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Rename order transitivity (#62)

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This commit is contained in:
Patrick Stevens
2019-11-02 19:05:52 +00:00
committed by GitHub
parent 763ddb8dbb
commit 1325236359
20 changed files with 220 additions and 220 deletions
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32
KeyValue/LinearStore/Implementation.agda
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@@ -42,11 +42,11 @@ module KeyValue.LinearStore.Implementation where
lookupReducedSucceedsAfterAdd {keyOrder = keyOrder} {min} k v record { firstEntry = firstEntry ; next = no } | inl (inl min<k) | inl (inl _) | inl (inl k<k) = exFalso (PartialOrder.irreflexive (TotalOrder.order keyOrder) k<k)
lookupReducedSucceedsAfterAdd {keyOrder = keyOrder} {min} k v record { firstEntry = firstEntry ; next = no } | inl (inl min<k) | inl (inl _) | inl (inr k<k) = exFalso (PartialOrder.irreflexive (TotalOrder.order keyOrder) k<k)
lookupReducedSucceedsAfterAdd {keyOrder = keyOrder} {min} k v record { firstEntry = firstEntry ; next = no } | inl (inl min<k) | inl (inl _) | inr p = refl
lookupReducedSucceedsAfterAdd {keyOrder = keyOrder} {min} k v record { firstEntry = firstEntry ; next = no } | inl (inl min<k) | inl (inr k<min) = exFalso (PartialOrder.irreflexive (TotalOrder.order keyOrder) (PartialOrder.transitive (TotalOrder.order keyOrder) min<k k<min))
lookupReducedSucceedsAfterAdd {keyOrder = keyOrder} {min} k v record { firstEntry = firstEntry ; next = no } | inl (inl min<k) | inl (inr k<min) = exFalso (PartialOrder.irreflexive (TotalOrder.order keyOrder) (PartialOrder.<Transitive (TotalOrder.order keyOrder) min<k k<min))
lookupReducedSucceedsAfterAdd {keyOrder = keyOrder} {min} k v record { firstEntry = firstEntry ; next = no } | inl (inl min<k) | inr min=k rewrite min=k = exFalso (PartialOrder.irreflexive (TotalOrder.order keyOrder) min<k)
lookupReducedSucceedsAfterAdd {keyOrder = keyOrder} {min} k v record { firstEntry = firstEntry ; next = (yes (newMin , (m ,, pr))) } | inl (inl min<k) with TotalOrder.totality keyOrder min k
lookupReducedSucceedsAfterAdd {keyOrder = keyOrder} {min} k v record { firstEntry = firstEntry ; next = (yes (newMin , (m ,, pr))) } | inl (inl min<k) | inl (inl _) = lookupReducedSucceedsAfterAdd {keyOrder = keyOrder} k v m
lookupReducedSucceedsAfterAdd {keyOrder = keyOrder} {min} k v record { firstEntry = firstEntry ; next = (yes (newMin , (m ,, pr))) } | inl (inl min<k) | inl (inr k<min) = exFalso (PartialOrder.irreflexive (TotalOrder.order keyOrder) (PartialOrder.transitive (TotalOrder.order keyOrder) min<k k<min))
lookupReducedSucceedsAfterAdd {keyOrder = keyOrder} {min} k v record { firstEntry = firstEntry ; next = (yes (newMin , (m ,, pr))) } | inl (inl min<k) | inl (inr k<min) = exFalso (PartialOrder.irreflexive (TotalOrder.order keyOrder) (PartialOrder.<Transitive (TotalOrder.order keyOrder) min<k k<min))
lookupReducedSucceedsAfterAdd {keyOrder = keyOrder} {min} k v record { firstEntry = firstEntry ; next = (yes (newMin , (m ,, pr))) } | inl (inl min<k) | inr min=k rewrite min=k = exFalso (PartialOrder.irreflexive (TotalOrder.order keyOrder) min<k)
lookupReducedSucceedsAfterAdd {keyOrder = keyOrder} {min} k v m | inl (inr k<min) with TotalOrder.totality keyOrder k k
