Nuprl Lemma : equipollent-exp
∀n,b:ℕ. ℕn ⟶ ℕb ~ ℕb^n
Proof
Definitions occuring in Statement :
equipollent: A ~ B,
exp: i^n,
int_seg: {i..j-},
nat: ℕ,
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,
uall: ∀[x:A]. B[x],
nat: ℕ,
ge: i ≥ j ,
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],
so_apply: x[s],
exp: i^n,
subtype_rel: A ⊆r B,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
guard: {T},
equipollent: A ~ B,
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
less_than': less_than'(a;b),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
subtract: n - m,
true: True,
biject: Bij(A;B;f),
inject: Inj(A;B;f),
respects-equality: respects-equality(S;T),
surject: Surj(A;B;f),
pi2: snd(t),
pi1: fst(t),
sq_type: SQType(T),
squash: ↓T,
nequal: a ≠ b ∈ T ,
assert: ↑b,
bnot: ¬bb
Lemmas referenced :
istype-nat,
equipollent_wf,
int_seg_wf,
subtract_wf,
exp_wf2,
nat_properties,
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,
nat_wf,
primrec0_lemma,
equipollent-void-domain,
primrec-unroll,
lt_int_wf,
equal-wf-base,
bool_wf,
int_subtype_base,
assert_wf,
less_than_wf,
le_int_wf,
le_wf,
bnot_wf,
uiff_transitivity,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
assert_functionality_wrt_uiff,
bnot_of_lt_int,
assert_of_le_int,
decidable__lt,
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,
biject_wf,
respects-equality-product,
respects-equality-trivial,
respects-equality-function,
istype-base,
pi2_wf,
pi1_wf,
decidable__equal_int,
subtype_base_sq,
int_seg_properties,
intformeq_wf,
int_formula_prop_eq_lemma,
istype-assert,
not_wf,
btrue_neq_bfalse,
iff_weakening_equal,
btrue_wf,
eq_int_eq_true,
istype-universe,
true_wf,
squash_wf,
equal_wf,
neg_assert_of_eq_int,
assert-bnot,
bool_subtype_base,
bool_cases_sqequal,
eq_int_wf,
assert_of_eq_int,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot,
primrec_wf,
equipollent-zero,
product_functionality_wrt_equipollent_right,
exp_wf4,
equipollent-multiply,
equipollent_transitivity
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :lambdaFormation_alt,
cut,
thin,
rename,
setElimination,
sqequalRule,
Error :functionIsType,
introduction,
extract_by_obid,
hypothesis,
Error :universeIsType,
sqequalHypSubstitution,
isectElimination,
functionEquality,
natural_numberEquality,
hypothesisEquality,
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,
Error :inhabitedIsType,
lambdaFormation,
isect_memberEquality,
voidEquality,
baseApply,
closedConclusion,
baseClosed,
applyEquality,
equalityTransitivity,
equalitySymmetry,
equalityElimination,
productElimination,
Error :equalityIstype,
independent_pairEquality,
Error :productIsType,
because_Cache,
addEquality,
minusEquality,
multiplyEquality,
productEquality,
sqequalBase,
applyLambdaEquality,
Error :functionExtensionality_alt,
instantiate,
cumulativity,
intEquality,
imageMemberEquality,
universeEquality,
imageElimination,
promote_hyp,
Error :equalityIsType1
Latex:
\mforall{}n,b:\mBbbN{}. \mBbbN{}n {}\mrightarrow{} \mBbbN{}b \msim{} \mBbbN{}b\^{}n
Date html generated:
2019_06_20-PM-02_17_19
Last ObjectModification:
2019_01_02-PM-00_32_04
Theory : equipollence!!cardinality!
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