Nuprl Lemma : finite-nat-seq-to-list-prop1
∀[f:finite-nat-seq()]
((||finite-nat-seq-to-list(f)|| = (fst(f)) ∈ ℕ) ∧ (∀i:ℕfst(f). (finite-nat-seq-to-list(f)[i] = ((snd(f)) i) ∈ ℕ)))
Proof
Definitions occuring in Statement :
finite-nat-seq-to-list: finite-nat-seq-to-list(f),
finite-nat-seq: finite-nat-seq(),
select: L[n],
length: ||as||,
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
pi1: fst(t),
pi2: snd(t),
all: ∀x:A. B[x],
and: P ∧ Q,
apply: f a,
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
finite-nat-seq: finite-nat-seq(),
finite-nat-seq-to-list: finite-nat-seq-to-list(f),
pi1: fst(t),
pi2: snd(t),
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,
all: ∀x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
select: L[n],
nil: [],
it: ⋅,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
decidable: Dec(P),
or: P ∨ Q,
cand: A c∧ B,
le: A ≤ B,
less_than': less_than'(a;b),
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
bool: 𝔹,
unit: Unit,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
nequal: a ≠ b ∈ T ,
squash: ↓T,
less_than: a < b,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
cons: [a / b]
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,
int_seg_wf,
nat_wf,
primrec0_lemma,
length_of_nil_lemma,
stuck-spread,
base_wf,
decidable__le,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
finite-nat-seq_wf,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
false_wf,
le_wf,
int_seg_properties,
lelt_wf,
subtype_rel_dep_function,
int_seg_subtype,
subtype_rel_self,
primrec-unroll,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
length-append,
length_of_cons_lemma,
itermAdd_wf,
int_term_value_add_lemma,
decidable__lt,
squash_wf,
true_wf,
select_append_front,
primrec_wf,
list_wf,
nil_wf,
append_wf,
cons_wf,
iff_weakening_equal,
non_neg_length,
select_wf,
length_wf_nat,
set_subtype_base,
int_subtype_base,
select_append_back,
length-singleton,
length_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
sqequalRule,
extract_by_obid,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
lambdaFormation,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
independent_functionElimination,
independent_pairEquality,
axiomEquality,
functionEquality,
baseClosed,
unionElimination,
because_Cache,
dependent_set_memberEquality,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
applyEquality,
functionExtensionality,
equalityElimination,
promote_hyp,
instantiate,
cumulativity,
addEquality,
imageElimination,
universeEquality,
imageMemberEquality,
minusEquality,
productEquality
Latex:
\mforall{}[f:finite-nat-seq()]
((||finite-nat-seq-to-list(f)|| = (fst(f)))
\mwedge{} (\mforall{}i:\mBbbN{}fst(f). (finite-nat-seq-to-list(f)[i] = ((snd(f)) i))))
Date html generated:
2017_04_20-AM-07_29_14
Last ObjectModification:
2017_02_27-PM-06_01_37
Theory : continuity
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