Nuprl Lemma : m-k-regular-monotone
∀[n,k:ℕ]. ∀[X:Type]. ∀[d:metric(X)]. ∀[s:ℕ ⟶ X]. m-k-regular(d;k;s) supposing m-k-regular(d;n;s) supposing n ≤ k
Proof
Definitions occuring in Statement :
m-k-regular: m-k-regular(d;k;s),
metric: metric(X),
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
le: A ≤ B,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
m-k-regular: m-k-regular(d;k;s),
all: ∀x:A. B[x],
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
and: P ∧ Q,
prop: ℙ,
nat: ℕ,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
ge: i ≥ j ,
decidable: Dec(P),
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
uiff: uiff(P;Q),
rdiv: (x/y),
req_int_terms: t1 ≡ t2
Lemmas referenced :
le_witness_for_triv,
m-k-regular_wf,
metric_wf,
istype-universe,
istype-le,
istype-nat,
mdist_wf,
radd_wf,
rdiv_wf,
int-to-real_wf,
rless-int,
nat_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
intformle_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
rless_wf,
rleq_functionality_wrt_implies,
rleq_weakening_equal,
radd_functionality_wrt_rleq,
rmul_preserves_rleq,
rmul_wf,
rinv_wf2,
itermSubtract_wf,
itermMultiply_wf,
rleq-int,
rleq_functionality,
req_transitivity,
rmul-rinv3,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalHypSubstitution,
lambdaFormation_alt,
hypothesis,
dependent_functionElimination,
thin,
hypothesisEquality,
because_Cache,
sqequalRule,
lambdaEquality_alt,
extract_by_obid,
isectElimination,
productElimination,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
functionIsTypeImplies,
inhabitedIsType,
universeIsType,
isect_memberEquality_alt,
isectIsTypeImplies,
functionIsType,
instantiate,
universeEquality,
setElimination,
rename,
applyEquality,
addEquality,
closedConclusion,
natural_numberEquality,
inrFormation_alt,
independent_functionElimination,
unionElimination,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
voidElimination,
independent_pairFormation
Latex:
\mforall{}[n,k:\mBbbN{}].
\mforall{}[X:Type]. \mforall{}[d:metric(X)]. \mforall{}[s:\mBbbN{} {}\mrightarrow{} X]. m-k-regular(d;k;s) supposing m-k-regular(d;n;s)
supposing n \mleq{} k
Date html generated:
2019_10_30-AM-06_58_46
Last ObjectModification:
2019_10_09-AM-09_39_24
Theory : reals
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