Nuprl Lemma : count_pairs_wf
∀[T:Type]. ∀[L:T List]. ∀[P:T ⟶ T ⟶ 𝔹]. (count(x < y in L | P[x;y]) ∈ ℕ)
Proof
Definitions occuring in Statement :
count_pairs: count(x < y in L | P[x; y]),
list: T List,
nat: ℕ,
bool: 𝔹,
uall: ∀[x:A]. B[x],
so_apply: x[s1;s2],
member: t ∈ T,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
count_pairs: count(x < y in L | P[x; y]),
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
so_lambda: λ2x y.t[x; y],
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
band: p ∧b q,
ifthenelse: if b then t else f fi ,
so_apply: x[s1;s2],
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
less_than: a < b,
squash: ↓T,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
prop: ℙ,
bfalse: ff,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
double_sum: sum(f[x; y] | x < n; y < m),
so_lambda: λ2x.t[x],
so_apply: x[s],
less_than': less_than'(a;b)
Lemmas referenced :
double_sum_wf,
eqtt_to_assert,
assert_of_lt_int,
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,
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,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
istype-less_than,
int_seg_wf,
length_wf,
istype-le,
bool_wf,
list_wf,
istype-universe,
non_neg_sum,
length_wf_nat,
sum_wf,
lt_int_wf,
assert_wf,
less_than_wf,
istype-false
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation_alt,
introduction,
cut,
dependent_set_memberEquality_alt,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
lambdaEquality_alt,
hypothesis,
inhabitedIsType,
lambdaFormation_alt,
unionElimination,
equalityElimination,
productElimination,
independent_isectElimination,
applyEquality,
setElimination,
rename,
imageElimination,
hypothesisEquality,
dependent_functionElimination,
natural_numberEquality,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
int_eqEquality,
Error :memTop,
independent_pairFormation,
universeIsType,
voidElimination,
equalityIstype,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
instantiate,
axiomEquality,
functionIsType,
isect_memberEquality_alt,
isectIsTypeImplies,
universeEquality,
closedConclusion,
cumulativity
Latex:
\mforall{}[T:Type]. \mforall{}[L:T List]. \mforall{}[P:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbB{}]. (count(x < y in L | P[x;y]) \mmember{} \mBbbN{})
Date html generated:
2020_05_20-AM-07_49_03
Last ObjectModification:
2020_01_22-PM-05_15_17
Theory : list!
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