Nuprl Lemma : count-all
∀[T:Type]. ∀[P:T ⟶ 𝔹]. ∀[L:T List]. count(P;L) ~ ||L|| supposing (∀x∈L.↑(P x))
Proof
Definitions occuring in Statement :
count: count(P;L),
l_all: (∀x∈L.P[x]),
length: ||as||,
list: T List,
assert: ↑b,
bool: 𝔹,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
guard: {T},
prop: ℙ,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
implies: P ⇒ Q,
not: ¬A,
top: Top,
and: P ∧ Q,
sq_type: SQType(T)
Lemmas referenced :
bool_wf,
list_wf,
l_member_wf,
assert_wf,
l_all_wf2,
length_wf,
filter_trivial,
count-length-filter,
int_formula_prop_wf,
int_formula_prop_not_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_and_lemma,
intformnot_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformand_wf,
satisfiable-full-omega-tt,
le_wf,
nat_properties,
count_wf,
decidable__le,
int_subtype_base,
set_subtype_base,
subtype_base_sq
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
instantiate,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
because_Cache,
independent_isectElimination,
sqequalRule,
hypothesis,
dependent_functionElimination,
unionElimination,
hypothesisEquality,
equalityTransitivity,
equalitySymmetry,
applyEquality,
lambdaEquality,
setElimination,
rename,
setEquality,
intEquality,
natural_numberEquality,
dependent_pairFormation,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
dependent_set_memberEquality,
independent_functionElimination,
sqequalAxiom,
functionEquality,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}[P:T {}\mrightarrow{} \mBbbB{}]. \mforall{}[L:T List]. count(P;L) \msim{} ||L|| supposing (\mforall{}x\mmember{}L.\muparrow{}(P x))
Date html generated:
2016_05_15-PM-03_40_28
Last ObjectModification:
2016_01_16-AM-10_52_04
Theory : general
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