Nuprl Lemma : seq-normalize-append
∀[n,m:ℕ]. ∀[s1,s2:Top]. (seq-normalize(n + m;seq-append(n;m;s1;s2)) ~ seq-append(n;m;s1;s2))
Proof
Definitions occuring in Statement :
seq-normalize: seq-normalize(n;s),
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),
seq-normalize: seq-normalize(n;s),
has-value: (a)↓,
member: t ∈ T,
subtype_rel: A ⊆r B,
nat: ℕ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
and: P ∧ Q,
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
less_than: a < b,
less_than': less_than'(a;b),
top: Top,
true: True,
squash: ↓T,
not: ¬A,
false: False,
prop: ℙ,
bfalse: ff,
exists: ∃x:A. B[x],
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,
decidable: Dec(P),
ge: i ≥ j ,
subtract: n - m,
nat_plus: ℕ+,
le: A ≤ B
Lemmas referenced :
set_subtype_base,
le_wf,
int_subtype_base,
has-value_wf_base,
is-exception_wf,
top_wf,
nat_wf,
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
less_than_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
iff_transitivity,
assert_wf,
bnot_wf,
not_wf,
iff_weakening_uiff,
assert_of_bnot,
bottom-sqle,
decidable__lt,
exception-not-value,
value-type-has-value,
int-value-type,
add-commutes,
add_functionality_wrt_le,
subtract_wf,
le_reflexive,
minus-one-mul,
zero-add,
one-mul,
add-mul-special,
add-associates,
two-mul,
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,
nat_properties,
set-value-type
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
cut,
thin,
sqequalSqle,
divergentSqle,
callbyvalueLess,
sqequalHypSubstitution,
hypothesis,
baseApply,
closedConclusion,
baseClosed,
hypothesisEquality,
applyEquality,
introduction,
extract_by_obid,
isectElimination,
intEquality,
lambdaEquality,
natural_numberEquality,
independent_isectElimination,
productElimination,
lessExceptionCases,
axiomSqleEquality,
exceptionSqequal,
sqleReflexivity,
because_Cache,
isect_memberFormation,
sqequalAxiom,
isect_memberEquality,
lambdaFormation,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
lessCases,
independent_pairFormation,
voidElimination,
voidEquality,
imageMemberEquality,
imageElimination,
independent_functionElimination,
setElimination,
rename,
dependent_pairFormation,
promote_hyp,
dependent_functionElimination,
instantiate,
cumulativity,
impliesFunctionality,
addEquality,
multiplyEquality,
minusEquality,
dependent_set_memberEquality
Latex:
\mforall{}[n,m:\mBbbN{}]. \mforall{}[s1,s2:Top]. (seq-normalize(n + m;seq-append(n;m;s1;s2)) \msim{} seq-append(n;m;s1;s2))
Date html generated:
2017_04_14-AM-07_26_47
Last ObjectModification:
2017_02_27-PM-02_56_21
Theory : bar-induction
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