Nuprl Lemma : fan-implies-bar-sep
∀[T:Type]. (Fan_d(T) ⇒ (∃size:ℕ. T ~ ℕsize) ⇒ BarSep(T;T))
Proof
Definitions occuring in Statement :
bar-separation: BarSep(T;S),
dfan: Fan_d(T),
equipollent: A ~ B,
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
natural_number: $n,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
bar-separation: BarSep(T;S),
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
member: t ∈ T,
nat: ℕ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
sq_type: SQType(T),
guard: {T},
prop: ℙ,
tbar: tbar(T;X),
iff: P ⇐⇒ Q,
and: P ∧ Q,
squash: ↓T,
true: True,
rev_implies: P ⇐ Q,
not: ¬A,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
false: False,
equipollent: A ~ B,
biject: Bij(A;B;f),
surject: Surj(A;B;f),
int_seg: {i..j-},
lelt: i ≤ j < k,
dfan: Fan_d(T),
so_lambda: λ2x.t[x],
so_apply: x[s],
ubar: ubar(T;X),
dbar: dbar(T;X),
dec-predicate: Decidable(X),
cand: A c∧ B,
jbar: jbar(T;S;X;Y),
subtype_rel: A ⊆r B,
le: A ≤ B,
less_than': less_than'(a;b),
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
nat_plus: ℕ+,
compose: f o g,
pi1: fst(t),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
bnot: ¬bb,
assert: ↑b,
pi2: snd(t),
sq_stable: SqStable(P),
less_than: a < b,
int_iseg: {i...j},
iseg: l1 ≤ l2,
select: L[n]
Lemmas referenced :
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
jbar_wf,
dec-predicate_wf,
list_wf,
istype-nat,
equipollent_wf,
int_seg_wf,
dfan_wf,
istype-universe,
equipollent-zero,
squash_wf,
true_wf,
istype-int,
iff_weakening_equal,
nat_properties,
decidable__le,
full-omega-unsat,
intformnot_wf,
intformle_wf,
itermConstant_wf,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
istype-le,
tbar_wf,
decidable__lt,
intformand_wf,
intformless_wf,
itermVar_wf,
intformeq_wf,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
istype-less_than,
length_wf,
unshuffle_wf,
firstn_wf,
map_wf,
pi1_wf,
pi2_wf,
decidable__or,
decidable__exists_int_seg,
itermMultiply_wf,
int_term_value_mul_lemma,
itermAdd_wf,
int_term_value_add_lemma,
length-unshuffle,
upto_wf,
map-length,
length_upto,
subtype_rel_function,
nat_wf,
int_seg_subtype_nat,
istype-false,
subtype_rel_self,
nequal_wf,
less_than_wf,
div_mul_cancel,
divide_wfa,
mul_com,
subtype_rel_list,
top_wf,
list_subtype_base,
set_subtype_base,
lelt_wf,
firstn_upto,
le_int_wf,
eqtt_to_assert,
assert_of_le_int,
int_seg_subtype,
eqff_to_assert,
bool_cases_sqequal,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
le_wf,
firstn_map,
unshuffle-map,
map-map,
sq_stable__le,
div_rem_sum,
rem_bounds_1,
length_wf_nat,
lt_int_wf,
assert_of_lt_int,
select_wf,
list_extensionality,
int_seg_properties,
length_firstn,
subtype_rel_sets_simple,
select-map,
select-upto,
select-firstn,
unshuffle-iseg,
firstn-iseg,
iseg_length,
firstn_append,
equal_wf,
decidable__all_length,
shuffle_wf,
length-shuffle,
unshuffle-shuffle,
eta_conv,
bnot_wf,
not_wf,
istype-assert,
bool_cases,
iff_transitivity,
assert_of_bnot,
decidable__exists_length,
decidable__all_int_seg,
decidable__not,
map_length_nat,
select-unshuffle,
not_over_exists,
equal-wf-base
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
Error :lambdaFormation_alt,
sqequalHypSubstitution,
productElimination,
thin,
cut,
introduction,
extract_by_obid,
dependent_functionElimination,
setElimination,
rename,
because_Cache,
hypothesis,
natural_numberEquality,
unionElimination,
instantiate,
isectElimination,
cumulativity,
intEquality,
independent_isectElimination,
independent_functionElimination,
Error :universeIsType,
hypothesisEquality,
Error :inhabitedIsType,
Error :functionIsType,
universeEquality,
sqequalRule,
Error :productIsType,
Error :inlFormation_alt,
applyEquality,
Error :lambdaEquality_alt,
imageElimination,
equalityTransitivity,
equalitySymmetry,
imageMemberEquality,
baseClosed,
Error :dependent_set_memberEquality_alt,
approximateComputation,
Error :dependent_pairFormation_alt,
Error :isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
int_eqEquality,
promote_hyp,
unionEquality,
productEquality,
multiplyEquality,
addEquality,
Error :unionIsType,
Error :equalityIstype,
sqequalBase,
closedConclusion,
functionExtensionality,
equalityElimination,
Error :inrFormation_alt,
Error :setIsType,
hyp_replacement,
applyLambdaEquality,
independent_pairEquality,
functionEquality,
baseApply
Latex:
\mforall{}[T:Type]. (Fan\_d(T) {}\mRightarrow{} (\mexists{}size:\mBbbN{}. T \msim{} \mBbbN{}size) {}\mRightarrow{} BarSep(T;T))
Date html generated:
2019_06_20-PM-02_47_21
Last ObjectModification:
2019_03_06-AM-11_06_01
Theory : fan-theorem
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