Nuprl Lemma : Wcmp_transitivity
∀[A:Type]. ∀[B:A ⟶ Type].
∀w1,w2,w3:W(A;a.B[a]).
((((w1 < w2) ⇒ (w2 ≤ w3) ⇒ (w1 < w3)) ∧ ((w1 ≤ w2) ⇒ (w2 < w3) ⇒ (w1 < w3)))
∧ ((w1 ≤ w2) ⇒ (w2 ≤ w3) ⇒ (w1 ≤ w3)))
Proof
Definitions occuring in Statement :
Wcmp: Wcmp(A;a.B[a];leq),
W: W(A;a.B[a]),
bfalse: ff,
btrue: tt,
uall: ∀[x:A]. B[x],
infix_ap: x f y,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
and: P ∧ Q,
implies: P ⇒ Q,
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
cand: A c∧ B,
infix_ap: x f y,
Wsup: Wsup(a;b),
Wcmp: Wcmp(A;a.B[a];leq),
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
btrue: tt,
guard: {T}
Lemmas referenced :
W-induction,
all_wf,
W_wf,
infix_ap_wf,
Wcmp_wf,
bfalse_wf,
btrue_wf,
Wsup_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
lambdaEquality,
applyEquality,
cumulativity,
because_Cache,
hypothesis,
productEquality,
functionEquality,
instantiate,
universeEquality,
independent_functionElimination,
lambdaFormation,
independent_pairFormation,
productElimination,
dependent_functionElimination,
dependent_pairFormation
Latex:
\mforall{}[A:Type]. \mforall{}[B:A {}\mrightarrow{} Type].
\mforall{}w1,w2,w3:W(A;a.B[a]).
((((w1 < w2) {}\mRightarrow{} (w2 \mleq{} w3) {}\mRightarrow{} (w1 < w3)) \mwedge{} ((w1 \mleq{} w2) {}\mRightarrow{} (w2 < w3) {}\mRightarrow{} (w1 < w3)))
\mwedge{} ((w1 \mleq{} w2) {}\mRightarrow{} (w2 \mleq{} w3) {}\mRightarrow{} (w1 \mleq{} w3)))
Date html generated:
2016_05_14-AM-06_16_01
Last ObjectModification:
2015_12_26-PM-00_04_39
Theory : co-recursion
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