Nuprl Lemma : glue-morph-comp
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[phi:{Gamma ⊢ _:𝔽}]. ∀[T:{Gamma, phi ⊢ _}]. ∀[w:{Gamma, phi ⊢ _:(T ⟶ A)}].
∀[I:fset(ℕ)]. ∀[rho:Gamma(I)]. ∀[J,K:fset(ℕ)]. ∀[f:J ⟶ I]. ∀[g:K ⟶ J]. ∀[u:glue-cube(Gamma;A;phi;T;w;I;rho)].
(glue-morph(Gamma;A;phi;T;w;J;f(rho);K;g;glue-morph(Gamma;A;phi;T;w;I;rho;J;f;u))
= glue-morph(Gamma;A;phi;T;w;I;rho;K;f ⋅ g;u)
∈ glue-cube(Gamma;A;phi;T;w;K;f ⋅ g(rho)))
Proof
Definitions occuring in Statement :
glue-morph: glue-morph(Gamma;A;phi;T;w;I;rho;J;f;u),
glue-cube: glue-cube(Gamma;A;phi;T;w;I;rho),
context-subset: Gamma, phi,
face-type: 𝔽,
cubical-fun: (A ⟶ B),
cubical-term: {X ⊢ _:A},
cubical-type: {X ⊢ _},
cube-set-restriction: f(s),
I_cube: A(I),
cubical_set: CubicalSet,
nh-comp: g ⋅ f,
names-hom: I ⟶ J,
fset: fset(T),
nat: ℕ,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
squash: ↓T,
prop: ℙ,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
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,
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
bdd-distributive-lattice: BoundedDistributiveLattice,
so_lambda: λ2x.t[x],
so_apply: x[s],
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬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),
context-subset: Gamma, phi,
glue-equations: glue-equations(Gamma;A;phi;T;w;I;rho;t;a)
Lemmas referenced :
equal-glue-cube,
cube-set-restriction_wf,
nh-comp_wf,
glue-morph_wf,
subtype_rel-equal,
glue-cube_wf,
equal_wf,
squash_wf,
true_wf,
istype-universe,
cube-set-restriction-comp,
subtype_rel_self,
iff_weakening_equal,
fl-eq_wf,
cubical-term-at_wf,
face-type_wf,
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,
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,
names-hom_wf,
I_cube_wf,
fset_wf,
nat_wf,
istype-cubical-term,
context-subset_wf,
cubical-fun_wf,
thin-context-subset,
cubical-type_wf,
cubical_set_wf,
iff_imp_equal_bool,
btrue_wf,
iff_functionality_wrt_iff,
istype-true,
face-term-at-restriction-eq-1,
cubical-type-at_wf,
I_cube_pair_redex_lemma,
cubical-type-ap-morph-comp-eq,
cube_set_restriction_pair_lemma,
bfalse_wf,
cubical-term-at-comp-is-1,
false_wf,
istype-void,
nh-comp-assoc,
cubical-type-ap-morph-comp
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
hypothesisEquality,
dependent_functionElimination,
hypothesis,
applyEquality,
independent_isectElimination,
instantiate,
lambdaEquality_alt,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeIsType,
universeEquality,
natural_numberEquality,
sqequalRule,
imageMemberEquality,
baseClosed,
productElimination,
independent_functionElimination,
inhabitedIsType,
lambdaFormation_alt,
unionElimination,
equalityElimination,
productEquality,
cumulativity,
isectEquality,
setElimination,
rename,
dependent_pairFormation_alt,
equalityIstype,
promote_hyp,
voidElimination,
independent_pairFormation,
hyp_replacement,
Error :memTop,
dependent_set_memberEquality_alt,
independent_pairEquality,
setIsType
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{}[I:fset(\mBbbN{})]. \mforall{}[rho:Gamma(I)]. \mforall{}[J,K:fset(\mBbbN{})]. \mforall{}[f:J {}\mrightarrow{} I].
\mforall{}[g:K {}\mrightarrow{} J]. \mforall{}[u:glue-cube(Gamma;A;phi;T;w;I;rho)].
(glue-morph(Gamma;A;phi;T;w;J;f(rho);K;g;glue-morph(Gamma;A;phi;T;w;I;rho;J;f;u))
= glue-morph(Gamma;A;phi;T;w;I;rho;K;f \mcdot{} g;u))
Date html generated:
2020_05_20-PM-05_40_12
Last ObjectModification:
2020_04_21-PM-05_53_49
Theory : cubical!type!theory
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