[Agda] struggling `with'

Christopher Jenkins cjenkin1 at trinity.edu
Sat Jan 3 19:05:32 CET 2015


Sergei,

It might be helpful if you posted full code examples using lpaste.net. It
has a nice syntax highlighting and is easier to extract to a file than copy
/ paste from email.

And I need to find out what is `inspect'.
>

"inspect" can be found living in Relation.Binary.Propositional equality,
and is a nifty little trick to remember that a with term w was originally
equal to f x. Here's an example:

http://lpaste.net/117700

When proving

filter-All :  ∀ {A : Set} → (p : A → Bool) → (xs : List A) →
             All (λ x → p x ≡ true) (filter p xs)

After doing analysis on xs and moving to a cons case, we find we need
to analyze p x in order to continue.

filter-All p (x ∷ xs) = {!!}
--Goal: All (λ x₁ → p x₁ ≡ true) (if p x then x ∷ filter p xs else filter p xs)

However after we get to the true case and are ready to construct an
element of All with _∷_, Agda seems to have forgotten why we were
here in first place!

filter-All p (x ∷ xs) with p x
filter-All p (x ∷ xs) | true = {!!} ∷ (filter-All p xs)
-- Goal: p x ≡ true

We can use inspect to remind Agda.

filter-All p (x ∷ xs) with p x | inspect p x
filter-All p (x ∷ xs) | true | Reveal_is_.[ eq ] = eq ∷ (filter-All p xs)


On Sat, Jan 3, 2015 at 7:19 AM, Sergei Meshveliani <mechvel at botik.ru> wrote:

> On Fri, 2015-01-02 at 20:47 +0100, Andrea Vezzosi wrote:
> > Only one match wasn't enough, but inspect and rewrite help to keep the
> > code short anyway :)
> >
>
> This looks good!
> Thank you very much.
> And I need to find out what is `inspect'.
>
> ------
> Sergei
>
>
> >   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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>



-- 
Christopher Jenkins
Computer Science 2013
Trinity University
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