Nuprl Lemma : name-morph-satisfies-comp
∀[I,J,K:fset(ℕ)]. ∀[psi:Point(face_lattice(I))]. ∀[f:J ⟶ I]. ∀[g:K ⟶ J]. uiff((f(psi) g) = 1;(psi f ⋅ g) = 1)
Proof
Definitions occuring in Statement :
name-morph-satisfies: (psi f) = 1,
face-presheaf: 𝔽,
face_lattice: face_lattice(I),
cube-set-restriction: f(s),
nh-comp: g ⋅ f,
names-hom: I ⟶ J,
lattice-point: Point(l),
fset: fset(T),
nat: ℕ,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
name-morph-satisfies: (psi f) = 1,
prop: ℙ,
squash: ↓T,
subtype_rel: A ⊆r B,
bounded-lattice-hom: Hom(l1;l2),
lattice-hom: Hom(l1;l2),
bdd-distributive-lattice: BoundedDistributiveLattice,
I_cube: A(I),
functor-ob: ob(F),
pi1: fst(t),
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,
so_lambda: λ2x.t[x],
so_apply: x[s],
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
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,
cube-set-restriction: f(s),
pi2: snd(t)
Lemmas referenced :
equal_wf,
squash_wf,
true_wf,
lattice-point_wf,
face_lattice_wf,
fl-morph_wf,
nh-comp_wf,
bounded-lattice-hom_wf,
bdd-distributive-lattice_wf,
fl-morph-restriction,
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,
iff_weakening_equal,
cube-set-restriction_wf,
face-presheaf_wf,
name-morph-satisfies_wf,
names-hom_wf,
subtype_rel_set,
bounded-lattice-structure_wf,
lattice-structure_wf,
lattice-axioms_wf,
bounded-lattice-structure-subtype,
bounded-lattice-axioms_wf,
uall_wf,
lattice-meet_wf,
lattice-join_wf,
nat_wf,
face-lattice-property,
free-dist-lattice-with-constraints-property
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
sqequalHypSubstitution,
hypothesis,
hyp_replacement,
thin,
equalitySymmetry,
sqequalRule,
applyEquality,
lambdaEquality,
imageElimination,
extract_by_obid,
isectElimination,
hypothesisEquality,
equalityTransitivity,
universeEquality,
because_Cache,
setElimination,
rename,
setEquality,
unionEquality,
productEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination,
productElimination,
independent_functionElimination,
axiomEquality,
independent_pairEquality,
isect_memberEquality,
instantiate,
cumulativity
Latex:
\mforall{}[I,J,K:fset(\mBbbN{})]. \mforall{}[psi:Point(face\_lattice(I))]. \mforall{}[f:J {}\mrightarrow{} I]. \mforall{}[g:K {}\mrightarrow{} J].
uiff((f(psi) g) = 1;(psi f \mcdot{} g) = 1)
Date html generated:
2017_10_05-AM-01_17_26
Last ObjectModification:
2017_07_28-AM-09_33_06
Theory : cubical!type!theory
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