Nuprl Lemma : decidable__exists_iseg
∀[T:Type]. ∀[P:(T List) ⟶ ℙ]. ((∀L:T List. Dec(P[L])) ⇒ (∀L:T List. Dec(∃L':T List. (L' ≤ L ∧ P[L']))))
Proof
Definitions occuring in Statement :
iseg: l1 ≤ l2,
list: T List,
decidable: Dec(P),
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
member: t ∈ T,
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
le: A ≤ B,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
prop: ℙ,
decidable: Dec(P),
or: P ∨ Q,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
guard: {T},
sq_type: SQType(T),
nat: ℕ,
ge: i ≥ j ,
less_than: a < b,
squash: ↓T,
cons: [a / b],
assert: ↑b,
ifthenelse: if b then t else f fi ,
bfalse: ff,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
int_iseg: {i...j},
cand: A c∧ B,
btrue: tt,
iseg: l1 ≤ l2,
less_than': less_than'(a;b),
true: True,
last: last(L),
subtract: n - m,
length: ||as||,
list_ind: list_ind,
nil: [],
it: ⋅
Lemmas referenced :
int_seg_properties,
full-omega-unsat,
intformand_wf,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
int_seg_wf,
decidable__equal_int,
subtract_wf,
subtype_base_sq,
set_subtype_base,
lelt_wf,
int_subtype_base,
intformnot_wf,
intformeq_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_subtract_lemma,
decidable__le,
decidable__lt,
istype-le,
istype-less_than,
subtype_rel_self,
non_neg_length,
nat_properties,
length_wf,
decidable__assert,
null_wf,
list-cases,
product_subtype_list,
null_cons_lemma,
last-lemma-sq,
pos_length,
iff_transitivity,
not_wf,
equal-wf-T-base,
list_wf,
assert_wf,
bnot_wf,
iff_weakening_uiff,
assert_of_null,
istype-assert,
nil_wf,
length_of_nil_lemma,
istype-void,
assert_of_bnot,
firstn_wf,
length_firstn,
le_wf,
decidable_wf,
iseg_wf,
primrec-wf2,
itermAdd_wf,
int_term_value_add_lemma,
istype-nat,
length_wf_nat,
istype-universe,
iseg_weakening,
null_nil_lemma,
iseg_nil,
last_wf,
cons_wf,
append_wf,
iseg_append,
append_back_nil,
iseg_append_iff,
iseg_single,
length_of_cons_lemma,
equal_wf,
list_extensionality,
length-append,
select_wf,
squash_wf,
true_wf,
select_append_front,
select_firstn,
iff_weakening_equal,
length_firstn_eq,
less_than_wf,
add_functionality_wrt_eq,
select_append_back,
nat_wf,
select-nthtl,
subtype_rel_list,
top_wf,
nth_tl_decomp,
nth_tl_is_nil,
minus-add,
minus-minus,
add-associates,
minus-one-mul,
add-swap,
add-mul-special,
add-commutes,
zero-add,
zero-mul,
select0,
hd_wf,
istype-false
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
setElimination,
rename,
productElimination,
hypothesis,
hypothesisEquality,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
Error :memTop,
sqequalRule,
independent_pairFormation,
universeIsType,
voidElimination,
unionElimination,
applyEquality,
instantiate,
cumulativity,
intEquality,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
dependent_set_memberEquality_alt,
because_Cache,
productIsType,
promote_hyp,
hypothesis_subsumption,
imageElimination,
baseClosed,
functionIsType,
equalityIstype,
functionEquality,
productEquality,
setIsType,
addEquality,
universeEquality,
inrFormation_alt,
inlFormation_alt,
hyp_replacement,
imageMemberEquality,
multiplyEquality
Latex:
\mforall{}[T:Type]. \mforall{}[P:(T List) {}\mrightarrow{} \mBbbP{}].
((\mforall{}L:T List. Dec(P[L])) {}\mRightarrow{} (\mforall{}L:T List. Dec(\mexists{}L':T List. (L' \mleq{} L \mwedge{} P[L']))))
Date html generated:
2020_05_19-PM-09_48_58
Last ObjectModification:
2019_12_31-PM-00_13_48
Theory : list_1
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