Nuprl Lemma : rmin_functionality_wrt_bdd-diff
∀[x1,x2,y1,y2:ℕ+ ⟶ ℤ]. (bdd-diff(y1;y2) ⇒ bdd-diff(x1;x2) ⇒ bdd-diff(rmin(x1;y1);rmin(x2;y2)))
Proof
Definitions occuring in Statement :
rmin: rmin(x;y),
bdd-diff: bdd-diff(f;g),
nat_plus: ℕ+,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
bdd-diff: bdd-diff(f;g),
exists: ∃x:A. B[x],
rmin: rmin(x;y),
member: t ∈ T,
nat: ℕ,
all: ∀x:A. B[x],
guard: {T},
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
and: P ∧ Q,
prop: ℙ,
le: A ≤ B,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
uiff: uiff(P;Q),
cand: A c∧ B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
imax_wf,
imax_nat,
nat_wf,
nat_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformeq_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_wf,
equal_wf,
le_wf,
less_than'_wf,
absval_wf,
subtract_wf,
imin_wf,
nat_plus_wf,
all_wf,
bdd-diff_wf,
imax_lb,
imax_ub,
le_functionality,
absval-imin-difference,
le_weakening
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
dependent_pairFormation,
dependent_set_memberEquality,
cut,
introduction,
extract_by_obid,
isectElimination,
setElimination,
rename,
hypothesisEquality,
hypothesis,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
dependent_functionElimination,
natural_numberEquality,
unionElimination,
independent_isectElimination,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
independent_functionElimination,
independent_pairEquality,
because_Cache,
applyEquality,
functionExtensionality,
axiomEquality,
functionEquality,
inrFormation,
inlFormation
Latex:
\mforall{}[x1,x2,y1,y2:\mBbbN{}\msupplus{} {}\mrightarrow{} \mBbbZ{}]. (bdd-diff(y1;y2) {}\mRightarrow{} bdd-diff(x1;x2) {}\mRightarrow{} bdd-diff(rmin(x1;y1);rmin(x2;y2)))
Date html generated:
2017_10_03-AM-08_22_20
Last ObjectModification:
2017_07_28-AM-07_22_16
Theory : reals
Home
Index