( V ametavariable) P P true. even in E)

Transcription

1 Propositional Calculus E Inference rules (3.1) Leibniz: (3.2) Transitivity: (3.3) Equanimity: P = Q E[V := P ]=E[V := Q] P = Q Q = R P = R P P Q Q ( V ametavariable) Derived inference rules (3.11) Redundant true : (3.91) Modus ponens: (4.8) Monotonicity: P P P true P P ) Q Q P Q E[V := P ] ) E[V := Q] true P (for parity of V even in E) (4.9) Antimonotonicity: P ) Q E[V := P ] ( E[V := Q] (for parity of V odd in E) Equivalence and true (3.5) Axiom, Associativity of : ((P Q) R) (P (Q R)) (3.6) Axiom, Symmetry of : P Q Q P (3.7) Axiom, Identity of : true Q Q (3.8) Reexivity of : P P (3.9) true (3.12) Metatheorem: Any two theorems are equivalent. Negation, inequivalence, and false (3.15) Axiom, Denition of false : false :true (3.17) Axiom, Distributivity of : over : :(P Q) :P Q (3.14) Axiom, Denition of 6 : (P 6 Q) :(P Q) (3.18) :P Q P :Q (3.19) Double negation: ::P P (3.16) Negation of false : :false true (3.20) (P 6 Q) :P Q (3.13) :P P false (3.21) Symmetry of 6 : (P 6 Q) (Q 6 P ) (3.22) Associativity of 6 : ((P 6 Q) 6 R) (P 6 (Q 6 R)) (3.23) Mutual associativity: ((P 6 Q) R) (P 6 (Q R))

2 (3.24) Mutual interchangeability: P 6 Q R P Q 6 R Disjunction (3.29) Axiom, Symmetry of _ : P _ Q Q _ P (3.30) Axiom, Associativity of _ : (P _ Q) _ R P _ (Q _ R) (3.31) Axiom, Idempotency of _ : P _ P P (3.32) Axiom, Distributivity of _ over : P _ (Q R) P _ Q P _ R (3.33) Axiom, Excluded Middle: P _:P true (3.34) Zero of _ : P _ true true (3.35) Identity of _ : P _ false P (3.36) Distributivity of _ over _ : P _ (Q _ R) (P _ Q) _ (P _ R) (3.37) P _ Q P _:Q P Conjunction (3.40) Axiom Golden rule : P ^ Q P Q P _ Q (3.41) Symmetry of ^ : P ^ Q Q ^ P (3.42) Associativity of ^ : (P ^ Q) ^ R P ^ (Q ^ R) (3.43) Idempotency of ^ : P ^ P P (3.44) Identity of ^ : P ^ true P (3.45) Zero of ^ : P ^ false false (3.46) Distributivity of ^ over ^ : P ^ (Q ^ R) (P ^ Q) ^ (P ^ R) (3.47) Contradiction: P ^:P false (3.48) Absorption: (a) P ^ (P _ Q) P (b) P _ (P ^ Q) P (3.49) Absorption: (a) P ^ (:P _ Q) P ^ Q (b) P _ (:P ^ Q) P _ Q (3.50) Distributivity of _ over ^ : P _ (Q ^ R) (P _ Q) ^ (P _ R) (3.51) Distributivity of ^ over _ : P ^ (Q _ R) (P ^ Q) _ (P ^ R) (3.52) De Morgan: (a) :(P ^ Q) :P _:Q (3.53) P ^ Q P ^:Q :P (b) :(P _ Q) :P ^:Q (3.54) P ^ (Q R) P ^ Q P ^ R P (3.55) P ^ (Q P ) P ^ Q (3.56) Replacement: (P Q) ^ (R P ) (P Q) ^ (R Q) (3.57) Denition of : P Q (P ^ Q) _ (:P ^:Q) (3.58) Exclusive or: P 6 Q (:P ^ Q) _ (P ^:Q)

3 (3.62) (P ^Q) ^ R P Q R P _Q Q_R R_P P _Q_R Implication (3.65) Axiom, Implication: P ) Q P _ Q Q (3.64) Axiom, Consequence: P ( Q Q ) P (3.66) Implication: P ) Q P ^ Q P (3.63) Implication: P ) Q :P _ Q (3.67) Contrapositive: P ) Q :Q ):P (3.68) P ) (Q R) P ^ Q P ^ R (3.69) Distributivity of ) (a) over : P ) (Q R) P ) Q P ) R (b) over _ : P ) Q _ R (P ) Q) _ (P ) R) (c) over ^ : P ) (Q ^ R) (P ) Q) ^ (P ) R) (d) over ) : P ) (Q ) R) (P ) Q) ) (P ) R) (3.70) Shunting: P ^ Q ) R P ) (Q ) R) (3.71) P ) (Q ) P ) (3.72) P ^ (P ) Q) P ^ Q (3.73) P ^ (Q ) P ) P (3.74) P _ (P ) Q) true (3.75) P _ (Q ) P ) Q ) P (3.76) P _ Q ) P ^ Q P Q (3.77) Reexivity of ) : P ) P true (3.78) Right zero of ) : P ) true true (3.79) Left identity of ) : true ) P P (3.80) P ) false :P (3.81) false ) P true (3.82) Weakening/strengthening: (a) P ) P _ Q (3.83) Modus ponens: P ^ (P ) Q) ) Q (b) P ^ Q ) P (3.84) (P ) R) ^ (Q ) R) (P _ Q ) R) (3.85) (P ) R) ^ (:P ) R) R (c) P ^ Q ) P _ Q (d) P _ (Q ^ R) ) P _ Q (e) P ^ Q ) P ^ (Q _ R) (3.86) Mutual implication: (P ) Q) ^ (Q ) P ) (P Q) (3.87) Antisymmetry: (P ) Q) ^ (Q ) P ) ) (P Q) (3.88) Transitivity of : (P Q) ^ (Q R) ) (P R) (3.89) Transitivity: (a) (P ) Q) ^ (Q ) R) ) (P ) R)

4 (b) (P Q) ^ (Q ) R) ) (P ) R) (c) (P ) Q) ^ (Q R) ) (P ) R) (??) Metatheorem Modus ponens: If P and P ) Q are theorems, so is Q Leibniz as an axiom (3.93) Axiom, Leibniz: X = Y ) E[V := X] = E[V := Y ] (3.94) Substitution: (a) X = Y ^ E[V := X] X = Y ^ E[V := Y ] E[V := Y ] (b) X = Y ) E[V := X] X = Y ) E[V := Y ] (c) Q ^ X = Y ) E[V := X] Q ^ X = Y ) (3.95) Replace by true : (a) P ) E[V := P ] P ) E[V := true] (b) Q ^ P ) E[V := P ] Q ^ P ) E[V := true] (3.96) Replace by false : (a) E[V := P ] ) P E[V := false] ) P (b) E[V := P ] ) P _ Q E[V := false] ) P _ Q (3.97) Replace by true : P ^ E[V := P ] P ^ E[V := true] (3.98) Replace by false : P _ E[V := P ] P _ E[V := false] (3.99) Boole: E[V := P ] (P ^ E[V := true]) _ (:P ^ E[V := false]) Monotonicity theorems (4.1) Monotonic _ : (P ) Q) ) (P _ R ) Q _ R) (4.2) Monotonic ^ : (P ) Q) ) (P ^ R ) Q ^ R) (4.3) Monotonic consequent: (P ) Q) ) ((R ) P ) ) (R ) Q)) (4.4) Antimonotonic : : (P ) Q) ) (:P (:Q) (4.5) Antimonotonic antecedent: (P ) Q) ) ((P ) R) ( (Q ) R)) (4.7) Metatheorem Montonicity: Provided P ) Q is a theorem and the parity of the position of V E is even, E[V := P ] ) E[V := Q]. Provided P ) Q is a theorem and the parity of the position of V E is odd, E[V := P ] ( E[V := Q]. (4.11) (P ) P 0 ) ) ((Q ) Q 0 ) ) (P ^ Q ) P 0 ^ Q 0 )) (4.12) (P _ Q _ R) ^ (P ) S) ^ (Q ) S) ^ (R ) S) ) S Proof techniques and strategies (3.10) Basic strategies: To prove P Q, transform P to Q, transform Q to P, or transform P Q to a previous theorem. (3.26) Heuristic: Identify applicable theorems by matching the structure of subexpressions with the theorems. (3.27) Principle: Structure proofs to avoid repeating the same subexpression on many lines. in

