| import OAI.Algebra.Finitistic.Ordinary.OrdinaryBlockModules |
| import OAI.Algebra.Finitistic.Ordinary.OrdinaryBalancedCongruence |
|
|
| namespace OAI |
| noncomputable section |
| namespace LittleFinitistic.OrdinarySimulation.FirstBlockTensor |
| universe u |
| variable (k L R O T Z : Type u) [Field k] [Ring L] [Ring R] [Algebra k L] [Algebra k R] |
| variable [AddCommGroup O] [Module k O] [Module R O] [Module Lᵐᵒᵖ O] |
| [IsScalarTower k R O] [IsScalarTower k Lᵐᵒᵖ O] [SMulCommClass R Lᵐᵒᵖ O] |
| variable [AddCommGroup T] [Module k T] [Module L T] [Module Rᵐᵒᵖ T] |
| [IsScalarTower k L T] [IsScalarTower k Rᵐᵒᵖ T] [SMulCommClass L Rᵐᵒᵖ T] |
| variable [AddCommGroup Z] [Module k Z] [Module L Z] [IsScalarTower k L Z] |
|
|
| abbrev source := LeftTensor k (L × R) (L × R) (OffDiagonal O T) (BlockModule.First k L R Z) |
| abbrev target := BlockModule.Second k L R (LeftTensor k L R O Z) |
|
|
| def forward : source k L R O T Z →ₗ[k] target k L R O Z := |
| BalancedTensor.lift ((BalancedTensor.mk k L O Z).compl₁₂ |
| (show OffDiagonal O T →ₗ[k] O from LinearMap.fst k O T) |
| (show BlockModule.First k L R Z →ₗ[k] Z from LinearMap.id)) (by |
| intro d x z |
| exact BalancedTensor.balance (k := k) (A := L) (M := O) (N := Z) d.1 x.1 (show Z from z)) |
|
|
| def inverse : target k L R O Z →ₗ[k] source k L R O T Z := |
| BalancedTensor.lift ((BalancedTensor.mk k (L × R) (OffDiagonal O T) (BlockModule.First k L R Z)).compl₁₂ |
| (show O →ₗ[k] OffDiagonal O T from LinearMap.inl k O T) |
| (show Z →ₗ[k] BlockModule.First k L R Z from LinearMap.id)) (by |
| intro a o z |
| have h := BalancedTensor.balance (k := k) (A := L × R) (M := OffDiagonal O T) |
| (N := BlockModule.First k L R Z) (a,0) (o,0) z |
| change BalancedTensor.mk k (L × R) (OffDiagonal O T) (BlockModule.First k L R Z) |
| (MulOpposite.op a • o,(0:Rᵐᵒᵖ) • (0:T)) (show BlockModule.First k L R Z from z) = |
| BalancedTensor.mk k (L × R) (OffDiagonal O T) (BlockModule.First k L R Z) (o,0) |
| (show BlockModule.First k L R Z from a • z) at h |
| rw [smul_zero] at h |
| exact h) |
|
|
| omit [Algebra k L] [Algebra k R] [Module R O] [IsScalarTower k R O] |
| [IsScalarTower k Lᵐᵒᵖ O] [SMulCommClass R Lᵐᵒᵖ O] [Module L T] |
| [IsScalarTower k L T] [IsScalarTower k Rᵐᵒᵖ T] [SMulCommClass L Rᵐᵒᵖ T] |
| [IsScalarTower k L Z] in |
| theorem other_block_zero (t : T) (z : BlockModule.First k L R Z) : |
| BalancedTensor.mk k (L × R) (OffDiagonal O T) (BlockModule.First k L R Z) (0,t) z = 0 := by |
| have h := BalancedTensor.balance (k := k) (A := L × R) (M := OffDiagonal O T) |
| (N := BlockModule.First k L R Z) (0,1) (0,t) z |
| change BalancedTensor.mk k (L × R) (OffDiagonal O T) (BlockModule.First k L R Z) |
| ((0:Lᵐᵒᵖ) • (0:O),(1:Rᵐᵒᵖ) • t) z = |
| BalancedTensor.mk k (L × R) (OffDiagonal O T) (BlockModule.First k L R Z) (0,t) ((0:L) • (show Z from z)) at h |
| erw [zero_smul, one_smul, zero_smul, map_zero] at h |
| exact h |
|
|
| omit [Module R O] [IsScalarTower k R O] [IsScalarTower k Lᵐᵒᵖ O] |
| [SMulCommClass R Lᵐᵒᵖ O] [Module L T] [IsScalarTower k L T] |
| [IsScalarTower k Rᵐᵒᵖ T] [SMulCommClass L Rᵐᵒᵖ T] [IsScalarTower k L Z] in |
| theorem inverse_forward (x : source k L R O T Z) : |
| inverse k L R O T Z (forward k L R O T Z x) = x := by |
| let F := (inverse k L R O T Z).comp (forward k L R O T Z) |
| have h : F = LinearMap.id := by |
| apply BalancedTensor.ext |
| intro x z |
| change BalancedTensor.mk k (L × R) (OffDiagonal O T) (BlockModule.First k L R Z) (x.1,0) z = |
| BalancedTensor.mk k (L × R) (OffDiagonal O T) (BlockModule.First k L R Z) x z |
| let q := (BalancedTensor.mk k (L × R) (OffDiagonal O T) (BlockModule.First k L R Z)).flip z |
| have hx : x = (x.1,0) + (0,x.2) := Prod.ext (add_zero _).symm (zero_add _).symm |
| exact ((congrArg q hx).trans ((q.map_add _ _).trans |
| ((congrArg (q (x.1,0) + ·) (other_block_zero k L R O T Z x.2 z)).trans (add_zero _)))).symm |
| exact congrArg (fun f => f x) h |
|
|
| omit [Module R O] [IsScalarTower k R O] [IsScalarTower k Lᵐᵒᵖ O] |
| [SMulCommClass R Lᵐᵒᵖ O] [Module L T] [IsScalarTower k L T] |
| [IsScalarTower k Rᵐᵒᵖ T] [SMulCommClass L Rᵐᵒᵖ T] [IsScalarTower k L Z] in |
| theorem forward_inverse (x : target k L R O Z) : |
| forward k L R O T Z (inverse k L R O T Z x) = x := by |
| have h : (forward k L R O T Z).comp (inverse k L R O T Z) = LinearMap.id := by |
| apply BalancedTensor.ext |
| intro o z |
| rfl |
| exact congrArg (fun f => f x) h |
|
|
| /-- Tensor on the first block selects O and has its actual output on the second block. -/ |
| def equivalence : source k L R O T Z ≃ₗ[L × R] target k L R O Z where |
| toFun := forward k L R O T Z |
| invFun := inverse k L R O T Z |
| left_inv := inverse_forward k L R O T Z |
| right_inv := forward_inverse k L R O T Z |
| map_add' := (forward k L R O T Z).map_add |
| map_smul' d x := by |
| let F := forward k L R O T Z |
| have h : F.comp (LeftTensor.action k (L × R) (L × R) (OffDiagonal O T) |
| (BlockModule.First k L R Z) d) = |
| (Algebra.lsmul k k (target k L R O Z) d).comp F := by |
| apply BalancedTensor.ext |
| intro x z |
| rfl |
| exact congrArg (fun f => f x) h |
|
|
| end LittleFinitistic.OrdinarySimulation.FirstBlockTensor |
|
|
| end |
|
|
| end OAI |
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