Nuprl Lemma : bijection_restriction
∀k:ℕ. ∀f:ℕk ⟶ ℕk.
Bij(ℕk;ℕk;f) ⇒ {(f ∈ ℕk - 1 ⟶ ℕk - 1) ∧ Bij(ℕk - 1;ℕk - 1;f)} supposing (f (k - 1)) = (k - 1) ∈ ℤ supposing 0 < k
Proof
Definitions occuring in Statement :
biject: Bij(A;B;f),
int_seg: {i..j-},
nat: ℕ,
less_than: a < b,
uimplies: b supposing a,
guard: {T},
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
member: t ∈ T,
apply: f a,
function: x:A ⟶ B[x],
subtract: n - m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x],
nat: ℕ,
implies: P ⇒ Q,
guard: {T},
and: P ∧ Q,
cand: A c∧ B,
prop: ℙ,
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
not: ¬A,
rev_implies: P ⇐ Q,
false: False,
uiff: uiff(P;Q),
subtract: n - m,
subtype_rel: A ⊆r B,
top: Top,
le: A ≤ B,
less_than': less_than'(a;b),
true: True,
less_than: a < b,
exists: ∃x:A. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
biject: Bij(A;B;f),
inject: Inj(A;B;f),
squash: ↓T,
surject: Surj(A;B;f)
Lemmas referenced :
member-less_than,
equal_wf,
int_seg_wf,
subtract_wf,
decidable__le,
false_wf,
not-le-2,
less-iff-le,
condition-implies-le,
minus-one-mul,
zero-add,
minus-one-mul-top,
nat_wf,
minus-add,
minus-minus,
add-associates,
add-swap,
add-commutes,
add_functionality_wrt_le,
add-zero,
le-add-cancel,
decidable__lt,
not-lt-2,
add-mul-special,
zero-mul,
le-add-cancel-alt,
lelt_wf,
biject_wf,
less_than_wf,
decidable__int_equal,
le-add-cancel2,
set_subtype_base,
int_subtype_base,
two-mul,
mul-distributes-right,
one-mul,
subtype_base_sq,
not-equal-2,
int_seg_properties,
nat_properties,
int_seg_subtype,
le_antisymmetry_iff,
less_than_transitivity1,
le_weakening,
less_than_irreflexivity,
equal_functionality_wrt_subtype_rel2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
setElimination,
rename,
hypothesisEquality,
hypothesis,
independent_isectElimination,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
independent_pairFormation,
intEquality,
applyEquality,
functionExtensionality,
because_Cache,
dependent_set_memberEquality,
dependent_functionElimination,
unionElimination,
voidElimination,
productElimination,
independent_functionElimination,
addEquality,
sqequalRule,
lambdaEquality,
isect_memberEquality,
voidEquality,
minusEquality,
functionEquality,
dependent_pairFormation,
sqequalIntensionalEquality,
promote_hyp,
multiplyEquality,
instantiate,
cumulativity,
applyLambdaEquality,
imageMemberEquality,
baseClosed
Latex:
\mforall{}k:\mBbbN{}. \mforall{}f:\mBbbN{}k {}\mrightarrow{} \mBbbN{}k.
Bij(\mBbbN{}k;\mBbbN{}k;f) {}\mRightarrow{} \{(f \mmember{} \mBbbN{}k - 1 {}\mrightarrow{} \mBbbN{}k - 1) \mwedge{} Bij(\mBbbN{}k - 1;\mBbbN{}k - 1;f)\} supposing (f (k - 1)) = (k - 1)
supposing 0 < k
Date html generated:
2017_04_14-AM-07_34_14
Last ObjectModification:
2017_02_27-PM-03_08_51
Theory : fun_1
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