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  • A B M i j A M j k B M i k B m m m λ m i j k A B A M j k B M i j A M i k B f c c f join i j k A M i j M j k A M i k A join m m id rawIApplicative RawIApplicative M rawIApplicative record pure return λ f

    Original URL path: http://nicolaspouillard.fr/publis/pouillard-pottier-fresh-look-agda-2010/html/Category.Monad.Indexed.html (2015-10-11)
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  • ι ο Ty data Constant Set where num Constant suc Constant Constant natFix τ Ty Constant module Make base Base where open Base base data Tm α Set where V x Name α Tm α t u Tm α Tm

    Original URL path: http://nicolaspouillard.fr/publis/pouillard-pottier-fresh-look-agda-2010/html/NotSoFresh.Examples.Term.DataTypes.html (2015-10-11)
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  • 4 field pure i A A F i i A i j k A B F i j A B F j k A F i k B rawFunctor i j RawFunctor F i j rawFunctor record λ g x pure g x private open module RF i j I RawFunctor rawFunctor i i j j public i j k A B F i j A F j k B

    Original URL path: http://nicolaspouillard.fr/publis/pouillard-pottier-fresh-look-agda-2010/html/Category.Applicative.Indexed.html (2015-10-11)
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  • contains some core definitions which are reexported by Algebra FunctionProperties They are placed here because Algebra FunctionProperties is a parameterised module and some of the parameters are irrelevant for these definitions module Algebra FunctionProperties Core where open import Level Unary

    Original URL path: http://nicolaspouillard.fr/publis/lfmtp2012-talk/html/Algebra.FunctionProperties.Core.html (2015-10-11)
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  • n 0 0 zero refl n 0 0 suc n begin suc n 0 refl n 0 0 n 0 n 0 0 n 0 Note that some modules contain generalised versions of specific instantiations of this module For instance the module Reasoning in Relation Binary PropositionalEquality is recommended for equational reasoning when the underlying equality is Relation Binary PropositionalEquality open import Relation Binary module Relation Binary EqReasoning s s

    Original URL path: http://nicolaspouillard.fr/publis/lfmtp2012-talk/html/Relation.Binary.EqReasoning.html (2015-10-11)
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  • P n CRec P n cRec builder RecursorBuilder CRec cRec builder P f zero tt cRec builder P f suc n f n ih ih where ih cRec builder P f n cRec Recursor CRec cRec build cRec builder Complete induction based on Rec RecStruct Rec WfRec mutual well founded Well founded well founded n acc well founded n well founded n Rec Acc n well founded zero well founded

    Original URL path: http://nicolaspouillard.fr/publis/lfmtp2012-talk/html/Induction.Nat.html (2015-10-11)
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  • Fin n eq suc n k to r suc q n lem n k q r eq begin suc n k solve 2 λ n k con 1 n k n con 1 k refl n k n suc k cong n eq n to r q n solve 3 λ n r q n r q n r con 1 q n refl n to r q to r suc q n Division A specification of integer division data DivMod Set where result divisor q r Fin divisor DivMod to r q divisor divisor Sometimes the following type is more usable functions in indices can be inconvenient data DivMod dividend divisor Set where result q r Fin divisor eq dividend to r q divisor DivMod dividend divisor Integer division with remainder Note that Induction Nat rec is used to establish termination of division The run time complexity of this implementation of integer division should be linear in the size of the dividend assuming that well founded recursion and the equality type are optimised properly see Inductive Families Need Not Store Their Indices Brady McBride McKinna TYPES 2003 divMod dividend divisor 0 False divisor 0 DivMod dividend divisor divMod m n 0 rec Pred dm m n 0 where Pred Set Pred dividend divisor 0 False divisor 0 DivMod dividend divisor 1 k n DivMod suc k n DivMod suc n k n 1 k n result q r eq result 1 q r lem n k q r eq dm dividend Rec Pred dividend Pred dividend dm m rec n dm m rec zero 0 dm zero rec suc n result 0 zero refl dm suc m rec suc n with compare m n dm suc m rec suc suc m k less m k result 0 r lem

    Original URL path: http://nicolaspouillard.fr/publis/lfmtp2012-talk/html/Data.Nat.DivMod.html (2015-10-11)
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  • λ y y x f y wfRec builder P f y rs y y x wfRec SubsetRecursor Acc WfRec wfRec subsetBuild wfRec builder Well founded induction for all elements assuming they are all accessible module All a A Set a Rel A a wf Well founded where wfRec builder RecursorBuilder WfRec wfRec builder P f x Some wfRec builder P f x wf x wfRec Recursor WfRec wfRec build wfRec builder It might be useful to establish proofs of Acc or Well founded using combinators such as the ones below see for instance Constructing Recursion Operators in Intuitionistic Type Theory by Lawrence C Paulson module Subrelation a A Set a Rel A a x y x y x y where accessible Acc Acc accessible acc rs acc λ y y x accessible rs y y x well founded Well founded Well founded well founded wf λ x accessible wf x module Inverse image A B Set Rel B f A B where accessible x Acc f x Acc on f x accessible acc rs acc λ y fy fx accessible rs f y fy fx well founded Well founded Well founded on f well founded wf λ x accessible wf f x module Transitive closure a A Set a Rel A a where infix 4 data Rel A a where x y x y x y x y trans x y z x y x y y z y z x z downwards closed x y Acc y x y Acc x downwards closed acc rs x y acc λ z z x rs z trans z x x y mutual accessible x Acc x Acc x accessible acc x acc accessible acc x accessible x Acc x WfRec Acc x accessible acc rs y y x accessible rs y

    Original URL path: http://nicolaspouillard.fr/publis/lfmtp2012-talk/html/Induction.WellFounded.html (2015-10-11)
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