Realizability and Strong Normalization for Heyting ... fileFederico Aschieri (joint work S. Berardi,...

Realizability and Strong Normalization forHeyting Arithmetic with EM1

Federico Aschieri(joint work S. Berardi, G. Birolo)

Equipe Plume, LIPENS de Lyon

Toulouse, 24 Avril 2013

Federico Aschieri (joint work S. Berardi, G. Birolo) Realizability and Strong Normalization for Heyting Arithmetic with EM1

Proof Terms

Variables: for terms x , y , z, . . . , for individuals α, β, . . .

Functions (→):λx u | tu

Pairs (∧):〈u, v〉 | π0u | π1u

Sums (∨):ι0(u) | ι1(u) | u[x .v1, y .v2]

Products (∀):λα u | tn (n individual)

Co-Products (∃):

(n, t) | u[(α, x).v ] (n individual)

Proof Terms

Variables: for terms x , y , z, . . . , for individuals α, β, . . .Functions (→):

λx u | tu

Pairs (∧):〈u, v〉 | π0u | π1u

Sums (∨):ι0(u) | ι1(u) | u[x .v1, y .v2]

Co-Products (∃):

Proof Terms

λx u | tu

Pairs (∧):〈u, v〉 | π0u | π1u

Sums (∨):ι0(u) | ι1(u) | u[x .v1, y .v2]

Co-Products (∃):

Proof Terms

λx u | tu

Pairs (∧):〈u, v〉 | π0u | π1u

Sums (∨):ι0(u) | ι1(u) | u[x .v1, y .v2]

Co-Products (∃):

Proof Terms

λx u | tu

Pairs (∧):〈u, v〉 | π0u | π1u

Sums (∨):ι0(u) | ι1(u) | u[x .v1, y .v2]

Co-Products (∃):

Proof Terms

λx u | tu

Pairs (∧):〈u, v〉 | π0u | π1u

Sums (∨):ι0(u) | ι1(u) | u[x .v1, y .v2]

Co-Products (∃):

Proof Terms(2)

Numerals:0,S0,SS0, . . .

Recursion (Induction):

rec u v t

Proof Terms(2)

Numerals:0,S0,SS0, . . .

Recursion (Induction):

rec u v t

Proof Terms (3)

(EM1):u ‖a v

Permutations:(u ‖a v)w 7→ uw ‖a vw

πi(u ‖a v) 7→ πiu ‖a πiv

(u ‖a v)[x .w1, y .w2] 7→ u[x .w1, y .w2] ‖a v [x .w1, y .w2]

(u ‖a v)[(α, x).w ] 7→ u[(α, x).w ] ‖a v [(α, x).w ]

Proof Terms (3)

(EM1):u ‖a v

Proof Terms (3)

(EM1):u ‖a v

Proof Terms (3)

(EM1):u ‖a v

Proof Terms (3)

(EM1):u ‖a v

Proof Terms (4)

(EM1):u ‖a v

H∀αPa

W∃α¬Pa

Exceptions:

(H∀αPa )n 7→ True, if P[n/α] = True

u ‖a v 7→ u, (if a does not occur free in u)

u ‖a v 7→ v [a := n] := v [W∃α¬Pa := (n,True)]

(if H∀αPa n occurs in u and P[n/α] = False)

Proof Terms (4)

(EM1):u ‖a v

H∀αPa

W∃α¬Pa

Exceptions:

Proof Terms (4)

(EM1):u ‖a v

H∀αPa

W∃α¬Pa

Exceptions:

Proof Terms (4)

(EM1):u ‖a v

H∀αPa

W∃α¬Pa

Exceptions:

u ‖a v 7→ u,

(if a does not occur free in u)

Proof Terms (4)

(EM1):u ‖a v

H∀αPa

W∃α¬Pa

Exceptions:

Proof Terms (4)

(EM1):u ‖a v

H∀αPa

W∃α¬Pa

Exceptions:

u ‖a v 7→ v [a := n]

