Nuprl Lemma : first_index_cons
∀[T:Type]. ∀[L:T List]. ∀[a:T]. ∀[P:T ⟶ 𝔹].
(index-of-first x in [a / L].P[x] ~ if P[a] then 1
if 0 <z index-of-first x in L.P[x] then index-of-first x in L.P[x] + 1
else 0
fi )
Proof
Definitions occuring in Statement :
first_index: index-of-first x in L.P[x],
cons: [a / b],
list: T List,
ifthenelse: if b then t else f fi ,
lt_int: i <z j,
bool: 𝔹,
uall: ∀[x:A]. B[x],
so_apply: x[s],
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n,
universe: Type,
sqequal: s ~ t
Definitions unfolded in proof :
first_index: index-of-first x in L.P[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
int_seg: {i..j-},
so_lambda: λ2x.t[x],
so_apply: x[s],
all: ∀x:A. B[x],
top: Top,
sq_type: SQType(T),
implies: P ⇒ Q,
guard: {T},
lelt: i ≤ j < k,
and: P ∧ Q,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
prop: ℙ,
ge: i ≥ j ,
le: A ≤ B,
select: L[n],
cons: [a / b],
less_than: a < b,
squash: ↓T,
subtype_rel: A ⊆r B,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
less_than': less_than'(a;b),
subtract: n - m
Lemmas referenced :
subtype_base_sq,
int_seg_wf,
length_wf,
cons_wf,
set_subtype_base,
lelt_wf,
int_subtype_base,
length_of_cons_lemma,
bool_wf,
list_wf,
search_succ,
length_wf_nat,
select_wf,
int_seg_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
non_neg_length,
decidable__lt,
intformless_wf,
itermAdd_wf,
int_formula_prop_less_lemma,
int_term_value_add_lemma,
select-cons-tl,
add-subtract-cancel,
search_wf,
equal_wf,
equal-wf-T-base,
assert_wf,
bnot_wf,
not_wf,
eqtt_to_assert,
uiff_transitivity,
eqff_to_assert,
assert_of_bnot,
false_wf,
lt_int_wf,
less_than_wf,
add-member-int_seg2,
int_seg_subtype,
subtract_wf,
add-is-int-iff,
itermSubtract_wf,
int_term_value_subtract_lemma,
le_int_wf,
le_wf,
assert_of_lt_int,
assert_functionality_wrt_uiff,
bnot_of_lt_int,
assert_of_le_int
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
thin,
instantiate,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
cumulativity,
natural_numberEquality,
addEquality,
hypothesisEquality,
hypothesis,
independent_isectElimination,
intEquality,
lambdaEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
sqequalAxiom,
functionEquality,
because_Cache,
universeEquality,
applyEquality,
functionExtensionality,
setElimination,
rename,
productElimination,
unionElimination,
dependent_pairFormation,
int_eqEquality,
independent_pairFormation,
computeAll,
imageElimination,
hyp_replacement,
applyLambdaEquality,
baseClosed,
lambdaFormation,
equalityElimination,
dependent_set_memberEquality,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion
Latex:
\mforall{}[T:Type]. \mforall{}[L:T List]. \mforall{}[a:T]. \mforall{}[P:T {}\mrightarrow{} \mBbbB{}].
(index-of-first x in [a / L].P[x] \msim{} if P[a] then 1
if 0 <z index-of-first x in L.P[x] then index-of-first x in L.P[x] + 1
else 0
fi )
Date html generated:
2017_10_01-AM-08_38_45
Last ObjectModification:
2017_07_26-PM-04_27_11
Theory : list!
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