Consistency proof of a feasible arithmetic inside a bounded arithmetic
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Consistency proof of a feasible arithmetic inside a bounded arithmetic
Consistency proof of a feasible arithmetic
inside a bounded arithmetic
Yoriyuki Yamagata
Sendai Logic Seminar, Oct. 31, 2014
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Consistency proof of a feasible arithmetic inside a bounded arithmetic
Motivation
Ultimate Goal
Conjecutre
S12 6= S2
where S i2, i = 1, 2, . . . are Buss’s hierarchies of bounded
arithmetics and S2 =⋃
i=1,2,... S i2
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Consistency proof of a feasible arithmetic inside a bounded arithmetic
Motivation
Approach using consistency statement
If S2 ` Con(S12 ), we have S1
2 6= S2.
Theorem (Pudlak, 1990)S2 6` Con(S1
2 )
QuestionCan we find a theory T such that S2 ` Con(T ) butS1
2 6` Con(T )?
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Consistency proof of a feasible arithmetic inside a bounded arithmetic
Motivation
Unprovability
Theorem (Buss and Ignjatovic 1995)
S12 6` Con(PV−)
PV− : Cook & Urquhart’s equational theory PV minusinductionBuss and Ignjatovic enrich PV− by propositional logic andBASIC e-axioms
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Consistency proof of a feasible arithmetic inside a bounded arithmetic
Motivation
Provability
Theorem (Beckmann 2002)
S12 ` Con(PV−)
if PV− is formulated without substitution
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Consistency proof of a feasible arithmetic inside a bounded arithmetic
Main results
Main result
Theorem
S12 ` Con(PV−)
Even if PV− is formulated with substitutionOur PV− is equational
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Consistency proof of a feasible arithmetic inside a bounded arithmetic
Consistency of PV-
Soundness theorem
Cook & Urquhart’s PV : language
We formulate PV as an equational theory of binary digits.
1. Constant : ε
2. Function symbols for all polynomial time functionss0, s1, ε
n, projni , . . .
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Consistency proof of a feasible arithmetic inside a bounded arithmetic
Consistency of PV-
Soundness theorem
Cook Urquhart’s PV : axioms
1. Recursive definition of functions
1.1 Composition1.2 Recursion
2. Equational axioms
3. Substitutiont(x) = u(x)
t(s) = u(s) (1)
4. PIND for all formulas
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Consistency proof of a feasible arithmetic inside a bounded arithmetic
Consistency of PV-
Soundness theorem
Consistency proof of PV− inside S12
Theorem (Consistency theorem)
S12 ` Con(PV−) (2)
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Consistency proof of a feasible arithmetic inside a bounded arithmetic
Consistency of PV-
Soundness theorem
Soundness theorem
TheoremLet
1. π `PV− 0 = 1
2. r ` t = u be a sub-proof of π
3. ρ : sequence of substitution such that tρ is closed
4. σ ` 〈t, ρ〉 ↓ v , α
Then,∃τ, τ ` 〈u, ρ〉 ↓ v , α (3)
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Consistency proof of a feasible arithmetic inside a bounded arithmetic
Consistency of PV-
Soundness theorem
Soundness theorem
Theorem (S12 )
Let
1. π `PV− 0 = 1, U > ||π||2. r ` t = u be a sub-proof of π
3. ρ : sequence of substitution such that tρ is closed and||ρ|| ≤ U − ||r ||
4. σ ` 〈t, ρ〉 ↓ v , α and |||σ|||,Σα∈αM(α) ≤ U − ||r ||Then,
∃τ, τ ` 〈u, ρ〉 ↓ v , α (4)
s.t. |||τ ||| ≤ |||σ|||+ ||r ||
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Consistency proof of a feasible arithmetic inside a bounded arithmetic
Consistency of PV-
Soundness theorem
Consistency of big-step semantics
Lemma (S12 )
There is no computation which proves 〈ε, ρ〉 ↓ s1ε
Proof.Immediate from the form of inference rules
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Consistency proof of a feasible arithmetic inside a bounded arithmetic
Consistency of PV-
Big-step semantics
Inference rules for 〈t, ρ〉 ↓ v
Definition (Substitution)
〈t, ρ2〉 ↓ v
〈x , ρ1[t/x ]ρ2〉 ↓ v (5)
where ρ1 does not contain a substitution to x .
