Nuprl Lemma : rminimum-shift
∀[k,n,m:ℤ]. ∀[x:Top]. rminimum(n;m;i.x[i]) ~ rminimum(n - k;m - k;i.x[i + k]) supposing n ≤ m
Proof
Definitions occuring in Statement :
rminimum: rminimum(n;m;k.x[k]),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
top: Top,
so_apply: x[s],
le: A ≤ B,
subtract: n - m,
add: n + m,
int: ℤ,
sqequal: s ~ t
Definitions unfolded in proof :
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
assert: ↑b,
bnot: ¬bb,
bfalse: ff,
uiff: uiff(P;Q),
it: ⋅,
unit: Unit,
bool: 𝔹,
btrue: tt,
ifthenelse: if b then t else f fi ,
subtract: n - m,
lt_int: i <z j,
ge: i ≥ j ,
and: P ∧ Q,
nat: ℕ,
guard: {T},
sq_type: SQType(T),
prop: ℙ,
top: Top,
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
implies: P ⇒ Q,
not: ¬A,
or: P ∨ Q,
decidable: Dec(P),
all: ∀x:A. B[x],
rminimum: rminimum(n;m;k.x[k]),
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
primrec-unroll,
int_term_value_add_lemma,
itermAdd_wf,
less_than_wf,
assert_wf,
iff_weakening_uiff,
assert-bnot,
bool_subtype_base,
bool_wf,
bool_cases_sqequal,
eqff_to_assert,
subtract-add-cancel,
assert_of_lt_int,
eqtt_to_assert,
lt_int_wf,
istype-top,
subtract-1-ge-0,
istype-less_than,
ge_wf,
int_formula_prop_less_lemma,
intformless_wf,
nat_properties,
istype-le,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_and_lemma,
itermConstant_wf,
intformle_wf,
intformand_wf,
decidable__le,
subtract_wf,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_subtract_lemma,
int_formula_prop_eq_lemma,
istype-void,
int_formula_prop_not_lemma,
istype-int,
itermVar_wf,
itermSubtract_wf,
intformeq_wf,
intformnot_wf,
full-omega-unsat,
decidable__equal_int,
int_subtype_base,
subtype_base_sq
Rules used in proof :
promote_hyp,
productElimination,
equalityElimination,
isectIsTypeImplies,
equalityIstype,
functionIsTypeImplies,
axiomSqEquality,
intWeakElimination,
rename,
setElimination,
lambdaFormation_alt,
inhabitedIsType,
independent_pairFormation,
dependent_set_memberEquality_alt,
equalitySymmetry,
equalityTransitivity,
universeIsType,
sqequalRule,
voidElimination,
isect_memberEquality_alt,
hypothesisEquality,
int_eqEquality,
lambdaEquality_alt,
dependent_pairFormation_alt,
independent_functionElimination,
approximateComputation,
natural_numberEquality,
unionElimination,
because_Cache,
dependent_functionElimination,
hypothesis,
independent_isectElimination,
intEquality,
cumulativity,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
instantiate,
thin,
cut,
introduction,
isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[k,n,m:\mBbbZ{}]. \mforall{}[x:Top]. rminimum(n;m;i.x[i]) \msim{} rminimum(n - k;m - k;i.x[i + k]) supposing n \mleq{} m
Date html generated:
2019_11_06-PM-00_30_22
Last ObjectModification:
2019_11_05-PM-00_10_22
Theory : reals
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