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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|---|---|---|---|---|---|---|---|---|---|---|
map_sum_le {ι : Type*} {s : finset ι} {f : ι → R} {g : Γ₀} (hf : ∀ i ∈ s, v (f i) ≤ g) :
v (∑ i in s, f i) ≤ g | begin
refine finset.induction_on s
(λ _, trans_rel_right (≤) v.map_zero zero_le') (λ a s has ih hf, _) hf,
rw finset.forall_mem_insert at hf, rw finset.sum_insert has,
exact v.map_add_le hf.1 (ih hf.2)
end | lemma | valuation.map_sum_le | ring_theory.valuation | src/ring_theory/valuation/basic.lean | [
"algebra.order.with_zero",
"ring_theory.ideal.operations"
] | [
"finset",
"finset.forall_mem_insert",
"finset.induction_on",
"ih",
"zero_le'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_sum_lt {ι : Type*} {s : finset ι} {f : ι → R} {g : Γ₀} (hg : g ≠ 0)
(hf : ∀ i ∈ s, v (f i) < g) : v (∑ i in s, f i) < g | begin
refine finset.induction_on s
(λ _, trans_rel_right (<) v.map_zero (zero_lt_iff.2 hg)) (λ a s has ih hf, _) hf,
rw finset.forall_mem_insert at hf, rw finset.sum_insert has,
exact v.map_add_lt hf.1 (ih hf.2)
end | lemma | valuation.map_sum_lt | ring_theory.valuation | src/ring_theory/valuation/basic.lean | [
"algebra.order.with_zero",
"ring_theory.ideal.operations"
] | [
"finset",
"finset.forall_mem_insert",
"finset.induction_on",
"ih"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_sum_lt' {ι : Type*} {s : finset ι} {f : ι → R} {g : Γ₀} (hg : 0 < g)
(hf : ∀ i ∈ s, v (f i) < g) : v (∑ i in s, f i) < g | v.map_sum_lt (ne_of_gt hg) hf | lemma | valuation.map_sum_lt' | ring_theory.valuation | src/ring_theory/valuation/basic.lean | [
"algebra.order.with_zero",
"ring_theory.ideal.operations"
] | [
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_pow : ∀ x (n:ℕ), v (x^n) = (v x)^n | v.to_monoid_with_zero_hom.to_monoid_hom.map_pow | lemma | valuation.map_pow | ring_theory.valuation | src/ring_theory/valuation/basic.lean | [
"algebra.order.with_zero",
"ring_theory.ideal.operations"
] | [
"map_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext_iff {v₁ v₂ : valuation R Γ₀} : v₁ = v₂ ↔ ∀ r, v₁ r = v₂ r | fun_like.ext_iff | lemma | valuation.ext_iff | ring_theory.valuation | src/ring_theory/valuation/basic.lean | [
"algebra.order.with_zero",
"ring_theory.ideal.operations"
] | [
"fun_like.ext_iff",
"valuation"
] | Deprecated. Use `fun_like.ext_iff`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_preorder : preorder R | preorder.lift v | def | valuation.to_preorder | ring_theory.valuation | src/ring_theory/valuation/basic.lean | [
"algebra.order.with_zero",
"ring_theory.ideal.operations"
] | [
"preorder.lift"
] | A valuation gives a preorder on the underlying ring. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
zero_iff [nontrivial Γ₀] (v : valuation K Γ₀) {x : K} :
v x = 0 ↔ x = 0 | map_eq_zero v | lemma | valuation.zero_iff | ring_theory.valuation | src/ring_theory/valuation/basic.lean | [
"algebra.order.with_zero",
"ring_theory.ideal.operations"
] | [
"map_eq_zero",
"nontrivial",
"valuation"
] | If `v` is a valuation on a division ring then `v(x) = 0` iff `x = 0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ne_zero_iff [nontrivial Γ₀] (v : valuation K Γ₀) {x : K} :
v x ≠ 0 ↔ x ≠ 0 | map_ne_zero v | lemma | valuation.ne_zero_iff | ring_theory.valuation | src/ring_theory/valuation/basic.lean | [
"algebra.order.with_zero",
"ring_theory.ideal.operations"
] | [
"map_ne_zero",
"ne_zero_iff",
"nontrivial",
"valuation"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unit_map_eq (u : Rˣ) :
(units.map (v : R →* Γ₀) u : Γ₀) = v u | rfl | theorem | valuation.unit_map_eq | ring_theory.valuation | src/ring_theory/valuation/basic.lean | [
"algebra.order.with_zero",
"ring_theory.ideal.operations"
] | [
"units.map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap {S : Type*} [ring S] (f : S →+* R) (v : valuation R Γ₀) :
valuation S Γ₀ | { to_fun := v ∘ f,
map_add_le_max' := λ x y, by simp only [comp_app, map_add, f.map_add],
.. v.to_monoid_with_zero_hom.comp f.to_monoid_with_zero_hom, } | def | valuation.comap | ring_theory.valuation | src/ring_theory/valuation/basic.lean | [
"algebra.order.with_zero",
"ring_theory.ideal.operations"
] | [
"ring",
"valuation"
] | A ring homomorphism `S → R` induces a map `valuation R Γ₀ → valuation S Γ₀`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comap_apply {S : Type*} [ring S] (f : S →+* R) (v : valuation R Γ₀) (s : S) :
v.comap f s = v (f s) | rfl | lemma | valuation.comap_apply | ring_theory.valuation | src/ring_theory/valuation/basic.lean | [
"algebra.order.with_zero",
"ring_theory.ideal.operations"
] | [
"ring",
"valuation"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_id : v.comap (ring_hom.id R) = v | ext $ λ r, rfl | lemma | valuation.comap_id | ring_theory.valuation | src/ring_theory/valuation/basic.lean | [
"algebra.order.with_zero",
"ring_theory.ideal.operations"
] | [
"ring_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_comp {S₁ : Type*} {S₂ : Type*} [ring S₁] [ring S₂] (f : S₁ →+* S₂) (g : S₂ →+* R) :
v.comap (g.comp f) = (v.comap g).comap f | ext $ λ r, rfl | lemma | valuation.comap_comp | ring_theory.valuation | src/ring_theory/valuation/basic.lean | [
