Nuprl Lemma : remove-combine-l-ordered-implies-member
∀[T:Type]
∀cmp:T ⟶ ℤ. ∀x:T. ∀l:T List.
(l-ordered(T;x,y.cmp x < cmp y;l) ⇒ (x ∈ remove-combine(cmp;l)) ⇒ (¬((cmp x) = 0 ∈ ℤ)))
Proof
Definitions occuring in Statement :
l-ordered: l-ordered(T;x,y.R[x; y];L),
remove-combine: remove-combine(cmp;l),
l_member: (x ∈ l),
list: T List,
less_than: a < b,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
so_lambda: λ2x.t[x],
implies: P ⇒ Q,
prop: ℙ,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
so_apply: x[s],
false: False,
iff: P ⇐⇒ Q,
and: P ∧ Q,
top: Top,
remove-combine: remove-combine(cmp;l),
list_ind: list_ind,
nil: [],
it: ⋅,
bool: 𝔹,
unit: Unit,
btrue: tt,
uiff: uiff(P;Q),
uimplies: b supposing a,
ifthenelse: if b then t else f fi ,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
bfalse: ff,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
nequal: a ≠ b ∈ T ,
less_than: a < b,
squash: ↓T
Lemmas referenced :
list_induction,
l-ordered_wf,
less_than_wf,
l_member_wf,
remove-combine_wf,
not_wf,
equal-wf-T-base,
list_wf,
false_wf,
true_wf,
l-ordered-nil-true,
remove-combine-nil,
nil_member,
nil_wf,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
satisfiable-full-omega-tt,
intformand_wf,
intformless_wf,
itermVar_wf,
intformeq_wf,
itermConstant_wf,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
lt_int_wf,
assert_of_lt_int,
and_wf,
or_wf,
cons_member,
cons_wf,
all_wf,
l-ordered-cons,
remove-combine-cons
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
thin,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
because_Cache,
sqequalRule,
lambdaEquality,
functionEquality,
cumulativity,
hypothesisEquality,
applyEquality,
functionExtensionality,
hypothesis,
dependent_functionElimination,
intEquality,
baseClosed,
independent_functionElimination,
voidElimination,
addLevel,
impliesFunctionality,
productElimination,
isect_memberEquality,
voidEquality,
levelHypothesis,
rename,
natural_numberEquality,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
independent_pairFormation,
computeAll,
promote_hyp,
instantiate,
dependent_set_memberEquality,
applyLambdaEquality,
setElimination,
imageElimination,
productEquality,
universeEquality
Latex:
\mforall{}[T:Type]
\mforall{}cmp:T {}\mrightarrow{} \mBbbZ{}. \mforall{}x:T. \mforall{}l:T List.
(l-ordered(T;x,y.cmp x < cmp y;l) {}\mRightarrow{} (x \mmember{} remove-combine(cmp;l)) {}\mRightarrow{} (\mneg{}((cmp x) = 0)))
Date html generated:
2018_05_21-PM-07_39_12
Last ObjectModification:
2017_07_26-PM-05_13_34
Theory : general
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