Nuprl Lemma : composition-type-lemma4
∀Gamma:j⊢. ∀phi:{Gamma ⊢ _:𝔽}. ∀A:{Gamma.𝕀 ⊢ _}. ∀u:{Gamma, phi.𝕀 ⊢ _:A}. ∀I,J:fset(ℕ). ∀f:J ⟶ I. ∀rho:Gamma(I).
∀K:fset(ℕ). ∀g:J,phi(f(rho))(K).
  ((u)<(s(f(rho));<new-name(J)>)> o iota((new-name(J)0) ⋅ g)
  = ((u)<(s(rho);<new-name(I)>)> o iota)subset-trans(I+new-name(I);J+new-name(J);f,new-name(I)=new-name(J);
                                                     s(phi(rho)))((new-name(J)0) ⋅ g)
  ∈ A(g((new-name(J)0)((s(f(rho));<new-name(J)>)))))
Proof
Definitions occuring in Statement : 
context-subset: Gamma, phi
, 
face-type: 𝔽
, 
interval-type: 𝕀
, 
cc-adjoin-cube: (v;u)
, 
cube-context-adjoin: X.A
, 
csm-ap-term: (t)s
, 
cubical-term-at: u(a)
, 
cubical-term: {X ⊢ _:A}
, 
cubical-type-at: A(a)
, 
cubical-type: {X ⊢ _}
, 
subset-trans: subset-trans(I;J;f;x)
, 
subset-iota: iota
, 
cubical-subset: I,psi
, 
face-presheaf: 𝔽
, 
csm-comp: G o F
, 
context-map: <rho>
, 
formal-cube: formal-cube(I)
, 
cube-set-restriction: f(s)
, 
I_cube: A(I)
, 
cubical_set: CubicalSet
, 
nc-e': g,i=j
, 
nc-0: (i0)
, 
nc-s: s
, 
new-name: new-name(I)
, 
add-name: I+i
, 
nh-comp: g ⋅ f
, 
names-hom: I ⟶ J
, 
dM_inc: <x>
, 
fset: fset(T)
, 
nat: ℕ
, 
all: ∀x:A. B[x]
, 
equal: s = t ∈ T
Definitions unfolded in proof : 
all: ∀x:A. B[x]
, 
member: t ∈ T
, 
uall: ∀[x:A]. B[x]
, 
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
, 
guard: {T}
, 
uimplies: b supposing a
, 
squash: ↓T
, 
prop: ℙ
, 
true: True
, 
subset-iota: iota
, 
csm-comp: G o F
, 
csm-ap: (s)x
, 
compose: f o g
, 
context-map: <rho>
, 
functor-arrow: arrow(F)
, 
cube-context-adjoin: X.A
, 
cc-adjoin-cube: (v;u)
, 
pi2: snd(t)
, 
implies: P 
⇒ Q
, 
interval-presheaf: 𝕀
, 
DeMorgan-algebra: DeMorganAlgebra
, 
so_lambda: λ2x.t[x]
, 
and: P ∧ Q
, 
so_apply: x[s]
, 
nat: ℕ
, 
ge: i ≥ j 
, 
decidable: Dec(P)
, 
or: P ∨ Q
, 
not: ¬A
, 
satisfiable_int_formula: satisfiable_int_formula(fmla)
, 
exists: ∃x:A. B[x]
, 
false: False
, 
names: names(I)
, 
iff: P 
⇐⇒ Q
, 
rev_implies: P 
⇐ Q
, 
nc-0: (i0)
, 
sq_type: SQType(T)
, 
name-morph-satisfies: (psi f) = 1
, 
bdd-distributive-lattice: BoundedDistributiveLattice
, 
bounded-lattice-hom: Hom(l1;l2)
, 
lattice-hom: Hom(l1;l2)
Lemmas referenced : 
composition-type-lemma3, 
context-subset-adjoin-subtype, 
interval-type_wf, 
I_cube_wf, 
cubical-subset_wf, 
cubical-term-at_wf, 
face-type_wf, 
cube-set-restriction_wf, 
subtype_rel_self, 
face-presheaf_wf2, 
names-hom_wf, 
fset_wf, 
nat_wf, 
cubical-term_wf, 
cube-context-adjoin_wf, 
context-subset_wf, 
cubical_set_cumulativity-i-j, 
cubical-type-cumulativity2, 
subtype_rel_transitivity, 
cubical-type_wf, 
cubical_set_wf, 
csm-ap-type-at, 
cubical-type-at_wf, 
squash_wf, 
true_wf, 
cc-adjoin-cube-restriction, 
interval-type-ap-morph, 
cc-adjoin-cube_wf, 
istype-cubical-type-at, 
new-name_wf, 
cubical-subset-I_cube, 
cube-set-restriction-comp, 
add-name_wf, 
nc-0_wf, 
nc-s_wf, 
f-subset-add-name, 
interval-type-at, 
I_cube_pair_redex_lemma, 
lattice-point_wf, 
dM_wf, 
subtype_rel_set, 
DeMorgan-algebra-structure_wf, 
lattice-structure_wf, 
