WordProd.lean`

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7 declarations, 0 contain sorry, 0 declared as axioms

Evaluation of words

A generating family P : Group.Generators G ι indexed by ι gives rise to an evaluation map P.wordProd : List (ι × Bool) → G.

This file is copied from Mathlib.Geometry.Group.WordProd (leanprover-community/mathlib4#43118). It is in Mathlib from v4.34.0.

TODO: delete this file when this repository moves to Mathlib v4.34.0.

Main definitions

Group.Generators.wordProd: the canonical map from a word List (ι × Bool) over a generating family ι to the corresponding group G. It sends each (i, true) to P.val i and (i, false) to (P.val i)⁻¹.

Imports (1)

Mathlib.GroupTheory.Generators

Imported by 2 files

def Group.Generators.wordProd The canonical map from a word List (ι × Bool) over a generating family ι to its corresponding group G line 49 source
Source documentation
The canonical map from a word List (ι × Bool) over a generating family ι to its corresponding group G.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Group, Group.Generators, List, Prod, Bool
Proof uses: MonoidHom, FreeGroup, Equiv, FreeGroup.lift, Group.Generators.val, FreeGroup.mk
9 instances, coercions and auxiliary constants left out.
```
def wordProd : G := FreeGroup.lift P.val (FreeGroup.mk l)
```
theorem Group.Generators.wordProd_surjective Every element of G is the product of some word over a generating family line 52 source
Source documentation
Every element of G is the product of some word over a generating family.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Group, Group.Generators, Function.Surjective, List, Prod, Bool, Group.Generators.wordProd
Proof uses: Function.Surjective.comp, FreeGroup, MonoidHom, Equiv, FreeGroup.lift, Group.Generators.val, FreeGroup.mk, Group.Generators.lift_val_surjective, Quot.mk_surjective, FreeGroup.Red.Step
9 instances, coercions and auxiliary constants left out.
```
theorem wordProd_surjective : Function.Surjective P.wordProd :=
…
```
lemma Group.Generators.wordProd_nil line 56 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Group, Group.Generators, Eq, Group.Generators.wordProd, List.nil, Prod, Bool, OfNat.ofNat
Proof uses: of_eq_true, Eq.trans, True, congrFun', congrArg, List, List.map, cond, Prod.snd, Group.Generators.val, Prod.fst, Inv.inv, List.prod, List.map_nil, eq_self
16 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma Group.Generators.wordProd_singleton line 60 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Group, Group.Generators, Bool, Eq, Group.Generators.wordProd, List.cons, Prod, Prod.mk, List.nil, cond, Group.Generators.val, Inv.inv
Proof uses: of_eq_true, Eq.trans, True, congrFun', congrArg, List.prod, List.map, Prod.snd, Prod.fst, List, List.map_cons, List.map_nil, mul_one, eq_self
14 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma Group.Generators.wordProd_cons line 63 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Group, Group.Generators, Bool, List, Prod, Eq, Group.Generators.wordProd, List.cons, Prod.mk, HMul.hMul, cond, Group.Generators.val, Inv.inv
Proof uses: of_eq_true, List.prod, List.map, Prod.snd, Prod.fst, Eq.trans, True, congrFun', congrArg, List.map_cons, eq_self
21 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma Group.Generators.wordProd_append line 66 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Group, Group.Generators, List, Prod, Bool, Eq, Group.Generators.wordProd, HAppend.hAppend, HMul.hMul
Proof uses: of_eq_true, List.prod, List.map, cond, Prod.snd, Group.Generators.val, Prod.fst, Inv.inv, Eq.trans, True, congrFun', congrArg, List.map_append, List.prod_append, OfNat.ofNat, eq_self
27 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma Group.Generators.wordProd_invRev line 69 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Group, Group.Generators, List, Prod, Bool, Eq, Group.Generators.wordProd, FreeGroup.invRev, Inv.inv
Proof uses: of_eq_true, List.prod, List.map, cond, Prod.snd, Group.Generators.val, Prod.fst, Eq.trans, True, congrFun', congrArg, map_inv, FreeGroup, MonoidHom, Equiv, FreeGroup.lift, FreeGroup.mk, eq_self
22 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.