[Merged by Bors] - chore: Generalise lemmas from finite groups to torsion elements #8342
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Many lemmas in `GroupTheory.OrderOfElement` were stated for elements of finite groups even though they work more generally for torsion elements of possibly infinite groups. This PR generalises those lemmas (and leaves convenience lemmas stated for finite groups), and fixes a bunch of names to use dot notation. ## Renames * `Function.eq_of_lt_minimalPeriod_of_iterate_eq` → `Function.iterate_injOn_Iio_minimalPeriod` * `Function.eq_iff_lt_minimalPeriod_of_iterate_eq` → `Function.iterate_eq_iterate_iff_of_lt_minimalPeriod` * `isOfFinOrder_iff_coe` → `Submonoid.isOfFinOrder_coe` * `orderOf_pos'` → `IsOfFinOrder.orderOf_pos` * `pow_eq_mod_orderOf` → `pow_mod_orderOf` (and turned around) * `pow_injective_of_lt_orderOf` → `pow_injOn_Iio_orderOf` * `mem_powers_iff_mem_range_order_of'` → `IsOfFinOrder.mem_powers_iff_mem_range_orderOf` * `orderOf_pow''` → `IsOfFinOrder.orderOf_pow` * `orderOf_pow_coprime` → `Nat.Coprime.orderOf_pow` * `zpow_eq_mod_orderOf` → `zpow_mod_orderOf` (and turned around) * `exists_pow_eq_one` → `isOfFinOrder_of_finite` * `pow_apply_eq_pow_mod_orderOf_cycleOf_apply` → `pow_mod_orderOf_cycleOf_apply` ## New lemmas * `IsOfFinOrder.powers_eq_image_range_orderOf` * `IsOfFinOrder.natCard_powers_le_orderOf` * `IsOfFinOrder.finite_powers` * `finite_powers` * `infinite_powers` * `Nat.card_submonoidPowers` * `IsOfFinOrder.mem_powers_iff_mem_zpowers` * `IsOfFinOrder.powers_eq_zpowers` * `IsOfFinOrder.mem_zpowers_iff_mem_range_orderOf` * `IsOfFinOrder.exists_pow_eq_one` ## Other changes * Move `decidableMemPowers`/`fintypePowers` to `GroupTheory.Submonoid.Membership` and `decidableMemZpowers`/`fintypeZpowers` to `GroupTheory.Subgroup.ZPowers`. * `finEquivPowers`, `finEquivZpowers`, `powersEquivPowers` and `zpowersEquivZpowers` now assume `IsOfFinTorsion x` instead of `Finite G`. * `isOfFinOrder_iff_pow_eq_one` now takes one less explicit argument. * Delete `Equiv.Perm.IsCycle.exists_pow_eq_one` since it was saying that a permutation over a finite type is torsion, but this is trivial since the group of permutation is itself finite, so we can use `isOfFinOrder_of_finite` instead.
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Many lemmas in `GroupTheory.OrderOfElement` were stated for elements of finite groups even though they work more generally for torsion elements of possibly infinite groups. This PR generalises those lemmas (and leaves convenience lemmas stated for finite groups), and fixes a bunch of names to use dot notation. ## Renames * `Function.eq_of_lt_minimalPeriod_of_iterate_eq` → `Function.iterate_injOn_Iio_minimalPeriod` * `Function.eq_iff_lt_minimalPeriod_of_iterate_eq` → `Function.iterate_eq_iterate_iff_of_lt_minimalPeriod` * `isOfFinOrder_iff_coe` → `Submonoid.isOfFinOrder_coe` * `orderOf_pos'` → `IsOfFinOrder.orderOf_pos` * `pow_eq_mod_orderOf` → `pow_mod_orderOf` (and turned around) * `pow_injective_of_lt_orderOf` → `pow_injOn_Iio_orderOf` * `mem_powers_iff_mem_range_order_of'` → `IsOfFinOrder.mem_powers_iff_mem_range_orderOf` * `orderOf_pow''` → `IsOfFinOrder.orderOf_pow` * `orderOf_pow_coprime` → `Nat.Coprime.orderOf_pow` * `zpow_eq_mod_orderOf` → `zpow_mod_orderOf` (and turned around) * `exists_pow_eq_one` → `isOfFinOrder_of_finite` * `pow_apply_eq_pow_mod_orderOf_cycleOf_apply` → `pow_mod_orderOf_cycleOf_apply` ## New lemmas * `IsOfFinOrder.powers_eq_image_range_orderOf` * `IsOfFinOrder.natCard_powers_le_orderOf` * `IsOfFinOrder.finite_powers` * `finite_powers` * `infinite_powers` * `Nat.card_submonoidPowers` * `IsOfFinOrder.mem_powers_iff_mem_zpowers` * `IsOfFinOrder.powers_eq_zpowers` * `IsOfFinOrder.mem_zpowers_iff_mem_range_orderOf` * `IsOfFinOrder.exists_pow_eq_one` ## Other changes * Move `decidableMemPowers`/`fintypePowers` to `GroupTheory.Submonoid.Membership` and `decidableMemZpowers`/`fintypeZpowers` to `GroupTheory.Subgroup.ZPowers`. * `finEquivPowers`, `finEquivZpowers`, `powersEquivPowers` and `zpowersEquivZpowers` now assume `IsOfFinTorsion x` instead of `Finite G`. * `isOfFinOrder_iff_pow_eq_one` now takes one less explicit argument. * Delete `Equiv.Perm.IsCycle.exists_pow_eq_one` since it was saying that a permutation over a finite type is torsion, but this is trivial since the group of permutation is itself finite, so we can use `isOfFinOrder_of_finite` instead.
