Nuprl Lemma : can-find-first2
∀[T:Type]. ∀L:T List. ∀P:{x:T| (x ∈ L)} ⟶ 𝔹. ((∃x:T [first-member(T;x;L;P)]) ∨ (∀x∈L.¬↑(P x)))
Proof
Definitions occuring in Statement :
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],
all: ∀x:A. B[x],
sq_exists: ∃x:A [B[x]],
not: ¬A,
or: P ∨ Q,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
prop: ℙ,
or: P ∨ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_exists: ∃x:A [B[x]],
first-member: first-member(T;x;L;P),
exists: ∃x:A. B[x],
and: P ∧ Q,
cand: A c∧ B,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
implies: P ⇒ Q,
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
less_than: a < b,
squash: ↓T,
l_member: (x ∈ l),
le: A ≤ B,
less_than': less_than'(a;b),
nat: ℕ,
ge: i ≥ j
Lemmas referenced :
can-find-first1-ext,
l_member_wf,
list-subtype,
bool_wf,
list_wf,
l_all_wf2,
not_wf,
assert_wf,
equal_functionality_wrt_subtype_rel2,
equal_wf,
select_wf,
int_seg_properties,
length_wf,
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,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
all_wf,
int_seg_wf,
exists_wf,
int_seg_subtype_nat,
false_wf,
less_than_wf,
nat_properties,
sq_exists_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setEquality,
cumulativity,
hypothesisEquality,
hypothesis,
dependent_functionElimination,
because_Cache,
equalityTransitivity,
equalitySymmetry,
rename,
functionEquality,
universeEquality,
unionElimination,
inlEquality,
sqequalRule,
lambdaEquality,
applyEquality,
functionExtensionality,
setElimination,
dependent_set_memberEquality,
inrEquality,
productElimination,
dependent_pairFormation,
independent_isectElimination,
independent_functionElimination,
independent_pairFormation,
productEquality,
natural_numberEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
imageElimination
Latex:
\mforall{}[T:Type]. \mforall{}L:T List. \mforall{}P:\{x:T| (x \mmember{} L)\} {}\mrightarrow{} \mBbbB{}. ((\mexists{}x:T [first-member(T;x;L;P)]) \mvee{} (\mforall{}x\mmember{}L.\mneg{}\muparrow{}(P x)))
Date html generated:
2018_05_21-PM-06_34_16
Last ObjectModification:
2017_07_26-PM-04_52_29
Theory : general
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