Nuprl Lemma : context-map-lemma2
∀[Gamma:j⊢]. ∀[phi:{Gamma ⊢ _:𝔽}]. ∀[I:fset(ℕ)]. ∀[rho:Gamma(I)].
(<(s(rho);<new-name(I)>)> ∈ I+new-name(I),s(phi(rho)) j⟶ Gamma, phi.𝕀)
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,
cubical-term-at: u(a),
cubical-term: {X ⊢ _:A},
cubical-subset: I,psi,
face-presheaf: 𝔽,
context-map: <rho>,
cube_set_map: A ⟶ B,
cube-set-restriction: f(s),
I_cube: A(I),
cubical_set: CubicalSet,
nc-s: s,
new-name: new-name(I),
add-name: I+i,
dM_inc: <x>,
fset: fset(T),
nat: ℕ,
uall: ∀[x:A]. B[x],
member: t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
implies: P ⇒ Q,
uimplies: b supposing 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,
cubical-term: {X ⊢ _:A},
interval-presheaf: 𝕀,
names-hom: I ⟶ J,
names: names(I),
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
compose: f o g,
DeMorgan-algebra: DeMorganAlgebra,
and: P ∧ Q,
guard: {T},
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),
pi2: snd(t),
type-cat: TypeCat,
cat-comp: cat-comp(C),
cc-adjoin-cube: (v;u),
cube-context-adjoin: X.A,
context-map: <rho>,
bdd-distributive-lattice: BoundedDistributiveLattice,
uiff: uiff(P;Q),
csm-adjoin: (s;u),
functor-arrow: arrow(F),
csm-ap: (s)x,
true: True,
name-morph-satisfies: (psi f) = 1,
squash: ↓T,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
context-subset: Gamma, phi,
cube-set-restriction: f(s)
Lemmas referenced :
I_cube_wf,
fset_wf,
nat_wf,
istype-cubical-term,
face-type_wf,
cubical_set_wf,
new-name_wf,
context-map-lemma1,
csm-adjoin_wf,
context-subset_wf,
cubical-subset_wf,
add-name_wf,
cube-set-restriction_wf,
face-presheaf_wf2,
nc-s_wf,
f-subset-add-name,
cubical-term-at_wf,
subtype_rel_self,
interval-type_wf,
cubical-subset-I_cube,
I_cube_pair_redex_lemma,
csm-ap-type-at,
interval-type-at,
trivial-member-add-name1,
fset-member_wf,
int-deq_wf,
strong-subtype-deq-subtype,
strong-subtype-set3,
le_wf,
istype-int,
strong-subtype-self,
names-hom_wf,
csm-interval-type,
interval-type-ap-morph,
cubical-subset-restriction,
nh-comp-sq,
dM-lift_wf2,
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,
subtype_rel_transitivity,
bounded-lattice-structure_wf,
bounded-lattice-axioms_wf,
equal_wf,
lattice-meet_wf,
lattice-join_wf,
DeMorgan-algebra-axioms_wf,
istype-cubical-type-at,
csm-ap-type_wf,
cube_set_map_wf,
cubical_set_cumulativity-i-j,
cubical-type-ap-morph_wf,
csm-equal,
cube-context-adjoin_wf,
arrow_pair_lemma,
name-morph-satisfies-comp,
face_lattice_wf,
lattice-1_wf,
cubical-term-at-morph,
nh-comp_wf,
iff_weakening_equal,
face-type-ap-morph,
squash_wf,
true_wf,
istype-universe,
cubical-type-cumulativity2,
cubical-type_wf,
cube-set-restriction-comp,
dM_inc_wf,
dM-lift-inc
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
sqequalHypSubstitution,
hypothesis,
universeIsType,
introduction,
extract_by_obid,
isectElimination,
thin,
hypothesisEquality,
instantiate,
inhabitedIsType,
lambdaFormation_alt,
setElimination,
rename,
equalityIstype,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
independent_functionElimination,
because_Cache,
independent_isectElimination,
applyEquality,
sqequalRule,
dependent_set_memberEquality_alt,
lambdaEquality_alt,
Error :memTop,
intEquality,
natural_numberEquality,
productEquality,
cumulativity,
isectEquality,
hyp_replacement,
functionIsType,
functionExtensionality,
productElimination,
imageElimination,
imageMemberEquality,
baseClosed,
universeEquality,
dependent_pairEquality_alt
Latex:
\mforall{}[Gamma:j\mvdash{}]. \mforall{}[phi:\{Gamma \mvdash{} \_:\mBbbF{}\}]. \mforall{}[I:fset(\mBbbN{})]. \mforall{}[rho:Gamma(I)].
(<(s(rho);<new-name(I)>)> \mmember{} I+new-name(I),s(phi(rho)) j{}\mrightarrow{} Gamma, phi.\mBbbI{})
Date html generated:
2020_05_20-PM-04_07_54
Last ObjectModification:
2020_04_17-PM-02_41_01
Theory : cubical!type!theory
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