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Idris2/libs/contrib/Syntax/PreorderReasoning/Generic.idr

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Division theorem (#695) Division Theorem. For every natural number `x` and positive natural number `n`, there is a unique decomposition: `x = q*n + r` with `q`,`r` natural and `r` < `n`. `q` is the quotient when dividing `x` by `n` `r` is the remainder when dividing `x` by `n`. This commit adds a proof for this fact, in case we want to reason about modular arithmetic (for example, when dealing with binary representations). A future, more systematic, development could perhaps follow: @clayrat 's (idris1) port of Coq's binary arithmetics: https://github.com/sbp/idris-bi/blob/master/src/Data/Bin/DivMod.idr https://github.com/sbp/idris-bi/blob/master/src/Data/Biz/DivMod.idr https://github.com/sbp/idris-bi/blob/master/src/Data/BizMod2/DivMod.idr In the process, it bulks up the stdlib with: + a generic PreorderReasoning module for arbitrary preorders, analogous for the equational reasoning module + some missing facts about Nat operations. + Refactor some Nat order properties using a 'reflect' function Co-authored-by: Ohad Kammar <ohad.kammar@ed.ac.uk> Co-authored-by: G. Allais <guillaume.allais@ens-lyon.org>
2020-10-06 15:09:02 +03:00
module Syntax.PreorderReasoning.Generic
Define and implement Relation interfaces (#1472) Co-authored-by: Guillaume ALLAIS <guillaume.allais@ens-lyon.org>
2021-07-09 11:06:27 +03:00
import Control.Relation
import Control.Order
Division theorem (#695) Division Theorem. For every natural number `x` and positive natural number `n`, there is a unique decomposition: `x = q*n + r` with `q`,`r` natural and `r` < `n`. `q` is the quotient when dividing `x` by `n` `r` is the remainder when dividing `x` by `n`. This commit adds a proof for this fact, in case we want to reason about modular arithmetic (for example, when dealing with binary representations). A future, more systematic, development could perhaps follow: @clayrat 's (idris1) port of Coq's binary arithmetics: https://github.com/sbp/idris-bi/blob/master/src/Data/Bin/DivMod.idr https://github.com/sbp/idris-bi/blob/master/src/Data/Biz/DivMod.idr https://github.com/sbp/idris-bi/blob/master/src/Data/BizMod2/DivMod.idr In the process, it bulks up the stdlib with: + a generic PreorderReasoning module for arbitrary preorders, analogous for the equational reasoning module + some missing facts about Nat operations. + Refactor some Nat order properties using a 'reflect' function Co-authored-by: Ohad Kammar <ohad.kammar@ed.ac.uk> Co-authored-by: G. Allais <guillaume.allais@ens-lyon.org>
2020-10-06 15:09:02 +03:00
infixl 0 ~~
infixl 0 <~
prefix 1 |~
infix 1 ...
public export
data Step : (leq : a -> a -> Type) -> a -> a -> Type where
[ fix #735 ] Make sure type constructors are fully applied
2020-10-16 00:44:30 +03:00
(...) : (y : a) -> x `leq` y -> Step leq x y
Division theorem (#695) Division Theorem. For every natural number `x` and positive natural number `n`, there is a unique decomposition: `x = q*n + r` with `q`,`r` natural and `r` < `n`. `q` is the quotient when dividing `x` by `n` `r` is the remainder when dividing `x` by `n`. This commit adds a proof for this fact, in case we want to reason about modular arithmetic (for example, when dealing with binary representations). A future, more systematic, development could perhaps follow: @clayrat 's (idris1) port of Coq's binary arithmetics: https://github.com/sbp/idris-bi/blob/master/src/Data/Bin/DivMod.idr https://github.com/sbp/idris-bi/blob/master/src/Data/Biz/DivMod.idr https://github.com/sbp/idris-bi/blob/master/src/Data/BizMod2/DivMod.idr In the process, it bulks up the stdlib with: + a generic PreorderReasoning module for arbitrary preorders, analogous for the equational reasoning module + some missing facts about Nat operations. + Refactor some Nat order properties using a 'reflect' function Co-authored-by: Ohad Kammar <ohad.kammar@ed.ac.uk> Co-authored-by: G. Allais <guillaume.allais@ens-lyon.org>
2020-10-06 15:09:02 +03:00
public export
[ fix #735 ] Make sure type constructors are fully applied
2020-10-16 00:44:30 +03:00
data FastDerivation : (leq : a -> a -> Type) -> (x : a) -> (y : a) -> Type where
(|~) : (x : a) -> FastDerivation leq x x
(<~) : {x, y : a}
-> FastDerivation leq x y -> {z : a} -> (step : Step leq y z)
