Nuprl Lemma : list-index-property
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[x:T]. ∀[L:T List]. L[outl(list-index(eq;L;x))] = x ∈ T supposing (x ∈ L)
Proof
Definitions occuring in Statement :
list-index: list-index(d;L;x),
l_member: (x ∈ l),
select: L[n],
list: T List,
deq: EqDecider(T),
outl: outl(x),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
nat: ℕ,
false: False,
ge: i ≥ j ,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
prop: ℙ,
or: P ∨ Q,
list-index: list-index(d;L;x),
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
select: L[n],
nil: [],
it: ⋅,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
isl: isl(x),
assert: ↑b,
ifthenelse: if b then t else f fi ,
bfalse: ff,
cons: [a / b],
le: A ≤ B,
less_than': less_than'(a;b),
colength: colength(L),
guard: {T},
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
less_than: a < b,
squash: ↓T,
decidable: Dec(P),
subtype_rel: A ⊆r B,
outl: outl(x),
btrue: tt,
true: True,
int_seg: {i..j-},
lelt: i ≤ j < k,
subtract: n - m,
bool: 𝔹,
unit: Unit,
eqof: eqof(d),
deq: EqDecider(T),
uiff: uiff(P;Q),
bnot: ¬bb
Lemmas referenced :
isl-list-index,
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
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,
istype-less_than,
list-cases,
list_ind_nil_lemma,
stuck-spread,
istype-base,
product_subtype_list,
colength-cons-not-zero,
colength_wf_list,
istype-le,
subtract-1-ge-0,
subtype_base_sq,
intformeq_wf,
int_formula_prop_eq_lemma,
set_subtype_base,
int_subtype_base,
spread_cons_lemma,
decidable__equal_int,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
itermAdd_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
int_term_value_add_lemma,
decidable__le,
le_wf,
list_ind_cons_lemma,
list-index_wf,
istype-nat,
l_member_wf,
list_wf,
deq_wf,
istype-universe,
select-cons-tl,
int_seg_properties,
decidable__lt,
add-associates,
add-swap,
add-commutes,
zero-add,
istype-true,
select_wf,
length_wf,
eqof_wf,
eqtt_to_assert,
safe-assert-deq,
eqff_to_assert,
bool_cases_sqequal,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
equal_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
dependent_functionElimination,
productElimination,
independent_functionElimination,
hypothesis,
lambdaFormation_alt,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
sqequalRule,
independent_pairFormation,
universeIsType,
axiomEquality,
functionIsTypeImplies,
inhabitedIsType,
unionElimination,
baseClosed,
promote_hyp,
hypothesis_subsumption,
equalityIstype,
because_Cache,
dependent_set_memberEquality_alt,
instantiate,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
imageElimination,
baseApply,
closedConclusion,
applyEquality,
intEquality,
sqequalBase,
isectIsTypeImplies,
universeEquality,
addEquality,
functionIsType,
equalityElimination,
cumulativity
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[x:T]. \mforall{}[L:T List].
L[outl(list-index(eq;L;x))] = x supposing (x \mmember{} L)
Date html generated:
2019_10_15-AM-10_24_24
Last ObjectModification:
2019_08_05-PM-02_04_47
Theory : decidable!equality
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