Nuprl Lemma : unglue-term_wf2
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[phi:{Gamma ⊢ _:𝔽}]. ∀[T:{Gamma, phi ⊢ _}]. ∀[w:{Gamma, phi ⊢ _:(T ⟶ A)}].
∀[b:{Gamma ⊢ _:Glue [phi ⊢→ (T;w)] A}].
(unglue(b) ∈ {Gamma ⊢ _:A[phi |⟶ app(w; b)]})
Proof
Definitions occuring in Statement :
unglue-term: unglue(b),
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]},
uimplies: b supposing a,
subtype_rel: A ⊆r B,
prop: ℙ,
context-subset: Gamma, phi,
all: ∀x:A. B[x],
unglue-term: unglue(b),
bdd-distributive-lattice: BoundedDistributiveLattice,
so_lambda: λ2x.t[x],
and: P ∧ Q,
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-app: app(w; u),
cubical-term-at: u(a),
same-cubical-type: Gamma ⊢ A = B
Lemmas referenced :
unglue-term_wf,
equal_wf,
thin-context-subset,
cubical-term-eqcd,
context-subset_wf,
context-subset-term-subtype,
istype-cubical-term,
glue-type_wf,
cubical-fun_wf,
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,
subset-cubical-term,
context-subset-is-subset,
lattice-1_wf,
face_lattice_wf,
btrue_wf,
iff_functionality_wrt_iff,
assert_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,
I_cube_wf,
fset_wf,
nat_wf,
cubical-app_wf_fun,
glue-type-constraint,
cubical-term-equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
equalitySymmetry,
dependent_set_memberEquality_alt,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
hypothesisEquality,
hypothesis,
universeIsType,
instantiate,
equalityTransitivity,
independent_isectElimination,
applyEquality,
sqequalRule,
functionExtensionality,
dependent_functionElimination,
Error :memTop,
setElimination,
rename,
cumulativity,
lambdaEquality_alt,
inhabitedIsType,
productEquality,
isectEquality,
independent_functionElimination,
productElimination,
independent_pairFormation,
lambdaFormation_alt,
natural_numberEquality,
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{}[b:\{Gamma \mvdash{} \_:Glue [phi \mvdash{}\mrightarrow{} (T;w)] A\}].
(unglue(b) \mmember{} \{Gamma \mvdash{} \_:A[phi |{}\mrightarrow{} app(w; b)]\})
Date html generated:
2020_05_20-PM-05_45_46
Last ObjectModification:
2020_04_21-PM-07_40_04
Theory : cubical!type!theory
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