Nuprl Lemma : glue-type-constraint
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[phi:{Gamma ⊢ _:𝔽}]. ∀[T:{Gamma, phi ⊢ _}]. ∀[w:{Gamma, phi ⊢ _:(T ⟶ A)}].
Gamma, phi ⊢ Glue [phi ⊢→ (T;w)] A = T
Proof
Definitions occuring in Statement :
glue-type: Glue [phi ⊢→ (T;w)] A,
same-cubical-type: Gamma ⊢ A = B,
context-subset: Gamma, phi,
face-type: 𝔽,
cubical-fun: (A ⟶ B),
cubical-term: {X ⊢ _:A},
cubical-type: {X ⊢ _},
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
same-cubical-type: Gamma ⊢ A = B,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
cubical-type: {X ⊢ _},
glue-type: Glue [phi ⊢→ (T;w)] A,
glue-cube: glue-cube(Gamma;A;phi;T;w;I;rho),
glue-morph: glue-morph(Gamma;A;phi;T;w;I;rho;J;f;u),
and: P ∧ Q,
all: ∀x:A. B[x],
context-subset: Gamma, phi,
bdd-distributive-lattice: BoundedDistributiveLattice,
so_lambda: λ2x.t[x],
prop: ℙ,
so_apply: x[s],
implies: P ⇒ Q,
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
true: True,
sq_type: SQType(T),
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],
eq_atom: x =a y,
bfalse: ff,
respects-equality: respects-equality(S;T),
squash: ↓T
Lemmas referenced :
cubical-type-equal2,
glue-type_wf,
subset-cubical-type,
context-subset-is-subset,
I_cube_wf,
context-subset_wf,
fset_wf,
nat_wf,
glue-cube_wf,
subset-I_cube,
names-hom_wf,
cube-set-restriction_wf,
istype-cubical-term,
cubical-fun_wf,
thin-context-subset,
cubical-type_wf,
face-type_wf,
cubical_set_wf,
I_cube_pair_redex_lemma,
subtype_base_sq,
bool_wf,
bool_subtype_base,
iff_imp_equal_bool,
fl-eq_wf,
cubical-term-at_wf,
lattice-1_wf,
face_lattice_wf,
btrue_wf,
iff_functionality_wrt_iff,
assert_wf,
equal_wf,
lattice-point_wf,
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,
true_wf,
iff_weakening_uiff,
assert-fl-eq,
iff_weakening_equal,
istype-true,
cubical_type_at_pair_lemma,
subtype_rel_self,
subset-cubical-term,
face-type-at,
respects-equality_weakening,
squash_wf,
istype-universe,
face-term-at-restriction-eq-1,
cubical_type_ap_morph_pair_lemma,
cube_set_restriction_pair_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
hypothesisEquality,
hypothesis,
applyEquality,
independent_isectElimination,
sqequalRule,
setElimination,
rename,
productElimination,
dependent_pairEquality_alt,
functionExtensionality,
dependent_functionElimination,
functionIsType,
universeIsType,
axiomEquality,
isect_memberEquality_alt,
isectIsTypeImplies,
inhabitedIsType,
instantiate,
Error :memTop,
cumulativity,
equalityTransitivity,
equalitySymmetry,
lambdaEquality_alt,
productEquality,
isectEquality,
independent_functionElimination,
independent_pairFormation,
lambdaFormation_alt,
natural_numberEquality,
dependent_set_memberEquality_alt,
equalityIstype,
imageElimination,
universeEquality,
imageMemberEquality,
baseClosed
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)\}].
Gamma, phi \mvdash{} Glue [phi \mvdash{}\mrightarrow{} (T;w)] A = T
Date html generated:
2020_05_20-PM-05_42_00
Last ObjectModification:
2020_04_21-PM-06_58_11
Theory : cubical!type!theory
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