alexkeizer
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Nov 21, 2023
Many lemmas in `GroupTheory.OrderOfElement` were stated for elements of finite groups even though they work more generally for torsion elements of possibly infinite groups. This PR generalises those lemmas (and leaves convenience lemmas stated for finite groups), and fixes a bunch of names to use dot notation. ## Renames * `Function.eq_of_lt_minimalPeriod_of_iterate_eq` → `Function.iterate_injOn_Iio_minimalPeriod` * `Function.eq_iff_lt_minimalPeriod_of_iterate_eq` → `Function.iterate_eq_iterate_iff_of_lt_minimalPeriod` * `isOfFinOrder_iff_coe` → `Submonoid.isOfFinOrder_coe` * `orderOf_pos'` → `IsOfFinOrder.orderOf_pos` * `pow_eq_mod_orderOf` → `pow_mod_orderOf` (and turned around) * `pow_injective_of_lt_orderOf` → `pow_injOn_Iio_orderOf` * `mem_powers_iff_mem_range_order_of'` → `IsOfFinOrder.mem_powers_iff_mem_range_orderOf` * `orderOf_pow''` → `IsOfFinOrder.orderOf_pow` * `orderOf_pow_coprime` → `Nat.Coprime.orderOf_pow` * `zpow_eq_mod_orderOf` → `zpow_mod_orderOf` (and turned around) * `exists_pow_eq_one` → `isOfFinOrder_of_finite` * `pow_apply_eq_pow_mod_orderOf_cycleOf_apply` → `pow_mod_orderOf_cycleOf_apply` ## New lemmas * `IsOfFinOrder.powers_eq_image_range_orderOf` * `IsOfFinOrder.natCard_powers_le_orderOf` * `IsOfFinOrder.finite_powers` * `finite_powers` * `infinite_powers` * `Nat.card_submonoidPowers` * `IsOfFinOrder.mem_powers_iff_mem_zpowers` * `IsOfFinOrder.powers_eq_zpowers` * `IsOfFinOrder.mem_zpowers_iff_mem_range_orderOf` * `IsOfFinOrder.exists_pow_eq_one` ## Other changes * Move `decidableMemPowers`/`fintypePowers` to `GroupTheory.Submonoid.Membership` and `decidableMemZpowers`/`fintypeZpowers` to `GroupTheory.Subgroup.ZPowers`. * `finEquivPowers`, `finEquivZpowers`, `powersEquivPowers` and `zpowersEquivZpowers` now assume `IsOfFinTorsion x` instead of `Finite G`. * `isOfFinOrder_iff_pow_eq_one` now takes one less explicit argument. * Delete `Equiv.Perm.IsCycle.exists_pow_eq_one` since it was saying that a permutation over a finite type is torsion, but this is trivial since the group of permutation is itself finite, so we can use `isOfFinOrder_of_finite` instead.