"algebra.order.with_zero",
"ring_theory.ideal.operations"
] | [
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map (f : Γ₀ →*₀ Γ'₀) (hf : monotone f) (v : valuation R Γ₀) :
valuation R Γ'₀ | { to_fun := f ∘ v,
map_add_le_max' := λ r s,
calc f (v (r + s)) ≤ f (max (v r) (v s)) : hf (v.map_add r s)
... = max (f (v r)) (f (v s)) : hf.map_max,
.. monoid_with_zero_hom.comp f v.to_monoid_with_zero_hom } | def | valuation.map | ring_theory.valuation | src/ring_theory/valuation/basic.lean | [
"algebra.order.with_zero",
"ring_theory.ideal.operations"
] | [
"monoid_with_zero_hom.comp",
"monotone",
"valuation"
] | A `≤`-preserving group homomorphism `Γ₀ → Γ'₀` induces a map `valuation R Γ₀ → valuation R Γ'₀`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_equiv (v₁ : valuation R Γ₀) (v₂ : valuation R Γ'₀) : Prop | ∀ r s, v₁ r ≤ v₁ s ↔ v₂ r ≤ v₂ s | def | valuation.is_equiv | ring_theory.valuation | src/ring_theory/valuation/basic.lean | [
"algebra.order.with_zero",
"ring_theory.ideal.operations"
] | [
"valuation"
] | Two valuations on `R` are defined to be equivalent if they induce the same preorder on `R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_neg (x : R) : v (-x) = v x | v.to_monoid_with_zero_hom.to_monoid_hom.map_neg x | lemma | valuation.map_neg | ring_theory.valuation | src/ring_theory/valuation/basic.lean | [
"algebra.order.with_zero",
"ring_theory.ideal.operations"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_sub_swap (x y : R) : v (x - y) = v (y - x) | v.to_monoid_with_zero_hom.to_monoid_hom.map_sub_swap x y | lemma | valuation.map_sub_swap | ring_theory.valuation | src/ring_theory/valuation/basic.lean | [
"algebra.order.with_zero",
"ring_theory.ideal.operations"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_sub (x y : R) : v (x - y) ≤ max (v x) (v y) | calc v (x - y) = v (x + -y) : by rw [sub_eq_add_neg]
... ≤ max (v x) (v $ -y) : v.map_add _ _
... = max (v x) (v y) : by rw map_neg | lemma | valuation.map_sub | ring_theory.valuation | src/ring_theory/valuation/basic.lean | [
"algebra.order.with_zero",
"ring_theory.ideal.operations"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_sub_le {x y g} (hx : v x ≤ g) (hy : v y ≤ g) : v (x - y) ≤ g | begin
rw sub_eq_add_neg,
exact v.map_add_le hx (le_trans (le_of_eq (v.map_neg y)) hy)
end | lemma | valuation.map_sub_le | ring_theory.valuation | src/ring_theory/valuation/basic.lean | [
"algebra.order.with_zero",
"ring_theory.ideal.operations"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_add_of_distinct_val (h : v x ≠ v y) : v (x + y) = max (v x) (v y) | begin
suffices : ¬v (x + y) < max (v x) (v y),
from or_iff_not_imp_right.1 (le_iff_eq_or_lt.1 (v.map_add x y)) this,
intro h',
wlog vyx : v y < v x,
{ refine this v h.symm _ (h.lt_or_lt.resolve_right vyx), rwa [add_comm, max_comm] },
rw max_eq_left_of_lt vyx at h',
apply lt_irrefl (v x),
calc v x = v ... | lemma | valuation.map_add_of_distinct_val | ring_theory.valuation | src/ring_theory/valuation/basic.lean | [
"algebra.order.with_zero",
"ring_theory.ideal.operations"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_add_eq_of_lt_right (h : v x < v y) : v (x + y) = v y | begin
convert v.map_add_of_distinct_val _,
{ symmetry, rw max_eq_right_iff, exact le_of_lt h },
{ exact ne_of_lt h }
end | lemma | valuation.map_add_eq_of_lt_right | ring_theory.valuation | src/ring_theory/valuation/basic.lean | [
"algebra.order.with_zero",
"ring_theory.ideal.operations"
] | [
"max_eq_right_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_add_eq_of_lt_left (h : v y < v x) : v (x + y) = v x | begin
rw add_comm, exact map_add_eq_of_lt_right _ h,
end | lemma | valuation.map_add_eq_of_lt_left | ring_theory.valuation | src/ring_theory/valuation/basic.lean | [
"algebra.order.with_zero",
"ring_theory.ideal.operations"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_eq_of_sub_lt (h : v (y - x) < v x) : v y = v x | begin
have := valuation.map_add_of_distinct_val v (ne_of_gt h).symm,
rw max_eq_right (le_of_lt h) at this,
simpa using this
end | lemma | valuation.map_eq_of_sub_lt | ring_theory.valuation | src/ring_theory/valuation/basic.lean | [
"algebra.order.with_zero",
"ring_theory.ideal.operations"
] | [
"valuation.map_add_of_distinct_val"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_one_add_of_lt (h : v x < 1) : v (1 + x) = 1 | begin
rw ← v.map_one at h,
simpa only [v.map_one] using v.map_add_eq_of_lt_left h
end | lemma | valuation.map_one_add_of_lt | ring_theory.valuation | src/ring_theory/valuation/basic.lean | [
"algebra.order.with_zero",
"ring_theory.ideal.operations"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_one_sub_of_lt (h : v x < 1) : v (1 - x) = 1 | begin
rw [← v.map_one, ← v.map_neg] at h,
rw sub_eq_add_neg 1 x,
simpa only [v.map_one, v.map_neg] using v.map_add_eq_of_lt_left h
end | lemma | valuation.map_one_sub_of_lt | ring_theory.valuation | src/ring_theory/valuation/basic.lean | [
"algebra.order.with_zero",
"ring_theory.ideal.operations"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_lt_val_iff (v : valuation K Γ₀) {x : K} (h : x ≠ 0) :
