Nuprl Lemma : face-and-eq-1
∀[Gamma:j⊢]
∀r,s:{Gamma ⊢ _:𝔽}. ∀I:fset(ℕ). ∀rho:Gamma(I).
((r ∧ s)(rho) = 1 ∈ Point(face_lattice(I))
⇐⇒ (r(rho) = 1 ∈ Point(face_lattice(I))) ∧ (s(rho) = 1 ∈ Point(face_lattice(I))))
Proof
Definitions occuring in Statement :
face-and: (a ∧ b),
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),
nat: ℕ,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
equal: s = t ∈ T,
lattice-1: 1,
lattice-point: Point(l)
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
subtype_rel: A ⊆r B,
bdd-distributive-lattice: BoundedDistributiveLattice,
so_lambda: λ2x.t[x],
prop: ℙ,
so_apply: x[s],
uimplies: b supposing a,
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,
rev_implies: P ⇐ Q
Lemmas referenced :
face-and-at,
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,
cubical-term-at_wf,
face-type_wf,
subtype_rel_self,
lattice-1_wf,
iff_weakening_uiff,
lattice-meet-eq-1,
bdd-distributive-lattice-subtype-bdd-lattice,
I_cube_wf,
fset_wf,
nat_wf,
cubical-term_wf,
cubical_set_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
lambdaFormation_alt,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
Error :memTop,
hypothesis,
independent_pairFormation,
productElimination,
productIsType,
equalityIstype,
universeIsType,
hypothesisEquality,
applyEquality,
instantiate,
lambdaEquality_alt,
productEquality,
cumulativity,
isectEquality,
because_Cache,
independent_isectElimination,
setElimination,
rename,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
promote_hyp,
dependent_functionElimination,
independent_pairEquality,
axiomEquality,
functionIsTypeImplies
Latex:
\mforall{}[Gamma:j\mvdash{}]
\mforall{}r,s:\{Gamma \mvdash{} \_:\mBbbF{}\}. \mforall{}I:fset(\mBbbN{}). \mforall{}rho:Gamma(I). ((r \mwedge{} s)(rho) = 1 \mLeftarrow{}{}\mRightarrow{} (r(rho) = 1) \mwedge{} (s(rho) = 1))
Date html generated:
2020_05_20-PM-02_41_09
Last ObjectModification:
2020_04_04-PM-04_49_15
Theory : cubical!type!theory
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