev's Sum Inequality**: When `f` and `g` antivary together (eg one is monotone, the
other is antitone), the scalar product of their sum is less than the size of the set times their
scalar product. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monovary.sum_smul_sum_le_card_smul_sum (hfg : monovary f g) :
(∑ i, f i) • ∑ i, g i ≤ fintype.card ι • ∑ i, f i • g i | (hfg.monovary_on _).sum_smul_sum_le_card_smul_sum | lemma | monovary.sum_smul_sum_le_card_smul_sum | algebra.order | src/algebra/order/chebyshev.lean | [
"algebra.big_operators.order",
"algebra.order.rearrangement",
"group_theory.perm.cycle.basic"
] | [
"fintype.card",
"monovary"
] | **Chebyshev's Sum Inequality**: When `f` and `g` monovary together (eg they are both
monotone/antitone), the scalar product of their sum is less than the size of the set times their
scalar product. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
antivary.card_smul_sum_le_sum_smul_sum (hfg : antivary f g) :
fintype.card ι • ∑ i, f i • g i ≤ (∑ i, f i) • ∑ i, g i | by convert (hfg.dual_right.monovary_on _).sum_smul_sum_le_card_smul_sum | lemma | antivary.card_smul_sum_le_sum_smul_sum | algebra.order | src/algebra/order/chebyshev.lean | [
"algebra.big_operators.order",
"algebra.order.rearrangement",
"group_theory.perm.cycle.basic"
] | [
"antivary",
"fintype.card"
] | **Chebyshev's Sum Inequality**: When `f` and `g` antivary together (eg one is monotone, the
other is antitone), the scalar product of their sum is less than the size of the set times their
scalar product. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monovary_on.sum_mul_sum_le_card_mul_sum (hfg : monovary_on f g s) :
(∑ i in s, f i) * (∑ i in s, g i) ≤ s.card * ∑ i in s, f i * g i | by { rw ←nsmul_eq_mul, exact hfg.sum_smul_sum_le_card_smul_sum } | lemma | monovary_on.sum_mul_sum_le_card_mul_sum | algebra.order | src/algebra/order/chebyshev.lean | [
"algebra.big_operators.order",
"algebra.order.rearrangement",
"group_theory.perm.cycle.basic"
] | [
"monovary_on"
] | **Chebyshev's Sum Inequality**: When `f` and `g` monovary together (eg they are both
monotone/antitone), the product of their sum is less than the size of the set times their scalar
product. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
antivary_on.card_mul_sum_le_sum_mul_sum (hfg : antivary_on f g s) :
(s.card : α) * ∑ i in s, f i * g i ≤ (∑ i in s, f i) * (∑ i in s, g i) | by { rw ←nsmul_eq_mul, exact hfg.card_smul_sum_le_sum_smul_sum } | lemma | antivary_on.card_mul_sum_le_sum_mul_sum | algebra.order | src/algebra/order/chebyshev.lean | [
"algebra.big_operators.order",
"algebra.order.rearrangement",
"group_theory.perm.cycle.basic"
] | [
"antivary_on"
] | **Chebyshev's Sum Inequality**: When `f` and `g` antivary together (eg one is monotone, the
other is antitone), the product of their sum is greater than the size of the set times their scalar
product. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sq_sum_le_card_mul_sum_sq : (∑ i in s, f i)^2 ≤ s.card * ∑ i in s, f i ^ 2 | by { simp_rw sq, exact (monovary_on_self _ _).sum_mul_sum_le_card_mul_sum } | lemma | sq_sum_le_card_mul_sum_sq | algebra.order | src/algebra/order/chebyshev.lean | [
"algebra.big_operators.order",
"algebra.order.rearrangement",
"group_theory.perm.cycle.basic"
] | [
"monovary_on_self"
] | Special case of **Chebyshev's Sum Inequality** or the **Cauchy-Schwarz Inequality**: The square
of the sum is less than the size of the set times the sum of the squares. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monovary.sum_mul_sum_le_card_mul_sum (hfg : monovary f g) :
(∑ i, f i) * (∑ i, g i) ≤ fintype.card ι * ∑ i, f i * g i | (hfg.monovary_on _).sum_mul_sum_le_card_mul_sum | lemma | monovary.sum_mul_sum_le_card_mul_sum | algebra.order | src/algebra/order/chebyshev.lean | [
"algebra.big_operators.order",
"algebra.order.rearrangement",
"group_theory.perm.cycle.basic"
] | [
"fintype.card",
"monovary"
] | **Chebyshev's Sum Inequality**: When `f` and `g` monovary together (eg they are both
monotone/antitone), the product of their sum is less than the size of the set times their scalar
product. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
antivary.card_mul_sum_le_sum_mul_sum (hfg : antivary f g) :
(fintype.card ι : α) * ∑ i, f i * g i ≤ (∑ i, f i) * (∑ i, g i) | (hfg.antivary_on _).card_mul_sum_le_sum_mul_sum | lemma | antivary.card_mul_sum_le_sum_mul_sum | algebra.order | src/algebra/order/chebyshev.lean | [
"algebra.big_operators.order",
"algebra.order.rearrangement",
"group_theory.perm.cycle.basic"
] | [
"antivary",
"fintype.card"
] | **Chebyshev's Sum Inequality**: When `f` and `g` antivary together (eg one is monotone, the
other is antitone), the product of their sum is less than the size of the set times their scalar
product. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sum_div_card_sq_le_sum_sq_div_card :
((∑ i in s, f i) / s.card) ^ 2 ≤ (∑ i in s, f i ^ 2) / s.card | begin
obtain rfl | hs := s.eq_empty_or_nonempty,
{ simp },
rw [←card_pos, ←@nat.cast_pos α] at hs,
rw [div_pow, div_le_div_iff (sq_pos_of_ne_zero _ hs.ne') hs, sq (s.card : α), mul_left_comm,
←mul_assoc],
exact mul_le_mul_of_nonneg_right (sq_sum_le_card_mul_sum_sq) hs.le,
end | lemma | sum_div_card_sq_le_sum_sq_div_card | algebra.order | src/algebra/order/chebyshev.lean | [
"algebra.big_operators.order",
"algebra.order.rearrangement",
"group_theory.perm.cycle.basic"
] | [
"div_le_div_iff",
"div_pow",
"mul_le_mul_of_nonneg_right",
"mul_left_comm",
"nat.cast_pos",
"sq_pos_of_ne_zero",
"sq_sum_le_card_mul_sum_sq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
conditionally_complete_linear_ordered_field (α : Type*)
extends linear_ordered_field α renaming max → sup min → inf, conditionally_complete_linear_order α | class | conditionally_complete_linear_ordered_field | algebra.order | src/algebra/order/complete_field.lean | [
"algebra.order.hom.ring",
"algebra.order.pointwise",
"analysis.special_functions.pow.real"
] | [
"conditionally_complete_linear_order",
"linear_ordered_field"
] | A field which is both linearly ordered and conditionally complete with respect to the order.
This axiomatizes the reals. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
conditionally_complete_linear_ordered_field.to_archimedean
[conditionally_complete_linear_ordered_field α] : archimedean α | archimedean_iff_nat_lt.2 begin
by_contra' h,
obtain ⟨x, h⟩ := h,
have := cSup_le (range_nonempty (coe : ℕ → α)) (forall_range_iff.2 $ λ n, le_sub_iff_add_le.2 $
le_cSup ⟨x, forall_range_iff.2 h⟩ ⟨n + 1, nat.cast_succ n⟩),
linarith,
end | instance | conditionally_complete_linear_ordered_field.to_archimedean | algebra.order | src/algebra/order/complete_field.lean | [
"algebra.order.hom.ring",
"algebra.order.pointwise",
"analysis.special_functions.pow.real"
] | [
"archimedean",
"cSup_le",
"conditionally_complete_linear_ordered_field",
"le_cSup",
"nat.cast_succ"
] | Any conditionally complete linearly ordered field is archimedean. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cut_map (a : α) : set β | (coe : ℚ → β) '' {t | ↑t < a} | def | linear_ordered_field.cut_map | algebra.order | src/algebra/order/complete_field.lean | [
"algebra.order.hom.ring",
"algebra.order.pointwise",
"analysis.special_functions.pow.real"
] | [] | The lower cut of rationals inside a linear ordered field that are less than a given element of
another linear ordered field. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cut_map_mono (h : a₁ ≤ a₂) : cut_map β a₁ ⊆ cut_map β a₂ | image_subset _ $ λ _, h.trans_lt' | lemma | linear_ordered_field.cut_map_mono | algebra.order | src/algebra/order/complete_field.lean | [
"algebra.order.hom.ring",
"algebra.order.pointwise",
"analysis.special_functions.pow.real"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_cut_map_iff : b ∈ cut_map β a ↔ ∃ q : ℚ, (q : α) < a ∧ (q : β) = b | iff.rfl | lemma | linear_ordered_field.mem_cut_map_iff | algebra.order | src/algebra/order/complete_field.lean | [
"algebra.order.hom.ring",
"algebra.order.pointwise",