1 < v x ↔ v x⁻¹ < 1 | by simpa using (inv_lt_inv₀ (v.ne_zero_iff.2 h) one_ne_zero).symm | lemma | valuation.one_lt_val_iff | ring_theory.valuation | src/ring_theory/valuation/basic.lean | [
"algebra.order.with_zero",
"ring_theory.ideal.operations"
] | [
"inv_lt_inv₀",
"one_ne_zero",
"valuation"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_add_subgroup (v : valuation R Γ₀) (γ : Γ₀ˣ) : add_subgroup R | { carrier := {x | v x < γ},
zero_mem' := by { have h := units.ne_zero γ, contrapose! h, simpa using h },
add_mem' := λ x y x_in y_in, lt_of_le_of_lt (v.map_add x y) (max_lt x_in y_in),
neg_mem' := λ x x_in, by rwa [set.mem_set_of_eq, map_neg] } | def | valuation.lt_add_subgroup | ring_theory.valuation | src/ring_theory/valuation/basic.lean | [
"algebra.order.with_zero",
"ring_theory.ideal.operations"
] | [
"add_subgroup",
"units.ne_zero",
"valuation"
] | The subgroup of elements whose valuation is less than a certain unit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
refl : v.is_equiv v | λ _ _, iff.refl _ | lemma | valuation.is_equiv.refl | ring_theory.valuation | src/ring_theory/valuation/basic.lean | [
"algebra.order.with_zero",
"ring_theory.ideal.operations"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm (h : v₁.is_equiv v₂) : v₂.is_equiv v₁ | λ _ _, iff.symm (h _ _) | lemma | valuation.is_equiv.symm | ring_theory.valuation | src/ring_theory/valuation/basic.lean | [
"algebra.order.with_zero",
"ring_theory.ideal.operations"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trans (h₁₂ : v₁.is_equiv v₂) (h₂₃ : v₂.is_equiv v₃) : v₁.is_equiv v₃ | λ _ _, iff.trans (h₁₂ _ _) (h₂₃ _ _) | lemma | valuation.is_equiv.trans | ring_theory.valuation | src/ring_theory/valuation/basic.lean | [
"algebra.order.with_zero",
"ring_theory.ideal.operations"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_eq {v' : valuation R Γ₀} (h : v = v') : v.is_equiv v' | by { subst h } | lemma | valuation.is_equiv.of_eq | ring_theory.valuation | src/ring_theory/valuation/basic.lean | [
"algebra.order.with_zero",
"ring_theory.ideal.operations"
] | [
"of_eq",
"valuation"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map {v' : valuation R Γ₀} (f : Γ₀ →*₀ Γ'₀) (hf : monotone f)
(inf : injective f) (h : v.is_equiv v') :
(v.map f hf).is_equiv (v'.map f hf) | let H : strict_mono f := hf.strict_mono_of_injective inf in
λ r s,
calc f (v r) ≤ f (v s) ↔ v r ≤ v s : by rw H.le_iff_le
... ↔ v' r ≤ v' s : h r s
... ↔ f (v' r) ≤ f (v' s) : by rw H.le_iff_le | lemma | valuation.is_equiv.map | ring_theory.valuation | src/ring_theory/valuation/basic.lean | [
"algebra.order.with_zero",
"ring_theory.ideal.operations"
] | [
"monotone",
"strict_mono",
"valuation"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap {S : Type*} [ring S] (f : S →+* R) (h : v₁.is_equiv v₂) :
(v₁.comap f).is_equiv (v₂.comap f) | λ r s, h (f r) (f s) | lemma | valuation.is_equiv.comap | ring_theory.valuation | src/ring_theory/valuation/basic.lean | [
"algebra.order.with_zero",
"ring_theory.ideal.operations"
] | [
"ring"
] | `comap` preserves equivalence. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
val_eq (h : v₁.is_equiv v₂) {r s : R} :
v₁ r = v₁ s ↔ v₂ r = v₂ s | by simpa only [le_antisymm_iff] using and_congr (h r s) (h s r) | lemma | valuation.is_equiv.val_eq | ring_theory.valuation | src/ring_theory/valuation/basic.lean | [
"algebra.order.with_zero",
"ring_theory.ideal.operations"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ne_zero (h : v₁.is_equiv v₂) {r : R} :
v₁ r ≠ 0 ↔ v₂ r ≠ 0 | begin
have : v₁ r ≠ v₁ 0 ↔ v₂ r ≠ v₂ 0 := not_iff_not_of_iff h.val_eq,
rwa [v₁.map_zero, v₂.map_zero] at this,
end | lemma | valuation.is_equiv.ne_zero | ring_theory.valuation | src/ring_theory/valuation/basic.lean | [
"algebra.order.with_zero",
"ring_theory.ideal.operations"
] | [
"ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_equiv_of_map_strict_mono [linear_ordered_comm_monoid_with_zero Γ₀]
[linear_ordered_comm_monoid_with_zero Γ'₀] [ring R] {v : valuation R Γ₀} (f : Γ₀ →*₀ Γ'₀)
(H : strict_mono f) :
is_equiv (v.map f (H.monotone)) v | λ x y, ⟨H.le_iff_le.mp, λ h, H.monotone h⟩ | lemma | valuation.is_equiv_of_map_strict_mono | ring_theory.valuation | src/ring_theory/valuation/basic.lean | [
"algebra.order.with_zero",
"ring_theory.ideal.operations"
] | [
"linear_ordered_comm_monoid_with_zero",
"ring",
"strict_mono",
"valuation"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_equiv_of_val_le_one [linear_ordered_comm_group_with_zero Γ₀]
[linear_ordered_comm_group_with_zero Γ'₀]
(v : valuation K Γ₀) (v' : valuation K Γ'₀) (h : ∀ {x:K}, v x ≤ 1 ↔ v' x ≤ 1) :
v.is_equiv v' | begin
intros x y,
by_cases hy : y = 0, { simp [hy, zero_iff], },
rw show y = 1 * y, by rw one_mul,
rw [← (inv_mul_cancel_right₀ hy x)],
iterate 2 {rw [v.map_mul _ y, v'.map_mul _ y]},
rw [v.map_one, v'.map_one],
split; intro H,
{ apply mul_le_mul_right',
replace hy := v.ne_zero_iff.mpr hy,
repla... | lemma | valuation.is_equiv_of_val_le_one | ring_theory.valuation | src/ring_theory/valuation/basic.lean | [
"algebra.order.with_zero",
"ring_theory.ideal.operations"
] | [
"inv_mul_cancel_right₀",
"le_of_le_mul_right",
"linear_ordered_comm_group_with_zero",
"mul_le_mul_right'",
"one_mul",
"valuation"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_equiv_iff_val_le_one
