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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