Nuprl Lemma : equipollent-nat-rationals
(our proof is constructive so we can compute the listing of rationals
see first-25-rationals)⋅
ℕ ~ ℚ
This theorem is one of freek's list of 100 theorems
Proof
Definitions occuring in Statement :
rationals: ℚ,
equipollent: A ~ B,
nat: ℕ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
prop: ℙ,
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
cand: A c∧ B,
subtract: n - m,
pi2: snd(t),
cons: [a / b],
select: L[n],
le: A ≤ B,
lelt: i ≤ j < k,
int_seg: {i..j-},
l_exists: (∃x∈L. P[x]),
subtype_rel: A ⊆r B,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
uimplies: b supposing a,
uiff: uiff(P;Q),
false: False,
or: P ∨ Q,
decidable: Dec(P),
guard: {T},
true: True,
less_than': less_than'(a;b),
squash: ↓T,
less_than: a < b,
nat_plus: ℕ+,
top: Top,
is-qrep: is-qrep(p),
has-value: (a)↓,
nat: ℕ,
bfalse: ff,
btrue: tt,
ifthenelse: if b then t else f fi ,
sq_type: SQType(T),
l_member: (x ∈ l),
ge: i ≥ j ,
int_upper: {i...},
equipollent: A ~ B,
surject: Surj(A;B;f),
inject: Inj(A;B;f),
biject: Bij(A;B;f)
Lemmas referenced :
equipollent_functionality_wrt_equipollent2,
nat_wf,
rationals_wf,
nat_plus_wf,
assert_wf,
is-qrep_wf,
equipollent-rationals-ext,
equipollent-nat-subset-ext,
decidable__assert,
list_wf,
l_member_wf,
not_wf,
le-add-cancel,
add-zero,
add-associates,
add_functionality_wrt_le,
add-commutes,
minus-one-mul-top,
zero-add,
minus-one-mul,
minus-add,
condition-implies-le,
not-lt-2,
imax-list_wf,
int_formula_prop_le_lemma,
intformle_wf,
decidable__le,
int_seg_properties,
select_wf,
le_wf,
length_wf,
lelt_wf,
pi2_wf,
map_wf,
cons_wf,
imax-list-ub,
equal_wf,
false_wf,
int_formula_prop_wf,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
intformeq_wf,
itermAdd_wf,
itermVar_wf,
itermConstant_wf,
intformless_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
add-is-int-iff,
decidable__lt,
nat_plus_properties,
less_than_wf,
length_wf_nat,
add_nat_plus,
length-map,
length_of_cons_lemma,
better-gcd-gcd,
assert_of_eq_int,
assert_of_bor,
iff_weakening_uiff,
iff_transitivity,
int_subtype_base,
equal-wf-base,
or_wf,
eq_int_wf,
bor_wf,
gcd_wf,
int-value-type,
value-type-has-value,
assoced_weakening,
absval_assoced,
assoced_functionality_wrt_assoced,
one_divs_any,
divides-iff-gcd-assoced,
absval_wf,
assoced_nelim,
assert_of_bnot,
eqff_to_assert,
assert_of_lt_int,
eqtt_to_assert,
bool_subtype_base,
bool_wf,
subtype_base_sq,
bool_cases,
int_term_value_minus_lemma,
itermMinus_wf,
minus-is-int-iff,
bnot_wf,
decidable__equal_int,
lt_int_wf,
absval_ifthenelse,
and_wf,
nat_properties,
select_member,
member_map,
l_exists_iff,
equal-wf-T-base,
cons_member,
ext-eq_weakening,
equipollent_weakening_ext-eq,
equipollent-nat-squared,
equipollent_functionality_wrt_equipollent,
equipollent-int-nat,
product_functionality_wrt_equipollent_left,
int_upper_wf,
biject_wf,
subtype_rel_sets,
int_upper_properties,
product_functionality_wrt_equipollent_right,
equipollent-int_upper-nat,
equipollent-product-com
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isectElimination,
thin,
hypothesis,
setEquality,
productEquality,
intEquality,
dependent_functionElimination,
productElimination,
independent_pairEquality,
hypothesisEquality,
independent_functionElimination,
sqequalRule,
lambdaEquality,
lambdaFormation,
because_Cache,
minusEquality,
imageElimination,
addEquality,
applyEquality,
int_eqEquality,
dependent_pairFormation,
approximateComputation,
independent_isectElimination,
closedConclusion,
baseApply,
promote_hyp,
pointwiseFunctionality,
unionElimination,
rename,
setElimination,
applyLambdaEquality,
equalitySymmetry,
equalityTransitivity,
baseClosed,
imageMemberEquality,
independent_pairFormation,
natural_numberEquality,
dependent_set_memberEquality,
voidEquality,
voidElimination,
isect_memberEquality,
orFunctionality,
callbyvalueReduce,
impliesFunctionality,
cumulativity,
instantiate,
inrFormation,
inlFormation,
functionExtensionality
Latex:
\mBbbN{} \msim{} \mBbbQ{}
Date html generated:
2018_05_21-PM-11_49_10
Last ObjectModification:
2017_08_09-PM-06_59_20
Theory : rationals
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