[linear_ordered_comm_group_with_zero Γ₀]
[linear_ordered_comm_group_with_zero Γ'₀]
(v : valuation K Γ₀) (v' : valuation K Γ'₀) :
v.is_equiv v' ↔ ∀ {x : K}, v x ≤ 1 ↔ v' x ≤ 1 | ⟨λ h x, by simpa using h x 1, is_equiv_of_val_le_one _ _⟩ | lemma | valuation.is_equiv_iff_val_le_one | ring_theory.valuation | src/ring_theory/valuation/basic.lean | [
"algebra.order.with_zero",
"ring_theory.ideal.operations"
] | [
"linear_ordered_comm_group_with_zero",
"valuation"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_equiv_iff_val_eq_one
[linear_ordered_comm_group_with_zero Γ₀]
[linear_ordered_comm_group_with_zero Γ'₀]
(v : valuation K Γ₀) (v' : valuation K Γ'₀) :
v.is_equiv v' ↔ ∀ {x : K}, v x = 1 ↔ v' x = 1 | begin
split,
{ intros h x,
simpa using @is_equiv.val_eq _ _ _ _ _ _ v v' h x 1 },
{ intros h, apply is_equiv_of_val_le_one, intros x,
split,
{ intros hx,
cases lt_or_eq_of_le hx with hx' hx',
{ have : v (1 + x) = 1,
{ rw ← v.map_one, apply map_add_eq_of_lt_left, simpa },
rw... | lemma | valuation.is_equiv_iff_val_eq_one | ring_theory.valuation | src/ring_theory/valuation/basic.lean | [
"algebra.order.with_zero",
"ring_theory.ideal.operations"
] | [
"linear_ordered_comm_group_with_zero",
"valuation"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_equiv_iff_val_lt_one
[linear_ordered_comm_group_with_zero Γ₀]
[linear_ordered_comm_group_with_zero Γ'₀]
(v : valuation K Γ₀) (v' : valuation K Γ'₀) :
v.is_equiv v' ↔ ∀ {x : K}, v x < 1 ↔ v' x < 1 | begin
split,
{ intros h x,
simp only [lt_iff_le_and_ne, and_congr ((is_equiv_iff_val_le_one _ _).1 h)
((is_equiv_iff_val_eq_one _ _).1 h).not] },
{ rw is_equiv_iff_val_eq_one,
intros h x,
by_cases hx : x = 0, { simp only [(zero_iff _).2 hx, zero_ne_one] },
split,
{ intro hh,
by_con... | lemma | valuation.is_equiv_iff_val_lt_one | ring_theory.valuation | src/ring_theory/valuation/basic.lean | [
"algebra.order.with_zero",
"ring_theory.ideal.operations"
] | [
"by_contra",
"inv_eq_iff_eq_inv",
"inv_one",
"linear_ordered_comm_group_with_zero",
"lt_iff_le_and_ne",
"lt_self_iff_false",
"map_inv₀",
"valuation",
"zero_ne_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_equiv_iff_val_sub_one_lt_one
[linear_ordered_comm_group_with_zero Γ₀]
[linear_ordered_comm_group_with_zero Γ'₀]
(v : valuation K Γ₀) (v' : valuation K Γ'₀) :
v.is_equiv v' ↔ ∀ {x : K}, v (x - 1) < 1 ↔ v' (x - 1) < 1 | begin
rw is_equiv_iff_val_lt_one,
exact (equiv.sub_right 1).surjective.forall
end | lemma | valuation.is_equiv_iff_val_sub_one_lt_one | ring_theory.valuation | src/ring_theory/valuation/basic.lean | [
"algebra.order.with_zero",
"ring_theory.ideal.operations"
] | [
"linear_ordered_comm_group_with_zero",
"valuation"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_equiv_tfae
[linear_ordered_comm_group_with_zero Γ₀]
[linear_ordered_comm_group_with_zero Γ'₀]
(v : valuation K Γ₀) (v' : valuation K Γ'₀) :
[v.is_equiv v',
∀ {x}, v x ≤ 1 ↔ v' x ≤ 1,
∀ {x}, v x = 1 ↔ v' x = 1,
∀ {x}, v x < 1 ↔ v' x < 1,
∀ {x}, v (x-1) < 1 ↔ v' (x-1) < 1].tfae | begin
tfae_have : 1 ↔ 2, { apply is_equiv_iff_val_le_one },
tfae_have : 1 ↔ 3, { apply is_equiv_iff_val_eq_one },
tfae_have : 1 ↔ 4, { apply is_equiv_iff_val_lt_one },
tfae_have : 1 ↔ 5, { apply is_equiv_iff_val_sub_one_lt_one },
tfae_finish
end | lemma | valuation.is_equiv_tfae | ring_theory.valuation | src/ring_theory/valuation/basic.lean | [
"algebra.order.with_zero",
"ring_theory.ideal.operations"
] | [
"linear_ordered_comm_group_with_zero",
"valuation"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
supp : ideal R | { carrier := {x | v x = 0},
zero_mem' := map_zero v,
add_mem' := λ x y hx hy, le_zero_iff.mp $
calc v (x + y) ≤ max (v x) (v y) : v.map_add x y
... ≤ 0 : max_le (le_zero_iff.mpr hx) (le_zero_iff.mpr hy),
smul_mem' := λ c x hx, calc v (c * x)
= v c * v x ... | def | valuation.supp | ring_theory.valuation | src/ring_theory/valuation/basic.lean | [
"algebra.order.with_zero",
"ring_theory.ideal.operations"
] | [
"ideal",
"map_mul",
"mul_zero"
] | The support of a valuation `v : R → Γ₀` is the ideal of `R` where `v` vanishes. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_supp_iff (x : R) : x ∈ supp v ↔ v x = 0 | iff.rfl | lemma | valuation.mem_supp_iff | ring_theory.valuation | src/ring_theory/valuation/basic.lean | [
"algebra.order.with_zero",
"ring_theory.ideal.operations"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_add_supp (a : R) {s : R} (h : s ∈ supp v) : v (a + s) = v a | begin
have aux : ∀ a s, v s = 0 → v (a + s) ≤ v a,
{ intros a' s' h', refine le_trans (v.map_add a' s') (max_le le_rfl _), simp [h'], },
apply le_antisymm (aux a s h),
calc v a = v (a + s + -s) : by simp
... ≤ v (a + s) : aux (a + s) (-s) (by rwa ←ideal.neg_mem_iff at h)
end | lemma | valuation.map_add_supp | ring_theory.valuation | src/ring_theory/valuation/basic.lean | [
"algebra.order.with_zero",
"ring_theory.ideal.operations"
] | [
"aux",
"le_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_supp {S : Type*} [comm_ring S] (f : S →+* R) :
supp (v.comap f) = ideal.comap f v.supp | ideal.ext $ λ x,
begin
rw [mem_supp_iff, ideal.mem_comap, mem_supp_iff],
refl,
end | lemma | valuation.comap_supp | ring_theory.valuation | src/ring_theory/valuation/basic.lean | [