"analysis.special_functions.pow.real"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mem_cut_map_iff [char_zero β] : (q : β) ∈ cut_map β a ↔ (q : α) < a | rat.cast_injective.mem_set_image | lemma | linear_ordered_field.coe_mem_cut_map_iff | algebra.order | src/algebra/order/complete_field.lean | [
"algebra.order.hom.ring",
"algebra.order.pointwise",
"analysis.special_functions.pow.real"
] | [
"char_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cut_map_self (a : α) : cut_map α a = Iio a ∩ range (coe : ℚ → α) | begin
ext,
split,
{ rintro ⟨q, h, rfl⟩,
exact ⟨h, q, rfl⟩ },
{ rintro ⟨h, q, rfl⟩,
exact ⟨q, h, rfl⟩ }
end | lemma | linear_ordered_field.cut_map_self | algebra.order | src/algebra/order/complete_field.lean | [
"algebra.order.hom.ring",
"algebra.order.pointwise",
"analysis.special_functions.pow.real"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cut_map_coe (q : ℚ) : cut_map β (q : α) = coe '' {r : ℚ | (r : β) < q} | by simp_rw [cut_map, rat.cast_lt] | lemma | linear_ordered_field.cut_map_coe | algebra.order | src/algebra/order/complete_field.lean | [
"algebra.order.hom.ring",
"algebra.order.pointwise",
"analysis.special_functions.pow.real"
] | [
"rat.cast_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cut_map_nonempty (a : α) : (cut_map β a).nonempty | nonempty.image _ $ exists_rat_lt a | lemma | linear_ordered_field.cut_map_nonempty | algebra.order | src/algebra/order/complete_field.lean | [
"algebra.order.hom.ring",
"algebra.order.pointwise",
"analysis.special_functions.pow.real"
] | [
"exists_rat_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cut_map_bdd_above (a : α) : bdd_above (cut_map β a) | begin
obtain ⟨q, hq⟩ := exists_rat_gt a,
exact ⟨q, ball_image_iff.2 $ λ r hr, by exact_mod_cast (hq.trans' hr).le⟩,
end | lemma | linear_ordered_field.cut_map_bdd_above | algebra.order | src/algebra/order/complete_field.lean | [
"algebra.order.hom.ring",
"algebra.order.pointwise",
"analysis.special_functions.pow.real"
] | [
"bdd_above",
"exists_rat_gt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cut_map_add (a b : α) : cut_map β (a + b) = cut_map β a + cut_map β b | begin
refine (image_subset_iff.2 $ λ q hq, _).antisymm _,
{ rw [mem_set_of_eq, ←sub_lt_iff_lt_add] at hq,
obtain ⟨q₁, hq₁q, hq₁ab⟩ := exists_rat_btwn hq,
refine ⟨q₁, q - q₁, _, _, add_sub_cancel'_right _ _⟩; try {norm_cast}; rwa coe_mem_cut_map_iff,
exact_mod_cast sub_lt_comm.mp hq₁q },
{ rintro _ ⟨_,... | lemma | linear_ordered_field.cut_map_add | algebra.order | src/algebra/order/complete_field.lean | [
"algebra.order.hom.ring",
"algebra.order.pointwise",
"analysis.special_functions.pow.real"
] | [
"exists_rat_btwn"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
induced_map (x : α) : β | Sup $ cut_map β x | def | linear_ordered_field.induced_map | algebra.order | src/algebra/order/complete_field.lean | [
"algebra.order.hom.ring",
"algebra.order.pointwise",
"analysis.special_functions.pow.real"
] | [
"induced_map"
] | The induced order preserving function from a linear ordered field to a conditionally complete
linear ordered field, defined by taking the Sup in the codomain of all the rationals less than the
input. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
induced_map_mono : monotone (induced_map α β) | λ a b h, cSup_le_cSup (cut_map_bdd_above β _) (cut_map_nonempty β _) (cut_map_mono β h) | lemma | linear_ordered_field.induced_map_mono | algebra.order | src/algebra/order/complete_field.lean | [
"algebra.order.hom.ring",
"algebra.order.pointwise",
"analysis.special_functions.pow.real"
] | [
"cSup_le_cSup",
"induced_map",
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
induced_map_rat (q : ℚ) : induced_map α β (q : α) = q | begin
refine cSup_eq_of_forall_le_of_forall_lt_exists_gt (cut_map_nonempty β q) (λ x h, _) (λ w h, _),
{ rw cut_map_coe at h,
obtain ⟨r, h, rfl⟩ := h,
exact le_of_lt h },
{ obtain ⟨q', hwq, hq⟩ := exists_rat_btwn h,
rw cut_map_coe,
exact ⟨q', ⟨_, hq, rfl⟩, hwq⟩ }
end | lemma | linear_ordered_field.induced_map_rat | algebra.order | src/algebra/order/complete_field.lean | [
"algebra.order.hom.ring",
"algebra.order.pointwise",
"analysis.special_functions.pow.real"
] | [
"cSup_eq_of_forall_le_of_forall_lt_exists_gt",
