Properties of OrderedCommRings#1303
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…onical embedding from the integers
…solving the equality ; use `_<+[_]`, `_≤·[_,_]`, etc. in equational reasoning ; shorten comment about `·CancelL<`, `·CancelR<` ; add `/2+/2≡mean` and `mean₊`
…s in `Field.Instances.Rationals`
…mmRing` ; add positive rationals and their properties
…into ocr-update
…keIsOrderedCommRingEquivFromMono`, add a new `makeIsOrderedCommRingEquiv` that is actually a specialized `make` (i.e. taking as inputs the individual proofs needed to build the OCR equivalence)
* typo * remove unnecessary imports, make some proofs ~~pointless~~ pointfree * make more proofs pointfree, rename `makeIsOrderedCommRingEquiv` → `makeIsOrderedCommRingEquivFromMono`, add a new `makeIsOrderedCommRingEquiv` that is actually a specialized `make` (i.e. taking as inputs the individual proofs needed to build the OCR equivalence)
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This PR introduces properties of Ordered Commutative Rings, including, but not limited to:
OrderedCommRingis a 1-GroupoidOrderedCommRingReasoningmodule, derived fromQuosetReasoning, with additional helpers for adding/multiplying the same element on both sides of inequalitiesTo prove the last point, this PR also includes the proof of the existence and uniqueness of a ring homomorphism from the fast integers into any other commutative ring.
The notion of OCR homomorphism is derived from homomorphisms of Ordered Heyting Fields, following Definition 4.3.1 of A.Booij PhD Thesis. In addition to requiring to reflect
<, we also ask to preserve≤.Although the latter is logically redundant, in specific cases it could be useful to provide that proof directly, rather than relying on the one derived from
reflect<. If not needed, this can be avoided using the helpermakeIsOrderedCommRingHomIf this addition is welcome, this PR would need to be merged before #1293, which will be updated to depend on these changes.