Nuprl Lemma : unglue-glue
∀[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)]}].
(unglue(glue [phi ⊢→ t] a) = a ∈ {Gamma ⊢ _:A})
Proof
Definitions occuring in Statement :
unglue-term: unglue(b)
,
glue-term: glue [phi ⊢→ t] 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]
,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x]
,
constrained-cubical-term: {Gamma ⊢ _:A[phi |⟶ t]}
,
glue-term: glue [phi ⊢→ t] a
,
unglue-term: unglue(b)
,
cubical-term-at: u(a)
,
member: t ∈ T
,
cubical-term: {X ⊢ _: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
,
all: ∀x:A. B[x]
,
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
,
pi2: snd(t)
,
context-subset: Gamma, phi
,
cubical-app: app(w; u)
Lemmas referenced :
fl-eq_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,
cubical-term-at_wf,
I_cube_wf,
fset_wf,
nat_wf,
cubical-term-equal,
unglue-term_wf,
glue-term_wf,
subset-cubical-term,
context-subset-is-subset,
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_pair_redex_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
sqequalHypSubstitution,
setElimination,
thin,
rename,
cut,
functionExtensionality,
sqequalRule,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
applyEquality,
hypothesis,
because_Cache,
inhabitedIsType,
lambdaFormation_alt,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_isectElimination,
instantiate,
lambdaEquality_alt,
productEquality,
cumulativity,
isectEquality,
universeIsType,
dependent_pairFormation_alt,
equalityIstype,
promote_hyp,
dependent_functionElimination,
independent_functionElimination,
voidElimination,
dependent_set_memberEquality_alt,
Error :memTop,
applyLambdaEquality
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)]\}].
(unglue(glue [phi \mvdash{}\mrightarrow{} t] a) = a)
Date html generated:
2020_05_20-PM-05_46_30
Last ObjectModification:
2020_04_21-PM-07_41_18
Theory : cubical!type!theory
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