Nuprl Lemma : int-seg-case_wf
∀[i,j:ℤ]. ∀[F,G:{i..j-} ⟶ ℙ]. ∀[d:∀k:{i..j-}. (F[k] ∨ G[k])].
(int-seg-case(i;j;d) ∈ (∃k:{i..j-}. F[k]) ∨ (∀k:{i..j-}. G[k]))
Proof
Definitions occuring in Statement :
int-seg-case: int-seg-case(i;j;d),
int_seg: {i..j-},
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
or: P ∨ Q,
member: t ∈ T,
function: x:A ⟶ B[x],
int: ℤ
Definitions unfolded in proof :
so_lambda: λ2x.t[x],
nat_plus: ℕ+,
bfalse: ff,
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
exposed-it: exposed-it,
le_int: i ≤z j,
lt_int: i <z j,
ifthenelse: if b then t else f fi ,
bnot: ¬bb,
label: ...$L... t,
sq_type: SQType(T),
gt: i > j,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
subtract: n - m,
uiff: uiff(P;Q),
sq_stable: SqStable(P),
le: A ≤ B,
ge: i ≥ j ,
nat: ℕ,
prop: ℙ,
subtype_rel: A ⊆r B,
so_apply: x[s],
exists: ∃x:A. B[x],
uimplies: b supposing a,
guard: {T},
lelt: i ≤ j < k,
cand: A c∧ B,
int_seg: {i..j-},
false: False,
implies: P ⇒ Q,
not: ¬A,
squash: ↓T,
true: True,
top: Top,
less_than': less_than'(a;b),
and: P ∧ Q,
less_than: a < b,
or: P ∨ Q,
decidable: Dec(P),
all: ∀x:A. B[x],
int-seg-case: int-seg-case(i;j;d),
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
minus-minus,
not-equal-2,
decidable__int_equal,
and_wf,
set_subtype_base,
omega-shadow,
mul-distributes-right,
two-mul,
one-mul,
le_reflexive,
le-add-cancel-alt,
not-lt-2,
add-is-int-iff,
assert_of_le_int,
true_wf,
squash_wf,
eqff_to_assert,
uiff_transitivity2,
assert_of_lt_int,
eqtt_to_assert,
uiff_transitivity,
bnot_wf,
le_wf,
le_int_wf,
assert_wf,
bool_wf,
equal-wf-base,
lt_int_wf,
istype-int,
subtype_rel_self,
int_subtype_base,
subtype_base_sq,
add-zero,
zero-mul,
add-mul-special,
not-gt-2,
subtract_nat_wf,
istype-nat,
primrec-unroll,
le-add-cancel2,
not-le-2,
istype-false,
subtract_wf,
decidable__le,
subtract-1-ge-0,
istype-le,
int_seg_wf,
le-add-cancel,
add-commutes,
add_functionality_wrt_le,
zero-add,
add-associates,
minus-one-mul-top,
add-swap,
minus-one-mul,
minus-add,
condition-implies-le,
less-iff-le,
sq_stable__le,
primrec0_lemma,
istype-less_than,
ge_wf,
nat_properties,
less_than_wf,
less_than_irreflexivity,
le_weakening2,
less_than_transitivity2,
less_than_transitivity1,
istype-void,
istype-top,
decidable__lt
Rules used in proof :
functionExtensionality,
dependent_pairEquality_alt,
inlEquality_alt,
equalityElimination,
closedConclusion,
baseApply,
universeEquality,
unionIsType,
functionIsType,
intEquality,
cumulativity,
instantiate,
multiplyEquality,
equalityIstype,
dependent_set_memberEquality_alt,
minusEquality,
addEquality,
functionIsTypeImplies,
equalitySymmetry,
equalityTransitivity,
axiomEquality,
intWeakElimination,
applyEquality,
universeIsType,
productIsType,
because_Cache,
independent_isectElimination,
rename,
setElimination,
lambdaEquality_alt,
inrEquality_alt,
independent_functionElimination,
productElimination,
imageElimination,
lambdaFormation_alt,
baseClosed,
imageMemberEquality,
natural_numberEquality,
voidElimination,
independent_pairFormation,
isectIsTypeImplies,
isect_memberEquality_alt,
inhabitedIsType,
axiomSqEquality,
isectElimination,
lessCases,
sqequalRule,
unionElimination,
hypothesis,
hypothesisEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[i,j:\mBbbZ{}]. \mforall{}[F,G:\{i..j\msupminus{}\} {}\mrightarrow{} \mBbbP{}]. \mforall{}[d:\mforall{}k:\{i..j\msupminus{}\}. (F[k] \mvee{} G[k])].
(int-seg-case(i;j;d) \mmember{} (\mexists{}k:\{i..j\msupminus{}\}. F[k]) \mvee{} (\mforall{}k:\{i..j\msupminus{}\}. G[k]))
Date html generated:
2019_10_15-AM-10_19_53
Last ObjectModification:
2019_10_02-PM-06_04_50
Theory : call!by!value_2
Home
Index