Nuprl Lemma : count_index_pairs_wf
∀[T:Type]. ∀[P:L:(T List) ⟶ ℕ||L|| - 1 ⟶ ℕ||L|| ⟶ 𝔹]. ∀[L:T List]. (count(i<j<||L|| : P L i j) ∈ ℕ)
Proof
Definitions occuring in Statement :
count_index_pairs: count(i<j<||L|| : P L i j),
length: ||as||,
list: T List,
int_seg: {i..j-},
nat: ℕ,
bool: 𝔹,
uall: ∀[x:A]. B[x],
member: t ∈ T,
function: x:A ⟶ B[x],
subtract: n - m,
natural_number: $n,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
and: P ∧ Q,
uiff: uiff(P;Q),
int_seg: {i..j-},
count_index_pairs: count(i<j<||L|| : P L i j),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
band: p ∧b q,
ifthenelse: if b then t else f fi ,
lelt: i ≤ j < k,
guard: {T},
decidable: Dec(P),
or: P ∨ Q,
less_than: a < b,
squash: ↓T,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
prop: ℙ,
bfalse: ff,
sq_type: SQType(T),
assert: ↑b,
bnot: ¬bb,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
nat: ℕ,
double_sum: sum(f[x; y] | x < n; y < m),
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
so_apply: x[s],
le: A ≤ B,
less_than': less_than'(a;b)
Lemmas referenced :
list_wf,
int_seg_wf,
subtract_wf,
length_wf,
bool_wf,
assert_of_lt_int,
length_wf_nat,
double_sum_wf,
lt_int_wf,
eqtt_to_assert,
int_seg_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermVar_wf,
itermSubtract_wf,
itermConstant_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_subtract_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
lelt_wf,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
eqff_to_assert,
assert-bnot,
not_functionality_wrt_iff,
assert_wf,
less_than_wf,
iff_transitivity,
iff_weakening_uiff,
assert_of_band,
le_wf,
non_neg_sum,
sum_wf,
istype-int,
istype-void,
int_subtype_base,
istype-false,
set_subtype_base
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalHypSubstitution,
hypothesis,
sqequalRule,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
universeIsType,
extract_by_obid,
isectElimination,
thin,
hypothesisEquality,
isect_memberEquality_alt,
because_Cache,
functionIsType,
natural_numberEquality,
universeEquality,
independent_isectElimination,
productElimination,
rename,
setElimination,
lambdaEquality,
lambdaFormation,
unionElimination,
equalityElimination,
applyEquality,
functionExtensionality,
cumulativity,
dependent_set_memberEquality,
independent_pairFormation,
dependent_functionElimination,
imageElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
promote_hyp,
instantiate,
productEquality,
dependent_set_memberEquality_alt,
lambdaEquality_alt,
inhabitedIsType,
lambdaFormation_alt,
dependent_pairFormation_alt,
productIsType,
equalityIsType1,
equalityIsType2,
baseApply,
closedConclusion,
baseClosed
Latex:
\mforall{}[T:Type]. \mforall{}[P:L:(T List) {}\mrightarrow{} \mBbbN{}||L|| - 1 {}\mrightarrow{} \mBbbN{}||L|| {}\mrightarrow{} \mBbbB{}]. \mforall{}[L:T List].
(count(i<j<||L|| : P L i j) \mmember{} \mBbbN{})
Date html generated:
2019_10_15-AM-10_58_12
Last ObjectModification:
2018_10_11-PM-05_31_17
Theory : list!
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