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Lemma2-4.agda
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Lemma2-4.agda
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open import Relation.Nullary.Core
open import Relation.Nullary.Decidable
open import Relation.Binary.Core
open import Relation.Binary.PropositionalEquality as
PropEq hiding (proof-irrelevance)
open import Data.Bool hiding (_≟_)
open import Data.Bool.Properties
open import Data.Empty
open import Data.Unit hiding (_≤?_; _≤_; _≟_)
open import Data.Product
open import Data.Nat renaming (_≟_ to _ℕ≟_)
--open Data.Nat.≤-Reasoning
open import Data.Nat.Properties
open SemiringSolver
open import Function hiding (_∘_)
open import Function.Equality hiding (cong; _∘_)
open import Function.Inverse hiding (sym)
open import Function.LeftInverse hiding (_∘_)
open import Misc
open import FinBijections
import Data.Fin as Fin
import Data.Fin.Props as Fin
module Lemma2-4 where
open import Lemma2-4-core public
neck⋆ : ∀ (e f : P) → {≥1 : True (1 ≤? ρ e f)} → Neck e
neck⋆ e f = ppneck (sppc e f)
neck-value : ∀ {e} → Neck e → Fin.Fin (1 + t) × Fin.Fin (1 + s)
neck-value {e} (x , y) = (Inverse.to (GP-L e) ⟨$⟩ x) ,
(Inverse.to (GP-P (el x)) ⟨$⟩ y)
neck-at : ∀ {e} → Fin.Fin (1 + t) × Fin.Fin (1 + s) → Neck e
neck-at {e} ( x , y ) = e₁ , e₂
where e₁ = Inverse.from (GP-L e) ⟨$⟩ x
e₂ = Inverse.from (GP-P (el e₁)) ⟨$⟩ y
neck-value-id₀ : ∀ {e} (x : Neck e) → neck-at ( neck-value x) ≡ x
neck-value-id₀ {e} ((e₁ ∶ e#e₁) , (e₂ ∶ e₁#e₂))
with neck-value ((e₁ ∶ e#e₁) , (e₂ ∶ e₁#e₂))
... | z rewrite Inverse.left-inverse-of (GP-L e) (e₁ ∶ e#e₁) |
Inverse.left-inverse-of (GP-P e₁) (e₂ ∶ e₁#e₂) = refl
neck-value-id₁ : ∀ {e} (x : Fin.Fin (1 + t) × Fin.Fin (1 + s)) →
neck-value ( neck-at {e} x) ≡ x
neck-value-id₁ {e} (x , y) with
neck-at {e} (x , y) | inspect (neck-at {e}) (x , y)
neck-value-id₁ {e} (x , y) | (._ ∶ e#e₁) , (._ ∶ e₁#e₂) | [ refl ]
rewrite Inverse.right-inverse-of (GP-L e) x |
Inverse.right-inverse-of (GP-P
