Reduce each expression as far as possible.

1. @@math_item

   content=\((\lambda x. (m\,\, x))\,\,j\)@@

2. @@math_item

   content=\((\lambda y. (y\,\, j))\,\,m\)@@

3. @@math_item

   content=\(((\lambda x.(\lambda y. (y\,\,(y\,\, x))))\,\, j)\,\,m\)@@

4. @@math_item

   content=\((\lambda y.(y\cnct{}j))(\lambda x.(m\cnct{}x))\)@@

5. @@math_item

   content=\((\lambda z.z) (\lambda y . y\,\, y) (\lambda x.x\,\, a)\)@@

6. @@math_item

   content=\((\lambda x\lambda y.x\,\, y\,\, y) (\lambda a.a) b\)@@

7. @@math_item

   content=\((\lambda x.x\,\, x) (\lambda y.y\,\, x) z\)@@

8. @@math_item

   content=\(((\lambda x.x\,\, x) (\lambda y.y)) (\lambda y.y)\)@@

9. @@math_item

   content=\((\lambda f.fx)g\)@@

10. @@math_item

    content=\((\lambda f.fx)ga\)@@

11. @@math_item

    content=\((\lambda f.fx)(ga)\)@@

12. @@math_item

    content=\((\lambda f\lambda x.fx)g a\)@@

13. @@math_item

    content=\((\lambda x\lambda y \lambda z.x(yz))f\)@@

Now some examples with logical forms:

1. @@math_item

   content=\((\lambda p.p\cnct{}john')(\lambda x.sleeps'x)\)@@

2. @@math_item

   content=\((\lambda p\lambda q.\forall x. p'x \to q'x)(\lambda x. student'x)(\lambda x. sleeps'x)\)@@

3. @@math_item

   content=\((\lambda p\lambda x.think'p\cnct{x})((\lambda p.p\cnct{}john')(\lambda x.sleeps'x))alice'\)@@

4. @@math_item

   content=\((\lambda p\lambda q.\exists x. p\cnct{x}\land q\cnct{x})(\lambda x. student' x)((\lambda p\lambda x.think'p\cnct{x})((\lambda p.p\cnct{}john')(\lambda x.sleeps'x)))\)@@

These are from @@exlink

lambdacalculator@@

1. @@math_item

   content=\((\x.a') b'\)@@

2. @@math_item

   content=\((\x.p'x \to \E x.r'b'\, x) a'\)@@

3. @@math_item

   content=\((\x\x.p'x \to q'x\,c') a'\, b'\)@@

4. @@math_item

   content=\((\x\E y.r'x\,y)y\)@@