Nuprl Lemma : id-graph-edge-implies-member
∀S:Id List. ∀G:Graph(S). ∀i:{i:Id| (i ∈ S)} . ∀j:Id. ((i⟶j)∈G ⇒ (j ∈ S))
Proof
Definitions occuring in Statement :
id-graph-edge: (i⟶j)∈G,
id-graph: Graph(S),
Id: Id,
l_member: (x ∈ l),
list: T List,
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]}
Definitions unfolded in proof :
id-graph-edge: (i⟶j)∈G,
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
id-graph: Graph(S),
subtype_rel: A ⊆r B,
uimplies: b supposing a,
so_lambda: λ2x.t[x],
so_apply: x[s],
l_member: (x ∈ l),
exists: ∃x:A. B[x],
cand: A c∧ B,
Id: Id,
sq_type: SQType(T),
guard: {T},
nat: ℕ,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
and: P ∧ Q,
sq_stable: SqStable(P),
squash: ↓T
Lemmas referenced :
equal_wf,
decidable__equal_Id,
sq_stable__l_member,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__le,
nat_properties,
select_wf,
atom2_subtype_base,
subtype_base_sq,
list_wf,
id-graph_wf,
set_wf,
subtype_rel_list,
Id_wf,
l_member_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
hypothesisEquality,
applyEquality,
setEquality,
independent_isectElimination,
lambdaEquality,
setElimination,
rename,
because_Cache,
productElimination,
instantiate,
cumulativity,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
natural_numberEquality,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
introduction,
imageMemberEquality,
baseClosed,
imageElimination
Latex:
\mforall{}S:Id List. \mforall{}G:Graph(S). \mforall{}i:\{i:Id| (i \mmember{} S)\} . \mforall{}j:Id. ((i{}\mrightarrow{}j)\mmember{}G {}\mRightarrow{} (j \mmember{} S))
Date html generated:
2016_05_14-PM-03_37_49
Last ObjectModification:
2016_01_14-PM-11_19_16
Theory : decidable!equality
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