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Consistency proof of a feasible arithmetic inside a bounded arithmetic
Consistency of PV-
Big-step semantics
Inference rules for 〈t, ρ〉 ↓ v
Definition (ε, s0, s1)
〈ε, ρ〉 ↓ ε (6)
〈t, ρ〉 ↓ v
〈si t, ρ〉 ↓ siv (7)
where i is either 0 or 1.
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Consistency proof of a feasible arithmetic inside a bounded arithmetic
Consistency of PV-
Big-step semantics
Inference rules for 〈t, ρ〉 ↓ v
Definition (Constant function)
〈εn(t1, . . . , tn), ρ〉 ↓ ε (8)
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Consistency proof of a feasible arithmetic inside a bounded arithmetic
Consistency of PV-
Big-step semantics
Inference rules for 〈t, ρ〉 ↓ v
Definition (Projection)
〈ti , ρ〉 ↓ vi〈projni (t1, . . . , tn), ρ〉 ↓ vi (9)
for i = 1, · · · , n.
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Consistency proof of a feasible arithmetic inside a bounded arithmetic
Consistency of PV-
Big-step semantics
Inference rules for 〈t, ρ〉 ↓ v
Definition (Composition)If f is defined by g(h(t)),
〈g(w), ρ〉 ↓ v 〈h(v), ρ〉 ↓ w 〈t, ρ〉 ↓ v
〈f (t1, . . . , tn), ρ〉 ↓ v (10)
t : members of t1, . . . , tn such that t are not numerals
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Consistency proof of a feasible arithmetic inside a bounded arithmetic
Consistency of PV-
Big-step semantics
Inference rules for 〈t, ρ〉 ↓ v
Definition (Recursion - ε)
〈gε(v1, . . . , vn), ρ〉 ↓ v 〈t, ρ〉 ↓ ε 〈t, ρ〉 ↓ v
〈f (t, t1, . . . , tn), ρ〉 ↓ v (11)
t : members ti of t1, . . . , tn such that ti are not numerals
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Consistency proof of a feasible arithmetic inside a bounded arithmetic
Consistency of PV-
Big-step semantics
Inference rules for 〈t, ρ〉 ↓ v
Definition (Recursion - s0, s1)
〈gi(v0,w , v), ρ〉 ↓ v 〈f (v0, v), ρ〉 ↓ w {〈t, ρ〉 ↓ siv0} 〈t, ρ〉 ↓ v
〈f (t, t1, . . . , tn), ρ〉 ↓ v(12)
i = 0, 1t : members ti of t1, . . . , tn such that ti are not numeralsThe clause {〈t, ρ〉 ↓ siv0} is there if t is not a numeral
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Consistency proof of a feasible arithmetic inside a bounded arithmetic
Consistency of PV-
Big-step semantics
Computation (sequence)
Definition
Computation sequence A DAG (or sequence with pointers)which is built by these inference rules
Computation statement A form 〈t, ρ〉 ↓ vt : main termρ : environmentv : value
Conclusion A computation statement which are not used forassumptions of inferences
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Consistency proof of a feasible arithmetic inside a bounded arithmetic
Consistency of PV-
Bounding Godel numbers by number of nodes
Upper bound of Godel number of main terms
Lemma (S12 )
M(σ) : the maximal size of main terms in σ.
M(σ) ≤ maxg∈Φ{(||g ||+ 1) · (C + ||g ||+ |||σ|||+ T )}
Φ = Base(t1, . . . , tm, ρ1, . . . , ρm)
T = ||t1||+ · · ·+ ||tm||+ ||ρ1||+ · · ·+ ||ρm||
t1, . . . , tm : main terms of conclusionsρ1, . . . , ρm : environments of conclusions σ.
