Nuprl Lemma : glue-term_wf
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[phi:{Gamma ⊢ _:𝔽}]. ∀[T:{Gamma, phi ⊢ _}]. ∀[w:{Gamma, phi ⊢ _:(T ⟶ A)}].
∀[t:{Gamma, phi ⊢ _:T}]. ∀[a:{Gamma ⊢ _:A[phi |⟶ app(w; t)]}].
(glue [phi ⊢→ t] a ∈ {Gamma ⊢ _:Glue [phi ⊢→ (T;w)] A})
Proof
Definitions occuring in Statement :
glue-term: glue [phi ⊢→ t] a,
glue-type: Glue [phi ⊢→ (T;w)] A,
constrained-cubical-term: {Gamma ⊢ _:A[phi |⟶ t]},
context-subset: Gamma, phi,
face-type: 𝔽,
cubical-app: app(w; u),
cubical-fun: (A ⟶ B),
cubical-term: {X ⊢ _:A},
cubical-type: {X ⊢ _},
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x],
member: t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
constrained-cubical-term: {Gamma ⊢ _:A[phi |⟶ t]},
subtype_rel: A ⊆r B,
glue-term: glue [phi ⊢→ t] a,
glue-type: Glue [phi ⊢→ (T;w)] A,
all: ∀x:A. B[x],
glue-cube: glue-cube(Gamma;A;phi;T;w;I;rho),
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,
context-subset: Gamma, phi,
glue-equations: glue-equations(Gamma;A;phi;T;w;I;rho;t;a),
squash: ↓T,
true: True,
csm-ap: (s)x,
subset-iota: iota,
cube-set-restriction: f(s),
pi2: snd(t),
cubical-app: app(w; u),
cubical-term-at: u(a),
cubical-term: {X ⊢ _:A},
glue-morph: glue-morph(Gamma;A;phi;T;w;I;rho;J;f;u)
Lemmas referenced :
constrained-cubical-term_wf,
cubical-type-cumulativity2,
cubical_set_cumulativity-i-j,
cubical-app_wf_fun,
context-subset_wf,
thin-context-subset,
istype-cubical-term,
cubical-fun_wf,
cubical-type_wf,
face-type_wf,
cubical_set_wf,
I_cube_wf,
fset_wf,
nat_wf,
cubical_type_at_pair_lemma,
fl-eq_wf,
cubical-term-at_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,
I_cube_pair_redex_lemma,
names-hom_wf,
cube-set-restriction_wf,
cube_set_restriction_pair_lemma,
cubical-type-at_wf,
squash_wf,
true_wf,
face-term-at-restriction-eq-1,
iff_weakening_equal,
cube-set-restriction-comp,
cubical-term-at-morph,
istype-universe,
cubical-term-at-comp-is-1,
csm-ap-type_wf,
subset-iota_wf2,
cubical-term-eqcd,
csm-ap-type-iota,
subset-cubical-type,
context-subset-is-subset,
csm-ap-type-at,
subtype_rel_wf,
context-subset-restriction,
glue-equations_wf,
cubical_type_ap_morph_pair_lemma,
equal-glue-cube,
glue-morph_wf,
istype-cubical-type-at,
glue-type_wf,
cubical-type-ap-morph_wf,
iff_imp_equal_bool,
btrue_wf,
iff_functionality_wrt_iff,
istype-true,
cubical-term-at-morph1
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
sqequalHypSubstitution,
setElimination,
thin,
rename,
hypothesis,
universeIsType,
instantiate,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
applyEquality,
sqequalRule,
because_Cache,
functionExtensionality,
dependent_functionElimination,
Error :memTop,
inhabitedIsType,
lambdaFormation_alt,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_isectElimination,
lambdaEquality_alt,
productEquality,
cumulativity,
isectEquality,
dependent_pairFormation_alt,
equalityIstype,
promote_hyp,
independent_functionElimination,
voidElimination,
dependent_set_memberEquality_alt,
independent_pairEquality,
setEquality,
independent_pairFormation,
setIsType,
hyp_replacement,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
universeEquality,
applyLambdaEquality,
functionIsType
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{}[t:\{Gamma, phi \mvdash{} \_:T\}]. \mforall{}[a:\{Gamma \mvdash{} \_:A[phi |{}\mrightarrow{} app(w; t)]\}].
(glue [phi \mvdash{}\mrightarrow{} t] a \mmember{} \{Gamma \mvdash{} \_:Glue [phi \mvdash{}\mrightarrow{} (T;w)] A\})
Date html generated:
2020_05_20-PM-05_43_20
Last ObjectModification:
2020_04_21-PM-07_02_40
Theory : cubical!type!theory
Home
Index