"algebra.order.with_zero",
"ring_theory.ideal.operations"
] | [
"comm_ring",
"ideal.comap",
"ideal.ext",
"ideal.mem_comap"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_valuation | valuation R (multiplicative Γ₀ᵒᵈ) | def | add_valuation | ring_theory.valuation | src/ring_theory/valuation/basic.lean | [
"algebra.order.with_zero",
"ring_theory.ideal.operations"
] | [
"multiplicative",
"valuation"
] | The type of `Γ₀`-valued additive valuations on `R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of : add_valuation R Γ₀ | { to_fun := f,
map_one' := h1,
map_zero' := h0,
map_add_le_max' := hadd,
map_mul' := hmul } | def | add_valuation.of | ring_theory.valuation | src/ring_theory/valuation/basic.lean | [
"algebra.order.with_zero",
"ring_theory.ideal.operations"
] | [
"add_valuation"
] | An alternate constructor of `add_valuation`, that doesn't reference `multiplicative Γ₀ᵒᵈ` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_apply : (of f h0 h1 hadd hmul) r = f r | rfl | theorem | add_valuation.of_apply | ring_theory.valuation | src/ring_theory/valuation/basic.lean | [
"algebra.order.with_zero",
"ring_theory.ideal.operations"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
valuation : valuation R (multiplicative Γ₀ᵒᵈ) | v | def | add_valuation.valuation | ring_theory.valuation | src/ring_theory/valuation/basic.lean | [
"algebra.order.with_zero",
"ring_theory.ideal.operations"
] | [
"multiplicative",
"valuation"
] | The `valuation` associated to an `add_valuation` (useful if the latter is constructed using
`add_valuation.of`). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
valuation_apply (r : R) :
v.valuation r = multiplicative.of_add (order_dual.to_dual (v r)) | rfl | lemma | add_valuation.valuation_apply | ring_theory.valuation | src/ring_theory/valuation/basic.lean | [
"algebra.order.with_zero",
"ring_theory.ideal.operations"
] | [
"multiplicative.of_add",
"order_dual.to_dual"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_zero : v 0 = ⊤ | v.map_zero | lemma | add_valuation.map_zero | ring_theory.valuation | src/ring_theory/valuation/basic.lean | [
"algebra.order.with_zero",
"ring_theory.ideal.operations"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_one : v 1 = 0 | v.map_one | lemma | add_valuation.map_one | ring_theory.valuation | src/ring_theory/valuation/basic.lean | [
"algebra.order.with_zero",
"ring_theory.ideal.operations"
] | [
"map_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_mul : ∀ x y, v (x * y) = v x + v y | v.map_mul | lemma | add_valuation.map_mul | ring_theory.valuation | src/ring_theory/valuation/basic.lean | [
"algebra.order.with_zero",
"ring_theory.ideal.operations"
] | [
"map_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_add : ∀ x y, min (v x) (v y) ≤ v (x + y) | v.map_add | lemma | add_valuation.map_add | ring_theory.valuation | src/ring_theory/valuation/basic.lean | [
"algebra.order.with_zero",
"ring_theory.ideal.operations"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_le_add {x y g} (hx : g ≤ v x) (hy : g ≤ v y) : g ≤ v (x + y) | v.map_add_le hx hy | lemma | add_valuation.map_le_add | ring_theory.valuation | src/ring_theory/valuation/basic.lean | [
"algebra.order.with_zero",
"ring_theory.ideal.operations"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_lt_add {x y g} (hx : g < v x) (hy : g < v y) : g < v (x + y) | v.map_add_lt hx hy | lemma | add_valuation.map_lt_add | ring_theory.valuation | src/ring_theory/valuation/basic.lean | [
"algebra.order.with_zero",
"ring_theory.ideal.operations"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_le_sum {ι : Type*} {s : finset ι} {f : ι → R} {g : Γ₀} (hf : ∀ i ∈ s, g ≤ v (f i)) :
g ≤ v (∑ i in s, f i) | v.map_sum_le hf | lemma | add_valuation.map_le_sum | ring_theory.valuation | src/ring_theory/valuation/basic.lean | [
"algebra.order.with_zero",
"ring_theory.ideal.operations"
] | [
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_lt_sum {ι : Type*} {s : finset ι} {f : ι → R} {g : Γ₀} (hg : g ≠ ⊤)
(hf : ∀ i ∈ s, g < v (f i)) : g < v (∑ i in s, f i) | v.map_sum_lt hg hf | lemma | add_valuation.map_lt_sum | ring_theory.valuation | src/ring_theory/valuation/basic.lean | [
"algebra.order.with_zero",
"ring_theory.ideal.operations"
] | [
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_lt_sum' {ι : Type*} {s : finset ι} {f : ι → R} {g : Γ₀} (hg : g < ⊤)
(hf : ∀ i ∈ s, g < v (f i)) : g < v (∑ i in s, f i) | v.map_sum_lt' hg hf | lemma | add_valuation.map_lt_sum' | ring_theory.valuation | src/ring_theory/valuation/basic.lean | [
"algebra.order.with_zero",
"ring_theory.ideal.operations"
] | [
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_pow : ∀ x (n:ℕ), v (x^n) = n • (v x) | v.map_pow | lemma | add_valuation.map_pow | ring_theory.valuation | src/ring_theory/valuation/basic.lean | [
"algebra.order.with_zero",
"ring_theory.ideal.operations"
] | [
"map_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext {v₁ v₂ : add_valuation R Γ₀} (h : ∀ r, v₁ r = v₂ r) : v₁ = v₂ | valuation.ext h | lemma | add_valuation.ext | ring_theory.valuation | src/ring_theory/valuation/basic.lean | [
"algebra.order.with_zero",
"ring_theory.ideal.operations"
