Nuprl Lemma : mklist_select
∀[T:Type]. ∀[n:ℕ]. ∀[f:ℕn ⟶ T]. ∀[i:ℕn]. (mklist(n;f)[i] = (f i) ∈ T)
Proof
Definitions occuring in Statement :
mklist: mklist(n;f),
select: L[n],
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
all: ∀x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
decidable: Dec(P),
or: P ∨ Q,
lelt: i ≤ j < k,
int_seg: {i..j-},
guard: {T},
less_than': less_than'(a;b),
le: A ≤ B,
so_apply: x[s],
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
mklist: mklist(n;f),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
select: L[n],
nil: [],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
squash: ↓T,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
subtract: n - m,
true: True
Lemmas referenced :
nat_properties,
full-omega-unsat,
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,
int_seg_wf,
decidable__le,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
nat_wf,
satisfiable-full-omega-tt,
int_seg_properties,
false_wf,
int_seg_subtype,
subtype_rel_dep_function,
primrec-unroll,
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
stuck-spread,
base_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
not_functionality_wrt_uiff,
assert_wf,
decidable__lt,
squash_wf,
true_wf,
select_append_front,
mklist_wf,
le_wf,
subtype_rel_function,
not-le-2,
condition-implies-le,
add-associates,
minus-add,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-mul-special,
zero-mul,
add-zero,
add-commutes,
le-add-cancel2,
subtype_rel_self,
cons_wf,
lelt_wf,
nil_wf,
mklist_length,
length_wf,
iff_weakening_equal,
int_subtype_base,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
select_append_back,
length_of_cons_lemma,
length_of_nil_lemma,
subtract-add-cancel,
select-cons-hd
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
lambdaFormation,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
axiomEquality,
functionEquality,
unionElimination,
because_Cache,
Error :universeIsType,
universeEquality,
cumulativity,
computeAll,
productElimination,
isect_memberFormation,
applyEquality,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
baseClosed,
promote_hyp,
instantiate,
imageElimination,
dependent_set_memberEquality,
addEquality,
minusEquality,
multiplyEquality,
functionExtensionality,
imageMemberEquality
Latex:
\mforall{}[T:Type]. \mforall{}[n:\mBbbN{}]. \mforall{}[f:\mBbbN{}n {}\mrightarrow{} T]. \mforall{}[i:\mBbbN{}n]. (mklist(n;f)[i] = (f i))
Date html generated:
2019_06_20-PM-01_31_32
Last ObjectModification:
2018_09_26-PM-06_08_54
Theory : list_1
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