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Lemma2-4-Inv.agda
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Lemma2-4-Inv.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.Empty
open import Data.Unit hiding (_≤?_; _≤_; _≟_)
open import Data.Product
open import Data.Bool hiding (_≟_)
open import Data.Nat hiding (compare) renaming (_≟_ to _ℕ≟_)
--open Data.Nat.≤-Reasoning
--open import Function.Inverse
open import Data.Sum
open import Data.Nat.Properties
open SemiringSolver
open import Function hiding (_∘_)
open import Function.Equality hiding (_∘_) renaming (cong to Icong)
open import Function.Inverse renaming (sym to Isym; zip to Izip; id to Iid)
open import Function.LeftInverse hiding (_∘_)
open import Function.Bijection hiding (_∘_)
import Data.Fin as F
import Data.Fin.Props as F
open import Function.Related.TypeIsomorphisms
import Relation.Binary.Sigma.Pointwise as SP
open import FinBijections
open import Data.Vec hiding ([_])
open import Misc
module Lemma2-4-Inv where
open import Lemma2-4 public
module 0<ρ<predn/2⁻¹ {e f : P} {≥1 : True (1 ≤? ρ e f)}
{<predn : True (suc (ρ e f) ≤? ⌈ (pred (n)) /2⌉)} where
<n : True (suc (ρ e f) ≤? ⌈ n /2⌉)
<n = fromWitness (begin
suc (ρ e f)
≤⟨ toWitness <predn ⟩
⌈ pred n /2⌉
≤⟨ ⌈n/2⌉-mono (≤⇒pred≤ n n m≤m) ⟩
(⌈ n /2⌉ ∎))
where open Data.Nat.≤-Reasoning
class-B : (nck : Neck e) → ρ (neck-e₂ nck) f ≡ ρ e f → e ≢ (neck-e₂ nck) →
el (proj₁ nck) ≡ el (proj₁ (neck⋆ e f {≥1})) ×
(neck-e₂ nck) ≢ (neck-e₂ (neck⋆ e f {≥1}))
class-B nck ρ≡ ≢e with pt (neck-e₂ nck) #? ln (el (proj₁ (neck⋆ e f {≥1})))
class-B nck ρ≡ ≢e | yes p with pt (neck-e₂ nck) ≟ pt (neck-e₂ (neck⋆ e f {≥1}))
class-B nck ρ≡ ≢e | yes p₁ | yes p = ⊥-elim (¬<-≡ (toWitness ≥1)
(sym (≡pred⇒zero (trans (sym ρ≡)
(0<ρ<n/2.class-A-ρ nck (pt-inj p))))))
class-B nck ρ≡ ≢e | yes p | no ¬p = cong el (cong proj₁
(neck!
{nck = nck}
{nck' = (proj₁ $ neck⋆ e f {≥1}) ,
(neck-e₂ nck) ∶ ((#sym p))} ≢e refl)) ,
(λ x → ¬p (cong pt x))
class-B nck ρ≡ ≢e | no ¬p = ⊥-elim
(≡suc⇒⊥ (trans
(sym (0<ρ<n/2.class-C₀-ρ {e} {f} {≥1} {<n}
nck (λ eq → ⊥-elim (¬p (helper
((#sym (neck-e₁#e₂ nck))) (cong ln eq))))
(λ e₂≡e → ≢e (sym e₂≡e))
(toWitness <predn))) ρ≡))
where helper : ∀ {x y z} → x # y → y ≡ z → x # z
helper # refl = #