] | [
"add_valuation",
"valuation.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext_iff {v₁ v₂ : add_valuation R Γ₀} : v₁ = v₂ ↔ ∀ r, v₁ r = v₂ r | valuation.ext_iff | lemma | add_valuation.ext_iff | ring_theory.valuation | src/ring_theory/valuation/basic.lean | [
"algebra.order.with_zero",
"ring_theory.ideal.operations"
] | [
"add_valuation",
"valuation.ext_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
top_iff [nontrivial Γ₀] (v : add_valuation K Γ₀) {x : K} :
v x = ⊤ ↔ x = 0 | v.zero_iff | lemma | add_valuation.top_iff | ring_theory.valuation | src/ring_theory/valuation/basic.lean | [
"algebra.order.with_zero",
"ring_theory.ideal.operations"
] | [
"add_valuation",
"nontrivial"
] | If `v` is an additive valuation on a division ring then `v(x) = ⊤` iff `x = 0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ne_top_iff [nontrivial Γ₀] (v : add_valuation K Γ₀) {x : K} :
v x ≠ ⊤ ↔ x ≠ 0 | v.ne_zero_iff | lemma | add_valuation.ne_top_iff | ring_theory.valuation | src/ring_theory/valuation/basic.lean | [
"algebra.order.with_zero",
"ring_theory.ideal.operations"
] | [
"add_valuation",
"nontrivial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap {S : Type*} [ring S] (f : S →+* R) (v : add_valuation R Γ₀) :
add_valuation S Γ₀ | v.comap f | def | add_valuation.comap | ring_theory.valuation | src/ring_theory/valuation/basic.lean | [
"algebra.order.with_zero",
"ring_theory.ideal.operations"
] | [
"add_valuation",
"ring"
] | A ring homomorphism `S → R` induces a map `add_valuation R Γ₀ → add_valuation S Γ₀`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comap_id : v.comap (ring_hom.id R) = v | v.comap_id | lemma | add_valuation.comap_id | ring_theory.valuation | src/ring_theory/valuation/basic.lean | [
"algebra.order.with_zero",
"ring_theory.ideal.operations"
] | [
"ring_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_comp {S₁ : Type*} {S₂ : Type*} [ring S₁] [ring S₂] (f : S₁ →+* S₂) (g : S₂ →+* R) :
v.comap (g.comp f) = (v.comap g).comap f | v.comap_comp f g | lemma | add_valuation.comap_comp | ring_theory.valuation | src/ring_theory/valuation/basic.lean | [
"algebra.order.with_zero",
"ring_theory.ideal.operations"
] | [
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map (f : Γ₀ →+ Γ'₀) (ht : f ⊤ = ⊤) (hf : monotone f) (v : add_valuation R Γ₀) :
add_valuation R Γ'₀ | v.map
{ to_fun := f,
map_mul' := f.map_add,
map_one' := f.map_zero,
map_zero' := ht } (λ x y h, hf h) | def | add_valuation.map | ring_theory.valuation | src/ring_theory/valuation/basic.lean | [
"algebra.order.with_zero",
"ring_theory.ideal.operations"
] | [
"add_valuation",
"monotone"
] | A `≤`-preserving, `⊤`-preserving group homomorphism `Γ₀ → Γ'₀` induces a map
`add_valuation R Γ₀ → add_valuation R Γ'₀`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_equiv (v₁ : add_valuation R Γ₀) (v₂ : add_valuation R Γ'₀) : Prop | v₁.is_equiv v₂ | def | add_valuation.is_equiv | ring_theory.valuation | src/ring_theory/valuation/basic.lean | [
"algebra.order.with_zero",
"ring_theory.ideal.operations"
] | [
"add_valuation"
] | Two additive valuations on `R` are defined to be equivalent if they induce the same
preorder on `R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_inv (v : add_valuation K Γ₀) {x : K} :
v x⁻¹ = - (v x) | map_inv₀ v.valuation x | lemma | add_valuation.map_inv | ring_theory.valuation | src/ring_theory/valuation/basic.lean | [
"algebra.order.with_zero",
"ring_theory.ideal.operations"
] | [
"add_valuation",
"map_inv",
"map_inv₀"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_neg (x : R) : v (-x) = v x | v.map_neg x | lemma | add_valuation.map_neg | ring_theory.valuation | src/ring_theory/valuation/basic.lean | [
"algebra.order.with_zero",
"ring_theory.ideal.operations"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_sub_swap (x y : R) : v (x - y) = v (y - x) | v.map_sub_swap x y | lemma | add_valuation.map_sub_swap | ring_theory.valuation | src/ring_theory/valuation/basic.lean | [
"algebra.order.with_zero",
"ring_theory.ideal.operations"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_sub (x y : R) : min (v x) (v y) ≤ v (x - y) | v.map_sub x y | lemma | add_valuation.map_sub | ring_theory.valuation | src/ring_theory/valuation/basic.lean | [
"algebra.order.with_zero",
"ring_theory.ideal.operations"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_le_sub {x y g} (hx : g ≤ v x) (hy : g ≤ v y) : g ≤ v (x - y) | v.map_sub_le hx hy | lemma | add_valuation.map_le_sub | ring_theory.valuation | src/ring_theory/valuation/basic.lean | [
"algebra.order.with_zero",
"ring_theory.ideal.operations"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_add_of_distinct_val (h : v x ≠ v y) : v (x + y) = min (v x) (v y) | v.map_add_of_distinct_val h | lemma | add_valuation.map_add_of_distinct_val | ring_theory.valuation | src/ring_theory/valuation/basic.lean | [
"algebra.order.with_zero",
"ring_theory.ideal.operations"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_eq_of_lt_sub (h : v x < v (y - x)) : v y = v x | v.map_eq_of_sub_lt h | lemma | add_valuation.map_eq_of_lt_sub | ring_theory.valuation | src/ring_theory/valuation/basic.lean | [
"algebra.order.with_zero",
