62 lines
2.7 KiB
Text
62 lines
2.7 KiB
Text
-- Authors: Floris van Doorn
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import algebra.ring .direct_sum ..heq ..move_to_lib
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open algebra group eq is_trunc sigma
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namespace algebra
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definition AddAbGroup_of_Ring [constructor] (R : Ring) : AddAbGroup :=
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AddAbGroup.mk R (add_ab_group_of_ring R)
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definition AddGroup_of_Ring [constructor] (R : Ring) : AddGroup :=
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AddGroup.mk R (add_group_of_add_ab_group R)
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definition ring_AddAbGroup_of_Ring [instance] (R : Ring) : ring (AddAbGroup_of_Ring R) :=
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Ring.struct R
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/- we give the following instance very high priority, otherwise type class inference would sometimes find the additive structure of R as the group structure. -/
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definition monoid_AddAbGroup_of_Ring [instance] [priority 3000] [constructor] (R : Ring) :
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monoid (Group_of_AbGroup (AddAbGroup_of_Ring R)) :=
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@monoid_of_ring _ (Ring.struct R)
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definition ring_right_action [constructor] {R : Ring} (r : R) :
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AddAbGroup_of_Ring R →a AddAbGroup_of_Ring R :=
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homomorphism.mk (λs, s * r) (λs s', !right_distrib)
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definition ring_of_ab_group [constructor] (G : Type) [ab_group G] (m : G → G → G) (o : G)
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(lm : Πg, m o g = g) (rm : Πg, m g o = g) (am : Πg₁ g₂ g₃, m (m g₁ g₂) g₃ = m g₁ (m g₂ g₃))
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(ld : Πg₁ g₂ g₃, m g₁ (g₂ * g₃) = m g₁ g₂ * m g₁ g₃)
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(rd : Πg₁ g₂ g₃, m (g₁ * g₂) g₃ = m g₁ g₃ * m g₂ g₃) : ring G :=
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ring.mk _ mul mul.assoc 1 one_mul mul_one inv mul.left_inv mul.comm m am o lm rm ld rd
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definition Ring_of_AbGroup [constructor] (G : AbGroup) (m : G → G → G) (o : G)
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(lm : Πg, m o g = g) (rm : Πg, m g o = g) (am : Πg₁ g₂ g₃, m (m g₁ g₂) g₃ = m g₁ (m g₂ g₃))
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(ld : Πg₁ g₂ g₃, m g₁ (g₂ * g₃) = m g₁ g₂ * m g₁ g₃)
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(rd : Πg₁ g₂ g₃, m (g₁ * g₂) g₃ = m g₁ g₃ * m g₂ g₃) : Ring :=
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Ring.mk G (ring_of_ab_group G m o lm rm am ld rd)
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/- graded ring -/
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structure graded_ring (M : Monoid) :=
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(R : M → AddAbGroup)
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(mul : Π⦃m m'⦄, R m → R m' → R (m * m'))
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(one : R 1)
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(mul_one : Π⦃m⦄ (r : R m), mul r one ==[R] r)
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(one_mul : Π⦃m⦄ (r : R m), mul one r ==[R] r)
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(mul_assoc : Π⦃m₁ m₂ m₃⦄ (r₁ : R m₁) (r₂ : R m₂) (r₃ : R m₃),
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mul (mul r₁ r₂) r₃ ==[R] mul r₁ (mul r₂ r₃))
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(mul_left_distrib : Π⦃m₁ m₂⦄ (r₁ : R m₁) (r₂ r₂' : R m₂),
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mul r₁ (r₂ + r₂') = mul r₁ r₂ + mul r₁ r₂')
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(mul_right_distrib : Π⦃m₁ m₂⦄ (r₁ r₁' : R m₁) (r₂ : R m₂),
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mul (r₁ + r₁') r₂ = mul r₁ r₂ + mul r₁' r₂)
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attribute graded_ring.R [coercion]
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infixl ` ** `:71 := graded_ring.mul
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-- definition ring_direct_sum {M : Monoid} (R : graded_ring M) : Ring :=
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-- Ring_of_AbGroup (dirsum R) _ (dirsum_incl R 1 (graded_ring.one R)) _ _ _ _ _
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end algebra
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