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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
1
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
2
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
3
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
4
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)
5
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
6
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
7
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)
8
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 ))
9
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
11
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)
12
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
13
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
14
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
15
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
16
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
17
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.
18
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)
19
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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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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