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Move Bool to live on its own (#118)

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Patrick Stevens
2020-04-18 17:14:39 +01:00
committed by GitHub
parent 264a5e2bd9
commit 84a146f72c
30 changed files with 270 additions and 123 deletions
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Setoids/Orders/Partial/Sequences.agda Normal file
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{-# OPTIONS --safe --warning=error --without-K --guardedness #-}
open import Numbers.Naturals.Semiring
open import Numbers.Naturals.Order
open import LogicalFormulae
open import Orders.Total.Definition
open import Orders.Partial.Definition
open import Setoids.Setoids
open import Functions
open import Sequences
open import Setoids.Orders.Partial.Definition
open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
module Setoids.Orders.Partial.Sequences {a b c : _} {A : Set a} {S : Setoid {a} {b} A} {_<_ : Rel {a} {c} A} (p : SetoidPartialOrder S _<_) where
open SetoidPartialOrder p
WeaklyIncreasing' : (Sequence A) → Set (b ⊔ c)
WeaklyIncreasing' s = (m n : ℕ) → (m <N n) → (index s m) <= (index s n)
WeaklyIncreasing : (Sequence A) → Set (b ⊔ c)
WeaklyIncreasing s = (m : ℕ) → (index s m) <= index s (succ m)
tailRespectsWeaklyIncreasing : (a : Sequence A) → WeaklyIncreasing a → WeaklyIncreasing (Sequence.tail a)
tailRespectsWeaklyIncreasing a incr m = incr (succ m)
weaklyIncreasingImplies' : (a : Sequence A) → WeaklyIncreasing a → WeaklyIncreasing' a
weaklyIncreasingImplies' a x zero (succ zero) m<n = x 0
weaklyIncreasingImplies' a x zero (succ (succ n)) m<n = <=Transitive (weaklyIncreasingImplies' a x zero (succ n) (succIsPositive n)) (x (succ n))
weaklyIncreasingImplies' a x (succ m) (succ n) m<n = weaklyIncreasingImplies' (Sequence.tail a) (tailRespectsWeaklyIncreasing a x) m n (canRemoveSuccFrom<N m<n)
weaklyIncreasing'Implies : (a : Sequence A) → WeaklyIncreasing' a → WeaklyIncreasing a
weaklyIncreasing'Implies a incr m = incr m (succ m) (le 0 refl)
StrictlyIncreasing' : (Sequence A) → Set (c)
StrictlyIncreasing' s = (m n : ℕ) → (m <N n) → (index s m) < (index s n)
StrictlyIncreasing : (Sequence A) → Set (c)
StrictlyIncreasing s = (m : ℕ) → (index s m) < index s (succ m)
tailRespectsStrictlyIncreasing : (a : Sequence A) → StrictlyIncreasing a → StrictlyIncreasing (Sequence.tail a)
tailRespectsStrictlyIncreasing a incr m = incr (succ m)
strictlyIncreasingImplies' : (a : Sequence A) → StrictlyIncreasing a → StrictlyIncreasing' a
strictlyIncreasingImplies' a x zero (succ zero) m<n = x 0
strictlyIncreasingImplies' a x zero (succ (succ n)) m<n = <Transitive (strictlyIncreasingImplies' a x zero (succ n) (succIsPositive n)) (x (succ n))
strictlyIncreasingImplies' a x (succ m) (succ n) m<n = strictlyIncreasingImplies' (Sequence.tail a) (tailRespectsStrictlyIncreasing a x) m n (canRemoveSuccFrom<N m<n)
strictlyIncreasing'Implies : (a : Sequence A) → StrictlyIncreasing' a → StrictlyIncreasing a
strictlyIncreasing'Implies a incr m = incr m (succ m) (le 0 refl)