(el (Inverse.from (GP-L e) ⟨$⟩ x))) y = refl
neck-value-bij : ∀ {e} → Inverse (PropEq.setoid (Neck e)) (PropEq.setoid (Fin.Fin (1 + t) × Fin.Fin (1 + s)))
neck-value-bij {e} = record {
to = record { _⟨$⟩_ = neck-value ; cong = cong neck-value } ;
from = record { _⟨$⟩_ = neck-at ; cong = cong neck-at } ;
inverse-of = record { left-inverse-of = neck-value-id₀ ;
right-inverse-of = neck-value-id₁ } }
-- In the below two lemmas we prove that to ensure λ < n
-- we need to ensure ρ < n / 2
module 0<ρ<n/2 {e f : P} {≥1 : True (1 ≤? ρ e f)}
{<n : True (suc (ρ e f) ≤? ⌈ n /2⌉)} where
-- e₂ ≡ e₂⋆
class-A-ρ : (nck : Neck e) → (neck-e₂ nck) ≡ (neck-e₂ (neck⋆ e f {≥1})) →
ρ (neck-e₂ nck) f ≡ pred (ρ e f)
class-A-ρ (proj₁ , ._ ∶ proof) refl = trans
(PropEq.sym (tailpp-ρ-shortest refl))
(lem-tailpp-ρ {ppc = sppc e f})
-- e₂ ≢ e₂⋆ , e₂ ≢ e , e₁ ≡ e₁⋆
class-B-ρ : (nck : Neck e) → el (proj₁ nck) ≡ el (proj₁ (neck⋆ e f {≥1})) →
(neck-e₂ nck) ≢ (neck-e₂ (neck⋆ e f {≥1})) →
neck-e₂ nck ≢ e → ρ (neck-e₂ nck) f ≡ ρ e f
class-B-ρ ((._ ∶ e#e₁) , (e₂ ∶ e₁#e₂)) refl e₂≢e₂⋆ c =
≤-≥⇒≡ is≤ (pred-mono (≰⇒> is≮))
where open Data.Nat.≤-Reasoning
e₂⋆ = (neck-e₂ (neck⋆ e f))
e₁⋆ = (neck-e₁ (neck⋆ e f))
is≤ : ρ e₂ f ≤ ρ e f
is≤ = begin
ρ e₂ f
≤⟨ sppc-ρ-shorter-than (((e₁⋆ ∶ (#sym e₁#e₂)) ,
proj₂ (neck⋆ e f)) ∷ sppc e₂⋆ f) ⟩
suc (ρ e₂⋆ f)
≡⟨ cong suc (class-A-ρ (neck⋆ e f) refl) ⟩
suc (pred (ρ e f))
≡⟨ suc∘pred ⟩
(ρ e f ∎)
is≮ : ρ e₂ f < ρ e f → ⊥
is≮ is< = ⊥-elim (e₂≢e₂⋆ (cong (λ x → neck-e₂ (ppneck (el x)
{lem-ρ'≥ρ ≥1})) z))
where ppc : ppchain e f
ppc = ((e₁⋆ ∶ e#e₁) , (e₂ ∶ e₁#e₂)) ∷ (sppc e₂ f)
<n/2 : ρ' ppc < ⌈ n /2⌉
<n/2 = begin
suc (suc (ρ e₂ f)) ≤⟨ s≤s is< ⟩
suc (ρ e f) ≤⟨ toWitness <n ⟩ (⌈ n /2⌉ ∎)
ppc-shortest : (ppc as-c) is-shortest
ppc-shortest rewrite lem-id₀ (sc (pt e₂) (pt f)) |
lem-2xρ-lambda {e₂} {f} |
lem-2xρ-lambda {e} {f} |
solve 1 (λ x →
(con 2 :+ x :+ (x :+ con 0)
:= con 1 :+ x :+
(con 1 :+ x :+
con 0)))
refl (ρ e₂ f)
= ≤-≥⇒≡ (m≤m {2} *-mono is<)
(m≤m {2} *-mono (sppc-ρ-shorter-than ppc))
