Nuprl Lemma : rel-exp-add-1-iff
∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ]. ∀a:ℕ+. ∀x,z:T. (x R^a z ⇐⇒ ∃y:T. ((x R^a - 1 y) ∧ (y R z)))
Proof
Definitions occuring in Statement :
rel_exp: R^n,
nat_plus: 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],
subtract: n - m,
natural_number: $n,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
infix_ap: x f y,
subtype_rel: A ⊆r B,
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
nat: ℕ,
nat_plus: ℕ+,
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,
so_apply: x[s],
cand: A c∧ B
Lemmas referenced :
equal_wf,
and_wf,
less_than_wf,
decidable__lt,
int_formula_prop_eq_lemma,
intformeq_wf,
rel_exp_iff,
nat_plus_wf,
le_wf,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_subtract_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
intformless_wf,
itermVar_wf,
itermSubtract_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__le,
nat_plus_properties,
subtract_wf,
infix_ap_wf,
exists_wf,
nat_plus_subtype_nat,
rel_exp_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
independent_pairFormation,
applyEquality,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
sqequalRule,
cumulativity,
lambdaEquality,
productEquality,
instantiate,
because_Cache,
universeEquality,
dependent_set_memberEquality,
setElimination,
rename,
natural_numberEquality,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
functionEquality,
productElimination,
independent_functionElimination,
inlFormation
Latex:
\mforall{}[T:Type]. \mforall{}[R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}].
\mforall{}a:\mBbbN{}\msupplus{}. \mforall{}x,z:T. (x R\^{}a z \mLeftarrow{}{}\mRightarrow{} \mexists{}y:T. ((x R\^{}a - 1 y) \mwedge{} (y R z)))
Date html generated:
2016_05_14-PM-03_56_12
Last ObjectModification:
2016_01_14-PM-11_11_47
Theory : relations2
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