OpenTheory Article File Format

Overview

An OpenTheory article file encodes a higher order logic theory. Articles take the form of text files using the UTF-8 encoding, and contain commands which are processed by a stack-based virtual machine. The result of reading an article is an import set Γ of assumptions and an export set Δ of theorems. The theory Γ ⊳ Δ encoded by the article consists of a proof that the theorems in Δ logically derive from the assumptions in Γ.

Types

The primitive types processed by the virtual machine are:

int
Integers, such as 0, 42 and -5.
string
Strings, such as "foo" and "".
name
Names in a hierarchical namespace, such as bool or Number.Natural.prime. Names are represented by the type string list * string, where the namespace is represented by the string list component and the local name is represented by the string component. For example, bool is represented by ([],"bool"), and Number.Natural.prime is represented by (["Number","Natural"],"prime"). The namespace represented by the empty string list is called the global namespace.
typeOp
Higher order logic type operators, such as bool and list.
type
Higher order logic types, such as α and bool list.
const
Higher order logic constants, such as T and =. Constants are identified by name.
var
Higher order logic term variables, such as x. Variables are identified by name and type.
term
Higher order logic terms, such as 13 and ∀x. x.
thm
Higher order logic theorems, such as ⊦ x = x and {x = y, y = T} ⊦ x. Theorems consist of a hypothesis term set and conclusion term, both of which are interpreted modulo α-equivalence.

When reading an article file, the virtual machine processes values of type object, which has the following ML-like definition:

datatype object =
        Num of int       (* An integer *)
| Name of name (* A name in a hierarchical namespace *)
| List of object list (* A list (or tuple) of objects *)
| TypeOp of typeOp (* A higher order logic type operator *)
| Type of type (* A higher order logic type *)
| Const of const (* A higher order logic constant *)
| Var of var (* A higher order logic term variable *)
| Term of term (* A higher order logic term *)
| Thm of thm (* A higher order logic theorem *)

Note: Two Thm objects are interchangeable if the theorems they contain are α-equivalent.

Virtual Machine

The virtual machine that reads in an article file is equipped with the following state:

Initially the stack, dictionary, assumptions and theorems are all empty.

The virtual machine reads the article file line by line. Every line in an article file is either a comment line or a command line.

When the virtual machine has finished processing all the lines in the article file, the assumptions and theorems are the result of reading the article (the stack and dictionary are discarded).

Commands

Here is the complete list of commands interpreted by the virtual machine:

  1. i — a decimal integer
  2. "s" — a quoted string
  3. absTerm
  4. absThm
  5. appTerm
  6. appThm
  7. assume
  8. axiom
  9. betaConv
  10. cons
  11. const
  12. constTerm
  13. deductAntisym
  14. def
  15. defineConst
  16. defineTypeOp
  17. eqMp
  18. nil
  19. opType
  20. pop
  21. ref
  22. refl
  23. remove
  24. subst
  25. thm
  26. typeOp
  27. var
  28. varTerm
  29. varType

The remainder of the section describes the precise effects of each command on the virtual machine state.

i — a decimal integer
A number command is a string that matches the following regular expression:

