Nuprl Lemma : last_with_property
∀[T:Type]
∀L:T List
∀[P:ℕ||L|| ⟶ ℙ]
((∀x:ℕ||L||. Dec(P x)) ⇒ (∃i:ℕ||L||. (P i)) ⇒ (∃i:ℕ||L||. ((P i) ∧ (∀j:ℕ||L||. ¬(P j) supposing i < j))))
Proof
Definitions occuring in Statement :
length: ||as||,
list: T List,
int_seg: {i..j-},
less_than: a < b,
decidable: Dec(P),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
exists: ∃x:A. B[x],
and: P ∧ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
less_than: a < b,
squash: ↓T,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
top: Top,
cand: A c∧ B,
subtype_rel: A ⊆r B,
interleaving_occurence: interleaving_occurence(T;L1;L2;L;f1;f2),
nat: ℕ,
ge: i ≥ j ,
sq_type: SQType(T)
Lemmas referenced :
interleaving_split,
exists_wf,
int_seg_wf,
length_wf,
all_wf,
decidable_wf,
list_wf,
int_seg_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
subtract_wf,
decidable__le,
itermSubtract_wf,
int_term_value_subtract_lemma,
lelt_wf,
subtype_rel_self,
less_than_wf,
isect_wf,
not_wf,
increasing_implies,
length_wf_nat,
nat_properties,
intformeq_wf,
int_formula_prop_eq_lemma,
subtype_base_sq,
int_subtype_base,
decidable__equal_int
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
dependent_functionElimination,
independent_functionElimination,
hypothesis,
productElimination,
natural_numberEquality,
sqequalRule,
lambdaEquality,
applyEquality,
functionEquality,
cumulativity,
universeEquality,
setElimination,
rename,
unionElimination,
imageElimination,
independent_isectElimination,
approximateComputation,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
functionExtensionality,
dependent_set_memberEquality,
because_Cache,
instantiate,
productEquality,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality
Latex:
\mforall{}[T:Type]
\mforall{}L:T List
\mforall{}[P:\mBbbN{}||L|| {}\mrightarrow{} \mBbbP{}]
((\mforall{}x:\mBbbN{}||L||. Dec(P x))
{}\mRightarrow{} (\mexists{}i:\mBbbN{}||L||. (P i))
{}\mRightarrow{} (\mexists{}i:\mBbbN{}||L||. ((P i) \mwedge{} (\mforall{}j:\mBbbN{}||L||. \mneg{}(P j) supposing i < j))))
Date html generated:
2019_10_15-AM-10_57_23
Last ObjectModification:
2018_09_17-PM-06_30_33
Theory : list!
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