lookupReducedSucceedsAfterAdd {keyOrder = keyOrder} {min} k v m | inl (inr k<min) | inl (inl k<k) = exFalso (PartialOrder.irreflexive (TotalOrder.order keyOrder) k<k)
@@ -65,9 +65,9 @@ module KeyValue.LinearStore.Implementation where
lookupReducedSucceedsAfterUnrelatedAdd {keyOrder = keyOrder} {min} k' v' k v pr m lookupReducedSucceeds | inl (inl min<k') with TotalOrder.totality keyOrder min k
lookupReducedSucceedsAfterUnrelatedAdd {keyOrder = keyOrder} {min} k' v' k v pr record { firstEntry = firstEntry ; next = no } () | inl (inl min<k') | inl (inl min<k)
lookupReducedSucceedsAfterUnrelatedAdd {keyOrder = keyOrder} {min} k' v' k v pr record { firstEntry = firstEntry ; next = (yes (a , (fst ,, snd))) } lookupReducedSucceeds | inl (inl min<k') | inl (inl min<k) = lookupReducedSucceedsAfterUnrelatedAdd {keyOrder = keyOrder} {a} k' v' k v pr fst lookupReducedSucceeds
lookupReducedSucceedsAfterUnrelatedAdd {keyOrder = keyOrder} {min} k' v' k v (inl x) m lookupReducedSucceeds | inl (inl min<k') | inl (inr k<min) = exFalso (PartialOrder.irreflexive (TotalOrder.order keyOrder) (PartialOrder.transitive (TotalOrder.order keyOrder) k<min (PartialOrder.transitive (TotalOrder.order keyOrder) min<k' x)))
lookupReducedSucceedsAfterUnrelatedAdd {keyOrder = keyOrder} {min} k' v' k v (inl x) m lookupReducedSucceeds | inl (inl min<k') | inl (inr k<min) = exFalso (PartialOrder.irreflexive (TotalOrder.order keyOrder) (PartialOrder.<Transitive (TotalOrder.order keyOrder) k<min (PartialOrder.<Transitive (TotalOrder.order keyOrder) min<k' x)))
lookupReducedSucceedsAfterUnrelatedAdd {keyOrder = keyOrder} {min} k' v' k v (inr k<k') m () | inl (inl min<k') | inl (inr k<min)
lookupReducedSucceedsAfterUnrelatedAdd {keyOrder = keyOrder} {.min} k' v' min v (inl x) m lookupReducedSucceeds | inl (inl min<k') | inr refl = exFalso (PartialOrder.irreflexive (TotalOrder.order keyOrder) (PartialOrder.transitive (TotalOrder.order keyOrder) x min<k'))
lookupReducedSucceedsAfterUnrelatedAdd {keyOrder = keyOrder} {.min} k' v' min v (inl x) m lookupReducedSucceeds | inl (inl min<k') | inr refl = exFalso (PartialOrder.irreflexive (TotalOrder.order keyOrder) (PartialOrder.<Transitive (TotalOrder.order keyOrder) x min<k'))
lookupReducedSucceedsAfterUnrelatedAdd {keyOrder = keyOrder} {.min} k' v' min v (inr x) record { firstEntry = v2 ; next = no } p | inl (inl min<k') | inr refl = applyEquality yes (yesInjective p)
lookupReducedSucceedsAfterUnrelatedAdd {keyOrder = keyOrder} {.min} k' v' min v (inr x) record { firstEntry = .v ; next = (yes (a , b)) } refl | inl (inl min<k') | inr refl = applyEquality yes refl
lookupReducedSucceedsAfterUnrelatedAdd {keyOrder = keyOrder} {min} k' v' k v pr m lookupReducedSucceeds | inl (inr k'<min) with TotalOrder.totality keyOrder k' k
@@ -75,17 +75,17 @@ module KeyValue.LinearStore.Implementation where
lookupReducedSucceedsAfterUnrelatedAdd {keyOrder = keyOrder} {min} k' v' k v pr m lookupReducedSucceeds | inl (inr k'<min) | inl (inl k'<k) | inl (inl min<k) = lookupReducedSucceeds
lookupReducedSucceedsAfterUnrelatedAdd {keyOrder = keyOrder} {min} k' v' k v pr m () | inl (inr k'<min) | inl (inl k'<k) | inl (inr k<min)
lookupReducedSucceedsAfterUnrelatedAdd {keyOrder = keyOrder} {min} k' v' k v pr m lookupReducedSucceeds | inl (inr k'<min) | inl (inl k'<k) | inr refl = lookupReducedSucceeds
