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 1813 Discrete Mathematics, Univ Oklahoma Copyright © 2000 by Rex Page
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 1813 Discrete Mathematics, Univ Oklahoma Copyright © 2000 by Rex Page
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} ab 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 1813 Discrete Mathematics, Univ Oklahoma Copyright © 2000 by Rex Page
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 ab 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 ab a b a |– c b |– c {E} c Or Elimination similar for b |– c subtree, except matching b in ab 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 1813 Discrete Mathematics, Univ Oklahoma Copyright © 2000 by Rex Page
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 1813 Discrete Mathematics, Univ Oklahoma Copyright © 2000 by Rex Page
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 1813 Discrete Mathematics, Univ Oklahoma Copyright © 2000 by Rex Page
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 1813 Discrete Mathematics, Univ Oklahoma Copyright © 2000 by Rex Page
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 1813 Discrete Mathematics, Univ Oklahoma Copyright © 2000 by Rex Page
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 1813 Discrete Mathematics, Univ Oklahoma Copyright © 2000 by Rex Page
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) {DeMF} (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 1813 Discrete Mathematics, Univ Oklahoma Copyright © 2000 by Rex Page
Lecture 6 - CS 1813 Discrete Math, University of Oklahoma 11/18/2018 End of Lecture CS 1813 Discrete Mathematics, Univ Oklahoma Copyright © 2000 by Rex Page