Nuprl Lemma : rel-immediate-rel-plus
∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ]. (SWellFounded(R x y) ⇒ (∀x,y:T. Dec(∃z:T. ((R x z) ∧ (R z y)))) ⇒ R+ => R!+)
Proof
Definitions occuring in Statement :
rel-immediate: R!,
strongwellfounded: SWellFounded(R[x; y]),
rel_plus: R+,
rel_implies: R1 => R2,
decidable: Dec(P),
uall: ∀[x:A]. B[x],
prop: ℙ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
strongwellfounded: SWellFounded(R[x; y]),
exists: ∃x:A. B[x],
rel_plus: R+,
rel_implies: R1 => R2,
infix_ap: x f y,
all: ∀x:A. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
nat_plus: ℕ+,
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,
so_lambda: λ2x.t[x],
so_apply: x[s],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
less_than': less_than'(a;b),
nat: ℕ,
rel_exp: R^n,
ge: i ≥ j ,
sq_type: SQType(T),
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
btrue: tt,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
bfalse: ff,
subtract: n - m,
eq_int: (i =z j),
less_than: a < b,
squash: ↓T,
true: True,
cand: A c∧ B,
rel-immediate: R!
Lemmas referenced :
nat_plus_subtype_nat,
nat_plus_properties,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
exists_wf,
nat_plus_wf,
rel_exp_wf,
all_wf,
decidable_wf,
strongwellfounded_wf,
int_seg_properties,
intformle_wf,
int_formula_prop_le_lemma,
int_seg_wf,
decidable__equal_int,
subtract_wf,
int_seg_subtype,
false_wf,
decidable__le,
itermSubtract_wf,
intformeq_wf,
int_term_value_subtract_lemma,
int_formula_prop_eq_lemma,
le_wf,
less_than_wf,
int_seg_subtype_nat,
rel-immediate_wf,
lelt_wf,
set_wf,
primrec-wf2,
nat_wf,
nat_properties,
itermAdd_wf,
int_term_value_add_lemma,
eq_int_wf,
assert_wf,
bnot_wf,
not_wf,
equal-wf-T-base,
subtype_base_sq,
int_subtype_base,
bool_cases,
bool_wf,
bool_subtype_base,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot,
infix_ap_wf,
rel_exp_add,
equal_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
sqequalRule,
cut,
hypothesis,
dependent_functionElimination,
hypothesisEquality,
applyEquality,
introduction,
extract_by_obid,
independent_functionElimination,
isectElimination,
setElimination,
rename,
natural_numberEquality,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
cumulativity,
functionExtensionality,
productEquality,
universeEquality,
because_Cache,
functionEquality,
addLevel,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
levelHypothesis,
hypothesis_subsumption,
dependent_set_memberEquality,
addEquality,
baseClosed,
instantiate,
impliesFunctionality,
hyp_replacement,
imageElimination,
imageMemberEquality
Latex:
\mforall{}[T:Type]. \mforall{}[R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}].
(SWellFounded(R x y) {}\mRightarrow{} (\mforall{}x,y:T. Dec(\mexists{}z:T. ((R x z) \mwedge{} (R z y)))) {}\mRightarrow{} R\msupplus{} => R!\msupplus{})
Date html generated:
2018_05_21-PM-07_42_32
Last ObjectModification:
2017_07_26-PM-05_20_16
Theory : general
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