
---------- ASSUMPTION
D, e |- e

D |- e(x)
---------  INST
D |- e(t)

----------  REFL
D |- t = t

D |- u = v
----------- SYM
D |- v = u

D |- u = v  D |- v = w
----------------------- TRAN
D |- u = w

D |- t1 = s1  D |- t2 = s2 ... D |- tn = sn
------------------------------------------- CON
D |- f(t1,...,tn) = f(s1,...,sn)

PRIMER:

1) Dokazati da u jednakosnoj logici vazi:

b = f(e, b) =  f(f(c,a), b) = f(c, f(a,b)) = f(c, e) = c

f(f(x,y),z) = f(x, f(y,z)),
f(e, x) = x, f(x, e) = x
f(a, b) = e, f(b, a) = e
f(a, c) = e, f(c, a) = e     |- b = c  TRAN

---------------------------------------------
f(f(x,y),z) = f(x, f(y,z)),
f(e, x) = x, f(x, e) = x
f(a, b) = e, f(b, a) = e
f(a, c) = e, f(c, a) = e     |- b = f(c, e) TRAN 

f(f(x,y),z) = f(x, f(y,z)),
f(e, x) = x, f(x, e) = x
f(a, b) = e, f(b, a) = e
f(a, c) = e, f(c, a) = e     |- f(c, e) = c
-----------------------------------------------
f(f(x,y),z) = f(x, f(y,z)),
f(e, x) = x, f(x, e) = x
f(a, b) = e, f(b, a) = e
f(a, c) = e, f(c, a) = e     |- b = f(c, f(a,b)) TRAN

f(f(x,y),z) = f(x, f(y,z)),
f(e, x) = x, f(x, e) = x
f(a, b) = e, f(b, a) = e
f(a, c) = e, f(c, a) = e     |- f(c, f(a,b)) = f(c, e)

f(f(x,y),z) = f(x, f(y,z)),
f(e, x) = x, f(x, e) = x
f(a, b) = e, f(b, a) = e
f(a, c) = e, f(c, a) = e     |- f(c, e) = c
--------------------------------------------

f(f(x,y),z) = f(x, f(y,z)),
f(e, x) = x, f(x, e) = x
f(a, b) = e, f(b, a) = e
f(a, c) = e, f(c, a) = e     |- b = f(f(c, a), b) TRAN

f(f(x,y),z) = f(x, f(y,z)),
f(e, x) = x, f(x, e) = x
f(a, b) = e, f(b, a) = e
f(a, c) = e, f(c, a) = e     |- f(f(c, a), b) = f(c, f(a,b))


f(f(x,y),z) = f(x, f(y,z)),
f(e, x) = x, f(x, e) = x
f(a, b) = e, f(b, a) = e
f(a, c) = e, f(c, a) = e     |- f(c, f(a,b)) = f(c, e)

f(f(x,y),z) = f(x, f(y,z)),
f(e, x) = x, f(x, e) = x
f(a, b) = e, f(b, a) = e
f(a, c) = e, f(c, a) = e     |- f(c, e) = c
--------------------------------------------

f(f(x,y),z) = f(x, f(y,z)),
f(e, x) = x, f(x, e) = x
f(a, b) = e, f(b, a) = e
f(a, c) = e, f(c, a) = e     |- b = f(e, b) SYM

f(f(x,y),z) = f(x, f(y,z)),
f(e, x) = x, f(x, e) = x
f(a, b) = e, f(b, a) = e
f(a, c) = e, f(c, a) = e     |- f(e,b) = f(f(c, a), b)


f(f(x,y),z) = f(x, f(y,z)),
f(e, x) = x, f(x, e) = x
f(a, b) = e, f(b, a) = e
f(a, c) = e, f(c, a) = e     |- f(f(c, a), b) = f(c, f(a,b))


f(f(x,y),z) = f(x, f(y,z)),
f(e, x) = x, f(x, e) = x
f(a, b) = e, f(b, a) = e
f(a, c) = e, f(c, a) = e     |- f(c, f(a,b)) = f(c, e)

f(f(x,y),z) = f(x, f(y,z)),
f(e, x) = x, f(x, e) = x
f(a, b) = e, f(b, a) = e
f(a, c) = e, f(c, a) = e     |- f(c, e) = c
--------------------------------------------

f(f(x,y),z) = f(x, f(y,z)),
f(e, x) = x, f(x, e) = x
f(a, b) = e, f(b, a) = e
f(a, c) = e, f(c, a) = e     |- f(e, b) = b INS

f(f(x,y),z) = f(x, f(y,z)),
f(e, x) = x, f(x, e) = x
f(a, b) = e, f(b, a) = e
f(a, c) = e, f(c, a) = e     |- f(e,b) = f(f(c, a), b)


