Nuprl Lemma : equipollent-sum
∀n:ℕ. ∀f:ℕn ⟶ ℕ. i:ℕn × ℕf[i] ~ ℕΣ(f[i] | i < n)
Proof
Definitions occuring in Statement :
equipollent: A ~ B,
sum: Σ(f[x] | x < k),
int_seg: {i..j-},
nat: ℕ,
so_apply: x[s],
all: ∀x:A. B[x],
function: x:A ⟶ B[x],
product: x:A × B[x],
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
nat: ℕ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
and: P ∧ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
sum: Σ(f[x] | x < k),
sum_aux: sum_aux(k;v;i;x.f[x]),
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
le: A ≤ B,
less_than': less_than'(a;b),
uiff: uiff(P;Q),
subtract: n - m,
true: True,
equipollent: A ~ B,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
sq_type: SQType(T),
less_than: a < b,
squash: ↓T,
bfalse: ff,
bnot: ¬bb,
assert: ↑b,
nequal: a ≠ b ∈ T ,
biject: Bij(A;B;f),
inject: Inj(A;B;f),
surject: Surj(A;B;f),
cand: A c∧ B,
outr: outr(x),
istype: istype(T),
isl: isl(x),
ge: i ≥ j
Lemmas referenced :
int_seg_wf,
subtract_wf,
istype-nat,
equipollent_wf,
sum_wf,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermSubtract_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_subtract_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
istype-le,
istype-less_than,
primrec-wf2,
all_wf,
nat_wf,
int_seg_properties,
satisfiable-full-omega-tt,
equipollent-zero,
subtype_rel_function,
int_seg_subtype,
istype-false,
not-le-2,
condition-implies-le,
add-associates,
minus-add,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-mul-special,
zero-mul,
add-zero,
add-commutes,
le-add-cancel2,
subtype_rel_self,
eq_int_wf,
eqtt_to_assert,
assert_of_eq_int,
subtype_base_sq,
int_subtype_base,
decidable__lt,
intformeq_wf,
int_formula_prop_eq_lemma,
eqff_to_assert,
bool_subtype_base,
bool_cases_sqequal,
bool_wf,
assert-bnot,
neg_assert_of_eq_int,
biject_wf,
product_subtype_base,
set_subtype_base,
lelt_wf,
assert_wf,
bnot_wf,
not_wf,
equal-wf-base,
istype-assert,
bool_cases,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot,
decidable__equal_int,
iff_imp_equal_bool,
btrue_wf,
bfalse_wf,
true_wf,
btrue_neq_bfalse,
equal_wf,
squash_wf,
istype-universe,
eq_int_eq_true,
iff_weakening_equal,
assert_elim,
union_subtype_base,
equal_functionality_wrt_subtype_rel2,
subtype_rel_product,
less_than_wf,
le_wf,
base_wf,
subtype_rel-equal,
equipollent_functionality_wrt_equipollent,
equipollent_weakening_ext-eq,
ext-eq_weakening,
equipollent_same,
union_functionality_wrt_equipollent,
istype-top,
assert_of_lt_int,
lt_int_wf,
sum-unroll,
equipollent-add,
nat_properties,
non_neg_sum
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :lambdaFormation_alt,
cut,
thin,
rename,
setElimination,
sqequalRule,
Error :functionIsType,
Error :universeIsType,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
natural_numberEquality,
hypothesisEquality,
hypothesis,
productEquality,
applyEquality,
because_Cache,
closedConclusion,
Error :dependent_set_memberEquality_alt,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
Error :dependent_pairFormation_alt,
Error :lambdaEquality_alt,
int_eqEquality,
Error :isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
Error :setIsType,
functionEquality,
Error :inhabitedIsType,
lambdaFormation,
functionExtensionality,
lambdaEquality,
productElimination,
dependent_pairFormation,
intEquality,
isect_memberEquality,
voidEquality,
computeAll,
addEquality,
minusEquality,
multiplyEquality,
equalityElimination,
Error :inrEquality_alt,
instantiate,
cumulativity,
equalityTransitivity,
equalitySymmetry,
Error :productIsType,
imageElimination,
Error :equalityIsType4,
baseApply,
baseClosed,
promote_hyp,
Error :inlEquality_alt,
Error :dependent_pairEquality_alt,
Error :equalityIstype,
unionEquality,
Error :unionIsType,
sqequalBase,
applyLambdaEquality,
universeEquality,
imageMemberEquality,
Error :equalityIsType3,
Error :equalityIsType1,
axiomSqEquality,
Error :isect_memberFormation_alt,
lessCases
Latex:
\mforall{}n:\mBbbN{}. \mforall{}f:\mBbbN{}n {}\mrightarrow{} \mBbbN{}. i:\mBbbN{}n \mtimes{} \mBbbN{}f[i] \msim{} \mBbbN{}\mSigma{}(f[i] | i < n)
Date html generated:
2019_06_20-PM-02_17_28
Last ObjectModification:
2018_11_24-PM-08_52_34
Theory : equipollence!!cardinality!
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