Nuprl Lemma : mFOL-induction
∀[P:mFOL() ─→ ℙ]
((∀name:Atom. ∀vars:ℤ List. P[name(vars)])
⇒ (∀knd:Atom. ∀left,right:mFOL(). (P[left] ⇒ P[right] ⇒ P[mFOconnect(knd;left;right)]))
⇒ (∀isall:𝔹. ∀var:ℤ. ∀body:mFOL(). (P[body] ⇒ P[mFOquant(isall;var;body)]))
⇒ {∀v:mFOL(). P[v]})
Proof
Definitions occuring in Statement :
mFOquant: mFOquant(isall;var;body),
mFOconnect: mFOconnect(knd;left;right),
mFOatomic: name(vars),
mFOL: mFOL(),
list: T List,
bool: 𝔹,
uall: ∀[x:A]. B[x],
prop: ℙ,
guard: {T},
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ─→ B[x],
int: ℤ,
atom: Atom
Lemmas :
uniform-comp-nat-induction,
all_wf,
mFOL_wf,
isect_wf,
le_wf,
mFOL_size_wf,
nat_wf,
less_than_wf,
mFOL-ext,
eq_atom_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_atom,
subtype_base_sq,
atom_subtype_base,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_atom,
subtract_wf,
decidable__le,
false_wf,
not-le-2,
less-iff-le,
condition-implies-le,
minus-one-mul,
zero-add,
minus-add,
minus-minus,
add-associates,
add-swap,
add-commutes,
add_functionality_wrt_le,
add-zero,
le-add-cancel,
subtract-is-less,
lelt_wf,
decidable__lt,
uall_wf,
int_seg_wf,
le_weakening,
mFOquant_wf,
mFOconnect_wf,
list_wf,
mFOatomic_wf
\mforall{}[P:mFOL() {}\mrightarrow{} \mBbbP{}]
((\mforall{}name:Atom. \mforall{}vars:\mBbbZ{} List. P[name(vars)])
{}\mRightarrow{} (\mforall{}knd:Atom. \mforall{}left,right:mFOL(). (P[left] {}\mRightarrow{} P[right] {}\mRightarrow{} P[mFOconnect(knd;left;right)]))
{}\mRightarrow{} (\mforall{}isall:\mBbbB{}. \mforall{}var:\mBbbZ{}. \mforall{}body:mFOL(). (P[body] {}\mRightarrow{} P[mFOquant(isall;var;body)]))
{}\mRightarrow{} \{\mforall{}v:mFOL(). P[v]\})
Date html generated:
2015_07_17-AM-07_53_49
Last ObjectModification:
2015_01_27-AM-10_07_04
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