Nuprl Lemma : seq-add_wf_consistent
∀T:Type. ∀R:n:ℕ ⟶ (ℕn ⟶ T) ⟶ T ⟶ ℙ. ∀n:ℕ. ∀s:R-consistent-seq(n). ∀t:T.
((R n s t) ⇒ (s.t@n ∈ R-consistent-seq(n + 1)))
Proof
Definitions occuring in Statement :
consistent-seq: R-consistent-seq(n),
seq-add: s.x@n,
int_seg: {i..j-},
nat: ℕ,
prop: ℙ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
apply: f a,
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n,
universe: Type
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
consistent-seq: R-consistent-seq(n),
uall: ∀[x:A]. B[x],
nat: ℕ,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
prop: ℙ,
so_apply: x[s],
int_seg: {i..j-},
lelt: i ≤ j < k,
guard: {T},
decidable: Dec(P),
or: P ∨ Q,
seq-add: s.x@n,
sq_stable: SqStable(P),
squash: ↓T,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
bfalse: ff,
exists: ∃x:A. B[x],
sq_type: SQType(T),
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
less_than: a < b,
subtract: n - m,
top: Top,
true: True
Lemmas referenced :
seq-add_wf,
int_seg_wf,
all_wf,
nat_wf,
int_seg_subtype_nat,
false_wf,
subtype_rel_dep_function,
subtype_rel_sets,
and_wf,
le_wf,
less_than_wf,
less_than_transitivity2,
le_weakening2,
consistent-seq_wf,
decidable__int_equal,
sq_stable__le,
equal_wf,
subtype_rel_self,
subtype_rel_wf,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
less_than_transitivity1,
le_weakening,
less_than_irreflexivity,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
iff_transitivity,
assert_wf,
bnot_wf,
not_wf,
iff_weakening_uiff,
assert_of_bnot,
le_antisymmetry_iff,
less-iff-le,
condition-implies-le,
add-associates,
minus-add,
minus-one-mul,
add-swap,
minus-one-mul-top,
zero-add,
add_functionality_wrt_le,
add-commutes,
le-add-cancel2,
decidable__lt,
not-lt-2,
not-equal-2,
le-add-cancel
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
sqequalHypSubstitution,
setElimination,
thin,
rename,
dependent_set_memberEquality,
introduction,
extract_by_obid,
isectElimination,
cumulativity,
hypothesisEquality,
functionExtensionality,
applyEquality,
natural_numberEquality,
hypothesis,
addEquality,
because_Cache,
sqequalRule,
lambdaEquality,
independent_isectElimination,
independent_pairFormation,
intEquality,
setEquality,
productElimination,
dependent_functionElimination,
functionEquality,
universeEquality,
unionElimination,
int_eqReduceTrueSq,
addLevel,
hyp_replacement,
equalitySymmetry,
levelHypothesis,
independent_functionElimination,
imageMemberEquality,
baseClosed,
imageElimination,
equalityTransitivity,
instantiate,
applyLambdaEquality,
equalityElimination,
voidElimination,
dependent_pairFormation,
promote_hyp,
impliesFunctionality,
int_eqReduceFalseSq,
isect_memberEquality,
voidEquality,
minusEquality
Latex:
\mforall{}T:Type. \mforall{}R:n:\mBbbN{} {}\mrightarrow{} (\mBbbN{}n {}\mrightarrow{} T) {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}. \mforall{}n:\mBbbN{}. \mforall{}s:R-consistent-seq(n). \mforall{}t:T.
((R n s t) {}\mRightarrow{} (s.t@n \mmember{} R-consistent-seq(n + 1)))
Date html generated:
2017_04_14-AM-07_26_34
Last ObjectModification:
2017_02_27-PM-02_55_55
Theory : bar-induction
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