class-C : (nck : Neck e) → neck-e₂ nck ≢ e → ρ (neck-e₂ nck) f ≡ 1 + ρ e f →
(neck-e₁ nck ≢ neck-e₁ (neck⋆ e f {≥1}))
class-C nck ≢e ρ≡ e≡e₁⋆ with ln (neck-e₁ nck) ≟ ln (neck-e₁ (neck⋆ e f {≥1}))
... | no ¬p = ¬p (cong ln e≡e₁⋆)
... | yes p
with (pt (neck-e₂ nck)) ≟ pt (neck-e₂ (neck⋆ e f {≥1}))
... | yes e≡e₂⋆ = helper (trans (sym ρ≡)
(0<ρ<n/2.class-A-ρ nck (pt-inj e≡e₂⋆)))
where helper : ∀ {x} → suc x ≡ pred x → ⊥
helper {zero} ()
helper {suc x} ()
... | no ¬p = ≡suc⇒⊥ (trans (sym ρ≡)
(0<ρ<n/2.class-B-ρ nck
(ln-inj p)
(λ x → ¬p (cong pt x)) ≢e))
module ρ≡1/2-predn⁻¹ {e f : P}
{ρ≡ : True (ρ e f ℕ≟ ⌈ (pred (n)) /2⌉)} {oddn : Odd n} where
≥1 : True (1 ≤? ρ e f )
≥1 = fromWitness (begin 1
≤⟨ s≤s z≤n ⟩ ⌈ pred n /2⌉
≡⟨ sym (toWitness ρ≡) ⟩ (ρ e f ∎))
where open Data.Nat.≤-Reasoning
<n : True (suc (ρ e f) ≤? ⌈ n /2⌉)
<n = fromWitness (begin suc (ρ e f)
≡⟨ cong suc (toWitness ρ≡) ⟩ suc ⌈ pred n /2⌉
≡⟨ cong suc (helper oddn) ⟩ suc (pred ⌈ n /2⌉)
≡⟨ refl ⟩ (⌈ n /2⌉ ∎))
where open Data.Nat.≤-Reasoning
helper : ∀ {x} → Odd x → ⌈ pred x /2⌉ ≡ pred ⌈ x /2⌉
helper {zero} ox = refl
helper {suc zero} ox = refl
helper {suc (suc m)} (oddSuc ox) = sym (lem-even⇒⌊≡⌋ (oddEven ox))
class-B : (nck : Neck e) → ρ (neck-e₂ nck) f ≡ ρ e f → e ≢ (neck-e₂ nck) →
(neck-e₂ nck) ≢ (neck-e₂ (neck⋆ e f {≥1}))
class-B nck ρ≡ e≢ e₂≡e₂⋆ = (¬<-≡ (toWitness ≥1)
(sym (≡pred⇒zero (trans (sym ρ≡)
(0<ρ<n/2.class-A-ρ {≥1 = ≥1}
{<n = <n}
nck e₂≡e₂⋆)))))
module ρ≡n/2⁻¹ {e f : P} {n-even : Even (n)}
{≡n/2 : True ( ρ e f ℕ≟ ⌈ (n) /2⌉ )} where
≥1 : True (1 ≤? ρ e f )
≥1 = fromWitness (begin 1 ≤⟨ s≤s z≤n ⟩ ⌈ n /2⌉
≡⟨ sym $ toWitness ≡n/2 ⟩
(ρ e f ∎))
where open Data.Nat.≤-Reasoning
class-A₀ : (nck : Neck e) → ρ (neck-e₂ nck) f ≡ pred ⌈ (n) /2⌉ →
(neck-e₂ nck) ≡ ρ≡n/2.e₂⋆ {n-even = n-even} {≡n/2 = ≡n/2} (proj₁ nck)
class-A₀ nck ρ≡ with pt (neck-e₂ nck) ≟ pt (ρ≡n/2.e₂⋆ {n-even = n-even}
{≡n/2 = ≡n/2} (proj₁ nck))
... | yes p = pt-inj p
... | no ¬p = ⊥-elim (¬<-≡ (s≤s z≤n)
(sym $ (≡pred⇒zero $
trans (sym (ρ≡n/2.class-A₁-ρ
nck (λ x → ¬p (cong pt x)))) ρ≡)))
class-A₁ : (nck : Neck e) → ρ (neck-e₂ nck) f ≡ ⌈ (n) /2⌉ →
(neck-e₂ nck) ≢ ρ≡n/2.e₂⋆ {n-even = n-even} {≡n/2 = ≡n/2} (proj₁ nck)
class-A₁ nck ρ≡ with pt (neck-e₂ nck) ≟ pt (ρ≡n/2.e₂⋆ {n-even = n-even}
{≡n/2 = ≡n/2} (proj₁ nck))
... | yes p = ⊥-elim (≡suc⇒⊥ (trans (sym ρ≡) (ρ≡n/2.class-A₀-ρ nck (pt-inj p))))
