Nuprl Lemma : Form-induction
∀[C:Type]. ∀[P:Form(C) ⟶ ℙ].
((∀name:Atom. P[Vname])
⇒ (∀value:C. P[Const(value)])
⇒ (∀var:Atom. ∀phi:Form(C). (P[phi] ⇒ P[{var | phi}]))
⇒ (∀left,right:Form(C). (P[left] ⇒ P[right] ⇒ P[left = right]))
⇒ (∀element,set:Form(C). (P[element] ⇒ P[set] ⇒ P[element ∈ set]))
⇒ (∀left,right:Form(C). (P[left] ⇒ P[right] ⇒ P[left ∧ right)]))
⇒ (∀left,right:Form(C). (P[left] ⇒ P[right] ⇒ P[left ∨ right]))
⇒ (∀body:Form(C). (P[body] ⇒ P[¬(body)]))
⇒ (∀var:Atom. ∀body:Form(C). (P[body] ⇒ P[∀var. body]))
⇒ (∀var:Atom. ∀body:Form(C). (P[body] ⇒ P[∃var. body]))
⇒ {∀v:Form(C). P[v]})
Proof
Definitions occuring in Statement :
FormExists: ∃var. body,
FormAll: ∀var. body,
FormNot: ¬(body),
FormOr: left ∨ right,
FormAnd: left ∧ right),
FormMember: element ∈ set,
FormEqual: left = right,
FormSet: {var | phi},
FormConst: Const(value),
FormVar: Vname,
Form: Form(C),
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],
atom: Atom,
universe: Type
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 ,
FormVar: Vname,
Form_size: Form_size(p),
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
bnot: ¬bb,
assert: ↑b,
FormConst: Const(value),
FormSet: {var | phi},
pi2: snd(t),
pi1: fst(t),
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,
FormEqual: left = right,
FormMember: element ∈ set,
FormAnd: left ∧ right),
FormOr: left ∨ right,
FormNot: ¬(body),
FormAll: ∀var. body,
FormExists: ∃var. body
Lemmas referenced :
uniform-comp-nat-induction,
all_wf,
Form_wf,
isect_wf,
le_wf,
Form_size_wf,
nat_wf,
less_than'_wf,
Form-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,
full-omega-unsat,
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,
FormExists_wf,
FormAll_wf,
FormNot_wf,
FormOr_wf,
FormAnd_wf,
FormMember_wf,
FormEqual_wf,
FormSet_wf,
FormConst_wf,
FormVar_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
sqequalRule,
lambdaEquality,
cumulativity,
hypothesisEquality,
hypothesis,
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,
atomEquality,
dependent_pairFormation,
applyLambdaEquality,
natural_numberEquality,
approximateComputation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidEquality,
independent_pairFormation,
dependent_set_memberEquality,
imageElimination,
functionEquality,
universeEquality
Latex:
\mforall{}[C:Type]. \mforall{}[P:Form(C) {}\mrightarrow{} \mBbbP{}].
((\mforall{}name:Atom. P[Vname])
{}\mRightarrow{} (\mforall{}value:C. P[Const(value)])
{}\mRightarrow{} (\mforall{}var:Atom. \mforall{}phi:Form(C). (P[phi] {}\mRightarrow{} P[\{var | phi\}]))
{}\mRightarrow{} (\mforall{}left,right:Form(C). (P[left] {}\mRightarrow{} P[right] {}\mRightarrow{} P[left = right]))
{}\mRightarrow{} (\mforall{}element,set:Form(C). (P[element] {}\mRightarrow{} P[set] {}\mRightarrow{} P[element \mmember{} set]))
{}\mRightarrow{} (\mforall{}left,right:Form(C). (P[left] {}\mRightarrow{} P[right] {}\mRightarrow{} P[left \mwedge{} right)]))
{}\mRightarrow{} (\mforall{}left,right:Form(C). (P[left] {}\mRightarrow{} P[right] {}\mRightarrow{} P[left \mvee{} right]))
{}\mRightarrow{} (\mforall{}body:Form(C). (P[body] {}\mRightarrow{} P[\mneg{}(body)]))
{}\mRightarrow{} (\mforall{}var:Atom. \mforall{}body:Form(C). (P[body] {}\mRightarrow{} P[\mforall{}var. body]))
{}\mRightarrow{} (\mforall{}var:Atom. \mforall{}body:Form(C). (P[body] {}\mRightarrow{} P[\mexists{}var. body]))
{}\mRightarrow{} \{\mforall{}v:Form(C). P[v]\})
Date html generated:
2018_05_21-PM-11_25_43
Last ObjectModification:
2017_10_13-PM-07_02_58
Theory : PZF
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