Nuprl Lemma : fl-all-hom_wf1
∀[I:fset(ℕ)]. ∀[i:ℕ].
(fl-all-hom(I;i) ∈ {g:Hom(face_lattice(I+i);face_lattice(I))|
(∀j:names(I)
((¬(j = i ∈ ℤ))
⇒ (((g (j=0)) = (j=0) ∈ Point(face_lattice(I)))
∧ ((g (j=1)) = (j=1) ∈ 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,
names: names(I),
bounded-lattice-hom: Hom(l1;l2),
lattice-0: 0,
lattice-point: Point(l),
fset: fset(T),
nat: ℕ,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q,
member: t ∈ T,
set: {x:A| B[x]} ,
apply: f a,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
fl-all-hom: fl-all-hom(I;i),
subtype_rel: A ⊆r B,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
true: True,
so_apply: x[s],
so_lambda: λ2x.t[x],
squash: ↓T,
false: False,
assert: ↑b,
bnot: ¬bb,
guard: {T},
sq_type: SQType(T),
or: P ∨ Q,
prop: ℙ,
exists: ∃x:A. B[x],
bfalse: ff,
bdd-distributive-lattice: BoundedDistributiveLattice,
uimplies: b supposing a,
and: P ∧ Q,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
implies: P ⇒ Q,
all: ∀x:A. B[x],
nat: ℕ,
names: names(I),
bounded-lattice-hom: Hom(l1;l2),
lattice-hom: Hom(l1;l2),
respects-equality: respects-equality(S;T),
fl1: (x=1),
fl0: (x=0),
face_lattice: face_lattice(I),
cand: A c∧ B,
ge: i ≥ j ,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top
Lemmas referenced :
istype-nat,
fset_wf,
nat_wf,
FL-meet-0-1,
iff_weakening_equal,
bdd-distributive-lattice-subtype-lattice,
lattice-meet-idempotent,
lattice-join_wf,
lattice-meet_wf,
uall_wf,
bounded-lattice-axioms_wf,
bounded-lattice-structure-subtype,
lattice-axioms_wf,
lattice-structure_wf,
bounded-lattice-structure_wf,
subtype_rel_set,
lattice-point_wf,
true_wf,
squash_wf,
fl1_wf,
not-added-name,
fl0_wf,
neg_assert_of_eq_int,
assert-bnot,
bool_subtype_base,
subtype_base_sq,
bool_cases_sqequal,
equal_wf,
eqff_to_assert,
bdd-distributive-lattice_wf,
lattice-0_wf,
assert_of_eq_int,
eqtt_to_assert,
bool_wf,
eq_int_wf,
face_lattice-deq_wf,
face_lattice_wf,
names-deq_wf,
add-name_wf,
names_wf,
fl-lift_wf,
bounded-lattice-hom_wf,
face-lattice_wf,
face-lattice0_wf,
ifthenelse_wf,
face_lattice-point-subtype,
f-subset-add-name,
respects-equality-face-lattice-point,
face-lattice1_wf,
trivial-member-add-name1,
fset-member_wf,
int-deq_wf,
names-subtype,
nat_properties,
full-omega-unsat,
intformand_wf,
intformeq_wf,
itermVar_wf,
intformnot_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_formula_prop_not_lemma,
int_formula_prop_wf,
set_subtype_base,
le_wf,
int_subtype_base,
assert_wf,
bnot_wf,
not_wf,
equal-wf-base,
eq_int_eq_true,
btrue_wf,
subtype_rel_self,
bfalse_wf,
assert_elim,
btrue_neq_bfalse,
istype-assert,
bool_cases,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
applyEquality,
hypothesis,
sqequalHypSubstitution,
sqequalRule,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
extract_by_obid,
isect_memberEquality_alt,
isectElimination,
thin,
hypothesisEquality,
isectIsTypeImplies,
inhabitedIsType,
universeIsType,
baseClosed,
imageMemberEquality,
natural_numberEquality,
productEquality,
universeEquality,
imageElimination,
voidElimination,
independent_functionElimination,
cumulativity,
instantiate,
dependent_functionElimination,
promote_hyp,
dependent_pairFormation,
independent_isectElimination,
productElimination,
equalityElimination,
unionElimination,
lambdaFormation,
rename,
setElimination,
lambdaEquality,
because_Cache,
lambdaEquality_alt,
setIsType,
functionIsType,
productIsType,
equalityIstype,
isectEquality,
dependent_set_memberEquality_alt,
lambdaFormation_alt,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
independent_pairFormation,
intEquality,
sqequalBase,
baseApply,
closedConclusion,
applyLambdaEquality
Latex:
\mforall{}[I:fset(\mBbbN{})]. \mforall{}[i:\mBbbN{}].
(fl-all-hom(I;i) \mmember{} \{g:Hom(face\_lattice(I+i);face\_lattice(I))|
(\mforall{}j:names(I). ((\mneg{}(j = i)) {}\mRightarrow{} (((g (j=0)) = (j=0)) \mwedge{} ((g (j=1)) = (j=1)))))
\mwedge{} ((g (i=0)) = 0)
\mwedge{} ((g (i=1)) = 0)\} )
Date html generated:
2019_11_04-PM-05_34_23
Last ObjectModification:
2018_12_13-PM-00_38_29
Theory : cubical!type!theory
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