5 (3.28) Heuristic Unfold-fold: To prove a theorem concerning an operator, remove it using its denition, manipulate, and reintroduce it using its denition. (3.38) Heuristic: To prove P Q, transform the expression with the most structure (either P or Q )into the other. (3.39) Principle: Structure proofs to minimize the number of rabbits. (3.61) Principle: Lemmas can provide structure, bring to light interesting facts, and ultimately shorten proofs. (3.60) Heuristic: Exploit the ability to parse theorems like the Golden rule in many dierent ways. (4.10) Deduction: To prove P ) Q, assume P and prove Q, with the variables and metavariables of P treated as constants. (4.12) Case analysis: To prove S,prove P _ Q _ R, P ) S, Q ) S, and R ) S. (4.13) Partial evaluation: To prove E[V := P ],prove E[V := true] and E[V := false]. (4.14) Mutual implication: To prove P Q,prove P ) Q and Q ) P. (4.19) Contradiction: To prove P,prove :P ) false (perhaps by assuming :P and proving false ). (4.22) Contrapositive: To prove P ) Q,prove :Q ):P. (5.2) Heuristic: To show that an English argument is sound, formalize it and prove that its formalization is a theorem. (5.3) Heuristic: To disprove an expression (prove that it is not a theorem), nd a counterexample: a state in which the expression is false. General Laws of Quantification For symmetric and associative binary operator? with identity u. (8.18) Leibniz: R ) P = Q (?x R : E[V := P ]) = (?x R : E[V := Q]) (8.19) Caveat. The axioms and theorems for all quantications hold only in cases in which all the individual quantications are dened or specied. Theorems (8.20) Axiom, Empty range: (?x false : P ) =u (8.21) Axiom, Identity accumulation: (?x R : u) =u (8.22) Axiom, One-point rule: Provided :occurs(`x' `E'), (?x x = E : P ) = P x E (8.23) Axiom, Distributivity: (?x R : P )? (?x R : Q) = (?x R : P?Q)

6 (8.24) Axiom, Range split: (?x R _ S : P )? (?x R ^ S : P ) = (?x R : P )? (?x S : P ) (8.25) Axiom, Interchange of dummies: Provided :occurs(`y' `R') and :occurs(`x' `Q'), (?x R :(?y Q : P )) = (?y Q :(?x R : P )) (8.26) Axiom, Nesting: Provided :occurs(`y' `R'), (?x y R ^ Q : P )=(?x R :(?y Q : P )) (8.27) Range split: Provided R ^ S false (?x R _ S : P ) = (?x R : P )? (?x S : P ) (8.28) Range split for idempotent? : (?x R _ S : P ) = (?x R : P )? (?x S : P ) (8.29) Dummy reordering: (?x y R : P )=(?y x R : P ) (8.30) Change of dummy: Provided :occurs(`y' `R P '), and f has an inverse, (?x R : P )=(?y R x f:y : P x f:y) (8.31) Dummy renaming: Provided :occurs(`y' `R P '), (?x R : P )=(?y R x y : P x y ) (8.32) Split o term (example): (?i 0 i<n+1:p ) = (?i 0 i<n: P )?P i n Theorems of the Predicate Calculus Universal quantification (9.3) Axiom, Distributivity of _ over 8 : Provided :occurs(`x' `P '), P _ (8x R : Q) (8x R : P _ Q) (9.4) Trading: (a) (8x R : P ) (8x : R ) P ) (b) (8x R : P ) (8x : R _ P P ) (c) (8x R : P ) (8x : R ^ P R) (d) (8x R : P ) (8x : :R _ P ) (9.5) Trading: (a) (8x Q ^ R : P ) (8x Q : R ) P ) (b) (8x Q ^ R : P ) (8x Q : R _ P P ) (c) (8x Q ^ R : P ) (8x Q : R ^ P R) (d) (8x Q ^ R : P ) (8x Q : :R _ P ) (9.6) 8-True body: (8x R : true) true (9.7) 8-False body: (8x : false) false (9.8) 8-Split-o-term: (8x : P ) (8x : P ) ^ P x E (9.9) Quantied freshy: Provided :occurs(`x' `P '), P ) (8x R : P ) (9.10) Provided :occurs(`x' `P '), P ^ (8x R : Q) ) (8x R : P ^ Q) (9.11) Distributivity of ^ over 8 : Provided :occurs(`x' `P '), :(8x : :R) ) ((8x R : P ^ Q) P ^ (8x R : Q)) (9.12) (8x R : P Q) ) ((8x R : P ) (8x R : Q))

