Nuprl Lemma : dl-valid-diamond-box-distibute
∀a:Prog. ∀phi,psi:Prop. (|= <a> phi ∧ [a] psi ⇒ |= <a> phi ∧ psi)
Proof
Definitions occuring in Statement :
dl-valid: |= phi,
dl-diamond: <x1> x,
dl-box: [x1] x,
dl-and: x1 ∧ x,
dl-prop: Prop,
dl-prog: Prog,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
dl-valid: |= phi,
member: t ∈ T,
dl-prop-sem: [|phi|],
dl-sem: dl-sem(K;n.R[n];m.P[m]),
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
top: Top,
so_apply: x[s],
so_lambda: so_lambda(x,y,z,w.t[x; y; z; w]),
so_apply: x[s1;s2;s3;s4],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
dl-prog-sem: [|alpha|],
and: P ∧ Q,
exists: ∃x:A. B[x],
prop: ℙ,
cand: A c∧ B,
subtype_rel: A ⊆r B,
guard: {T}
Lemmas referenced :
istype-void,
istype-atom,
istype-universe,
dl-valid_wf,
dl-prog-sem_wf,
subtype_rel_self,
dl-prop-sem_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
sqequalHypSubstitution,
cut,
hypothesis,
dependent_functionElimination,
thin,
hypothesisEquality,
sqequalRule,
introduction,
extract_by_obid,
isectElimination,
isect_memberEquality_alt,
voidElimination,
productElimination,
universeIsType,
functionIsType,
universeEquality,
because_Cache,
instantiate,
inhabitedIsType,
dependent_pairFormation_alt,
independent_pairFormation,
productIsType,
applyEquality,
lambdaEquality_alt,
independent_functionElimination
Latex:
\mforall{}a:Prog. \mforall{}phi,psi:Prop. (|= <a> phi \mwedge{} [a] psi {}\mRightarrow{} |= <a> phi \mwedge{} psi)
Date html generated:
2019_10_15-AM-11_45_07
Last ObjectModification:
2019_03_26-AM-11_28_38
Theory : dynamic!logic
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