Nuprl Lemma : int-seg-case-monotone
∀[i,j:ℤ]. ∀[F,G:{i..j-} ⟶ ℙ]. ∀[d:∀k:{i..j-}. (F[k] ∨ G[k])].
∀k:{i..j + 1-}. ((↑isl(int-seg-case(i;k;d))) ⇒ (∀k':{k..j + 1-}. (↑isl(int-seg-case(i;k';d)))))
Proof
Definitions occuring in Statement :
int-seg-case: int-seg-case(i;j;d),
int_seg: {i..j-},
assert: ↑b,
isl: isl(x),
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
or: P ∨ Q,
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
btrue: tt,
it: ⋅,
unit: Unit,
exposed-it: exposed-it,
le_int: i ≤z j,
lt_int: i <z j,
bnot: ¬bb,
label: ...$L... t,
sq_type: SQType(T),
nat_plus: ℕ+,
bfalse: ff,
int-seg-case: int-seg-case(i;j;d),
sq_stable: SqStable(P),
guard: {T},
ge: i ≥ j ,
nat: ℕ,
bool: 𝔹,
ifthenelse: if b then t else f fi ,
assert: ↑b,
isl: isl(x),
prop: ℙ,
subtype_rel: A ⊆r B,
true: True,
less_than': less_than'(a;b),
top: Top,
subtract: n - m,
uimplies: b supposing a,
uiff: uiff(P;Q),
false: False,
rev_implies: P ⇐ Q,
not: ¬A,
iff: P ⇐⇒ Q,
or: P ∨ Q,
decidable: Dec(P),
le: A ≤ B,
cand: A c∧ B,
so_apply: x[s],
so_lambda: λ2x.t[x],
squash: ↓T,
less_than: a < b,
and: P ∧ Q,
lelt: i ≤ j < k,
int_seg: {i..j-},
implies: P ⇒ Q,
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
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_int_wf,
assert_wf,
bool_wf,
equal-wf-base,
lt_int_wf,
subtype_base_sq,
primrec-unroll,
mul-associates,
mul-distributes,
omega-shadow,
mul-distributes-right,
two-mul,
one-mul,
le_reflexive,
le_weakening2,
istype-top,
zero-mul,
add-mul-special,
not-le-2,
decidable__le,
sq_stable__le,
subtract_nat_wf,
subtract_wf,
istype-nat,
subtract-1-ge-0,
int_subtype_base,
set_subtype_base,
add-is-int-iff,
add-zero,
ge_wf,
less_than_irreflexivity,
less_than_transitivity1,
nat_properties,
istype-true,
bfalse_wf,
btrue_wf,
subtype_rel_self,
less_than_wf,
le_wf,
and_wf,
subtype_rel_sets_simple,
or_wf,
subtype_rel_dep_function,
istype-less_than,
istype-le,
le-add-cancel2,
add-commutes,
add_functionality_wrt_le,
zero-add,
minus-one-mul-top,
add-swap,
minus-one-mul,
minus-add,
istype-void,
add-associates,
condition-implies-le,
less-iff-le,
not-lt-2,
istype-false,
decidable__lt,
int-seg-case_wf,
istype-assert,
int_seg_wf
Rules used in proof :
equalityElimination,
cumulativity,
instantiate,
multiplyEquality,
axiomSqEquality,
lessCases,
imageMemberEquality,
baseClosed,
closedConclusion,
baseApply,
intWeakElimination,
universeEquality,
isectIsTypeImplies,
functionIsTypeImplies,
axiomEquality,
equalitySymmetry,
equalityTransitivity,
equalityIstype,
unionIsType,
functionIsType,
inhabitedIsType,
intEquality,
unionEquality,
productIsType,
minusEquality,
isect_memberEquality_alt,
natural_numberEquality,
addEquality,
independent_isectElimination,
independent_functionElimination,
voidElimination,
unionElimination,
dependent_functionElimination,
independent_pairFormation,
dependent_set_memberEquality_alt,
applyEquality,
lambdaEquality_alt,
sqequalRule,
because_Cache,
imageElimination,
productElimination,
hypothesis,
hypothesisEquality,
rename,
setElimination,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
universeIsType,
lambdaFormation_alt,
cut,
introduction,
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])].
\mforall{}k:\{i..j + 1\msupminus{}\}. ((\muparrow{}isl(int-seg-case(i;k;d))) {}\mRightarrow{} (\mforall{}k':\{k..j + 1\msupminus{}\}. (\muparrow{}isl(int-seg-case(i;k';d)))))
Date html generated:
2019_10_15-AM-10_19_56
Last ObjectModification:
2019_10_02-PM-11_11_31
Theory : call!by!value_2
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