[Agda] struggling `with'
Andrea Vezzosi
sanzhiyan at gmail.com
Fri Jan 2 20:47:40 CET 2015
Only one match wasn't enough, but inspect and rewrite help to keep the
code short anyway :)
ins-kv∘ins-ku-eq _ k _ _ [] _ with k ≟ k
... | yes _ = =pn-refl
... | no k≉k = ⊥-elim $ k≉k ≈refl
ins-kv∘ins-ku-eq comb k u v ((k' , w) ∷ ps) sym-assoc-comb = prove
where
ins = insertWithKey comb
kv = (k , v)
ku = (k , u)
k'w = (k' , w)
ckuw = comb k u w
ckvu = comb k v u
prove : ins kv (ins ku (k'w ∷ ps)) =pn ins (k , comb k v u) (k'w ∷ ps)
prove with k ≟ k' | PE.inspect (_≟_ k) k'
prove | yes p | PE.[ eq ] rewrite eq = e0 ∷pn =pn-refl
where
e0 : (k' , comb k v ckuw) =p (k' , comb k ckvu w)
e0 = (≈refl , sym-assoc-comb)
prove | no ¬p | PE.[ eq ] rewrite eq = =p-refl ∷pn e0
where
e0 : (ins kv (ins ku ps)) =pn (ins (k , ckvu) ps)
e0 = ins-kv∘ins-ku-eq comb k u v ps sym-assoc-comb
Cheers,
Andrea
On Fri, Jan 2, 2015 at 8:15 PM, Sergei Meshveliani <mechvel at botik.ru> wrote:
>
> On Fri, 2015-01-02 at 16:51 +0100, Andrea Vezzosi wrote:
>> It's really hard to see what's going on without being able to load the
>> code, and with a lot of it omitted.
>>
>> We need at least the definition of "ins" and the original type of
>> "foo", then we might be able to figure out a better type for the two
>> alternatives, so that the splitting on (k ≟ k') only needs to be done
>> once.
>>
>
> Here follows the full code. It occurs small.
>
> I am grateful to anyone who shows how to write
> ins-kv∘ins-ku-eq
> in a nicer way.
>
> ------
> Sergei
>
>
>
> --**********************************************************************
> module AssocList where
> open import Level using (Level)
> open import Function using (_$_; case_of_)
> open import Relation.Nullary using (¬_; Dec; yes; no)
> open import Relation.Unary using (Decidable)
> open import Relation.Binary using
> (Rel; _⇒_; Reflexive; Symmetric; Transitive; IsEquivalence;
> module IsEquivalence; Setoid; module Setoid; module DecSetoid;
> DecSetoid
> )
> open import Relation.Binary.PropositionalEquality as PE using (_≡_)
> import Relation.Binary.EqReasoning as EqR
> open import Data.Empty using (⊥; ⊥-elim)
> open import Data.Product using (_×_; _,_)
> open import Relation.Binary.Product.Pointwise using (_×-setoid_)
> open import Data.List using (List; []; _∷_)
> open import Relation.Binary.List.Pointwise as Pointwise using ()
> renaming ([] to []pn; _∷_ to _∷pn_)
>
> -----------------------------------------------------------------------
> module _ {α α= β β= : Level} (keyDSetoid : DecSetoid α α=)
> (valSetoid : Setoid β β=)
> where
> open DecSetoid keyDSetoid using (_≈_; _≟_; setoid) renaming
> (Carrier to K; isEquivalence to kEquiv)
> open IsEquivalence kEquiv using ()
> renaming (refl to ≈refl; sym to ≈sym; trans to ≈trans)
>
> open Setoid valSetoid using () renaming (Carrier to V; _≈_ to _=v_;
> isEquivalence to vEquiv)
> open IsEquivalence vEquiv using ()
> renaming (refl to =v-refl; sym to =v-sym; trans to =v-trans;
> reflexive to =v-reflexive)
>
> pairSetoid = setoid ×-setoid valSetoid -- for K × V
> open Setoid pairSetoid using () renaming (Carrier to KV; _≈_ to _=p_;
> isEquivalence to pEquiv)
> open IsEquivalence pEquiv using ()
> renaming (refl to =p-refl; sym to =p-sym; trans to =p-trans;
> reflexive to =p-reflexive)
>
> point-p-setoid = Pointwise.setoid pairSetoid
> open Setoid point-p-setoid using ()
> renaming (_≈_ to _=pn_; isEquivalence to pnEquiv)
> -- "pn" stands for "pointwise for List KV"
>
> open IsEquivalence pnEquiv using ()
> renaming (refl to =pn-refl; reflexive to =pn-reflexive;
> sym to =pn-sym)
> open module EqR-pn = EqR point-p-setoid
> renaming (begin_ to begin-pn_; _∎ to _end-pn; _≈⟨_⟩_ to _=pn[_]_)
>
>
> -------------------------------------------------------------------
> Pairs : Set _
> Pairs = List KV
>
> CombineKVV : Set _ -- (α ⊔ β)
> CombineKVV = K → V → V → V
>
> ----------------------------------------------------------------------
> insertWithKey : CombineKVV → KV → Pairs → Pairs
> -- \key newV oldV |→ resV
>
> insertWithKey _ p [] = p ∷ []
> insertWithKey comb (k , new) ((k' , old) ∷ ps) =
> case k ≟ k'
> of \
> { (yes _) → (k' , comb k new old) ∷ ps
> ; (no _) → (k' , old) ∷ (insertWithKey comb (k , new) ps) }
>
> ----------------------------------------------------------------------
> ins-kv∘ins-ku-eq :
> (comb : CombineKVV) → ∀ k u v ps →
> (∀ {w} → comb k v (comb k u w) =v comb k (comb k v u) w) →
> let ins = insertWithKey comb
> in
> ins (k , v) (ins (k , u) ps) =pn ins (k , comb k v u) ps
>
> -- Example: it fits comb _ u v = f u v with any Associative f.
