Nuprl Lemma : equipollent-nat-list-as-product
ℕ ~ k:ℕ × (ℕ^k)
Proof
Definitions occuring in Statement :
power-type: (T^k),
equipollent: A ~ B,
nat: ℕ,
product: x:A × B[x]
Definitions unfolded in proof :
exists: ∃x:A. B[x],
equipollent: A ~ B,
member: t ∈ T,
uall: ∀[x:A]. B[x],
nat: ℕ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
prop: ℙ,
subtype_rel: A ⊆r B,
power-type: (T^k),
eq_int: (i =z j),
bfalse: ff,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
ge: i ≥ j ,
int_upper: {i...},
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
so_lambda: λ2x.t[x],
so_apply: x[s],
pi1: fst(t),
subtract: n - m,
inv_funs: InvFuns(A;B;f;g),
tidentity: Id{T},
identity: Id,
compose: f o g,
nequal: a ≠ b ∈ T ,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
equipollent-nat-powered3,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
false_wf,
le_wf,
it_wf,
subtype_rel_self,
equal-wf-base,
power-type_wf,
nat_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
int_upper_subtype_nat,
nat_properties,
nequal-le-implies,
zero-add,
coded-pair_wf,
subtract_wf,
int_upper_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermSubtract_wf,
itermVar_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_wf,
itermAdd_wf,
int_term_value_add_lemma,
biject_wf,
fun_with_inv_is_bij2,
code-pair_wf,
exists_wf,
subtract-add-cancel,
inv_funs_wf,
add_nat_wf,
pi1_wf_top,
int_upper_wf,
set_subtype_base,
int_subtype_base,
add-associates,
add-swap,
add-commutes,
add-is-int-iff,
intformeq_wf,
int_formula_prop_eq_lemma,
decidable__equal_int,
add-subtract-cancel,
code-coded-pair,
assert_wf,
bnot_wf,
not_wf,
equal-wf-T-base,
bool_cases,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot,
equal-unit,
unit_wf2,
nequal_wf,
coded-code-pair
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
productElimination,
thin,
dependent_pairFormation,
lambdaEquality,
isectElimination,
setElimination,
rename,
because_Cache,
hypothesis,
natural_numberEquality,
lambdaFormation,
unionElimination,
equalityElimination,
sqequalRule,
independent_isectElimination,
dependent_pairEquality,
dependent_set_memberEquality,
equalityTransitivity,
equalitySymmetry,
independent_pairFormation,
hypothesisEquality,
applyEquality,
intEquality,
baseClosed,
promote_hyp,
dependent_functionElimination,
instantiate,
cumulativity,
independent_functionElimination,
voidElimination,
hypothesis_subsumption,
int_eqEquality,
isect_memberEquality,
voidEquality,
computeAll,
productEquality,
addEquality,
functionExtensionality,
functionEquality,
independent_pairEquality,
applyLambdaEquality,
pointwiseFunctionality,
baseApply,
closedConclusion,
hyp_replacement,
spreadEquality,
impliesFunctionality
Latex:
\mBbbN{} \msim{} k:\mBbbN{} \mtimes{} (\mBbbN{}\^{}k)
Date html generated:
2018_05_21-PM-08_14_47
Last ObjectModification:
2017_07_26-PM-05_49_30
Theory : general
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