1 Propositional Logic - Axioms and Inference

Rules

Axioms

Axiom 1.1 [Commutativity]

(p ∧ q) = (q ∧ p)(p ∨ q) = (q ∨ p)(p = q) = (q = p)

Axiom 1.2 [Associativity]

p ∧ (q ∧ r) = (p ∧ q) ∧ rp ∨ (q ∨ r) = (p ∨ q) ∨ r

Axiom 1.3 [Distributivity]

p ∨ (q ∧ r) = (p ∨ q) ∧ (p ∨ r)p ∧ (q ∨ r) = (p ∧ q) ∨ (p ∧ r)

Axiom 1.4 [De Morgan]

¬(p ∧ q) = ¬p ∨ ¬q¬(p ∨ q) = ¬p ∧ ¬q

Axiom 1.5 [Negation]

¬¬p = p

Axiom 1.6 [Excluded Middle]

p ∨ ¬p = T

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Axiom 1.7 [Contradiction]

p ∧ ¬p = F

Axiom 1.8 [Implication]

p ⇒ q = ¬p ∨ q

Axiom 1.9 [Equality]

(p = q) = (p ⇒ q) ∧ (q ⇒ p)

Axiom 1.10 [or-simplification]

p ∨ p = pp ∨ T = Tp ∨ F = p

p ∨ (p ∧ q) = p

Axiom 1.11 [and-simplification]

p ∧ p = pp ∧ T = pp ∧ F = F

p ∧ (p ∨ q) = p

Axiom 1.12 [Identity]

p = p

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Inference Rulesp1 = p2 , p2 = p3

p1 = p3Transitivity

p1 = p2E(p1) = E(p2) , E(p2) = E(p1)

Substitution

q1 , q2 , . . . , qn , q1 ∧ q2 ∧ . . . ∧ qn ⇒ (p1 = p2)

E(p1) = E(p2) , E(p2) = E(p1)Conditional Substitution

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2 Propositional Logic - Derived Theorems

Equivalence and Truth

Theorem 2.1 [Associativity of = ]

((p = q) = r) = (p = (q = r))

Theorem 2.2 [Identity of = ]

(T = p) = p

Theorem 2.3 [Truth]

T

Negation, Inequivalence, and False

Theorem 2.4 [Definition of F ]

F = ¬T

Theorem 2.5 [Distributivity of ¬ over = ]

¬(p = q) = (¬p = q)(¬p = q) = (p = ¬q)

Theorem 2.6 [Negation of F ]

¬F = T

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Theorem 2.7 [Definition of ¬]

(¬p = p) = F¬p = (p = F )

Disjunction

Theorem 2.8 [Distributivity of ∨ over = ]

(p ∨ (q = r)) = ((p ∨ q) = (p ∨ r))((p ∨ (q = r)) = (p ∨ q)) = (p ∨ r)

Theorem 2.9 [Distributivity of ∨ over ∨ ]

p ∨ (q ∨ r) = (p ∨ q) ∨ (p ∨ r)

Conjunction

Theorem 2.10 [Mutual definition of ∧ and ∨ ]

(p ∧ q) = (p = (q = (p ∨ q)))(p ∧ q) = ((p = q) = (p ∨ q))

((p ∧ q) = p) = (q = (p ∨ q))((p ∧ q) = (p = q)) = (p ∨ q)(((p ∧ q) = p) = q) = (p ∨ q)

Theorem 2.11 [Distributivity of ∧ over ∧ ]

p ∧ (q ∧ r) = (p ∧ q) ∧ (p ∧ r)

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Theorem 2.12 [Absorption]

p ∧ (¬p ∨ q) = p ∧ qp ∨ (¬p ∧ q) = p ∨ q

Theorem 2.13 [Distributivity of ∧ over = ]

(p ∧ q) = ((p ∧ ¬q) = ¬p)((p ∧ q) = (p ∧ ¬q)) = ¬p

p ∧ (q = p) = (p ∧ q)

Theorem 2.14 [Replacement]

(p = q) ∧ (r = p) = (p = q) ∧ (r = q)

Theorem 2.15 [Definition of = ]

(p = q) = (p ∧ q) ∨ (¬p ∧ ¬q)

Theorem 2.16 [Exclusive or]

¬(p = q) = (¬p ∧ q) ∨ (p ∧ ¬q)

Implication

Theorem 2.17 [Definition of Implication]

