Nuprl Lemma : fl_all_com
∀[I:fset(ℕ)]. ∀[i,j:ℕ]. ∀[phi:Point(face_lattice(I+i+j))]. ((∀i.(∀j.phi)) = (∀j.(∀i.phi)) ∈ Point(face_lattice(I)))
Proof
Definitions occuring in Statement :
fl_all: (∀i.phi),
face_lattice: face_lattice(I),
add-name: I+i,
lattice-point: Point(l),
fset: fset(T),
nat: ℕ,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
and: P ∧ Q,
cand: A c∧ B,
all: ∀x:A. B[x],
compose: f o g,
fl_all: (∀i.phi),
subtype_rel: A ⊆r B,
bdd-distributive-lattice: BoundedDistributiveLattice,
so_lambda: λ2x.t[x],
prop: ℙ,
so_apply: x[s],
implies: P ⇒ Q,
names: names(I),
nat: ℕ,
bounded-lattice-hom: Hom(l1;l2),
lattice-hom: Hom(l1;l2),
squash: ↓T,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
top: Top,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
false: False,
label: ...$L... t,
nequal: a ≠ b ∈ T ,
ge: i ≥ j ,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
sq_stable: SqStable(P),
fl-all-hom: fl-all-hom(I;i),
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
Lemmas referenced :
face_lattice-hom-equal,
add-name_wf,
names_wf,
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,
nat_wf,
fset_wf,
compose-bounded-lattice-hom,
bdd-distributive-lattice-subtype-bdd-lattice,
fl-all-hom_wf1,
bounded-lattice-hom_wf,
all_wf,
not_wf,
fl0_wf,
names-subtype,
f-subset-add-name,
fl1_wf,
trivial-member-add-name1,
fset-member_wf,
int-deq_wf,
strong-subtype-deq-subtype,
strong-subtype-set3,
le_wf,
strong-subtype-self,
lattice-0_wf,
subtype_rel-equal,
squash_wf,
true_wf,
add-name-com,
bdd-distributive-lattice_wf,
iff_weakening_equal,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
fl_all-0,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
fl_all-fl0,
deq_wf,
nat_properties,
decidable__equal_int,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformeq_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
sq_stable__fset-member,
decidable__le,
intformle_wf,
itermConstant_wf,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
not-added-name,
fl_all_wf,
fl_all-fl1,
face-lattice-property,
free-dist-lattice-with-constraints-property
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
independent_isectElimination,
lambdaFormation,
sqequalRule,
independent_pairFormation,
because_Cache,
applyEquality,
instantiate,
lambdaEquality,
productEquality,
cumulativity,
universeEquality,
isect_memberFormation,
isect_memberEquality,
axiomEquality,
setElimination,
rename,
setEquality,
functionEquality,
intEquality,
dependent_functionElimination,
dependent_set_memberEquality,
natural_numberEquality,
equalityTransitivity,
equalitySymmetry,
imageElimination,
imageMemberEquality,
baseClosed,
productElimination,
independent_functionElimination,
unionElimination,
equalityElimination,
voidElimination,
voidEquality,
dependent_pairFormation,
promote_hyp,
int_eqEquality,
computeAll,
hyp_replacement,
applyLambdaEquality
Latex:
\mforall{}[I:fset(\mBbbN{})]. \mforall{}[i,j:\mBbbN{}]. \mforall{}[phi:Point(face\_lattice(I+i+j))]. ((\mforall{}i.(\mforall{}j.phi)) = (\mforall{}j.(\mforall{}i.phi)))
Date html generated:
2017_10_05-AM-01_16_30
Last ObjectModification:
2017_07_28-AM-09_32_40
Theory : cubical!type!theory
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