Nuprl Lemma : equipollent-product
∀n:ℕ. ∀f:ℕn ⟶ ℕ. i:ℕn ⟶ ℕf[i] ~ ℕΠ(f[i] | i < n)
Proof
Definitions occuring in Statement :
equipollent: A ~ B,
int-prod: Π(f[x] | x < k),
int_seg: {i..j-},
nat: ℕ,
so_apply: x[s],
all: ∀x:A. B[x],
function: x:A ⟶ B[x],
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
nat: ℕ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
and: P ∧ Q,
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
less_than': less_than'(a;b),
int-prod: Π(f[x] | x < k),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
nequal: a ≠ b ∈ T ,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
equipollent: A ~ B,
less_than: a < b,
biject: Bij(A;B;f),
inject: Inj(A;B;f),
surject: Surj(A;B;f),
pi1: fst(t),
pi2: snd(t),
squash: ↓T
Lemmas referenced :
all_wf,
int_seg_wf,
subtract_wf,
nat_wf,
equipollent_wf,
int-prod_wf,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermSubtract_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
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,
le_wf,
set_wf,
less_than_wf,
primrec-wf2,
int_prod0_lemma,
equipollent-singletons,
singleton-type-one,
singleton-type-void-domain,
int_seg_properties,
subtype_rel_dep_function,
int_seg_subtype,
false_wf,
subtype_rel_self,
primrec-unroll,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
intformeq_wf,
int_formula_prop_eq_lemma,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
decidable__lt,
lelt_wf,
int-prod_wf_nat,
equipollent_functionality_wrt_equipollent2,
equipollent_inversion,
equipollent-multiply,
product_functionality_wrt_equipollent_left,
biject_wf,
decidable__equal_int,
int_subtype_base
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
thin,
rename,
setElimination,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
functionEquality,
natural_numberEquality,
hypothesisEquality,
hypothesis,
sqequalRule,
lambdaEquality,
because_Cache,
applyEquality,
functionExtensionality,
dependent_set_memberEquality,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
independent_functionElimination,
productElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
instantiate,
cumulativity,
multiplyEquality,
productEquality,
independent_pairEquality,
applyLambdaEquality,
hyp_replacement,
imageElimination
Latex:
\mforall{}n:\mBbbN{}. \mforall{}f:\mBbbN{}n {}\mrightarrow{} \mBbbN{}. i:\mBbbN{}n {}\mrightarrow{} \mBbbN{}f[i] \msim{} \mBbbN{}\mPi{}(f[i] | i < n)
Date html generated:
2017_04_17-AM-09_32_09
Last ObjectModification:
2017_02_27-PM-05_32_55
Theory : equipollence!!cardinality!
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