f(f(x,y),z) = f(x, f(y,z)),
f(e, x) = x, f(x, e) = x
f(a, b) = e, f(b, a) = e
f(a, c) = e, f(c, a) = e     |- f(f(c, a), b) = f(c, f(a,b))


f(f(x,y),z) = f(x, f(y,z)),
f(e, x) = x, f(x, e) = x
f(a, b) = e, f(b, a) = e
f(a, c) = e, f(c, a) = e     |- f(c, f(a,b)) = f(c, e)

f(f(x,y),z) = f(x, f(y,z)),
f(e, x) = x, f(x, e) = x
f(a, b) = e, f(b, a) = e
f(a, c) = e, f(c, a) = e     |- f(c, e) = c
--------------------------------------------

f(f(x,y),z) = f(x, f(y,z)),
f(e, x) = x, f(x, e) = x
f(a, b) = e, f(b, a) = e
f(a, c) = e, f(c, a) = e     |- f(e, x) = x  ASSUMPTION

f(f(x,y),z) = f(x, f(y,z)),
f(e, x) = x, f(x, e) = x
f(a, b) = e, f(b, a) = e
f(a, c) = e, f(c, a) = e     |- f(e,b) = f(f(c, a), b) CON


f(f(x,y),z) = f(x, f(y,z)),
f(e, x) = x, f(x, e) = x
f(a, b) = e, f(b, a) = e
f(a, c) = e, f(c, a) = e     |- f(f(c, a), b) = f(c, f(a,b))


f(f(x,y),z) = f(x, f(y,z)),
f(e, x) = x, f(x, e) = x
f(a, b) = e, f(b, a) = e
f(a, c) = e, f(c, a) = e     |- f(c, f(a,b)) = f(c, e)

f(f(x,y),z) = f(x, f(y,z)),
f(e, x) = x, f(x, e) = x
f(a, b) = e, f(b, a) = e
f(a, c) = e, f(c, a) = e     |- f(c, e) = c
--------------------------------------------

f(f(x,y),z) = f(x, f(y,z)),
f(e, x) = x, f(x, e) = x
f(a, b) = e, f(b, a) = e
f(a, c) = e, f(c, a) = e     |- e = f(c, a)  (SYM, ASS)

f(f(x,y),z) = f(x, f(y,z)),
f(e, x) = x, f(x, e) = x
f(a, b) = e, f(b, a) = e
f(a, c) = e, f(c, a) = e     |- b = b  (REFL)


f(f(x,y),z) = f(x, f(y,z)),
f(e, x) = x, f(x, e) = x
f(a, b) = e, f(b, a) = e
f(a, c) = e, f(c, a) = e     |- f(f(c, a), b) = f(c, f(a,b)) INS


f(f(x,y),z) = f(x, f(y,z)),
f(e, x) = x, f(x, e) = x
f(a, b) = e, f(b, a) = e
f(a, c) = e, f(c, a) = e     |- f(c, f(a,b)) = f(c, e)

f(f(x,y),z) = f(x, f(y,z)),
f(e, x) = x, f(x, e) = x
f(a, b) = e, f(b, a) = e
f(a, c) = e, f(c, a) = e     |- f(c, e) = c
--------------------------------------------

f(f(x,y),z) = f(x, f(y,z)),
f(e, x) = x, f(x, e) = x
f(a, b) = e, f(b, a) = e
f(a, c) = e, f(c, a) = e     |- f(f(x, y), z) = f(x, f(y,z)) ASSUMPTION


f(f(x,y),z) = f(x, f(y,z)),
f(e, x) = x, f(x, e) = x
f(a, b) = e, f(b, a) = e
f(a, c) = e, f(c, a) = e     |- f(c, f(a,b)) = f(c, e) CON

f(f(x,y),z) = f(x, f(y,z)),
f(e, x) = x, f(x, e) = x
f(a, b) = e, f(b, a) = e
f(a, c) = e, f(c, a) = e     |- f(c, e) = c
--------------------------------------------

f(f(x,y),z) = f(x, f(y,z)),
f(e, x) = x, f(x, e) = x
f(a, b) = e, f(b, a) = e
f(a, c) = e, f(c, a) = e     |- c = c  (REFL)

f(f(x,y),z) = f(x, f(y,z)),
f(e, x) = x, f(x, e) = x
f(a, b) = e, f(b, a) = e
f(a, c) = e, f(c, a) = e     |- f(a,b) = e ASSUMPTION

f(f(x,y),z) = f(x, f(y,z)),
f(e, x) = x, f(x, e) = x
f(a, b) = e, f(b, a) = e
f(a, c) = e, f(c, a) = e     |- f(c, e) = c  INS
--------------------------------------------

f(f(x,y),z) = f(x, f(y,z)),
f(e, x) = x, f(x, e) = x
f(a, b) = e, f(b, a) = e
f(a, c) = e, f(c, a) = e     |- f(x, e) = x ASSUMPTION
--------------------------------------------
KRAJ




