Nuprl Lemma : add-plus-1-div-2-implies-lt
∀[n,m:ℕ]. n < m supposing n < ((n + m) + 1) ÷ 2
Proof
Definitions occuring in Statement :
nat: ℕ,
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
divide: n ÷ m,
add: n + m,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
nat: ℕ,
ge: i ≥ j ,
all: ∀x:A. B[x],
true: True,
nequal: a ≠ b ∈ T ,
not: ¬A,
implies: P ⇒ Q,
sq_type: SQType(T),
guard: {T},
false: False,
prop: ℙ,
decidable: Dec(P),
or: P ∨ Q,
less_than: a < b,
squash: ↓T,
and: P ∧ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
int_nzero: ℤ-o,
uiff: uiff(P;Q),
nat_plus: ℕ+,
less_than': less_than'(a;b),
subtype_rel: A ⊆r B,
le: A ≤ B,
iff: P ⇐⇒ Q,
subtract: n - m,
isOdd: isOdd(n),
rev_uimplies: rev_uimplies(P;Q),
ifthenelse: if b then t else f fi ,
btrue: tt,
rev_implies: P ⇐ Q,
bfalse: ff
Lemmas referenced :
nat_properties,
decidable__lt,
subtype_base_sq,
int_subtype_base,
equal-wf-base,
true_wf,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermMultiply_wf,
itermVar_wf,
itermConstant_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_mul_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
div_rem_sum,
nequal_wf,
decidable__equal_int,
add-is-int-iff,
intformeq_wf,
itermSubtract_wf,
itermAdd_wf,
int_formula_prop_eq_lemma,
int_term_value_subtract_lemma,
int_term_value_add_lemma,
false_wf,
less_than_wf,
subtract_wf,
subtract-is-int-iff,
rem_bounds_1,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
le_wf,
decidable__or,
intformor_wf,
int_formula_prop_or_lemma,
rem_add1,
iff_weakening_equal,
member-less_than,
nat_wf,
eq_int_wf,
add-zero,
assert_of_eq_int,
modulus_wf_int_mod,
int-subtype-int_mod,
modulus-is-rem,
assert_wf,
bnot_wf,
not_wf,
equal-wf-T-base,
bool_cases,
bool_wf,
bool_subtype_base,
eqtt_to_assert,
eqff_to_assert,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot,
isOdd-add,
not-same-parity-implies-even-odd,
even-iff-not-odd,
not_assert_elim,
isOdd_wf,
and_wf,
equal_wf,
assert_elim,
btrue_neq_bfalse
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
dependent_functionElimination,
multiplyEquality,
natural_numberEquality,
divideEquality,
addEquality,
addLevel,
lambdaFormation,
instantiate,
cumulativity,
intEquality,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
voidElimination,
baseClosed,
because_Cache,
unionElimination,
imageElimination,
productElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
isect_memberEquality,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
dependent_set_memberEquality,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
hyp_replacement,
Error :applyLambdaEquality,
remainderEquality,
imageMemberEquality,
applyEquality,
equalityEquality,
impliesFunctionality,
setEquality
Latex:
\mforall{}[n,m:\mBbbN{}]. n < m supposing n < ((n + m) + 1) \mdiv{} 2
Date html generated:
2016_10_25-AM-10_59_26
Last ObjectModification:
2016_07_12-AM-07_07_09
Theory : general
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