Logic: Lecture 12, 13 September 2012
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A "sequent" G |- phi consists of a list of formulas G (the "antecedent") and the conclusion phi (the "succedent").

The intention is that from the assumptions G, we can prove phi.

As usual, we expect soundness to hold so G |- phi implies G |= phi.


Sequent Calculus
-----------------

A formal system for derivations using sequents


Basic axioms/rules

                  
(Assumption)     -----------   if phi in G
                   G |- phi

                   G  |- phi
(Antecdent)      -----------   if G subset of G'  
                   G' |- phi   (note that G, G' are lists, but the 
                                condition is on G, G' as sets)

                   G, phi |- psi,    G, ~phi |- psi
(Proof by cases)  ---------------------------------
                            G |- psi


                   G, ~phi |- psi,   G, ~phi |- ~psi
(Contradiction)   ----------------------------------
                            G |- phi


                   G, phi |- chi,    G, psi |- chi
(Or, Antecedent)  --------------------------------
                      G, (phi or psi) |- chi


                   G |- phi                 G |- phi
(Or, Succedent)  ------------------ ,      ------------------
                   G |- phi or psi          G |- psi or phi

   
                   G, phi[x|->y] |- psi 
(E, Antecedent)   ----------------------,    y not free in {G, Ex phi, psi}
                   G,  Ex phi    |- psi     


                   G |- phi[x|->t]
(E, Succedent)   ---------------------
                   G | Ex phi
                   

(Equality)        ----------
                    t = t

                   G         |- phi[x |-> t]
(Substitution)    ----------------------------
                   G, t = t' |- phi[x |-> t']



Typical proofs
---------------

1. (TND, Excluded Middle)   |- phi or ~phi

   a . phi |-  phi         (Assumption)
v   b.  phi |-  phi or ~phi (Or, Succedent from a)
   c. ~phi |- ~phi         (Assumption)
   d. ~phi |-  phi or ~phi (Or, Succedent from c)
   e       |-  phi or ~Phi (Proof by cases, b and e)

2. (Ctr', Second Contradiction rule)      

   G |- psi,  G |- ~psi
   --------------------
        G |- phi

   a.  G       |- psi      (Premise)
   b.  G       |- ~psi     (Premise)
   c.  G, ~phi |- psi      (Antecdent, from a)
   d.  G, ~phi |- ~psi     (Antecdent, from b)
   e.  G       |- phi      (Contradiction, c and d)

3. (Ch, Chain rule)
     
   G |- phi,     G, phi |- psi
   --------------------------
           G |- psi

   a. G       |- phi       (Premise)
   b. G, phi  |- psi       (Premise)
   c. G, ~phi |- phi       (Antecedent, from a)
   d. G, ~phi |- ~phi      (Assumption)
   e. G, ~phi |- psi       (Ctr', c and d)
   e. G,      |- psi       (Proof by cases, b and e)

4. (Contrapositive, 4 variants)

   G,  phi |- psi    G, ~phi |- ~psi   G, ~phi |- psi   G, phi |- ~psi 
   ---------------,  ---------------,  --------------,  --------------
   G, ~psi |- ~phi    G, psi |- phi    G, ~psi |- phi   G, psi |- ~phi

   a. G, phi        |- psi       (Premise)
   b. G, ~psi, phi  |- psi       (Antecedent, from a)
   c. G, ~psi, phi  |- ~psi      (Assumption)
   d. G, ~psi, phi  |- ~phi      (Ctr', b and c)
   e. G, ~psi, ~phi |- ~phi      (Assumption)
   f. G, ~psi       |- ~phi      (Proof by Cases, d and e)

5. G |- phi or psi,  G |- ~phi
   ---------------------------
          G |- psi

   a. G             |- phi or psi  (Premise)
   b. G             |- ~phi        (Premise)
   c. G, psi        |- psi         (Assumption)
   d. G, phi        |- ~phi        (Antecdent, from b)
   e. G, phi        |- phi         (Assumption)
   f. G, phi        |- psi         (Ctr', d and e)
   g. G, phi or psi |- psi         (Or, Antecedent f)
   h. G             |- psi         (Chain, a and g)

6. Modus ponens

   G |- phi -> psi,  G |- phi
   -------------------------
         G |- psi

   a. G              |- ~phi or psi  (Premise)
   b. G              |- phi          (Premise)
   c. G, psi         |- psi          (Assumption)
   d. G, ~phi        |- ~phi         (Antecdent, from b)
   e. G, ~phi        |- ~phi         (Assumption)
   f. G, ~phi        |- psi          (Ctr', d and e)
   g. G, ~phi or psi |- psi          (Or, Antecedent f)
   h. G              |- psi          (Chain, a and g)

7. Equality axioms

   Symmetry: t0 = t1 |- t1 = t0

   a.          |- t0 = t0  (Equality)
   b.  t0 = t1 |- t1 = t0  (Substitution, from a using phi(x): x = t0)

   Transitivity:  t0 = t1, t1 = t2 |- t0 = t2

   a. t0 = t1          |- t0 = t1   (Assumption)
   b. t0 = t1, t1 = t2 |- t0 = t2   (Substition, from a using
                                     phi(x): t0 = x)

   Relations:  R(t0,..,tn), t0 = t'0, ..., tn = t'n |- R(t'0,..,t'n)

   Proof for the case R(t0,t1)

   a. R(t0,t1)                     |- R(t0,t1)    (Assumption)
   b. R(t0,t1), t0 = t'0           |- R(t'0,t1)   (Substitution, a, 
                                                   phi(x): R(x,t1))
   c. R(t0,t1), t0 = t'0, t1 = t'1 |- R(t'0,t'1)  (Substitution, b,
                                                   phi(x): R(t'0,x))

   Functions: t0 = t'0, ..., tn = t'n |- f(t0,..,tn) = f(t'0,..,t'n)

   Proof for the case f(t0,t1)

   a.                              |- f(t0,t1) = f(t0,t1)  (Assumption)
   b. t0 = t'0                     |- f(t0,t1) = f(t'0,t1) 
                                      (Substitution, a, 
                                       phi(x) : f(t0,t1) = f(x,t1)

   c. t0 = t'0, t1 = t'1           |- f(t0,t1) = f(t'0,t'1) 
                                      (Substitution, b, 
                                       phi(x) : f(t0,t1) = f(t'0,x)

                                                 
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