An LKProof ending with a conjunction on the left:
(π) A, B, Γ :- Δ -------------- A ∧ B, Γ :- Δ
An LKProof ending with a conjunction on the left:
(π) A, B, Γ :- Δ -------------- A ∧ B, Γ :- Δ
The subproof π.
The index of A.//<editor-fold desc="Base proof classes">
The index of B.
An LKProof ending with a conjunction on the right:
(π1) (π2) Γ :- Δ, A Π :- Λ, B -------------------------- Γ, Π :- Δ, Λ, A∧B
An LKProof ending with a conjunction on the right:
(π1) (π2) Γ :- Δ, A Π :- Λ, B -------------------------- Γ, Π :- Δ, Λ, A∧B
The proof π1.
The index of A.
The proof π2
The index of B.
An LKProof deriving a sequent from two other sequents:
(π1) (π2) Γ :- Δ Γ' :- Δ' ------------------ Π :- Λ
An LKProof deriving a sequent from two other sequents:
(π1) (π2) Γ :- Δ Γ' :- Δ' ------------------ Π :- Λ
An LKProof ending with a left contraction:
(π) A, A, Γ :- Δ -------------- A, Γ :- Δ
An LKProof ending with a left contraction:
(π) A, A, Γ :- Δ -------------- A, Γ :- Δ
The subproof π.
The index of one occurrence of A.
The index of the other occurrence of A.
An LKProof ending with a right contraction:
(π) Γ :- Δ, A, A -------------- Γ :- Δ, A
An LKProof ending with a right contraction:
(π) Γ :- Δ, A, A -------------- Γ :- Δ, A
The subproof π.
The index of one occurrence of A.
The index of the other occurrence of A.
An LKProof ending with a cut:
(π1) (π2) Γ :- Δ, A A, Π :- Λ ------------------------ Γ, Π :- Δ, Λ
An LKProof ending with a cut:
(π1) (π2) Γ :- Δ, A A, Π :- Λ ------------------------ Γ, Π :- Δ, Λ
The proof π1.
The index of A in π1.
The proof π2.
The index of A in π2.
An LKProof ending with an existential quantifier on the left:
(π) A[x\α], Γ :- Δ ---------------∀:r ∃x.A Γ :- ΔThis rule is only applicable if the eigenvariable condition is satisfied: α must not occur freely in Γ :- Δ.
An LKProof ending with an existential quantifier on the left:
(π) A[x\α], Γ :- Δ ---------------∀:r ∃x.A Γ :- ΔThis rule is only applicable if the eigenvariable condition is satisfied: α must not occur freely in Γ :- Δ.
The proof π.
The index of A[x\α].
The variable α.
The variable x.
An LKProof ending with an existential quantifier on the right:
(π) Γ :- Δ, A[x\t] ----------------∃:r Γ :- Δ, ∃x.A
An LKProof ending with an existential quantifier on the right:
(π) Γ :- Δ, A[x\t] ----------------∃:r Γ :- Δ, ∃x.A
The proof π.
The index of A[x\t].
The formula A.
The term t.
The variable x.
An LKProof ending with a universal quantifier on the left:
(π) A[x\t], Γ :- Δ ----------------∀:l ∀x.A, Γ :- Δ
An LKProof ending with a universal quantifier on the left:
(π) A[x\t], Γ :- Δ ----------------∀:l ∀x.A, Γ :- Δ
The proof π.
The index of A[x\t].
The formula A.
The term t.
The variable x.
An LKProof ending with a universal quantifier on the right:
(π) Γ :- Δ, A[x\α] ---------------∀:r Γ :- Δ, ∀x.AThis rule is only applicable if the eigenvariable condition is satisfied: α must not occur freely in Γ :- Δ.
An LKProof ending with a universal quantifier on the right:
(π) Γ :- Δ, A[x\α] ---------------∀:r Γ :- Δ, ∀x.AThis rule is only applicable if the eigenvariable condition is satisfied: α must not occur freely in Γ :- Δ.
The proof π.
The index of A[x\α].
The variable α.
The variable x.
