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216fb31 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 | import OAI.Algebra.Finitistic.Bar.LiteralMultibarCoordinates
namespace OAI
noncomputable section
namespace DerivedBar.RecursiveTotal
open DerivedBar.Multibar
/-- Remove the zero input degree from the list of bar lengths. -/
def moduleBlocks : (r : ℕ) → {n : ℕ} → (t : Index (r+1) n) → Blocks r
| 0, _, ⟨p,_⟩ => p.val.1
| r+1, _, ⟨p,t⟩ => (p.val.1, moduleBlocks r t)
theorem total_moduleBlocks (r n : ℕ) (t : Index (r+1) n) :
total r (moduleBlocks r t) + inputDegree (r+1) t = n := by
induction r generalizing n with
| zero => rcases t with ⟨⟨⟨i,q⟩,h⟩,t⟩; exact h
| succ r ih =>
rcases t with ⟨⟨⟨i,q⟩,h⟩,t⟩
have ht := ih q t
change i + total r (moduleBlocks r t) + inputDegree (r+1) t = n
omega
theorem pattern_moduleBlocks (r n : ℕ) (t : Index (r+1) n) :
pattern r (moduleBlocks r t) = leadingPattern r n t := by
induction r generalizing n with
| zero =>
rcases t with ⟨⟨⟨i,q⟩,h⟩,t⟩
simp [moduleBlocks, pattern, leadingPattern, innerPattern]
| succ r ih =>
rcases t with ⟨⟨⟨i,q⟩,h⟩,t⟩
change List.replicate i false ++ true :: pattern r (moduleBlocks r t) = _
rw [ih q t]
rfl
/-- The recursive total index attached to a block tuple, with input degree zero. -/
def moduleIndex : (r : ℕ) → (b : Blocks r) → Index (r+1) (total r b)
| 0, i => ⟨⟨⟨i,0⟩,by exact Nat.add_zero i⟩,()⟩
| r+1, (i,b) => ⟨⟨⟨i,total r b⟩,rfl⟩,moduleIndex r b⟩
@[simp] theorem inputDegree_moduleIndex (r : ℕ) (b : Blocks r) :
inputDegree (r+1) (moduleIndex r b) = 0 := by
induction r with
| zero => rfl
| succ r ih => rcases b with ⟨i,b⟩; exact ih b
@[simp] theorem moduleBlocks_moduleIndex (r : ℕ) (b : Blocks r) :
moduleBlocks r (moduleIndex r b) = b := by
induction r with
| zero => rfl
| succ r ih => rcases b with ⟨i,b⟩; change (i,moduleBlocks r (moduleIndex r b)) = _; rw [ih]
@[simp] theorem inputDegree_castIndex {r n m : ℕ} (h : n=m) (t : Index r n) :
inputDegree r (castIndex h t) = inputDegree r t := by subst m; rfl
@[simp] theorem moduleBlocks_castIndex {r n m : ℕ} (h : n=m) (t : Index (r+1) n) :
moduleBlocks r (castIndex h t) = moduleBlocks r t := by subst m; rfl
/-- All indices are recovered from the blocks and input degree; no quotient of
block tensors or identification of distinct patterns is introduced. -/
theorem moduleBlocks_injective (r n : ℕ) (t s : Index (r+1) n)
(ht : inputDegree (r+1) t=0) (hs : inputDegree (r+1) s=0)
(h : moduleBlocks r t=moduleBlocks r s) : t=s := by
induction r generalizing n with
| zero =>
rcases t with ⟨⟨⟨i,q⟩,hq⟩,t⟩
rcases s with ⟨⟨⟨j,p⟩,hp⟩,s⟩
change q=0 at ht
change p=0 at hs
subst q p
have hij : i=j := by omega
subst j
cases t; cases s; rfl
| succ r ih =>
rcases t with ⟨⟨⟨i,q⟩,hq⟩,t⟩
rcases s with ⟨⟨⟨j,p⟩,hp⟩,s⟩
change (i,moduleBlocks r t) = (j,moduleBlocks r s) at h
have hij := congrArg Prod.fst h
dsimp at hij
subst j
have hqp : q=p := by omega
subst p
have hts := ih q t s ht hs (congrArg Prod.snd h)
subst s
rfl
/-- Exact module-input indexing by the manuscript's tuples of r+1 block lengths. -/
def moduleBlockEquiv (r n : ℕ) :
{t : Index (r+1) n // inputDegree (r+1) t=0} ≃
{b : Blocks r // total r b=n} where
toFun t := ⟨moduleBlocks r t.val, by
have h := total_moduleBlocks r n t.val
rw [t.property, Nat.add_zero] at h
exact h⟩
invFun b := ⟨castIndex b.property (moduleIndex r b.val), by simp⟩
left_inv t := by
apply Subtype.ext
apply moduleBlocks_injective r n _ _ (by simp) t.property
simp
right_inv b := by apply Subtype.ext; simp
/-- Exact full-degree shape index, including the shift by the number of X slots. -/
def moduleShapeEquiv (r n : ℕ) :
{t : Index (r+1) n // inputDegree (r+1) t=0} ≃
Multibar.Indexed.Shape (r+n) r :=
(moduleBlockEquiv r n).trans (Multibar.Indexed.blockDegreeEquiv r n)
end DerivedBar.RecursiveTotal
end
end OAI
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