Nuprl Lemma : glue-equations_wf
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[phi:{Gamma ⊢ _:𝔽}]. ∀[T:{Gamma, phi ⊢ _}]. ∀[w:{Gamma, phi ⊢ _:(T ⟶ A)}].
∀[I:fset(ℕ)]. ∀[rho:Gamma(I)]. ∀[t:J:fset(ℕ) ⟶ f:{f:J ⟶ I| phi(f(rho)) = 1 ∈ Point(face_lattice(J))} ⟶ T(f(rho))].
∀[a:A(rho)].
(glue-equations(Gamma;A;phi;T;w;I;rho;t;a) ∈ ℙ)
Proof
Definitions occuring in Statement :
glue-equations: glue-equations(Gamma;A;phi;T;w;I;rho;t;a),
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 ⊢ _},
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,
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
equal: s = t ∈ T,
lattice-1: 1,
lattice-point: Point(l)
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
prop: ℙ,
all: ∀x:A. B[x],
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,
context-subset: Gamma, phi,
bdd-distributive-lattice: BoundedDistributiveLattice,
so_lambda: λ2x.t[x],
and: P ∧ Q,
so_apply: x[s],
uimplies: b supposing a,
glue-equations: glue-equations(Gamma;A;phi;T;w;I;rho;t;a),
implies: P ⇒ Q,
squash: ↓T,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
cubical-fun: (A ⟶ B),
cubical-fun-family: cubical-fun-family(X; A; B; I; a)
Lemmas referenced :
fset_wf,
nat_wf,
names-hom_wf,
equal_wf,
lattice-point_wf,
face_lattice_wf,
cubical-term-at_wf,
face-type_wf,
cube-set-restriction_wf,
lattice-1_wf,
subtype_rel_self,
I_cube_pair_redex_lemma,
subtype_rel_set,
bounded-lattice-structure_wf,
lattice-structure_wf,
lattice-axioms_wf,
bounded-lattice-structure-subtype,
bounded-lattice-axioms_wf,
lattice-meet_wf,
lattice-join_wf,
cubical-type-at_wf,
context-subset_wf,
istype-cubical-type-at,
I_cube_wf,
istype-cubical-term,
cubical-fun_wf,
thin-context-subset,
cubical-type_wf,
cubical_set_wf,
all_wf,
cubical-term-at-comp-is-1,
nh-comp_wf,
cubical-type-ap-morph_wf,
subtype_rel-equal,
cube_set_restriction_pair_lemma,
squash_wf,
true_wf,
cube-set-restriction-comp,
iff_weakening_equal,
cubical_type_at_pair_lemma,
cube-set-restriction-id,
nh-id_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
functionEquality,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
setEquality,
hypothesisEquality,
because_Cache,
applyEquality,
sqequalRule,
lambdaFormation_alt,
universeIsType,
setElimination,
rename,
dependent_functionElimination,
Error :memTop,
dependent_set_memberEquality_alt,
equalityIstype,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
instantiate,
lambdaEquality_alt,
productEquality,
cumulativity,
isectEquality,
independent_isectElimination,
independent_functionElimination,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
universeEquality,
productElimination,
setIsType,
hyp_replacement
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{}[t:J:fset(\mBbbN{})
{}\mrightarrow{} f:\{f:J {}\mrightarrow{} I| phi(f(rho)) = 1\}
{}\mrightarrow{} T(f(rho))]. \mforall{}[a:A(rho)].
(glue-equations(Gamma;A;phi;T;w;I;rho;t;a) \mmember{} \mBbbP{})
Date html generated:
2020_05_20-PM-05_38_35
Last ObjectModification:
2020_04_21-PM-05_19_16
Theory : cubical!type!theory
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