ppc-irred : irred (ppc as-c)
ppc-irred = λ {a} {b} x →
shortest-irred {pt e} {pt f} (ppc as-c)
ppc-shortest {a} {b} x
z = A₂-ρ (ppc ∶ (<n/2 , ppc-irred ))
(sppc e f ∶ (toWitness <n , sppc-irred))
class-C₀-ρ : (nck : Neck e) → .(neck-e₁ nck ≢ neck-e₁ (neck⋆ e f)) →
neck-e₂ nck ≢ e → (ρ e f) < ⌈ (pred (n)) /2⌉ →
ρ (neck-e₂ nck) f ≡ 1 + ρ e f
class-C₀-ρ ((e₁ ∶ e#e₁) , (e₂ ∶ e₁#e₂)) neck≢ e₂≢e <n/2 =
≤-≥⇒≡ is≤ (≰⇒> is≮)
where open Data.Nat.≤-Reasoning
.e₁#e : ln e₁ # pt e
e₁#e = #sym e#e₁
.e₂#e₁ : pt e₂ # ln e₁
e₂#e₁ = #sym e₁#e₂
is≤ : ρ e₂ f ≤ 1 + ρ e f
is≤ = sppc-ρ-shorter-than (((e₁ ∶ (#sym e₁#e₂)) ,
(e ∶ (#sym e#e₁))) ∷ sppc e f)
e₁≢e₁⋆ : ∀ {x y} {ρxy≥1 : True (1 ≤? ρ x y)} →
(neck≢ : e₁ ≢ neck-e₁ (neck⋆ x y)) →
(ln e₁) ≢ ln (neck-e₁ (neck⋆ x y))
e₁≢e₁⋆ neck≢ = λ x → neck≢ (ln-inj x)
¬e₁#e₁⋆ : ∀ {x y} {ρxy≥1 : True (1 ≤? ρ x y)} →
(neck≢ : e₁ ≢ neck-e₁ (neck⋆ x y)) →
(ln e₁) # ln (neck-e₁ (neck⋆ x y)) → ⊥
¬e₁#e₁⋆ {x} {y} {ρxy≥1} neck≢ = λ x →
IP-pt (((e₁≢e₁⋆ {ρxy≥1 = ρxy≥1}) neck≢)) x
¬e₁#neck : ∀ {x y} {ρxy≥1 : True (1 ≤? ρ x y)} →
(neck≢ : e₁ ≢ neck-e₁ (neck⋆ x y {ρxy≥1})) →
(ln e₁) # neck (sc (pt x) (pt y)) → ⊥
¬e₁#neck {a} {b} {ρxy≥1} neck≢ = λ x →
((¬e₁#e₁⋆ {ρxy≥1 = ρxy≥1}) (neck≢))
(subst (_#_ (ln e₁))
(PropEq.sym (lem-neckl (sppc a b)))
(subst (λ x₁ → ln e₁ # neck x₁)
(PropEq.sym (lem-id₀ (sc (pt a) (pt b)))) x))
c₂ : chain (ln e₁) (pt f)
c₂ = _∷_ (ln e₁) {{(λ ())}} (sc (pt e) (pt f))
c₂-len : len c₂ < n
c₂-len = begin
2 + lambda (pt e) (pt f)
≡⟨ cong (_+_ 2) lem-2xρ-lambda ⟩
2 + 2 * ρ e f ≡⟨
solve 1 (λ y → con 2 :+ con 2 :* y :=
con 2 :* (con 1 :+ y)) refl
(ρ e f)
⟩
2 * (1 + ρ e f) ≤⟨ m≤m {2} *-mono <n/2 ⟩
2 * ⌈ (pred (n)) /2⌉ ≤⟨ lem-2x⌈n/2⌉ ⟩ (n ∎)
.c₂-irred : irred c₂
c₂-irred = irred-∷ (ln e₁) (sc (pt e) (pt f) )
{fromWitness
(begin 2 ≤⟨ m≤m {2} *-mono toWitness ≥1 ⟩
_ ≡⟨ PropEq.sym lem-2xρ-lambda ⟩
(lambda (pt e) (pt f) ∎))}
(¬e₁#neck {ρxy≥1 = ≥1} (neck≢))
(shortest-irred (sc (pt e) (pt f)) refl)
ρ<⇒λ< : ∀ {x y x' y'} → ρ x y ≤ ρ x' y' →