lookupReducedSucceedsAfterUnrelatedAdd {keyOrder = keyOrder} {min} k' v' k v (inl x) m lookupReducedSucceeds | inl (inr k'<min) | inl (inr k<k') = exFalso (PartialOrder.irreflexive (TotalOrder.order keyOrder) (PartialOrder.transitive (TotalOrder.order keyOrder) x k<k'))
lookupReducedSucceedsAfterUnrelatedAdd {keyOrder = keyOrder} {min} k' v' k v (inl x) m lookupReducedSucceeds | inl (inr k'<min) | inl (inr k<k') = exFalso (PartialOrder.irreflexive (TotalOrder.order keyOrder) (PartialOrder.<Transitive (TotalOrder.order keyOrder) x k<k'))
lookupReducedSucceedsAfterUnrelatedAdd {keyOrder = keyOrder} {min} k' v' k v (inr x) m lookupReducedSucceeds | inl (inr k'<min) | inl (inr k<k') with TotalOrder.totality keyOrder min k
lookupReducedSucceedsAfterUnrelatedAdd {keyOrder = keyOrder} {min} k' v' k v (inr _) m lookupReducedSucceeds | inl (inr k'<min) | inl (inr k<k') | inl (inl x) = exFalso (PartialOrder.irreflexive (TotalOrder.order keyOrder) (PartialOrder.transitive (TotalOrder.order keyOrder) x (PartialOrder.transitive (TotalOrder.order keyOrder) k<k' k'<min)))
lookupReducedSucceedsAfterUnrelatedAdd {keyOrder = keyOrder} {min} k' v' k v (inr _) m lookupReducedSucceeds | inl (inr k'<min) | inl (inr k<k') | inl (inl x) = exFalso (PartialOrder.irreflexive (TotalOrder.order keyOrder) (PartialOrder.<Transitive (TotalOrder.order keyOrder) x (PartialOrder.<Transitive (TotalOrder.order keyOrder) k<k' k'<min)))
lookupReducedSucceedsAfterUnrelatedAdd {keyOrder = keyOrder} {min} k' v' k v (inr _) m () | inl (inr k'<min) | inl (inr k<k') | inl (inr x)
lookupReducedSucceedsAfterUnrelatedAdd {keyOrder = keyOrder} {min} k' v' k v (inr _) m lookupReducedSucceeds | inl (inr k'<min) | inl (inr k<k') | inr x rewrite x = exFalso (PartialOrder.irreflexive (TotalOrder.order keyOrder) (PartialOrder.transitive (TotalOrder.order keyOrder) k<k' k'<min))
lookupReducedSucceedsAfterUnrelatedAdd {keyOrder = keyOrder} {min} k' v' k v (inr _) m lookupReducedSucceeds | inl (inr k'<min) | inl (inr k<k') | inr x rewrite x = exFalso (PartialOrder.irreflexive (TotalOrder.order keyOrder) (PartialOrder.<Transitive (TotalOrder.order keyOrder) k<k' k'<min))
lookupReducedSucceedsAfterUnrelatedAdd {keyOrder = keyOrder} {min} k' v' k v (inl x) m lookupReducedSucceeds | inl (inr k'<min) | inr k'=k rewrite k'=k = exFalso (PartialOrder.irreflexive (TotalOrder.order keyOrder) x)
lookupReducedSucceedsAfterUnrelatedAdd {keyOrder = keyOrder} {min} k' v' k v (inr x) m lookupReducedSucceeds | inl (inr k'<min) | inr k'=k rewrite k'=k = exFalso (PartialOrder.irreflexive (TotalOrder.order keyOrder) x)
lookupReducedSucceedsAfterUnrelatedAdd {keyOrder = keyOrder} {min} k' v' k v pr m lookupReducedSucceeds | inr refl with TotalOrder.totality keyOrder min k
lookupReducedSucceedsAfterUnrelatedAdd {keyOrder = keyOrder} {k'} k' v' k v (inl _) record { firstEntry = firstEntry ; next = no } () | inr refl | inl (inl min<k)
lookupReducedSucceedsAfterUnrelatedAdd {keyOrder = keyOrder} {k'} k' v' k v (inl _) record { firstEntry = firstEntry ; next = (yes (a , (fst ,, snd))) } lookupReducedSucceeds | inr refl | inl (inl min<k) = lookupReducedSucceeds
lookupReducedSucceedsAfterUnrelatedAdd {keyOrder = keyOrder} {k'} k' v' k v (inr x) m lookupReducedSucceeds | inr refl | inl (inl min<k) = exFalso (PartialOrder.irreflexive (TotalOrder.order keyOrder) (PartialOrder.transitive (TotalOrder.order keyOrder) min<k x))
