Nuprl Lemma : sub-spread-transitive
∀[Pos:Type]. ∀[Mv:Pos ⟶ Type]. Trans(Spread(Pos;a.Mv[a]);s',s.s' ≤ s)
Proof
Definitions occuring in Statement :
sub-spread: s' ≤ s,
Spread: Spread(Pos;a.Mv[a]),
trans: Trans(T;x,y.E[x; y]),
uall: ∀[x:A]. B[x],
so_apply: x[s],
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
trans: Trans(T;x,y.E[x; y]),
all: ∀x:A. B[x],
implies: P ⇒ Q,
sub-spread: s' ≤ s,
exists: ∃x:A. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
nat: ℕ,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
top: Top,
and: P ∧ Q,
int_seg: {i..j-},
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
less_than: a < b,
less_than': less_than'(a;b),
true: True,
squash: ↓T,
lelt: i ≤ j < k,
guard: {T},
bfalse: ff,
subtype_rel: A ⊆r B,
sq_type: SQType(T),
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
le: A ≤ B,
subgame: subgame(g;p;n),
eq_int: (i =z j),
subtract: n - m,
nequal: a ≠ b ∈ T ,
isl: isl(x)
Lemmas referenced :
sub-spread_wf,
istype-universe,
Spread_wf,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermAdd_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_add_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
le_wf,
lt_int_wf,
eqtt_to_assert,
assert_of_lt_int,
istype-top,
int_seg_properties,
istype-less_than,
eqff_to_assert,
int_subtype_base,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
less_than_wf,
subtract_wf,
itermSubtract_wf,
intformless_wf,
int_term_value_subtract_lemma,
int_formula_prop_less_lemma,
decidable__lt,
int_seg_wf,
subgame_wf,
MoveChoice_wf,
ge_wf,
istype-false,
unit_wf2,
subtract-1-ge-0,
equal_wf,
squash_wf,
true_wf,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
set_subtype_base,
lelt_wf,
eq_int_wf,
bnot_wf,
not_wf,
equal-wf-base,
istype-assert,
bool_cases,
assert_of_eq_int,
iff_transitivity,
assert_of_bnot,
neg_assert_of_eq_int,
subtype_rel_weakening,
spread-ext,
add-member-int_seg2,
btrue_neq_bfalse,
btrue_wf,
bfalse_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
Error :lambdaFormation_alt,
sqequalHypSubstitution,
productElimination,
thin,
Error :universeIsType,
cut,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
sqequalRule,
Error :lambdaEquality_alt,
applyEquality,
hypothesis,
Error :inhabitedIsType,
Error :functionIsType,
universeEquality,
Error :dependent_pairFormation_alt,
Error :dependent_set_memberEquality_alt,
addEquality,
setElimination,
rename,
dependent_functionElimination,
natural_numberEquality,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
int_eqEquality,
Error :isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
because_Cache,
equalityElimination,
lessCases,
axiomSqEquality,
imageMemberEquality,
baseClosed,
imageElimination,
Error :productIsType,
equalityTransitivity,
equalitySymmetry,
Error :equalityIsType2,
baseApply,
closedConclusion,
promote_hyp,
instantiate,
cumulativity,
Error :equalityIsType1,
intWeakElimination,
axiomEquality,
Error :functionIsTypeImplies,
Error :inlEquality_alt,
applyLambdaEquality,
hyp_replacement,
unionEquality,
Error :functionExtensionality_alt,
intEquality,
Error :equalityIsType4,
functionEquality,
productEquality,
minusEquality,
Error :inrEquality_alt
Latex:
\mforall{}[Pos:Type]. \mforall{}[Mv:Pos {}\mrightarrow{} Type]. Trans(Spread(Pos;a.Mv[a]);s',s.s' \mleq{} s)
Date html generated:
2019_06_20-PM-02_02_51
Last ObjectModification:
2018_10_12-AM-10_42_59
Theory : spread
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