Nuprl Lemma : causal_order_reflexive
∀[T:Type]. ∀L:T List. ∀[R:ℕ||L|| ⟶ ℕ||L|| ⟶ ℙ]. ∀[P:ℕ||L|| ⟶ ℙ]. (Refl(ℕ||L||)(R _1 _2) ⇒ causal_order(L;R;P;P))
Proof
Definitions occuring in Statement :
causal_order: causal_order(L;R;P;Q),
length: ||as||,
list: T List,
refl: Refl(T;x,y.E[x; y]),
int_seg: {i..j-},
uall: ∀[x:A]. B[x],
prop: ℙ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type
Definitions unfolded in proof :
causal_order: causal_order(L;R;P;Q),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
exists: ∃x:A. B[x],
member: t ∈ T,
and: P ∧ Q,
cand: A c∧ B,
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
top: Top,
prop: ℙ,
subtype_rel: A ⊆r B,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
refl: Refl(T;x,y.E[x; y])
Lemmas referenced :
int_seg_properties,
length_wf,
decidable__le,
full-omega-unsat,
intformnot_wf,
intformle_wf,
itermVar_wf,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
le_wf,
subtype_rel_self,
int_seg_wf,
refl_wf,
list_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation_alt,
lambdaFormation,
dependent_pairFormation,
hypothesisEquality,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
hypothesis,
setElimination,
rename,
productElimination,
dependent_functionElimination,
because_Cache,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
productEquality,
applyEquality,
instantiate,
universeEquality,
functionIsType,
universeIsType,
inhabitedIsType
Latex:
\mforall{}[T:Type]
\mforall{}L:T List
\mforall{}[R:\mBbbN{}||L|| {}\mrightarrow{} \mBbbN{}||L|| {}\mrightarrow{} \mBbbP{}]. \mforall{}[P:\mBbbN{}||L|| {}\mrightarrow{} \mBbbP{}]. (Refl(\mBbbN{}||L||)(R $_{1}$ $\mbackslash{}\000Cff5f{2}$) {}\mRightarrow{} causal\_order(L;R;P;P))
Date html generated:
2019_10_15-AM-10_57_34
Last ObjectModification:
2018_09_27-AM-09_52_41
Theory : list!
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