Nuprl Lemma : less-iff-le
∀x,y:ℤ. uiff(x < y;(1 + x) ≤ y)
Proof
Definitions occuring in Statement :
less_than: a < b,
uiff: uiff(P;Q),
le: A ≤ B,
all: ∀x:A. B[x],
add: n + m,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
prop: ℙ,
uall: ∀[x:A]. B[x],
false: False,
implies: P ⇒ Q,
not: ¬A,
le: A ≤ B,
member: t ∈ T,
uimplies: b supposing a,
and: P ∧ Q,
uiff: uiff(P;Q),
all: ∀x:A. B[x],
rev_uimplies: rev_uimplies(P;Q),
subtype_rel: A ⊆r B,
squash: ↓T,
cand: A c∧ B,
less_than: a < b,
or: P ∨ Q,
true: True,
less_than': less_than'(a;b),
guard: {T},
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
top: Top,
bfalse: ff,
exists: ∃x:A. B[x],
sq_type: SQType(T),
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
le_wf,
less_than_wf,
less_than'_wf,
le-iff-less-or-equal,
int_subtype_base,
equal-wf-base,
lt_int_wf,
eqtt_to_assert,
assert_of_lt_int,
istype-top,
istype-void,
eqff_to_assert,
bool_subtype_base,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
iff_transitivity,
assert_wf,
bnot_wf,
not_wf,
iff_weakening_uiff,
assert_of_bnot,
istype-assert,
zero-add,
le_reflexive,
add_functionality_wrt_lt,
less_than_transitivity1
Rules used in proof :
intEquality,
equalitySymmetry,
equalityTransitivity,
axiomEquality,
hypothesis,
natural_numberEquality,
addEquality,
isectElimination,
extract_by_obid,
voidElimination,
hypothesisEquality,
dependent_functionElimination,
lambdaEquality,
independent_pairEquality,
thin,
productElimination,
sqequalHypSubstitution,
sqequalRule,
cut,
introduction,
isect_memberFormation,
independent_pairFormation,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
independent_isectElimination,
lessCases,
applyEquality,
closedConclusion,
baseApply,
baseClosed,
imageMemberEquality,
inlFormation,
inrFormation,
lessDiscrete,
Error :inhabitedIsType,
Error :lambdaFormation_alt,
unionElimination,
equalityElimination,
because_Cache,
Error :isect_memberFormation_alt,
axiomSqEquality,
Error :isect_memberEquality_alt,
Error :isectIsTypeImplies,
imageElimination,
independent_functionElimination,
Error :dependent_pairFormation_alt,
Error :equalityIsType4,
promote_hyp,
instantiate,
cumulativity,
Error :functionIsType,
Error :equalityIsType1
Latex:
\mforall{}x,y:\mBbbZ{}. uiff(x < y;(1 + x) \mleq{} y)
Date html generated:
2019_06_20-AM-11_23_03
Last ObjectModification:
2018_10_16-PM-02_47_37
Theory : arithmetic
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