Nuprl Lemma : composition-type-lemma3
∀Gamma:j⊢. ∀phi:{Gamma ⊢ _:𝔽}. ∀A:{Gamma.𝕀 ⊢ _}. ∀u:{Gamma, phi.𝕀 ⊢ _:A}. ∀I,J:fset(ℕ). ∀f:J ⟶ I. ∀a:Gamma(I).
((u)<(s(f(a));<new-name(J)>)> o iota
= ((u)<(s(a);<new-name(I)>)> o iota)subset-trans(I+new-name(I);J+new-name(J);f,new-name(I)=new-name(J);s(phi(a)))
∈ {J+new-name(J),s(phi(f(a))) ⊢ _:(A)<(s(f(a));<new-name(J)>)> o iota})
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},
csm-ap-type: (AF)s,
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-s: s,
new-name: new-name(I),
add-name: I+i,
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],
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
interval-presheaf: 𝕀,
names: names(I),
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
squash: ↓T,
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,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
context-map: <rho>,
subset-iota: iota,
csm-comp: G o F,
compose: f o g,
cube-context-adjoin: X.A,
cc-adjoin-cube: (v;u),
subset-trans: subset-trans(I;J;f;x),
pi2: snd(t),
bdd-distributive-lattice: BoundedDistributiveLattice,
uiff: uiff(P;Q),
name-morph-satisfies: (psi f) = 1,
bounded-lattice-hom: Hom(l1;l2),
lattice-hom: Hom(l1;l2),
context-subset: Gamma, phi,
DeMorgan-algebra: DeMorganAlgebra,
nc-e': g,i=j,
sq_type: SQType(T),
names-hom: I ⟶ J,
cube_set_map: A ⟶ B,
psc_map: A ⟶ B,
nat-trans: nat-trans(C;D;F;G),
cat-ob: cat-ob(C),
op-cat: op-cat(C),
spreadn: spread4,
cube-cat: CubeCat,
fset: fset(T),
quotient: x,y:A//B[x; y],
cat-arrow: cat-arrow(C),
type-cat: TypeCat,
cat-comp: cat-comp(C)
Lemmas referenced :
context-subset-adjoin-subtype,
interval-type_wf,
cc-adjoin-cube_wf,
add-name_wf,
new-name_wf,
cube-set-restriction_wf,
nc-s_wf,
f-subset-add-name,
interval-type-at,
I_cube_pair_redex_lemma,
dM_inc_wf,
trivial-member-add-name1,
fset-member_wf,
nat_wf,
int-deq_wf,
strong-subtype-deq-subtype,
strong-subtype-set3,
le_wf,
istype-int,
strong-subtype-self,
I_cube_wf,
equal_wf,
face-presheaf_wf2,
cubical-term-at_wf,
face-type_wf,
subtype_rel_self,
fl-morph-restriction,
nc-e'_wf,
iff_weakening_equal,
nc-e'-lemma3,
face-term-at-restriction,
nh-comp_wf,
cube-set-restriction-comp,
cubical-subset_wf,
squash_wf,
true_wf,
context-map-lemma2,
names-hom_wf,
fset_wf,
istype-cubical-term,
cube-context-adjoin_wf,
context-subset_wf,
cubical_set_cumulativity-i-j,
thin-context-subset-adjoin,
cubical-type-cumulativity2,
cubical-type_wf,
cubical_set_wf,
csm-equal,
cubical-subset-I_cube,
arrow_pair_lemma,
istype-cubical-type-at,
name-morph-satisfies-comp,
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,
lattice-meet_wf,
lattice-join_wf,
fl-morph_wf,
nh-comp-assoc,
istype-universe,
face-type-ap-morph,
cubical-term-at-morph,
fl-morph-comp2,
interval-type-ap-morph,
member_wf,
dM_wf,
DeMorgan-algebra-structure_wf,
DeMorgan-algebra-structure-subtype,
subtype_rel_transitivity,
DeMorgan-algebra-axioms_wf,
dM-lift-inc,
nh-comp-sq,
subtype_base_sq,
bool_wf,
bool_subtype_base,
eq_int_eq_true_intro,
btrue_wf,
csm-ap-comp-term,
cube_set_map_wf,
subset-trans_wf,
cubical-term_wf,
csm-ap-type_wf,
equal_functionality_wrt_subtype_rel2,
csm-ap-term_wf,
subtype_rel_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
hypothesisEquality,
hypothesis,
isect_memberFormation_alt,
dependent_functionElimination,
applyEquality,
sqequalRule,
independent_isectElimination,
Error :memTop,
dependent_set_memberEquality_alt,
lambdaEquality_alt,
setElimination,
rename,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
universeIsType,
intEquality,
natural_numberEquality,
axiomEquality,
instantiate,
imageElimination,
imageMemberEquality,
baseClosed,
productElimination,
independent_functionElimination,
functionExtensionality,
dependent_pairEquality_alt,
productEquality,
cumulativity,
isectEquality,
universeEquality,
equalityIstype,
hyp_replacement,
independent_pairFormation,
productIsType,
applyLambdaEquality
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{}a:Gamma(I).
((u)<(s(f(a));<new-name(J)>)> o iota
= ((u)<(s(a);<new-name(I)>)> o iota)subset-trans(I+new-name(I);J+new-name(J);
f,new-name(I)=new-name(J);s(phi(a))))
Date html generated:
2020_05_20-PM-04_09_10
Last ObjectModification:
2020_04_17-PM-03_54_03
Theory : cubical!type!theory
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