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Raphaël Cauderlier
math_transfer
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cc9cac58
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cc9cac58
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Apr 15, 2017
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Raphael Cauderlier
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Readme Update
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@@ 53,15 +53,17 @@ The following tools are required to compile the MathTransfer library:
Sukerujo is an alternative Dedukti parser used by FoCaLiZe
 Zenon Modulo version 0.4.3
 Zenon Modulo version 0.4.3
Focalide
 Dktactics
Dktactics is a tactic language for Dedukti

Dkt
ransfer

Zenon T
ransfer
Dktransfer is a transfer tactic for Dedukti
Zenon Transfer is a patched version of Zenon Modulo that does not
perform proof search by itself (unlike Zenon Modulo) but uses
Dktactics to prove transfer theorems.
Moreover, for the interoperability example, the following extra tools
are required to import logical developments from other proof systems:
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@@ 133,42 +135,44 @@ the natural numbers. They can be imported from any arithmetic library.
The logic used in the MathTransfer library is an higherorder logic
seen as an extension of multisorted firstorder logic.
**** TODO detail the description of the logic
*** Arithmetic
***
Natural Number
Arithmetic
The directory [[./lib/arith][lib/arith]] contains the definitions of arithmetical
structures, that is mathematical structures about natural numbers and
integers.
structures, that is mathematical structures about natural numbers.
**** Natural numbers
The file [[./lib/arith/naturals.fcl][lib/arith/naturals.fcl]] defines the following species
**** Definitions
The file [[./lib/arith/naturals.fcl][lib/arith/definitions/naturals.fcl]] defines the following species
representing mathematical structures built from natural numbers with
various operations and relations:
 Peano axioms
 Zenon and Successor
The first building block of the arithmetic library is the =NatDecl=
species in file =naturals.fcl=. This species requires two
functions: =zero= of type =Self= and =succ= of type =Self > Self=
but does not specify them.
 Unit
=one= is a constant of type =Self= specified as being equal to the
successor of zero.
 Binary notation
Peano axiomatization of natural numbers forms the first building
block of the arithmetic library: the =NatDecl= species in file
=naturals.fcl=. This species requires two functions: =zero= of type
=Self= and =succ= of type =Self > Self= and three axioms:
=zero_succ= expressing that =zero= is the successor of no natural
number, =succ_inj= expressing the injectivity of =succ= and =ind=
expressing the induction axiom. We see =ind= as a single
firstorder axiom by representing predicate (and function)
application as a binary operation =@=.
The function =bit0= computes the double of a natural number
(=bit0(zero) = zero=, =bit0(succ(n)) = succ(succ(bit0(n)))=). Hence
it corresponds to adding a =0= bit in binary notation.
 Reasoning by case (all numbers are zero or successors)
The function =bit1= adds a =1= bit in binary notation so =bit1(n) =
succ(bit0(n))=.
In FoCaLiZe, firstorder statements are easier to use than
higherorder ones because the firstorder theorem prover Zenon can
assist the user in the first case and not in the second case.
 Predecessor
An inductive argument that does not actually use the induction
hypothesis is the same thing than reasoning by case on a natural
number and this can be stated as a firstorder theorem: any natural
number is either zero or the successor of some natural number.
The predecessor of =n= is =m= if =n= is =succ(m)= and =zero=
if =n= is =zero=.
 Addition
...
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@@ 189,69 +193,118 @@ integers.
=le= is a binary relation over natural numbers defined by the following axiom:
+ =le(m, n) <> (ex p : Self, plus(m, p) = n)=
 Subtraction
=minus= is a binary operation over natural numbers defined by the
following axioms:
+ =minus(zero, n) = zero=
+ =minus(succ(n), zero) = succ(n)=
+ =minus(succ(m), succ(n)) = minus(m, n)=
 Strict ordering
=lt= is a binary relation over natural numbers defined by the following axiom:
+ =lt(m, n) <> le(succ(m), n)=
 Divisibility
**** Morphisms
=divides= is a binary relation over natural numbers defined by the following axiom:
+ =divides(m, n) <> (ex p : Self, times(m, p) = n)=
The files in the directory [[./lib/arith/morphisms][lib/arith/morphisms]] define isomorphisms
between representations of natural numbers. Three representations of
morphisms are proposed:
 Strict Divisibility
 The relational representation, suitable for automatic transfer, is
proposed in file [[./lib/arith/morphisms/natmorph_rel.fcl][lib/arith/morphisms/natmorph_rel.fcl]]. An
isomorphism is there defines as a relation =morph= of type =A >
Self > prop= that is assumed
+ functional: it preserves equality
+ injective: it reflects equality (its inverse is functional)
+ surjective
+ total: its inverse is also surjective
=sd= is a binary relation over natural numbers defined by the following axiom:
+ =sd(m, n) <> (divides(m, n) /\ lt(m, n) /\ lt(succ(zero), m))=
Moreover, it preserves and reflects all the operation of the
structure.
