Nuprl Lemma : glue-term-constraint
∀[Gamma:j⊢]. ∀[phi:{Gamma ⊢ _:𝔽}]. ∀[T:{Gamma, phi ⊢ _}]. ∀[t:{Gamma, phi ⊢ _:T}]. ∀[A,a:Top].
Gamma, phi ⊢ glue [phi ⊢→ t] a=t:T
Proof
Definitions occuring in Statement :
glue-term: glue [phi ⊢→ t] a,
same-cubical-term: X ⊢ u=v:A,
context-subset: Gamma, phi,
face-type: 𝔽,
cubical-term: {X ⊢ _:A},
cubical-type: {X ⊢ _},
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x],
top: Top
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
same-cubical-term: X ⊢ u=v:A,
member: t ∈ T,
uimplies: b supposing a,
context-subset: Gamma, phi,
all: ∀x:A. B[x],
glue-term: glue [phi ⊢→ t] a,
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,
bdd-distributive-lattice: BoundedDistributiveLattice,
so_lambda: λ2x.t[x],
prop: ℙ,
and: P ∧ Q,
so_apply: x[s],
implies: P ⇒ Q,
assert: ↑b,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
true: True,
sq_type: SQType(T),
cubical-term-at: u(a)
Lemmas referenced :
I_cube_wf,
context-subset_wf,
fset_wf,
nat_wf,
cubical-term-equal,
istype-top,
istype-cubical-term,
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,
subtype_rel_self,
lattice-point_wf,
face_lattice_wf,
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,
lattice-1_wf,
btrue_wf,
iff_functionality_wrt_iff,
assert_wf,
true_wf,
iff_weakening_uiff,
assert-fl-eq,
iff_weakening_equal,
istype-true
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
equalitySymmetry,
cut,
functionExtensionality,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
because_Cache,
equalityTransitivity,
independent_isectElimination,
universeIsType,
instantiate,
dependent_functionElimination,
Error :memTop,
sqequalRule,
cumulativity,
applyEquality,
lambdaEquality_alt,
productEquality,
isectEquality,
setElimination,
rename,
inhabitedIsType,
independent_functionElimination,
productElimination,
independent_pairFormation,
lambdaFormation_alt,
natural_numberEquality,
equalityIstype
Latex:
\mforall{}[Gamma:j\mvdash{}]. \mforall{}[phi:\{Gamma \mvdash{} \_:\mBbbF{}\}]. \mforall{}[T:\{Gamma, phi \mvdash{} \_\}]. \mforall{}[t:\{Gamma, phi \mvdash{} \_:T\}]. \mforall{}[A,a:Top].
Gamma, phi \mvdash{} glue [phi \mvdash{}\mrightarrow{} t] a=t:T
Date html generated:
2020_05_20-PM-05_43_51
Last ObjectModification:
2020_04_21-PM-07_02_13
Theory : cubical!type!theory
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