lookupReducedSucceedsAfterUnrelatedAdd {keyOrder = keyOrder} {k'} k' v' k v (inr x) m lookupReducedSucceeds | inr refl | inl (inl min<k) = exFalso (PartialOrder.irreflexive (TotalOrder.order keyOrder) (PartialOrder.<Transitive (TotalOrder.order keyOrder) min<k x))
lookupReducedSucceedsAfterUnrelatedAdd {keyOrder = keyOrder} {k'} k' v' k v (inl x) m () | inr refl | inl (inr k<min)
lookupReducedSucceedsAfterUnrelatedAdd {keyOrder = keyOrder} {k'} k' v' k v (inr x) m () | inr refl | inl (inr k<min)
lookupReducedSucceedsAfterUnrelatedAdd {keyOrder = keyOrder} {.min} .min v' min v (inl x) m lookupReducedSucceeds | inr refl | inr refl = exFalso (PartialOrder.irreflexive (TotalOrder.order keyOrder) x)
@@ -116,10 +116,10 @@ module KeyValue.LinearStore.Implementation where
lookupReducedFailsAfterUnrelatedAdd {keyOrder = keyOrder} {min} .k v' k (inr x) record { firstEntry = firstEntry ; next = next } lookupReducedFails | inl (inr k'<min) | inl (inl min<k) | inr refl = exFalso (PartialOrder.irreflexive (TotalOrder.order keyOrder) x)
lookupReducedFailsAfterUnrelatedAdd {keyOrder = keyOrder} {min} k' v' k pr m lookupReducedFails | inl (inr k'<min) | inl (inr k<min) with TotalOrder.totality keyOrder k' k
lookupReducedFailsAfterUnrelatedAdd {keyOrder = keyOrder} {min} k' v' k (inl _) m lookupReducedFails | inl (inr k'<min) | inl (inr k<min) | inl (inl k'<k) with TotalOrder.totality keyOrder min k
lookupReducedFailsAfterUnrelatedAdd {keyOrder = keyOrder} {min} k' v' k (inl _) m lookupReducedFails | inl (inr k'<min) | inl (inr k<min) | inl (inl k'<k) | inl (inl min<k) = exFalso (PartialOrder.irreflexive (TotalOrder.order keyOrder) (PartialOrder.transitive (TotalOrder.order keyOrder) k<min min<k))
lookupReducedFailsAfterUnrelatedAdd {keyOrder = keyOrder} {min} k' v' k (inl _) m lookupReducedFails | inl (inr k'<min) | inl (inr k<min) | inl (inl k'<k) | inl (inl min<k) = exFalso (PartialOrder.irreflexive (TotalOrder.order keyOrder) (PartialOrder.<Transitive (TotalOrder.order keyOrder) k<min min<k))
lookupReducedFailsAfterUnrelatedAdd {keyOrder = keyOrder} {min} k' v' k (inl _) m lookupReducedFails | inl (inr k'<min) | inl (inr k<min) | inl (inl k'<k) | inl (inr k<min') = refl
lookupReducedFailsAfterUnrelatedAdd {keyOrder = keyOrder} {min} k' v' k (inl _) m lookupReducedFails | inl (inr k'<min) | inl (inr k<min) | inl (inl k'<k) | inr refl = exFalso (PartialOrder.irreflexive (TotalOrder.order keyOrder) k<min)
lookupReducedFailsAfterUnrelatedAdd {keyOrder = keyOrder} {min} k' v' k (inr x) m lookupReducedFails | inl (inr k'<min) | inl (inr k<min) | inl (inl k'<k) = exFalso (PartialOrder.irreflexive (TotalOrder.order keyOrder) (PartialOrder.transitive (TotalOrder.order keyOrder) k'<k x))
lookupReducedFailsAfterUnrelatedAdd {keyOrder = keyOrder} {min} k' v' k (inr x) m lookupReducedFails | inl (inr k'<min) | inl (inr k<min) | inl (inl k'<k) = exFalso (PartialOrder.irreflexive (TotalOrder.order keyOrder) (PartialOrder.<Transitive (TotalOrder.order keyOrder) k'<k x))
lookupReducedFailsAfterUnrelatedAdd {keyOrder = keyOrder} {min} k' v' k pr m lookupReducedFails | inl (inr k'<min) | inl (inr k<min) | inl (inr k<k') = refl
lookupReducedFailsAfterUnrelatedAdd {keyOrder = keyOrder} {min} .k v' k (inl x) m lookupReducedFails | inl (inr k'<min) | inl (inr k<min) | inr refl = exFalso (PartialOrder.irreflexive (TotalOrder.order keyOrder) x)
