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f205175 | 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 | import Mathlib
namespace OAI
section
noncomputable section
open Set Function Module
open scoped Classical BigOperators
namespace CriticalPercolation.LinearGrowth
variable {K L T i J : Type*} [Field K] [Field L] [Field T] [Fintype i]
lemma pi_span_map_top (f : K →+* T) (v : J → i → K)
(hv : Submodule.span K (Set.range v)=⊤) :
Submodule.span T (Set.range (fun j k => f (v j k)))=⊤ := by
let P := Submodule.span T (Set.range (fun j k => f (v j k)))
have hm (x : i → K) : (fun k => f (x k))∈P := by
have hx : x∈Submodule.span K (Set.range v) := hv ▸ Submodule.mem_top
induction hx using Submodule.span_induction with
| mem x hx => obtain ⟨j,rfl⟩ := hx; exact Submodule.subset_span ⟨j,rfl⟩
| zero => simpa only [Pi.zero_def,Pi.zero_apply,map_zero] using P.zero_mem
| add x y hx hy ihx ihy => simpa only [Pi.add_def,Pi.add_apply,map_add] using P.add_mem ihx ihy
| smul c x hx ih => simpa only [Pi.smul_def,Pi.smul_apply,smul_eq_mul,map_mul] using P.smul_mem (f c) ih
apply top_unique
rw [←(Pi.basisFun T i).span_eq]
apply Submodule.span_le.mpr
rintro x ⟨k,rfl⟩
have he : (Pi.basisFun T i) k=(fun l => f ((Pi.single k (1:K) : i → K) l)) := by
ext l
simp [Pi.basisFun_apply,Pi.single_apply]
rw [he]
exact hm (Pi.single k 1)
lemma pi_span_descends (f : L →+* T) (w : J → i → L)
(hw : Submodule.span T (Set.range (fun j k => f (w j k)))=⊤) :
Submodule.span L (Set.range w)=⊤ := by
by_contra hn
obtain ⟨φ,hφ,hker⟩ := (Submodule.span L (Set.range w)).exists_le_ker_of_lt_top (lt_top_iff_ne_top.mpr hn)
let c (k : i) := φ (Pi.single k 1)
have hφsum (x : i → L) : φ x=∑ k, x k*c k := by
conv_lhs => rw [←(Pi.basisFun L i).sum_repr x]
simp [c,Pi.basisFun_apply,Pi.basisFun_repr, map_sum,map_smul,smul_eq_mul]
let ψ : (i → T) →ₗ[T] T := {
toFun x := ∑ k, x k*f (c k)
map_add' x y := by simp [add_mul,Finset.sum_add_distrib]
map_smul' a x := by simp [smul_eq_mul,Finset.mul_sum,mul_assoc] }
have hker' : Submodule.span T (Set.range (fun j k => f (w j k)))≤ψ.ker := by
apply Submodule.span_le.mpr
rintro x ⟨j,rfl⟩
change (∑ k, f (w j k)*f (c k))=0
simp_rw [←map_mul]
rw [←map_sum,←hφsum]
have he : φ (w j)=0 := hker (Submodule.subset_span ⟨j,rfl⟩)
rw [he,map_zero]
have hψ : ψ=0 := LinearMap.ker_eq_top.mp (top_unique (hw ▸ hker'))
apply hφ
apply LinearMap.ext
intro x
change φ x=0
rw [hφsum]
apply Finset.sum_eq_zero
intro k hk
have he := congrArg (fun a : (i → T) →ₗ[T] T => a (Pi.single k 1)) hψ
have hc : c k=0 := f.injective (by simpa [ψ,Pi.single_apply] using he)
rw [hc,mul_zero]
lemma pi_span_transfer (f : K →+* T) (e : L →+* T)
(v : J → i → K) (w : J → i → L)
(hm : ∀ j k, f (v j k)=e (w j k))
(hv : Submodule.span K (Set.range v)=⊤) : Submodule.span L (Set.range w)=⊤ := by
apply pi_span_descends e w
simpa only [hm] using pi_span_map_top f v hv
end CriticalPercolation.LinearGrowth
end
end
end OAI
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