Nuprl Lemma : lexico_well_fnd
∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ]. (WellFnd{i}(T;a,b.R[a;b]) ⇒ WellFnd{i}(T List;as,bs.as lexico(T; a,b.R[a;b]) bs))
Proof
Definitions occuring in Statement :
lexico: lexico(T; a,b.lt[a; b]),
list: T List,
wellfounded: WellFnd{i}(A;x,y.R[x; y]),
uall: ∀[x:A]. B[x],
prop: ℙ,
infix_ap: x f y,
so_apply: x[s1;s2],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
infix_ap: x f y,
so_lambda: λ2x.t[x],
so_apply: x[s],
wellfounded: WellFnd{i}(A;x,y.R[x; y]),
guard: {T},
or: P ∨ Q,
cons: [a / b],
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
length: ||as||,
list_ind: list_ind,
nil: [],
it: ⋅,
false: False,
not: ¬A,
nat: ℕ,
lexico: lexico(T; a,b.lt[a; b]),
select: L[n],
uimplies: b supposing a,
top: Top,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
squash: ↓T,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
sq_type: SQType(T),
uiff: uiff(P;Q),
cand: A c∧ B,
subtract: n - m,
less_than: a < b
Lemmas referenced :
wellfounded_wf,
list_wf,
equal-wf-T-base,
nat_wf,
length_wf_nat,
int_subtype_base,
lexico_wf,
set_wf,
less_than_wf,
primrec-wf2,
equal_wf,
infix_ap_wf,
list-cases,
product_subtype_list,
all_wf,
false_wf,
length_wf,
nil_wf,
length_of_nil_lemma,
le_wf,
stuck-spread,
base_wf,
non_neg_length,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
int_seg_properties,
intformeq_wf,
int_formula_prop_eq_lemma,
length_of_cons_lemma,
nat_properties,
itermAdd_wf,
int_term_value_add_lemma,
product_well_fnd,
inv_image_ind_a,
or_wf,
subtype_rel_dep_function,
hd_wf,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
squash_wf,
true_wf,
length_tl,
iff_weakening_equal,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
tl_wf,
listp_properties,
decidable__equal_int,
subtype_base_sq,
reduce_hd_cons_lemma,
add-is-int-iff,
decidable__lt,
lelt_wf,
reduce_tl_nil_lemma,
reduce_tl_cons_lemma,
int_seg_wf,
select_wf,
select-cons-tl,
select_cons_tl,
add-member-int_seg2,
add-associates,
add-swap,
add-commutes,
zero-add,
add-subtract-cancel,
int_seg_subtype,
subtype_rel_self,
decidable__equal_nat
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
sqequalRule,
lambdaEquality,
applyEquality,
functionExtensionality,
hypothesis,
functionEquality,
universeEquality,
rename,
setElimination,
setEquality,
baseApply,
closedConclusion,
baseClosed,
because_Cache,
intEquality,
natural_numberEquality,
instantiate,
dependent_functionElimination,
unionElimination,
promote_hyp,
hypothesis_subsumption,
productElimination,
equalityTransitivity,
equalitySymmetry,
dependent_set_memberEquality,
independent_pairFormation,
voidEquality,
independent_functionElimination,
independent_isectElimination,
isect_memberEquality,
voidElimination,
dependent_pairFormation,
int_eqEquality,
computeAll,
applyLambdaEquality,
productEquality,
independent_pairEquality,
imageElimination,
imageMemberEquality,
inlFormation,
addLevel,
hyp_replacement,
pointwiseFunctionality,
levelHypothesis,
inrFormation,
addEquality
Latex:
\mforall{}[T:Type]. \mforall{}[R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}].
(WellFnd\{i\}(T;a,b.R[a;b]) {}\mRightarrow{} WellFnd\{i\}(T List;as,bs.as lexico(T; a,b.R[a;b]) bs))
Date html generated:
2018_05_21-PM-08_37_20
Last ObjectModification:
2017_07_26-PM-06_01_37
Theory : general
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