Nuprl Lemma : funinv-unique
∀[n:ℕ]. ∀[f:{f:ℕn ⟶ ℕn| Inj(ℕn;ℕn;f)} ]. ∀[g:ℕn ⟶ ℕn].
inv(f) = g ∈ {f:ℕn ⟶ ℕn| Inj(ℕn;ℕn;f)} supposing (f o g) = (λx.x) ∈ (ℕn ⟶ ℕn)
Proof
Definitions occuring in Statement :
funinv: inv(f),
inject: Inj(A;B;f),
compose: f o g,
int_seg: {i..j-},
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
set: {x:A| B[x]} ,
lambda: λx.A[x],
function: x:A ⟶ B[x],
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
nat: ℕ,
prop: ℙ,
squash: ↓T,
so_lambda: λ2x.t[x],
so_apply: x[s],
inject: Inj(A;B;f),
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
implies: P ⇒ Q,
compose: f o g,
and: P ∧ Q,
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top
Lemmas referenced :
funinv_wf2,
inject_wf,
int_seg_wf,
set_wf,
equal-wf-T-base,
compose_wf,
nat_wf,
equal_wf,
funinv-property,
int_seg_properties,
lelt_wf,
nat_properties,
decidable__equal_int,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformeq_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_formula_prop_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
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
setElimination,
rename,
dependent_set_memberEquality,
hypothesis,
natural_numberEquality,
because_Cache,
functionExtensionality,
applyEquality,
lambdaEquality,
sqequalRule,
imageMemberEquality,
baseClosed,
imageElimination,
functionEquality,
isect_memberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
setEquality,
independent_functionElimination,
hyp_replacement,
Error :applyLambdaEquality,
productElimination,
intEquality,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[f:\{f:\mBbbN{}n {}\mrightarrow{} \mBbbN{}n| Inj(\mBbbN{}n;\mBbbN{}n;f)\} ]. \mforall{}[g:\mBbbN{}n {}\mrightarrow{} \mBbbN{}n]. inv(f) = g supposing (f o g) = (\mlambda{}x.x)
Date html generated:
2016_10_21-AM-09_59_48
Last ObjectModification:
2016_07_12-AM-05_14_19
Theory : int_2
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