Nuprl Lemma : fl-all-hom_wf
∀[I:fset(ℕ)]. ∀[i:{i:ℕ| ¬i ∈ I} ].
(fl-all-hom(I;i) ∈ {g:Hom(face_lattice(I+i);face_lattice(I))|
(∀x:Point(face_lattice(I)). ((g x) = x ∈ Point(face_lattice(I))))
∧ ((g (i=0)) = 0 ∈ Point(face_lattice(I)))
∧ ((g (i=1)) = 0 ∈ Point(face_lattice(I)))} )
Proof
Definitions occuring in Statement :
fl-all-hom: fl-all-hom(I;i),
fl1: (x=1),
fl0: (x=0),
face_lattice: face_lattice(I),
add-name: I+i,
bounded-lattice-hom: Hom(l1;l2),
lattice-0: 0,
lattice-point: Point(l),
fset-member: a ∈ s,
fset: fset(T),
int-deq: IntDeq,
nat: ℕ,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
not: ¬A,
and: P ∧ Q,
member: t ∈ T,
set: {x:A| B[x]} ,
apply: f a,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
squash: ↓T,
and: P ∧ Q,
cand: A c∧ B,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
bdd-distributive-lattice: BoundedDistributiveLattice,
so_lambda: λ2x.t[x],
prop: ℙ,
so_apply: x[s],
uimplies: b supposing a,
bounded-lattice-hom: Hom(l1;l2),
lattice-hom: Hom(l1;l2),
names: names(I),
nat: ℕ,
guard: {T},
implies: P ⇒ Q,
not: ¬A,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
true: True
Lemmas referenced :
fl-all-hom_wf1,
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,
uall_wf,
equal_wf,
lattice-meet_wf,
lattice-join_wf,
all_wf,
face_lattice-point-subtype,
add-name_wf,
f-subset-add-name,
fl0_wf,
trivial-member-add-name1,
fset-member_wf,
nat_wf,
int-deq_wf,
fl1_wf,
set_wf,
not_wf,
strong-subtype-deq-subtype,
strong-subtype-set3,
le_wf,
strong-subtype-self,
fset_wf,
face-lattice-hom-is-id,
face_lattice_hom_subtype,
squash_wf,
true_wf,
deq_wf,
nat_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
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,
names_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
setElimination,
rename,
applyLambdaEquality,
sqequalRule,
imageMemberEquality,
baseClosed,
imageElimination,
dependent_set_memberEquality,
equalityTransitivity,
equalitySymmetry,
productElimination,
lambdaFormation,
applyEquality,
instantiate,
lambdaEquality,
productEquality,
cumulativity,
universeEquality,
because_Cache,
independent_isectElimination,
independent_pairFormation,
dependent_functionElimination,
intEquality,
natural_numberEquality,
independent_functionElimination,
addLevel,
hyp_replacement,
unionElimination,
dependent_pairFormation,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
levelHypothesis
Latex:
\mforall{}[I:fset(\mBbbN{})]. \mforall{}[i:\{i:\mBbbN{}| \mneg{}i \mmember{} I\} ].
(fl-all-hom(I;i) \mmember{} \{g:Hom(face\_lattice(I+i);face\_lattice(I))|
(\mforall{}x:Point(face\_lattice(I)). ((g x) = x)) \mwedge{} ((g (i=0)) = 0) \mwedge{} ((g (i=1)) = 0)\} \000C)
Date html generated:
2017_10_05-AM-01_15_41
Last ObjectModification:
2017_07_28-AM-09_32_11
Theory : cubical!type!theory
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