"exists_rat_btwn",
"induced_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
induced_map_zero : induced_map α β 0 = 0 | by exact_mod_cast induced_map_rat α β 0 | lemma | linear_ordered_field.induced_map_zero | algebra.order | src/algebra/order/complete_field.lean | [
"algebra.order.hom.ring",
"algebra.order.pointwise",
"analysis.special_functions.pow.real"
] | [
"induced_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
induced_map_one : induced_map α β 1 = 1 | by exact_mod_cast induced_map_rat α β 1 | lemma | linear_ordered_field.induced_map_one | algebra.order | src/algebra/order/complete_field.lean | [
"algebra.order.hom.ring",
"algebra.order.pointwise",
"analysis.special_functions.pow.real"
] | [
"induced_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
induced_map_nonneg (ha : 0 ≤ a) : 0 ≤ induced_map α β a | (induced_map_zero α _).ge.trans $ induced_map_mono _ _ ha | lemma | linear_ordered_field.induced_map_nonneg | algebra.order | src/algebra/order/complete_field.lean | [
"algebra.order.hom.ring",
"algebra.order.pointwise",
"analysis.special_functions.pow.real"
] | [
"induced_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_lt_induced_map_iff : (q : β) < induced_map α β a ↔ (q : α) < a | begin
refine ⟨λ h, _, λ hq, _⟩,
{ rw ←induced_map_rat α at h,
exact (induced_map_mono α β).reflect_lt h },
{ obtain ⟨q', hq, hqa⟩ := exists_rat_btwn hq,
apply lt_cSup_of_lt (cut_map_bdd_above β a) (coe_mem_cut_map_iff.mpr hqa),
exact_mod_cast hq }
end | lemma | linear_ordered_field.coe_lt_induced_map_iff | algebra.order | src/algebra/order/complete_field.lean | [
"algebra.order.hom.ring",
"algebra.order.pointwise",
"analysis.special_functions.pow.real"
] | [
"exists_rat_btwn",
"induced_map",
"lt_cSup_of_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_induced_map_iff : b < induced_map α β a ↔ ∃ q : ℚ, b < q ∧ (q : α) < a | ⟨λ h, (exists_rat_btwn h).imp $ λ q, and.imp_right coe_lt_induced_map_iff.1,
λ ⟨q, hbq, hqa⟩, hbq.trans $ by rwa coe_lt_induced_map_iff⟩ | lemma | linear_ordered_field.lt_induced_map_iff | algebra.order | src/algebra/order/complete_field.lean | [
"algebra.order.hom.ring",
"algebra.order.pointwise",
"analysis.special_functions.pow.real"
] | [
"and.imp_right",
"exists_rat_btwn",
"induced_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
induced_map_self (b : β) : induced_map β β b = b | eq_of_forall_rat_lt_iff_lt $ λ q, coe_lt_induced_map_iff | lemma | linear_ordered_field.induced_map_self | algebra.order | src/algebra/order/complete_field.lean | [
"algebra.order.hom.ring",
"algebra.order.pointwise",
"analysis.special_functions.pow.real"
] | [
"eq_of_forall_rat_lt_iff_lt",
"induced_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
induced_map_induced_map (a : α) :
induced_map β γ (induced_map α β a) = induced_map α γ a | eq_of_forall_rat_lt_iff_lt $ λ q,
by rw [coe_lt_induced_map_iff, coe_lt_induced_map_iff, iff.comm, coe_lt_induced_map_iff] | lemma | linear_ordered_field.induced_map_induced_map | algebra.order | src/algebra/order/complete_field.lean | [
"algebra.order.hom.ring",
"algebra.order.pointwise",
"analysis.special_functions.pow.real"
] | [
"eq_of_forall_rat_lt_iff_lt",
"induced_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
induced_map_inv_self (b : β) : induced_map γ β (induced_map β γ b) = b | by rw [induced_map_induced_map, induced_map_self] | lemma | linear_ordered_field.induced_map_inv_self | algebra.order | src/algebra/order/complete_field.lean | [
"algebra.order.hom.ring",
"algebra.order.pointwise",
"analysis.special_functions.pow.real"
] | [
"induced_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
induced_map_add (x y : α) : induced_map α β (x + y) = induced_map α β x + induced_map α β y | begin
rw [induced_map, cut_map_add],
exact cSup_add (cut_map_nonempty β x) (cut_map_bdd_above β x) (cut_map_nonempty β y)
(cut_map_bdd_above β y),
end | lemma | linear_ordered_field.induced_map_add | algebra.order | src/algebra/order/complete_field.lean | [
"algebra.order.hom.ring",
"algebra.order.pointwise",
"analysis.special_functions.pow.real"
] | [
