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)
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)
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)
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)
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)
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)
Federico Aschieri (joint work S. Berardi, G. Birolo) Realizability and Strong Normalization for Heyting Arithmetic with EM1
Proof Terms(2)
Numerals:0,S0,SS0, . . .
Recursion (Induction):
rec u v t
Federico Aschieri (joint work S. Berardi, G. Birolo) Realizability and Strong Normalization for Heyting Arithmetic with EM1
Proof Terms(2)
Numerals:0,S0,SS0, . . .
Recursion (Induction):
rec u v t
Federico Aschieri (joint work S. Berardi, G. Birolo) Realizability and Strong Normalization for Heyting Arithmetic with EM1
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 ]
Federico Aschieri (joint work S. Berardi, G. Birolo) Realizability and Strong Normalization for Heyting Arithmetic with EM1
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 ]
Federico Aschieri (joint work S. Berardi, G. Birolo) Realizability and Strong Normalization for Heyting Arithmetic with EM1
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 ]
Federico Aschieri (joint work S. Berardi, G. Birolo) Realizability and Strong Normalization for Heyting Arithmetic with EM1
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 ]
Federico Aschieri (joint work S. Berardi, G. Birolo) Realizability and Strong Normalization for Heyting Arithmetic with EM1
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 ]
Federico Aschieri (joint work S. Berardi, G. Birolo) Realizability and Strong Normalization for Heyting Arithmetic with EM1
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)
Federico Aschieri (joint work S. Berardi, G. Birolo) Realizability and Strong Normalization for Heyting Arithmetic with EM1
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)
Federico Aschieri (joint work S. Berardi, G. Birolo) Realizability and Strong Normalization for Heyting Arithmetic with EM1
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)
Federico Aschieri (joint work S. Berardi, G. Birolo) Realizability and Strong Normalization for Heyting Arithmetic with EM1
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)
Federico Aschieri (joint work S. Berardi, G. Birolo) Realizability and Strong Normalization for Heyting Arithmetic with EM1
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)
Federico Aschieri (joint work S. Berardi, G. Birolo) Realizability and Strong Normalization for Heyting Arithmetic with EM1
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)
Federico Aschieri (joint work S. Berardi, G. Birolo) Realizability and Strong Normalization for Heyting Arithmetic with EM1
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)
Federico Aschieri (joint work S. Berardi, G. Birolo) Realizability and Strong Normalization for Heyting Arithmetic with EM1
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)
Federico Aschieri (joint work S. Berardi, G. Birolo) Realizability and Strong Normalization for Heyting Arithmetic with EM1
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
Federico Aschieri (joint work S. Berardi, G. Birolo) Realizability and Strong Normalization for Heyting Arithmetic with EM1
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
Federico Aschieri (joint work S. Berardi, G. Birolo) Realizability and Strong Normalization for Heyting Arithmetic with EM1
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
Federico Aschieri (joint work S. Berardi, G. Birolo) Realizability and Strong Normalization for Heyting Arithmetic with EM1
Logical Interpretation
∀αPP[n/α]
...
∀αPP[m/α]
...
∀αP...
C
∃α¬P...C
C
Federico Aschieri (joint work S. Berardi, G. Birolo) Realizability and Strong Normalization for Heyting Arithmetic with EM1
Logical Interpretation (2)
∀αPP[n/α]
...
∀αPP[m/α]
...C
∃α¬P...C
C
Converts to:
Π0
P[n/α]
...
Π1
P[m/α]
...C
Federico Aschieri (joint work S. Berardi, G. Birolo) Realizability and Strong Normalization for Heyting Arithmetic with EM1
Logical Interpretation (3)
∀αPP[n/α] = False
...
∀αPP[m/α]
...C
∃α¬P...C
C
Converts to:
Π¬P[n/α]
∃α¬P...C
Federico Aschieri (joint work S. Berardi, G. Birolo) Realizability and Strong Normalization for Heyting Arithmetic with EM1
Realizability
t A
t may contain free hypotheses H∀αPa
Federico Aschieri (joint work S. Berardi, G. Birolo) Realizability and Strong Normalization for Heyting Arithmetic with EM1
Realizability
t A
t may contain free hypotheses H∀αPa
Federico Aschieri (joint work S. Berardi, G. Birolo) Realizability and Strong Normalization for Heyting Arithmetic with EM1
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]
Federico Aschieri (joint work S. Berardi, G. Birolo) Realizability and Strong Normalization for Heyting Arithmetic with EM1
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]
Federico Aschieri (joint work S. Berardi, G. Birolo) Realizability and Strong Normalization for Heyting Arithmetic with EM1
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]
Federico Aschieri (joint work S. Berardi, G. Birolo) Realizability and Strong Normalization for Heyting Arithmetic with EM1
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.
Federico Aschieri (joint work S. Berardi, G. Birolo) Realizability and Strong Normalization for Heyting Arithmetic with EM1
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.
Federico Aschieri (joint work S. Berardi, G. Birolo) Realizability and Strong Normalization for Heyting Arithmetic with EM1
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.
Federico Aschieri (joint work S. Berardi, G. Birolo) Realizability and Strong Normalization for Heyting Arithmetic with EM1
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.
Federico Aschieri (joint work S. Berardi, G. Birolo) Realizability and Strong Normalization for Heyting Arithmetic with EM1
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.
Federico Aschieri (joint work S. Berardi, G. Birolo) Realizability and Strong Normalization for Heyting Arithmetic with EM1
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
Federico Aschieri (joint work S. Berardi, G. Birolo) Realizability and Strong Normalization for Heyting Arithmetic with EM1
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
Federico Aschieri (joint work S. Berardi, G. Birolo) Realizability and Strong Normalization for Heyting Arithmetic with EM1
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
Federico Aschieri (joint work S. Berardi, G. Birolo) Realizability and Strong Normalization for Heyting Arithmetic with EM1
Soundness, Strong Normalization, Witness Extraction
HA + EM1 ` t : A =⇒ t A
t A =⇒ t ∈ SN
t ∃αP =⇒ t 7→∗ (n,u) ∧ P[n/α]
Federico Aschieri (joint work S. Berardi, G. Birolo) Realizability and Strong Normalization for Heyting Arithmetic with EM1
Soundness, Strong Normalization, Witness Extraction
HA + EM1 ` t : A =⇒ t A
t A =⇒ t ∈ SN
t ∃αP =⇒ t 7→∗ (n,u) ∧ P[n/α]
Federico Aschieri (joint work S. Berardi, G. Birolo) Realizability and Strong Normalization for Heyting Arithmetic with EM1
Soundness, Strong Normalization, Witness Extraction
HA + EM1 ` t : A =⇒ t A
t A =⇒ t ∈ SN
t ∃αP =⇒ t 7→∗ (n,u) ∧ P[n/α]
Federico Aschieri (joint work S. Berardi, G. Birolo) Realizability and Strong Normalization for Heyting Arithmetic with EM1
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)
Federico Aschieri (joint work S. Berardi, G. Birolo) Realizability and Strong Normalization for Heyting Arithmetic with EM1
Top Related