Nuprl Lemma : bar-val-stable
∀[T:Type]. ∀[n:ℕ]. ∀[x:bar-base(T)].
∀[m:ℤ]. bar-val(m;x) = bar-val(n;x) ∈ (T?) supposing n ≤ m supposing ↑isl(bar-val(n;x))
Proof
Definitions occuring in Statement :
bar-val: bar-val(n;x),
bar-base: bar-base(T),
nat: ℕ,
assert: ↑b,
isl: isl(x),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
le: A ≤ B,
unit: Unit,
union: left + right,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
guard: {T},
uimplies: b supposing a,
prop: ℙ,
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
bar-val: bar-val(n;x),
isl: isl(x),
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
eq_int: (i =z j),
subtract: n - m,
bfalse: ff,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
not: ¬A,
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
top: Top,
true: True,
exposed-bfalse: exposed-bfalse,
bool: 𝔹,
unit: Unit,
it: ⋅,
exists: ∃x:A. B[x],
sq_type: SQType(T),
bnot: ¬bb,
squash: ↓T,
nequal: a ≠ b ∈ T
Lemmas referenced :
nat_properties,
less_than_transitivity1,
less_than_irreflexivity,
ge_wf,
less_than_wf,
le_wf,
assert_wf,
isl_wf,
unit_wf2,
bar-val_wf,
bar-base_wf,
bar-base-subtype,
true_wf,
false_wf,
equal_wf,
decidable__le,
subtract_wf,
not-ge-2,
less-iff-le,
condition-implies-le,
minus-one-mul,
zero-add,
minus-one-mul-top,
minus-add,
minus-minus,
add-associates,
add-swap,
add-commutes,
add_functionality_wrt_le,
add-zero,
le-add-cancel,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
le_weakening,
squash_wf,
not-le-2,
not-equal-2,
le-add-cancel-alt,
iff_weakening_equal,
le_weakening2,
nat_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
lambdaFormation,
natural_numberEquality,
independent_isectElimination,
independent_functionElimination,
voidElimination,
sqequalRule,
lambdaEquality,
dependent_functionElimination,
isect_memberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
intEquality,
cumulativity,
because_Cache,
applyEquality,
unionEquality,
unionElimination,
inlEquality,
dependent_set_memberEquality,
independent_pairFormation,
productElimination,
addEquality,
voidEquality,
minusEquality,
equalityElimination,
dependent_pairFormation,
promote_hyp,
instantiate,
imageElimination,
imageMemberEquality,
baseClosed,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}[n:\mBbbN{}]. \mforall{}[x:bar-base(T)].
\mforall{}[m:\mBbbZ{}]. bar-val(m;x) = bar-val(n;x) supposing n \mleq{} m supposing \muparrow{}isl(bar-val(n;x))
Date html generated:
2017_04_14-AM-07_45_59
Last ObjectModification:
2017_02_27-PM-03_16_37
Theory : co-recursion
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