Nuprl Lemma : csm-cubical-subset-type-lemma
∀[Gamma,Delta:j⊢]. ∀[sigma:Delta j⟶ Gamma]. ∀[A:{Gamma ⊢ _}]. ∀[I:fset(ℕ)]. ∀[i:{i:ℕ| ¬i ∈ I} ]. ∀[rho:Delta(I+i)].
∀[J:fset(ℕ)]. ∀[f:J ⟶ I].
(((A)sigma(f((i1)(rho))) = A(f((i1)((sigma)rho))) ∈ Type) ∧ ((A)sigma(f((i0)(rho))) = A(f((i0)((sigma)rho))) ∈ Type))
Proof
Definitions occuring in Statement :
csm-ap-type: (AF)s,
cubical-type-at: A(a),
cubical-type: {X ⊢ _},
csm-ap: (s)x,
cube_set_map: A ⟶ B,
cube-set-restriction: f(s),
I_cube: A(I),
cubical_set: CubicalSet,
nc-1: (i1),
nc-0: (i0),
add-name: I+i,
names-hom: I ⟶ J,
fset-member: a ∈ s,
fset: fset(T),
int-deq: IntDeq,
nat: ℕ,
uall: ∀[x:A]. B[x],
not: ¬A,
and: P ∧ Q,
set: {x:A| B[x]} ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
and: P ∧ Q,
cand: A c∧ B,
squash: ↓T,
prop: ℙ,
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,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
true: True,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
csm-ap-type-at,
cubical-type-at_wf,
equal_wf,
squash_wf,
true_wf,
istype-universe,
I_cube_wf,
csm-ap_wf,
cube-set-restriction_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,
nc-1_wf,
names-hom_wf,
fset_wf,
nat_wf,
csm-ap-restriction,
subtype_rel_self,
iff_weakening_equal,
nc-0_wf,
istype-nat,
fset-member_wf,
int-deq_wf,
strong-subtype-deq-subtype,
strong-subtype-set3,
le_wf,
strong-subtype-self,
istype-void,
cube_set_map_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
Error :memTop,
hypothesis,
applyEquality,
lambdaEquality_alt,
imageElimination,
because_Cache,
hypothesisEquality,
instantiate,
equalityTransitivity,
equalitySymmetry,
universeIsType,
universeEquality,
dependent_set_memberEquality_alt,
setElimination,
rename,
dependent_functionElimination,
natural_numberEquality,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
int_eqEquality,
independent_pairFormation,
voidElimination,
inhabitedIsType,
imageMemberEquality,
baseClosed,
productElimination,
independent_pairEquality,
axiomEquality,
isect_memberEquality_alt,
isectIsTypeImplies,
setIsType,
functionIsType,
intEquality
Latex:
\mforall{}[Gamma,Delta:j\mvdash{}]. \mforall{}[sigma:Delta j{}\mrightarrow{} Gamma]. \mforall{}[A:\{Gamma \mvdash{} \_\}]. \mforall{}[I:fset(\mBbbN{})]. \mforall{}[i:\{i:\mBbbN{}| \mneg{}i \mmember{} I\} ].
\mforall{}[rho:Delta(I+i)]. \mforall{}[J:fset(\mBbbN{})]. \mforall{}[f:J {}\mrightarrow{} I].
(((A)sigma(f((i1)(rho))) = A(f((i1)((sigma)rho))))
\mwedge{} ((A)sigma(f((i0)(rho))) = A(f((i0)((sigma)rho)))))
Date html generated:
2020_05_20-PM-03_44_43
Last ObjectModification:
2020_04_09-AM-10_58_40
Theory : cubical!type!theory
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