Nuprl Lemma : composition-type-lemma1
∀[Gamma:j⊢]. ∀[A:{Gamma.𝕀 ⊢ _}]. ∀[I:fset(ℕ)]. ∀[rho:Gamma(I)].
((A)[0(𝕀)](rho) = A((new-name(I)0)((s(rho);<new-name(I)>))) ∈ Type)
Proof
Definitions occuring in Statement :
interval-0: 0(𝕀),
interval-type: 𝕀,
csm-id-adjoin: [u],
cc-adjoin-cube: (v;u),
cube-context-adjoin: X.A,
csm-ap-type: (AF)s,
cubical-type-at: A(a),
cubical-type: {X ⊢ _},
cube-set-restriction: f(s),
I_cube: A(I),
cubical_set: CubicalSet,
nc-0: (i0),
nc-s: s,
new-name: new-name(I),
add-name: I+i,
dM_inc: <x>,
fset: fset(T),
nat: ℕ,
uall: ∀[x:A]. B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
squash: ↓T,
prop: ℙ,
interval-0: 0(𝕀),
csm-id-adjoin: [u],
csm-ap: (s)x,
csm-id: 1(X),
csm-adjoin: (s;u),
cc-adjoin-cube: (v;u),
all: ∀x:A. B[x],
uimplies: b supposing a,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
lattice-point: Point(l),
record-select: r.x,
dM: dM(I),
free-DeMorgan-algebra: free-DeMorgan-algebra(T;eq),
mk-DeMorgan-algebra: mk-DeMorgan-algebra(L;n),
record-update: r[x := v],
ifthenelse: if b then t else f fi ,
eq_atom: x =a y,
bfalse: ff,
free-DeMorgan-lattice: free-DeMorgan-lattice(T;eq),
free-dist-lattice: free-dist-lattice(T; eq),
mk-bounded-distributive-lattice: mk-bounded-distributive-lattice,
mk-bounded-lattice: mk-bounded-lattice(T;m;j;z;o),
btrue: tt,
cubical-type-at: A(a),
pi1: fst(t),
interval-type: 𝕀,
constant-cubical-type: (X),
I_cube: A(I),
functor-ob: ob(F),
interval-presheaf: 𝕀,
names: names(I),
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
nc-0: (i0),
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A,
nequal: a ≠ b ∈ T ,
satisfiable_int_formula: satisfiable_int_formula(fmla)
Lemmas referenced :
I_cube_wf,
fset_wf,
nat_wf,
cubical-type_wf,
cube-context-adjoin_wf,
cubical_set_cumulativity-i-j,
interval-type_wf,
cubical_set_wf,
csm-ap-type-at,
cubical-type-at_wf,
squash_wf,
true_wf,
cc-adjoin-cube-restriction,
cc-adjoin-cube_wf,
istype-cubical-type-at,
equal_wf,
istype-universe,
cube-set-restriction-comp,
add-name_wf,
new-name_wf,
nc-s_wf,
f-subset-add-name,
nc-0_wf,
subtype_rel_self,
iff_weakening_equal,
cube-set-restriction-when-id,
nh-comp_wf,
s-comp-nc-0-new,
interval-type-ap-morph,
dM0_wf,
dM-lift-inc,
trivial-member-add-name1,
fset-member_wf,
int-deq_wf,
strong-subtype-deq-subtype,
strong-subtype-set3,
le_wf,
istype-int,
strong-subtype-self,
dM0-sq-empty,
eq_int_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
eq_int_eq_true,
btrue_wf,
not_assert_elim,
btrue_neq_bfalse,
full-omega-unsat,
intformnot_wf,
intformeq_wf,
itermVar_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_formula_prop_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
hypothesis,
universeIsType,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
inhabitedIsType,
instantiate,
applyEquality,
Error :memTop,
lambdaEquality_alt,
imageElimination,
equalityTransitivity,
equalitySymmetry,
because_Cache,
dependent_functionElimination,
universeEquality,
setElimination,
rename,
independent_isectElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
productElimination,
independent_functionElimination,
dependent_set_memberEquality_alt,
intEquality,
lambdaFormation_alt,
unionElimination,
equalityElimination,
dependent_pairFormation_alt,
equalityIstype,
promote_hyp,
cumulativity,
voidElimination,
approximateComputation,
int_eqEquality
Latex:
\mforall{}[Gamma:j\mvdash{}]. \mforall{}[A:\{Gamma.\mBbbI{} \mvdash{} \_\}]. \mforall{}[I:fset(\mBbbN{})]. \mforall{}[rho:Gamma(I)].
((A)[0(\mBbbI{})](rho) = A((new-name(I)0)((s(rho);<new-name(I)>))))
Date html generated:
2020_05_20-PM-04_06_59
Last ObjectModification:
2020_04_10-AM-03_43_47
Theory : cubical!type!theory
Home
Index