Nuprl Lemma : face-1-in-context-subset
∀[G:j⊢]. ∀[phi:{G ⊢ _:𝔽}]. (phi = 1(𝔽) ∈ {G, phi ⊢ _:𝔽})
Proof
Definitions occuring in Statement :
context-subset: Gamma, phi,
face-1: 1(𝔽),
face-type: 𝔽,
cubical-term: {X ⊢ _:A},
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
context-subset: Gamma, phi,
all: ∀x:A. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
face-1: 1(𝔽),
cubical-term-at: u(a),
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,
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,
bdd-distributive-lattice: BoundedDistributiveLattice,
so_lambda: λ2x.t[x],
prop: ℙ,
and: P ∧ Q,
so_apply: x[s],
guard: {T},
implies: P ⇒ Q
Lemmas referenced :
I_cube_pair_redex_lemma,
I_cube_wf,
context-subset_wf,
fset_wf,
nat_wf,
cubical-term-equal,
face-type_wf,
subset-cubical-term,
context-subset-is-subset,
cubical-term_wf,
cubical_set_wf,
subtype_rel_self,
cubical-type-at_wf_face-type,
equal_functionality_wrt_subtype_rel2,
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
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
functionExtensionality,
sqequalHypSubstitution,
introduction,
extract_by_obid,
dependent_functionElimination,
thin,
Error :memTop,
hypothesis,
isectElimination,
hypothesisEquality,
applyEquality,
because_Cache,
independent_isectElimination,
sqequalRule,
equalityTransitivity,
equalitySymmetry,
universeIsType,
instantiate,
setElimination,
rename,
lambdaEquality_alt,
productEquality,
cumulativity,
isectEquality,
independent_functionElimination
Latex:
\mforall{}[G:j\mvdash{}]. \mforall{}[phi:\{G \mvdash{} \_:\mBbbF{}\}]. (phi = 1(\mBbbF{}))
Date html generated:
2020_05_20-PM-03_06_49
Last ObjectModification:
2020_04_04-PM-05_23_20
Theory : cubical!type!theory
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