Nuprl Lemma : enumerate-increases
∀[P:ℕ ⟶ 𝔹]. ∀[n,m:ℕ]. enumerate(P;n) < enumerate(P;m) supposing n < m supposing ∀n:ℕ. ∃k:ℕ. ((↑(P k)) ∧ (n ≤ k))
Proof
Definitions occuring in Statement :
enumerate: enumerate(P;n),
nat: ℕ,
assert: ↑b,
bool: 𝔹,
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
le: A ≤ B,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
nat: ℕ,
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
and: P ∧ Q,
assert: ↑b,
bnot: ¬bb,
guard: {T},
sq_type: SQType(T),
or: P ∨ Q,
bfalse: ff,
prop: ℙ,
not: ¬A,
false: False,
satisfiable_int_formula: satisfiable_int_formula(fmla),
ge: i ≥ j ,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
implies: P ⇒ Q,
top: Top,
enumerate: enumerate(P;n),
so_apply: x[s],
so_lambda: λ2x.t[x],
has-value: (a)↓,
decidable: Dec(P),
nequal: a ≠ b ∈ T ,
true: True,
subtract: n - m,
le: A ≤ B,
less_than: a < b,
squash: ↓T,
cand: A c∧ B
Lemmas referenced :
istype-less_than,
member-less_than,
enumerate_wf,
istype-nat,
istype-assert,
istype-le,
bool_wf,
nat_wf,
add-subtract-cancel,
neg_assert_of_eq_int,
assert-bnot,
bool_subtype_base,
subtype_base_sq,
bool_cases_sqequal,
equal_wf,
eqff_to_assert,
int_formula_prop_wf,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_and_lemma,
intformle_wf,
itermConstant_wf,
itermVar_wf,
itermAdd_wf,
intformeq_wf,
intformand_wf,
satisfiable-full-omega-tt,
nat_properties,
assert_of_eq_int,
eqtt_to_assert,
eq_int_wf,
primrec-unroll,
set_wf,
assert_wf,
int-value-type,
value-type-has-value,
le_wf,
int_formula_prop_not_lemma,
intformnot_wf,
decidable__le,
mu_wf,
assert_elim,
add-zero,
zero-mul,
zero-add,
add-commutes,
add-mul-special,
add-swap,
minus-one-mul,
minus-add,
add-associates,
int_subtype_base,
set_subtype_base,
add-is-int-iff,
int_term_value_subtract_lemma,
itermSubtract_wf,
subtract_wf,
int_formula_prop_less_lemma,
intformless_wf,
decidable__lt,
full-omega-unsat,
istype-int,
ge_wf,
subtract-1-ge-0,
less_than_functionality,
le_weakening,
le_weakening2,
squash_wf,
less_than_wf,
and_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
hypothesis,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
sqequalRule,
isect_memberEquality_alt,
independent_isectElimination,
applyEquality,
lambdaEquality_alt,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
isectIsTypeImplies,
because_Cache,
functionIsType,
productIsType,
universeIsType,
lambdaFormation,
independent_functionElimination,
cumulativity,
instantiate,
promote_hyp,
computeAll,
independent_pairFormation,
dependent_functionElimination,
intEquality,
int_eqEquality,
lambdaEquality,
dependent_pairFormation,
productElimination,
equalityElimination,
unionElimination,
voidEquality,
voidElimination,
isect_memberEquality,
natural_numberEquality,
addEquality,
setEquality,
functionExtensionality,
callbyvalueReduce,
dependent_set_memberEquality,
multiplyEquality,
baseClosed,
closedConclusion,
baseApply,
applyLambdaEquality,
lambdaFormation_alt,
intWeakElimination,
approximateComputation,
dependent_pairFormation_alt,
Error :memTop,
functionIsTypeImplies,
dependent_set_memberEquality_alt,
imageElimination,
functionEquality,
productEquality,
hyp_replacement,
equalityIstype,
imageMemberEquality
Latex:
\mforall{}[P:\mBbbN{} {}\mrightarrow{} \mBbbB{}]
\mforall{}[n,m:\mBbbN{}]. enumerate(P;n) < enumerate(P;m) supposing n < m
supposing \mforall{}n:\mBbbN{}. \mexists{}k:\mBbbN{}. ((\muparrow{}(P k)) \mwedge{} (n \mleq{} k))
Date html generated:
2020_05_20-AM-08_10_51
Last ObjectModification:
2020_01_04-PM-11_12_55
Theory : general
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