[Merged by Bors] - feat: port algebra.group.pi #1088
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* Data.Int.Cast.Lemmas * Data.Nat.Cast.Basic
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* change `Pi.Single` -> `Pi.single` * add `to_additive` #aligns
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Dec 18, 2022
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mathlib3port tracking sha: `b3f25363ae62cb169e72cd6b8b1ac97bacf21ca7` ### Porting notes * Moved `AddMonoidWithOne` and `AddGroupWithOne` here from their original files. * Corrected the name of `Pi.Single` to `Pi.single` in `Data.Pi.Algebra`, on which this depends. (Note: this change was only done in the final two commits, which can be split off into a subsequent PR for a good history if need be.) * Replaced explicit data fields by sourcing previously-defined instances where possible, as per [this zulip discussion](https://leanprover.zulipchat.com/#narrow/stream/287929-mathlib4/topic/not.20porting.20pi_instance) * Changed a name: `update_eq_div_mulSingle` => `update_eq_div_mul_mulSingle`, as it involves a multiplication of two `mulSingle`s. (Note: the additive version in mathlib is `update_eq_sub_add_single`, and `to_additive` "agrees" that `update_eq_div_mul_mulSingle` is the right name in mathlib4.) ### Review questions * Are we sure about using instances as sources where possible? Just want to double-check that it won't cause any problems. * If so, should the "highest-up" or "lowest-down" instances be used for sourcing when diamonds occur? E.g. `Foo` extends `Bar0`, `Bar1`, and both have relevant instances `bar0`, `bar1`. However, `Bar1` extends `Baz`, and we also have a relevant instance `baz` which suffices for everything not in `Bar0`. Should the instance for `Foo` start `{ bar0, bar1 with ... }` or `{ bar0, baz with ... }`? Does it matter? * Should the data `mul := (· * ·)`, `zero := (0 : ∀ i, f i)` remain explicit or be obtained by sourcing `Pi.instMul`, `Pi.instZero` respectively?
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mathlib3port tracking sha:
b3f25363ae62cb169e72cd6b8b1ac97bacf21ca7Porting notes
AddMonoidWithOneandAddGroupWithOnehere from their original files.Pi.SingletoPi.singleinData.Pi.Algebra, on which this depends. (Note: this change was only done in the final two commits, which can be split off into a subsequent PR for a good history if need be.)update_eq_div_mulSingle=>update_eq_div_mul_mulSingle, as it involves a multiplication of twomulSingles. (Note: the additive version in mathlib isupdate_eq_sub_add_single, andto_additive"agrees" thatupdate_eq_div_mul_mulSingleis the right name in mathlib4.)Review questions
FooextendsBar0,Bar1, and both have relevant instancesbar0,bar1. However,Bar1extendsBaz, and we also have a relevant instancebazwhich suffices for everything not inBar0. Should the instance forFoostart{ bar0, bar1 with ... }or{ bar0, baz with ... }? Does it matter?mul := (· * ·),zero := (0 : ∀ i, f i)remain explicit or be obtained by sourcingPi.instMul,Pi.instZerorespectively?