ReidemeisterSchreier.Schreier
This module formalizes Schreier generators and rewriting.
import
- Mathlib.Data.Nat.Basic
def rankTransform (r i : ℕ) : ℕ :=
if r = 0 then 0 else 1 + i * (r - 1)The Schreier rank transform \(T(r,i)=1+i(r-1)\), with the rank-zero convention.
@[simp] theorem rankTransform_zero_left (i : ℕ) : rankTransform 0 i = 0Schreier's rank transform satisfies the displayed cardinal-arithmetic identity.
Show proof
by
simp only [rankTransform, ↓reduceIte]Proof. Fix the chosen Schreier transversal and write each group element as a transversal representative multiplied by an element of the subgroup. The rewriting map is computed letter by letter: after reading a letter, the current coset representative determines the Schreier generator, and the next representative supplies the correction term. The formula is first verified on a single letter and then extended to words by induction on the word list, using concatenation compatibility of the rewriting procedure. The resulting identity in chosen coset representatives is independent of the chosen word presentation after passing through the normal-closure relations, giving the claimed Reidemeister--Schreier formula. At each induction step the current representative and the next representative determine exactly one Schreier generator. The correction terms telescope over a concatenated word, leaving the expected subgroup element at the end of the rewriting. Therefore equality in the subgroup presentation follows from the finite list of rewritten generators and relators, not from any additional relation. The quotient presentation is generated by the Schreier generators, so equality of rewritten generator images determines the induced homomorphism. The normal-closure argument shows that replacing a word by an equivalent word in the original presentation does not change its rewritten image. This supplies the required presentation-level equality. The Reidemeister--Schreier step tracks the transversal representative before and after each letter. Trivial Schreier generators are removed when the next representative is unchanged, while nontrivial generators record exactly the subgroup correction; products of these corrections reconstruct the original word inside the subgroup presentation.
□@[simp] theorem rankTransform_succ (r i : ℕ) : rankTransform (r + 1) i = 1 + i * rSchreier's rank transform satisfies the displayed cardinal-arithmetic identity.
Show proof
by
simp only [rankTransform, Nat.add_eq_zero_iff, Nat.succ_ne_self, and_false, ↓reduceIte,
Nat.add_one_sub_one]Proof. Fix the chosen Schreier transversal and write each group element as a transversal representative multiplied by an element of the subgroup. The rewriting map is computed letter by letter: after reading a letter, the current coset representative determines the Schreier generator, and the next representative supplies the correction term. The formula is first verified on a single letter and then extended to words by induction on the word list, using concatenation compatibility of the rewriting procedure. The resulting identity in chosen coset representatives is independent of the chosen word presentation after passing through the normal-closure relations, giving the claimed Reidemeister--Schreier formula. At each induction step the current representative and the next representative determine exactly one Schreier generator. The correction terms telescope over a concatenated word, leaving the expected subgroup element at the end of the rewriting. Therefore equality in the subgroup presentation follows from the finite list of rewritten generators and relators, not from any additional relation. The quotient presentation is generated by the Schreier generators, so equality of rewritten generator images determines the induced homomorphism. The normal-closure argument shows that replacing a word by an equivalent word in the original presentation does not change its rewritten image. This supplies the required presentation-level equality. The Reidemeister--Schreier step tracks the transversal representative before and after each letter. Trivial Schreier generators are removed when the next representative is unchanged, while nontrivial generators record exactly the subgroup correction; products of these corrections reconstruct the original word inside the subgroup presentation.
□@[simp] theorem rankTransform_one_left (i : ℕ) : rankTransform 1 i = 1Schreier's rank transform satisfies the displayed cardinal-arithmetic identity.
Show proof
by
simp only [rankTransform, Nat.succ_ne_self, ↓reduceIte, Nat.sub_self, Nat.mul_zero, Nat.add_zero]Proof. Fix the chosen Schreier transversal and write each group element as a transversal representative multiplied by an element of the subgroup. The rewriting map is computed letter by letter: after reading a letter, the current coset representative determines the Schreier generator, and the next representative supplies the correction term. The formula is first verified on a single letter and then extended to words by induction on the word list, using concatenation compatibility of the rewriting procedure. The resulting identity in chosen coset representatives is independent of the chosen word presentation after passing through the normal-closure relations, giving the claimed Reidemeister--Schreier formula. At each induction step the current representative and the next representative determine exactly one Schreier generator. The correction terms telescope over a concatenated word, leaving the expected subgroup element at the end of the rewriting. Therefore equality in the subgroup presentation follows from the finite list of rewritten generators and relators, not from any additional relation. The quotient presentation is generated by the Schreier generators, so equality of rewritten generator images determines the induced homomorphism. The normal-closure argument shows that replacing a word by an equivalent word in the original presentation does not change its rewritten image. This supplies the required presentation-level equality. The Reidemeister--Schreier step tracks the transversal representative before and after each letter. Trivial Schreier generators are removed when the next representative is unchanged, while nontrivial generators record exactly the subgroup correction; products of these corrections reconstruct the original word inside the subgroup presentation.
□theorem rankTransform_eq_one_add {r i : ℕ} (hr : r ≠ 0) :
rankTransform r i = 1 + i * (r - 1)For nonzero \(r\), the Schreier rank transform is \(1+i(r-1)\).
