Nuprl Lemma : dl-prog-sem_functionality
∀alpha:Prog
∀[K:Type]. ∀[R:ℕ ⟶ K ⟶ K ⟶ ℙ]. ∀[P,P':ℕ ⟶ K ⟶ ℙ].
((∀n∈dl-prop-atoms() prog(alpha).∀k:K. (P[n] k ⇐⇒ P'[n] k)) ⇒ (∀k,k':K. ([|alpha|] k k' ⇐⇒ [|alpha|] k k')))
Proof
Definitions occuring in Statement :
dl-prop-atoms: dl-prop-atoms(),
dl-prog-sem: [|alpha|],
dl-prog-obj: prog(x),
dl-prog: Prog,
l_all: (∀x∈L.P[x]),
nat: ℕ,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
so_lambda: λ2x.t[x],
prop: ℙ,
so_apply: x[s],
nat: ℕ,
ge: i ≥ j ,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
and: P ∧ Q,
dl-prog-sem: [|alpha|],
dl-same-sem: dl-same-sem(x;K;r;s),
dl-kind: dl-kind(d),
mobj-kind: mobj-kind(x),
pi1: fst(t),
dl-prog-obj: prog(x)
Lemmas referenced :
dl-sem_functionality1,
dl-prog-obj_wf,
dl-prog_wf,
l_all_wf,
nat_wf,
dl-prop-atoms_wf,
iff_wf,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
istype-le,
istype-nat,
l_member_wf,
istype-universe
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
hypothesis,
universeIsType,
isect_memberFormation_alt,
isectElimination,
independent_functionElimination,
applyEquality,
sqequalRule,
lambdaEquality_alt,
functionEquality,
dependent_set_memberEquality_alt,
setElimination,
rename,
natural_numberEquality,
unionElimination,
independent_isectElimination,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
because_Cache,
setIsType,
inhabitedIsType,
functionIsType,
universeEquality,
instantiate,
productElimination,
tokenEquality
Latex:
\mforall{}alpha:Prog
\mforall{}[K:Type]. \mforall{}[R:\mBbbN{} {}\mrightarrow{} K {}\mrightarrow{} K {}\mrightarrow{} \mBbbP{}]. \mforall{}[P,P':\mBbbN{} {}\mrightarrow{} K {}\mrightarrow{} \mBbbP{}].
((\mforall{}n\mmember{}dl-prop-atoms() prog(alpha).\mforall{}k:K. (P[n] k \mLeftarrow{}{}\mRightarrow{} P'[n] k))
{}\mRightarrow{} (\mforall{}k,k':K. ([|alpha|] k k' \mLeftarrow{}{}\mRightarrow{} [|alpha|] k k')))
Date html generated:
2019_10_15-AM-11_44_03
Last ObjectModification:
2019_03_26-AM-11_55_52
Theory : dynamic!logic
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