Proved sorted permutations are equal #2748
Conversation
| injective⇒nonStrictlyContractive : ∀ (f : Fin n → Fin m) → Injective _≡_ _≡_ f → | ||
| ∀ i → ¬ (∀ j → j ≤ i → f j < i) | ||
| injective⇒nonStrictlyContractive f f-injective i j≤i⇒fj<i = | ||
| ℕ.n≮n (toℕ i) (injective⇒≤ h-injective) | ||
| where | ||
| h : Fin′ (suc i) → Fin′ i | ||
| h k = lower (f (inject! k)) (j≤i⇒fj<i _ (ℕ.s≤s⁻¹ (inject!-< k))) | ||
|
|
||
| h-injective : Injective _≡_ _≡_ h | ||
| h-injective = inject!-injective ∘ f-injective ∘ lower-injective _ _ | ||
|
|
||
| injective⇒existsPivot : ∀ (f : Fin n → Fin m) → Injective _≡_ _≡_ f → | ||
| ∀ (i : Fin n) → ∃ λ (j : Fin n) → j ≤ i × i ≤ f j | ||
| injective⇒existsPivot {n = suc n} f f-injective i with any? (λ j → j ≤? i ×-dec i ≤? f j) | ||
| ... | yes result = result | ||
| ... | no ¬result = contradiction | ||
| strictlyContractive | ||
| (injective⇒nonStrictlyContractive f f-injective i) | ||
| where | ||
| strictlyContractive : ∀ j → j ≤ i → f j < i | ||
| strictlyContractive j j≤i with i ≤? f j | ||
| ... | yes i≤fj = contradiction (j , j≤i , i≤fj) ¬result | ||
| ... | no i≰fj = ℕ.≰⇒> i≰fj | ||
|
|
There was a problem hiding this comment.
Downstream, after resolving the issues around #2744, I'd hope that these could be read off from a single appeal to min? to construct a suitable minimal counterexample (for existsPivot) or else rule out the Universal case branch by contradiction with nonStrictlyContractive. See also #2746 for the use of pigeonhole to prove injective⇒≤, so some simplification in the proof of nonStrictlyContractive might also be possible?
| inject!-< : (k : Fin′ i) → inject! k < i | ||
| lower-injective : lower i i<n ≡ lower j j<n → i ≡ j | ||
| injective⇒nonStrictlyContractive : ∀ (f : Fin n → Fin m) → Injective _≡_ _≡_ f → ∀ i → ¬ (∀ j → j ≤ i → f j < i) | ||
| injective⇒existsPivot : ∀ (f : Fin n → Fin m) → Injective _≡_ _≡_ f → ∀ (i : Fin n) → ∃ λ (j : Fin n) → j ≤ i × i ≤ f j |
There was a problem hiding this comment.
Any particular reason (beyond 'redundant' documentation for the sake of the reader) to include the types of i and j here? They're inferrable form the declared type of f, or not?
| @@ -121,6 +121,10 @@ lower₁ {zero} zero ne = contradiction refl ne | |||
| lower₁ {suc n} zero _ = zero | |||
| lower₁ {suc n} (suc i) ne = suc (lower₁ i (ne ∘ cong suc)) | |||
|
|
|||
| lower : ∀ (i : Fin m) → .(toℕ i ℕ.< n) → Fin n | |||
There was a problem hiding this comment.
As on previous iterations: suggest deprecating/redefining lower₁ in terms of this new definition, to avoid redundant duplication?
UPDATED: oh, I see... it would involve importing Data.Nat.Properties in order to fix up the precondition, and that might be considered A Bad Thing...
| ∀ {i j} → toℕ i ≡ toℕ j → | ||
| lookup ys j ≤ lookup xs i | ||
| ↗↭↗⇒≤ {xs} {ys} xs↭ys xs↗ ys↗ {i} {j} i≡j | ||
| with Fin.injective⇒existsPivot _ (inverseʳ⇒injective _ (Inverse.inverseʳ (toFin xs↭ys))) i |
There was a problem hiding this comment.
| with Fin.injective⇒existsPivot _ (inverseʳ⇒injective _ (Inverse.inverseʳ (toFin xs↭ys))) i | |
| with Fin.injective⇒existsPivot _ (inverseʳ⇒injective _ (Inverse.inverseʳ (onIndices xs↭ys))) i |
???
There was a problem hiding this comment.
Also, there's no need for with here: the pattern is irrefutable, and computable with a pattern-matching let, so...
| with Fin.injective⇒existsPivot _ (inverseʳ⇒injective _ (Inverse.inverseʳ (toFin xs↭ys))) i | ||
| ... | (k , k≤i , i≤π[k]) = begin | ||
| lookup ys j ≤⟨ lookup-mono-≤ O ys↗ (P.subst (ℕ._≤ _) i≡j i≤π[k]) ⟩ | ||
| lookup ys (toFin xs↭ys ⟨$⟩ʳ k) ≈⟨ toFin-lookup xs↭ys k ⟨ |
There was a problem hiding this comment.
Ditto.
| lookup ys (toFin xs↭ys ⟨$⟩ʳ k) ≈⟨ toFin-lookup xs↭ys k ⟨ | |
| lookup ys (onIndices xs↭ys ⟨$⟩ʳ k) ≈⟨ onIndices-lookup xs↭ys k ⟨ |
| lower {suc _} {suc n} zero leq = zero | ||
| lower {suc _} {suc n} (suc i) leq = suc (lower i (ℕ.s≤s⁻¹ leq)) |
There was a problem hiding this comment.
| lower {suc _} {suc n} zero leq = zero | |
| lower {suc _} {suc n} (suc i) leq = suc (lower i (ℕ.s≤s⁻¹ leq)) | |
| lower {n = suc _} zero leq = zero | |
| lower {n = suc _} (suc i) leq = suc (lower i (ℕ.s≤s⁻¹ leq)) |
I don't think there's any need to involve the m binding in the definition, and named telescopes are (typically) more robust in the face of modifications to variable declarations etc.
| lower-injective {suc _} {suc n} zero zero eq = refl | ||
| lower-injective {suc _} {suc n} (suc i) (suc j) eq = |
There was a problem hiding this comment.
Ditto re: avoiding the m binding
| lower-injective {suc _} {suc n} zero zero eq = refl | |
| lower-injective {suc _} {suc n} (suc i) (suc j) eq = | |
| lower-injective {n = suc n} zero zero eq = refl | |
| lower-injective {n = suc n} (suc i) (suc j) eq = |
| ``` | ||
|
|
||
| * In `Data.Fin.Permutation`: | ||
| ```agda | ||
| cast-id : .(m ≡ n) → Permutation m n | ||
| swap : Permutation m n → Permutation (suc (suc m)) (suc (suc n)) | ||
| inject!-injective : swap : Permutation m n → Permutation (suc (suc m)) (suc (suc n)) |
The main result. Would like to try and sneak this into v2.3 for @onestruggler.
Have also added an explanation of the choice of propositional permutation in
SortingAlgorithmand a lemma that proves the setoid version.