Nuprl Lemma : select-cons
∀[x,L:Top]. ∀[i:ℤ]. ([x / L][i] ~ if i ≤z 0 then x else L[i - 1] fi )
Proof
Definitions occuring in Statement :
select: L[n],
cons: [a / b],
le_int: i ≤z j,
ifthenelse: if b then t else f fi ,
uall: ∀[x:A]. B[x],
top: Top,
subtract: n - m,
natural_number: $n,
int: ℤ,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
prop: ℙ,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A,
decidable: Dec(P),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
subtract: n - m,
subtype_rel: A ⊆r B,
top: Top,
le: A ≤ B,
less_than': less_than'(a;b),
true: True
Lemmas referenced :
le_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_le_int,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
le_wf,
top_wf,
select-cons-hd,
select-cons-tl,
decidable__lt,
false_wf,
not-lt-2,
not-le-2,
condition-implies-le,
minus-add,
minus-zero,
add-zero,
add-commutes,
zero-add,
add_functionality_wrt_le,
le-add-cancel
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
natural_numberEquality,
hypothesis,
lambdaFormation,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_isectElimination,
sqequalRule,
dependent_pairFormation,
promote_hyp,
dependent_functionElimination,
instantiate,
because_Cache,
independent_functionElimination,
voidElimination,
sqequalAxiom,
intEquality,
isect_memberEquality,
independent_pairFormation,
addEquality,
applyEquality,
lambdaEquality,
voidEquality,
minusEquality
Latex:
\mforall{}[x,L:Top]. \mforall{}[i:\mBbbZ{}]. ([x / L][i] \msim{} if i \mleq{}z 0 then x else L[i - 1] fi )
Date html generated:
2017_04_14-AM-08_36_58
Last ObjectModification:
2017_02_27-PM-03_28_52
Theory : list_0
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