Nuprl Lemma : seq-append-assoc
∀[n,m,k:ℕ]. ∀[s1,s2,s3:Top].
(seq-append(n;m + k;s1;seq-append(m;k;s2;s3)) ~ seq-append(n + m;k;seq-append(n;m;s1;s2);s3))
Proof
Definitions occuring in Statement :
seq-append: seq-append(n;m;s1;s2),
nat: ℕ,
uall: ∀[x:A]. B[x],
top: Top,
add: n + m,
sqequal: s ~ t
Definitions unfolded in proof :
seq-append: seq-append(n;m;s1;s2),
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
top: Top,
nat: ℕ,
all: ∀x:A. B[x],
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
less_than: a < b,
less_than': less_than'(a;b),
true: True,
squash: ↓T,
not: ¬A,
false: False,
bfalse: ff,
exists: ∃x:A. B[x],
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
prop: ℙ,
ge: i ≥ j ,
subtract: n - m,
nat_plus: ℕ+,
le: A ≤ B,
decidable: Dec(P),
has-value: (a)↓
Lemmas referenced :
istype-int,
istype-top,
istype-nat,
istype-void,
lt_int_wf,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
set_subtype_base,
int_subtype_base,
bool_subtype_base,
bool_cases_sqequal,
subtype_base_sq,
iff_transitivity,
assert_wf,
bnot_wf,
not_wf,
less_than_wf,
iff_weakening_uiff,
assert_of_bnot,
istype-less_than,
istype-assert,
le_wf,
bool_wf,
add_functionality_wrt_le,
subtract_wf,
le_reflexive,
minus-one-mul,
zero-add,
one-mul,
add-mul-special,
add-associates,
two-mul,
add-commutes,
mul-distributes-right,
zero-mul,
less-iff-le,
add-zero,
not-lt-2,
minus-one-mul-top,
add-swap,
omega-shadow,
mul-distributes,
minus-add,
mul-associates,
mul-swap,
mul-commutes,
le-add-cancel,
nat_properties,
decidable__lt,
has-value_wf_base,
is-exception_wf,
bottom-sqle
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
Error :isect_memberFormation_alt,
introduction,
cut,
thin,
Error :lambdaFormation_alt,
Error :equalityIsType4,
extract_by_obid,
hypothesis,
hypothesisEquality,
because_Cache,
sqequalHypSubstitution,
axiomSqEquality,
Error :inhabitedIsType,
Error :isect_memberEquality_alt,
isectElimination,
Error :isectIsTypeImplies,
voidElimination,
setElimination,
rename,
multiplyEquality,
minusEquality,
natural_numberEquality,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_isectElimination,
lessCases,
independent_pairFormation,
imageMemberEquality,
baseClosed,
imageElimination,
independent_functionElimination,
addEquality,
Error :dependent_pairFormation_alt,
baseApply,
closedConclusion,
applyEquality,
intEquality,
Error :lambdaEquality_alt,
promote_hyp,
dependent_functionElimination,
instantiate,
Error :functionIsType,
Error :universeIsType,
Error :equalityIsType1,
cumulativity,
Error :dependent_set_memberEquality_alt,
sqequalSqle,
divergentSqle,
callbyvalueLess,
sqleReflexivity,
lessExceptionCases,
axiomSqleEquality,
exceptionSqequal
Latex:
\mforall{}[n,m,k:\mBbbN{}]. \mforall{}[s1,s2,s3:Top].
(seq-append(n;m + k;s1;seq-append(m;k;s2;s3)) \msim{} seq-append(n + m;k;seq-append(n;m;s1;s2);s3))
Date html generated:
2019_06_20-AM-11_28_36
Last ObjectModification:
2018_10_27-AM-11_38_11
Theory : bar-induction
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