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Consistency proof of a feasible arithmetic inside a bounded arithmetic
Consistency of PV-
Bounding Godel numbers by number of nodes
Size of environments
Lemma (S12 )
Assume σ ` 〈t1, ρ1〉 ↓ v1, . . . , 〈tm, ρm〉 ↓ vm.If 〈t, ρ〉 ↓ v ∈ σ then ρ is a subsequence of one of theρ1, . . . , ρm.
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Consistency proof of a feasible arithmetic inside a bounded arithmetic
Consistency of PV-
Bounding Godel numbers by number of nodes
Size of value
Lemma (S12 )
If σ contains 〈t, ρ〉 ↓ v , then ||v || ≤ |||σ|||.
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Consistency proof of a feasible arithmetic inside a bounded arithmetic
Consistency of PV-
Bounding Godel numbers by number of nodes
Size of arguments
Lemma (S12 )
σ contains 〈t, ρ〉 ↓ v and u is a subterm of ts.t. u is a numeralThen,
||u|| ≤ max{|||σ|||,T} (13)
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Consistency proof of a feasible arithmetic inside a bounded arithmetic
Consistency of PV-
Bounding Godel numbers by number of nodes
Proof of bound for Godel numbers of main terms
σ = σ1, . . . σnM(σi) be the size of the main term of σi .Prove
M(σi) ≤ maxg∈Φ{(||g ||+ 1) · (C + ||g ||+ |||σ|||+ T )} (14)
by induction on n − iIf σi is a conclusion, M(σi) ≤ TAssume σi is an assumption of σj , j > i .
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Consistency proof of a feasible arithmetic inside a bounded arithmetic
Consistency of PV-
Bounding Godel numbers by number of nodes
Proof of bound for Godel numbers of main terms
Assume that M(σj) satisfies induction hypothesisWe analyze different cases of the inference which derives σj
〈t, ρ〉 ↓ v
〈x , ρ′[t/x ]ρ〉 ↓ v (15)
Since ||t|| ≤ ||ρ′[t/x ]ρ|| ≤ T , we are done
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Consistency proof of a feasible arithmetic inside a bounded arithmetic
Consistency of PV-
Bounding Godel numbers by number of nodes
Proof of bound for Godel numbers of main terms
〈f1(v1), ρ〉 ↓ w1, . . . , 〈fl1(v1,w), ρ〉 ↓ wl1 , 〈tk1 , ρ〉 ↓ v21 , . . . , 〈tkl2 , ρ〉 ↓ v
2l2
〈f (t), ρ〉 ↓ v(16)
v 1, v 2,w , v are numeralstk1 , . . . , tkl2 are a subsequence t.f1, . . . , fl1 ∈ Φthe elements of v 1 come from either v 2 or t that are numerals.
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Consistency proof of a feasible arithmetic inside a bounded arithmetic
Consistency of PV-
Bounding Godel numbers by number of nodes
Proof of bound for Godel numbers of main terms
||tk1||, . . . , ||tkl2 || ≤ ||f (t)|| (17)
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Consistency proof of a feasible arithmetic inside a bounded arithmetic
Consistency of PV-
Bounding Godel numbers by number of nodes
Proof of bound for Godel numbers of main terms
||fk(v 1)|| ≤ ||fk ||+ ar(fk) · (C + |||σ|||+ T ) (18)
≤ maxg∈Φ{||g ||+ ar(g) · (C + |||σ|||+ T )} (19)
≤ maxg∈Φ{(||g ||+ 1) · (C + |||σ|||+ T )} (20)
where t1 are terms ti s.t. tiρ is a numeral
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Consistency proof of a feasible arithmetic inside a bounded arithmetic
Consistency of PV-
Bounding Godel numbers by number of nodes
Polynomial bound of number of symbols in
computation sequence
CorollaryThere is a polynomial p such that
||σ|| ≤ p(|||σ|||, ||α||) (21)
where ||α|| are conclusions of σ
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Consistency proof of a feasible arithmetic inside a bounded arithmetic
Consistency of PV-
Bounding Godel numbers by number of nodes
Proof of Corollary
Proof.