lambda (pt x) (pt y) ≤ lambda (pt x') (pt y')
ρ<⇒λ< {x} {y} {x'} {y'} ρ<
rewrite lem-2xρ-lambda {x} {y} |
lem-2xρ-lambda {x'} {y'} = m≤m {2} *-mono ρ<
is≮ : ρ e₂ f ≤ ρ e f → ⊥
is≮ is< with (neck (sc (pt e₂) (pt f))) ≟ (ln e₁)
is≮ is< | yes p = helper (proj₁ (proj₂ c)) (cong len (cong el z))
where c : Σ (chain (ln e₁) (pt f)) (λ t → (len t < len c₂) ×
irred t)
c rewrite (PropEq.sym p) = (tail (sc (pt e₂) (pt f))) ,
s≤s (begin
len (tail (sc (pt e₂) (pt f)))
≡⟨ (lem-tail-len {c = sc (pt e₂) (pt f)}) ⟩
pred (lambda (pt e₂) (pt f))
≤⟨ ≤⇒pred≤ _ _ m≤m ⟩
lambda (pt e₂) (pt f)
≤⟨ ρ<⇒λ< is< ⟩
(len (sc (pt e) (pt f)) ∎)) ,
(shortest-irred (tail (sc (pt e₂) (pt f)))
(tail-shortest refl))
z = A₂ ((proj₁ c) ∶
((<-trans (proj₁ (proj₂ c)) c₂-len) ,
proj₂ (proj₂ c))) (c₂ ∶ (c₂-len ,
c₂-irred))
helper : ∀ {x y} → x < y → x ≡ y → ⊥
helper {zero} () refl
helper {suc x} q refl = helper (pred-mono q) refl
is≮ is< | no ¬p with 1 ≤? ρ e₂ f
is≮ is< | no ¬p | yes p = e₂≢e (pt-inj (cong
(λ x → neck (el x)) z))
where c : Σ' (chain (ln e₁) (pt f)) (λ t → (len t < n) × irred t)
c = _∷_ (ln e₁) {{(λ ())}} (sc (pt e₂) (pt f)) ∶
((begin
2 + lambda (pt e₂) (pt f) ≤⟨ s≤s (s≤s (ρ<⇒λ< is<)) ⟩
2 + lambda (pt e) (pt f) ≤⟨ c₂-len ⟩ (n ∎)) ,
irred-∷ (ln e₁) (sc (pt e₂) (pt f))
{fromWitness
(begin 2 ≤⟨ m≤m {2} *-mono p ⟩
_ ≡⟨ PropEq.sym lem-2xρ-lambda ⟩
(lambda (pt e₂) (pt f) ∎))}
(λ x → ¬e₁#neck {ρxy≥1 = fromWitness p}
(λ x₁ → ¬p (trans
(trans (cong neck (PropEq.sym
(lem-id₀ (sc (pt e₂) (pt f)))))
(PropEq.sym (lem-neckl (sppc e₂ f))))
(cong ln (PropEq.sym x₁))))
x)
(shortest-irred (sc (pt e₂) (pt f)) refl))
z = A₂ c (c₂ ∶ (c₂-len , c₂-irred))
is≮ is< | no ¬p₁ | no ¬p rewrite PropEq.sym (ρ≡0⇒e≡f {e₂} {f}
(<1⇒≡0 (≰⇒> ¬p))) =
helper {1} {len (proj₁ c)}
(begin
2 ≤⟨ m≤m {2} *-mono toWitness ≥1 ⟩
2 * ρ e f
≡⟨ PropEq.sym lem-2xρ-lambda ⟩
lambda (pt e) (pt f)
≤⟨ ≤′⇒≤ (≤′-step ≤′-refl) ⟩
suc _
≡⟨ PropEq.sym (cong len (cong el z)) ⟩
(len (proj₁ c) ∎))
(PropEq.sym (proj₁ (proj₂ c)))
