Nuprl Lemma : evodd-induction2
∀[Q:b:𝔹 ⟶ (pw-evenodd() b) ⟶ ℙ]
(Q[tt;evodd-zero()]
⇒ (∀b:𝔹. ∀x:pw-evenodd() b. (Q[b;x] ⇒ Q[¬bb;evodd-succ(x)]))
⇒ (∀b:𝔹. ∀n:pw-evenodd() b. Q[b;n]))
Proof
Definitions occuring in Statement :
evodd-succ: evodd-succ(n),
evodd-zero: evodd-zero(),
pw-evenodd: pw-evenodd(),
bnot: ¬bb,
btrue: tt,
bool: 𝔹,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x],
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s1;s2],
so_apply: x[s],
unit: Unit,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
squash: ↓T,
guard: {T},
true: True,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
sq_type: SQType(T),
evodd-zero: evodd-zero(),
pw-evenodd: pw-evenodd(),
so_lambda: λ2x y.t[x; y],
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
btrue: tt,
evodd-succ: evodd-succ(n)
Lemmas referenced :
evodd-induction,
all_wf,
bool_wf,
bnot_wf,
unit_wf2,
pw-evenodd_wf,
equal-wf-T-base,
evodd-succ_wf,
subtype_rel-equal,
equal_wf,
bnot_bnot_elim,
iff_weakening_equal,
btrue_wf,
evodd-zero_wf,
subtype_base_sq,
bool_subtype_base,
pW-sup_wf,
squash_wf,
true_wf,
param-W_wf,
void_wf,
subtype_rel_dep_function,
subtype_rel_self
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
lambdaFormation,
independent_functionElimination,
unionElimination,
sqequalRule,
equalityElimination,
voidEquality,
lambdaEquality,
applyEquality,
functionExtensionality,
voidElimination,
functionEquality,
unionEquality,
baseClosed,
because_Cache,
universeEquality,
independent_isectElimination,
instantiate,
imageElimination,
natural_numberEquality,
imageMemberEquality,
equalityTransitivity,
equalitySymmetry,
productElimination,
cumulativity,
dependent_functionElimination,
addLevel,
hyp_replacement,
levelHypothesis,
inlEquality,
axiomEquality,
inrEquality
Latex:
\mforall{}[Q:b:\mBbbB{} {}\mrightarrow{} (pw-evenodd() b) {}\mrightarrow{} \mBbbP{}]
(Q[tt;evodd-zero()]
{}\mRightarrow{} (\mforall{}b:\mBbbB{}. \mforall{}x:pw-evenodd() b. (Q[b;x] {}\mRightarrow{} Q[\mneg{}\msubb{}b;evodd-succ(x)]))
{}\mRightarrow{} (\mforall{}b:\mBbbB{}. \mforall{}n:pw-evenodd() b. Q[b;n]))
Date html generated:
2017_04_14-AM-07_43_21
Last ObjectModification:
2017_02_27-PM-03_14_09
Theory : co-recursion
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