"ring_theory.ideal.operations"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
refl : v.is_equiv v | valuation.is_equiv.refl | lemma | add_valuation.is_equiv.refl | ring_theory.valuation | src/ring_theory/valuation/basic.lean | [
"algebra.order.with_zero",
"ring_theory.ideal.operations"
] | [
"valuation.is_equiv.refl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm (h : v₁.is_equiv v₂) : v₂.is_equiv v₁ | h.symm | lemma | add_valuation.is_equiv.symm | ring_theory.valuation | src/ring_theory/valuation/basic.lean | [
"algebra.order.with_zero",
"ring_theory.ideal.operations"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trans (h₁₂ : v₁.is_equiv v₂) (h₂₃ : v₂.is_equiv v₃) : v₁.is_equiv v₃ | h₁₂.trans h₂₃ | lemma | add_valuation.is_equiv.trans | ring_theory.valuation | src/ring_theory/valuation/basic.lean | [
"algebra.order.with_zero",
"ring_theory.ideal.operations"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_eq {v' : add_valuation R Γ₀} (h : v = v') : v.is_equiv v' | valuation.is_equiv.of_eq h | lemma | add_valuation.is_equiv.of_eq | ring_theory.valuation | src/ring_theory/valuation/basic.lean | [
"algebra.order.with_zero",
"ring_theory.ideal.operations"
] | [
"add_valuation",
"of_eq",
"valuation.is_equiv.of_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map {v' : add_valuation R Γ₀} (f : Γ₀ →+ Γ'₀) (ht : f ⊤ = ⊤) (hf : monotone f)
(inf : injective f) (h : v.is_equiv v') :
(v.map f ht hf).is_equiv (v'.map f ht hf) | h.map
{ to_fun := f,
map_mul' := f.map_add,
map_one' := f.map_zero,
map_zero' := ht } (λ x y h, hf h) inf | lemma | add_valuation.is_equiv.map | ring_theory.valuation | src/ring_theory/valuation/basic.lean | [
"algebra.order.with_zero",
"ring_theory.ideal.operations"
] | [
"add_valuation",
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap {S : Type*} [ring S] (f : S →+* R) (h : v₁.is_equiv v₂) :
(v₁.comap f).is_equiv (v₂.comap f) | h.comap f | lemma | add_valuation.is_equiv.comap | ring_theory.valuation | src/ring_theory/valuation/basic.lean | [
"algebra.order.with_zero",
"ring_theory.ideal.operations"
] | [
"ring"
] | `comap` preserves equivalence. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
val_eq (h : v₁.is_equiv v₂) {r s : R} :
v₁ r = v₁ s ↔ v₂ r = v₂ s | h.val_eq | lemma | add_valuation.is_equiv.val_eq | ring_theory.valuation | src/ring_theory/valuation/basic.lean | [
"algebra.order.with_zero",
"ring_theory.ideal.operations"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ne_top (h : v₁.is_equiv v₂) {r : R} :
v₁ r ≠ ⊤ ↔ v₂ r ≠ ⊤ | h.ne_zero | lemma | add_valuation.is_equiv.ne_top | ring_theory.valuation | src/ring_theory/valuation/basic.lean | [
"algebra.order.with_zero",
"ring_theory.ideal.operations"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
supp : ideal R | v.supp | def | add_valuation.supp | ring_theory.valuation | src/ring_theory/valuation/basic.lean | [
"algebra.order.with_zero",
"ring_theory.ideal.operations"
] | [
"ideal"
] | The support of an additive valuation `v : R → Γ₀` is the ideal of `R` where `v x = ⊤` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_supp_iff (x : R) : x ∈ supp v ↔ v x = ⊤ | v.mem_supp_iff x | lemma | add_valuation.mem_supp_iff | ring_theory.valuation | src/ring_theory/valuation/basic.lean | [
"algebra.order.with_zero",
"ring_theory.ideal.operations"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_add_supp (a : R) {s : R} (h : s ∈ supp v) : v (a + s) = v a | v.map_add_supp a h | lemma | add_valuation.map_add_supp | ring_theory.valuation | src/ring_theory/valuation/basic.lean | [
"algebra.order.with_zero",
"ring_theory.ideal.operations"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
valuation.extend_to_localization : valuation B Γ | let f := is_localization.to_localization_map S B,
h : ∀ s : S, is_unit (v.1.to_monoid_hom s) := λ s, is_unit_iff_ne_zero.2 (hS s.2) in
{ map_zero' := by convert f.lift_eq _ 0; simp,
map_add_le_max' := λ x y, begin
obtain ⟨a,b,s,rfl,rfl⟩ : ∃ (a b : A) (s : S), f.mk' a s = x ∧ f.mk' b s = y,
{ obtain ⟨a,s,r... | def | valuation.extend_to_localization | ring_theory.valuation | src/ring_theory/valuation/extend_to_localization.lean | [
"ring_theory.localization.at_prime",
"ring_theory.valuation.basic"
] | [
"is_localization.mk'_spec",
"is_localization.to_localization_map",
"is_unit",
"max_mul_mul_right",
"mul_le_mul_right'",
"submonoid.coe_mul",
"valuation"
] | We can extend a valuation `v` on a ring to a localization at a submonoid of
the complement of `v.supp`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
valuation.extend_to_localization_apply_map_apply (a : A) :
v.extend_to_localization hS B (algebra_map A B a) = v a | submonoid.localization_map.lift_eq _ _ a | lemma | valuation.extend_to_localization_apply_map_apply | ring_theory.valuation | src/ring_theory/valuation/extend_to_localization.lean | [
"ring_theory.localization.at_prime",
"ring_theory.valuation.basic"
] | [
"algebra_map",
"submonoid.localization_map.lift_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integer : subring R | { carrier := { x | v x ≤ 1 },
one_mem' := le_of_eq v.map_one,
mul_mem' := λ x y hx hy, trans_rel_right (≤) (v.map_mul x y) (mul_le_one' hx hy),
zero_mem' := trans_rel_right (≤) v.map_zero zero_le_one,