where c : Σ (chain (ln e₁) (pt f)) (λ t → (len t ≡ 1) × irred t)
c rewrite PropEq.sym (ρ≡0⇒e≡f {e₂} {f}
(<1⇒≡0 (≰⇒> ¬p))) = _∷_ (ln e₁)
{{(λ ())}}
[ pt e₂ ] , refl , tt
z : _
z = (A₂ ((proj₁ c) ∶ (((begin
suc (len (proj₁ c))
≡⟨ cong suc (proj₁ (proj₂ c)) ⟩
2 ≤⟨ s≤s (s≤s z≤n) ⟩ (n ∎)) ,
proj₂ (proj₂ c)))) (c₂ ∶ (c₂-len , c₂-irred)))
helper : ∀ {x y} → x < y → x ≡ y → ⊥
helper {zero} () refl
helper {suc x} q refl = helper (pred-mono q) refl
class-C₁-ρ : (nck : Neck e) → neck-e₁ nck ≢ neck-e₁ (neck⋆ e f) →
neck-e₂ nck ≢ neck-e₂ (neck⋆ e f) →
neck-e₂ nck ≢ e → (ρ e f) ≡ ⌈ (pred (n)) /2⌉ →
ρ (neck-e₂ nck) f ≡ ρ e f
class-C₁-ρ ((e₁ ∶ e#e₁) , (e₂ ∶ e₁#e₂)) neck≢ e₂≢e₂⋆ e₂≢e ≡n/2 =
≤-≥⇒≡ is≤ (pred-mono (≰⇒> is≮))
where open Data.Nat.≤-Reasoning
e₂⋆ = (neck-e₂ (neck⋆ e f))
e₁⋆ = (neck-e₁ (neck⋆ e f))
is≤ : ρ e₂ f ≤ ρ e f
is≤ = begin
ρ e₂ f
≤⟨ A₁'-ρ ⟩
⌊ n /2⌋
≡⟨ PropEq.sym ≡n/2 ⟩
(ρ e f ∎)
is≮ : ρ e₂ f < ρ e f → ⊥
is≮ is< = ⊥-elim (e₂≢e₂⋆ (cong (λ x → neck-e₂ (ppneck (el x)
{lem-ρ'≥ρ ≥1})) z))
where ppc : ppchain e f
ppc = ((e₁ ∶ e#e₁) , (e₂ ∶ e₁#e₂)) ∷ sppc e₂ f
<n/2 : ρ' ppc < ⌈ n /2⌉
<n/2 = begin
suc (suc (ρ e₂ f)) ≤⟨ s≤s is< ⟩
suc (ρ e f) ≤⟨ toWitness <n ⟩ (⌈ n /2⌉ ∎)
ppc-shortest : (ppc as-c) is-shortest
ppc-shortest rewrite lem-id₀ (sc (pt e₂) (pt f)) |
lem-2xρ-lambda {e₂} {f} |
lem-2xρ-lambda {e} {f} |
solve 1 (λ x →
(con 2 :+ x :+ (x :+ con 0)
:= con 1 :+ x :+
(con 1 :+ x :+ con 0)))
refl (ρ e₂ f)
= ≤-≥⇒≡ (m≤m {2} *-mono is<)
(m≤m {2} *-mono
(sppc-ρ-shorter-than ppc))
ppc-irred : irred (ppc as-c)
ppc-irred = λ {a} {b} x → shortest-irred {pt e} {pt f}
(ppc as-c) ppc-shortest {a} {b} x
z = A₂-ρ (ppc ∶ (<n/2 , ppc-irred ))
(sppc e f ∶ (toWitness <n , sppc-irred))
module ρ≡n/2 {e f : P} {n-even : Even (n)}
{≡n/2 : True ( ρ e f ℕ≟ ⌈ (n) /2⌉ )} where
open import Lemma2-1
private
λ≡n : lambda (pt e) (pt f) ≡ n
λ≡n = PropEq.sym (
begin
n
≡⟨ PropEq.sym (lem-2x⌈n/2⌉-even n-even) ⟩
2 * ⌈ n /2⌉
≡⟨ cong (_*_ 2) (sym $ toWitness ≡n/2) ⟩
2 * ρ e f
≡⟨ PropEq.sym lem-2xρ-lambda ⟩
(lambda (pt e) (pt f) ∎))