"induced_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_induced_map_mul_self_of_mem_cut_map (ha : 0 < a) (b : β) (hb : b ∈ cut_map β (a * a)) :
b ≤ induced_map α β a * induced_map α β a | begin
obtain ⟨q, hb, rfl⟩ := hb,
obtain ⟨q', hq', hqq', hqa⟩ := exists_rat_pow_btwn two_ne_zero hb (mul_self_pos.2 ha.ne'),
transitivity (q' : β)^2,
exact_mod_cast hqq'.le,
rw pow_two at ⊢ hqa,
exact mul_self_le_mul_self (by exact_mod_cast hq'.le) (le_cSup (cut_map_bdd_above β a) $
coe_mem_cut_map_iff.2... | lemma | linear_ordered_field.le_induced_map_mul_self_of_mem_cut_map | algebra.order | src/algebra/order/complete_field.lean | [
"algebra.order.hom.ring",
"algebra.order.pointwise",
"analysis.special_functions.pow.real"
] | [
"induced_map",
"le_cSup",
"lt_of_mul_self_lt_mul_self",
"mul_self_le_mul_self",
"pow_two",
"two_ne_zero"
] | Preparatory lemma for `induced_ring_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_mem_cut_map_mul_self_of_lt_induced_map_mul_self (ha : 0 < a) (b : β)
(hba : b < induced_map α β a * induced_map α β a) :
∃ c ∈ cut_map β (a * a), b < c | begin
obtain hb | hb := lt_or_le b 0,
{ refine ⟨0, _, hb⟩,
rw [←rat.cast_zero, coe_mem_cut_map_iff, rat.cast_zero],
exact mul_self_pos.2 ha.ne' },
obtain ⟨q, hq, hbq, hqa⟩ := exists_rat_pow_btwn two_ne_zero hba (hb.trans_lt hba),
rw ←cast_pow at hbq,
refine ⟨(q^2 : ℚ), coe_mem_cut_map_iff.2 _, hbq⟩,
... | lemma | linear_ordered_field.exists_mem_cut_map_mul_self_of_lt_induced_map_mul_self | algebra.order | src/algebra/order/complete_field.lean | [
"algebra.order.hom.ring",
"algebra.order.pointwise",
"analysis.special_functions.pow.real"
] | [
"induced_map",
"lt_of_mul_self_lt_mul_self",
"mul_self_lt_mul_self",
"pow_two",
"rat.cast_zero",
"two_ne_zero"
] | Preparatory lemma for `induced_ring_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
induced_add_hom : α →+ β | ⟨induced_map α β, induced_map_zero α β, induced_map_add α β⟩ | def | linear_ordered_field.induced_add_hom | algebra.order | src/algebra/order/complete_field.lean | [
"algebra.order.hom.ring",
"algebra.order.pointwise",
"analysis.special_functions.pow.real"
] | [] | `induced_map` as an additive homomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
induced_order_ring_hom : α →+*o β | { monotone' := induced_map_mono _ _,
..(induced_add_hom α β).mk_ring_hom_of_mul_self_of_two_ne_zero -- reduce to the case of x = y
begin
-- reduce to the case of 0 < x
suffices : ∀ x, 0 < x →
induced_add_hom α β (x * x) = induced_add_hom α β x * induced_add_hom α β x,
{ rintro x,
... | def | linear_ordered_field.induced_order_ring_hom | algebra.order | src/algebra/order/complete_field.lean | [
"algebra.order.hom.ring",
"algebra.order.pointwise",
"analysis.special_functions.pow.real"
] | [
"cSup_eq_of_forall_le_of_forall_lt_exists_gt",
"mul_neg",
"mul_zero",
"neg_mul"
] | `induced_map` as an `order_ring_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
induced_order_ring_iso : β ≃+*o γ | { inv_fun := induced_map γ β,
left_inv := induced_map_inv_self _ _,
right_inv := induced_map_inv_self _ _,
map_le_map_iff' := λ x y, begin
refine ⟨λ h, _, λ h, induced_map_mono _ _ h⟩,
simpa [induced_order_ring_hom, add_monoid_hom.mk_ring_hom_of_mul_self_of_two_ne_zero,
induced_add_hom] using induce... | def | linear_ordered_field.induced_order_ring_iso | algebra.order | src/algebra/order/complete_field.lean | [
"algebra.order.hom.ring",
"algebra.order.pointwise",
"analysis.special_functions.pow.real"
] | [
"add_monoid_hom.mk_ring_hom_of_mul_self_of_two_ne_zero",
"induced_map",
"inv_fun"
] | The isomorphism of ordered rings between two conditionally complete linearly ordered fields. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_induced_order_ring_iso : ⇑(induced_order_ring_iso β γ) = induced_map β γ | rfl | lemma | linear_ordered_field.coe_induced_order_ring_iso | algebra.order | src/algebra/order/complete_field.lean | [
"algebra.order.hom.ring",
"algebra.order.pointwise",
"analysis.special_functions.pow.real"
] | [
"induced_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
induced_order_ring_iso_symm :
