1. Lecture 6 CS 1813 – Discrete Mathematics
Lecture 6 - CS 1813 Discrete Math, University of Oklahoma
11/18/2018
Lecture 6 CS 1813 – Discrete Mathematics
toBe  (toBe)
and
Other Conundrums
CS Discrete Mathematics, Univ Oklahoma
Copyright © 2000 by Rex Page

2. The Discharge When Do You Do It?
When you cite one of the following rules
I discharge 1 assumption
E discharge 2 assumptions
RAA discharge 1 assumption
Why these particular rules?
They have turnstiles among their premises
No other rules have this sequents as premises
No other rule triggers a discharge
When you cite any other rule or theorem to justify a deduction in a proof, do not discharge any assumption
CS Discrete Mathematics, Univ Oklahoma
Copyright © 2000 by Rex Page

3. The Discharge What Assumption Do You Do It To?
Lecture 6 - CS 1813 Discrete Math, University of Oklahoma
11/18/2018
The Discharge What Assumption Do You Do It To?
Find conclusion of sequent triggering discharge
In subtree above conclusion, find leaf that matches rule
Discharge all identical assumptions in the subtree
In the end, remaining leaves are theorem premises
a |– b
 {I}
ab
discharge
{IL}
a  b
{E}
{I} a
a  False
False
(a  b)  False
triggers 1 discharge
Before clicking to bring up orange animation pointing out False/b connection,
ask a student from class to identify the conclusion of the sequent triggering the discharge.
Before proceeding to the “Discharge all identical” line,
ask a student in class to identify the subtree above the conclusion of the sequent triggering the discharge.
In this example, the proposition “a” occurs as an assumption at only one point in the subtree
corresponding to the sequent triggering the discharge.
But, in other examples, the assumption a could occur in several places.
Before, going to the “In the end” line, ask a student in class to identify places in this example
where assumptions occur. Get the student to recognize that assumptions correspond to leaves in the tree.
Therefore, the propositions residing at leaves that remain undischarged at the end of the proof
are the assumptions of the theorem proved by the tree-proof.
CS Discrete Mathematics, Univ Oklahoma
Copyright © 2000 by Rex Page

4. How to Match for Discharge
Lecture 6 - CS 1813 Discrete Math, University of Oklahoma
11/18/2018
How to Match for Discharge
discharged assumption in
a |– b subtree must
be identical to the formula
corresponding to a in ab
Implication Introduction
a |– b
 {I}
a  b
I rule
discharged assumption in
a |– c subtree must
be identical to the formula
corresponding to a in ab
a  b a |– c b |– c
{E} c
Or Elimination
similar for b |– c subtree,
except matching b in ab
Note: RAA rule not illustrated in any proofs given so far.
Comes up later in this lecture.
For now, just observe formal structure of discharges in these 3 rules.
a |– False
{RAA} a
Reductio ad Absurdum
discharged assumption in
a |– False subtree must
be identical to a where
a is formula below line
CS Discrete Mathematics, Univ Oklahoma
Copyright © 2000 by Rex Page

5. Tree A Mathematical Entity
Informal definition
tree: a branch or a leaf
branch: an item of information, together with a collection of trees
More formal definition
tree ::= branch | leaf
leaf ::= data
branch ::= (data, tree-sequence)
tree-sequence ::= sequence in which each component is a tree
data and sequence not defined
assumed to be terms already understood
(1, [(2, [3, 4] ), 5] ) 2 1 4 3 5
CS Discrete Mathematics, Univ Oklahoma
Copyright © 2000 by Rex Page

6. Representation of Trees an example
leaf
data Proof = Assume Prop |
AndI (Proof, Proof) Prop |
AndEL Proof Prop |
AndER Proof Prop |
OrIL Proof Prop |
OrIR Proof Prop |
OrE (Proof, Proof, Proof) Prop |
ImpI Proof Prop |
ImpE (Proof, Proof) Prop |
ID Proof Prop |
CTR Proof Prop |
RAA Proof Prop |
Use Theorem [Proof] Prop
a Proof
is a tree
“sequence” of trees .
data
branch
CS Discrete Mathematics, Univ Oklahoma
Copyright © 2000 by Rex Page

7. A Proof Using Contradiction
a  b, a |– b (disjunctive syllogism)
False
{CTR} a
use CTR rule
Plan
Derive b from a
Derive b from b
Cite E
a  b a |– c b |– c
{E} c
 
a  b b b
{E} b a
 a
{ +&- }
remaining assumptions
discharge what?
False
{CTR}
{ID}
CS Discrete Mathematics, Univ Oklahoma
Copyright © 2000 by Rex Page

8. CS 1813 Discrete Mathematics, Univ Oklahoma
Reductio ad Absurdum
(a)
{ F} a
Double Negation Fwd
(a) |– a
Plan
Derive False from a, given (a)
Conclude a (citing RAA)
a |– False
{RAA} a
a (a)

What rule for this?
{ +&- }
remaining assumption
False
{RAA} a
discharge what?
CS Discrete Mathematics, Univ Oklahoma
Copyright © 2000 by Rex Page

9. Law of the Excluded Middle |– (a)  a
Plan
Derive False from ((a)  a)
Conclude (a)  a, citing RAA
a |– False
{RAA} a
{()ER} a
((a)  a)
((a)  a)
{()EL}
(a)

False
{ +&- }
{RAA}
(a)  a
discharge what?
(a  b)
{()ER} b
anything else?
conclusion
What assumptions remain?
CS Discrete Mathematics, Univ Oklahoma
Copyright © 2000 by Rex Page

10. DeMorgan’s Law —  Forward (a  b) |– (a)  (b)
Plan
Derive a  b from ((a)  (b))
Note conflict with assumption
Conclude (a)  (b), citing RAA
(a  b)
{DeMF}
(a)(b)
DeMorgan And Fwd
Discharge?
((a)  (b))
{()EL}
(a)
((a)  (b))
{()ER}
(b)
{F} b
{F} a
{I}
a  b
(a  b)

False
{ +&- }
{RAA}
(a)  (b)
CS Discrete Mathematics, Univ Oklahoma
Copyright © 2000 by Rex Page

11. Lecture 6 - CS 1813 Discrete Math, University of Oklahoma
11/18/2018
End of Lecture
CS Discrete Mathematics, Univ Oklahoma
Copyright © 2000 by Rex Page