Nuprl Lemma : extend-face-term-morph
∀[I:fset(ℕ)]. ∀[phi:Point(face_lattice(I))]. ∀[u:{I,phi ⊢ _:𝔽}]. ∀[J:fset(ℕ)]. ∀[f:J ⟶ I].
((extend-face-term(I;phi;u))<f> = extend-face-term(J;(phi)<f>;(u)subset-trans(I;J;f;phi)) ∈ Point(face_lattice(J)))
Proof
Definitions occuring in Statement :
extend-face-term: extend-face-term(I;phi;u),
face-type: 𝔽,
csm-ap-term: (t)s,
cubical-term: {X ⊢ _:A},
subset-trans: subset-trans(I;J;f;x),
cubical-subset: I,psi,
fl-morph: <f>,
face_lattice: face_lattice(I),
names-hom: I ⟶ J,
lattice-point: Point(l),
fset: fset(T),
nat: ℕ,
uall: ∀[x:A]. B[x],
apply: f a,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
bounded-lattice-hom: Hom(l1;l2),
lattice-hom: Hom(l1;l2),
bdd-distributive-lattice: BoundedDistributiveLattice,
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,
I_cube: A(I),
functor-ob: ob(F),
pi1: fst(t),
face-presheaf: 𝔽,
and: P ∧ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
top: Top,
uimplies: b supposing a,
cand: A c∧ B,
cubical-type-at: A(a),
csm-ap-type: (AF)s,
face-type: 𝔽,
constant-cubical-type: (X),
true: True,
squash: ↓T,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
cubical-subset: I,psi,
names-cat: NamesCat,
rep-sub-sheaf: rep-sub-sheaf(C;X;P),
all: ∀x:A. B[x],
cube-set-restriction: f(s),
pi2: snd(t),
fl-morph: <f>,
fl-lift: fl-lift(T;eq;L;eqL;f0;f1),
face-lattice-property,
free-dist-lattice-with-constraints-property,
lattice-extend-wc: lattice-extend-wc(L;eq;eqL;f;ac),
lattice-extend: lattice-extend(L;eq;eqL;f;ac),
lattice-fset-join: \/(s),
reduce: reduce(f;k;as),
list_ind: list_ind,
fset-image: f"(s),
f-union: f-union(domeq;rngeq;s;x.g[x]),
list_accum: list_accum,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
cubical-term-at: u(a),
subset-trans: subset-trans(I;J;f;x),
csm-ap-term: (t)s,
csm-ap: (s)x
Lemmas referenced :
extend-face-term-unique,
fl-morph_wf,
bounded-lattice-hom_wf,
face_lattice_wf,
bdd-distributive-lattice_wf,
csm-ap-term_wf,
cubical-subset_wf,
subtype_rel_self,
fset_wf,
names_wf,
assert_wf,
fset-antichain_wf,
union-deq_wf,
names-deq_wf,
fset-all_wf,
fset-contains-none_wf,
face-lattice-constraints_wf,
face-type_wf,
subset-trans_wf,
csm-face-type,
extend-face-term_wf,
set_wf,
names-hom_wf,
name-morph-satisfies_wf,
nat_wf,
cubical-term_wf,
lattice-point_wf,
subtype_rel_set,
bounded-lattice-structure_wf,
lattice-structure_wf,
lattice-axioms_wf,
bounded-lattice-structure-subtype,
bounded-lattice-axioms_wf,
uall_wf,
equal_wf,
lattice-meet_wf,
lattice-join_wf,
lattice-hom-le,
bdd-distributive-lattice-subtype-bdd-lattice,
extend-face-term-le,
cubical-term-at_wf,
csm-ap-type_wf,
squash_wf,
true_wf,
fl-morph-comp2,
iff_weakening_equal,
I_cube_pair_redex_lemma,
cat_arrow_triple_lemma,
extend-face-term-property,
nh-comp_wf,
name-morph-satisfies-comp,
face-lattice-property,
free-dist-lattice-with-constraints-property
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
hypothesis,
lambdaEquality,
setElimination,
rename,
sqequalRule,
because_Cache,
setEquality,
unionEquality,
productEquality,
isect_memberEquality,
voidElimination,
voidEquality,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
independent_pairFormation,
instantiate,
cumulativity,
universeEquality,
natural_numberEquality,
imageElimination,
imageMemberEquality,
baseClosed,
productElimination,
independent_functionElimination,
dependent_functionElimination,
dependent_set_memberEquality,
hyp_replacement
Latex:
\mforall{}[I:fset(\mBbbN{})]. \mforall{}[phi:Point(face\_lattice(I))]. \mforall{}[u:\{I,phi \mvdash{} \_:\mBbbF{}\}]. \mforall{}[J:fset(\mBbbN{})]. \mforall{}[f:J {}\mrightarrow{} I].
((extend-face-term(I;phi;u))<f> = extend-face-term(J;(phi)<f>(u)subset-trans(I;J;f;phi)))
Date html generated:
2017_10_05-AM-07_34_06
Last ObjectModification:
2017_03_03-AM-00_58_10
Theory : cubical!type!theory
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