Nuprl Lemma : funinv-property
∀[n:ℕ]. ∀[f:{f:ℕn ⟶ ℕn| Inj(ℕn;ℕn;f)} ]. ∀[x:ℕn]. (((f (inv(f) x)) = x ∈ ℤ) ∧ ((inv(f) (f x)) = x ∈ ℕn))
Proof
Definitions occuring in Statement :
funinv: inv(f),
inject: Inj(A;B;f),
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
and: P ∧ Q,
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 :
uall: ∀[x:A]. B[x],
member: t ∈ T,
and: P ∧ Q,
nat: ℕ,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
int_seg: {i..j-},
all: ∀x:A. B[x],
uimplies: b supposing a,
squash: ↓T,
cand: A c∧ B,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
inject: Inj(A;B;f),
lelt: i ≤ j < k,
ge: i ≥ j ,
le: A ≤ B,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
false: False,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
sq_stable: SqStable(P)
Lemmas referenced :
funinv_wf2,
int_seg_wf,
set_wf,
inject_wf,
nat_wf,
equal_wf,
squash_wf,
injection-is-surjection,
surject_wf,
true_wf,
iff_weakening_equal,
int_seg_properties,
nat_properties,
decidable__le,
lelt_wf,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformnot_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_not_lemma,
int_formula_prop_wf,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
sq_stable__and,
sq_stable__equal,
funinv_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
setElimination,
rename,
sqequalRule,
productElimination,
independent_pairEquality,
axiomEquality,
hypothesis,
natural_numberEquality,
isect_memberEquality,
because_Cache,
functionEquality,
lambdaEquality,
functionExtensionality,
applyEquality,
intEquality,
dependent_set_memberEquality,
setEquality,
dependent_functionElimination,
independent_isectElimination,
applyLambdaEquality,
imageMemberEquality,
baseClosed,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
independent_functionElimination,
independent_pairFormation,
unionElimination,
voidElimination,
dependent_pairFormation,
int_eqEquality,
voidEquality,
computeAll,
lambdaFormation
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[f:\{f:\mBbbN{}n {}\mrightarrow{} \mBbbN{}n| Inj(\mBbbN{}n;\mBbbN{}n;f)\} ]. \mforall{}[x:\mBbbN{}n]. (((f (inv(f) x)) = x) \mwedge{} ((inv(f) (f x)) = x))
Date html generated:
2017_04_14-AM-09_19_17
Last ObjectModification:
2017_02_27-PM-03_55_22
Theory : int_2
Home
Index