From d5d004ad6cd7d2697a6c1a61d4e306df6841b28c Mon Sep 17 00:00:00 2001 From: Sayantan Khan Date: Wed, 3 May 2017 03:41:34 +0530 Subject: [PATCH] almost done --- .../homological_lemmas.hlean | 156 +++++++++++++++--- 1 file changed, 133 insertions(+), 23 deletions(-) diff --git a/hott/algebra/homological_algebra/homological_lemmas.hlean b/hott/algebra/homological_algebra/homological_lemmas.hlean index 277769416..e6e3d0a00 100644 --- a/hott/algebra/homological_algebra/homological_lemmas.hlean +++ b/hott/algebra/homological_algebra/homological_lemmas.hlean @@ -17,17 +17,76 @@ open abelian_chain_complex open abelian_chain_complex.ab_exact_chain_complex open abelian_chain_complex.exact_chain_map +set_option unifier.max_steps 1000000 + /- Auxiliary lemmas. May be moved somewhere else. -/ -lemma transport_equality : Π {G₁ G₂ : AbGroup}, Π (φ : homomorphism (G₁) (G₂)), Π {x y : G₁}, +lemma transport_equality_hom : Π {G₁ G₂ : AbGroup}, Π (φ : homomorphism (G₁) (G₂)), Π {x y : G₁}, (x = y) → (φ(x) = φ(y)) := λ G₁ G₂ φ x, @eq.rec(_)(x)(_) (eq.refl(group_fun φ(x))) +lemma transport_equality_left_mul : Π {G : AbGroup}, Π (l : G), Π {x y : G}, + (x = y) → (l*x = l*y) := + λ G l x, @eq.rec(_)(x)(_) (eq.refl(l*x)) + +lemma transport_equality_right_mul : Π {G : AbGroup}, Π (r : G), Π {x y : G}, + (x = y) → (x*r = y*r) := + λ G r x, @eq.rec(_)(x)(_) (eq.refl(x*r)) + +lemma mul_inverse : Π {G : AbGroup}, Π (x : G), (x⁻¹)*x = group.one(G) := λ G x, ab_group.mul_left_inv x + +lemma inverse_inverse : Π {G : AbGroup}, Π (x : G), (x⁻¹)⁻¹ = x := + λ G x, inv_inv(x) + +lemma mul_commutative : Π {G : AbGroup}, Π (x y : G), x*y = y*x := + λ G x y, ab_group.mul_comm x y + +lemma mul_associative : Π {G : AbGroup}, Π (x y z : G), (x*y)*z = x*(y*z) := + λ G x y z, ab_group.mul_assoc x y z + +lemma mul_respects_one : Π {G : AbGroup}, Π (x : G), + 1*x = x := λ G x, ab_group.one_mul x + +lemma cancelling : Π {G : AbGroup}, Π (x y : G), (x = y) → (y⁻¹*x = 1) := + λ G x y proofEq, + eq.trans (transport_equality_left_mul (y⁻¹) (proofEq)) (mul_inverse(y)) + lemma hom_respects_one : Π {G₁ G₂ : AbGroup}, Π (φ : homomorphism (G₁) (G₂)), Π (x : G₁), (x = group.one(G₁)) → (φ(x) = group.one(G₂)) := - λ G₁ G₂ φ x proofOfOne, eq.trans (transport_equality (φ)(proofOfOne)) (respect_one φ) + λ G₁ G₂ φ x proofOfOne, eq.trans (transport_equality_hom (φ)(proofOfOne)) (respect_one φ) + +lemma hom_respects_mul : Π {G₁ G₂ : AbGroup}, Π (φ : homomorphism (G₁) (G₂)), Π (x y : G₁), + φ(x*y) = φ(x)*φ(y) := + λ G₁ G₂ φ x y, to_respect_mul φ x y + +lemma hom_respects_inv : Π {G₁ G₂ : AbGroup}, Π (φ : homomorphism (G₁) (G₂)), Π (x : G₁), + φ(x⁻¹) = (φ(x))⁻¹ := + λ G₁ G₂ φ x, to_respect_inv φ x + +lemma hom_respects_cancelling : Π {G₁ G₂ : AbGroup}, Π (φ : homomorphism (G₁) (G₂)), Π {x y : G₁}, + (φ(x) = φ(y)) → φ(y⁻¹*x) = 1 := + λ G₁ G₂ φ x y proofEq, + eq.trans + (eq.trans (hom_respects_mul (φ) (y⁻¹) (x)) (transport_equality_right_mul (group_fun (φ) (x)) (hom_respects_inv(φ)(y)))) + (eq.trans (transport_equality_left_mul((group_fun (φ) (y))⁻¹)(proofEq)) + (mul_inverse(group_fun (φ) (y)))) + +axiom combine_terms : Π {G₁ G₂ : AbGroup}, Π (φ : homomorphism (G₁) (G₂)), Π (a c : G₁), Π (b : G₂), + φ(a) = b⁻¹*φ(c) → b = φ(a⁻¹*c) + +-- lemma combine_terms : Π {G₁ G₂ : AbGroup}, Π (φ : homomorphism (G₁) (G₂)), Π (a b : G₁), Π (c : G₂), +-- φ(a)*φ(b) = c → c = φ(a*b) := +-- λ G₁ G₂ φ a b c proofEq, +-- eq.symm (eq.trans (hom_respects_mul (φ) (a) (b)) (proofEq)) + +-- lemma take_over : Π {G₁ G₂ : AbGroup}, Π (φ : homomorphism (G₁) (G₂)), Π (a c : G₁), Π (b : G₂), +-- φ(a) = b⁻¹ * φ(c) → b*φ(a) = φ(c) := +-- λ G₁ G₂ φ a c b proofEq, +-- eq.trans +-- (transport_equality_right_mul (group_fun (φ) (a)) (eq.symm (inverse_inverse(b)))) +-- (transport_equality_left_mul ((b⁻¹)⁻¹) (proofEq)) /- Simple lemma showing surjective and injective imply bijective. @@ -108,7 +167,7 @@ lemma lFlLemmaThree : Π {C₁ C₂ : ab_exact_chain_complex}, Π (M : exact_cha ((group_map(M)(z-1-1)(c)) = (boundary_map(C₂)(z-1)(b'))) → (boundary_map(C₁)(z-1-1)(c) = group.one(chain_group(C₁)(z-1-1-1))) := λ C₁ C₂ M z proofSurj_m proofSurj_p proofInj_q b' c proofEq, lFlLemmaTwo (M) (z) (proofSurj_m) (proofSurj_p) (proofInj_q) (c) - (eq.trans (transport_equality (boundary_map(C₂)(z-1-1)) (proofEq)) + (eq.trans (transport_equality_hom (boundary_map(C₂)(z-1-1)) (proofEq)) (boundary_of_boundary(C₂)(z-1)(b'))) lemma lFlLemmaFour : Π {C₁ C₂ : ab_exact_chain_complex}, Π (M : exact_chain_map (C₁) (C₂)), Π (z : ℤ), @@ -121,7 +180,7 @@ lemma lFlLemmaFour : Π {C₁ C₂ : ab_exact_chain_complex}, Π (M : exact_chai (pr₁(exactness(C₁)(z-1)(c) (lFlLemmaThree (M) (z) (proofSurj_m) (proofSurj_p) (proofInj_q) (b') (c) (proofEq) ))) ( eq.trans - (transport_equality (group_map(M)(z-1-1)) (pr₂(exactness(C₁)(z-1)(c) (lFlLemmaThree (M) (z) (proofSurj_m) (proofSurj_p) (proofInj_q) (b') (c) (proofEq) )))) + (transport_equality_hom (group_map(M)(z-1-1)) (pr₂(exactness(C₁)(z-1)(c) (lFlLemmaThree (M) (z) (proofSurj_m) (proofSurj_p) (proofInj_q) (b') (c) (proofEq) )))) (proofEq) ) ) @@ -140,22 +199,73 @@ lemma lFlLemmaFive : Π {C₁ C₂ : ab_exact_chain_complex}, Π (M : exact_chai (pr₁(lFlLemmaFour (M) (z) (proofSurj_m) (proofSurj_p) (proofInj_q) (b'))) (pr₂(lFlLemmaFour (M) (z) (proofSurj_m) (proofSurj_p) (proofInj_q) (b'))) ) --- lemma lFlLemmaThree : Π {C₁ C₂ : ab_exact_chain_complex}, Π (M : exact_chain_map (C₁) (C₂)), Π (z : ℤ), --- surjective_map (group_map(M)(z)) → surjective_map (group_map(M)(z-1-1)) → injective_map (group_map(M)(z-1-1-1)) → --- (Π (b' : chain_group(C₂)(z-1)), (Σ (b : chain_group(C₁)(z-1)), boundary_map(C₂)(z-1)(b') = boundary_map(C₂)(z-1)(group_map(M)(z-1)(b)))) := --- λ C₁ C₂ M z proofSurj_m proofSurj_p proofInj_q b', --- ( --- (λ c proofcGoesZero, --- sigma.mk --- (sorry) --- (sorry) --- ) --- (pr₁(lFlLemmaOne (M) (z) (proofSurj_m) (proofSurj_p) (proofInj_q) (b'))) --- ( --- lFlLemmaTwo (M) (z) (proofSurj_m) (proofSurj_p) (proofInj_q) (pr₁(lFlLemmaOne (M) (z) (proofSurj_m) (proofSurj_p) (proofInj_q) (b'))) --- (eq.trans --- (eq.symm (commutes (M) (z-1-1) (pr₁(lFlLemmaOne (M) (z) (proofSurj_m) (proofSurj_p) (proofInj_q) (b'))))) --- (transport_equality (boundary_map(C₂)(z-1-1)) (pr₂(lFlLemmaOne (M) (z) (proofSurj_m) (proofSurj_p) (proofInj_q) (b')))) --- ) --- ) --- ) + +lemma lFlLemmaSix : Π {C₁ C₂ : ab_exact_chain_complex}, Π (M : exact_chain_map (C₁) (C₂)), Π (z : ℤ), + surjective_map (group_map(M)(z)) → surjective_map (group_map(M)(z-1-1)) → injective_map (group_map(M)(z-1-1-1)) → + Π (b' : chain_group(C₂)(z-1)), Σ (b : chain_group(C₁)(z-1)), boundary_map(C₂)(z-1)((b'⁻¹)*group_map(M)(z-1)(b)) = 1 := + λ C₁ C₂ M z proofSurj_m proofSurj_p proofInj_q b', + ( + (λ b proofEq, + sigma.mk (b) + (hom_respects_cancelling (boundary_map(C₂)(z-1)) (proofEq)) + ) + (pr₁(lFlLemmaFive (M) (z) (proofSurj_m) (proofSurj_p) (proofInj_q) (b'))) + (pr₂(lFlLemmaFive (M) (z) (proofSurj_m) (proofSurj_p) (proofInj_q) (b'))) + ) + +lemma lFlLemmaSeven : Π {C₁ C₂ : ab_exact_chain_complex}, Π (M : exact_chain_map (C₁) (C₂)), Π (z : ℤ), + surjective_map (group_map(M)(z)) → surjective_map (group_map(M)(z-1-1)) → injective_map (group_map(M)(z-1-1-1)) → + Π (b' : chain_group(C₂)(z-1)), Π (b : chain_group(C₁)(z-1)), boundary_map(C₂)(z-1)((b'⁻¹)*group_map(M)(z-1)(b)) = 1 → + Σ (a : chain_group(C₁)(z)), boundary_map(C₂)(z)(group_map(M)(z)(a)) = (b'⁻¹)*group_map(M)(z-1)(b) := + λ C₁ C₂ M z proofSurj_m proofSurj_p proofInj_q b' b proofZero, + ( + (λ a' proofPreImage, + sigma.mk + (pr₁(surjective_map.get_preimage(proofSurj_m)(a'))) + (eq.trans + (transport_equality_hom (boundary_map(C₂)(z)) (pr₂(surjective_map.get_preimage(proofSurj_m)(a')))) + (proofPreImage) + ) + ) + (pr₁(exactness(C₂)(z)((b'⁻¹)*group_map(M)(z-1)(b))(proofZero))) + (pr₂(exactness(C₂)(z)((b'⁻¹)*group_map(M)(z-1)(b))(proofZero))) + ) + +lemma lFlLemmaEight : Π {C₁ C₂ : ab_exact_chain_complex}, Π (M : exact_chain_map (C₁) (C₂)), Π (z : ℤ), + surjective_map (group_map(M)(z)) → surjective_map (group_map(M)(z-1-1)) → injective_map (group_map(M)(z-1-1-1)) → + Π (b' : chain_group(C₂)(z-1)), Π (b : chain_group(C₁)(z-1)), boundary_map(C₂)(z-1)((b'⁻¹)*group_map(M)(z-1)(b)) = 1 → + Σ (a : chain_group(C₁)(z)), group_map(M)(z-1)(boundary_map(C₁)(z)(a)) = (b'⁻¹)*group_map(M)(z-1)(b) := + λ C₁ C₂ M z proofSurj_m proofSurj_p proofInj_q b' b proofZero, + ( + (λ a proofBottomRight, + sigma.mk + (a) + (eq.trans (eq.symm (commutes(M)(z)(a))) (proofBottomRight)) + ) + (pr₁(lFlLemmaSeven (M) (z) (proofSurj_m) (proofSurj_p) (proofInj_q) (b') (b) (proofZero))) + (pr₂(lFlLemmaSeven (M) (z) (proofSurj_m) (proofSurj_p) (proofInj_q) (b') (b) (proofZero))) + ) + +lemma lFlLemmaNine : Π {C₁ C₂ : ab_exact_chain_complex}, Π (M : exact_chain_map (C₁) (C₂)), Π (z : ℤ), + surjective_map (group_map(M)(z)) → surjective_map (group_map(M)(z-1-1)) → injective_map (group_map(M)(z-1-1-1)) → + Π (b' : chain_group(C₂)(z-1)), Σ (j : chain_group(C₁)(z-1)), group_map(M)(z-1)(j) = b' := + λ C₁ C₂ M z proofSurj_m proofSurj_p proofInj_q b', + ( + (λ b proofEq1, + (λ a proofEq2, + sigma.mk + ((boundary_map(C₁)(z)(a))⁻¹*b) + (eq.symm (combine_terms (group_map(M)(z-1)) (boundary_map(C₁)(z)(a)) (b) (b') (proofEq2))) + ) + (pr₁ (lFlLemmaEight (M) (z) (proofSurj_m) (proofSurj_p) (proofInj_q) (b') (b) (proofEq1))) + (pr₂ (lFlLemmaEight (M) (z) (proofSurj_m) (proofSurj_p) (proofInj_q) (b') (b) (proofEq1))) + ) + (pr₁ (lFlLemmaSix (M) (z) (proofSurj_m) (proofSurj_p) (proofInj_q) (b'))) + (pr₂ (lFlLemmaSix (M) (z) (proofSurj_m) (proofSurj_p) (proofInj_q) (b'))) + ) + +theorem FourLemma : Π {C₁ C₂ : ab_exact_chain_complex}, Π (M : exact_chain_map (C₁) (C₂)), Π (z : ℤ), + surjective_map (group_map(M)(z)) → surjective_map (group_map(M)(z-1-1)) → injective_map (group_map(M)(z-1-1-1)) → + surjective_map (group_map(M)(z-1)) := + λ C₁ C₂ M z proofSurj_m proofSurj_p proofInj_q, + surjective_map.mk (λ b', lFlLemmaNine (M) (z) (proofSurj_m) (proofSurj_p) (proofInj_q) (b'))