prove some properties about is_built_from
One property in this commit which we will use is the resulting short exact sequence if you have only two nontrivial subgroups.
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Floris van Doorn
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1 changed files with 84 additions and 26 deletions
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@ -280,6 +280,7 @@ section
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induction φ₁ with φ₁ q₁, induction φ₂ with φ₂ q₂, esimp at p, induction p,
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exact ap (homomorphism.mk φ₁) !is_prop.elim
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end
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end
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section
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@ -403,6 +404,11 @@ end
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(h : Π(r : R) g, φ (r • g) = r • φ g) : M₁ ≃lm M₂ :=
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isomorphism.mk (lm_homomorphism_of_group_homomorphism φ h) (group.isomorphism.is_equiv_to_hom φ)
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definition trivial_homomorphism [constructor] (M₁ M₂ : LeftModule R) : M₁ →lm M₂ :=
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lm_homomorphism_of_group_homomorphism (group.trivial_add_homomorphism M₁ M₂)
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(λs m, (smul_zero s)⁻¹)
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section
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local attribute pSet_of_LeftModule [coercion]
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definition is_exact_mod (f : M₁ →lm M₂) (f' : M₂ →lm M₃) : Type :=
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@ -493,31 +499,6 @@ end
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is_built_from.mk (λn, part H (n+1)) (λn, ses H (n+1)) isomorphism.rfl (pred (n₀ H))
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(λs Hle, HD' H _ (le_succ_of_pred_le Hle))
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definition isomorphism_of_is_contr_part (H : is_built_from D E) (n : ℕ) (HE : is_contr (E n)) :
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part H n ≃lm part H (n+1) :=
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isomorphism_of_is_contr_left (ses H n) HE
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definition is_contr_submodules (H : is_built_from D E) (HD : is_contr D) (n : ℕ) :
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is_contr (part H n) :=
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begin
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induction n with n IH,
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{ exact is_trunc_equiv_closed_rev -2 (equiv_of_isomorphism (e0 H)) _ },
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{ exact is_contr_right_of_short_exact_mod (ses H n) IH }
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end
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definition is_contr_quotients (H : is_built_from D E) (HD : is_contr D) (n : ℕ) :
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is_contr (E n) :=
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begin
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apply is_contr_left_of_short_exact_mod (ses H n),
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exact is_contr_submodules H HD n
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end
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definition is_contr_total (H : is_built_from D E) (HE : Πn, is_contr (E n)) : is_contr D :=
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have is_contr (part H 0), from
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nat.rec_down (λn, is_contr (part H n)) _ (HD' H _ !le.refl)
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(λn H2, is_contr_middle_of_short_exact_mod (ses H n) (HE n) H2),
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is_contr_equiv_closed (equiv_of_isomorphism (e0 H)) _
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definition is_built_from_isomorphism (e : D ≃lm D') (f : Πn, E n ≃lm E' n)
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(H : is_built_from D E) : is_built_from D' E' :=
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⦃is_built_from, H, e0 := e0 H ⬝lm e,
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@ -527,9 +508,86 @@ end
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is_built_from D' E :=
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⦃is_built_from, H, e0 := e0 H ⬝lm e⦄
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definition isomorphism_of_is_contr_submodule (H : is_built_from D E) (n : ℕ)
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(HE : is_contr (E n)) : part H n ≃lm part H (n+1) :=
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isomorphism_of_is_contr_left (ses H n) HE
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definition isomorphism_of_is_contr_submodules_range (H : is_built_from D E) {n k : ℕ}
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(Hnk : n ≤ k) (HE : Πl, n ≤ l → l < k → is_contr (E l)) : part H n ≃lm part H k :=
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begin
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induction Hnk with k Hnk IH,
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{ reflexivity },
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{ refine IH (λl Hnl Hlk, HE l Hnl (lt.step Hlk)) ⬝lm
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isomorphism_of_is_contr_submodule H k (HE k Hnk !self_lt_succ), }
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end
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definition is_contr_quotients_of_is_contr_total (H : is_built_from D E) (HD : is_contr D) (n : ℕ) :
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is_contr (part H n) :=
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begin
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induction n with n IH,
