Nuprl Lemma : comp-nc-1-subset-I_cube
∀[I:fset(ℕ)]. ∀[i:{i:ℕ| ¬i ∈ I} ]. ∀[phi:𝔽(I)]. ∀J:fset(ℕ). ∀[f:I,phi(J)]. ((i1) ⋅ f ∈ I+i,s(phi)(J))
Proof
Definitions occuring in Statement :
cubical-subset: I,psi,
face-presheaf: 𝔽,
cube-set-restriction: f(s),
I_cube: A(I),
nc-1: (i1),
nc-s: s,
add-name: I+i,
nh-comp: g ⋅ f,
fset-member: a ∈ s,
fset: fset(T),
int-deq: IntDeq,
nat: ℕ,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
not: ¬A,
member: t ∈ T,
set: {x:A| B[x]}
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
and: P ∧ Q,
nat: ℕ,
ge: i ≥ j ,
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,
prop: ℙ,
subtype_rel: A ⊆r B,
I_cube: A(I),
functor-ob: ob(F),
pi1: fst(t),
face-presheaf: 𝔽,
lattice-point: Point(l),
record-select: r.x,
face_lattice: face_lattice(I),
face-lattice: face-lattice(T;eq),
free-dist-lattice-with-constraints: free-dist-lattice-with-constraints(T;eq;x.Cs[x]),
constrained-antichain-lattice: constrained-antichain-lattice(T;eq;P),
mk-bounded-distributive-lattice: mk-bounded-distributive-lattice,
mk-bounded-lattice: mk-bounded-lattice(T;m;j;z;o),
record-update: r[x := v],
ifthenelse: if b then t else f fi ,
eq_atom: x =a y,
bfalse: ff,
btrue: tt,
bdd-distributive-lattice: BoundedDistributiveLattice,
so_lambda: λ2x.t[x],
so_apply: x[s],
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
squash: ↓T,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
cubical-subset-I_cube-member,
member-cubical-subset-I_cube,
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,
cube-set-restriction_wf,
face-presheaf_wf,
nc-s_wf,
f-subset-add-name,
nh-comp_wf,
nc-1_wf,
name-morph-satisfies-comp,
subtype_rel_self,
lattice-point_wf,
face_lattice_wf,
subtype_rel_set,
bounded-lattice-structure_wf,
lattice-structure_wf,
lattice-axioms_wf,
bounded-lattice-structure-subtype,
bounded-lattice-axioms_wf,
equal_wf,
lattice-meet_wf,
lattice-join_wf,
name-morph-satisfies_wf,
names-hom_wf,
I_cube_wf,
cubical-subset_wf,
small_cubical_set_subtype,
istype-nat,
fset-member_wf,
nat_wf,
int-deq_wf,
strong-subtype-deq-subtype,
strong-subtype-set3,
le_wf,
strong-subtype-self,
istype-void,
fset_wf,
squash_wf,
true_wf,
istype-universe,
nh-comp-assoc,
iff_weakening_equal,
nh-id-right,
s-comp-nc-1
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
productElimination,
dependent_set_memberEquality_alt,
setElimination,
rename,
hypothesis,
dependent_functionElimination,
natural_numberEquality,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
Error :memTop,
sqequalRule,
independent_pairFormation,
universeIsType,
voidElimination,
applyEquality,
because_Cache,
instantiate,
productEquality,
cumulativity,
isectEquality,
hyp_replacement,
equalitySymmetry,
imageElimination,
equalityTransitivity,
imageMemberEquality,
baseClosed,
inhabitedIsType,
setIsType,
functionIsType,
intEquality,
universeEquality
Latex:
\mforall{}[I:fset(\mBbbN{})]. \mforall{}[i:\{i:\mBbbN{}| \mneg{}i \mmember{} I\} ]. \mforall{}[phi:\mBbbF{}(I)].
\mforall{}J:fset(\mBbbN{}). \mforall{}[f:I,phi(J)]. ((i1) \mcdot{} f \mmember{} I+i,s(phi)(J))
Date html generated:
2020_05_20-PM-03_44_31
Last ObjectModification:
2020_01_06-PM-04_07_42
Theory : cubical!type!theory
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