M(σ) ≤ maxg∈Φ{(||g ||+ 1) · (C + ||g ||+ |||σ|||+ T )} (22)
≤ (||α||+ 1) · (C + ||α||+ |||σ|||+ T ) (23)
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Consistency proof of a feasible arithmetic inside a bounded arithmetic
Consistency of PV-
Bounding Godel numbers by number of nodes
Polynomial bound of Godel number of
computation sequence
Definition
`B α⇔def ∃σ, |||σ||| ≤ B ∧ σ ` α (24)
Corollary`B α is a Σb
1-formula
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Consistency proof of a feasible arithmetic inside a bounded arithmetic
Consistency of PV-
Soundness of defining axioms
Backward lemma
Lemma (S12 )
If f has a defining axiom
f (c(t), t) = tf ,c(t, f (t, t), t) (25)
c(t) : ε, s0t, s1t, t
`B 〈f (c(t), t), ρ〉 ↓ v , α
⇒`B+||(25)|| 〈tf ,c(t, f (t, t), t), ρ〉 ↓ v , α (26)
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Consistency proof of a feasible arithmetic inside a bounded arithmetic
Consistency of PV-
Soundness of defining axioms
Forward lemma
Lemma (S12 )
`B 〈tf ,c(t, f (t, t), t), ρ〉 ↓ v , α
⇒`B+||(25)|| 〈f (c(t), t), ρ〉 ↓B v , α (27)
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Consistency proof of a feasible arithmetic inside a bounded arithmetic
Consistency of PV-
Soundness of defining axioms
Fusion lemma
Lemma (S12 )
σ `n α (28)
〈f (v), ρ〉 ↓ v , 〈t, ρ〉 ↓ v ∈ σ (29)
v are numeralsThen ∃τ such that
τ ` 〈f (t), ρ〉 ↓ v , α (30)
τ |n = σ (31)
|||τ ||| ≤ |||σ|||+ 1 (32)
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Consistency proof of a feasible arithmetic inside a bounded arithmetic
Consistency of PV-
Soundness of defining axioms
Constructor lemma
Lemma (S12 )
〈ε, ρ〉 ↓ v ∈ σ ⇒ v ≡ ε
〈si t, ρ〉 ↓ v ∈ σ ⇒ v ≡ siv0, 〈t, ρ〉 ↓ v0 ∈ σ
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Consistency proof of a feasible arithmetic inside a bounded arithmetic
Consistency of PV-
Soundness of defining axioms
Proof of Backward lemmaFor the case that f = g(h1(x), . . . , hn(x))
〈g(w), ρ〉 ↓ v 〈h(v), ρ〉 ↓ w 〈t, ρ〉 ↓ v
〈f (t1, . . . , tn), ρ〉 ↓ v
By Fusion lemma, τ1 ` 〈g(w), ρ〉 ↓ v , 〈h(t), ρ〉 ↓ w where|||τ1||| ≤ |||σ|||+ #hBy Fusion lemma τ ` 〈g(h(t)), ρ〉 ↓ v
|||τ ||| ≤ |||τ1|||+ 1
≤ |||σ|||+ #h + 1
≤ |||σ|||+ ||g(h(t))||
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Consistency proof of a feasible arithmetic inside a bounded arithmetic
Consistency of PV-
Soundness of defining axioms
Proof of Forward lemma
α,
β1, 〈t0, ρ〉 ↓ v
〈h1(t), ρ〉 ↓ w1 , . . .
〈g(h(t)), ρ〉 ↓ v
is transformed to
αg(w),
β1
〈h1(v), ρ〉 ↓ w1 , . . . , 〈t0, ρ〉 ↓ v
〈f (t), ρ〉 ↓ v .
