Nuprl Lemma : cond_safety_and
∀[A:Type]. ∀[P,Q:(A List) ⟶ ℙ].
(safety(A;x.P[x]) ⇒ (∀tr1,tr2:A List. (tr1 ≤ tr2 ⇒ P[tr2] ⇒ Q[tr2] ⇒ Q[tr1])) ⇒ safety(A;x.P[x] ∧ Q[x]))
Proof
Definitions occuring in Statement :
safety: safety(A;tr.P[tr]),
iseg: l1 ≤ l2,
list: T List,
uall: ∀[x:A]. B[x],
prop: ℙ,
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 :
safety: safety(A;tr.P[tr]),
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
and: P ∧ Q,
cand: A c∧ B,
member: t ∈ T,
prop: ℙ,
so_apply: x[s],
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
guard: {T}
Lemmas referenced :
subtype_rel_self,
iseg_wf,
list_wf,
all_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation_alt,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
cut,
independent_pairFormation,
hypothesis,
productEquality,
applyEquality,
hypothesisEquality,
instantiate,
introduction,
extract_by_obid,
isectElimination,
universeEquality,
because_Cache,
lambdaEquality,
functionEquality,
inhabitedIsType,
functionIsType,
universeIsType,
dependent_functionElimination,
independent_functionElimination
Latex:
\mforall{}[A:Type]. \mforall{}[P,Q:(A List) {}\mrightarrow{} \mBbbP{}].
(safety(A;x.P[x])
{}\mRightarrow{} (\mforall{}tr1,tr2:A List. (tr1 \mleq{} tr2 {}\mRightarrow{} P[tr2] {}\mRightarrow{} Q[tr2] {}\mRightarrow{} Q[tr1]))
{}\mRightarrow{} safety(A;x.P[x] \mwedge{} Q[x]))
Date html generated:
2019_10_15-AM-10_54_15
Last ObjectModification:
2018_09_27-AM-10_45_49
Theory : list!
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