Nuprl Lemma : rel-exp-add-iff
∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ]. ∀a,b:ℕ. ∀x,z:T. (x R^a + b z ⇐⇒ ∃y:T. ((x R^a y) ∧ (y R^b z)))
Proof
Definitions occuring in Statement :
rel_exp: R^n,
nat: ℕ,
uall: ∀[x:A]. B[x],
prop: ℙ,
infix_ap: x f y,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
function: x:A ⟶ B[x],
add: n + m,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
nat: ℕ,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
and: P ∧ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
infix_ap: x f y,
cand: A c∧ B,
le: A ≤ B,
less_than': less_than'(a;b),
rel_exp: R^n,
eq_int: (i =z j),
subtract: n - m,
ifthenelse: if b then t else f fi ,
btrue: tt,
sq_type: SQType(T),
guard: {T},
uiff: uiff(P;Q),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
bfalse: ff,
subtype_rel: A ⊆r B,
bool: 𝔹,
unit: Unit,
it: ⋅
Lemmas referenced :
infix_ap_wf,
rel_exp_wf,
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,
intformless_wf,
int_formula_prop_less_lemma,
all_wf,
nat_wf,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
exists_wf,
set_wf,
less_than_wf,
primrec-wf2,
zero-add,
false_wf,
eq_int_wf,
intformeq_wf,
int_formula_prop_eq_lemma,
assert_wf,
bnot_wf,
not_wf,
equal-wf-T-base,
bool_cases,
subtype_base_sq,
bool_wf,
bool_subtype_base,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot,
general_arith_equation1,
equal-wf-base,
int_subtype_base,
uiff_transitivity,
equal_wf,
rel_exp_add
Rules used in proof :
cut,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
thin,
instantiate,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
cumulativity,
hypothesisEquality,
because_Cache,
universeEquality,
dependent_set_memberEquality,
addEquality,
natural_numberEquality,
setElimination,
rename,
hypothesis,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
functionExtensionality,
applyEquality,
functionEquality,
productEquality,
equalityTransitivity,
equalitySymmetry,
baseClosed,
productElimination,
independent_functionElimination,
impliesFunctionality,
baseApply,
closedConclusion,
equalityElimination
Latex:
\mforall{}[T:Type]. \mforall{}[R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}].
\mforall{}a,b:\mBbbN{}. \mforall{}x,z:T. (x R\^{}a + b z \mLeftarrow{}{}\mRightarrow{} \mexists{}y:T. ((x R\^{}a y) \mwedge{} (y rel\_exp(T; R; b) z)))
Date html generated:
2017_04_17-AM-09_28_17
Last ObjectModification:
2017_02_27-PM-05_29_07
Theory : relations2
Home
Index