where open ≡-Reasoning
chain-with-e₁ : (l#e : L# e) → chain (pt e) (pt f)
chain-with-e₁ l#e = proj₁ $ Inverse.from (I e (pt f) λ≡n) ⟨$⟩ l#e
lem-chain-with-e₁ : ∀ {l#e} → neck (chain-with-e₁ l#e) ≡ ln (el l#e)
lem-chain-with-e₁ {l#e} = refl
len-cwc≡n : (l#e : L# e) → len (chain-with-e₁ l#e) ≡ n
len-cwc≡n l#e = proj₂ $ Inverse.from (I e (pt f) λ≡n) ⟨$⟩ l#e
ppc-with-e₁ : (l#e : L# e) → ppchain e f
ppc-with-e₁ l#e = chain-with-e₁ l#e as-ppc
ρef≥1 : ρ e f ≥ 1
ρef≥1 = begin 1 ≤⟨ s≤s z≤n ⟩ ⌈ n /2⌉ ≡⟨ sym $ toWitness ≡n/2 ⟩ (ρ e f ∎)
where open Data.Nat.≤-Reasoning
ρ'ppc≡n : (l#e : L# e) → ρ' (ppc-with-e₁ l#e) ≡ ⌈ (n) /2⌉
ρ'ppc≡n l#e rewrite PropEq.sym (lem-lambda/2-ρ {e} {f}) |
PropEq.sym (lem-len/2-ρ {ppc = ppc-with-e₁ l#e}) |
lem-id₀ (sc (pt e) (pt f)) |
lem-id₀ (chain-with-e₁ l#e) =
trans (cong ⌊_/2⌋ lenc≡n) (lem-even⇒⌊≡⌋ n-even)
where lenc≡n = proj₂ $ Inverse.from (I e (pt f) λ≡n) ⟨$⟩ l#e
ρ'ppc≥1 : (l#e : L# e) → ρ' (ppc-with-e₁ l#e) ≥ 1
ρ'ppc≥1 l#e = begin 1 ≤⟨ s≤s z≤n ⟩ ⌈ (n) /2⌉ ≡⟨ PropEq.sym (ρ'ppc≡n l#e)
⟩ (ρ' (ppc-with-e₁ l#e) ∎)
where open Data.Nat.≤-Reasoning
ppc-shortest : (l#e : L# e) → (ppc-with-e₁ l#e) is-ρ-shortest
ppc-shortest nck = trans (ρ'ppc≡n nck) (sym $ toWitness ≡n/2)
lem-e₂⋆# : ∀ {l#e} → (ln $ el $ proj₁ (ppneck (ppc-with-e₁ l#e)
{fromWitness (ρ'ppc≥1 l#e)})) ≡
(ln $ el $ l#e)
lem-e₂⋆# {l#e} = subst (λ z → ln (neck-e₁ (ppneck (ppc-with-e₁ l#e)
{fromWitness (ρ'ppc≥1 l#e)})) ≡ neck z)
(lem-id₀ (chain-with-e₁ l#e))
(lem-neckl (ppc-with-e₁ l#e))
e₂⋆# : (l#e : L# e) → P# (el l#e)
e₂⋆# l#e = (neck-e₂ $ ppneck (ppc-with-e₁ l#e)
{fromWitness (ρ'ppc≥1 l#e)}) ∶
subst (λ z →
(z # pt (neck-e₂ $ ppneck (ppc-with-e₁ l#e)
{fromWitness (ρ'ppc≥1 l#e)})))
(lem-e₂⋆# {l#e})
(neck-e₁#e₂ $ ppneck (ppc-with-e₁ l#e))
e₂⋆ : (l#e : L# e) → P
e₂⋆ l#e = el $ e₂⋆# l#e
class-A₀-ρ : (nck : Neck e) → (neck-e₂ nck) ≡ (e₂⋆ (proj₁ nck)) →
ρ (neck-e₂ nck) f ≡ pred ⌈ (n) /2⌉
class-A₀-ρ nck eq = begin
ρ e₂ f
≡⟨ cong ((Function.flip ρ) f) eq ⟩
ρ (e₂⋆ (proj₁ nck)) f