(p ⇒ q) = ((p ∨ q) = q)((p ⇒ q) = (p ∨ q)) = q

(p ⇒ q) = ((p ∧ q) = p)((p ⇒ q) = (p ∧ q)) = p

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Theorem 2.18 [Contrapositive]

(p ⇒ q) = (¬q ⇒ ¬p)

Theorem 2.19 [Distributivity of ⇒ over = ]

p ⇒ (q = r) = ((p ⇒ q) = (p ⇒ r))

Theorem 2.20 [Shunting]

p ∧ q ⇒ r = p ⇒ (q ⇒ r)

Theorem 2.21 [Elimination/Introduction of ⇒ ]

p ∧ (p ⇒ q) = p ∧ qp ∧ (q ⇒ p) = pp ∨ (p ⇒ q) = Tp ∨ (q ⇒ p) = ¬q ∨ p

(p ∨ q) ⇒ (p ∧ q) = (p = q)p ⇒ F = ¬pF ⇒ p = T

Theorem 2.22 [Right Zero of ⇒ ]

(p ⇒ T ) = T

Theorem 2.23 [Left Identity of ⇒ ]

(T ⇒ p) = p

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Theorem 2.24 [Weakening/Strengthening]

p ⇒ p ∨ qp ∧ q ⇒ pp ∧ q ⇒ p ∨ q

p ∨ (q ∧ r) ⇒ p ∨ qp ∧ q ⇒ p ∧ (q ∨ r)

Theorem 2.25 [Modus Ponens]

p ∧ (p ⇒ q) ⇒ q

Theorem 2.26 [Proof by Cases]

(p ⇒ r) ∧ (q ⇒ r) = (p ∨ q ⇒ r)(p ⇒ r) ∧ (¬p ⇒ r) = r

Theorem 2.27 [Mutual Implication]

(p ⇒ q) ∧ (q ⇒ p) = (p = q)

Theorem 2.28 [Antisymmetry]

(p ⇒ q) ∧ (q ⇒ p) ⇒ (p = q)

Theorem 2.29 [Transitivity]

(p ⇒ q) ∧ (q ⇒ r) ⇒ (p ⇒ r)(p = q) ∧ (q ⇒ r) ⇒ (p ⇒ r)(p ⇒ q) ∧ (q = r) ⇒ (p ⇒ r)

Theorem 2.30 [Monotonicity of ∨ ]

(p ⇒ q) ⇒ (p ∨ r ⇒ q ∨ r)

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Theorem 2.31 [Monotonicity of ∧ ]

(p ⇒ q) ⇒ (p ∧ r ⇒ q ∧ r)

Substitution

Theorem 2.32 [Leibniz]

(e = f) ⇒ (E(e) = E(f))

Theorem 2.33 [Substitution]

(e = f) ∧ E(e) = (e = f) ∧ E(f)(e = f) ⇒ E(e) = (e = f) ⇒ E(f)

q ∧ (e = f) ⇒ E(e) = q ∧ (e = f) ⇒ E(f)

Theorem 2.34 [Replace by T ]

p ∧ E(p) = p ∧ E(T )p ⇒ E(p) = p ⇒ E(T )

q ∧ p ⇒ E(p) = q ∧ p ⇒ E(T )

Theorem 2.35 [Replace by F ]

p ∨ E(p) = p ∨ E(F )E(p) ⇒ p = E(F ) ⇒ p

E(p) ⇒ p ∨ q = E(F ) ⇒ p ∨ q

Theorem 2.36 [Shannon]

E(p) = (p ∧ E(T )) ∨ (¬p ∧ E(F ))

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3 Propositional Logic - Examples and Exer-

cises

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4 Predicate Logic - Axioms

Axiom 4.1 [Definition of ∃]

(m ≥ n) ⇒

∃i : m ≤ i < n : pi=F

(m < n) ⇒

∃i : m ≤ i < n : pi=

∃i : m ≤ i < n− 1 : pi∨

pn−1

Axiom 4.2 [Definition of ∀]

(m ≥ n) ⇒

∀i : m ≤ i < n : pi=T

(m < n) ⇒

∀i : m ≤ i < n : pi=

∀i : m ≤ i < n− 1 : pi∧

pn−1

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Axiom 4.3 [Range Split]

