Nuprl Lemma : face-forall-1
∀[Gamma:j⊢]. ∀[phi:{Gamma.𝕀 ⊢ _:𝔽}]. (Gamma ⊢ ∀ phi) = 1(𝔽) ∈ {Gamma ⊢ _:𝔽} supposing phi = 1(𝔽) ∈ {Gamma.𝕀 ⊢ _:𝔽}
Proof
Definitions occuring in Statement :
face-forall: (∀ phi),
face-1: 1(𝔽),
face-type: 𝔽,
interval-type: 𝕀,
cube-context-adjoin: X.A,
cubical-term: {X ⊢ _:A},
cubical_set: CubicalSet,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
true: True,
squash: ↓T,
prop: ℙ,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
face-1: 1(𝔽),
face-forall: (∀ phi),
cubical-term-at: u(a),
all: ∀x:A. B[x],
bdd-distributive-lattice: BoundedDistributiveLattice,
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
Lemmas referenced :
face-1_wf,
cube-context-adjoin_wf,
cubical_set_cumulativity-i-j,
interval-type_wf,
cubical-term_wf,
face-type_wf,
cubical_set_wf,
equal_wf,
squash_wf,
true_wf,
istype-universe,
face-forall_wf,
subtype_rel_self,
iff_weakening_equal,
I_cube_wf,
fset_wf,
nat_wf,
cubical-term-equal,
fl_all-1,
lattice-1_wf,
face_lattice_wf,
cubical-type-at_wf_face-type
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
hypothesis,
equalityIstype,
inhabitedIsType,
hypothesisEquality,
thin,
instantiate,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
applyEquality,
sqequalRule,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
universeIsType,
because_Cache,
natural_numberEquality,
lambdaEquality_alt,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination,
productElimination,
independent_functionElimination,
functionExtensionality,
dependent_functionElimination,
Error :memTop,
setElimination,
rename
Latex:
\mforall{}[Gamma:j\mvdash{}]. \mforall{}[phi:\{Gamma.\mBbbI{} \mvdash{} \_:\mBbbF{}\}]. (Gamma \mvdash{} \mforall{} phi) = 1(\mBbbF{}) supposing phi = 1(\mBbbF{})
Date html generated:
2020_05_20-PM-02_49_59
Last ObjectModification:
2020_04_04-PM-05_03_35
Theory : cubical!type!theory
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