add_mem' := λ x y hx hy, le_trans (v.map_add x y) (max_le hx hy),
neg_mem' := λ x hx, trans_rel_right (≤) (v.... | def | valuation.integer | ring_theory.valuation | src/ring_theory/valuation/integers.lean | [
"ring_theory.valuation.basic"
] | [
"subring",
"zero_le_one"
] | The ring of integers under a given valuation is the subring of elements with valuation ≤ 1. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
integers : Prop | (hom_inj : function.injective (algebra_map O R))
(map_le_one : ∀ x, v (algebra_map O R x) ≤ 1)
(exists_of_le_one : ∀ ⦃r⦄, v r ≤ 1 → ∃ x, algebra_map O R x = r) | structure | valuation.integers | ring_theory.valuation | src/ring_theory/valuation/integers.lean | [
"ring_theory.valuation.basic"
] | [
"algebra_map"
] | Given a valuation v : R → Γ₀ and a ring homomorphism O →+* R, we say that O is the integers of v
if f is injective, and its range is exactly `v.integer`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
integer.integers : v.integers v.integer | { hom_inj := subtype.coe_injective,
map_le_one := λ r, r.2,
exists_of_le_one := λ r hr, ⟨⟨r, hr⟩, rfl⟩ } | theorem | valuation.integer.integers | ring_theory.valuation | src/ring_theory/valuation/integers.lean | [
"ring_theory.valuation.basic"
] | [
"subtype.coe_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_of_is_unit {x : O} (hx : is_unit x) : v (algebra_map O R x) = 1 | let ⟨u, hu⟩ := hx in le_antisymm (hv.2 _) $
by { rw [← v.map_one, ← (algebra_map O R).map_one, ← u.mul_inv, ← mul_one (v (algebra_map O R x)),
hu, (algebra_map O R).map_mul, v.map_mul], exact mul_le_mul_left' (hv.2 (u⁻¹ : units O)) _ } | lemma | valuation.integers.one_of_is_unit | ring_theory.valuation | src/ring_theory/valuation/integers.lean | [
"ring_theory.valuation.basic"
] | [
"algebra_map",
"is_unit",
"map_mul",
"map_one",
"mul_le_mul_left'",
"mul_one",
"units"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_unit_of_one {x : O} (hx : is_unit (algebra_map O R x)) (hvx : v (algebra_map O R x) = 1) :
is_unit x | let ⟨u, hu⟩ := hx in
have h1 : v u ≤ 1, from hu.symm ▸ hv.2 x,
have h2 : v (u⁻¹ : Rˣ) ≤ 1,
by rw [← one_mul (v _), ← hvx, ← v.map_mul, ← hu, u.mul_inv, hu, hvx, v.map_one],
let ⟨r1, hr1⟩ := hv.3 h1, ⟨r2, hr2⟩ := hv.3 h2 in
⟨⟨r1, r2, hv.1 $ by rw [ring_hom.map_mul, ring_hom.map_one, hr1, hr2, units.mul_inv],
hv.1 $ ... | lemma | valuation.integers.is_unit_of_one | ring_theory.valuation | src/ring_theory/valuation/integers.lean | [
"ring_theory.valuation.basic"
] | [
"algebra_map",
"is_unit",
"one_mul",
"ring_hom.map_mul",
"ring_hom.map_one",
"units.inv_mul",
"units.mul_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_of_dvd {x y : O} (h : x ∣ y) : v (algebra_map O R y) ≤ v (algebra_map O R x) | let ⟨z, hz⟩ := h in by { rw [← mul_one (v (algebra_map O R x)), hz, ring_hom.map_mul, v.map_mul],
exact mul_le_mul_left' (hv.2 z) _ } | lemma | valuation.integers.le_of_dvd | ring_theory.valuation | src/ring_theory/valuation/integers.lean | [
"ring_theory.valuation.basic"
] | [
"algebra_map",
"mul_le_mul_left'",
"mul_one",
"ring_hom.map_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dvd_of_le {x y : O} (h : v (algebra_map O F x) ≤ v (algebra_map O F y)) : y ∣ x | classical.by_cases (λ hy : algebra_map O F y = 0, have hx : x = 0,
from hv.1 $ (algebra_map O F).map_zero.symm ▸
(v.zero_iff.1 $ le_zero_iff.1 (v.map_zero ▸ hy ▸ h)),
hx.symm ▸ dvd_zero y) $
λ hy : algebra_map O F y ≠ 0,
have v ((algebra_map O F y)⁻¹ * algebra_map O F x) ≤ 1,
by { rw [← v.map_one, ← inv_m... | lemma | valuation.integers.dvd_of_le | ring_theory.valuation | src/ring_theory/valuation/integers.lean | [
"ring_theory.valuation.basic"
] | [
"algebra_map",
"dvd_zero",
"inv_mul_cancel",
"map_mul",
"mul_inv_cancel_left₀",
"mul_le_mul_left'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dvd_iff_le {x y : O} : x ∣ y ↔ v (algebra_map O F y) ≤ v (algebra_map O F x) | ⟨hv.le_of_dvd, hv.dvd_of_le⟩ | lemma | valuation.integers.dvd_iff_le | ring_theory.valuation | src/ring_theory/valuation/integers.lean | [
"ring_theory.valuation.basic"
] | [
"algebra_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_iff_dvd {x y : O} : v (algebra_map O F x) ≤ v (algebra_map O F y) ↔ y ∣ x | ⟨hv.dvd_of_le, hv.le_of_dvd⟩ | lemma | valuation.integers.le_iff_dvd | ring_theory.valuation | src/ring_theory/valuation/integers.lean | [
"ring_theory.valuation.basic"
] | [
"algebra_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_of_integral {x : R} (hx : is_integral O x) : x ∈ v.integer | let ⟨p, hpm, hpx⟩ := hx in le_of_not_lt $ λ (hvx : 1 < v x), begin
rw [hpm.as_sum, eval₂_add, eval₂_pow, eval₂_X, eval₂_finset_sum, add_eq_zero_iff_eq_neg] at hpx,
replace hpx := congr_arg v hpx, refine ne_of_gt _ hpx,
rw [v.map_neg, v.map_pow],
refine v.map_sum_lt' (zero_lt_one.trans_le (one_le_pow_of_one_le' ... | lemma | valuation.integers.mem_of_integral | ring_theory.valuation | src/ring_theory/valuation/integral.lean | [
"ring_theory.integrally_closed",
"ring_theory.valuation.integers"
] | [
"is_integral",
"mul_lt_mul₀",
"one_le_pow_of_one_le'",
"one_mul",
"pow_lt_pow₀"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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