(m1 ≤ m2 ≤ m3) ⇒

∀i : m1 ≤ i < m2 : pi∧

∀i : m2 ≤ i < m3 : pi

=∀i : m1 ≤ i < m3 : pi

(m1 ≤ m2) ∧ (n1 ≥ n2) ⇒

∀i : m1 ≤ i < n1 : pi∧

∀i : m2 ≤ i < n2 : pi

=∀i : m1 ≤ i < n1 : pi

(m1 ≤ m2 ≤ m3) ⇒

∃i : m1 ≤ i < m2 : pi∨

∃i : m2 ≤ i < m3 : pi

=∃i : m1 ≤ i < m3 : pi

(m1 ≤ m2) ∧ (n1 ≥ n2) ⇒

∃i : m1 ≤ i < n1 : pi∨

∃i : m2 ≤ i < n2 : pi

=∃i : m1 ≤ i < n1 : pi

Axiom 4.4 [Interchange of Dummies]

∀i : m1 ≤ i < n1 : (∀j : m2 ≤ j < n2 : pi,j)=

∀j : m2 ≤ j < n2 : (∀i : m1 ≤ i < n1 : pi,j)

∃i : m1 ≤ i < n1 : (∃j : m2 ≤ j < n2 : pi,j)=

∃j : m2 ≤ j < n2 : (∃i : m1 ≤ i < n1 : pi,j)

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Axiom 4.5 [Dummy Renaming]

∀i : m ≤ i < n : pi = ∀j : m ≤ j < n : pj

Axiom 4.6 [Distributivity of ∨ over ∀ ]

(p ∨ (∀i : m ≤ i < n : qi)) = ∀i : m ≤ i < n : (p ∨ qi)

Axiom 4.7 [Distributivity of ∧ over ∀ ]

(m < n) ⇒

p ∧ (∀i : m ≤ i < n : qi)=

∀i : m ≤ i < n : p ∧ qi

∀i : m ≤ i < n : pi∧

∀i : m ≤ i < n : qi

= ∀i : m ≤ i < n : (pi ∧ qi)

Axiom 4.8 [Distributivity of ∧ over ∃ ]

(p ∧ (∃i : m ≤ i < n : qi)) = ∃i : m ≤ i < n : (p ∧ qi)

Axiom 4.9 [Distributivity of ∨ over ∃ ]

(m < n) ⇒

p ∨ (∃i : m ≤ i < n : qi)=

∃i : m ≤ i < n : (p ∨ qi)

∃i : m ≤ i < n : pi∨

∃i : m ≤ i < n : qi

= ∃i : m ≤ i < n : (pi ∨ qi)

Axiom 4.10 [Universality of T ]

∀i : m ≤ i < n : T = T

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Axiom 4.11 [Existence of F ]

∃i : m ≤ i < n : F = F

Axiom 4.12 [Generalized De Morgan]

¬ (∃i : m ≤ i < n : pi) = ∀i : m ≤ i < n : ¬pi¬ (∀i : m ≤ i < n : pi) = ∃i : m ≤ i < n : ¬pi

Axiom 4.13 [Trading]

(m ≤ i < n) ⇒ pi = ∀i : m ≤ i < n : pi(m ≤ i < n) ∧ pi ⇒ ∃i : m ≤ i < n : pi

Axiom 4.14 [Definition of Numerical Quantification]

(m ≥ n) ⇒

N i : m ≤ i < n : pi=0

(m < n) ∧ ¬pn−1 ⇒

N i : m ≤ i < n : pi=

N i : m ≤ i < n− 1 : pi

(m < n) ∧ pn−1 ⇒

N i : m ≤ i < n : pi=

N i : m ≤ i < n− 1 : pi+1

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Axiom 4.15 [Definition of Σ]

(m ≥ n) ⇒

Σi : m ≤ i < n : ei=0

(m < n) ⇒

Σi : m ≤ i < n : ei=

Σi : m ≤ i < n− 1 : ei+

en−1

Axiom 4.16 [Definition of Π]

(m ≥ n) ⇒

Πi : m ≤ i < n : ei=1

(m < n) ⇒

Πi : m ≤ i < n : ei=

Πi : m ≤ i < n− 1 : ei∗

en−1

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5 Predicate Logic - Derived Theorems

Theorem 5.1 [Definition of ∃]

(m ≥ n) ⇒

∃i : m < i ≤ n : pi=F

(m < n) ⇒

∃i : m < i ≤ n : pi=

∃i : m+ 1 < i ≤ n : pi∨

pm+1

Theorem 5.2 [Definition of ∀]

(m ≥ n) ⇒

∀i : m < i ≤ n : pi=T

(m < n) ⇒

∀i : m < i ≤ n : pi=

∀i : m+ 1 < i ≤ n : pi∧

pm+1

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Theorem 5.3 [Definition of Σ]

