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

[Martin Stone Davis]

1000 apologies for not doing this before asking in the first place: I've
*considerably* shortened the code.

  module Map0WIP'' {K : Set} where

    open import Data.Product
    open import Data.Maybe
    open import Relation.Binary
    open import Relation.Binary.PropositionalEquality

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

      get : K → Maybe K → K
      get 𝑘 nothing = 𝑘
      get 𝑘 (just k₀) with k₀ ≟ 𝑘
      ... | kk = k₀

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


--
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 Sat, Jan 16, 2016 at 11:20 AM, Martin Stone Davis <
martin.stone.davis at gmail.com> wrote:

> 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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