Nuprl Lemma : l_index_wf
∀[T:Type]. ∀[dT:EqDecider(T)]. ∀[L:T List]. ∀[x:T]. index(L;x) ∈ ℕ||L|| supposing (x ∈ L)
Proof
Definitions occuring in Statement :
l_index: index(L;x),
l_member: (x ∈ l),
length: ||as||,
list: T List,
deq: EqDecider(T),
int_seg: {i..j-},
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
member: t ∈ T,
natural_number: $n,
universe: Type
Definitions unfolded in proof :
l_index: index(L;x),
uall: ∀[x:A]. B[x],
member: t ∈ T,
deq: EqDecider(T),
int_seg: {i..j-},
uimplies: b supposing a,
guard: {T},
lelt: i ≤ j < k,
and: P ∧ Q,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
implies: P ⇒ Q,
not: ¬A,
top: Top,
prop: ℙ,
less_than: a < b,
squash: ↓T,
l_member: (x ∈ l),
nat: ℕ,
le: A ≤ B,
cand: A c∧ B,
eqof: eqof(d),
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
ge: i ≥ j
Lemmas referenced :
nat_properties,
assert_wf,
safe-assert-deq,
lelt_wf,
deq_wf,
list_wf,
l_member_wf,
int_seg_wf,
int_formula_prop_less_lemma,
intformless_wf,
decidable__lt,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__le,
length_wf,
int_seg_properties,
select_wf,
length_wf_nat,
mu-bound
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
lambdaEquality,
applyEquality,
setElimination,
rename,
cumulativity,
independent_isectElimination,
natural_numberEquality,
productElimination,
dependent_functionElimination,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
because_Cache,
imageElimination,
universeEquality,
isect_memberFormation,
introduction,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
dependent_set_memberEquality
Latex:
\mforall{}[T:Type]. \mforall{}[dT:EqDecider(T)]. \mforall{}[L:T List]. \mforall{}[x:T]. index(L;x) \mmember{} \mBbbN{}||L|| supposing (x \mmember{} L)
Date html generated:
2016_05_14-PM-03_31_54
Last ObjectModification:
2016_01_14-PM-11_20_23
Theory : decidable!equality
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