(m ≥ n) ⇒

Σi : m < i ≤ n : ei=0

(m < n) ⇒

Σi : m < i ≤ n : ei=

Σi : m+ 1 < i ≤ n : ei+

em+1

Theorem 5.4 [Definition of Π]

(m ≥ n) ⇒

Πi : m < i ≤ n : ei=1

(m < n) ⇒

Πi : m < i ≤ n : ei=

Πi : m+ 1 < i ≤ n : ei∗

em+1

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6 Some Simple Laws of Arithmetic

Throughout this compendium, we assume the validity of all “simple” arith-metic rules. Examples of such rules are all simplification rules, e.g. =

2 + 3 = 5x+ x = 2 ∗ xx+ y − y = x(x/3) ∗ 3 = x0 ∗ x = 01 ∗ x = xx ∗ x = x2

0x = 01x = 1

(2 ∗ x+ 10 = 20) = (x = 5)(x+ y < 2 ∗ y) = (x < y)(x+ y = x+ z) = (y = z)

x ∗ (y + 1)− (x+ z) = (x ∗ y − z)

Following is a collection of theorems that might be used. The list is notexhaustive but intend to show the level of complexity that you can specifytheorems on.

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Theorems on < and ≤

(x < y) = (y > x)

(x < y) ⇒ ¬(y = x) ∧ ¬(y < x)

(x < y) ⇒ (x ≤ y)

(x < y) = (x ≤ y) ∧ (x 6= y)

(x < y) ∧ (y ≤ z) ⇒ (x < z)

(x ≤ y) = ¬(x > y)

(x ≤ x) = T

(x ≤ y) ∧ (y ≤ z) ⇒ (x ≤ z)

(x ≤ y) ∧ (y < z) ⇒ (x < z)

(x ≤ y) ∧ ¬(x < y) ⇒ (x < y)

(x ≤ y − 1) = (x < y)

(x ≤ y) ∨ (y ≤ x) = T

(x ≤ y) ∨ (y < x) = T

(x ≤ y) = (x < y) ∨ (x = y)

Theorems on properties about + and −

(x < y) ⇒ (x < y + 1)

(x < y + 1) = (x ≤ y)

(x < y) ⇒ (z − y < z − x)

(0 < x) = (−x < 0)

(x− 1 < x) = T

(x ≤ y − 1) = (x < y)

(x ≤ y) = (x− 1 < y)

(x1 < y1) ∧ (x2 < y2) ⇒ (x1 + x2 < y1 + y2)

(x1 ≤ y1) ∧ (x2 ≤ y2) ⇒ (x1 + x2 ≤ y1 + y2)

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Theorems on properties about ∗ and /

(0 < x) = (0 < 2 ∗ x)

(0 < x) = (x < 2 ∗ x)

(0 < x) = (x÷ 2 < x)

(0 ≤ x/2) = (0 ≤ x)

(x = 0) ⇒ (x ∗ y = 0)

2 ∗ (x/2) = x

Theorems on equivalence relation

(x = x) = T

(x = y) = (x ≤ y) ∧ (y ≤ x)

Theorems about odd(n) and even(n)

odd(x) ⇒ ((x− 1)÷ 2 = (x− 1)/2)

even(x) ⇒ (x÷ 2 = x/2)

odd(x+ 2 ∗ y) = odd(x)

even(x+ 2 ∗ y) = even(x)

odd(x) = ¬even(x)

odd(x) ⇒ ((x ≥ 1) = (x ≥ 0))

odd(x) ∧ (x = 0) = F

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7 Predicate Logic - Examples and Exercises

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8 Arrays - Axioms

8.1 Axioms

Axiom 8.1 [Assignment to Array Element]

((b; i : e) [j] = f) =

(i = j) ⇒ (e = f)∧

(i 6= j) ⇒ (b[j] = f)

Axiom 8.2 [Definition of Arithmetic Relations]

(b[i : j] = x) = (∀k : i ≤ k < j + 1 : b[k] = x)(b[i : j] < x) = (∀k : i ≤ k < j + 1 : b[k] < x)(b[i : j] > x) = (∀k : i ≤ k < j + 1 : b[k] > x)(b[i : j] ≤ x) = (∀k : i ≤ k < j + 1 : b[k] ≤ x)(b[i : j] ≥ x) = (∀k : i ≤ k < j + 1 : b[k] ≥ x)(b[i : j] 6= x) = (∀k : i ≤ k < j + 1 : b[k] 6= x)

x ∈ b[i : j] = (∃k : i ≤ k < j + 1 : x = b[k])

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