(induced_order_ring_iso β γ).symm = induced_order_ring_iso γ β | rfl | lemma | linear_ordered_field.induced_order_ring_iso_symm | algebra.order | src/algebra/order/complete_field.lean | [
"algebra.order.hom.ring",
"algebra.order.pointwise",
"analysis.special_functions.pow.real"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
induced_order_ring_iso_self : induced_order_ring_iso β β = order_ring_iso.refl β | order_ring_iso.ext induced_map_self | lemma | linear_ordered_field.induced_order_ring_iso_self | algebra.order | src/algebra/order/complete_field.lean | [
"algebra.order.hom.ring",
"algebra.order.pointwise",
"analysis.special_functions.pow.real"
] | [
"order_ring_iso.ext",
"order_ring_iso.refl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ring_hom_monotone (hR : ∀ r : R, 0 ≤ r → ∃ s : R, s^2 = r) (f : R →+* S) : monotone f | (monotone_iff_map_nonneg f).2 $ λ r h, by { obtain ⟨s, rfl⟩ := hR r h, rw map_pow, apply sq_nonneg } | lemma | ring_hom_monotone | algebra.order | src/algebra/order/complete_field.lean | [
"algebra.order.hom.ring",
"algebra.order.pointwise",
"analysis.special_functions.pow.real"
] | [
"map_pow",
"monotone",
"monotone_iff_map_nonneg",
"sq_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
real.ring_hom.unique : unique (ℝ →+* ℝ) | { default := ring_hom.id ℝ,
uniq := λ f, congr_arg order_ring_hom.to_ring_hom
(subsingleton.elim ⟨f, ring_hom_monotone (λ r hr, ⟨real.sqrt r, sq_sqrt hr⟩) f⟩ default), } | instance | real.ring_hom.unique | algebra.order | src/algebra/order/complete_field.lean | [
"algebra.order.hom.ring",
"algebra.order.pointwise",
"analysis.special_functions.pow.real"
] | [
"ring_hom.id",
"ring_hom_monotone",
"unique"
] | There exists no nontrivial ring homomorphism `ℝ →+* ℝ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_euclidean : Prop | (map_lt_map_iff' : ∀ {x y}, abv x < abv y ↔ x ≺ y) | structure | absolute_value.is_euclidean | algebra.order | src/algebra/order/euclidean_absolute_value.lean | [
"algebra.order.absolute_value",
"algebra.euclidean_domain.instances"
] | [] | An absolute value `abv : R → S` is Euclidean if it is compatible with the
`euclidean_domain` structure on `R`, namely `abv` is strictly monotone with respect to the well
founded relation `≺` on `R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_lt_map_iff {x y : R} (h : abv.is_euclidean) : abv x < abv y ↔ x ≺ y | map_lt_map_iff' h | lemma | absolute_value.is_euclidean.map_lt_map_iff | algebra.order | src/algebra/order/euclidean_absolute_value.lean | [
"algebra.order.absolute_value",
"algebra.euclidean_domain.instances"
] | [
"map_lt_map_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_mod_lt (h : abv.is_euclidean) (a : R) {b : R} (hb : b ≠ 0) :
abv (a % b) < abv b | h.map_lt_map_iff.mpr (euclidean_domain.mod_lt a hb) | lemma | absolute_value.is_euclidean.sub_mod_lt | algebra.order | src/algebra/order/euclidean_absolute_value.lean | [
"algebra.order.absolute_value",
"algebra.euclidean_domain.instances"
] | [
"euclidean_domain.mod_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_is_euclidean : is_euclidean (absolute_value.abs : absolute_value ℤ ℤ) | { map_lt_map_iff' := λ x y, show abs x < abs y ↔ nat_abs x < nat_abs y,
by rw [abs_eq_nat_abs, abs_eq_nat_abs, coe_nat_lt] } | lemma | absolute_value.abs_is_euclidean | algebra.order | src/algebra/order/euclidean_absolute_value.lean | [
"algebra.order.absolute_value",
"algebra.euclidean_domain.instances"
] | [
"absolute_value",
"absolute_value.abs"
] | `abs : ℤ → ℤ` is a Euclidean absolute value | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
floor_semiring (α) [ordered_semiring α] | (floor : α → ℕ)
(ceil : α → ℕ)
(floor_of_neg {a : α} (ha : a < 0) : floor a = 0)
(gc_floor {a : α} {n : ℕ} (ha : 0 ≤ a) : n ≤ floor a ↔ (n : α) ≤ a)
(gc_ceil : galois_connection ceil coe) | class | 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"
] | [
"galois_connection",
"ordered_semiring"
] | A `floor_semiring` is an ordered semiring over `α` with a function
`floor : α → ℕ` satisfying `∀ (n : ℕ) (x : α), n ≤ ⌊x⌋ ↔ (n : α) ≤ x)`.
