Nuprl Lemma : evodd-induction2-ext
∀[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 :
member: t ∈ T,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
ifthenelse: if b then t else f fi ,
evodd-induction2,
evodd-induction,
param-W-induction,
uall: ∀[x:A]. B[x],
so_lambda: so_lambda(x,y,z,w.t[x; y; z; w]),
so_apply: x[s1;s2;s3;s4],
top: Top,
uimplies: b supposing a,
strict4: strict4(F),
and: P ∧ Q,
all: ∀x:A. B[x],
implies: P ⇒ Q,
has-value: (a)↓,
prop: ℙ,
guard: {T},
or: P ∨ Q,
squash: ↓T,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
evodd-induction2,
lifting-strict-decide,
has-value_wf_base,
base_wf,
is-exception_wf,
evodd-induction,
param-W-induction
Rules used in proof :
introduction,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
instantiate,
extract_by_obid,
hypothesis,
sqequalRule,
thin,
sqequalHypSubstitution,
equalityTransitivity,
equalitySymmetry,
isectElimination,
baseClosed,
isect_memberEquality,
voidElimination,
voidEquality,
independent_isectElimination,
independent_pairFormation,
lambdaFormation,
callbyvalueApply,
baseApply,
closedConclusion,
hypothesisEquality,
applyExceptionCases,
inrFormation,
imageMemberEquality,
imageElimination,
inlFormation,
because_Cache
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:
2018_05_21-PM-00_05_32
Last ObjectModification:
2018_05_19-AM-07_00_31
Theory : co-recursion
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