[Agda] Re: How to solve goals that involve a vertical bar?

[Martin Stone Davis]

FWIW, here is a shorter version of the same problem:

  module Map0WIP' {K : Set} where
    open import Data.Product
    open import Data.Sum
    open import Data.Maybe
    open import Data.Empty
    open import Relation.Nullary.Negation
    open import Relation.Nullary
    open import Relation.Binary
    open import Relation.Binary.PropositionalEquality

    record _∉_ (𝑘 : K) (m₀ : Maybe K) : Set where
      inductive
      field
        un∉ : m₀ ≡ nothing ⊎
                ∃ λ m₁ → 𝑘 ∉ m₁ ×
                ∃ λ k₀ → k₀ ≢ 𝑘 ×
                ∃ λ (k₀∉m₁ : k₀ ∉ m₁)
                → m₀ ≡ just k₀

    _∈_ : (𝑘 : K) (m₀ : Maybe K) → Set
    𝑘 ∈ m₀ = ¬ 𝑘 ∉ m₀

    here : ∀ {k₀ : K} → k₀ ∈ just k₀
    here record { un∉ = (inj₁ ()) }
    here record { un∉ = inj₂ (_ , _ , _ , k₀≢k₀ , _ , refl) } = ⊥-elim
(k₀≢k₀ refl)

    module _ {isDecEquivalence : IsDecEquivalence {A = K} _≡_} where
      open IsDecEquivalence isDecEquivalence using (_≟_)

      get : ∀ {𝑘 : K} {m₀ : Maybe K} (𝑘∈m₀ : 𝑘 ∈ m₀) → K
      get {𝑘} {m₀ = nothing} 𝑘∈m₀ = 𝑘
      get {𝑘} {just k₀} 𝑘∈m₀ with k₀ ≟ 𝑘
      get {𝑘} {just k₀} 𝑘∈m₀ | kk = k₀

      put : (k₀ : K) → (m₁ : Maybe K) → k₀ ∉ m₁ → ∃ λ (m₀ : Maybe K) → ∃ λ
(k₀∈m₀ : k₀ ∈ m₀) → get k₀∈m₀ ≡ k₀
      put k₀ m₁ k₀∉m₁ with k₀ ≟ k₀
      put k₀ m₁ k₀∉m₁ | kk = just k₀ , here , {!!}


--
Martin Stone Davis

Postal/Residential:
1223 Ferry St
Apt 5
Eugene, OR 97401
Talk / Text / Voicemail: (310) 699-3578 <3106993578>
Electronic Mail: martin.stone.davis at gmail.com
Website: martinstonedavis.com

On Fri, Jan 15, 2016 at 7:05 PM, Martin Stone Davis <
martin.stone.davis at gmail.com> wrote:

> In the hole of the function 'put' (see code below), Agda reports that the
> goal is
>
> (get {α} {K} V {isDecEquivalence} {k₀}
>>        {.Sandbox.Record.Map0WIP.recCon-NOT-PRINTED
>>         (.Sandbox.Record.Map0WIP.recCon-NOT-PRINTED
>>          (inj₂ (k₀ , v₀ , m₁ , k₀∉m₁)))}
>>        here
>>        | k₀ ≟ k₀)
>>       ≡ v₀
>>
>
> I have tried invoking a with-abstraction on k₀ ≟ k₀ but to no avail: the
> vertical bar remains. TIA for any help resolving this.
>
> CODE BEGINS
>     open import Level
>     open import Relation.Binary.Core
>     open import Data.Product
>     open import Data.Sum
>     open import Data.Unit.Base
>
>     record Maybe {α} (A : Set α) : Set α where
>       field
>         unmaybe : ⊤ ⊎ A
>
>     nothing : ∀ {α} {A : Set α} → Maybe A
>     nothing = record { unmaybe = inj₁ tt }
>
>     just : ∀ {α} {A : Set α} (a : A) → Maybe A
>     just a = record { unmaybe = inj₂ a }
>
>     mutual
>       record Map : Set α where
>         inductive
>         field
>           unmap : Maybe (∃ λ k₀ → V k₀ × ∃ λ m₁ → k₀ ∉ m₁)
>
>       record _∉_ (𝑘 : K) (m₀ : Map) : Set α where
>         inductive
>         field
>           un∉ : m₀ ≡ record { unmap = nothing } ⊎
>                   ∃ λ m₁ → 𝑘 ∉ m₁ ×
>                   ∃ λ k₀ → k₀ ≢ 𝑘 ×
>                   ∃ λ (k₀∉m₁ : k₀ ∉ m₁) → ∃ λ v₀
>                   → m₀ ≡ record { unmap = just (k₀ , v₀ , m₁ , k₀∉m₁) }
>
>     open import Data.Empty
>     open import Relation.Nullary.Negation
>     open import Relation.Nullary
>
>     _∈_ : (𝑘 : K) (m₀ : Map) → Set α
>     𝑘 ∈ m₀ = ¬ 𝑘 ∉ m₀
>
>     pattern ∅ = record { unmap = record { unmaybe = inj₁ tt } }
>     pattern M⟦_+_⋆_∣_⟧ m₁ k₀ v₀ k₀∉m₁ = record { unmap = record { unmaybe
> = inj₂ (k₀ , v₀ , m₁ , k₀∉m₁) } }
>     pattern M⟦_+_∣_⟧ m₁ v₀ k₀∉m₁ = record { unmap = record { unmaybe =
> inj₂ (_ , v₀ , m₁ , k₀∉m₁) } }
>     pattern M⟦_⋆_∣_⟧ k₀ v₀ k₀∉m₁ = record { unmap = record { unmaybe =
> inj₂ (k₀ , v₀ , _ , k₀∉m₁) } }
>     pattern M⟦_∣_⟧ v₀ k₀∉m₁ = record { unmap = record { unmaybe = inj₂ (_
> , v₀ , _ , k₀∉m₁) } }
>
>     pattern ∉∅ = record { un∉ = (inj₁ refl) }
>     pattern ¬∉∅ = record { un∉ = (inj₁ ()) }
>     pattern ∉⟦_/_⟧ 𝑘∉m₁ k₀≢𝑘  = record { un∉ = inj₂ (_ , 𝑘∉m₁ , _ ,
> k₀≢𝑘 , _ , _ , refl) }
>
>     here : ∀ {k₀ : K} {v₀ : V k₀} {m₁ : Map} {k₀∉m₁ : k₀ ∉ m₁} → k₀ ∈ M⟦
> v₀ ∣ k₀∉m₁ ⟧
>     here ¬∉∅
>     here ∉⟦ _ / k₀≢k₀ ⟧ = ⊥-elim (k₀≢k₀ refl)
>
>     infixl 40 _⊂_∣_
>     _⊂_∣_ : Map → Map → (K → Set α) → Set α
>     m ⊂ m' ∣ c = ∀ {𝑘} → c 𝑘 → 𝑘 ∉ m' → 𝑘 ∉ m
>
>     shrink : ∀ {k₀ v₀ m₁ k₀∉m₁} → M⟦ m₁ + v₀ ∣ k₀∉m₁ ⟧ ⊂ m₁ ∣ λ 𝑘 → k₀ ≢
> 𝑘
>     shrink k₀≢𝑘 ∉∅ = ∉⟦ ∉∅ / k₀≢𝑘 ⟧
>     shrink k₀≢𝑘 ∉⟦ 𝑘∉m₀ / k₁≢𝑘 ⟧ = ∉⟦ shrink k₁≢𝑘 𝑘∉m₀ / k₀≢𝑘 ⟧
>
>     somewhere : ∀ {𝑘 k₀ v₀ m₁ k₀∉m₁} → 𝑘 ∈ M⟦ m₁ + v₀ ∣ k₀∉m₁ ⟧ → k₀ ≢
> 𝑘 → 𝑘 ∈ m₁
>     somewhere 𝑘∈m₀ k₀≢𝑘 𝑘∉m₁ = contradiction (shrink k₀≢𝑘 𝑘∉m₁) 𝑘∈m₀
>
>     open import Relation.Binary
>     open import Relation.Binary.PropositionalEquality
>     module _ {isDecEquivalence : IsDecEquivalence {A = K} _≡_} where
>       open IsDecEquivalence isDecEquivalence using (_≟_)
>
>       get : ∀ {𝑘 : K} {m₀ : Map} (𝑘∈m₀ : 𝑘 ∈ m₀) → V 𝑘
>       get {m₀ = ∅} 𝑘∈m₀ = contradiction ∉∅ 𝑘∈m₀
>       get {𝑘} {M⟦ m₁ + k₀ ⋆ v₀ ∣ k₀∉m₁ ⟧} 𝑘∈m₀ with k₀ ≟ 𝑘
>       get {𝑘} {M⟦ m₁ + k₀ ⋆ v₀ ∣ k₀∉m₁ ⟧} 𝑘∈m₀ | yes k₀≡𝑘 rewrite k₀≡𝑘
> = v₀
>       get {𝑘} {M⟦ m₁ + k₀ ⋆ v₀ ∣ k₀∉m₁ ⟧} 𝑘∈m₀ | no k₀≢𝑘 = get
> (somewhere 𝑘∈m₀ k₀≢𝑘)
>
>       put : (k₀ : K) → (v₀ : V k₀) (m₁ : Map) → k₀ ∉ m₁ → ∃ λ (m₀ : Map) →
> ∃ λ (k₀∈m₀ : k₀ ∈ m₀) → get k₀∈m₀ ≡ v₀
>       put k₀ v₀ m₁ k₀∉m₁ = M⟦ m₁ + k₀ ⋆ v₀ ∣ k₀∉m₁ ⟧ , here , {!!}
>
> CODE ENDS
>
> --
> Martin Stone Davis
>
> Postal/Residential:
> 1223 Ferry St
> Apt 5
> Eugene, OR 97401
> Talk / Text / Voicemail: (310) 699-3578 <3106993578>
> Electronic Mail: martin.stone.davis at gmail.com
> Website: martinstonedavis.com
>
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