Nuprl Lemma : fill_from_comp_wf1
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[comp:Gamma ⊢ CompOp(A)].
(fill_from_comp(Gamma;A;comp) ∈ I:fset(ℕ)
⟶ i:{i:ℕ| ¬i ∈ I}
⟶ rho:Gamma(I+i)
⟶ phi:𝔽(I)
⟶ u:{I+i,s(phi) ⊢ _:(A)<rho> o iota}
⟶ a0:cubical-path-0(Gamma;A;I;i;rho;phi;u)
⟶ let j = new-name(I+i) in
cubical-path-1(Gamma;A;I+i;j;m(i;j)(rho);fl-join(I+i;s(phi);(i=0));fillterm(Gamma;A;I;i;j;rho;a0;u)))
Proof
Definitions occuring in Statement :
fill_from_comp: fill_from_comp(Gamma;A;comp),
fillterm: fillterm(Gamma;A;I;i;j;rho;a0;u),
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-term: {X ⊢ _:A},
csm-ap-type: (AF)s,
cubical-type: {X ⊢ _},
subset-iota: iota,
cubical-subset: I,psi,
fl-join: fl-join(I;x;y),
face-presheaf: 𝔽,
fl0: (x=0),
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-m: m(i;j),
nc-s: s,
new-name: new-name(I),
add-name: I+i,
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,
fill_from_comp: fill_from_comp(Gamma;A;comp),
nat: ℕ,
ge: i ≥ j ,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
and: P ∧ Q,
prop: ℙ,
has-value: (a)↓,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
composition-op: Gamma ⊢ CompOp(A),
face-presheaf: 𝔽,
I_cube: A(I),
names: names(I),
let: let
Lemmas referenced :
new-name_wf,
add-name_wf,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
istype-int,
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,
value-type-has-value,
nat_wf,
not_wf,
fset-member_wf,
int-deq_wf,
set-value-type,
istype-nat,
le_wf,
int-value-type,
ob_pair_lemma,
fl0_wf,
trivial-member-add-name1,
cube-set-restriction_wf,
nc-m_wf,
fl-join_wf,
face-presheaf_wf2,
nc-s_wf,
f-subset-add-name,
fillterm_wf,
cubical-path-0-fillterm,
cubical-path-0_wf,
cubical-type-cumulativity2,
cubical_set_cumulativity-i-j,
cubical-term_wf,
cubical-subset_wf,
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,
istype-void,
fset_wf,
composition-op_wf,
cubical-type_wf,
cubical_set_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
lambdaEquality_alt,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
dependent_set_memberEquality_alt,
setElimination,
rename,
hypothesis,
dependent_functionElimination,
natural_numberEquality,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
int_eqEquality,
Error :memTop,
sqequalRule,
independent_pairFormation,
universeIsType,
voidElimination,
inhabitedIsType,
lambdaFormation_alt,
callbyvalueReduce,
setEquality,
applyEquality,
because_Cache,
intEquality,
instantiate,
equalityIstype,
equalityTransitivity,
equalitySymmetry,
setIsType,
functionIsType,
axiomEquality,
isect_memberEquality_alt,
isectIsTypeImplies
Latex:
\mforall{}[Gamma:j\mvdash{}]. \mforall{}[A:\{Gamma \mvdash{} \_\}]. \mforall{}[comp:Gamma \mvdash{} CompOp(A)].
(fill\_from\_comp(Gamma;A;comp) \mmember{} I:fset(\mBbbN{})
{}\mrightarrow{} i:\{i:\mBbbN{}| \mneg{}i \mmember{} I\}
{}\mrightarrow{} rho:Gamma(I+i)
{}\mrightarrow{} phi:\mBbbF{}(I)
{}\mrightarrow{} u:\{I+i,s(phi) \mvdash{} \_:(A)<rho> o iota\}
{}\mrightarrow{} a0:cubical-path-0(Gamma;A;I;i;rho;phi;u)
{}\mrightarrow{} let j = new-name(I+i) in
cubical-path-1(Gamma;A;I+i;j;m(i;j)(rho);fl-join(I+i;s(phi);(i=0));
fillterm(Gamma;A;I;i;j;rho;a0;u)))
Date html generated:
2020_05_20-PM-03_54_22
Last ObjectModification:
2020_04_09-PM-03_32_46
Theory : cubical!type!theory
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