Nuprl Lemma : equipollent-nat-powered3
∃f:n:ℕ ⟶ ℕ ⟶ (ℕ^n + 1). ∀n:ℕ. ∃g:(ℕ^n + 1) ⟶ ℕ. InvFuns(ℕ;(ℕ^n + 1);f n;g)
Proof
Definitions occuring in Statement :
power-type: (T^k),
inv_funs: InvFuns(A;B;f;g),
nat: ℕ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
apply: f a,
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n
Definitions unfolded in proof :
exists: ∃x:A. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
prop: ℙ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
nat: ℕ,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
implies: P ⇒ Q,
not: ¬A,
top: Top,
and: P ∧ Q,
so_apply: x[s]
Lemmas referenced :
biject-inverse2,
inv_funs_wf,
le_wf,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
itermAdd_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__le,
nat_properties,
power-type_wf,
exists_wf,
all_wf,
nat_wf,
equipollent-nat-powered2
Rules used in proof :
cut,
lemma_by_obid,
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
productElimination,
thin,
dependent_pairFormation,
hypothesisEquality,
lambdaFormation,
hypothesis,
isectElimination,
sqequalRule,
lambdaEquality,
functionEquality,
because_Cache,
dependent_set_memberEquality,
addEquality,
setElimination,
rename,
natural_numberEquality,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
applyEquality,
independent_functionElimination
Latex:
\mexists{}f:n:\mBbbN{} {}\mrightarrow{} \mBbbN{} {}\mrightarrow{} (\mBbbN{}\^{}n + 1). \mforall{}n:\mBbbN{}. \mexists{}g:(\mBbbN{}\^{}n + 1) {}\mrightarrow{} \mBbbN{}. InvFuns(\mBbbN{};(\mBbbN{}\^{}n + 1);f n;g)
Date html generated:
2016_05_15-PM-06_07_23
Last ObjectModification:
2016_01_16-PM-00_45_06
Theory : general
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