Nuprl Lemma : min-increasing-sequence-prop1
∀b:ℕ ⟶ ℕ. ∀n,x,k:ℕ. ((min-increasing-sequence(b;n;x) = (inl k) ∈ (ℕ?)) ⇒ (x ≤ (b k)))
Proof
Definitions occuring in Statement :
min-increasing-sequence: min-increasing-sequence(a;n;k),
nat: ℕ,
le: A ≤ B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
unit: Unit,
apply: f a,
function: x:A ⟶ B[x],
inl: inl x,
union: left + right,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
le: A ≤ B,
decidable: Dec(P),
or: P ∨ Q,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
isl: isl(x),
min-increasing-sequence: min-increasing-sequence(a;n;k),
exposed-bfalse: exposed-bfalse,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q
Lemmas referenced :
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
istype-less_than,
le_witness_for_triv,
unit_wf2,
min-increasing-sequence_wf,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
istype-le,
union_subtype_base,
nat_wf,
set_subtype_base,
le_wf,
int_subtype_base,
unit_subtype_base,
subtract-1-ge-0,
istype-nat,
btrue_neq_bfalse,
bfalse_wf,
btrue_wf,
primrec0_lemma,
primrec-unroll,
lt_int_wf,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
less_than_wf,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
le_int_wf,
assert_of_le_int
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
Error :dependent_pairFormation_alt,
Error :lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
Error :isect_memberEquality_alt,
voidElimination,
sqequalRule,
independent_pairFormation,
Error :universeIsType,
productElimination,
equalityTransitivity,
equalitySymmetry,
Error :functionIsTypeImplies,
Error :inhabitedIsType,
Error :equalityIstype,
Error :unionIsType,
because_Cache,
Error :dependent_set_memberEquality_alt,
unionElimination,
applyEquality,
intEquality,
baseApply,
closedConclusion,
baseClosed,
sqequalBase,
Error :functionIsType,
applyLambdaEquality,
Error :equalityIsType4,
Error :productIsType,
equalityElimination,
promote_hyp,
instantiate,
cumulativity
Latex:
\mforall{}b:\mBbbN{} {}\mrightarrow{} \mBbbN{}. \mforall{}n,x,k:\mBbbN{}. ((min-increasing-sequence(b;n;x) = (inl k)) {}\mRightarrow{} (x \mleq{} (b k)))
Date html generated:
2019_06_20-PM-03_07_13
Last ObjectModification:
2018_12_06-PM-11_57_09
Theory : continuity
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