Nuprl Lemma : face-forall-and
∀[Gamma:j⊢]. ∀[phi,psi:{Gamma.𝕀 ⊢ _:𝔽}]. ((Gamma ⊢ ∀ (phi ∧ psi)) = ((∀ phi) ∧ (∀ psi)) ∈ {Gamma ⊢ _:𝔽})
Proof
Definitions occuring in Statement :
face-forall: (∀ phi),
face-and: (a ∧ b),
face-type: 𝔽,
interval-type: 𝕀,
cube-context-adjoin: X.A,
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],
member: t ∈ T,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
interval-presheaf: 𝕀,
all: ∀x:A. B[x],
names: names(I),
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
face-forall: (∀ phi),
face-and: (a ∧ b),
cubical-term-at: u(a),
cubical-term: {X ⊢ _: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
Lemmas referenced :
I_cube_wf,
fset_wf,
nat_wf,
cubical-term-equal,
face-type_wf,
face-forall_wf,
face-and_wf,
cube-context-adjoin_wf,
interval-type_wf,
cubical-term_wf,
cubical_set_cumulativity-i-j,
cubical_set_wf,
interval-type-at,
I_cube_pair_redex_lemma,
dM_inc_wf,
add-name_wf,
new-name_wf,
trivial-member-add-name1,
fset-member_wf,
int-deq_wf,
strong-subtype-deq-subtype,
strong-subtype-set3,
le_wf,
istype-int,
strong-subtype-self,
fl_all-meet,
cc-adjoin-cube_wf,
cube-set-restriction_wf,
nc-s_wf,
f-subset-add-name,
subtype_rel_self,
cubical-type-at_wf_face-type
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
functionExtensionality,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
instantiate,
applyEquality,
because_Cache,
sqequalRule,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
inhabitedIsType,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
universeIsType,
Error :memTop,
dependent_functionElimination,
dependent_set_memberEquality_alt,
lambdaEquality_alt,
setElimination,
rename,
intEquality,
natural_numberEquality
Latex:
\mforall{}[Gamma:j\mvdash{}]. \mforall{}[phi,psi:\{Gamma.\mBbbI{} \mvdash{} \_:\mBbbF{}\}]. ((Gamma \mvdash{} \mforall{} (phi \mwedge{} psi)) = ((\mforall{} phi) \mwedge{} (\mforall{} psi)))
Date html generated:
2020_05_20-PM-02_51_11
Last ObjectModification:
2020_04_04-PM-05_05_52
Theory : cubical!type!theory
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