This page is designed to be (eventually) a full reference manual for the ghilbert proof verifier.
Ghilbert distinguishes between interfaces and modules and different commands are valid in interfaces or modules. Interfaces live in a .ghi file and proof module live in a .gh file. Interfaces contain axioms and statements to be proved; modules prove those statements. A lemma which is proven in a module but which is not likely to be useful in contexts other than the given proof should not be listed in any interface.
Anything from # to the end of the line is a comment (that is, it is ignored by Ghilbert).
Interfaces contain:
param (identifier page (parameter1 parameter2 ...) prefix) When interfaces with one or more param commands in them are imported or exported, the import or export command must be given a parameter list matching the parameters of the interface. This does not mean that parameters generated from exactly the same page with the correct prefix have to be used (in that case, parameter lists would be somewhat pointless). Instead, any parameters providing the kinds, terms and definitions as the specified page with prefix can be used.
kind (new-kind) For example, kind (formula) says there is a kind called formula.
tvar (kind new-variable-1 new-variable-2...)
var (kind new-variable-1 new-variable-2...) For example, tvar (formula p q r) declares variables p, q and r, all formulas.
term (kind (new-term argument-1 argument-2...) freemap)
For example, term (formula (¬ p)) defines a term called ¬ of kind formula which takes one argument, which is a formula (assuming p had been declared with tvar (formula p)). The name of the term is the first thing in the list, and although the ghilbert web site will sometimes typeset it differently, that is a display feature only; in editing proofs the term comes first. For example, in a proof you might mention (∧ p q), which might get typeset as p ∧ q.
The arguments of a term each have a kind and binding. By binding, we mean that the argument is either a term (if the variable supplied to term was declared with tvar), or the term being defined binds this variable. For example, consider the following example:
kind (formula) tvar (formula φ ψ) kind (object) var (object x y) term (formula (∀ x φ))
In this case ∀ is considered to bind the variable which is the first argument. Ghilbert applies the same treatment to any term with a binding variable; ∀ is not a special case for the verifier. The effect of this treatment is that, for example, y is not considered free in ∀ y ψ, which affects the verifier's processing of freeness constraints.
The freemap consists of entries of the form (variable term). A freemap entry indicates that if the variable occurs free in the specified term, it is considered free in the term being defined. For example;
term (formula (subst A x φ) (x A))
This means that x is considered free in, for example, (subst (x + x) x φ) (because x occurs free in A, which in this case is x + x). If a variable is not mentioned in the freemap, it is considered bound in the term being defined.
stmt (new-statement (freeness-constraints) (hypotheses) (consequent)) For example, one version of the famous modus ponens states that from the hypotheses p and p → q, one can conclude q:
stmt (applyModusPonens () (p (→ p q)) q) :*Freeness constraints (a list of constraints), for example:
((φ x)) ::means that x is not free in φ. The first variable in a constraint must be a term variable; the others are binding variables. All binding variables are implicitly considered distinct from each other, so there is no need for a constraint like (x y) (where x and y are binding variables). A variable satisfies the constraint if it does not appear in the term, and also if it bound in that term (see the discussion under the term command).
import (identifier page (parameter1 parameter2 ...) prefix) For example:
import (NICOD Interface:Nicod_axioms () "") As you can see, interfaces with no parameters are imported with empty parameter lists. If the interface being imported has parameters, you need to first import the interfaces to be used as parameters, and then supply their identifiers as parameter1 and so on. For example:
import (CLASSICAL Interface:Classical_propositional_calculus () "")
import (PRINCIPIA Interface:Axioms_of_first-order_logic (CLASSICAL) "") where the second interface expects Interface:Classical_propositional_calculus or a compatible interface as its only parameter.
The prefix allows the names in the interface to differ from the name in the module (which may be necessary to avoid naming conflicts with some of the imported interfaces, for example). The prefix can be either a string or "" for the empty string.
defthm (theorem-name kind term constraints hypotheses conclusion proof...)
Define a new term of kind kind whose name and arguments are described by term. Also prove a theorem which is specified by constrants, hypotheses, and conclusion, and name it theorem-name.
Unlike a regular proof, the conclusion need not exactly match what is on the proof stack at the end of proof. Everything must exactly match except for uses of the new term being defined, which must follow these rules:
• The matched subexpression from the proof stack must be a term expression, not just a variable. The subexpression's kind must agree with kind.
• Any variable in the matched subexpression that is not one of the corresponding definition term's use's actual arguments is called a definition dummy variable. Each definition dummy variable must also be a proof dummy variable, that is, it must not occur in the hypotheses or conclusion of the theorem.
• Every such matching subexpression is identical. (Specifically, even definition dummy variables must be identical in all the expansions.)
• All definition dummy variables in the definiens must be binding variables, and must not occur explicitly free in the definiens.
• Every actual argument variable of the term use must occur in the matched subexpression.
The conclusion must mention the term being defined (with the same arguments as in term). The conclusion may contain more than one use of the new term being defined, but in that case each occurrence must have the same arguments.
thm (new-theorem (freeness-constraints) (hypotheses) (consequent) proof)
This represents the proof of a theorem.
kindbind (existing-name new-name) For example, if you have a kind called formula and you want .formula to refer to the same kind, specify
kindbind (formula .formula)
export (identifier page (parameter1 parameter2 ...) prefix) See import for a description of identifier, page, parameters, and prefix. For the export to succeed, the proof module must have:
Some of the error messages you may encounter are:
Term freemap mismatch with verify context
This means that you are trying to export to a term, but the thing you are exporting does not agree with the term's freemap. See the discussion of freemaps under term above.
Definition dummy 'x' is not a proof dummy
The variable x appears in the term being defined and also in the conclusion or hypotheses. See the discussion under defthm above.
term xxx not known
This means that xxx is the first item in an expression, but it was not defined by term or defthm. One common cause of this error is writing, for example (p → q) instead of (→ p q).
term must be id, found (x y)
Similar to the previous error, but the first item in the expression is not even an identifier. For example, in ((→ p q)) (with too many parentheses), the term is (→ p q), not → as intended.
Other related systems, which may help you understand ghilbert (especially until this manual is a bit more complete) are: