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
Definitions unfolded in proof :
uall: ∀[x:A]. B[x]
,
implies: P
⇒ Q
,
guard: {T}
,
so_lambda: λ2x.t[x]
,
member: t ∈ T
,
uimplies: b supposing a
,
subtype_rel: A ⊆r B
,
nat: ℕ
,
prop: ℙ
,
so_apply: x[s]
,
all: ∀x:A. B[x]
,
le: A ≤ B
,
and: P ∧ Q
,
not: ¬A
,
false: False
,
ext-eq: A ≡ B
,
bool: 𝔹
,
unit: Unit
,
it: ⋅
,
btrue: tt
,
uiff: uiff(P;Q)
,
sq_type: SQType(T)
,
eq_atom: x =a y
,
ifthenelse: if b then t else f fi
,
mFOatomic: name(vars)
,
mFOL_size: mFOL_size(p)
,
spreadn: spread3,
bfalse: ff
,
exists: ∃x:A. B[x]
,
or: P ∨ Q
,
bnot: ¬bb
,
assert: ↑b
,
mFOconnect: mFOconnect(knd;left;right)
,
cand: A c∧ B
,
ge: i ≥ j
,
decidable: Dec(P)
,
satisfiable_int_formula: satisfiable_int_formula(fmla)
,
top: Top
,
int_seg: {i..j-}
,
lelt: i ≤ j < k
,
less_than: a < b
,
squash: ↓T
,
mFOquant: mFOquant(isall;var;body)
Lemmas referenced :
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,
nat_properties,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
intformle_wf,
itermAdd_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_le_lemma,
int_term_value_add_lemma,
int_formula_prop_wf,
subtract_wf,
decidable__le,
itermSubtract_wf,
int_term_value_subtract_lemma,
lelt_wf,
uall_wf,
int_seg_wf,
mFOquant_wf,
mFOconnect_wf,
list_wf,
mFOatomic_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
sqequalRule,
lambdaEquality,
hypothesis,
hypothesisEquality,
applyEquality,
because_Cache,
setElimination,
rename,
functionExtensionality,
independent_functionElimination,
productElimination,
independent_pairEquality,
dependent_functionElimination,
voidElimination,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
hypothesis_subsumption,
tokenEquality,
unionElimination,
equalityElimination,
independent_isectElimination,
instantiate,
cumulativity,
atomEquality,
dependent_pairFormation,
independent_pairFormation,
applyLambdaEquality,
natural_numberEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidEquality,
computeAll,
dependent_set_memberEquality,
imageElimination,
functionEquality,
universeEquality
Latex:
\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:
2018_05_21-PM-10_21_32
Last ObjectModification:
2017_07_26-PM-06_37_56
Theory : minimal-first-order-logic
Home
Index