[Agda] Re: How to solve goals that involve a vertical bar?
Martin Stone Davis
martin.stone.davis at gmail.com
Sun Jan 17 07:31:04 CET 2016
I've solved my own problem: it turned out that I needed a supporting lemma.
The below sketches the idea: lemma-hard cannot be proved without the
supporting lemma-easy.
module VerticalBarProblemSolved where
open import Data.Bool.Base using (Bool ; true ; false)
open import Data.Product using (∃ ; _,_)
open import Relation.Nullary using (yes ; no)
open import Relation.Binary using (IsDecEquivalence)
open import Relation.Binary.PropositionalEquality using (_≡_ ; refl)
open import Data.Nat.Base using (ℕ ; suc)
open import Data.Empty using (⊥-elim)
module _ {isDecEquivalence : IsDecEquivalence {A = ℕ} _≡_} where
open IsDecEquivalence isDecEquivalence using (_≟_)
sucIffTrue : ℕ → Bool → ℕ
sucIffTrue n true = suc n
sucIffTrue n false with n ≟ n
sucIffTrue n false | yes refl = n
sucIffTrue n false | no n≢n = n
lemma-easy : (n : ℕ) → sucIffTrue n false ≡ n
lemma-easy n with n ≟ n
lemma-easy n | yes refl = refl
lemma-easy n | no n≢n = refl
lemma-hard : (n : ℕ) → ∃ λ (b : Bool) → sucIffTrue n b ≡ n
lemma-hard n = false , lemma-easy n
--
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 7:49 PM, Martin Stone Davis <
martin.stone.davis at gmail.com> wrote:
> Oops. Please ignore all the prior emails. It turns out that my second and
> third emails were cases in which case-splitting on the with-abstraction
> makes the vertical-bar go away!
>
> Sorry about that. I will come up with a simpler example in which the
> problem arises before I post again.
>
> --
> 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 7:35 PM, Martin Stone Davis <
> martin.stone.davis at gmail.com> wrote:
>
>> 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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