mirror of
https://github.com/Smaug123/agdaproofs
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68 lines
2.0 KiB
Agda
68 lines
2.0 KiB
Agda
{-# OPTIONS --safe --warning=error --without-K #-}
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open import LogicalFormulae
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open import Orders.Total.Definition
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open import Orders.Partial.Definition
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open import Setoids.Setoids
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open import Setoids.Orders.Partial.Definition
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open import Setoids.Orders.Total.Definition
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open import Functions.Definition
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open import Sets.EquivalenceRelations
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open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
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module Setoids.Orders.Total.Lemmas {a b : _} {A : Set a} {S : Setoid {a} {b} A} {c : _} {_<_ : A → A → Set c} {P : SetoidPartialOrder S _<_} (T : SetoidTotalOrder P) where
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open SetoidTotalOrder T
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open SetoidPartialOrder P
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open Setoid S
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open Equivalence eq
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maxInequalitiesR : {a b c : A} → (a < b) → (a < c) → (a < max b c)
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maxInequalitiesR {a} {b} {c} a<b a<c with totality b c
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... | inl (inl x) = a<c
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... | inl (inr x) = a<b
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... | inr x = a<c
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minInequalitiesR : {a b c : A} → (a < b) → (a < c) → (a < min b c)
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minInequalitiesR {a} {b} {c} a<b a<c with totality b c
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... | inl (inl x) = a<b
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... | inl (inr x) = a<c
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... | inr x = a<b
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maxInequalitiesL : {a b c : A} → (a < c) → (b < c) → (max a b < c)
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maxInequalitiesL {a} {b} {c} a<b a<c with totality a b
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... | inl (inl x) = a<c
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... | inl (inr x) = a<b
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... | inr x = a<c
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minInequalitiesL : {a b c : A} → (a < c) → (b < c) → (min a b < c)
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minInequalitiesL {a} {b} {c} a<b a<c with totality a b
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... | inl (inl x) = a<b
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... | inl (inr x) = a<c
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... | inr x = a<b
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minLessL : (a b : A) → min a b <= a
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minLessL a b with totality a b
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... | inl (inl x) = inr reflexive
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... | inl (inr x) = inl x
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... | inr x = inr reflexive
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minLessR : (a b : A) → min a b <= b
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minLessR a b with totality a b
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... | inl (inl x) = inl x
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... | inl (inr x) = inr reflexive
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... | inr x = inr x
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maxGreaterL : (a b : A) → a <= max a b
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maxGreaterL a b with totality a b
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... | inl (inl x) = inl x
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... | inl (inr x) = inr reflexive
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... | inr x = inr x
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maxGreaterR : (a b : A) → b <= max a b
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maxGreaterR a b with totality a b
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... | inl (inl x) = inr reflexive
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... | inl (inr x) = inl x
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... | inr x = inr reflexive
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