Nuprl Lemma : pi_term-induction
∀[P:pi_term() ⟶ ℙ]
(P[pizero()]
⇒ (∀pre:pi_prefix(). ∀body:pi_term(). (P[body] ⇒ P[picomm(pre;body)]))
⇒ (∀left,right:pi_term(). (P[left] ⇒ P[right] ⇒ P[pioption(left;right)]))
⇒ (∀left,right:pi_term(). (P[left] ⇒ P[right] ⇒ P[pipar(left;right)]))
⇒ (∀body:pi_term(). (P[body] ⇒ P[pirep(body)]))
⇒ (∀name:Name. ∀body:pi_term(). (P[body] ⇒ P[pinew(name;body)]))
⇒ {∀v:pi_term(). P[v]})
Proof
Definitions occuring in Statement :
pinew: pinew(name;body),
pirep: pirep(body),
pipar: pipar(left;right),
pioption: pioption(left;right),
picomm: picomm(pre;body),
pizero: pizero(),
pi_term: pi_term(),
pi_prefix: pi_prefix(),
name: Name,
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]
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 ,
pizero: pizero(),
pi_term_size: pi_term_size(p),
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
bnot: ¬bb,
assert: ↑b,
picomm: picomm(pre;body),
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,
pioption: pioption(left;right),
pipar: pipar(left;right),
pirep: pirep(body),
pinew: pinew(name;body)
Latex:
\mforall{}[P:pi\_term() {}\mrightarrow{} \mBbbP{}]
(P[pizero()]
{}\mRightarrow{} (\mforall{}pre:pi\_prefix(). \mforall{}body:pi\_term(). (P[body] {}\mRightarrow{} P[picomm(pre;body)]))
{}\mRightarrow{} (\mforall{}left,right:pi\_term(). (P[left] {}\mRightarrow{} P[right] {}\mRightarrow{} P[pioption(left;right)]))
{}\mRightarrow{} (\mforall{}left,right:pi\_term(). (P[left] {}\mRightarrow{} P[right] {}\mRightarrow{} P[pipar(left;right)]))
{}\mRightarrow{} (\mforall{}body:pi\_term(). (P[body] {}\mRightarrow{} P[pirep(body)]))
{}\mRightarrow{} (\mforall{}name:Name. \mforall{}body:pi\_term(). (P[body] {}\mRightarrow{} P[pinew(name;body)]))
{}\mRightarrow{} \{\mforall{}v:pi\_term(). P[v]\})
Date html generated:
2016_05_17-AM-11_22_35
Last ObjectModification:
2016_01_18-AM-07_49_35
Theory : event-logic-applications
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