-> FastDerivation leq x z
Division theorem (#695) Division Theorem. For every natural number `x` and positive natural number `n`, there is a unique decomposition: `x = q*n + r` with `q`,`r` natural and `r` < `n`. `q` is the quotient when dividing `x` by `n` `r` is the remainder when dividing `x` by `n`. This commit adds a proof for this fact, in case we want to reason about modular arithmetic (for example, when dealing with binary representations). A future, more systematic, development could perhaps follow: @clayrat 's (idris1) port of Coq's binary arithmetics: https://github.com/sbp/idris-bi/blob/master/src/Data/Bin/DivMod.idr https://github.com/sbp/idris-bi/blob/master/src/Data/Biz/DivMod.idr https://github.com/sbp/idris-bi/blob/master/src/Data/BizMod2/DivMod.idr In the process, it bulks up the stdlib with: + a generic PreorderReasoning module for arbitrary preorders, analogous for the equational reasoning module + some missing facts about Nat operations. + Refactor some Nat order properties using a 'reflect' function Co-authored-by: Ohad Kammar <ohad.kammar@ed.ac.uk> Co-authored-by: G. Allais <guillaume.allais@ens-lyon.org>
2020-10-06 15:09:02 +03:00
[ fix #735 ] Make sure type constructors are fully applied
2020-10-16 00:44:30 +03:00
public export
Change PreorderReasoning arguments to 0
2021-02-12 18:36:38 +03:00
CalcWith : Preorder dom leq => {0 x : dom} -> {0 y : dom} -> FastDerivation leq x y -> x `leq` y
Define and implement Relation interfaces (#1472) Co-authored-by: Guillaume ALLAIS <guillaume.allais@ens-lyon.org>
2021-07-09 11:06:27 +03:00
CalcWith (|~ x) = reflexive {rel = leq}
CalcWith ((<~) der (z ... step)) = transitive {rel = leq} (CalcWith der) step
Division theorem (#695) Division Theorem. For every natural number `x` and positive natural number `n`, there is a unique decomposition: `x = q*n + r` with `q`,`r` natural and `r` < `n`. `q` is the quotient when dividing `x` by `n` `r` is the remainder when dividing `x` by `n`. This commit adds a proof for this fact, in case we want to reason about modular arithmetic (for example, when dealing with binary representations). A future, more systematic, development could perhaps follow: @clayrat 's (idris1) port of Coq's binary arithmetics: https://github.com/sbp/idris-bi/blob/master/src/Data/Bin/DivMod.idr https://github.com/sbp/idris-bi/blob/master/src/Data/Biz/DivMod.idr https://github.com/sbp/idris-bi/blob/master/src/Data/BizMod2/DivMod.idr In the process, it bulks up the stdlib with: + a generic PreorderReasoning module for arbitrary preorders, analogous for the equational reasoning module + some missing facts about Nat operations. + Refactor some Nat order properties using a 'reflect' function Co-authored-by: Ohad Kammar <ohad.kammar@ed.ac.uk> Co-authored-by: G. Allais <guillaume.allais@ens-lyon.org>
2020-10-06 15:09:02 +03:00
public export
Change PreorderReasoning arguments to 0
2021-02-12 18:36:38 +03:00
(~~) : {0 x : dom} -> {0 y : dom}
[ fix #735 ] Make sure type constructors are fully applied
2020-10-16 00:44:30 +03:00
-> FastDerivation leq x y -> {z : dom} -> (step : Step Equal y z)
-> FastDerivation leq x z
Division theorem (#695) Division Theorem. For every natural number `x` and positive natural number `n`, there is a unique decomposition: `x = q*n + r` with `q`,`r` natural and `r` < `n`. `q` is the quotient when dividing `x` by `n` `r` is the remainder when dividing `x` by `n`. This commit adds a proof for this fact, in case we want to reason about modular arithmetic (for example, when dealing with binary representations). A future, more systematic, development could perhaps follow: @clayrat 's (idris1) port of Coq's binary arithmetics: https://github.com/sbp/idris-bi/blob/master/src/Data/Bin/DivMod.idr https://github.com/sbp/idris-bi/blob/master/src/Data/Biz/DivMod.idr https://github.com/sbp/idris-bi/blob/master/src/Data/BizMod2/DivMod.idr In the process, it bulks up the stdlib with: + a generic PreorderReasoning module for arbitrary preorders, analogous for the equational reasoning module + some missing facts about Nat operations. + Refactor some Nat order properties using a 'reflect' function Co-authored-by: Ohad Kammar <ohad.kammar@ed.ac.uk> Co-authored-by: G. Allais <guillaume.allais@ens-lyon.org>
2020-10-06 15:09:02 +03:00
(~~) der (z ... Refl) = der
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