lookupReducedFailsAfterUnrelatedAdd {keyOrder = keyOrder} {min} .k v' k (inr x) m lookupReducedFails | inl (inr k'<min) | inl (inr k<min) | inr refl = exFalso (PartialOrder.irreflexive (TotalOrder.order keyOrder) x)
@@ -133,13 +133,13 @@ module KeyValue.LinearStore.Implementation where
countReducedBehavesWhenAddingNotPresent : {a b c : _} {keys : Set a} {values : Set b} {keyOrder : TotalOrder {_} {c} keys} {min : keys} → (k : keys) → (v : values) → (m : ReducedMap keys values keyOrder min) → (lookupReduced {keyOrder = keyOrder} m k ≡ no) → countReduced (addReducedMap k v m) ≡ succ (countReduced m)
countReducedBehavesWhenAddingNotPresent {keyOrder = keyOrder} {min} k v m lookupReducedFails with TotalOrder.totality keyOrder k min
countReducedBehavesWhenAddingNotPresent {keyOrder = keyOrder} {min} k v record { firstEntry = firstEntry ; next = next } lookupReducedFails | inl (inl k<min) with TotalOrder.totality keyOrder min k
countReducedBehavesWhenAddingNotPresent {keyOrder = keyOrder} {min} k v record { firstEntry = firstEntry ; next = next } lookupReducedFails | inl (inl k<min) | inl (inl min<k) = exFalso (PartialOrder.irreflexive (TotalOrder.order keyOrder) (PartialOrder.transitive (TotalOrder.order keyOrder) min<k k<min))
countReducedBehavesWhenAddingNotPresent {keyOrder = keyOrder} {min} k v record { firstEntry = firstEntry ; next = next } lookupReducedFails | inl (inl k<min) | inl (inl min<k) = exFalso (PartialOrder.irreflexive (TotalOrder.order keyOrder) (PartialOrder.<Transitive (TotalOrder.order keyOrder) min<k k<min))
countReducedBehavesWhenAddingNotPresent {keyOrder = keyOrder} {min} k v record { firstEntry = firstEntry ; next = next } lookupReducedFails | inl (inl k<min) | inl (inr _) = refl
countReducedBehavesWhenAddingNotPresent {keyOrder = keyOrder} {min} k v record { firstEntry = firstEntry ; next = next } lookupReducedFails | inl (inl k<min) | inr refl = exFalso (PartialOrder.irreflexive (TotalOrder.order keyOrder) k<min)
countReducedBehavesWhenAddingNotPresent {keyOrder = keyOrder} {min} k v record { firstEntry = firstEntry ; next = next } lookupReducedFails | inl (inr min<k) with TotalOrder.totality keyOrder min k
countReducedBehavesWhenAddingNotPresent {keyOrder = keyOrder} {min} k v record { firstEntry = firstEntry ; next = no } lookupReducedFails | inl (inr min<k) | inl (inl _) = refl
countReducedBehavesWhenAddingNotPresent {keyOrder = keyOrder} {min} k v record { firstEntry = firstEntry ; next = (yes (a , b)) } lookupReducedFails | inl (inr min<k) | inl (inl _) = applyEquality succ (countReducedBehavesWhenAddingNotPresent {keyOrder = keyOrder} k v (_&&_.fst b) lookupReducedFails)
countReducedBehavesWhenAddingNotPresent {keyOrder = keyOrder} {min} k v record { firstEntry = firstEntry ; next = next } lookupReducedFails | inl (inr min<k) | inl (inr k<min) = exFalso (PartialOrder.irreflexive (TotalOrder.order keyOrder) (PartialOrder.transitive (TotalOrder.order keyOrder) min<k k<min))
countReducedBehavesWhenAddingNotPresent {keyOrder = keyOrder} {min} k v record { firstEntry = firstEntry ; next = next } lookupReducedFails | inl (inr min<k) | inl (inr k<min) = exFalso (PartialOrder.irreflexive (TotalOrder.order keyOrder) (PartialOrder.<Transitive (TotalOrder.order keyOrder) min<k k<min))
