We have now defined the parts of a constructive logic. We have:
We have defined display forms for each of these operators, and included tactics for automating their application.
At this point, we can put together a complete constructive first order logic. The process is pretty simple: we simple define a new module that extends each of the individual theories.
In the interface, we just include each of the module for the operators.
1. Create the constructive interface in the file Fol_ctheory:
extends Fol_type extends Fol_true extends Fol_false extends Fol_implies extends Fol_and extends Fol_or extends Fol_not extends Fol_univ extends Fol_all extends Fol_exists
The implementation includes the same theories as the interface. In addition, we establish display precedence relations that cross the module boundaries.
2. Create the implementation with the extends statements of the interface, and the following relations between the precedences in the file fol_ctheory.ml:
open Fol_implies open Fol_and open Fol_or open Fol_not open Fol_all open Fol_exists prec prec_implies < prec_and prec prec_implies < prec_or prec prec_or < prec_and prec prec_and < prec_not prec prec_all < prec_implies prec prec_exists < prec_implies
At this point, we have a complete theory. Before going on to the classical theory, let's prove a simple theorem.
3. Create a theorem object:
interactive or_choice 'H : sequent ['ext] { 'H >- "type"{'A} } --> sequent ['ext] { 'H >- "type"{'B} } --> sequent ['ext] { 'H >- "type"{'C} } --> sequent ['ext] { 'H >- (('A or 'B) => 'C) => 'A => 'C }
To prove this, compile the FOL directory, and compile the editor with the new module. We can perform the initial steps of the proof using the autoT tactic, which produces the following goal.
The first subgoal is not proved because it requires thinning of the hypothesis ('A or 'B) => 'c. This can be performed in a single step (open the Fol_struct module first).
The second subgoal requires an argument. First, we decompose the hypothesis, to get the two cases.
The second subgoal can be proved with the trivialT tactic, which uses the nthHypT structural rule to prove the goal. For the first goal, we split the conclusion to get the the following goals.
These two subgoals have trivial proof using autoT.
