Nuprl Lemma : bdd-diff-add
∀[f1,f2,g1,g2:ℕ+ ⟶ ℤ]. (bdd-diff(f1;f2) ⇒ bdd-diff(g1;g2) ⇒ bdd-diff(λn.(f1[n] + g1[n]);λn.(f2[n] + g2[n])))
Proof
Definitions occuring in Statement :
bdd-diff: bdd-diff(f;g),
nat_plus: ℕ+,
uall: ∀[x:A]. B[x],
so_apply: x[s],
implies: P ⇒ Q,
lambda: λx.A[x],
function: x:A ⟶ B[x],
add: n + m,
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],
member: t ∈ T,
nat: ℕ,
ge: i ≥ j ,
all: ∀x:A. B[x],
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,
so_apply: x[s],
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
nat_plus: ℕ+,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
guard: {T}
Lemmas referenced :
nat_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
le_wf,
less_than'_wf,
nat_plus_wf,
all_wf,
absval_wf,
subtract_wf,
bdd-diff_wf,
nat_wf,
nat_plus_properties,
less_than_wf,
decidable__equal_int,
intformeq_wf,
itermSubtract_wf,
itermMinus_wf,
int_formula_prop_eq_lemma,
int_term_value_subtract_lemma,
int_term_value_minus_lemma,
add-is-int-iff,
subtract-is-int-iff,
false_wf,
and_wf,
equal_wf,
le_functionality,
le_weakening,
int-triangle-inequality,
add_functionality_wrt_le
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
dependent_pairFormation,
dependent_set_memberEquality,
addEquality,
setElimination,
rename,
cut,
hypothesisEquality,
hypothesis,
introduction,
extract_by_obid,
isectElimination,
dependent_functionElimination,
natural_numberEquality,
unionElimination,
independent_isectElimination,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
independent_pairEquality,
because_Cache,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
applyEquality,
functionExtensionality,
functionEquality,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
baseClosed,
setEquality,
hyp_replacement,
Error :applyLambdaEquality
Latex:
\mforall{}[f1,f2,g1,g2:\mBbbN{}\msupplus{} {}\mrightarrow{} \mBbbZ{}].
(bdd-diff(f1;f2) {}\mRightarrow{} bdd-diff(g1;g2) {}\mRightarrow{} bdd-diff(\mlambda{}n.(f1[n] + g1[n]);\mlambda{}n.(f2[n] + g2[n])))
Date html generated:
2016_10_26-AM-09_02_44
Last ObjectModification:
2016_07_12-AM-08_13_01
Theory : reals
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