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Consistency proof of a feasible arithmetic inside a bounded arithmetic
Consistency of PV-
Soundness of defining axioms
Proof of soundness theorem - defining axioms
f (c(t), t) = tf ,c(t, f (t, t), t)
By Backward and Forward lemma,
`B 〈f (c(t), t), ρ〉 ↓ v , α⇒`B+||r || 〈tf ,c(t, f (t, t), t), ρ〉 ↓ v , α
`B 〈tf ,c(t, f (t, t), t), ρ〉 ↓ v , α⇒`B+||r || 〈f (c(t), t), ρ〉 ↓ v , α
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Consistency proof of a feasible arithmetic inside a bounded arithmetic
Consistency of PV-
Substitution lemma
Substitution I
Lemma (S12 )
U ,V : large integers
σ ` 〈t1[u/x ], ρ〉 ↓ v1, . . . , 〈tm[u/x ], ρ〉 ↓ vm, α (33)
|||σ||| ≤ U − ||t1[u/x ]|| − · · · − ||tm[u/x ]|| (34)
M(σ) ≤ V (35)
Then,
∃τ, τ ` 〈t1, [u/x ]ρ〉 ↓ v1 . . . , 〈tm, [u/x ]ρ〉 ↓ vm, α (36)
|||τ ||| ≤ |||σ|||+ ||t1[u/x ]||+ · · ·+ ||tm[u/x ]|| (37)
M(τ) ≤ M(σ) (38)
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Consistency proof of a feasible arithmetic inside a bounded arithmetic
Consistency of PV-
Substitution lemma
Substitution IILemma (S1
2 )U ,V : large integers
σ ` 〈t1, [u/x ]ρ〉 ↓ v1, . . . , 〈tm, [u/x ]ρ〉 ↓ vm, α (39)
|||σ||| ≤ U − ||t1[u/x ]|| − · · · − ||tm[u/x ]|| (40)
M(σ) ≤ V (41)
Then,
∃τ, τ ` 〈t1[u/x ], ρ〉 ↓ v1 . . . , 〈tm[u/x ], ρ〉 ↓ vm, α (42)
|||τ ||| ≤ |||σ|||+ ||t1[u/x ]||+ · · ·+ ||tm[u/x ]|| (43)
M(τ) ≤ M(σ) · ||u|| (44)
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Consistency proof of a feasible arithmetic inside a bounded arithmetic
Consistency of PV-
Substitution lemma
Proof of Substitution I
By induction on ||t1[u/x ]||+ · · ·+ ||tm[u/x ]||Sub-induction on the length of σCase analysis of the last inference of σ
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Consistency proof of a feasible arithmetic inside a bounded arithmetic
Consistency of PV-
Substitution lemma
Proof of Substitution I
〈s, ρ1〉 ↓ v
〈y , ρ′[s/y ]ρ1〉 ↓ vR
Two possibility:
1. t1 ≡ x and u ≡ y
2. t1 ≡ y
2 is impossible
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Consistency proof of a feasible arithmetic inside a bounded arithmetic
Consistency of PV-
Substitution lemma
Proof of Substitution I
.... σ1
〈s, ρ1〉 ↓ v
〈x [y/x ], ρ′[s/y ]ρ1〉 ↓ vR
M(σ1) ≤ M(σ), we can apply IH to σ1
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Consistency proof of a feasible arithmetic inside a bounded arithmetic
Consistency of PV-
Substitution lemma
Proof of Substitution I
Let τ be .... τ1
〈s, ρ1〉 ↓ v
〈y , [s/y ]ρ1〉 ↓ v
〈x , [y/x ][s/y ]ρ1〉 ↓ v
|||τ ||| ≤ |||τ1|||+ 2
≤ |||σ1|||+ ||t2[u/x ]||+ · · ·+ ||tm[u/x ]||+ 2
≤ |||σ1|||+ ||t1[u/x ]||+ · · ·+ ||tm[u/x ]||+ 1