0|[-]?[1-9][0-9]*

Treat the command string as an integer represented in decimal, and convert to the integer i. Push the object Num i onto the stack.
Stack:     Before:     stack
After: Num i
:: stack
"s" — a quoted string
A name command is a string that matches the regular expression NAME, which is defined using the following grammar:
    NAME   ::=   ["] NAMESPACE COMPONENT ["]
    NAMESPACE   ::=   (COMPONENT [.])*
    COMPONENT   ::=   ([^."\] | [\][."\])*
A string matching the regular expression COMPONENT is processed to a string value by converting all occurrences of \c to c. A string matching the regular expression NAMESPACE is processed to a string list value by processing its constituent COMPONENT strings in order. A string matching the regular expression NAME is processed to a string list * string value by discarding its outer double quotes and processing its constituent NAMESPACE and COMPONENT strings. A name command is executed by processing the NAME to (ns,s), and pushing the object Name (ns,s) onto the stack.
Stack:     Before:     stack
After: Name (ns,s)
:: stack
absTerm
Pop a term b; pop a variable v; push the lambda abstraction term λv. b onto the stack.
Stack:     Before:     Term b
:: Var v
:: stack
After: Term (λv. b)
:: stack
absThm
Pop a theorem Γ ⊦ t = u; pop a variable v; push the theorem Γ ⊦ (λv. t) = (λv. u) onto the stack.
Stack:     Before:     Thm (Γ ⊦ t = u)
:: Var v
:: stack
After: Thm (Γ ⊦ (λv. t) = (λv. u))
:: stack
Constraint: The variable v must not be free in Γ.
appTerm
Pop a term x; pop a term f; push the function application term f x onto the stack.
Stack:     Before:     Term x
:: Term f
:: stack
After: Term (f x)
:: stack
Constraint: The term f must have a type of the form σ → τ, and the term x must have a matching type σ.
appThm
Pop a theorem Δ ⊦ x = y; pop a theorem Γ ⊦ f = g; push the theorem Γ∪Δ ⊦ f x = g y onto the stack.
Stack:     Before:     Thm (Δ ⊦ x = y)
:: Thm (Γ ⊦ f = g)
:: stack
After: Thm (Γ∪Δ ⊦ f x = g y)
:: stack
Constraint: The term f must have a type of the form σ → τ, and the term x must have a matching type σ.
assume
Pop a term φ; push the theorem {φ} ⊦ φ onto the stack.
Stack:     Before:     Term φ
:: stack
After: Thm ({φ} ⊦ φ)
:: stack
Constraint: The term φ must have type bool.
axiom
Pop a term φ; pop a list of terms Γ; push the theorem Γ ⊦ φ onto the stack and add it to the article assumptions.
Stack:     Before:     Term φ
:: List [Term t1, ..., Term tn]
:: stack
After: Thm ({t1, ..., tn} ⊦ φ)
:: stack
Assumptions:     Before:     assumptions
After: assumptions
∪ {{t1, ..., tn} ⊦ φ}
Constraint: The terms t1, ..., tn and φ must have type bool.
betaConv
Pop a term (λv. t) u; push the theorem ⊦ (λv. t) u = t[u/v] onto the stack.
Stack:     Before:     Term ((λv. t) u)
:: stack
After: Thm (⊦ (λv. t) u = t[u/v])
:: stack
cons
Pop a list t; pop an object h; push the list h :: t onto the stack.
Stack:     Before:     List t
:: h
:: stack
After: List (h :: t)
:: stack
const
Pop a name n; push a new constant c with name n onto the stack.
Stack:     Before:     Name n
:: stack
After: Const c
:: stack
constTerm
Pop a type ty; pop a constant c; push a constant term t with constant c and type ty.
Stack:     Before:     Type ty
:: Const c
:: stack
After: Term t
:: stack
deductAntisym
Pop a theorem Δ ⊦ ψ; pop a theorem Γ ⊦ φ; push the theorem (Γ - {ψ}) ∪ (Δ - {φ}) ⊦ φ = ψ onto the stack.
Stack:     Before:     Thm (Δ ⊦ ψ)
:: Thm (Γ ⊦ φ)
:: stack
After: Thm ((Γ - {ψ}) ∪ (Δ - {φ}) ⊦ φ = ψ)
:: stack
def
Pop a number k; peek an object x on the stack; update the dictionary so that key k maps to object x.
Stack:     Before:     Num k
:: x
:: stack
After: x
:: stack
Dictionary:     Before:     dict
After: dict[k → x]
defineConst
Pop a term t; pop a name n; push a new constant c with name n; push the theorem ⊦ c = t onto the stack.
Stack:     Before:     Term t
:: Name n
:: stack
After: Thm (⊦ c = t)
:: Const c
:: stack
Constraint: The term t must not have any free term variables.
Constraint: Every type variable that appears in the type of a subterm of t must also appear in the type of t.
defineTypeOp
Pop a theorem ⊦ φ t; pop a list of names A; pop a name rep; pop a name abs; pop a name n; push a new type operator op with name n; push a new constant abs with name abs; push a new constant rep with name rep; push the theorem ⊦ abs (rep a) = a; push the theorem ⊦ φ r = (rep (abs r) = r) onto the stack.
