Nuprl Lemma : equal-glue-cube
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}].
∀phi:{Gamma ⊢ _:𝔽}
∀[T:{Gamma, phi ⊢ _}]. ∀[w:{Gamma, phi ⊢ _:(T ⟶ A)}].
∀I:fset(ℕ). ∀rho:Gamma(I). ∀u,v:glue-cube(Gamma;A;phi;T;w;I;rho).
u = v ∈ glue-cube(Gamma;A;phi;T;w;I;rho)
supposing if (phi(rho)==1)
then u = v ∈ T(rho)
else u = v ∈ (J:fset(ℕ) ⟶ f:{f:J ⟶ I| phi(f(rho)) = 1 ∈ Point(face_lattice(J))} ⟶ T(f(rho)) × A(rho))
fi
Proof
Definitions occuring in Statement :
glue-cube: glue-cube(Gamma;A;phi;T;w;I;rho),
context-subset: Gamma, phi,
face-type: 𝔽,
cubical-fun: (A ⟶ B),
cubical-term-at: u(a),
cubical-term: {X ⊢ _:A},
cubical-type-at: A(a),
cubical-type: {X ⊢ _},
fl-eq: (x==y),
face_lattice: face_lattice(I),
cube-set-restriction: f(s),
I_cube: A(I),
cubical_set: CubicalSet,
names-hom: I ⟶ J,
fset: fset(T),
nat: ℕ,
ifthenelse: if b then t else f fi ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
product: x:A × B[x],
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],
glue-cube: glue-cube(Gamma;A;phi;T;w;I;rho),
member: t ∈ T,
subtype_rel: A ⊆r B,
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,
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
bdd-distributive-lattice: BoundedDistributiveLattice,
so_lambda: λ2x.t[x],
prop: ℙ,
so_apply: x[s],
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
true: True,
context-subset: Gamma, phi
Lemmas referenced :
fl-eq_wf,
cubical-term-at_wf,
face-type_wf,
subtype_rel_self,
lattice-point_wf,
face_lattice_wf,
lattice-1_wf,
eqtt_to_assert,
assert-fl-eq,
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,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
glue-cube_wf,
I_cube_wf,
fset_wf,
nat_wf,
istype-cubical-term,
context-subset_wf,
cubical-fun_wf,
thin-context-subset,
cubical-type_wf,
cubical_set_wf,
iff_imp_equal_bool,
btrue_wf,
iff_functionality_wrt_iff,
true_wf,
iff_weakening_equal,
istype-true,
names-hom_wf,
cube-set-restriction_wf,
istype-cubical-type-at,
I_cube_pair_redex_lemma,
glue-equations_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
sqequalHypSubstitution,
cut,
introduction,
extract_by_obid,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
applyEquality,
sqequalRule,
because_Cache,
inhabitedIsType,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_isectElimination,
instantiate,
lambdaEquality_alt,
productEquality,
cumulativity,
isectEquality,
universeIsType,
setElimination,
rename,
dependent_pairFormation_alt,
equalityIstype,
promote_hyp,
dependent_functionElimination,
independent_functionElimination,
voidElimination,
independent_pairFormation,
natural_numberEquality,
productIsType,
functionIsType,
setIsType,
Error :memTop,
dependent_set_memberEquality_alt
Latex:
\mforall{}[Gamma:j\mvdash{}]. \mforall{}[A:\{Gamma \mvdash{} \_\}].
\mforall{}phi:\{Gamma \mvdash{} \_:\mBbbF{}\}
\mforall{}[T:\{Gamma, phi \mvdash{} \_\}]. \mforall{}[w:\{Gamma, phi \mvdash{} \_:(T {}\mrightarrow{} A)\}].
\mforall{}I:fset(\mBbbN{}). \mforall{}rho:Gamma(I). \mforall{}u,v:glue-cube(Gamma;A;phi;T;w;I;rho).
u = v supposing if (phi(rho)==1) then u = v else u = v fi
Date html generated:
2020_05_20-PM-05_39_07
Last ObjectModification:
2020_04_21-PM-05_17_32
Theory : cubical!type!theory
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