Nuprl Lemma : rel_exp_add-ext
∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ]. ∀m,n:ℕ. ∀[x,y,z:T]. ((x R^m y) ⇒ (y R^n z) ⇒ (x R^m + n z))
Proof
Definitions occuring in Statement :
rel_exp: R^n,
nat: ℕ,
uall: ∀[x:A]. B[x],
prop: ℙ,
infix_ap: x f y,
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
add: n + m,
universe: Type
Definitions unfolded in proof :
member: t ∈ T,
rel_exp_add,
complete_nat_ind_with_y,
complete_nat_measure_ind,
genrec: genrec,
bool_cases,
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],
so_lambda: λ2x.t[x],
top: Top,
so_apply: x[s],
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,
eq_int: (i =z j),
btrue: tt,
bfalse: ff,
any: any x,
subtract: n - m
Lemmas referenced :
rel_exp_add,
lifting-strict-decide,
top_wf,
equal_wf,
has-value_wf_base,
base_wf,
is-exception_wf,
lifting-strict-int_eq,
complete_nat_ind_with_y,
complete_nat_measure_ind,
bool_cases
Rules used in proof :
introduction,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
instantiate,
extract_by_obid,
hypothesis,
sqequalRule,
thin,
sqequalHypSubstitution,
isectElimination,
baseClosed,
isect_memberEquality,
voidElimination,
voidEquality,
independent_isectElimination,
independent_pairFormation,
lambdaFormation,
callbyvalueDecide,
hypothesisEquality,
equalityTransitivity,
equalitySymmetry,
unionEquality,
unionElimination,
sqleReflexivity,
dependent_functionElimination,
independent_functionElimination,
baseApply,
closedConclusion,
decideExceptionCases,
inrFormation,
because_Cache,
imageMemberEquality,
imageElimination,
exceptionSqequal,
inlFormation
Latex:
\mforall{}[T:Type]. \mforall{}[R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}].
\mforall{}m,n:\mBbbN{}. \mforall{}[x,y,z:T]. ((x R\^{}m y) {}\mRightarrow{} (y R\^{}n z) {}\mRightarrow{} (x rel\_exp(T; R; m + n) z))
Date html generated:
2017_04_14-AM-07_38_19
Last ObjectModification:
2017_02_27-PM-03_10_11
Theory : relations
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