Nuprl Lemma : insert-int-comm
∀T:Type. ∀a,b:T. ((T ⊆r ℤ) ⇒ (∀L:T List. (insert-int(b;insert-int(a;L)) = insert-int(a;insert-int(b;L)) ∈ (T List))))
Proof
Definitions occuring in Statement :
insert-int: insert-int(x;l),
list: T List,
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
so_apply: x[s],
implies: P ⇒ Q,
top: Top,
prop: ℙ,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
ifthenelse: if b then t else f fi ,
false: False,
guard: {T},
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
not: ¬A,
squash: ↓T,
decidable: Dec(P),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
le: A ≤ B,
less_than': less_than'(a;b),
true: True
Lemmas referenced :
list_induction,
all_wf,
equal-wf-base,
list_wf,
list_subtype_base,
int_subtype_base,
insert_int_nil_lemma,
nil_wf,
insert-int-cons,
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
less_than_transitivity2,
le_weakening2,
less_than_irreflexivity,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
less_than_wf,
cons_wf,
squash_wf,
true_wf,
decidable__equal_int,
false_wf,
not-equal-2,
not-lt-2,
add_functionality_wrt_le,
add-associates,
add-commutes,
le-add-cancel,
add-swap,
insert-int_wf,
subtype_rel_self,
iff_weakening_equal,
less-iff-le,
subtype_rel_list,
subtype_rel_wf
Rules used in proof :
cut,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
intEquality,
sqequalRule,
lambdaEquality,
hypothesis,
baseApply,
closedConclusion,
baseClosed,
hypothesisEquality,
applyEquality,
because_Cache,
independent_isectElimination,
independent_functionElimination,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
rename,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
dependent_pairFormation,
promote_hyp,
instantiate,
cumulativity,
imageElimination,
universeEquality,
independent_pairFormation,
addEquality,
natural_numberEquality,
imageMemberEquality
Latex:
\mforall{}T:Type. \mforall{}a,b:T.
((T \msubseteq{}r \mBbbZ{}) {}\mRightarrow{} (\mforall{}L:T List. (insert-int(b;insert-int(a;L)) = insert-int(a;insert-int(b;L)))))
Date html generated:
2017_09_29-PM-05_48_41
Last ObjectModification:
2017_07_26-PM-01_36_58
Theory : list_0
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