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{ exact is_trunc_equiv_closed_rev -2 (equiv_of_isomorphism (e0 H)) _ },
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{ exact is_contr_right_of_short_exact_mod (ses H n) IH }
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end
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definition is_contr_quotients_of_is_contr_submodules (H : is_built_from D E) {n : ℕ}
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(HE : Πk, n ≤ k → is_contr (E k)) : is_contr (part H n) :=
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begin
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refine is_contr_equiv_closed_rev _ (HD' H (max n (n₀ H)) !le_max_right),
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apply equiv_of_isomorphism,
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apply isomorphism_of_is_contr_submodules_range H !le_max_left,
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intros l Hnl Hl', exact HE l Hnl
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end
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/- alternate direct proof -/
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-- nat.rec_down (λk, is_contr (part H (n + k))) _ (HD' H _ !nat.le_add_left)
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-- (λk H2, is_contr_middle_of_short_exact_mod (ses H (n + k)) (HE (n + k) !nat.le_add_right)
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-- proof H2 qed)
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definition isomorphism_of_is_contr_submodules (H : is_built_from D E) {n : ℕ}
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(HE : Πk, n < k → is_contr (E k)) : E n ≃lm part H n :=
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isomorphism_of_is_contr_right (ses H n) (is_contr_quotients_of_is_contr_submodules H HE)
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-- definition is_contr_quotients_of_is_contr_submodules1 (H : is_built_from D E) (n : ℕ)
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-- (HE : Πk, n ≤ k → is_contr (E k)) : is_contr (part H n) :=
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-- nat.rec_down (λk, is_contr (part H (n + k))) _ (HD' H _ !nat.le_add_left)
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-- (λk H2, is_contr_middle_of_short_exact_mod (ses H (n + k)) (HE (n + k) !nat.le_add_right)
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-- proof H2 qed)
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definition isomorphism_zero_of_is_built_from (H : is_built_from D E) (p : n₀ H = 1) : E 0 ≃lm D :=
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isomorphism_of_is_contr_right (ses H 0) (HD' H 1 (le_of_eq p)) ⬝lm e0 H
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definition isomorphism_total_of_is_contr_submodules (H : is_built_from D E) {n : ℕ}
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(HE : Πk, k < n → is_contr (E k)) : D ≃lm part H n :=
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(e0 H)⁻¹ˡᵐ ⬝lm isomorphism_of_is_contr_submodules_range H !zero_le (λk H0k Hkn, HE k Hkn)
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definition isomorphism_of_is_contr_submodules_but_one (H : is_built_from D E) {n : ℕ}
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(HE : Πk, k ≠ n → is_contr (E k)) : D ≃lm E n :=
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(e0 H)⁻¹ˡᵐ ⬝lm
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isomorphism_of_is_contr_submodules_range H !zero_le (λk H0k Hkn, HE k (ne_of_lt Hkn)) ⬝lm
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(isomorphism_of_is_contr_submodules H (λk Hk, HE k (ne.symm (ne_of_lt Hk))))⁻¹ˡᵐ
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definition short_exact_mod_of_is_contr_submodules (H : is_built_from D E) {n m : ℕ} (Hnm : n < m)
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(HE : Πk, k ≠ n → k ≠ m → is_contr (E k)) : short_exact_mod (E n) D (E m) :=
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begin
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refine short_exact_mod_isomorphism isomorphism.rfl _ _ (ses H n),
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{ exact isomorphism_total_of_is_contr_submodules H (λk Hk, HE k (ne_of_lt Hk)
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(ne_of_lt (lt.trans Hk Hnm))) },
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{ exact isomorphism_of_is_contr_submodules H
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(λk Hk, HE k (ne.symm (ne_of_lt (lt.trans Hnm Hk))) (ne.symm (ne_of_lt Hk))) ⬝lm
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(isomorphism_of_is_contr_submodules_range H Hnm
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(λk Hnk Hkm, HE k (ne.symm (ne_of_lt Hnk)) (ne_of_lt Hkm)))⁻¹ˡᵐ }
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end
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definition is_contr_submodules (H : is_built_from D E) (HD : is_contr D) (n : ℕ) :
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is_contr (E n) :=
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begin
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apply is_contr_left_of_short_exact_mod (ses H n),
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exact is_contr_quotients_of_is_contr_total H HD n
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end
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definition is_contr_total (H : is_built_from D E) (HE : Πn, is_contr (E n)) : is_contr D :=
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have is_contr (part H 0), from is_contr_quotients_of_is_contr_submodules H (λn H, HE n),
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is_contr_equiv_closed (equiv_of_isomorphism (e0 H)) _
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section int
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open int
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definition left_module_int_of_ab_group [constructor] (A : Type) [add_ab_group A] : left_module rℤ A :=
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