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| import valuation.field | |
| import valuation.canonical | |
| /-! | |
| # The valuation on the completion of the valuation field | |
| In this file we extend the canonical valuation on the the valuation field of (R, v) to its | |
| completion (for the topological field structure coming from the valuation). | |
| -/ | |
| noncomputable theory | |
| open valuation uniform_space | |
| local attribute [instance] valued.subgroups_basis valued.uniform_space | |
| variables {Γ₀ : Type*} [linear_ordered_comm_group_with_zero Γ₀] | |
| variables {R : Type*} [comm_ring R] (v : valuation R Γ₀) | |
| local notation `hat` K := completion K | |
| namespace valuation_field | |
| /--The valuation field of a valuation is naturally a valued field.-/ | |
| instance : valued (valuation_field v) := | |
| ⟨value_monoid v, valuation_field.canonical_valuation v⟩ | |
| /--The topology on the valuation field of a valuation.-/ | |
| instance : topological_space (valuation_field v) := | |
| subgroups_basis.topology _ | |
| /--The valuation field of a valuation is a topological ring.-/ | |
| instance : topological_ring (valuation_field v) := | |
| ring_filter_basis.is_topological_ring _ rfl | |
| /--The valuation field of a valuation is a uniform additive group.-/ | |
| instance : uniform_add_group (valuation_field v) := | |
| topological_add_group_is_uniform | |
| -- This instance shouldn't be necessary. Why doesn't the general machinery step in? | |
| /--The completion of the valuation field of a valuation is a commutative ring.-/ | |
| instance : comm_ring (hat v.valuation_field) := | |
| completion.comm_ring _ | |
| -- The following instance is even more suspicious. | |
| /--The completion of the valuation field of a valuation is a ring.-/ | |
| instance : ring (hat v.valuation_field) := | |
| comm_ring.to_ring _ | |
| end valuation_field | |
| /--The valuation on the completion of the valuation field of a valuation.-/ | |
| def valuation_on_completion (v : valuation R Γ₀) : | |
| valuation (hat v.valuation_field) (value_monoid v) := | |
| valued.extension_valuation | |
| /--The completion of the valuation field of a valuation is a valued field.-/ | |
| instance valuation_field.completion_valued : valued (completion $ valuation_field v) := | |
| ⟨value_monoid v, valuation_on_completion v⟩ | |