7 (9.13) Provided :occurs(`x' `Q'), Q ) (8x R : P ) (8x R : Q ) P ) (9.14) Monotonic 8-body: (8x R : P ) Q) ) ((8x R : P ) ) (8x R : Q)) (9.15) Antimonotonic 8-range: (8x :R : P ) Q) ) ((8x P : R) ( (8x Q : R)) (9.17) Instantiation: (8x : P ) ) P x E (9.18) Provided :occurs(`x' `P '), (8x : P ) P (9.21) Metatheorem: P is a theorem i (8x : P ) is a theorem. Existential quantification (9.22) Axiom, Generalized De Morgan: (9x R : P ) :(8x R : :P ) (9.23) Generalized De Morgan: (a) :(9x R : :P ) (8x R : P ) (b) :(9x R : P ) (8x R : :P ) (c) (9x R : :P ) :(8x R : P ) (9.25) Distributivity of ^ over 9 : Provided :occurs(`x' `P '), P ^ (9x R : Q) (9x R : P ^ Q) (9.26) Trading: (9x R : P ) (9x : R ^ P ) (9.27) Trading: (9x Q ^ R : P ) (9x Q : R ^ P ) (9.28) 9-False body: (9x R : false) false (9.29) 9-True body: (9x : true) true (9.30) 9-Split-o-term: (9x : P ) (9x : P ) _ P x E (9.31) Quantied freshy: Provided :occurs(`x' `P '), (9x R : P ) ) P (9.32) Provided :occurs(`x' `P '), (9x R : P _ Q) ) P _ (9x R : Q) (9.33) Distributivity of _ over 9 : Provided :occurs(`x' `P '), (9x : R) ) ((9x R : P _ Q) P _ (9x R : Q)) (9.34) (8x R : P Q) ) ((9x R : P ) (9x R : Q)) (9.35) Provided :occurs(`x' `Q'), (9x R : P ) ) Q (8x R : P ) Q) (9.36) Monotonic 9-body: (8x R : P ) Q) ) ((9x R : P ) ) (9x R : Q)) (9.37) Monotonic 9-range: (8x R : P ) Q) ) ((9x P : R) ) (9x Q : R)) (9.38) 9-Introduction: P x E ) (9x : P ) (9.39) Provided :occurs(`x' `P '), (9x : P ) P (9.40) Interchange of quantications: Provided :occurs(`y' `R') and :occurs(`x' `Q'), (9x R :(8y Q : P )) ) (8y Q :(9x R : P ))

8 (9.41) Metatheorem Witness: Provided :occurs(`^x' `P Q R'), (9x R : P ) ) Q is a theorem i (R ^ P ) x^x ) Q is a theorem. Set Theory Set comprehension and enumeration (10.2) Set enumeration: fe 0 ::: E n;1g = fx x = E 0 x = E n;1 : xg (10.3) Axiom, Universe: U: set(t) =fx:t true : xg (10.4) Axiom, Set membership: Provided :occurs(`x' `F '), F 2 fx R : Eg (9x R : F = E) (10.5) Axiom, Extensionality: S = T (8x : x 2 S x 2 T ) (10.6) S = fx x 2 S : xg (10.7) Empty range: fx false : P g = fg (10.8) Empty set: (v 2 fx false : P g) = (v 2 fg) = false (10.9) Singleton membership : e 2 feg e = E (10.10) One-point rule: Provided :occurs(`x' `E'), fx x = E : P g = fp [x:= E]g (10.11) Dummy reordering: fx y R : P g = fy x R : P g (10.12) Change of dummy: Provided :occurs(`y' `R P ') and f has an inverse, fx R : P g = fy R[x:= f:y]:p [x:= f:y]g (10.13) Dummy renaming: Provided :occurs(`y' `R P '), fx R : P g = fy R[x:= y] :P [x:= y]g (10.14) Provided :occurs(`y' `R E'), fx R : Eg = fy (9x R : y = E)g (10.15) x 2 fx Rg R (10.16) Principle of comprehension. To each predicate R corresponds the set fx:t Rg, which contains the objects in t that satisfy characteristic predicate R of the set. (10.17) fx Qg = fx Rg (8x : Q R) (10.18) Metatheorem: fx Qg = fx Rg is valid i Q R is valid. Operations on sets (10.20) Axiom, Size: # S = (x x 2 S :1) (10.21) Axiom, Subset: S T (8x x 2 S : x 2 T ) (10.22) Axiom, Proper subset: S T S T ^ S 6= T (10.23) Axiom, Superset: T S S T (10.24) Axiom, Proper superset: T S S T (10.25) Axiom, Complement: x 2 S x 62 S (for x in U ) (10.26) Axiom, Union: v 2 S [ T v 2 S _ v 2 T

