Nuprl Lemma : decidable-bar-rec_wf
∀[B,Q:n:ℕ ⟶ (ℕn ⟶ ℕ) ⟶ ℙ]. ∀[bar:∀s:ℕ ⟶ ℕ. (↓∃n:ℕ. B[n;s])]. ∀[dec:∀n:ℕ. ∀s:ℕn ⟶ ℕ. (B[n;s] ∨ (¬B[n;s]))].
∀[base:∀n:ℕ. ∀s:ℕn ⟶ ℕ. (B[n;s] ⇒ Q[n;s])]. ∀[ind:∀n:ℕ. ∀s:ℕn ⟶ ℕ. ((∀m:ℕ. Q[n + 1;s.m@n]) ⇒ Q[n;s])].
(decidable-bar-rec(dec;base;ind;0;seq-normalize(0;⊥)) ∈ Q[0;seq-normalize(0;⊥)])
Proof
Definitions occuring in Statement :
decidable-bar-rec: decidable-bar-rec(dec;base;ind;n;s),
seq-normalize: seq-normalize(n;s),
seq-add: s.x@n,
int_seg: {i..j-},
nat: ℕ,
bottom: ⊥,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
squash: ↓T,
implies: P ⇒ Q,
or: P ∨ Q,
member: t ∈ T,
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n
Definitions unfolded in proof :
less_than': less_than'(a;b),
le: A ≤ B,
subtype_rel: A ⊆r B,
and: P ∧ Q,
prop: ℙ,
top: Top,
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
uimplies: b supposing a,
or: P ∨ Q,
decidable: Dec(P),
ge: i ≥ j ,
so_apply: x[s1;s2],
implies: P ⇒ Q,
nat: ℕ,
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
squash: ↓T,
int_seg: {i..j-},
lelt: i ≤ j < k,
guard: {T},
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
decidable-bar-rec: decidable-bar-rec(dec;base;ind;n;s),
seq-add: s.x@n
Lemmas referenced :
subtype_rel_self,
istype-false,
int_seg_subtype_nat,
subtype_rel_function,
squash_wf,
nat_wf,
seq-add_wf,
istype-le,
int_formula_prop_le_lemma,
int_formula_prop_and_lemma,
intformle_wf,
intformand_wf,
decidable__le,
int_formula_prop_wf,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_formula_prop_eq_lemma,
istype-void,
int_formula_prop_not_lemma,
istype-int,
itermConstant_wf,
itermVar_wf,
itermAdd_wf,
intformeq_wf,
intformnot_wf,
full-omega-unsat,
decidable__equal_int,
nat_properties,
int_seg_wf,
istype-nat,
seq-normalize_wf,
int_seg_properties,
equal_wf,
true_wf,
istype-universe,
seq-normalize-equal,
iff_weakening_equal,
not_wf
Rules used in proof :
universeEquality,
Error :lambdaFormation_alt,
productEquality,
Error :unionIsType,
Error :inhabitedIsType,
Error :isectIsTypeImplies,
independent_pairFormation,
Error :dependent_set_memberEquality_alt,
voidElimination,
Error :isect_memberEquality_alt,
int_eqEquality,
Error :lambdaEquality_alt,
Error :dependent_pairFormation_alt,
independent_functionElimination,
approximateComputation,
independent_isectElimination,
unionElimination,
addEquality,
dependent_functionElimination,
applyEquality,
because_Cache,
hypothesisEquality,
rename,
setElimination,
natural_numberEquality,
thin,
isectElimination,
Error :universeIsType,
extract_by_obid,
Error :functionIsType,
equalitySymmetry,
equalityTransitivity,
axiomEquality,
sqequalRule,
hypothesis,
sqequalHypSubstitution,
cut,
introduction,
Error :isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
imageElimination,
SquashedBarInduction,
instantiate,
productElimination,
imageMemberEquality,
baseClosed,
Error :functionExtensionality_alt,
applyLambdaEquality,
intEquality,
functionExtensionality,
functionEquality,
unionEquality,
Error :equalityIstype
Latex:
\mforall{}[B,Q:n:\mBbbN{} {}\mrightarrow{} (\mBbbN{}n {}\mrightarrow{} \mBbbN{}) {}\mrightarrow{} \mBbbP{}]. \mforall{}[bar:\mforall{}s:\mBbbN{} {}\mrightarrow{} \mBbbN{}. (\mdownarrow{}\mexists{}n:\mBbbN{}. B[n;s])]. \mforall{}[dec:\mforall{}n:\mBbbN{}. \mforall{}s:\mBbbN{}n {}\mrightarrow{} \mBbbN{}.
(B[n;s] \mvee{} (\mneg{}B[n;s]))].
\mforall{}[base:\mforall{}n:\mBbbN{}. \mforall{}s:\mBbbN{}n {}\mrightarrow{} \mBbbN{}. (B[n;s] {}\mRightarrow{} Q[n;s])]. \mforall{}[ind:\mforall{}n:\mBbbN{}. \mforall{}s:\mBbbN{}n {}\mrightarrow{} \mBbbN{}.
((\mforall{}m:\mBbbN{}. Q[n + 1;s.m@n]) {}\mRightarrow{} Q[n;s])].
(decidable-bar-rec(dec;base;ind;0;seq-normalize(0;\mbot{})) \mmember{} Q[0;seq-normalize(0;\mbot{})])
Date html generated:
2019_06_20-PM-03_05_20
Last ObjectModification:
2019_01_09-PM-02_33_02
Theory : continuity
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