≤ |||σ|||+ ||t1[u/x ]||+ · · ·+ ||tm[u/x ]||
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Consistency proof of a feasible arithmetic inside a bounded arithmetic
Consistency of PV-
Substitution lemma
Proof of Substitution I
The case that R is not a substitutionWe consider two cases
1. t1[u/x ] ≡ f (u1[u/x ], . . . , ul [u/x ])
2. t1[u/x ] ≡ u
We only consider the first caseAssume u1[u/x ], . . . , ul [u/x ] is partitioned intou1
1 [u/x ], . . . , u1l1[u/x ] which are not numeral, and numerals
u21 [u/x ], . . . , u2
l2[u/x ]
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Consistency proof of a feasible arithmetic inside a bounded arithmetic
Consistency of PV-
Substitution lemma
Proof of Substitution I
β, 〈u11 [u/x ], ρ〉 ↓ v 1
1 , . . . , 〈u1l1[u/x ], ρ〉 ↓ v 1
l1
〈t1[u/x ], ρ〉 ↓ vR
|||σ0||| ≤ |||σ||| − 1 + l1
≤ U − ||t1[u/x ]|| − · · · − ||tm[u/x ]|| − 1 + l1
≤ U − ||u11 [u/x ]|| − · · · − ||u1
l1[u/x ]|| − · · ·− ||t2[u/x ]|| − · · · − ||tm[u/x ]||
σ0 : The computation which is obtained by removing the lastelement of σ and make all 〈uj [u/x ], ρ〉 ↓ v 1
j conclusions
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Consistency proof of a feasible arithmetic inside a bounded arithmetic
Consistency of PV-
Substitution lemma
Proof of Substitution I
τ0 ` 〈u11 , [u/x ]ρ〉 ↓ v 1
1 , . . . , 〈u1l1 , [u/x ]ρ〉 ↓ v 1
l1 , . . . (45)
|||τ0||| ≤ |||σ0|||+ ||u11 [u/x ]||+ · · ·+ ||u1
l1[u/x ]||+||t2[u/x ]||+ · · ·+ ||tm[u/x ]|| (46)
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Consistency proof of a feasible arithmetic inside a bounded arithmetic
Consistency of PV-
Substitution lemma
Proof of Substitution I
We add the proof of 〈u2k2 , [u/x ]ρ〉 ↓ v 1
k2 for all k2 = 1, . . . , l2
whose numbers of nodes are bounded by ||u2k2[u/x ]||+ 1
τ1 ` 〈u1, [u/x ]ρ〉 ↓ v1, . . . , 〈ul , [u/x ]ρ〉 ↓ vl , . . . (47)
|||τ1||| ≤ |||σ0|||+ ||u1[u/x ]||+ · · ·+ ||ul [u/x ]||+ l2 + ||t2[u/x ]||+ · · ·+ ||tm[u/x ]|| (48)
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Consistency proof of a feasible arithmetic inside a bounded arithmetic
Consistency of PV-
Substitution lemma
Proof of Substitution I
τ ` 〈t1, [u/x ]ρ〉 ↓ v1, . . . , 〈tm, [u/x ]ρ〉 ↓ vm, α (49)
|||τ ||| ≤ |||τ1|||+ 1
≤ |||σ0|||+ ||u1[u/x ]||+ · · ·+ ||uk [u/x ]||+l2 + ||t2[u/x ]||+ · · ·+ ||tm[u/x ]||+ 1
≤ |||σ|||+ l1 + ||u1[u/x ]||+ · · ·+ ||ul [u/x ]||+l2 + ||t2[u/x ]||+ · · ·+ ||tm[u/x ]||+ 1
≤ |||σ|||+ ||t1[u/x ]||+ · · ·+ ||tm[u/x ]||
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Consistency proof of a feasible arithmetic inside a bounded arithmetic
Consistency of PV-
Substitution lemma
Proof of Substitution II