:= v [W∃α¬Pa := (n,True)]

Proof Terms (4)

(EM1):u ‖a v

H∀αPa

W∃α¬Pa

Exceptions:

Proof Terms (4)

(EM1):u ‖a v

H∀αPa

W∃α¬Pa

Exceptions:

Typing Rules

Γ,a : ∀αNP ` u : C Γ,a : ∃αN¬P ` v : CΓ ` u ‖a v : C

P atomic

Γ,a : ∀αNP ` H∀αPa : ∀αNP

Γ,a : ∃αN¬P ` W∃α¬Pa : ∃αN¬P

Typing Rules

P atomic

Typing Rules

P atomic

Logical Interpretation

∀αPP[n/α]

∀αPP[m/α]

∀αP...

∃α¬P...C

Logical Interpretation (2)

∀αPP[n/α]

∀αPP[m/α]

∃α¬P...C

Converts to:

P[n/α]

P[m/α]

Logical Interpretation (3)

∀αPP[n/α] = False

∀αPP[m/α]

∃α¬P...C

Converts to:

Π¬P[n/α]

∃α¬P...C

Realizability

t may contain free hypotheses H∀αPa

Realizability

t may contain free hypotheses H∀αPa

Realizability (2)

t A → B if and only if for all u, if u A, then tu B

t A ∧ B if and only if π0t A and π1t B

t ∀αNA if and only if for every numeral n, tn A[n/α]

Realizability (2)

Realizability (3)

t A ∨ B if and only if one of the following holds:

i) t = ι0(u) and u A or t = ι1(u) and u B;

ii) t = u ‖a v and u A ∨ B and v [a := m] A ∨ B forevery numeral m;(v [a := m] = v [W∃α¬P

a := (m,H∀α.α=0a S0)] if m is not a

witness for P)

iii) t /∈ NF is neutral and for all t ′, t 7→ t ′ implies t ′ A ∨ B.

t ∃αNA if and only if one of the following holds:

i) t = (n,u) for some numeral n and u A[n/α];

ii) t = u ‖a v and u ∃αNA and v [a := m] ∃αNA forevery numeral m

iii) t /∈ NF is neutral and for all t ′, t 7→ t ′ implies t ′ ∃αNA.

Realizability (3)

witness for P)

Realizability (3)

witness for P)

Realizability (3)

witness for P)

Realizability (3)

witness for P)

Realizability (4)

t P if and only if one of the following holds:

i) t ∈ NF and P = False implies t contains a subterm H∀αQa n

with Q[n/α] = False;

ii) t = u ‖a v and u P and v [a := m] P for every numeralm;

iii) t /∈ NF is neutral and for all t ′, t 7→ t ′ implies t ′ P

Realizability (4)

Soundness, Strong Normalization, Witness Extraction

HA + EM1 ` t : A =⇒ t A

t A =⇒ t ∈ SN

t ∃αP =⇒ t 7→∗ (n,u) ∧ P[n/α]

HA + EM1 ` t : A =⇒ t A

t A =⇒ t ∈ SN

t ∃αP =⇒ t 7→∗ (n,u) ∧ P[n/α]

HA + EM1 ` t : A =⇒ t A

t A =⇒ t ∈ SN

t ∃αP =⇒ t 7→∗ (n,u) ∧ P[n/α]

Disjunction Property and Existence Property

EM−1 ( Q atomic) :

Γ,a : ∀αNP ` u : ∃βQ Γ,a : ∃αN¬P ` v : ∃βQΓ ` u ‖a v : ∃βQ

HA + EM−1 ` A ∨ B =⇒ HA + EM−1 ` A or HA + EM−1 ` B

HA + EM−1 ` ∃αA =⇒ HA + EM−1 ` A(n)

Realizability and Strong Normalization for Heyting ... fileFederico Aschieri (joint work S. Berardi,...

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