 Primality
 The functional representation, easier to instantiate with functional
proof systems, is proposed in file
[[./lib/arith/morphisms/natmorph.fcl][lib/arith/morphisms/natmorph.fcl]]. An isomorphism is a function
=morph= of type =Self > B= that preserves =zero= and =succ= . Peano
axioms are assumed in both =Self= and =B=. Injectivity,
surjectivity, and preservation of all arithmetic operations are
proved (using manual Dedukti instantiation of the induction axiom).
=prime= is a predicate over natural numbers defined by the following axiom:
+ =prime(p) <> lt(succ(zero), p) /\ all d : Self, ~(sd(d, p))=
 The reversed functional representation is similar but the morphism
is seen as a function of type =A > Self=.
The file
**** Theorems
**** TODO Integers
The file [[./lib/arith/theorems/natthms.fcl][lib/arith/theorems/natthms.fcl]] extend the hierarchy of
natural number structures by requiring common properties of arithmetic
operations. The statements are taken from OpenTheory standard library.
**** Transfer
The file [[./lib/arith/transfer/nattransfer.fcl][lib/arith/transfer/nattransfer.fcl]] combines the morphism
hierarchy with the theorem hierachy. If the theorems of file
[[./lib/arith/theorems/natthms.fcl][lib/arith/theorems/natthms.fcl]] hold for a collection =A= and if the
current species is isomorphic to =A= then the theorems also hold for
the current species.
These transfer theorems are automatically proved by a Meta Dedukti
tactic defined in the directory [[./lib/arith/transfer/meta][lib/arith/transfer/meta/]]. The calls to
this tactic are produced by Zenon Transfer.
* The Interoperability Example
* TODO Related work
The directory [[./example][example]] illustrates how the MathTransfer library can be
used for interoperability of proof systems. The chosen systems for
this example are Coq and HOL (more precisely, the OpenTheory proof
format for proof assistants of the HOL family).
The theorem proved using Coq and HOL in combination is a correctness
proof of the Sieve of Eratosthenes. The algorithm and the proof are
written in Coq and the arithmetic results required are imported from
OpenTheory.
**
Automating proofs of transfer theorems
**
Logic Combination
*** In Isabelle/HOL
The directory [[./example/logic][example/logic]] contains the Dedukti combination of the
logics of Coq (defined in file [[./example/logic/Coq.dk][Coq.dk]]) and HOL (defined in file
[[./lib/logic/hol.dk][hol.dk]]). The combination is defined in file [[./example/logic/hol_to_coq.dk][hol_to_coq.dk]], it relies
on a definition of inhabited types in Coq (file [[./example/logic/holtypes.v][holtypes.v]]).
*** In Coq
For more details on this logic combination, see
[[https://arxiv.org/pdf/1507.08721.pdf]].
**
* In Dedukti
**
Extenstion of the hierarchies
** TODO Proof reuse
To state the lemmas that we need to transfer from HOL to Coq, we first
extend the hierarchies of mathematical structures to axiomatize
divisibility and primality. More precisely, in file [[./example/arith/natural_full.fcl][natural_full.fcl]],
we define the following operations:
 Divisibility
Nicolas Magaud
=divides= is a binary relation over natural numbers defined by the following axiom:
+ =divides(m, n) <> (ex p : Self, times(m, p) = n)=
 Strict Divisibility
=sd= is a binary relation over natural numbers defined by the following axiom:
+ =sd(m, n) <> (divides(m, n) /\ lt(m, n) /\ lt(succ(zero), m))=
 Primality
=prime= is a predicate over natural numbers defined by the following axiom:
+ =prime(p) <> lt(succ(zero), p) /\ all d : Self, ~(sd(d, p))=
Planning?
We then require two properties:
 if =m= divides =n= and 1 < =n=, then =m= is smaller or equal to =n=,
 if =n= is not 1, then =n= has a prime divisor.
** TODO Univalent foundations
We then extend the morphism hierarchy to the new operations and
transfer both theorems.
Ongoing program of founding mathematics on type theory using the
univalent axiom. This work does not assume an extra axiom. Can be
related to computational content of univalence.
** Instantiations
* Future work
In file [[./example/arith/natural_hol.fcl][natural_hol.fcl]], we fully instantiate the hierarchies with the
definitions and proofs of OpenTheory. In file [[./example/arith/natural_coq.fcl][natural_coq.fcl]], we
instantiate the definition hierarchy with the definitions of Coq and
use transfer to instantiate the theorem hierarchy. Finally, both
instantiations are combined in file [[./example/arith/final_coll.fcl][final_coll.fcl]].
Better distinction between bool and Prop: because of the use of
OpenTheory stdlib, we tend to confuse bool and Prop. Decidable
operations on natural numbers should be in bool.
** Final Proof
Setoids: The development would be a bit more general if it did not
depend on equality. This would make the library more compatible with
FoCaLiZe stdlib.
The Coq proof, with the OpenTheory lemmas as hypotheses, is done in
file [[./example/arith/sieve.v][sieve.v]]. The missing bits are added in Dedukti in file [[./example/arith/final.dk][final.dk]].
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