Nuprl Lemma : fan-bar-sep
∀[T:Type]. (Fan(T) ⇒ (∃size:ℕ. T ~ ℕsize) ⇒ BarSep(T;T))
Proof
Definitions occuring in Statement :
altbarsep: BarSep(T;S),
altfan: Fan(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 :
remainder: n rem m,
eq_int: (i =z j),
assert: ↑b,
bnot: ¬bb,
bfalse: ff,
ifthenelse: if b then t else f fi ,
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
sq_stable: SqStable(P),
altubar: uniformBar(X),
rev_uimplies: rev_uimplies(P;Q),
uiff: uiff(P;Q),
altjbar: jbar(X;Y),
so_apply: x[s],
nequal: a ≠ b ∈ T ,
int_nzero: ℤ-o,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
less_than: a < b,
nat_plus: ℕ+,
altfan: Fan(T),
top: Top,
satisfiable_int_formula: satisfiable_int_formula(fmla),
lelt: i ≤ j < k,
ge: i ≥ j ,
int_seg: {i..j-},
surject: Surj(A;B;f),
biject: Bij(A;B;f),
equipollent: A ~ B,
false: False,
less_than': less_than'(a;b),
le: A ≤ B,
not: ¬A,
rev_implies: P ⇐ Q,
true: True,
squash: ↓T,
and: P ∧ Q,
iff: P ⇐⇒ Q,
altbar: bar(X),
prop: ℙ,
guard: {T},
sq_type: SQType(T),
uimplies: b supposing a,
or: P ∨ Q,
decidable: Dec(P),
nat: ℕ,
member: t ∈ T,
exists: ∃x:A. B[x],
all: ∀x:A. B[x],
altbarsep: BarSep(T;S),
implies: P ⇒ Q,
uall: ∀[x:A]. B[x]
Lemmas referenced :
rem_invariant,
equal_wf,
less_than_wf,
iff_weakening_uiff,
div-cancel2,
rem-exact,
multiply-is-int-iff,
neg_assert_of_eq_int,
assert-bnot,
bool_subtype_base,
bool_cases_sqequal,
eqff_to_assert,
div_bounds_1,
assert_of_lt_int,
lt_int_wf,
assert_of_eq_int,
eqtt_to_assert,
eq_int_wf,
remainder_wfa,
exp_wf4,
not-all-finite,
decidable__assert,
decidable__exists_int_seg,
le_weakening2,
sq_stable__le,
int_seg_subtype,
assert_wf,
exists_wf,
decidable__all_fun,
ext-eq_weakening,
equipollent_weakening_ext-eq,
function_functionality_wrt_equipollent_right,
equipollent_functionality_wrt_equipollent,
equipollent_inversion,
equipollent_functionality_wrt_equipollent2,
equipollent_same,
exp_wf2,
false_wf,
add-is-int-iff,
add_nat_wf,
equipollent-exp,
subtype_rel_self,
nat_wf,
subtype_rel_function,
istype-assert,
assert-b-exists,
assert_of_bor,
div-cancel3,
int_term_value_add_lemma,
itermAdd_wf,
int_term_value_mul_lemma,
itermMultiply_wf,
int_seg_properties,
rem_bounds_1,
div_rem_sum,
istype-false,
nequal_wf,
divide_wfa,
int_seg_subtype_nat,
divide_wf,
b-exists_wf,
bor_wf,
istype-less_than,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_and_lemma,
intformeq_wf,
itermVar_wf,
intformless_wf,
intformand_wf,
decidable__lt,
int_formula_prop_wf,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
itermConstant_wf,
intformle_wf,
intformnot_wf,
full-omega-unsat,
decidable__le,
nat_properties,
altbar_wf,
istype-le,
istype-void,
iff_weakening_equal,
istype-int,
true_wf,
squash_wf,
equipollent-zero,
istype-universe,
altfan_wf,
equipollent_wf,
istype-nat,
bool_wf,
int_seg_wf,
altjbar_wf,
int_subtype_base,
subtype_base_sq,
decidable__equal_int
Rules used in proof :
hyp_replacement,
functionExtensionality,
equalityElimination,
functionEquality,
baseApply,
pointwiseFunctionality,
applyLambdaEquality,
Error :inrFormation_alt,
addEquality,
multiplyEquality,
sqequalBase,
Error :equalityIstype,
closedConclusion,
promote_hyp,
int_eqEquality,
Error :isect_memberEquality_alt,
Error :dependent_pairFormation_alt,
approximateComputation,
voidElimination,
independent_pairFormation,
Error :dependent_set_memberEquality_alt,
baseClosed,
imageMemberEquality,
equalitySymmetry,
equalityTransitivity,
imageElimination,
Error :lambdaEquality_alt,
applyEquality,
Error :inlFormation_alt,
universeEquality,
Error :productIsType,
sqequalRule,
Error :functionIsType,
Error :inhabitedIsType,
hypothesisEquality,
Error :universeIsType,
independent_functionElimination,
independent_isectElimination,
intEquality,
cumulativity,
isectElimination,
instantiate,
unionElimination,
natural_numberEquality,
hypothesis,
because_Cache,
rename,
setElimination,
dependent_functionElimination,
extract_by_obid,
introduction,
cut,
thin,
productElimination,
sqequalHypSubstitution,
Error :lambdaFormation_alt,
Error :isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[T:Type]. (Fan(T) {}\mRightarrow{} (\mexists{}size:\mBbbN{}. T \msim{} \mBbbN{}size) {}\mRightarrow{} BarSep(T;T))
Date html generated:
2019_06_20-PM-02_46_36
Last ObjectModification:
2019_06_06-PM-01_21_50
Theory : fan-theorem
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