Nuprl Lemma : strongwf-implies
∀[T:Type]. ∀[R:T ⟶ T ⟶ Type]. (SWellFounded(R[x;y]) ⇒ WellFnd{i}(T;x,y.R[x;y]))
Proof
Definitions occuring in Statement :
strongwellfounded: SWellFounded(R[x; y]),
wellfounded: WellFnd{i}(A;x,y.R[x; y]),
uall: ∀[x:A]. B[x],
so_apply: x[s1;s2],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
wellfounded: WellFnd{i}(A;x,y.R[x; y]),
strongwellfounded: SWellFounded(R[x; y]),
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
guard: {T},
exists: ∃x:A. B[x],
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s1;s2],
subtype_rel: A ⊆r B,
so_apply: x[s],
nat: ℕ,
all: ∀x:A. B[x],
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
decidable: Dec(P),
or: P ∨ Q,
le: A ≤ B,
less_than': less_than'(a;b),
ge: i ≥ j ,
less_than: a < b,
squash: ↓T
Lemmas referenced :
all_wf,
exists_wf,
nat_wf,
less_than_wf,
int_seg_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
int_seg_wf,
decidable__equal_int,
subtract_wf,
int_seg_subtype,
false_wf,
decidable__le,
intformnot_wf,
itermSubtract_wf,
intformeq_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
int_formula_prop_eq_lemma,
le_wf,
equal_wf,
int_seg_subtype_nat,
decidable__lt,
lelt_wf,
set_wf,
primrec-wf2,
nat_properties,
itermAdd_wf,
int_term_value_add_lemma
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
cut,
introduction,
extract_by_obid,
isectElimination,
cumulativity,
hypothesisEquality,
lambdaEquality,
functionEquality,
applyEquality,
functionExtensionality,
hypothesis,
universeEquality,
because_Cache,
setElimination,
rename,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
unionElimination,
addLevel,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
levelHypothesis,
hypothesis_subsumption,
dependent_set_memberEquality,
independent_functionElimination,
addEquality,
imageElimination
Latex:
\mforall{}[T:Type]. \mforall{}[R:T {}\mrightarrow{} T {}\mrightarrow{} Type]. (SWellFounded(R[x;y]) {}\mRightarrow{} WellFnd\{i\}(T;x,y.R[x;y]))
Date html generated:
2017_04_17-AM-09_26_36
Last ObjectModification:
2017_02_27-PM-05_27_54
Theory : relations2
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