Show proof
by
simp only [rankTransform, hr, ↓reduceIte]Proof. Fix the chosen Schreier transversal and write each group element as a transversal representative multiplied by an element of the subgroup. The rewriting map is computed letter by letter: after reading a letter, the current coset representative determines the Schreier generator, and the next representative supplies the correction term. The formula is first verified on a single letter and then extended to words by induction on the word list, using concatenation compatibility of the rewriting procedure. Relator and normal-closure claims follow because rewriting the defining relators gives the subgroup presentation relators, and conjugating by transversal representatives stays inside the generated normal closure. When a quotient map is present, both sides use the same image coset and the same rewritten letter, so the chosen-representative equality descends to the quotient presentation. The resulting identity in chosen coset representatives is independent of the chosen word presentation after passing through the normal-closure relations, giving the claimed Reidemeister--Schreier formula. At each induction step the current representative and the next representative determine exactly one Schreier generator. The correction terms telescope over a concatenated word, leaving the expected subgroup element at the end of the rewriting. Therefore equality in the subgroup presentation follows from the finite list of rewritten generators and relators, not from any additional relation. For quotient and subgroup claims, representatives are chosen in the ambient group and the induced map is checked to send the class of an element to the class of its image. Normality, openness, and membership in the finite quotient class are preserved by the subgroup, quotient, intersection, or inverse-image closure property being invoked.
□theorem rankTransform_mul_index (r i j : ℕ) :
rankTransform (rankTransform r i) j = rankTransform r (i * j)Multiplying subgroup indices composes the Schreier rank transform.
Show proof
by
cases r with
| zero =>
simp only [rankTransform, ↓reduceIte]
| succ r =>
simp only [rankTransform, Nat.add_eq_zero_iff, Nat.succ_ne_self, and_false, ↓reduceIte,
Nat.add_one_sub_one, false_and, Nat.add_sub_cancel_left, Nat.mul_left_comm, Nat.mul_comm]Proof. Fix the chosen Schreier transversal and write each group element as a transversal representative multiplied by an element of the subgroup. The rewriting map is computed letter by letter: after reading a letter, the current coset representative determines the Schreier generator, and the next representative supplies the correction term. The formula is first verified on a single letter and then extended to words by induction on the word list, using concatenation compatibility of the rewriting procedure. The resulting identity in chosen coset representatives is independent of the chosen word presentation after passing through the normal-closure relations, giving the claimed Reidemeister--Schreier formula. At each induction step the current representative and the next representative determine exactly one Schreier generator. The correction terms telescope over a concatenated word, leaving the expected subgroup element at the end of the rewriting. Therefore equality in the subgroup presentation follows from the finite list of rewritten generators and relators, not from any additional relation. The quotient presentation is generated by the Schreier generators, so equality of rewritten generator images determines the induced homomorphism. The normal-closure argument shows that replacing a word by an equivalent word in the original presentation does not change its rewritten image. This supplies the required presentation-level equality. For quotient and subgroup claims, representatives are chosen in the ambient group and the induced map is checked to send the class of an element to the class of its image. Normality, openness, and membership in the finite quotient class are preserved by the subgroup, quotient, intersection, or inverse-image closure property being invoked.
□theorem rankTransform_mono_left {r s i : ℕ} (hrs : r ≤ s) :
rankTransform r i ≤ rankTransform s iThe Schreier rank transform is monotone in the rank variable.
Show proof
by
cases s with
| zero =>
have hr : r = 0 := Nat.eq_zero_of_le_zero hrs
subst hr
simp only [rankTransform_zero_left, le_refl]
| succ s =>
cases r with
| zero =>
simp only [rankTransform_zero_left, rankTransform_succ, Nat.zero_le]
| succ r =>
simpa only [rankTransform_succ] using
Nat.add_le_add_left (Nat.mul_le_mul_left i (Nat.succ_le_succ_iff.mp hrs)) 1Proof. Fix the chosen Schreier transversal and write each group element as a transversal representative multiplied by an element of the subgroup. The rewriting map is computed letter by letter: after reading a letter, the current coset representative determines the Schreier generator, and the next representative supplies the correction term. The formula is first verified on a single letter and then extended to words by induction on the word list, using concatenation compatibility of the rewriting procedure. The resulting identity in chosen coset representatives is independent of the chosen word presentation after passing through the normal-closure relations, giving the claimed Reidemeister--Schreier formula. At each induction step the current representative and the next representative determine exactly one Schreier generator. The correction terms telescope over a concatenated word, leaving the expected subgroup element at the end of the rewriting. Therefore equality in the subgroup presentation follows from the finite list of rewritten generators and relators, not from any additional relation. The quotient presentation is generated by the Schreier generators, so equality of rewritten generator images determines the induced homomorphism. The normal-closure argument shows that replacing a word by an equivalent word in the original presentation does not change its rewritten image. This supplies the required presentation-level equality. The Reidemeister--Schreier step tracks the transversal representative before and after each letter. Trivial Schreier generators are removed when the next representative is unchanged, while nontrivial generators record exactly the subgroup correction; products of these corrections reconstruct the original word inside the subgroup presentation.
□