Nuprl Lemma : first_index_property
∀[T:Type]. ∀[P:T ⟶ 𝔹]. ∀[L:T List].
(↑P[L[index-of-first x in L.P[x] - 1]]) ∧ (¬(∃x∈firstn(index-of-first x in L.P[x] - 1;L). ↑P[x]))
supposing 0 < index-of-first x in L.P[x]
Proof
Definitions occuring in Statement :
first_index: index-of-first x in L.P[x],
firstn: firstn(n;as),
l_exists: (∃x∈L. P[x]),
select: L[n],
list: T List,
assert: ↑b,
bool: 𝔹,
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
not: ¬A,
and: P ∧ Q,
function: x:A ⟶ B[x],
subtract: n - m,
natural_number: $n,
universe: Type
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
so_apply: x[s],
int_seg: {i..j-},
uimplies: b supposing a,
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,
implies: P ⇒ Q,
not: ¬A,
top: Top,
prop: ℙ,
less_than: a < b,
squash: ↓T,
first_index: index-of-first x in L.P[x],
iff: P ⇐⇒ Q,
cand: A c∧ B,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
ge: i ≥ j ,
rev_implies: P ⇐ Q,
le: A ≤ B,
nat: ℕ
Lemmas referenced :
btrue_neq_bfalse,
assert_elim,
equal_wf,
and_wf,
not_assert_elim,
member-firstn,
l_exists_iff,
assert_witness,
bool_wf,
list_wf,
less_than_wf,
l_member_wf,
assert_wf,
firstn_wf,
l_exists_wf,
int_term_value_add_lemma,
itermAdd_wf,
lelt_wf,
nat_wf,
le_wf,
nat_properties,
search_wf,
non_neg_length,
int_term_value_subtract_lemma,
itermSubtract_wf,
first_index_wf,
subtract_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,
search_property
Rules used in proof :
cut,
lemma_by_obid,
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
dependent_functionElimination,
thin,
isectElimination,
hypothesisEquality,
hypothesis,
lambdaEquality,
applyEquality,
cumulativity,
setElimination,
rename,
independent_isectElimination,
natural_numberEquality,
productElimination,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
because_Cache,
imageElimination,
equalityTransitivity,
equalitySymmetry,
setEquality,
addEquality,
introduction,
functionEquality,
universeEquality,
isect_memberFormation,
independent_pairEquality,
independent_functionElimination,
lambdaFormation,
dependent_set_memberEquality
Latex:
\mforall{}[T:Type]. \mforall{}[P:T {}\mrightarrow{} \mBbbB{}]. \mforall{}[L:T List].
(\muparrow{}P[L[index-of-first x in L.P[x] - 1]]) \mwedge{} (\mneg{}(\mexists{}x\mmember{}firstn(index-of-first x in L.P[x] - 1;L). \muparrow{}P[x]))
supposing 0 < index-of-first x in L.P[x]
Date html generated:
2016_05_15-PM-04_11_58
Last ObjectModification:
2016_01_16-AM-11_09_00
Theory : general
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