countReducedBehavesWhenAddingNotPresent {keyOrder = keyOrder} {min} k v record { firstEntry = firstEntry ; next = next } lookupReducedFails | inl (inr min<k) | inr refl = exFalso (PartialOrder.irreflexive (TotalOrder.order keyOrder) min<k)
countReducedBehavesWhenAddingNotPresent {keyOrder = keyOrder} {min} k v m lookupReducedFails | inr refl with TotalOrder.totality keyOrder min min
countReducedBehavesWhenAddingNotPresent {keyOrder = keyOrder} {k} k v m lookupReducedFails | inr refl | inl (inl min<min) = exFalso (PartialOrder.irreflexive (TotalOrder.order keyOrder) min<min)
@@ -150,7 +150,7 @@ module KeyValue.LinearStore.Implementation where
countReducedBehavesWhenAddingPresent : {a b c : _} {keys : Set a} {values : Set b} {keyOrder : TotalOrder {_} {c} keys} {min : keys} → (k : keys) → (v v' : values) → (m : ReducedMap keys values keyOrder min) → (lookupReduced {keyOrder = keyOrder} m k ≡ yes v') → countReduced (addReducedMap k v m) ≡ countReduced m
countReducedBehavesWhenAddingPresent {keyOrder = keyOrder} {min} k v v' m lookupReducedSucceeds with TotalOrder.totality keyOrder k min
countReducedBehavesWhenAddingPresent {keyOrder = keyOrder} {min} k v v' m lookupReducedSucceeds | inl (inl k<min) with TotalOrder.totality keyOrder min k
countReducedBehavesWhenAddingPresent {keyOrder = keyOrder} {min} k v v' m lookupReducedSucceeds | inl (inl k<min) | inl (inl min<k) = exFalso (PartialOrder.irreflexive (TotalOrder.order keyOrder) (PartialOrder.transitive (TotalOrder.order keyOrder) k<min min<k))
countReducedBehavesWhenAddingPresent {keyOrder = keyOrder} {min} k v v' m lookupReducedSucceeds | inl (inl k<min) | inl (inl min<k) = exFalso (PartialOrder.irreflexive (TotalOrder.order keyOrder) (PartialOrder.<Transitive (TotalOrder.order keyOrder) k<min min<k))
countReducedBehavesWhenAddingPresent {keyOrder = keyOrder} {min} k v v' m () | inl (inl k<min) | inl (inr _)
countReducedBehavesWhenAddingPresent {keyOrder = keyOrder} {min} k v v' m lookupReducedSucceeds | inl (inl k<min) | inr q = exFalso (PartialOrder.irreflexive (TotalOrder.order keyOrder) (identityOfIndiscernablesLeft (TotalOrder._<_ keyOrder) k<min (equalityCommutative q)))
countReducedBehavesWhenAddingPresent {keyOrder = keyOrder} {min} k v v' m lookupReducedSucceeds | inl (inr min<k) with TotalOrder.totality keyOrder min k
@@ -198,7 +198,7 @@ module KeyValue.LinearStore.Implementation where
lookupReducedWhenLess : {a b c : _} {keyDom : Set a} {values : Set b} {keyOrder : TotalOrder {_} {c} keyDom} {min : keyDom} → (m : ReducedMap keyDom values keyOrder min) → (k : keyDom) → (TotalOrder._<_ keyOrder k min) → (lookupReduced m k ≡ no)
lookupReducedWhenLess {keyOrder = keyOrder} {min} m k k<min with TotalOrder.totality keyOrder min k
lookupReducedWhenLess {keyOrder = keyOrder} {min} m k k<min | inl (inl min<k) = exFalso (PartialOrder.irreflexive (TotalOrder.order keyOrder) (PartialOrder.transitive (TotalOrder.order keyOrder) k<min min<k))
lookupReducedWhenLess {keyOrder = keyOrder} {min} m k k<min | inl (inl min<k) = exFalso (PartialOrder.irreflexive (TotalOrder.order keyOrder) (PartialOrder.<Transitive (TotalOrder.order keyOrder) k<min min<k))
lookupReducedWhenLess {keyOrder = keyOrder} {min} m k k<min | inl (inr _) = refl
lookupReducedWhenLess {keyOrder = keyOrder} {min} m k k<min | inr min=k rewrite min=k = exFalso (PartialOrder.irreflexive (TotalOrder.order keyOrder) k<min)