>
> ins-kv∘ins-ku-eq _ k _ _ [] _ with k ≟ k
> ... | yes _ = =pn-refl
> ... | no k≉k = ⊥-elim $ k≉k ≈refl
>
> ins-kv∘ins-ku-eq comb k u v ((k' , w) ∷ ps) sym-assoc-comb =
> prove (k ≟ k')
> where
> ins = insertWithKey comb
> kv = (k , v)
> ku = (k , u)
> k'w = (k' , w)
> ckuw = comb k u w
> ckvu = comb k v u
>
> -------------------------------------------------------------------
> case≈ : k ≈ k' →
> ins kv (ins ku (k'w ∷ ps)) =pn ins (k , ckvu) (k'w ∷ ps)
>
> case≈ k≈k' =
> begin-pn
> ins kv (ins ku (k'w ∷ ps)) =pn[ =pn-reflexive $
> PE.cong (ins kv) e1
> ]
> ins kv ((k' , ckuw) ∷ ps) =pn[ =pn-reflexive e2 ]
> (k' , comb k v ckuw) ∷ ps =pn[ e0 ∷pn =pn-refl ]
> (k' , comb k ckvu w) ∷ ps =pn[ =pn-reflexive $ PE.sym e3 ]
> ins (k , ckvu) (k'w ∷ ps)
> end-pn
> where
> e0 : (k' , comb k v ckuw) =p (k' , comb k ckvu w)
> e0 = (≈refl , sym-assoc-comb)
>
> e1 : ins ku ((k' , w) ∷ ps) ≡ (k' , ckuw) ∷ ps
> e1 with k ≟ k'
> ... | yes _ = PE.refl
> ... | no k≉k' = ⊥-elim $ k≉k' k≈k'
>
> e2 : ins kv ((k' , ckuw) ∷ ps) ≡ (k' , comb k v ckuw) ∷ ps
> e2 with k ≟ k'
> ... | yes _ = PE.refl
> ... | no k≉k' = ⊥-elim $ k≉k' k≈k'
>
> e3 : ins (k , ckvu) (k'w ∷ ps) ≡ (k' , comb k ckvu w) ∷ ps
> e3 with k ≟ k'
> ... | yes _ = PE.refl
> ... | no k≉k' = ⊥-elim $ k≉k' k≈k'
>
> -------------------------------------------------------------------
> case≉ : ¬ k ≈ k' →
> ins kv (ins ku (k'w ∷ ps)) =pn ins (k , ckvu) (k'w ∷ ps)
>
> case≉ k≉k' =
> begin-pn
> ins kv (ins ku (k'w ∷ ps)) =pn[ =pn-reflexive $
> PE.cong (ins kv) e1 ]
> ins kv (k'w ∷ (ins ku ps)) =pn[ =pn-reflexive e2 ]
> k'w ∷ (ins kv $ ins ku ps) =pn[ =p-refl ∷pn e0 ]
> k'w ∷ (ins (k , ckvu) ps) =pn[ =pn-reflexive e3 ]
> ins (k , ckvu) (k'w ∷ ps)
> end-pn
> where
> e0 : (ins kv (ins ku ps)) =pn (ins (k , ckvu) ps)
> e0 = ins-kv∘ins-ku-eq comb k u v ps sym-assoc-comb
>
> e1 : ins ku (k'w ∷ ps) ≡ k'w ∷ (ins ku ps)
> e1 with k ≟ k'
> ... | no _ = PE.refl
> ... | yes k≈k' = ⊥-elim $ k≉k' k≈k'
>
> e2 : ins kv (k'w ∷ ins ku ps) ≡ k'w ∷ (ins kv $ ins ku ps)
> e2 with k ≟ k'
> ... | no _ = PE.refl
> ... | yes k≈k' = ⊥-elim $ k≉k' k≈k'
>
> e3 : k'w ∷ (ins (k , ckvu) ps) ≡ ins (k , ckvu) (k'w ∷ ps)
> e3 with k ≟ k'
> ... | no _ = PE.refl
> ... | yes k≈k' = ⊥-elim $ k≉k' k≈k'
>
> -----------------------------------------------------------------
> prove : Dec (k ≈ k') → ins kv (ins ku (k'w ∷ ps)) =pn
> ins (k , comb k v u) (k'w ∷ ps)
> prove (no k≉k') = case≉ k≉k'
> prove (yes k≈k') = case≈ k≈k'
>
> -------------------------------------------------------------------------------
>
>
>
>
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