Nuprl Lemma : l-ordered-remove-combine
∀T:Type. ∀R:T ⟶ T ⟶ ℙ. ∀cmp:T ⟶ ℤ. ∀L:T List.
(l-ordered(T;x,y.R[x;y];L) ⇒ l-ordered(T;x,y.R[x;y];remove-combine(cmp;L)))
Proof
Definitions occuring in Statement :
l-ordered: l-ordered(T;x,y.R[x; y];L),
remove-combine: remove-combine(cmp;l),
list: T List,
prop: ℙ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
int: ℤ,
universe: Type
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,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
so_apply: x[s],
true: True,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
top: Top,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
uimplies: b supposing a,
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
cand: A c∧ B,
not: ¬A,
subtype_rel: A ⊆r B
Lemmas referenced :
list_induction,
l-ordered_wf,
remove-combine_wf,
list_wf,
true_wf,
l-ordered-nil-true,
remove-combine-nil,
nil_wf,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
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,
l-ordered-cons,
less_than_wf,
remove-combine-implies-member,
l_member_wf,
all_wf,
remove-combine-cons,
cons_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
functionEquality,
cumulativity,
applyEquality,
functionExtensionality,
hypothesis,
dependent_functionElimination,
independent_functionElimination,
natural_numberEquality,
addLevel,
impliesFunctionality,
because_Cache,
productElimination,
isect_memberEquality,
voidElimination,
voidEquality,
rename,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
dependent_pairFormation,
promote_hyp,
instantiate,
independent_pairFormation,
productEquality,
universeEquality,
intEquality
Latex:
\mforall{}T:Type. \mforall{}R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}. \mforall{}cmp:T {}\mrightarrow{} \mBbbZ{}. \mforall{}L:T List.
(l-ordered(T;x,y.R[x;y];L) {}\mRightarrow{} l-ordered(T;x,y.R[x;y];remove-combine(cmp;L)))
Date html generated:
2018_05_21-PM-07_37_50
Last ObjectModification:
2017_07_26-PM-05_12_08
Theory : general
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