Note that many lemmas require a `linear_order`. Please see the above `TODO`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
floor : α → ℕ | floor_semiring.floor | def | nat.floor | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [] | `⌊a⌋₊` is the greatest natural `n` such that `n ≤ a`. If `a` is negative, then `⌊a⌋₊ = 0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ceil : α → ℕ | floor_semiring.ceil | def | nat.ceil | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [] | `⌈a⌉₊` is the least natural `n` such that `a ≤ n` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
floor_nat : (nat.floor : ℕ → ℕ) = id | rfl | lemma | nat.floor_nat | 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 | |
ceil_nat : (nat.ceil : ℕ → ℕ) = id | rfl | lemma | nat.ceil_nat | 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 | |
le_floor_iff (ha : 0 ≤ a) : n ≤ ⌊a⌋₊ ↔ (n : α) ≤ a | floor_semiring.gc_floor ha | lemma | nat.le_floor_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 | |
le_floor (h : (n : α) ≤ a) : n ≤ ⌊a⌋₊ | (le_floor_iff $ n.cast_nonneg.trans h).2 h | lemma | nat.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 (ha : 0 ≤ a) : ⌊a⌋₊ < n ↔ a < n | lt_iff_lt_of_le_iff_le $ le_floor_iff ha | lemma | nat.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_lt_one (ha : 0 ≤ a) : ⌊a⌋₊ < 1 ↔ a < 1 | (floor_lt ha).trans $ by rw nat.cast_one | lemma | nat.floor_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"
] | [
"nat.cast_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_of_floor_lt (h : ⌊a⌋₊ < n) : a < n | lt_of_not_le $ λ h', (le_floor h').not_lt h | lemma | nat.lt_of_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_of_not_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_one_of_floor_lt_one (h : ⌊a⌋₊ < 1) : a < 1 | by exact_mod_cast lt_of_floor_lt h | lemma | nat.lt_one_of_floor_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 | |
floor_le (ha : 0 ≤ a) : (⌊a⌋₊ : α) ≤ a | (le_floor_iff ha).1 le_rfl | lemma | nat.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"
] | [
"le_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_succ_floor (a : α) : a < ⌊a⌋₊.succ | lt_of_floor_lt $ nat.lt_succ_self _ | lemma | nat.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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_floor_add_one (a : α) : a < ⌊a⌋₊ + 1 | by simpa using lt_succ_floor a | lemma | nat.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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
floor_coe (n : ℕ) : ⌊(n : α)⌋₊ = n | eq_of_forall_le_iff $ λ a, by { rw [le_floor_iff, nat.cast_le], exact n.cast_nonneg } | lemma | nat.floor_coe | 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",
"nat.cast_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
floor_zero : ⌊(0 : α)⌋₊ = 0 | by rw [← nat.cast_zero, floor_coe] | lemma | nat.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"
] | [
"nat.cast_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
floor_one : ⌊(1 : α)⌋₊ = 1 | by rw [←nat.cast_one, floor_coe] | lemma | nat.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_of_nonpos (ha : a ≤ 0) : ⌊a⌋₊ = 0 | ha.lt_or_eq.elim floor_semiring.floor_of_neg $ by { rintro rfl, exact floor_zero } | lemma | nat.floor_of_nonpos | 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 → ℕ) | λ a b h, begin
obtain ha | ha := le_total a 0,
{ rw floor_of_nonpos ha,
exact nat.zero_le _ },
{ exact le_floor ((floor_le ha).trans h) }
end | lemma | nat.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 | |
le_floor_iff' (hn : n ≠ 0) : n ≤ ⌊a⌋₊ ↔ (n : α) ≤ a | begin
obtain ha | ha := le_total a 0,
{ rw floor_of_nonpos ha,
exact iff_of_false (nat.pos_of_ne_zero hn).not_le
(not_le_of_lt $ ha.trans_lt $ cast_pos.2 $ nat.pos_of_ne_zero hn) },
{ exact le_floor_iff ha }
end | lemma | nat.le_floor_iff' | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [
"iff_of_false",
"not_le_of_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_le_floor_iff (x : α) : 1 ≤ ⌊x⌋₊ ↔ 1 ≤ x | by exact_mod_cast (@le_floor_iff' α _ _ x 1 one_ne_zero) | lemma | nat.one_le_floor_iff | algebra.order | src/algebra/order/floor.lean | [
"data.int.lemmas",
"data.set.intervals.group",
"data.set.lattice",
"tactic.abel",
"tactic.linarith",
"tactic.positivity"
] | [
"one_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
