Nuprl Lemma : select-as-reduce
∀[n:ℕ]. ∀[L:Top List].
L[||L|| - n + 1]
~ outr(reduce(λu,x. case x of inl(i) => if (i =z n) then inr u else inl (i + 1) fi | inr(u) => x;inl 0;L))
supposing n < ||L||
Proof
Definitions occuring in Statement :
select: L[n],
length: ||as||,
reduce: reduce(f;k;as),
list: T List,
nat: ℕ,
outr: outr(x),
ifthenelse: if b then t else f fi ,
eq_int: (i =z j),
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
top: Top,
lambda: λx.A[x],
decide: case b of inl(x) => s[x] | inr(y) => t[y],
inr: inr x ,
inl: inl x,
subtract: n - m,
add: n + m,
natural_number: $n,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
top: Top,
and: P ∧ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
guard: {T},
or: P ∨ Q,
select: L[n],
nil: [],
it: ⋅,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
cons: [a / b],
colength: colength(L),
decidable: Dec(P),
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
bool: 𝔹,
unit: Unit,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
bnot: ¬bb,
assert: ↑b,
le: A ≤ B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
int_seg: {i..j-},
lelt: i ≤ j < k,
outr: outr(x)
Lemmas referenced :
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
less_than_wf,
equal-wf-T-base,
nat_wf,
colength_wf_list,
top_wf,
less_than_transitivity1,
less_than_irreflexivity,
list-cases,
reduce_nil_lemma,
length_of_nil_lemma,
stuck-spread,
base_wf,
product_subtype_list,
spread_cons_lemma,
intformeq_wf,
itermAdd_wf,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
le_wf,
equal_wf,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
decidable__equal_int,
reduce_cons_lemma,
length_of_cons_lemma,
list_wf,
length_wf,
le_int_wf,
bool_wf,
assert_wf,
lt_int_wf,
bnot_wf,
eqtt_to_assert,
assert_of_le_int,
eq_int_wf,
assert_of_eq_int,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
add-is-int-iff,
false_wf,
not_wf,
uiff_transitivity,
assert_functionality_wrt_uiff,
bnot_of_le_int,
assert_of_lt_int,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot,
general_arith_equation1,
select_cons_tl_sq,
decidable__lt,
lelt_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
thin,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
independent_functionElimination,
sqequalAxiom,
applyEquality,
because_Cache,
unionElimination,
baseClosed,
promote_hyp,
hypothesis_subsumption,
productElimination,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
dependent_set_memberEquality,
addEquality,
instantiate,
cumulativity,
imageElimination,
equalityElimination,
pointwiseFunctionality,
baseApply,
closedConclusion,
impliesFunctionality
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[L:Top List].
L[||L|| - n + 1] \msim{} outr(reduce(\mlambda{}u,x. case x
of inl(i) =>
if (i =\msubz{} n) then inr u else inl (i + 1) fi
| inr(u) =>
x;inl 0;L))
supposing n < ||L||
Date html generated:
2018_05_21-PM-06_51_51
Last ObjectModification:
2017_07_26-PM-04_58_00
Theory : general
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