Nuprl Lemma : pi-comp_wf1
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[B:{Gamma.A ⊢ _}]. ∀[cA:Gamma ⊢ CompOp(A)]. ∀[cB:Gamma.A ⊢ CompOp(B)].
(pi-comp(Gamma;A;B;cA;cB) ∈ I:fset(ℕ)
⟶ i:{i:ℕ| ¬i ∈ I}
⟶ rho:Gamma(I+i)
⟶ phi:𝔽(I)
⟶ mu:{I+i,s(phi) ⊢ _:(ΠA B)<rho> o iota}
⟶ lambda:cubical-path-0(Gamma;ΠA B;I;i;rho;phi;mu)
⟶ J:fset(ℕ)
⟶ f:J ⟶ I
⟶ u1:A(f((i1)(rho)))
⟶ let j = new-name(J) in
let nu = pi-comp-nu(Gamma;A;cA;I;i;rho;J;f;u1;j) in
cubical-path-1(Gamma.A;B;J;j;(f,i=j(rho);nu);f(phi);pi-comp-app(Gamma;A;I;i;rho;phi;mu;J;f;j;nu)))
Proof
Definitions occuring in Statement :
pi-comp: pi-comp(Gamma;A;B;cA;cB),
pi-comp-app: pi-comp-app(Gamma;A;I;i;rho;phi;mu;J;f;j;nu),
pi-comp-nu: pi-comp-nu(Gamma;A;cA;I;i;rho;J;f;u1;j),
composition-op: Gamma ⊢ CompOp(A),
cubical-path-1: cubical-path-1(Gamma;A;I;i;rho;phi;u),
cubical-path-0: cubical-path-0(Gamma;A;I;i;rho;phi;u),
cubical-pi: ΠA B,
cc-adjoin-cube: (v;u),
cube-context-adjoin: X.A,
cubical-term: {X ⊢ _:A},
csm-ap-type: (AF)s,
cubical-type-at: A(a),
cubical-type: {X ⊢ _},
subset-iota: iota,
cubical-subset: I,psi,
face-presheaf: 𝔽,
csm-comp: G o F,
context-map: <rho>,
formal-cube: formal-cube(I),
cube-set-restriction: f(s),
I_cube: A(I),
cubical_set: CubicalSet,
nc-e': g,i=j,
nc-1: (i1),
nc-s: s,
new-name: new-name(I),
add-name: I+i,
names-hom: I ⟶ J,
fset-member: a ∈ s,
fset: fset(T),
int-deq: IntDeq,
nat: ℕ,
let: let,
uall: ∀[x:A]. B[x],
not: ¬A,
member: t ∈ T,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
pi-comp: pi-comp(Gamma;A;B;cA;cB),
all: ∀x:A. B[x],
implies: P ⇒ Q,
has-value: (a)↓,
subtype_rel: A ⊆r B,
prop: ℙ,
uimplies: b supposing a,
so_lambda: λ2x.t[x],
so_apply: x[s],
nat: ℕ,
let: let,
composition-op: Gamma ⊢ CompOp(A),
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
and: P ∧ Q
Lemmas referenced :
new-name_wf,
value-type-has-value,
nat_wf,
not_wf,
fset-member_wf,
int-deq_wf,
set-value-type,
istype-nat,
le_wf,
istype-int,
int-value-type,
cc-adjoin-cube_wf,
add-name_wf,
cube-set-restriction_wf,
nc-e'_wf,
pi-comp-nu_wf,
subtype_rel-equal,
cubical-type-at_wf,
nc-r_wf,
trivial-member-add-name1,
nc-r'_wf,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
istype-le,
face-presheaf_wf2,
pi-comp-app_wf,
pi-comp-lambda_wf,
cubical_set_cumulativity-i-j,
cubical-type-cumulativity2,
cube-context-adjoin_wf,
nc-1_wf,
names-hom_wf,
cubical-path-0_wf,
cubical-pi_wf,
cubical-term_wf,
cubical-subset_wf,
nc-s_wf,
f-subset-add-name,
csm-ap-type_wf,
cubical-type-cumulativity,
csm-comp_wf,
formal-cube_wf1,
subset-iota_wf,
context-map_wf,
I_cube_wf,
strong-subtype-deq-subtype,
strong-subtype-set3,
strong-subtype-self,
fset_wf,
composition-op_wf,
cubical-type_wf,
cubical_set_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
functionExtensionality,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
inhabitedIsType,
lambdaFormation_alt,
rename,
callbyvalueReduce,
setEquality,
applyEquality,
because_Cache,
independent_isectElimination,
lambdaEquality_alt,
intEquality,
natural_numberEquality,
setElimination,
dependent_functionElimination,
dependent_set_memberEquality_alt,
unionElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
int_eqEquality,
Error :memTop,
independent_pairFormation,
universeIsType,
voidElimination,
instantiate,
equalityIstype,
equalityTransitivity,
equalitySymmetry,
axiomEquality,
isect_memberEquality_alt,
isectIsTypeImplies
Latex:
\mforall{}[Gamma:j\mvdash{}]. \mforall{}[A:\{Gamma \mvdash{} \_\}]. \mforall{}[B:\{Gamma.A \mvdash{} \_\}]. \mforall{}[cA:Gamma \mvdash{} CompOp(A)].
\mforall{}[cB:Gamma.A \mvdash{} CompOp(B)].
(pi-comp(Gamma;A;B;cA;cB) \mmember{} I:fset(\mBbbN{})
{}\mrightarrow{} i:\{i:\mBbbN{}| \mneg{}i \mmember{} I\}
{}\mrightarrow{} rho:Gamma(I+i)
{}\mrightarrow{} phi:\mBbbF{}(I)
{}\mrightarrow{} mu:\{I+i,s(phi) \mvdash{} \_:(\mPi{}A B)<rho> o iota\}
{}\mrightarrow{} lambda:cubical-path-0(Gamma;\mPi{}A B;I;i;rho;phi;mu)
{}\mrightarrow{} J:fset(\mBbbN{})
{}\mrightarrow{} f:J {}\mrightarrow{} I
{}\mrightarrow{} u1:A(f((i1)(rho)))
{}\mrightarrow{} let j = new-name(J) in
let nu = pi-comp-nu(Gamma;A;cA;I;i;rho;J;f;u1;j) in
cubical-path-1(Gamma.A;B;J;j;(f,i=j(rho);nu);f(phi);
pi-comp-app(Gamma;A;I;i;rho;phi;mu;J;f;j;nu)))
Date html generated:
2020_05_20-PM-03_59_40
Last ObjectModification:
2020_04_09-PM-08_18_05
Theory : cubical!type!theory
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