Nuprl Lemma : subtype-l_all
∀[T:Type]. ∀[L:T List]. ∀[P,Q:{x:T| (x ∈ L)} ⟶ ℙ].
(∀x∈L.P[x]) ⊆r (∀x∈L.Q[x]) supposing ∀x:T. ((x ∈ L) ⇒ (P[x] ⊆r Q[x]))
Proof
Definitions occuring in Statement :
l_all: (∀x∈L.P[x]),
l_member: (x ∈ l),
list: T List,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
l_all: (∀x∈L.P[x]),
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
int_seg: {i..j-},
guard: {T},
lelt: i ≤ j < k,
and: P ∧ Q,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
implies: P ⇒ Q,
not: ¬A,
top: Top,
prop: ℙ,
less_than: a < b,
squash: ↓T
Lemmas referenced :
subtype_rel_wf,
l_member_wf,
all_wf,
select_member,
int_formula_prop_less_lemma,
intformless_wf,
decidable__lt,
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,
int_seg_properties,
length_wf,
int_seg_wf,
select_wf,
subtype_rel_dep_function
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaEquality,
sqequalHypSubstitution,
hypothesisEquality,
applyEquality,
lemma_by_obid,
isectElimination,
thin,
because_Cache,
sqequalRule,
independent_isectElimination,
hypothesis,
natural_numberEquality,
lambdaFormation,
dependent_functionElimination,
setElimination,
rename,
productElimination,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
imageElimination,
independent_functionElimination,
axiomEquality,
cumulativity,
functionEquality,
dependent_set_memberEquality,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}[T:Type]. \mforall{}[L:T List]. \mforall{}[P,Q:\{x:T| (x \mmember{} L)\} {}\mrightarrow{} \mBbbP{}].
(\mforall{}x\mmember{}L.P[x]) \msubseteq{}r (\mforall{}x\mmember{}L.Q[x]) supposing \mforall{}x:T. ((x \mmember{} L) {}\mRightarrow{} (P[x] \msubseteq{}r Q[x]))
Date html generated:
2016_05_14-AM-07_47_30
Last ObjectModification:
2016_01_15-AM-08_34_32
Theory : list_1
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