Nuprl Lemma : find-first_wf
∀[T:Type]. ∀[L:T List]. ∀[P:{x:T| (x ∈ L)} ⟶ 𝔹]. (find-first(P;L) ∈ (∃x:T [first-member(T;x;L;P)]) ∨ (∀x∈L.¬↑(P x)))
Proof
Definitions occuring in Statement :
find-first: find-first(P;L),
first-member: first-member(T;x;L;P),
l_all: (∀x∈L.P[x]),
l_member: (x ∈ l),
list: T List,
assert: ↑b,
bool: 𝔹,
uall: ∀[x:A]. B[x],
sq_exists: ∃x:A [B[x]],
not: ¬A,
or: P ∨ Q,
member: t ∈ T,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
find-first: find-first(P;L),
prop: ℙ,
can-find-first-ext,
all: ∀x:A. B[x],
implies: P ⇒ Q,
first-member: first-member(T;x;L;P),
subtype_rel: A ⊆r B,
or: P ∨ Q,
so_lambda: λ2x.t[x],
and: P ∧ Q,
int_seg: {i..j-},
uimplies: b supposing a,
guard: {T},
lelt: i ≤ j < k,
decidable: Dec(P),
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
less_than: a < b,
squash: ↓T,
l_member: (x ∈ l),
le: A ≤ B,
less_than': less_than'(a;b),
cand: A c∧ B,
nat: ℕ,
ge: i ≥ j ,
so_apply: x[s],
sq_exists: ∃x:A [B[x]]
Lemmas referenced :
istype-universe,
l_member_wf,
bool_wf,
list_wf,
can-find-first-ext,
sq_exists_wf,
exists_wf,
int_seg_wf,
length_wf,
equal_wf,
select_wf,
int_seg_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
assert_wf,
int_seg_subtype_nat,
istype-false,
less_than_wf,
nat_properties,
not_wf,
select_member,
le_wf,
l_all_wf2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
sqequalHypSubstitution,
hypothesis,
functionIsType,
setIsType,
introduction,
extract_by_obid,
isectElimination,
thin,
hypothesisEquality,
universeIsType,
universeEquality,
sqequalRule,
instantiate,
inhabitedIsType,
lambdaFormation_alt,
equalityIsType1,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
independent_functionElimination,
applyEquality,
lambdaEquality_alt,
isectIsType,
unionIsType,
natural_numberEquality,
productEquality,
because_Cache,
setElimination,
rename,
independent_isectElimination,
productElimination,
unionElimination,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
imageElimination,
productIsType,
dependent_set_memberEquality_alt,
functionEquality
Latex:
\mforall{}[T:Type]. \mforall{}[L:T List]. \mforall{}[P:\{x:T| (x \mmember{} L)\} {}\mrightarrow{} \mBbbB{}].
(find-first(P;L) \mmember{} (\mexists{}x:T [first-member(T;x;L;P)]) \mvee{} (\mforall{}x\mmember{}L.\mneg{}\muparrow{}(P x)))
Date html generated:
2019_10_15-AM-11_08_07
Last ObjectModification:
2018_10_09-PM-03_14_28
Theory : general
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