Résolution Quelques équivalences logiques (rappel) D`autres

∀x. A ≡ ¬∃x. ¬A
¬∀x. A ≡ ∃x. ¬A
∃x. A ≡ ¬∀x. ¬A
¬∃x. A ≡ ∀x. ¬A
∀x. (A∧B)≡ ∀x. A ∧ ∀x. B
∃x. (A∨B)≡ ∃x. A ∨ ∃x. B
∃x. (A→B)≡ ∀x. A → ∃x. B
∀x. ∀y. A ≡ ∀y. ∀x. A
∃x. ∃y. A ≡ ∃y. ∃x. A
x /∈V I(A)
∀x. A ≡ ∃x. A ≡A
∀x. (A∧B)≡A∧ ∀x. B
∃x. (A∧B)≡A∧ ∃x. B
∀x. (A∨B)≡A∨ ∀x. B
∃x. (A∨B)≡A∨ ∃x. B
∃x. (A→B)≡A→ ∃x. B
∀x. (A→B)≡A→ ∀x. B
∃x. (B→A)≡ ∀x. B →A
∀x. (B→A)≡ ∃x. B →A
G
Q1x1. . . QnxnA Qi
∀ ∃ A
G G0
G≡G0

G
∀x1. . . ∀xn∃xn+1Qn+2xn+2Qn+ixn+iAf
n
∀x1. . . ∀xnQn+2xn+2Qn+ixn+iA{xn+1/f(x1, . . . , xn)}
G
G G0
G G0
G n
∃G n
G0G
G n ∃G0n
G0∃
G G0
r(t1, . . . , tn)
¬r(t1, . . . , tn)
L1∨. . . ∨Lq
Li⊥
G
CG
V I(C1)∩V I(C2) = ∅C1, C2∈ CGC16=C2
GCG
D∨r(s1, . . . , sn)C∨ ¬r(t1, . . . , tn)
σ(D∨C)(coupure)
σ{s1. =t1, . . . , sn. =tn}

D∨L∨L0
σ(D∨L)(factorisation)
L=r(s1, . . . , sn)L=¬r(s1, . . . , sn)
L0=r(t1, . . . , tn)L0=¬r(t1, . . . , tn)
σ{s1. =t1, . . . , sn. =tn}
r(s1, . . . , sn)r(t1, . . . , tn)
r(s1, . . . , sn)¬r(t1, . . . , tn)
⊥
∆`RF
F∆ ∆ `R⊥
⊥∆
∆`RF
∆|=F∆`R⊥∆
∆|=F
∆`RF∆ ∆ `R⊥
Σ
Σ Σ
Σ
Σ
f∈ΣFn
I(f)(t1, . . . , tn) = f(t1, . . . , tn)
p∈ΣPn
I(p)Sp
I(p)(t1, . . . , tn) = Vp(t1, . . . , tn)∈ Sp

x1, . . . , xnt
IσI
τ={x1/t1, . . . , xn/tn}d1. . . dn[ti]I,σ =di
[t]I,σ[x1:=d1]...[xn:=dn]= [τ(t)]I,σ
x1, . . . , xnG
IσI
τ={x1/t1, . . . , xn/tn}d1. . . dn
[ti]I,σ =di[G]I,σ[x1:=d1]...[xn:=dn]= [τ(G)]I,σ
G=r(x1, x2)τ={x1/a, x2/s(a)}
I(r)(n, m) = Vn < m I(a) = 0 I(s)(n) = n+ 1
C
I I C
B0, B1, B2, . . .
Σ
B0
BiV F
BiBi+1
A1
q(a), q(b), r(a), r(b)
A2
q(a), q(b), q(s(a)), q(s(b)), q(s(s(a))), q(s(s(b))), . . .
AC
n A
CA n
C
n A
A1A2
{¬r(x)∨q(x), q(a), r(a)}
⊥ ∈ C
C

ACC
A
A
A
C={¬r(x)∨q(s(x)), r(a),¬q(s(a))}
CC
C
CC
C1C2C0
1C0 2
C1C2C0
res
C0
1C0 2Cres
C0
res Cres
Cres C1C2
∆|=G
∆`RG∆ ∆ `R⊥