Nuprl Lemma : face-or-list-eq-1
∀[Gamma:j⊢]
∀L:{Gamma ⊢ _:𝔽} List. ∀I:fset(ℕ). ∀rho:Gamma(I).
(\/(L)(rho) = 1 ∈ Point(face_lattice(I)) ⇐⇒ (∃x∈L. x(rho) = 1 ∈ Point(face_lattice(I))))
Proof
Definitions occuring in Statement :
face-or-list: \/(L),
face-type: 𝔽,
cubical-term-at: u(a),
cubical-term: {X ⊢ _:A},
face_lattice: face_lattice(I),
I_cube: A(I),
cubical_set: CubicalSet,
fset: fset(T),
l_exists: (∃x∈L. P[x]),
list: T List,
nat: ℕ,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
equal: s = t ∈ T,
lattice-1: 1,
lattice-point: Point(l)
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
prop: ℙ,
cubical-type-at: A(a),
pi1: fst(t),
face-type: 𝔽,
constant-cubical-type: (X),
I_cube: A(I),
functor-ob: ob(F),
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,
and: P ∧ Q,
so_apply: x[s],
uimplies: b supposing a,
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
cubical-term-at: u(a),
face-or-list: \/(L),
face-0: 0(𝔽),
not: ¬A,
false: False,
l_exists: (∃x∈L. P[x]),
exists: ∃x:A. B[x],
select: L[n],
nil: [],
it: ⋅,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
satisfiable_int_formula: satisfiable_int_formula(fmla),
less_than: a < b,
squash: ↓T,
decidable: Dec(P),
or: P ∨ Q
Lemmas referenced :
list_induction,
cubical-term_wf,
face-type_wf,
all_wf,
fset_wf,
nat_wf,
I_cube_wf,
iff_wf,
equal_wf,
lattice-point_wf,
face_lattice_wf,
cubical-term-at_wf,
face-or-list_wf,
lattice-1_wf,
l_exists_wf,
l_member_wf,
subtype_rel_self,
subtype_rel_set,
bounded-lattice-structure_wf,
lattice-structure_wf,
lattice-axioms_wf,
bounded-lattice-structure-subtype,
bounded-lattice-axioms_wf,
lattice-meet_wf,
lattice-join_wf,
list_wf,
cubical_set_wf,
reduce_nil_lemma,
lattice-0_wf,
l_exists_wf_nil,
istype-void,
face-lattice-0-not-1,
stuck-spread,
istype-base,
length_of_nil_lemma,
int_seg_properties,
full-omega-unsat,
intformand_wf,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
reduce_cons_lemma,
exists_wf,
int_seg_wf,
length_wf,
select_wf,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
decidable__lt,
l_exists_cons,
face-or_wf,
cons_wf,
face-or-eq-1
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
cut,
thin,
instantiate,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
sqequalRule,
lambdaEquality_alt,
applyEquality,
cumulativity,
universeIsType,
universeEquality,
because_Cache,
setElimination,
rename,
productEquality,
isectEquality,
independent_isectElimination,
setIsType,
independent_functionElimination,
functionIsType,
productIsType,
equalityIstype,
inhabitedIsType,
dependent_functionElimination,
Error :memTop,
independent_pairFormation,
equalityTransitivity,
equalitySymmetry,
voidElimination,
productElimination,
baseClosed,
natural_numberEquality,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
closedConclusion,
imageElimination,
unionElimination,
promote_hyp,
unionIsType,
inlFormation_alt,
inrFormation_alt
Latex:
\mforall{}[Gamma:j\mvdash{}]
\mforall{}L:\{Gamma \mvdash{} \_:\mBbbF{}\} List. \mforall{}I:fset(\mBbbN{}). \mforall{}rho:Gamma(I). (\mbackslash{}/(L)(rho) = 1 \mLeftarrow{}{}\mRightarrow{} (\mexists{}x\mmember{}L. x(rho) = 1))
Date html generated:
2020_05_20-PM-02_42_30
Last ObjectModification:
2020_04_04-PM-04_56_31
Theory : cubical!type!theory
Home
Index