Stack:     Before:     Thm (⊦ φ t)
:: List [Name α1, ..., Name αk]
:: Name rep
:: Name abs
:: Name n
:: stack
After: Thm (⊦ φ r = (rep (abs r) = r))
:: Thm (⊦ abs (rep a) = a)
:: Const rep
:: Const abs
:: TypeOp op
:: stack
Constraint: The predicate φ must not contain any free term variables.
Constraint: A must list all the type variables in φ precisely once.
eqMp
Pop a theorem Γ ⊦ φ; pop a theorem Δ ⊦ φ' = ψ; push the theorem Γ ∪ Δ ⊦ ψ onto the stack.
Stack:     Before:     Thm (Γ ⊦ φ)
:: Thm (Δ ⊦ φ' = ψ)
:: stack
After: Thm (Γ ∪ Δ ⊦ ψ)
:: stack
Constraint: The terms φ and φ' must be α-equivalent.
nil
Push the empty list [] onto the stack.
Stack:     Before:     stack
After: List []
:: stack
opType
Pop a list of types tys; pop a type operator op; push a type with type operator op and arguments tys.
Stack:     Before:     List [Type ty1, ..., Type tyn]
:: TypeOp op
:: stack
After: Type ((ty1, ..., tyn) op)
:: stack
pop
Pop an object from the top of the stack.
Stack:     Before:     x
:: stack
After: stack
ref
Pop a number k from the stack; look up key k in the dictionary to get an object x; push the object x onto the stack.
Stack:     Before:     Num k
:: stack
After: dict[k]
:: stack
Constraint: The number k must be in the domain of the dictionary.
Note: This command reads the dictionary, but does not change it.
refl
Pop a term t; push the theorem ⊦ t = t onto the stack.
Stack:     Before:     Term t
:: stack
After: Thm (⊦ t = t)
:: stack
remove
Pop a number k from the stack; look up key k in the dictionary to get an object x; push the object x onto the stack; delete the entry for key k from the dictionary.
Stack:     Before:     Num k
:: stack
After: dict[k]
:: stack
Dictionary:     Before:     dict
After: dict[entry k deleted]
Constraint: The number k must be in the domain of the dictionary.
Note: This command is exactly the same as the ref command, except that it also deletes the entry from the dictionary.
subst
Pop a theorem Γ ⊦ φ; pop a substitution σ; push the theorem Γ[σ] ⊦ φ[σ] onto the stack.
Stack:     Before:     Thm θ
:: List [List [List [Name α1, Type ty1], ..., List [Name αm, Type tym]],
         List [List [Var v1, Term t1], ..., List [Var vn, Term tn]]]
:: stack
After: Thm ((θ[ty1/α1, ..., tym/αm])[t1/v1, ..., tn/vn])
:: stack
Constraint: The names αi must be in the global namespace (i.e., be of the form ([],si)).
Note: Both the hypothesis set and conclusion of the theorem is instantiated by this rule.
Note: The type variables are instantiated first, followed by the term variables.
Note: Bound variables will be renamed if necessary to prevent distinct variables becoming identical after the instantiation.
thm
Pop a term φ; pop a list of terms Γ; pop a theorem Δ ⊦ ψ from the stack; α-convert the theorem Δ ⊦ ψ to Γ ⊦ φ and add it to the article theorems.
Stack:     Before:     Term φ
:: List [Term t1, ..., Term tn]
:: Thm (Δ ⊦ ψ)
:: stack
After: stack
Theorems:     Before:     theorems
After: theorems
∪ {{t1, ..., tn} ⊦ φ}
Constraint: The terms φ and ψ must be α-equivalent.
Constraint: The term set Δ must be α-equivalent to a subset of Γ.
typeOp
Pop a name n; push a new type operator op with name n onto the stack.
Stack:     Before:     Name n
:: stack
After: TypeOp op
:: stack
var
Pop a type ty; pop a name n; push a term variable v with name n and type ty onto the stack.
Stack:     Before:     Type ty
:: Name n
:: stack
After: Var v
:: stack
Constraint: The name n must be in the global namespace (i.e., be of the form ([],s)).
varTerm
Pop a term variable v; push a variable term t with variable v onto the stack.
Stack:     Before:     Var v
:: stack
After: Term t
:: stack
varType
Pop a name n; push a type variable ty with name n onto the stack.
Stack:     Before:     Name n
:: stack
After: Type ty
:: stack
Constraint: The name n must be in the global namespace (i.e., be of the form ([],s)).
Note: The OpenTheory standard theory library adopts the HOL Light convention of naming type variables with capital letters, and theorem provers are expected to map them to local conventions as part of processing articles. For example, in HOL4 it would be natural to map an OpenTheory type variable with name "A" to a local type variable with name "'a".