Nuprl Lemma : equipollent-nat-powered
∀n:ℕ. ℕ ~ (ℕ^n + 1)
Proof
Definitions occuring in Statement :
power-type: (T^k),
equipollent: A ~ B,
nat: ℕ,
all: ∀x:A. B[x],
add: n + m,
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,
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,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
ge: i ≥ j ,
power-type: (T^k),
eq_int: (i =z j),
subtract: n - m,
ifthenelse: if b then t else f fi ,
bfalse: ff,
btrue: tt,
equipollent: A ~ B,
bool: 𝔹,
unit: Unit,
it: ⋅,
subtype_rel: A ⊆r B,
uiff: uiff(P;Q),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
inv_funs: InvFuns(A;B;f;g),
tidentity: Id{T},
identity: Id,
compose: f o g,
guard: {T},
squash: ↓T,
true: True,
sq_type: SQType(T),
code-pair: code-pair(a;b),
triangular-num: t(n)
Lemmas referenced :
equipollent_wf,
nat_wf,
power-type_wf,
subtract_wf,
subtract-add-cancel,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_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_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
le_wf,
set_wf,
less_than_wf,
primrec-wf2,
nat_properties,
itermAdd_wf,
int_term_value_add_lemma,
equipollent-type-unit-pair,
eq_int_wf,
bool_wf,
uiff_transitivity,
equal-wf-base,
int_subtype_base,
assert_wf,
eqtt_to_assert,
assert_of_eq_int,
intformeq_wf,
int_formula_prop_eq_lemma,
iff_transitivity,
bnot_wf,
not_wf,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot,
general_arith_equation1,
equal_wf,
coded-pair_wf,
fun_with_inv_is_bij2,
add-subtract-cancel,
biject-inverse,
biject_wf,
code-pair_wf,
inv_funs_wf,
code-coded-pair,
subtype_base_sq,
set_subtype_base,
squash_wf,
true_wf,
iff_weakening_equal,
product_subtype_base,
decidable__equal_int,
coded-code-pair,
zero-le-nat
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
thin,
rename,
setElimination,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesis,
because_Cache,
dependent_set_memberEquality,
addEquality,
hypothesisEquality,
natural_numberEquality,
sqequalRule,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
productElimination,
equalityElimination,
baseApply,
closedConclusion,
baseClosed,
applyEquality,
independent_functionElimination,
equalityTransitivity,
equalitySymmetry,
impliesFunctionality,
productEquality,
spreadEquality,
independent_pairEquality,
functionExtensionality,
instantiate,
cumulativity,
applyLambdaEquality,
imageElimination,
universeEquality,
imageMemberEquality
Latex:
\mforall{}n:\mBbbN{}. \mBbbN{} \msim{} (\mBbbN{}\^{}n + 1)
Date html generated:
2018_05_21-PM-08_14_26
Last ObjectModification:
2017_07_26-PM-05_49_18
Theory : general
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