Nuprl Lemma : poss-maj-invariant
∀T:Type. ∀eq:EqDecider(T). ∀L:T List. ∀x:T.
let n,z = poss-maj(eq;L;x)
in ((count(eq z;L) - count(λt.(¬b(eq z t));L)) ≤ n)
∧ (∀y:T. ((¬↑(eq z y)) ⇒ (n ≤ (count(λt.(¬b(eq y t));L) - count(eq y;L)))))
Proof
Definitions occuring in Statement :
poss-maj: poss-maj(eq;L;x),
count: count(P;L),
list: T List,
deq: EqDecider(T),
bnot: ¬bb,
assert: ↑b,
le: A ≤ B,
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q,
apply: f a,
lambda: λx.A[x],
spread: spread def,
subtract: n - m,
universe: Type
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
implies: P ⇒ Q,
not: ¬A,
top: Top,
prop: ℙ,
decidable: Dec(P),
or: P ∨ Q,
subtype_rel: A ⊆r B,
le: A ≤ B,
less_than': less_than'(a;b),
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,
uiff: uiff(P;Q),
rev_implies: P ⇐ Q,
int_iseg: {i...j},
cand: A c∧ B,
so_lambda: λ2x.t[x],
deq: EqDecider(T),
so_apply: x[s],
count: count(P;L),
poss-maj: poss-maj(eq;L;x),
list_accum: list_accum,
nil: [],
it: ⋅,
subtract: n - m,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
bool: 𝔹,
unit: Unit,
btrue: tt,
eqof: eqof(d),
sq_type: SQType(T),
bnot: ¬bb,
int_upper: {i...},
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
int_seg_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
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,
int_seg_subtype,
false_wf,
decidable__le,
intformnot_wf,
itermSubtract_wf,
intformeq_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
int_formula_prop_eq_lemma,
le_wf,
length_wf,
non_neg_length,
nat_properties,
decidable__lt,
lelt_wf,
less_than_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,
assert_of_null,
iff_weakening_uiff,
assert_of_bnot,
firstn_wf,
length_firstn,
all_wf,
count_wf,
equal_wf,
set_wf,
primrec-wf2,
nat_wf,
itermAdd_wf,
int_term_value_add_lemma,
length_wf_nat,
deq_wf,
reduce_nil_lemma,
last_wf,
list_accum_append,
subtype_rel_list,
top_wf,
list_accum_cons_lemma,
list_accum_nil_lemma,
poss-maj_wf,
bool_wf,
eqtt_to_assert,
safe-assert-deq,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
eq_int_wf,
assert_of_eq_int,
neg_assert_of_eq_int,
int_upper_subtype_nat,
nequal-le-implies,
zero-add,
cons_wf,
nil_wf,
subtract-is-int-iff,
le_weakening2,
int_upper_properties,
count-append,
count-single,
eqof_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
natural_numberEquality,
because_Cache,
hypothesisEquality,
hypothesis,
setElimination,
rename,
productElimination,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
unionElimination,
addLevel,
applyEquality,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
levelHypothesis,
hypothesis_subsumption,
dependent_set_memberEquality,
cumulativity,
imageElimination,
independent_functionElimination,
promote_hyp,
baseClosed,
impliesFunctionality,
productEquality,
functionEquality,
addEquality,
universeEquality,
equalityElimination,
instantiate,
pointwiseFunctionality,
baseApply,
closedConclusion,
impliesLevelFunctionality,
hyp_replacement
Latex:
\mforall{}T:Type. \mforall{}eq:EqDecider(T). \mforall{}L:T List. \mforall{}x:T.
let n,z = poss-maj(eq;L;x)
in ((count(eq z;L) - count(\mlambda{}t.(\mneg{}\msubb{}(eq z t));L)) \mleq{} n)
\mwedge{} (\mforall{}y:T. ((\mneg{}\muparrow{}(eq z y)) {}\mRightarrow{} (n \mleq{} (count(\mlambda{}t.(\mneg{}\msubb{}(eq y t));L) - count(eq y;L)))))
Date html generated:
2017_04_17-AM-09_08_31
Last ObjectModification:
2017_02_27-PM-05_18_12
Theory : decidable!equality
Home
Index