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  • these parameters two tactics are returned prove and solve For an example of the use of this module see Algebra RingSolver module Relation Binary Reflection e a s Expr Set e A Set a Sem Setoid a s var n Fin n Expr n n Expr n Vec A n Setoid Carrier Sem correct n e Expr n ρ e ρ Setoid Sem e ρ where open import Data Vec N ary open import Data Product import Relation Binary EqReasoning as Eq open Setoid Sem open Eq Sem If two normalised expressions are semantically equal then their non normalised forms are also equal prove n ρ Vec A n e e e ρ e ρ e ρ e ρ prove ρ e e hyp begin e ρ sym correct e ρ e ρ hyp e ρ correct e ρ e ρ Applies the function to all possible variables close A Set e n N ary n Expr n A A close n f f Vec map var allFin n A variant of prove which should in many cases be easier to use because variables and environments are handled in a less explicit way If the type signature of solve is a bit daunting then it may be helpful to instantiate n with a small natural number and normalise the remainder of the type solve n f N ary n Expr n Expr n Expr n Eq n curry proj close n f curry proj close n f Eq n curry proj close n f curry proj close n f solve n f hyp curry cong proj close n f proj close n f λ ρ prove ρ proj close n f proj close n f curry cong proj close n f proj close n f Eq to Eq n hyp

    Original URL path: http://nicolaspouillard.fr/publis/lfmtp2012-talk/html/Relation.Binary.Reflection.html (2015-10-11)
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  • Relation Nullary p P Set p Dec P Bool yes true no false True p P Set p Dec P Set True Q T Q False p P Set p Dec P Set False Q T not Q Gives a witness to the truth toWitness p P Set p Q Dec P True Q P toWitness Q yes p p toWitness Q no Establishes a truth given a witness fromWitness p P Set p Q Dec P P True Q fromWitness Q yes p const fromWitness Q no p p map p q P Set p Q Set q P Q Dec P Dec Q map P Q yes p yes Equivalence to P Q p map P Q no p no p Equivalence from P Q map p q P Set p Q Set q P Q Q P Dec P Dec Q map P Q Q P map equivalence P Q Q P If a decision procedure returns yes then we can extract the proof using from yes From yes p P Set p Dec P Set p From yes P P yes P From yes no Lift from yes p P Set p p Dec P From

    Original URL path: http://nicolaspouillard.fr/publis/lfmtp2012-talk/html/Relation.Nullary.Decidable.html (2015-10-11)
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  • k r 1 x 2 k Digits Digit b is the type of digits in base b Digit Set Digit b Fin b Some specific digit kinds Decimal Digit 10 Bit Digit 2 Some named digits 0b Bit 0b zero 1b Bit 1b suc zero Showing digits The characters used to show the first 16 digits digitChars Vec Char 16 digitChars 0 1 2 3 4 5 6 7 8 9 a b c d e f showDigit shows digits in base 16 showDigit base base 16 True base 16 Digit base Char showDigit base 16 base 16 d Vec lookup Fin inject d toWitness base 16 digitChars Digit expansions fromDigits takes a digit expansion of a natural number starting with the least significant digit and returns the corresponding natural number fromDigits base List Fin base fromDigits 0 fromDigits base d ds to d fromDigits ds base toDigits b n yields the digits of n in base b starting with the least significant digit Note that the list of digits is always non empty This function should be linear in n if optimised properly see Data Nat DivMod data Digits base Set where digits ds List Fin base Digits base

    Original URL path: http://nicolaspouillard.fr/publis/lfmtp2012-talk/html/Data.Digit.html (2015-10-11)
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  • indexedSetoid To Identity and composition id a a A B Setoid a a A A id record Fun id cong Fun id infixr 9 a a A B Setoid a a b b B B Setoid b b c c C B Setoid c c B C A B A C f g record Fun f g cong Fun cong f cong g Constant equality preserving function const a a A B Setoid a a b b B B Setoid b b B Setoid Carrier B A B const B B b record Fun const b cong Fun const B Setoid refl B Function setoids Dependent setoid f f t t From B Setoid f f I Setoid B Setoid Carrier From t t B Setoid setoid From To record Carrier Π From To λ f g x y x y f x g y isEquivalence record refl λ f cong f sym λ f g x y To sym f g From sym x y trans λ f g g h x y To trans f g From refl g h x y where open module From B Setoid From using renaming to open module To I Setoid To

    Original URL path: http://nicolaspouillard.fr/publis/lfmtp2012-talk/html/Function.Equality.html (2015-10-11)
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  • with η equality which is often nice to have but sometimes it is convenient to be able to stop unfolding see Hidden types below data Unit Set where unit Unit Hidden types Hidden values Hidden a Set a Set a Hidden A Unit A The hide function can be used to hide function applications Note that the type checker doesn t see that hide f x contains the application f

    Original URL path: http://nicolaspouillard.fr/publis/lfmtp2012-talk/html/Data.Unit.Core.html (2015-10-11)
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  • Rel A Rel A Irreflexive Antisymmetric Asymmetric irr antisym asym irrefl antisym λ x y y x irrefl antisym x y y x x y asym antisym a A Set a Rel A Rel A Asymmetric Antisymmetric asym antisym asym x y y x elim asym x y y x asym irr a A Set a Rel A Rel A Respects Symmetric Asymmetric Irreflexive asym irr resp sym asym x y x y x y asym x y y x where y y y y y y proj resp x y x y y x y x y x proj resp sym x y y y total refl a A Set a Rel A Rel A Respects Symmetric Total total refl resp sym total refl where refl refl x y x y with total x y inj x y x y inj y x proj resp x y proj resp sym x y y x total dec dec a A Set a Rel A Rel A Antisymmetric Total Decidable Decidable total dec dec refl antisym total dec where dec Decidable dec x y with total x y inj x y yes x y inj y x with x y yes x y yes refl x y no x y no λ x y x y antisym x y y x tri asym a A Set a Rel A Rel A Trichotomous Asymmetric tri asym tri x y x y x y with tri x y tri x y x y x y tri x y x y x y tri x y x y x y tri irr a A Set a Rel A Rel A Respects Symmetric Trichotomous Irreflexive tri irr resp sym tri asym irr resp sym tri asym tri tri dec a A Set a Rel A

    Original URL path: http://nicolaspouillard.fr/publis/lfmtp2012-talk/html/Relation.Binary.Consequences.html (2015-10-11)
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  • Homogeneous Rel i a I Set i I Set a Level Set Rel A REL A A Simple properties of indexed binary relations Reflexivity Reflexive i a I Set i A I Set a Rel A Set Reflexive i B Reflexive i Symmetry Symmetric i a I Set i A I Set a Rel A Set Symmetric i j B Sym i j Transitivity Transitive i a I Set i

    Original URL path: http://nicolaspouillard.fr/publis/lfmtp2012-talk/html/Relation.Binary.Indexed.Core.html (2015-10-11)
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  • B Set a b where constructor mkBijectiveREL field injectiveREL InjectiveREL A B surjectiveREL SurjectiveREL A B BijectiveRel a A Set a Rel A Set BijectiveRel A BijectiveREL A A private probably already defined somewhere on a A Set a f A A Rel A Rel A on f i j f i f j module Pres Bij Props a A Set a f A A f inj i j f

    Original URL path: http://nicolaspouillard.fr/publis/lfmtp2012-talk/html/Relation.Binary.Bijection.html (2015-10-11)
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