Nuprl Lemma : TC-ind-ext
∀[Dom:Type]. ∀[B:Dom ⟶ ℙ]. ∀[R:Dom ⟶ Dom ⟶ ℙ].
  ((∀x,y:Dom.  ((R x y) 
⇒ (B x) 
⇒ (B y))) 
⇒ (∀x,y:Dom.  (TC(λa,b.R a b)(x,y) 
⇒ (B x) 
⇒ (B y))))
Proof
Definitions occuring in Statement : 
TC: TC(λx,y.F[x; y])(a,b)
, 
uall: ∀[x:A]. B[x]
, 
prop: ℙ
, 
all: ∀x:A. B[x]
, 
implies: P 
⇒ Q
, 
apply: f a
, 
function: x:A ⟶ B[x]
, 
universe: Type
Definitions unfolded in proof : 
member: t ∈ T
, 
TC-ind, 
transitive-closure-induction, 
transitive-closure-minimal, 
spreadn: spread3, 
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 y.t[x; y]
, 
so_apply: x[s1;s2]
Lemmas referenced : 
TC-ind, 
transitive-closure-induction, 
transitive-closure-minimal, 
is-exception_wf, 
base_wf, 
has-value_wf_base, 
lifting-strict-spread
Rules used in proof : 
introduction, 
cut, 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
instantiate, 
extract_by_obid, 
hypothesis, 
sqequalHypSubstitution, 
sqequalRule, 
lemma_by_obid, 
isectElimination, 
thin, 
baseClosed, 
isect_memberEquality, 
voidElimination, 
voidEquality, 
independent_isectElimination, 
independent_pairFormation, 
lambdaFormation, 
callbyvalueApply, 
baseApply, 
closedConclusion, 
hypothesisEquality, 
applyExceptionCases, 
inrFormation, 
imageMemberEquality, 
imageElimination, 
exceptionSqequal, 
inlFormation, 
equalityTransitivity, 
equalitySymmetry
Latex:
\mforall{}[Dom:Type].  \mforall{}[B:Dom  {}\mrightarrow{}  \mBbbP{}].  \mforall{}[R:Dom  {}\mrightarrow{}  Dom  {}\mrightarrow{}  \mBbbP{}].
    ((\mforall{}x,y:Dom.    ((R  x  y)  {}\mRightarrow{}  (B  x)  {}\mRightarrow{}  (B  y)))  {}\mRightarrow{}  (\mforall{}x,y:Dom.    (TC(\mlambda{}a,b.R  a  b)(x,y)  {}\mRightarrow{}  (B  x)  {}\mRightarrow{}  (B  y))))
Date html generated:
2016_05_16-AM-09_07_52
Last ObjectModification:
2016_01_17-AM-09_53_25
Theory : first-order!and!ancestral!logic
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