floor_lt' (hn : n ≠ 0) : ⌊a⌋₊ < n ↔ a < n | lt_iff_lt_of_le_iff_le $ le_floor_iff' hn | lemma | nat.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_pos : 0 < ⌊a⌋₊ ↔ 1 ≤ a | by { convert le_floor_iff' nat.one_ne_zero, exact cast_one.symm } | lemma | nat.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 | |
pos_of_floor_pos (h : 0 < ⌊a⌋₊) : 0 < a | (le_or_lt a 0).resolve_left (λ ha, lt_irrefl 0 $ by rwa floor_of_nonpos ha at h) | lemma | nat.pos_of_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 | |
lt_of_lt_floor (h : n < ⌊a⌋₊) : ↑n < a | (nat.cast_lt.2 h).trans_le $ floor_le (pos_of_floor_pos $ (nat.zero_le n).trans_lt h).le | lemma | nat.lt_of_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_le_of_le (h : a ≤ n) : ⌊a⌋₊ ≤ n | le_imp_le_iff_lt_imp_lt.2 lt_of_lt_floor h | lemma | nat.floor_le_of_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_one_of_le_one (h : a ≤ 1) : ⌊a⌋₊ ≤ 1 | floor_le_of_le $ h.trans_eq $ nat.cast_one.symm | lemma | nat.floor_le_one_of_le_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_eq_zero : ⌊a⌋₊ = 0 ↔ a < 1 | by { rw [←lt_one_iff, ←@cast_one α], exact floor_lt' nat.one_ne_zero } | lemma | nat.floor_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
floor_eq_iff (ha : 0 ≤ a) : ⌊a⌋₊ = n ↔ ↑n ≤ a ∧ a < ↑n + 1 | by rw [←le_floor_iff ha, ←nat.cast_one, ←nat.cast_add, ←floor_lt ha, nat.lt_add_one_iff,
le_antisymm_iff, and.comm] | lemma | nat.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"
] | [
"nat.lt_add_one_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
floor_eq_iff' (hn : n ≠ 0) : ⌊a⌋₊ = n ↔ ↑n ≤ a ∧ a < ↑n + 1 | by rw [← le_floor_iff' hn, ← nat.cast_one, ← nat.cast_add, ← floor_lt' (nat.add_one_ne_zero n),
nat.lt_add_one_iff, le_antisymm_iff, and.comm] | lemma | nat.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"
] | [
"nat.cast_add",
"nat.cast_one",
"nat.lt_add_one_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
floor_eq_on_Ico (n : ℕ) : ∀ a ∈ (set.Ico n (n+1) : set α), ⌊a⌋₊ = n | λ a ⟨h₀, h₁⟩, (floor_eq_iff $ n.cast_nonneg.trans h₀).mpr ⟨h₀, h₁⟩ | lemma | nat.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 : ℕ) : ∀ a ∈ (set.Ico n (n+1) : set α), (⌊a⌋₊ : α) = n | λ x hx, by exact_mod_cast floor_eq_on_Ico n x hx | lemma | nat.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_zero : (floor : α → ℕ) ⁻¹' {0} = Iio 1 | ext $ λ a, floor_eq_zero | lemma | nat.preimage_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 | |
preimage_floor_of_ne_zero {n : ℕ} (hn : n ≠ 0) : (floor : α → ℕ) ⁻¹' {n} = Ico n (n + 1) | ext $ λ a, floor_eq_iff' hn | lemma | nat.preimage_floor_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gc_ceil_coe : galois_connection (ceil : α → ℕ) coe | floor_semiring.gc_ceil | lemma | nat.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⌉₊ ≤ n ↔ a ≤ n | gc_ceil_coe _ _ | lemma | nat.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 | |
lt_ceil : n < ⌈a⌉₊ ↔ (n : α) < a | lt_iff_lt_of_le_iff_le ceil_le | lemma | nat.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 : n + 1 ≤ ⌈a⌉₊ ↔ (n : α) < a | by rw [← nat.lt_ceil, nat.add_one_le_iff] | lemma | nat.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"
] | [
"nat.add_one_le_iff",
"nat.lt_ceil"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_le_ceil_iff : 1 ≤ ⌈a⌉₊ ↔ 0 < a | by rw [← zero_add 1, nat.add_one_le_ceil_iff, nat.cast_zero] | lemma | nat.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"
] | [
"nat.add_one_le_ceil_iff",
"nat.cast_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ceil_le_floor_add_one (a : α) : ⌈a⌉₊ ≤ ⌊a⌋₊ + 1 | by { rw [ceil_le, nat.cast_add, nat.cast_one], exact (lt_floor_add_one a).le } | lemma | nat.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"
] | [
"nat.cast_add",
"nat.cast_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_ceil (a : α) : a ≤ ⌈a⌉₊ | ceil_le.1 le_rfl | lemma | nat.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"
] | [
"le_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ceil_int_cast {α : Type*} [linear_ordered_ring α]
[floor_semiring α] (z : ℤ) : ⌈(z : α)⌉₊ = z.to_nat | eq_of_forall_ge_iff $ λ a, by { simp, norm_cast } | lemma | nat.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",
"floor_semiring",
"linear_ordered_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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