Nuprl Lemma : monotone-bar-induction4
∀B,Q:n:ℕ ⟶ (ℕn ⟶ ℕ) ⟶ ℙ.
((∀n:ℕ. ∀s:ℕn ⟶ ℕ. (B[n;s] ⇒ ⇃(Q[n;s])))
⇒ (∀n:ℕ. ∀s:ℕn ⟶ ℕ. ((∀m:ℕ. ⇃(Q[n + 1;s.m@n])) ⇒ ⇃(Q[n;s])))
⇒ (∀alpha:ℕ ⟶ ℕ
⇃(∃n:ℕ. (B[n;alpha] ∧ (∀m:{n...}. ∀s:ℕm ⟶ ℕ. ((alpha = s ∈ (ℕm ⟶ ℕ)) ⇒ B[m;s] ⇒ (∀k:ℕ. B[m + 1;s.k@m]))))))
⇒ ⇃(Q[0;λx.⊥]))
Proof
Definitions occuring in Statement :
quotient: x,y:A//B[x; y],
seq-add: s.x@n,
int_upper: {i...},
int_seg: {i..j-},
nat: ℕ,
bottom: ⊥,
prop: ℙ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
true: True,
lambda: λx.A[x],
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
and: P ∧ Q,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
nat: ℕ,
uimplies: b supposing a,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
so_lambda: λ2x.t[x],
int_upper: {i...},
so_apply: x[s1;s2],
sq_stable: SqStable(P),
squash: ↓T,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
so_apply: x[s],
int_seg: {i..j-},
lelt: i ≤ j < k,
so_lambda: λ2x y.t[x; y],
guard: {T},
isl: isl(x),
ext2Baire: ext2Baire(n;f;d),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
true: True,
outl: outl(x),
less_than: a < b,
subtract: n - m,
seq-add: s.x@n,
nequal: a ≠ b ∈ T ,
seq-adjoin: s++t,
seq-append: seq-append(n;m;s1;s2),
cand: A c∧ B,
quotient: x,y:A//B[x; y]
Lemmas referenced :
strong-continuity-rel,
subtype_rel_function,
nat_wf,
int_seg_wf,
int_seg_subtype_nat,
istype-false,
subtype_rel_self,
all_wf,
int_upper_wf,
equal_wf,
upper_subtype_nat,
sq_stable__le,
nat_properties,
int_upper_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
istype-le,
seq-add_wf,
istype-int_upper,
prop-truncation-quot,
int_seg_properties,
intformless_wf,
int_formula_prop_less_lemma,
istype-nat,
quotient_wf,
exists_wf,
true_wf,
equiv_rel_true,
unit_wf2,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
union_subtype_base,
set_subtype_base,
lelt_wf,
int_subtype_base,
unit_subtype_base,
istype-assert,
btrue_wf,
bfalse_wf,
subtype_rel_union,
basic_bar_induction,
assert_wf,
decidable__assert,
seq-adjoin_wf,
ext2Baire_wf,
squash_wf,
bool_wf,
isl_wf,
lt_int_wf,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
less_than_wf,
istype-less_than,
le_wf,
btrue_neq_bfalse,
decidable__lt,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
minus-one-mul,
add-commutes,
add-associates,
add-mul-special,
zero-mul,
zero-add,
int_seg_subtype,
add-is-int-iff,
primrec-wf2,
add-zero,
not-le-2,
condition-implies-le,
minus-add,
add-swap,
minus-one-mul-top,
add_functionality_wrt_le,
le-add-cancel2,
add-member-int_seg2,
eq_int_wf,
assert_of_eq_int,
decidable__equal_nat,
neg_assert_of_eq_int,
istype-top,
quotient-member-eq,
member_wf,
istype-universe
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :lambdaFormation_alt,
rename,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
Error :lambdaEquality_alt,
productEquality,
applyEquality,
hypothesisEquality,
isectElimination,
hypothesis,
because_Cache,
natural_numberEquality,
setElimination,
independent_isectElimination,
sqequalRule,
independent_pairFormation,
functionEquality,
closedConclusion,
independent_functionElimination,
imageMemberEquality,
baseClosed,
imageElimination,
Error :dependent_set_memberEquality_alt,
addEquality,
unionElimination,
approximateComputation,
Error :dependent_pairFormation_alt,
int_eqEquality,
Error :isect_memberEquality_alt,
voidElimination,
Error :universeIsType,
Error :functionIsType,
productElimination,
Error :inhabitedIsType,
Error :productIsType,
Error :equalityIstype,
universeEquality,
Error :unionIsType,
instantiate,
equalityTransitivity,
equalitySymmetry,
intEquality,
baseApply,
sqequalBase,
hyp_replacement,
Error :functionExtensionality_alt,
equalityElimination,
promote_hyp,
cumulativity,
applyLambdaEquality,
Error :setIsType,
minusEquality,
functionExtensionality,
int_eqReduceTrueSq,
int_eqReduceFalseSq,
lessCases,
Error :isect_memberFormation_alt,
axiomSqEquality,
Error :isectIsTypeImplies,
pointwiseFunctionality,
pertypeElimination
Latex:
\mforall{}B,Q:n:\mBbbN{} {}\mrightarrow{} (\mBbbN{}n {}\mrightarrow{} \mBbbN{}) {}\mrightarrow{} \mBbbP{}.
((\mforall{}n:\mBbbN{}. \mforall{}s:\mBbbN{}n {}\mrightarrow{} \mBbbN{}. (B[n;s] {}\mRightarrow{} \00D9(Q[n;s])))
{}\mRightarrow{} (\mforall{}n:\mBbbN{}. \mforall{}s:\mBbbN{}n {}\mrightarrow{} \mBbbN{}. ((\mforall{}m:\mBbbN{}. \00D9(Q[n + 1;s.m@n])) {}\mRightarrow{} \00D9(Q[n;s])))
{}\mRightarrow{} (\mforall{}alpha:\mBbbN{} {}\mrightarrow{} \mBbbN{}
\00D9(\mexists{}n:\mBbbN{}
(B[n;alpha]
\mwedge{} (\mforall{}m:\{n...\}. \mforall{}s:\mBbbN{}m {}\mrightarrow{} \mBbbN{}. ((alpha = s) {}\mRightarrow{} B[m;s] {}\mRightarrow{} (\mforall{}k:\mBbbN{}. B[m + 1;s.k@m]))))))
{}\mRightarrow{} \00D9(Q[0;\mlambda{}x.\mbot{}]))
Date html generated:
2019_06_20-PM-02_55_53
Last ObjectModification:
2018_12_06-PM-11_35_00
Theory : continuity
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