... | no ¬p = λ eq → ¬p (cong pt eq)
module 0<ρ<n/2⁻¹ {e f : P} {≥1 : True (1 ≤? ρ e f)}
{<n : True (suc (ρ e f) ≤? ⌈ n /2⌉)} where
class-A : (nck : Neck e) → ρ (neck-e₂ nck) f ≡ pred (ρ e f) →
e ≢ (neck-e₂ nck) →(neck-e₂ nck) ≡ (neck-e₂ (neck⋆ e f {≥1}))
class-A nck ρ≡ ≢e with pt (neck-e₂ nck) ≟ pt (neck-e₂ (neck⋆ e f {≥1})) |
ln (neck-e₁ nck) ≟ ln (neck-e₁ (neck⋆ e f {≥1}))
... | yes p | yes p₁ = pt-inj p
... | yes p | no ¬p = ⊥-elim (¬p
(cong ln
(cong neck-e₁
(neck! {nck = nck}
{nck' = (proj₁ $ neck⋆ e f {≥1}) ,
neck-e₂ (neck⋆ e f {≥1}) ∶
neck-e₁#e₂ (neck⋆ e f {≥1})} ≢e (pt-inj p)))))
... | no ¬p | yes p = ⊥-elim (¬<-≡ (toWitness ≥1)
(sym (≡pred⇒zero
(sym
(trans (sym ρ≡)
(0<ρ<n/2.class-B-ρ nck
(ln-inj p)
(λ x → ¬p (cong pt x))
(λ x → ≢e (sym x))))))))
... | no ¬p | no ¬p₁ with parity n
... | isEven x = ⊥-elim (helper
(trans (sym
(0<ρ<n/2.class-C₀-ρ nck
(λ x₁ → ¬p₁ (cong ln x₁))
(λ x₁ → ≢e (sym x₁)) helper₀)) ρ≡))
where ρ<predn/2 : ∀ {x} → Even (suc (suc (suc x))) →
⌈ (suc (suc x)) /2⌉ ≡ ⌈ (suc (suc (suc x))) /2⌉
ρ<predn/2 {zero} (evenSuc ())
ρ<predn/2 {suc zero} en = refl
ρ<predn/2 {suc (suc x₁)} (evenSuc en) = cong suc (ρ<predn/2 en)
helper₀ : ρ e f < ⌈ suc (suc nn) /2⌉
helper₀ = begin suc (ρ e f) ≤⟨ toWitness <n ⟩ ⌈ n /2⌉
≡⟨ sym (ρ<predn/2 x) ⟩
(⌈ pred n /2⌉ ∎)
where open Data.Nat.≤-Reasoning
helper : ∀ {x} → suc x ≡ pred x → ⊥
helper {zero} ()
helper {suc x} ()
... | isOdd x with suc (ρ e f) ≤? ⌈ (pred n) /2⌉
... | yes p = ⊥-elim
(helper
(trans
(sym
(0<ρ<n/2.class-C₀-ρ nck (λ x₁ → ¬p₁ (cong ln x₁))
(λ x₁ → ≢e (sym x₁)) p)) ρ≡))
where helper : ∀ {x} → suc x ≡ pred x → ⊥
helper {zero} ()
helper {suc x} ()
class-A nck ρ≡ ≢e | no ¬p₁ | no ¬p₂ | isOdd x | no ¬p =
⊥-elim (¬<-≡ (toWitness ≥1) (sym (≡pred⇒zero
(sym (trans (sym ρ≡)
(0<ρ<n/2.class-C₁-ρ nck (λ x₁ → ¬p₂ (cong ln x₁))
(λ x₁ → ¬p₁ (cong pt x₁))
(λ x₁ → ≢e (sym x₁)) ρ≡predn/2))))))
where ρ≡predn/2 : (ρ e f) ≡ ⌈ (pred n) /2⌉
ρ≡predn/2 = ≤-≥⇒≡ (begin ρ e f ≤⟨ pred-mono (toWitness <n) ⟩ pred ⌈ n /2⌉
≡⟨ sym (helper x) ⟩
(⌈ pred n /2⌉ ∎)) (pred-mono (≰⇒> ¬p))
where open Data.Nat.≤-Reasoning
helper : ∀ {x} → Odd x → ⌈ pred x /2⌉ ≡ pred ⌈ x /2⌉
helper {zero} ox = refl
helper {suc zero} ox = refl
helper {suc (suc m)} (oddSuc ox) = sym (lem-even⇒⌊≡⌋ (oddEven ox))