Nuprl Lemma : funinv_wf
∀[n,m:ℕ]. ∀[f:{f:ℕn ⟶ ℕm| Surj(ℕn;ℕm;f)} ]. (inv(f) ∈ {g:ℕm ⟶ ℕn| Inj(ℕm;ℕn;g) ∧ (∀x:ℕm. ((f (g x)) = x ∈ ℤ))} )
Proof
Definitions occuring in Statement :
funinv: inv(f),
surject: Surj(A;B;f),
inject: Inj(A;B;f),
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
and: P ∧ Q,
member: t ∈ T,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
funinv: inv(f),
so_apply: x[s],
so_lambda: λ2x.t[x],
uiff: uiff(P;Q),
prop: ℙ,
top: Top,
false: False,
satisfiable_int_formula: satisfiable_int_formula(fmla),
implies: P ⇒ Q,
not: ¬A,
or: P ∨ Q,
decidable: Dec(P),
and: P ∧ Q,
lelt: i ≤ j < k,
ge: i ≥ j ,
guard: {T},
exists: ∃x:A. B[x],
uimplies: b supposing a,
nat: ℕ,
int_seg: {i..j-},
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
surject: Surj(A;B;f),
member: t ∈ T,
uall: ∀[x:A]. B[x],
cand: A c∧ B,
inject: Inj(A;B;f)
Lemmas referenced :
nat_wf,
surject_wf,
set_wf,
exists_wf,
assert_wf,
assert_of_eq_int,
equal_wf,
int_formula_prop_wf,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
intformeq_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__equal_int,
nat_properties,
int_seg_properties,
int_seg_wf,
eq_int_wf,
mu-bound-property+,
inject_wf,
all_wf,
decidable__le,
intformle_wf,
itermConstant_wf,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
lelt_wf,
and_wf
Rules used in proof :
functionEquality,
axiomEquality,
cumulativity,
addLevel,
independent_pairFormation,
voidEquality,
voidElimination,
isect_memberEquality,
intEquality,
int_eqEquality,
independent_functionElimination,
approximateComputation,
unionElimination,
applyLambdaEquality,
equalitySymmetry,
equalityTransitivity,
dependent_pairFormation,
productElimination,
independent_isectElimination,
natural_numberEquality,
sqequalRule,
because_Cache,
applyEquality,
lambdaEquality,
isectElimination,
extract_by_obid,
hypothesisEquality,
dependent_functionElimination,
hypothesis,
lambdaFormation,
sqequalHypSubstitution,
rename,
thin,
setElimination,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
dependent_set_memberEquality,
productEquality
Latex:
\mforall{}[n,m:\mBbbN{}]. \mforall{}[f:\{f:\mBbbN{}n {}\mrightarrow{} \mBbbN{}m| Surj(\mBbbN{}n;\mBbbN{}m;f)\} ].
(inv(f) \mmember{} \{g:\mBbbN{}m {}\mrightarrow{} \mBbbN{}n| Inj(\mBbbN{}m;\mBbbN{}n;g) \mwedge{} (\mforall{}x:\mBbbN{}m. ((f (g x)) = x))\} )
Date html generated:
2019_06_20-PM-01_17_35
Last ObjectModification:
2018_08_25-AM-08_20_11
Theory : int_2
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