An LKProof ending with an implication on the left:
(π1) (π2) Γ :- Δ, A B, Π :- Λ -------------------------- A→B, Γ, Π :- Δ, Λ
An LKProof ending with an implication on the left:
(π1) (π2) Γ :- Δ, A B, Π :- Λ -------------------------- A→B, Γ, Π :- Δ, Λ
The proof π1.
The index of A.
The proof π2
The index of B.
An LKProof ending with an implication on the right:
(π) A, Γ :- Δ, B -------------- Γ :- Δ, A → B
An LKProof ending with an implication on the right:
(π) A, Γ :- Δ, B -------------- Γ :- Δ, A → B
The subproof π.
The index of A.
The index of B.
An LKProof consisting of a single sequent:
--------ax Γ :- Δ
An LKProof consisting of a single sequent:
--------ax Γ :- Δ
An LKProof consisting of a logical axiom:
--------ax A :- Awith A atomic.
An LKProof consisting of a logical axiom:
--------ax A :- Awith A atomic.
The atom A.
An LKProof ending with a negation on the left:
(π) Γ :- Δ, A -----------¬:l ¬A, Γ :- Δ
An LKProof ending with a negation on the left:
(π) Γ :- Δ, A -----------¬:l ¬A, Γ :- Δ
The proof π.
The index of A in the succedent.
An LKProof ending with a negation on the right:
(π) A, Γ :- Δ -----------¬:r Γ :- Δ, ¬A
An LKProof ending with a negation on the right:
(π) A, Γ :- Δ -----------¬:r Γ :- Δ, ¬A
The proof π.
The index of A in the antecedent.
This class models the connection of formula occurrences between two sequents in a proof.
An LKProof ending with a disjunction on the left:
(π1) (π2) A, Γ :- Δ B, Π :- Λ -------------------------- A∨B, Γ, Π :- Δ, Λ
An LKProof ending with a disjunction on the left:
(π1) (π2) A, Γ :- Δ B, Π :- Λ -------------------------- A∨B, Γ, Π :- Δ, Λ
The proof π1.
The index of A.
The proof π2
The index of B.
An LKProof ending with a disjunction on the right:
(π) Γ :- Δ, A, B -------------- Γ :- Δ, A ∨ B
An LKProof ending with a disjunction on the right:
(π) Γ :- Δ, A, B -------------- Γ :- Δ, A ∨ B
The subproof π.
The index of A.
The index of B.
Class for convenient construction of proofs.
Class for convenient construction of proofs. Allows you to write proofs post-order style (à la Bussproofs). Example:
(ProofBuilder c LogicalAxiom(A) c LogicalAxiom(B) u (WeakeningLeftRule(_, C)) b (AndRightRule(_,_, And(A, B)) qed)The constructor is private, so the only way to instantiate this class is by using the ProofBuilder object. This means that the stack will always be empty in the beginning.
An LKProof consisting of a reflexivity axiom:
------------ax :- s = swith s a term.
An LKProof consisting of a reflexivity axiom:
------------ax :- s = swith s a term.
The term s.
An LKProof deriving a sequent from another sequent:
(π) Γ :- Δ ---------- Γ' :- Δ'
An LKProof deriving a sequent from another sequent:
(π) Γ :- Δ ---------- Γ' :- Δ'
An LKProof ending with a left weakening:
(π) Γ :- Δ ---------w:l A, Γ :- Δ
An LKProof ending with a left weakening:
(π) Γ :- Δ ---------w:l A, Γ :- Δ
The subproof π.
The formula A.
An LKProof ending with a right weakening:
(π) Γ :- Δ ---------w:r Γ :- Δ, A
An LKProof ending with a right weakening:
(π) Γ :- Δ ---------w:r Γ :- Δ, A
The subproof π.
The formula A.
An LKProof introducing ⊥ on the left:
--------⊥:l ⊥ :-
An LKProof introducing ⊥ on the left:
--------⊥:l ⊥ :-
An LKProof introducing ⊤ on the right:
--------⊤:r :- ⊤
An LKProof introducing ⊤ on the right:
--------⊤:r :- ⊤