lattice-axioms_wf, 
bounded-lattice-structure-subtype, 
DeMorgan-algebra-structure-subtype, 
bounded-lattice-structure_wf, 
bounded-lattice-axioms_wf, 
equal_wf, 
lattice-meet_wf, 
lattice-join_wf, 
DeMorgan-algebra-axioms_wf, 
nat_properties, 
decidable__le, 
full-omega-unsat, 
intformand_wf, 
intformnot_wf, 
intformle_wf, 
itermConstant_wf, 
itermVar_wf, 
istype-int, 
int_formula_prop_and_lemma, 
int_formula_prop_not_lemma, 
int_formula_prop_le_lemma, 
int_term_value_constant_lemma, 
int_term_value_var_lemma, 
int_formula_prop_wf, 
istype-le, 
nh-comp_wf, 
trivial-member-add-name1, 
fset-member_wf, 
int-deq_wf, 
strong-subtype-deq-subtype, 
strong-subtype-set3, 
le_wf, 
strong-subtype-self, 
dM-lift_wf2, 
istype-universe, 
dM-lift-inc, 
iff_weakening_equal, 
nh-comp-sq, 
dM0-sq-empty, 
subtype_base_sq, 
bool_wf, 
bool_subtype_base, 
eq_int_eq_true_intro, 
btrue_wf, 
dM0_wf, 
dM-lift-0, 
csm-ap-type_wf, 
csm-comp_wf, 
formal-cube_wf1, 
subset-iota_wf, 
context-map_wf, 
dM_inc_wf, 
name-morph-satisfies_wf, 
face-type-at, 
face_lattice_wf, 
fl-morph_wf, 
lattice-1_wf, 
cube_set_restriction_pair_lemma, 
nh-id-right, 
fl-morph-comp2, 
nh-comp-assoc, 
s-comp-nc-0
Rules used in proof : 
cut, 
introduction, 
extract_by_obid, 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
lambdaFormation_alt, 
hypothesis, 
sqequalHypSubstitution, 
dependent_functionElimination, 
thin, 
hypothesisEquality, 
isectElimination, 
applyLambdaEquality, 
universeIsType, 
instantiate, 
applyEquality, 
sqequalRule, 
inhabitedIsType, 
cumulativity, 
because_Cache, 
independent_isectElimination, 
equalityTransitivity, 
equalitySymmetry, 
lambdaEquality_alt, 
hyp_replacement, 
Error :memTop, 
imageElimination, 
natural_numberEquality, 
imageMemberEquality, 
baseClosed, 
setElimination, 
rename, 
equalityIstype, 
independent_functionElimination, 
productEquality, 
isectEquality, 
dependent_set_memberEquality_alt, 
unionElimination, 
approximateComputation, 
dependent_pairFormation_alt, 
int_eqEquality, 
independent_pairFormation, 
voidElimination, 
intEquality, 
universeEquality, 
productElimination
Latex:
\mforall{}Gamma:j\mvdash{}.  \mforall{}phi:\{Gamma  \mvdash{}  \_:\mBbbF{}\}.  \mforall{}A:\{Gamma.\mBbbI{}  \mvdash{}  \_\}.  \mforall{}u:\{Gamma,  phi.\mBbbI{}  \mvdash{}  \_:A\}.  \mforall{}I,J:fset(\mBbbN{}).  \mforall{}f:J  {}\mrightarrow{}  I.
\mforall{}rho:Gamma(I).  \mforall{}K:fset(\mBbbN{}).  \mforall{}g:J,phi(f(rho))(K).
    ((u)<(s(f(rho));<new-name(J)>)>  o  iota((new-name(J)0)  \mcdot{}  g)
    =  ((u)<(s(rho);<new-name(I)>)>  o  iota)subset-trans(I+new-name(I);J+new-name(J);
                                                                                                          f,new-name(I)=new-name(J);
                                                                                                          s(phi(rho)))((new-name(J)0)  \mcdot{}  g))
Date html generated:
2020_05_20-PM-04_09_29
Last ObjectModification:
2020_04_11-PM-06_37_25
Theory : cubical!type!theory
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