grunweg
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Dec 15, 2023
Many lemmas in `GroupTheory.OrderOfElement` were stated for elements of finite groups even though they work more generally for torsion elements of possibly infinite groups. This PR generalises those lemmas (and leaves convenience lemmas stated for finite groups), and fixes a bunch of names to use dot notation. ## Renames * `Function.eq_of_lt_minimalPeriod_of_iterate_eq` → `Function.iterate_injOn_Iio_minimalPeriod` * `Function.eq_iff_lt_minimalPeriod_of_iterate_eq` → `Function.iterate_eq_iterate_iff_of_lt_minimalPeriod` * `isOfFinOrder_iff_coe` → `Submonoid.isOfFinOrder_coe` * `orderOf_pos'` → `IsOfFinOrder.orderOf_pos` * `pow_eq_mod_orderOf` → `pow_mod_orderOf` (and turned around) * `pow_injective_of_lt_orderOf` → `pow_injOn_Iio_orderOf` * `mem_powers_iff_mem_range_order_of'` → `IsOfFinOrder.mem_powers_iff_mem_range_orderOf` * `orderOf_pow''` → `IsOfFinOrder.orderOf_pow` * `orderOf_pow_coprime` → `Nat.Coprime.orderOf_pow` * `zpow_eq_mod_orderOf` → `zpow_mod_orderOf` (and turned around) * `exists_pow_eq_one` → `isOfFinOrder_of_finite` * `pow_apply_eq_pow_mod_orderOf_cycleOf_apply` → `pow_mod_orderOf_cycleOf_apply` ## New lemmas * `IsOfFinOrder.powers_eq_image_range_orderOf` * `IsOfFinOrder.natCard_powers_le_orderOf` * `IsOfFinOrder.finite_powers` * `finite_powers` * `infinite_powers` * `Nat.card_submonoidPowers` * `IsOfFinOrder.mem_powers_iff_mem_zpowers` * `IsOfFinOrder.powers_eq_zpowers` * `IsOfFinOrder.mem_zpowers_iff_mem_range_orderOf` * `IsOfFinOrder.exists_pow_eq_one` ## Other changes * Move `decidableMemPowers`/`fintypePowers` to `GroupTheory.Submonoid.Membership` and `decidableMemZpowers`/`fintypeZpowers` to `GroupTheory.Subgroup.ZPowers`. * `finEquivPowers`, `finEquivZpowers`, `powersEquivPowers` and `zpowersEquivZpowers` now assume `IsOfFinTorsion x` instead of `Finite G`. * `isOfFinOrder_iff_pow_eq_one` now takes one less explicit argument. * Delete `Equiv.Perm.IsCycle.exists_pow_eq_one` since it was saying that a permutation over a finite type is torsion, but this is trivial since the group of permutation is itself finite, so we can use `isOfFinOrder_of_finite` instead.
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Many lemmas in
GroupTheory.OrderOfElementwere stated for elements of finite groups even though they work more generally for torsion elements of possibly infinite groups. This PR generalises those lemmas (and leaves convenience lemmas stated for finite groups), and fixes a bunch of names to use dot notation.Renames
Function.eq_of_lt_minimalPeriod_of_iterate_eq→Function.iterate_injOn_Iio_minimalPeriodFunction.eq_iff_lt_minimalPeriod_of_iterate_eq→Function.iterate_eq_iterate_iff_of_lt_minimalPeriodisOfFinOrder_iff_coe→Submonoid.isOfFinOrder_coeorderOf_pos'→IsOfFinOrder.orderOf_pospow_eq_mod_orderOf→pow_mod_orderOf(and turned around)pow_injective_of_lt_orderOf→pow_injOn_Iio_orderOfmem_powers_iff_mem_range_order_of'→IsOfFinOrder.mem_powers_iff_mem_range_orderOforderOf_pow''→IsOfFinOrder.orderOf_poworderOf_pow_coprime→Nat.Coprime.orderOf_powzpow_eq_mod_orderOf→zpow_mod_orderOf(and turned around)exists_pow_eq_one→isOfFinOrder_of_finitepow_apply_eq_pow_mod_orderOf_cycleOf_apply→pow_mod_orderOf_cycleOf_applyNew lemmas
IsOfFinOrder.powers_eq_image_range_orderOfIsOfFinOrder.natCard_powers_le_orderOfIsOfFinOrder.finite_powersfinite_powersinfinite_powersNat.card_submonoidPowersIsOfFinOrder.mem_powers_iff_mem_zpowersIsOfFinOrder.powers_eq_zpowersIsOfFinOrder.mem_zpowers_iff_mem_range_orderOfIsOfFinOrder.exists_pow_eq_oneOther changes
decidableMemPowers/fintypePowerstoGroupTheory.Submonoid.MembershipanddecidableMemZpowers/fintypeZpowerstoGroupTheory.Subgroup.ZPowers.finEquivPowers,finEquivZpowers,powersEquivPowersandzpowersEquivZpowersnow assumeIsOfFinTorsion xinstead ofFinite G.isOfFinOrder_iff_pow_eq_onenow takes one less explicit argument.Equiv.Perm.IsCycle.exists_pow_eq_onesince it was saying that a permutation over a finite type is torsion, but this is trivial since the group of permutation is itself finite, so we can useisOfFinOrder_of_finiteinstead.I need this for Kneser's theorem