9 (10.27) Axiom, Intersection: v 2 S \ T v 2 S ^ v 2 T (10.28) Axiom, Dierence: v 2 S ; T v 2 S ^ v 62 T (10.29) Axiom, Power set: v 2 PS v S (10.31) Metatheorem: For any set expressions E s and F s, (a) E s = F s is valid i E p F p is valid, (b) E s F s is valid i E p ) F p is valid, (c) E s = U is valid i E p is valid. Properties of complement (10.32) fx:t P g = fx: t :P g (10.33) S = S (10.34) U = fg (10.35) fg= U Properties of union and intersection (10.36){(10.40): [ is symmetric, associative, idempotent, has zero U, and has identity fg. (10.41) Weakening: S S [ T (10.42) Excluded middle: S [S = U (10.43){(10.47): \ is symmetric, associative, idempotent, has zero fg, and has identity U. (10.48) Strengthening: S \ T S (10.49) Contradiction: S \S = fg (10.50) Distributivity of [ over \ : S [ (T \ V ) = (S [ T ) \ (S [ V ) (10.51) Distributivity of \ over [ : S \ (T [ V ) = (S \ T ) [ (S \ V ) (10.52) De Morgan: (a) (S [ T ) = S \T (b) (S \ T ) = S [T (10.53) S T ^ W V ) (S [ W ) (T [ V ) (10.54) S T ^ W V ) (S \ W ) (T \ V ) (10.55) S T S [ T = T (10.56) S T S \ T = S (10.57) S [ T = U (8x x 2 U : x 62 S ) x 2 T ) (10.58) S \ T = fg (8x : x 2 S ) x 62 T ) (10.59) Range split: fx P _ Q : Eg = fx P : Eg [fx Q : Eg Properties of set difference (10.60) S ; T = S \T (10.61) S ; T S (10.62) S ;fg= S

10 (10.63) S ; U = (10.64) fg;s = fg (10.65) U ; S = S (10.66) S \ (T ; S) = fg (10.67) S [ (T ; S) = S [ T (10.68) S ; (T [ V ) = (S ; T ) \ (S ; V ) (10.69) S ; (T \ V ) = (S ; T ) [ (S ; V ) (10.70) (S [ T ) ; V = (S ; V ) [ (T ; V ) (10.71) (S \ T ) ; V = (S ; V ) \ (T ; V ) (10.72) (S ; T ) = S [ T Properties of subset (10.73) (8x : P ) Q) fx P gfx Qg (10.74) Antisymmetry: S T ^ T S S = T (10.75) Reexivity: S S (10.76) Transitivity: S T ^ T V ) S V (10.77) fgs (10.78) S T S T ^ :(T S) (10.79) S T S T ^ (9x x 2 T : x 62 S) (10.80) S T S T _ S = T (10.81) S 6 S (10.82) S T ) S T (10.83) S T ) T 6 S (10.84) S T ) T 6 S (10.85) S T ^:(V T ) ) :(V S) (10.86) T U ^:(T S) ) :(U S) (10.87) Transitivity: (a) S T ^ T V ) S V (b) S T ^ T V ) S V (c) S T ^ T V ) S V Properties of powerset (10.88) Pf g = ff gg (10.89) S 2 PS (10.90) # (PS) =2 # S (for nite set S ) (10.91) Monotonic P : S T )PS PT (10.95) Axiom of Choice: For t a type, there exists a function f :set(t)! t such that for any nonempty set S, f:s 2 S.

11 Mathematical Induction (11.3) Mathematical Induction over N: For all P, P:0 ^ (8n:N :(8i 0 i n : P:i) ) P (n + 1)) ) (8n:N : P:n) (11.7) Heuristic: In proving (8i 0 i n : P:i) ) P (n +1) by assuming (8i 0 i n : P:i), manipulate or restate P (n + 1) in order to expose at least one of P:0 ::: P:n. (11.16) Mathematical Induction over hu i : For all P, (8x : P:x) (8x :(8y y x : P:y) ) P:x) (11.18) Denition of well-founded: hu i is well-founded i, for all S, S 6= fg (9x : x 2 S ^ (8y y x : y=2 S)) (11.19) Theorem: hu i is well founded i it admits induction. (11.21) Axiom, Finite chain property: (8x :(8y y x : DCF :y) ) DCF :x) (11.23) Theorem: hu i is well founded i every decreasing chain is nite.