Nuprl Lemma : before_all_imp_count_zero
∀s:QOSet. ∀a:|s|. ∀cs:|s| List. ((↑(∀bc(:|s|) ∈ cs. (c <b a))) ⇒ ((a #∈ cs) = 0 ∈ ℤ))
Proof
Definitions occuring in Statement :
count: a #∈ as,
ball: ball,
list: T List,
assert: ↑b,
all: ∀x:A. B[x],
implies: P ⇒ Q,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T,
qoset: QOSet,
set_blt: a <b b,
set_car: |p|
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
implies: P ⇒ Q,
prop: ℙ,
qoset: QOSet,
dset: DSet,
so_apply: x[s],
top: Top,
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
and: P ∧ Q,
ball: ball,
iff: P ⇐⇒ Q,
uiff: uiff(P;Q),
uimplies: b supposing a,
rev_implies: P ⇐ Q,
b2i: b2i(b),
infix_ap: x f y,
bool: 𝔹,
unit: Unit,
it: ⋅,
bfalse: ff,
not: ¬A,
false: False,
guard: {T},
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x]
Lemmas referenced :
list_induction,
assert_wf,
ball_wf,
set_car_wf,
set_blt_wf,
equal-wf-T-base,
count_wf,
list_wf,
ball_nil_lemma,
count_nil_lemma,
true_wf,
ball_cons_lemma,
count_cons_lemma,
band_wf,
qoset_wf,
iff_transitivity,
set_lt_wf,
iff_weakening_uiff,
assert_of_band,
assert_of_set_lt,
set_eq_wf,
bool_wf,
uiff_transitivity,
equal_wf,
eqtt_to_assert,
assert_of_dset_eq,
bnot_wf,
not_wf,
eqff_to_assert,
assert_of_bnot,
set_lt_transitivity_2,
set_leq_weakening_eq,
set_lt_irreflexivity,
decidable__equal_int,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformeq_wf,
itermAdd_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
because_Cache,
sqequalRule,
lambdaEquality,
functionEquality,
dependent_functionElimination,
setElimination,
rename,
hypothesis,
hypothesisEquality,
intEquality,
baseClosed,
independent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
natural_numberEquality,
productEquality,
independent_pairFormation,
productElimination,
independent_isectElimination,
applyEquality,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
impliesFunctionality,
dependent_pairFormation,
int_eqEquality,
computeAll
Latex:
\mforall{}s:QOSet. \mforall{}a:|s|. \mforall{}cs:|s| List. ((\muparrow{}(\mforall{}\msubb{}c(:|s|) \mmember{} cs. (c <\msubb{} a))) {}\mRightarrow{} ((a \#\mmember{} cs) = 0))
Date html generated:
2017_10_01-AM-10_01_34
Last ObjectModification:
2017_03_03-PM-01_03_44
Theory : polynom_2
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