Nuprl Lemma : howard-bar-rec_wf
∀B,Q:n:ℕ ⟶ (ℕn ⟶ ℕ) ⟶ ℙ. ∀bar:∀s:ℕ ⟶ ℕ. ⇃(∃n:ℕ. B[n;s]). ∀mon:∀n:ℕ. ∀m:ℕn. ∀s:ℕn ⟶ ℕ. (B[m;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]).
⇃(Q[0;seq-normalize(0;⊥)])
Proof
Definitions occuring in Statement :
quotient: x,y:A//B[x; y],
seq-normalize: seq-normalize(n;s),
seq-add: s.x@n,
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,
true: True,
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n
Definitions unfolded in proof :
so_lambda: λ2x y.t[x; y],
lelt: i ≤ j < k,
guard: {T},
int_seg: {i..j-},
less_than': less_than'(a;b),
le: A ≤ B,
subtype_rel: A ⊆r B,
so_apply: x[s],
and: P ∧ Q,
top: Top,
not: ¬A,
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
uimplies: b supposing a,
or: P ∨ Q,
decidable: Dec(P),
ge: i ≥ j ,
so_apply: x[s1;s2],
implies: P ⇒ Q,
nat: ℕ,
so_lambda: λ2x.t[x],
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T,
all: ∀x:A. B[x],
isl: isl(x),
squash: ↓T,
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
true: True,
howard-bar-rec: howard-bar-rec(M;mon;base;ind;n;s),
bfalse: ff,
sq_stable: SqStable(P),
ext2Baire: ext2Baire(n;f;d),
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
sq_type: SQType(T),
bnot: ¬bb,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
seq-add: s.x@n
Lemmas referenced :
equiv_rel_true,
true_wf,
exists_wf,
quotient_wf,
subtype_rel_self,
int_formula_prop_less_lemma,
intformless_wf,
int_seg_properties,
int_seg_subtype,
subtype_rel_dep_function,
false_wf,
int_seg_subtype_nat,
seq-add_wf,
le_wf,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
itermAdd_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__le,
nat_properties,
int_seg_wf,
nat_wf,
all_wf,
strong-continuity-rel-unique-pair,
subtype_rel_function,
istype-false,
istype-nat,
implies-quotient-true,
ext2Baire_wf,
full-omega-unsat,
istype-int,
istype-void,
istype-le,
unit_wf2,
equal_wf,
assert_wf,
equal-wf-base,
set_subtype_base,
int_subtype_base,
seq-normalize_wf,
istype-assert,
btrue_wf,
bfalse_wf,
seq-normalize-equal,
isl_wf,
sq_stable__le,
le_weakening2,
lt_int_wf,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
less_than_wf,
istype-less_than
Rules used in proof :
cumulativity,
productElimination,
universeEquality,
computeAll,
independent_pairFormation,
voidEquality,
voidElimination,
isect_memberEquality,
intEquality,
int_eqEquality,
dependent_pairFormation,
independent_isectElimination,
unionElimination,
dependent_functionElimination,
addEquality,
dependent_set_memberEquality,
hypothesisEquality,
functionExtensionality,
applyEquality,
because_Cache,
rename,
setElimination,
natural_numberEquality,
functionEquality,
lambdaEquality,
sqequalRule,
hypothesis,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
Error :lambdaEquality_alt,
Error :lambdaFormation_alt,
Error :functionIsType,
Error :inhabitedIsType,
closedConclusion,
unionEquality,
productEquality,
Error :dependent_set_memberEquality_alt,
approximateComputation,
independent_functionElimination,
Error :dependent_pairFormation_alt,
Error :isect_memberEquality_alt,
Error :universeIsType,
Error :inlEquality_alt,
Error :dependent_pairEquality_alt,
Error :equalityIstype,
equalityTransitivity,
equalitySymmetry,
Error :unionIsType,
Error :productIsType,
imageElimination,
SquashedBarInduction,
hyp_replacement,
applyLambdaEquality,
instantiate,
imageMemberEquality,
baseClosed,
Error :functionExtensionality_alt,
equalityElimination,
promote_hyp
Latex:
\mforall{}B,Q:n:\mBbbN{} {}\mrightarrow{} (\mBbbN{}n {}\mrightarrow{} \mBbbN{}) {}\mrightarrow{} \mBbbP{}. \mforall{}bar:\mforall{}s:\mBbbN{} {}\mrightarrow{} \mBbbN{}. \00D9(\mexists{}n:\mBbbN{}. B[n;s]). \mforall{}mon:\mforall{}n:\mBbbN{}. \mforall{}m:\mBbbN{}n. \mforall{}s:\mBbbN{}n {}\mrightarrow{} \mBbbN{}.
(B[m;s] {}\mRightarrow{} 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]).
\00D9(Q[0;seq-normalize(0;\mbot{})])
Date html generated:
2019_06_20-PM-03_06_11
Last ObjectModification:
2019_01_13-AM-09_04_24
Theory : continuity
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