Nuprl Lemma : dl-box-choose
∀a,b:Prog. ∀phi:Prop. ([a ⋃ b] phi ⇐⇒ [a] phi ∧ [b] phi)
Proof
Definitions occuring in Statement :
dl-equiv: (phi ⇐⇒ psi),
dl-box: [x1] x,
dl-and: x1 ∧ x,
dl-choose: x1 ⋃ x,
dl-prop: Prop,
dl-prog: Prog,
all: ∀x:A. B[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
dl-equiv: (phi ⇐⇒ psi),
and: P ∧ Q,
dl-valid: |= phi,
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],
member: t ∈ T,
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|],
implies: P ⇒ Q,
cand: A c∧ B,
or: P ∨ Q,
subtype_rel: A ⊆r B,
prop: ℙ,
guard: {T}
Lemmas referenced :
dl-ind-dl-implies,
istype-void,
dl-ind-dl-box,
dl-ind-dl-choose,
dl-ind-dl-and,
dl-prog-sem_wf,
istype-nat,
subtype_rel_self,
dl-prop-sem_wf,
istype-universe,
dl-prop_wf,
dl-prog_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
independent_pairFormation,
sqequalRule,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
isect_memberEquality_alt,
voidElimination,
hypothesis,
dependent_functionElimination,
hypothesisEquality,
independent_functionElimination,
inlFormation_alt,
universeIsType,
applyEquality,
lambdaEquality_alt,
instantiate,
universeEquality,
because_Cache,
inrFormation_alt,
functionIsType,
unionIsType,
productElimination,
unionElimination,
productIsType,
inhabitedIsType
Latex:
\mforall{}a,b:Prog. \mforall{}phi:Prop. ([a \mcup{} b] phi \mLeftarrow{}{}\mRightarrow{} [a] phi \mwedge{} [b] phi)
Date html generated:
2019_10_15-AM-11_44_41
Last ObjectModification:
2019_03_27-AM-00_34_44
Theory : dynamic!logic
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