@@ -220,7 +220,7 @@ module KeyValue.LinearStore.Implementation where
lookupCertainReduced {keyOrder = keyOrder} {min} record { firstEntry = firstEntry ; next = (yes (a , (fst ,, snd))) } k record { index = (succ index) ; index<m = index<m ; isHere = isHere } | inl (inl min<k) = lookupCertainReduced {keyOrder = keyOrder} fst k record { index = index ; index<m = canRemoveSuccFrom<N index<m ; isHere = isHere }
lookupCertainReduced {keyOrder = keyOrder} {min} record { firstEntry = firstEntry ; next = (yes (a , (fst ,, snd))) } k record { index = zero ; index<m = _ ; isHere = isHere } | inl (inr k<min) rewrite isHere = exFalso (PartialOrder.irreflexive (TotalOrder.order keyOrder) k<min)
lookupCertainReduced {keyOrder = keyOrder} {min} record { firstEntry = firstEntry ; next = (yes (a , (fst ,, snd))) } k record { index = (succ index) ; index<m = index<m ; isHere = isHere } | inl (inr k<min) with TotalOrder.totality keyOrder a k
lookupCertainReduced {keyOrder = keyOrder} {min} record { firstEntry = firstEntry ; next = (yes (a , (fst ,, snd))) } k record { index = (succ index) ; index<m = index<m ; isHere = isHere } | inl (inr k<min) | inl (inl a<k) = exFalso (PartialOrder.irreflexive (TotalOrder.order keyOrder) (PartialOrder.transitive (TotalOrder.order keyOrder) k<min (PartialOrder.transitive (TotalOrder.order keyOrder) snd a<k)))
lookupCertainReduced {keyOrder = keyOrder} {min} record { firstEntry = firstEntry ; next = (yes (a , (fst ,, snd))) } k record { index = (succ index) ; index<m = index<m ; isHere = isHere } | inl (inr k<min) | inl (inl a<k) = exFalso (PartialOrder.irreflexive (TotalOrder.order keyOrder) (PartialOrder.<Transitive (TotalOrder.order keyOrder) k<min (PartialOrder.<Transitive (TotalOrder.order keyOrder) snd a<k)))
lookupCertainReduced {values = values} {keyOrder = keyOrder} {min} record { firstEntry = firstEntry ; next = (yes (a , (fst ,, snd))) } k record { index = (succ index) ; index<m = index<m ; isHere = isHere } | inl (inr k<min) | inl (inr k<a) = exFalso h
where
f : Sg values (λ v → lookupReduced fst k ≡ yes v)
@@ -232,7 +232,7 @@ module KeyValue.LinearStore.Implementation where
h : False
h with f
h | a , b rewrite g = noIsNotYes b
lookupCertainReduced {keyOrder = keyOrder} {min} record { firstEntry = firstEntry ; next = (yes (a , (fst ,, snd))) } k record { index = (succ index) ; index<m = index<m ; isHere = isHere } | inl (inr k<min) | inr refl = exFalso (PartialOrder.irreflexive (TotalOrder.order keyOrder) (PartialOrder.transitive (TotalOrder.order keyOrder) snd k<min))
lookupCertainReduced {keyOrder = keyOrder} {min} record { firstEntry = firstEntry ; next = (yes (a , (fst ,, snd))) } k record { index = (succ index) ; index<m = index<m ; isHere = isHere } | inl (inr k<min) | inr refl = exFalso (PartialOrder.irreflexive (TotalOrder.order keyOrder) (PartialOrder.<Transitive (TotalOrder.order keyOrder) snd k<min))
lookupCertainReduced {keyOrder = keyOrder} {min} record { firstEntry = firstEntry ; next = (yes (a , (fst ,, snd))) } k pr | inr min=k = firstEntry , refl
lookupCertain : {a b c : _} {keyDom : Set a} {values : Set b} {keyOrder : TotalOrder {_} {c} keyDom} → (m : Map keyDom values keyOrder) → (k : keyDom) → (vecContains (keys m) k) → Sg values (λ v → lookup m k ≡ yes v)