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],
int_nzero: ℤ-o,
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,
nat_plus: ℕ+,
le: A ≤ B
Lemmas referenced :
nat_properties,
decidable__le,
divide_wfa,
subtype_base_sq,
int_subtype_base,
nequal_wf,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermAdd_wf,
itermVar_wf,
itermConstant_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
mul_preserves_le,
istype-le,
div_rem_sum2,
rem_bounds_1,
decidable__lt,
istype-less_than,
itermMultiply_wf,
itermSubtract_wf,
int_term_value_mul_lemma,
int_term_value_subtract_lemma,
member-less_than,
istype-nat
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
dependent_functionElimination,
addEquality,
natural_numberEquality,
closedConclusion,
Error :dependent_set_memberEquality_alt,
Error :lambdaFormation_alt,
instantiate,
cumulativity,
intEquality,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
voidElimination,
Error :equalityIstype,
Error :inhabitedIsType,
baseClosed,
sqequalBase,
Error :universeIsType,
unionElimination,
imageElimination,
productElimination,
approximateComputation,
Error :dependent_pairFormation_alt,
Error :lambdaEquality_alt,
int_eqEquality,
Error :isect_memberEquality_alt,
sqequalRule,
independent_pairFormation,
because_Cache,
Error :isectIsTypeImplies
Latex:
\mforall{}[n,m:\mBbbN{}]. n < m supposing n < ((n + m) + 1) \mdiv{} 2
Date html generated:
2019_06_20-PM-02_30_58
Last ObjectModification:
2019_03_06-AM-10_54_02
Theory : num_thy_1
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