Nuprl Lemma : seq-append1-assoc
∀[n,n1:ℕ]. ∀[s,s1,t:Top].
(λm.if m=n + n1 then t else (seq-append(n;n1;s;s1) m) ~ seq-append(n;n1 + 1;s;λm.if m=n1 then t else (s1 m)))
Proof
Definitions occuring in Statement :
seq-append: seq-append(n;m;s1;s2),
nat: ℕ,
uall: ∀[x:A]. B[x],
top: Top,
int_eq: if a=b then c else d,
apply: f a,
lambda: λx.A[x],
add: n + m,
natural_number: $n,
sqequal: s ~ t
Definitions unfolded in proof :
seq-append: seq-append(n;m;s1;s2),
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],
decidable: Dec(P),
or: P ∨ Q,
less_than: a < b,
less_than': less_than'(a;b),
top: Top,
true: True,
squash: ↓T,
not: ¬A,
implies: P ⇒ Q,
false: False,
prop: ℙ,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
bfalse: ff,
exists: ∃x:A. B[x],
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,
subtract: n - m,
le: A ≤ B,
ge: i ≥ j ,
nat_plus: ℕ+
Lemmas referenced :
set_subtype_base,
le_wf,
istype-int,
int_subtype_base,
decidable__lt,
istype-void,
decidable__int_equal,
less_than_wf,
lt_int_wf,
eqtt_to_assert,
assert_of_lt_int,
eq_int_wf,
subtract_wf,
assert_of_eq_int,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
iff_transitivity,
assert_wf,
bnot_wf,
not_wf,
equal_wf,
iff_weakening_uiff,
assert_of_bnot,
not-lt-2,
condition-implies-le,
minus-add,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-commutes,
le_antisymmetry_iff,
add_functionality_wrt_le,
add-associates,
le-add-cancel,
has-value_wf_base,
is-exception_wf,
less-iff-le,
not-equal-2,
zero-add,
le-add-cancel2,
bottom-sqle,
exception-not-value,
value-type-has-value,
int-value-type,
equal-wf-base-T,
nat_wf,
set-value-type,
add-mul-special,
two-mul,
mul-distributes-right,
zero-mul,
add-zero,
one-mul,
le_reflexive,
omega-shadow,
mul-associates,
top_wf,
nat_properties
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
cut,
thin,
sqequalSqle,
divergentSqle,
callbyvalueIntEq,
sqequalHypSubstitution,
hypothesis,
baseApply,
closedConclusion,
baseClosed,
hypothesisEquality,
applyEquality,
introduction,
extract_by_obid,
isectElimination,
intEquality,
Error :lambdaEquality_alt,
natural_numberEquality,
independent_isectElimination,
productElimination,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
setElimination,
rename,
unionElimination,
because_Cache,
lessCases,
Error :isect_memberFormation_alt,
axiomSqEquality,
Error :inhabitedIsType,
Error :isect_memberEquality_alt,
Error :universeIsType,
independent_pairFormation,
voidElimination,
imageMemberEquality,
Error :lambdaFormation_alt,
imageElimination,
independent_functionElimination,
int_eqReduceTrueSq,
int_eqReduceFalseSq,
sqleReflexivity,
addEquality,
equalityElimination,
Error :dependent_pairFormation_alt,
Error :equalityIsType1,
promote_hyp,
instantiate,
cumulativity,
Error :equalityIsType2,
minusEquality,
int_eqExceptionCases,
axiomSqleEquality,
exceptionSqequal,
callbyvalueLess,
lessExceptionCases,
multiplyEquality,
Error :dependent_set_memberEquality_alt
Latex:
\mforall{}[n,n1:\mBbbN{}]. \mforall{}[s,s1,t:Top].
(\mlambda{}m.if m=n + n1 then t else (seq-append(n;n1;s;s1) m) \msim{} seq-append(n;n1 + 1;s;\mlambda{}m.if m=n1
then t
else (s1 m)))
Date html generated:
2019_06_20-AM-11_28_46
Last ObjectModification:
2018_09_28-PM-10_41_38
Theory : bar-induction
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