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Course: CAS 734, Fall 2009School: St. Mary MD
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and Computing Software 734 Formalized Mathematics Fall 2006 Exercise 7 50 pts. Due 11 December 2006 Assigned: 13 November 2006 Revised: 13 November 2006 Part 1 Starting from scratch, create an imps theory of Peano arithmetic called peano1 with the constants 0, S, +, and and the following axioms: 1. 0 is not a successor. 2. S is injective. 3. Induction. 4. The two axioms that specify + recursively. 5. The two...
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and Computing Software 734 Formalized Mathematics Fall 2006
Exercise 7 50 pts. Due 11 December 2006
Assigned: 13 November 2006 Revised: 13 November 2006
Part 1
Starting from scratch, create an imps theory of Peano arithmetic called peano1 with the constants 0, S, +, and and the following axioms: 1. 0 is not a successor. 2. S is injective. 3. Induction. 4. The two axioms that specify + recursively. 5. The two axioms that specify recursively.
Part 2
Prove the following theorems in peano1: 1. 0 is the additive identity. 2. S(0) is the multiplicative identity. 3. + is associative. 4. is associative. 5. + is commutative. is 6. commutative. 1
Part 3
Create a compound macete that reduces any ground expression of peano1 (i.e., any expression containing no variables and no constants other than 0, S, +, and ) to an expression of the form S m (0) (i.e., S applied to 0 m times). Give several examples expressed as theorems to show that it works.
Part 4
Create the following three...
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