https://github.com/light2802/theorem-prover

https://github.com/light2802/theorem-prover

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• Host: GitHub
• URL: https://github.com/light2802/theorem-prover
• Owner: light2802
• Created: 2021-11-09T20:16:03.000Z (over 4 years ago)
• Default Branch: master
• Last Pushed: 2021-11-09T20:18:26.000Z (over 4 years ago)
• Last Synced: 2025-06-03T21:30:09.255Z (about 1 year ago)
• Language: Python
• Size: 6.84 KB
• Stars: 0
• Watchers: 1
• Forks: 0
• Open Issues: 0
• Metadata Files:
  • Readme: README.md

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README

          
An automated theorem prover for first-order logic. For any provable formula, this program is guaranteed to find the proof (eventually). However, as a consequence of the negative answer to Hilbert's *Entscheidungsproblem*, there are some unprovable formulae that will cause this program to loop forever.

Some notes:

* The proof steps are shown as [sequents](http://en.wikipedia.org/wiki/Sequent).

* The actual theorem prover is in `prover.py`. The command-line interface (including the parser) is in `main.py`. `language.py` contains boilerplate classes used to represent logical formulae.

* The system will not accept a lemma unless it can be proven. An axiom is admitted without proof.

* This is only a pedagogical tool. It is too slow to be used for anything practical.

To get started, run `main.py`:

    $ ./main.py

    Terms:

      x               (variable)

      f(term, ...)    (function)

    Formulae:

      P(term)        (predicate)

      not P          (complement)

      P or Q         (disjunction)

      P and Q        (conjunction)

      P implies Q    (implication)

      forall x. P    (universal quantification)

      exists x. P    (existential quantification)

    Enter formulae at the prompt. The following commands are also available for manipulating axioms:

      axioms              (list axioms)

      lemmas              (list lemmas)

      axiom      (add an axiom)

      lemma      (prove and add a lemma)

      remove     (remove an axiom or lemma)

      reset               (remove all axioms and lemmas)

    >

Example session:

    > P or not P

    0. ⊢ (P ∨ ¬P)

    1. ⊢ P, ¬P

    2. P ⊢ P

    Formula proven: (P ∨ ¬P).

    > P and not P

    0. ⊢ (P ∧ ¬P)

    1. ⊢ P

    Formula unprovable: (P ∧ ¬P).

    > forall x. P(x) implies (Q(x) implies P(x))

    0. ⊢ (∀x. (P(x) → (Q(x) → P(x))))

    1. ⊢ (P(v1) → (Q(v1) → P(v1)))

    2. P(v1) ⊢ (Q(v1) → P(v1))

    3. Q(v1), P(v1) ⊢ P(v1)

    Formula proven: (∀x. (P(x) → (Q(x) → P(x)))).

    > exists x. (P(x) implies forall y. P(y))

    0. ⊢ (∃x. (P(x) → (∀y. P(y))))

    1. ⊢ (P(t1) → (∀y. P(y))), (∃x. (P(x) → (∀y. P(y))))

    2. P(t1) ⊢ (∀y. P(y)), (∃x. (P(x) → (∀y. P(y))))

    3. P(t1) ⊢ (∀y. P(y)), (P(t2) → (∀y. P(y))), (∃x. (P(x) → (∀y. P(y))))

    4. P(t1) ⊢ (P(t2) → (∀y. P(y))), (∃x. (P(x) → (∀y. P(y)))), P(v1)

    5. P(t1), P(t2) ⊢ (∀y. P(y)), (∃x. (P(x) → (∀y. P(y)))), P(v1)

    6. P(t1), P(t2) ⊢ (∀y. P(y)), (P(t3) → (∀y. P(y))), (∃x. (P(x) → (∀y. P(y)))), P(v1)

    7. P(t1), P(t2) ⊢ (P(t3) → (∀y. P(y))), P(v2), (∃x. (P(x) → (∀y. P(y)))), P(v1)

    8. P(t3), P(t1), P(t2) ⊢ (∀y. P(y)), P(v2), (∃x. (P(x) → (∀y. P(y)))), P(v1)

      t3 = v1

    Formula proven: (∃x. (P(x) → (∀y. P(y)))).

    > axiom forall x. Equals(x, x)

    Axiom added: (∀x. Equals(x, x)).

    > axioms

    (∀x. Equals(x, x))

    > lemma Equals(a, a)

    0. (∀x. Equals(x, x)) ⊢ Equals(a, a)

    1. Equals(t1, t1), (∀x. Equals(x, x)) ⊢ Equals(a, a)

      t1 = a

    Lemma proven: Equals(a, a).

    > lemmas

    Equals(a, a)

    > remove forall x. Equals(x, x)

    Axiom removed: (∀x. Equals(x, x)).

    This lemma was proven using that axiom and was also removed:

      Equals(a, a)