Similar to the proof of Substitution I
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Consistency proof of a feasible arithmetic inside a bounded arithmetic
Consistency of PV-
Substitution lemma
Proof of soundness theorem - substitution.... r1
t = ut[t0/x ] = u[t0/x ]
〈t[t0/x ], ρ〉 ↓B v (Assumption)
〈t, [t0/x ]ρ〉 ↓B+||t[t0/x]|| v (Substitution I)
Since ||[t0/x ]ρ|| ≤ U − ||r ||+ ||[t0/x ]|| ≤ U − ||r1||
〈u, [t0/x ]ρ〉 ↓B+||t[t0/x]||+||r1|| v (IH)
〈u[t0/x ], ρ〉 ↓B+||t[t0/x]||+||r1||+||u[t0/x]|| v (Substitution II)
Since B + ||t[t0/x ]||+ ||r1||+ ||u[t0/x ]|| ≤ B + ||r ||
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Consistency proof of a feasible arithmetic inside a bounded arithmetic
Consistency of PV-
Equality axioms
Equality axioms - transitivity.... r1
t = u
.... r2u = w
t = w
By induction hypothesis,
σ `B 〈t, ρ〉 ↓ v ⇒ ∃τ, τ `B+||r1|| 〈u, ρ〉 ↓ v (50)
Since
||ρ|| ≤ U − ||r || ≤ U − ||r2|||||τ ||| ≤ U − ||r ||+ ||r1||
≤ U − ||r2||
By induction hypothesis, ∃δ, δ `B+||r1||+||r2|| 〈w , ρ〉 ↓ v
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Consistency proof of a feasible arithmetic inside a bounded arithmetic
Consistency of PV-
Equality axioms
Equality axioms - substitution
.... r1
t = uf (t) = f (u)
σ : computation of 〈f (t), ρ〉 ↓ v .We only consider that the case t is not a numeral
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Consistency proof of a feasible arithmetic inside a bounded arithmetic
Consistency of PV-
Equality axioms
Proof of soundness of equality axiomsσ has a form
〈f1(v0, v), ρ〉 ↓ w1 , . . . , 〈t, ρ〉 ↓ v0, 〈tk1 , ρ〉 ↓ v1, . . .
〈f (t, t2, . . . , tk), ρ〉 ↓ v
∃σ1, s.t. σ1 ` 〈t, ρ〉 ↓ v0, 〈f (t, t2, . . . , tk), ρ〉 ↓ v , α
|||σ1||| ≤ |||σ|||+ 1 (51)
≤ U − ||r ||+ 1 (52)
≤ U − ||r1|| (53)
and
||f (t, t2, . . . , tk)||+ Σα∈αM(α) ≤ U − ||r ||+ ||f (t, t2, . . . , tk)||≤ U − ||r1|| (54)
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Consistency proof of a feasible arithmetic inside a bounded arithmetic
Consistency of PV-
Equality axioms
Proof of soundness of equality axioms
By IH, ∃τ1 s.t. τ1 ` 〈u, ρ〉 ↓ v0, 〈f (t, t2, . . . , tk), ρ〉 ↓ v , α and|||τ1||| ≤ |||σ1|||+ ||r1||
〈f1(v0, v), ρ〉 ↓ w1 , . . . , 〈t, ρ〉 ↓ v0, 〈tk1 , ρ〉 ↓ v1, . . .
〈f (t, t2, . . . , tk), ρ〉 ↓ v
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Consistency proof of a feasible arithmetic inside a bounded arithmetic
Consistency of PV-
Equality axioms
Proof of soundness of equality axioms
〈f1(v0, v), ρ〉 ↓ w1 , . . . , 〈u, ρ〉 ↓ v0, 〈tk1 , ρ〉 ↓ v1, . . .
〈f (u, t2, . . . , tk), ρ〉 ↓ v
Let this derivation be τ
|||τ ||| ≤ |||σ1|||+ ||r1||+ 1
≤ |||σ|||+ ||r1||+ 2
≤ |||σ|||+ ||r ||