≡⟨ PropEq.sym (tailpp-ρ-shortest (ppc-shortest
(proj₁ nck))) ⟩
ρ' (tailpp (ppc-with-e₁ (proj₁ nck)))
≡⟨ lem-tailpp-ρ {ppc = ppc-with-e₁ (proj₁ nck)} ⟩
pred (ρ' (ppc-with-e₁ (proj₁ nck)))
≡⟨ cong pred (ρ'ppc≡n (proj₁ nck)) ⟩
(pred ⌈ n /2⌉ ∎)
where open ≡-Reasoning
e₂ = neck-e₂ nck
lem-e₂⋆ : ∀ {l#e} → e₂⋆ l#e ≢ e
lem-e₂⋆ {l#e} eq = ¬<-≡ ρef≥1 (sym (≡pred⇒zero (trans (class-A₀-ρ
(l#e , e ∶ (#sym $ pf l#e)) (sym eq))
(cong pred $ (sym $ toWitness ≡n/2)))))
class-A₁-ρ : (nck : Neck e) → (neck-e₂ nck) ≢ (e₂⋆ (proj₁ nck)) →
ρ (neck-e₂ nck) f ≡ ⌈ (n) /2⌉
class-A₁-ρ nck neq = ≤-≥⇒≡ (begin _ ≤⟨ A₁'-ρ ⟩ ⌊ n /2⌋
≡⟨ lem-even⇒⌊≡⌋ n-even ⟩ (⌈ n /2⌉ ∎))
(≰⇒> (λ x → ¬is< (s≤s x)))
where open Data.Nat.≤-Reasoning
e₁ = neck-e₁ nck
e₂ = neck-e₂ nck
ppc : ppchain e f
ppc = nck ∷ (sppc e₂ f)
¬is< : suc (ρ (neck-e₂ nck) f) ≤ ⌈ (n) /2⌉ → ⊥
¬is< is< with suc (suc (ρ e₂ f)) ≤? ⌈ (n) /2⌉
¬is< is< | yes p = ¬<-≡ ρef<n/2 ((toWitness ≡n/2))
where ρef<n/2 : ρ e f < ⌈ (n) /2⌉
ρef<n/2 = begin
suc (ρ e f)
≤⟨ s≤s (sppc-ρ-shorter-than ppc) ⟩
suc (suc (ρ e₂ f))
≤⟨ p ⟩ (⌈ n /2⌉ ∎)
¬is< is< | no ¬p = neq e₂≡e₂⋆
where ρ'ppc≡n/2 : ρ' ppc ≡ ⌈ (n) /2⌉
ρ'ppc≡n/2 = ≤-≥⇒≡ is< (≰⇒> (λ x → ¬p (s≤s x)))
len-ppc-as-c≡n : len (ppc as-c) ≡ n
len-ppc-as-c≡n rewrite
lem-id₀ (sc (pt e₂) (pt f)) |
(lem-2xρ-lambda {e₂} {f}) |
sym (lem-2x⌈n/2⌉-even n-even) =
trans (solve 1
(λ x → con 2 :+ x :+ (x :+ con 0) :=
con 1 :+ x :+ (con 1 :+ x :+ con 0))
refl (ρ e₂ f)) (cong (_*_ 2) ρ'ppc≡n/2)
c≡ : ppc as-c ≡ chain-with-e₁ (proj₁ nck)
c≡ = cong proj₁ (sym
(Inverse.left-inverse-of (I e (pt f) λ≡n)
(ppc as-c , len-ppc-as-c≡n)))
ppc≡ : ppc ≡ ppc-with-e₁ (proj₁ nck)
ppc≡ = trans (sym (lem-id₁ ppc)) (cong _as-ppc c≡)
e₂≡e₂⋆ : e₂ ≡ (e₂⋆ (proj₁ nck))
e₂≡e₂⋆ = cong neck-e₂ (cong (λ x → ppneck-gen x
{fromWitness ρef≥1}) ppc≡)
lem-neck⋆ : ∀ {e f ≥1 <n} → neck-e₂ (neck⋆ e f {≥1}) ≢ e
lem-neck⋆ {e} {f} {≥1} {<n} = λ eq → ¬<-≡ (toWitness ≥1)
(sym
(≡pred⇒zero
(subst (λ z → ρ z f ≡ pred (ρ e f)) eq
(0<ρ<n/2.class-A-ρ {<n = <n} (neck⋆ e f) refl))))