Nuprl Lemma : WfdSpread-induction
∀[Pos:Type]
((∀a,b:Pos. Dec(a = b ∈ Pos))
⇒ (∀[Mv:Pos ⟶ Type]. ∀[P:WfdSpread(Pos;a.Mv[a]) ⟶ ℙ].
((∀a:Pos. ∀f:Mv[a] ⟶ WfdSpread(Pos;a.Mv[a]). ((∀m:Mv[a]. P[f m]) ⇒ P[mkW(a;f)]))
⇒ (∀w:WfdSpread(Pos;a.Mv[a]). P[w]))))
Proof
Definitions occuring in Statement :
mkW: mkW(a;f),
WfdSpread: WfdSpread(Pos;a.Mv[a]),
decidable: Dec(P),
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
so_apply: x[s],
so_lambda: λ2x y.t[x; y],
uimplies: b supposing a,
nat: ℕ,
so_apply: x[s1;s2],
prop: ℙ,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
and: P ∧ Q,
squash: ↓T,
W_sel: W_sel(w;n;s),
subgame: subgame(g;p;n),
ifthenelse: if b then t else f fi ,
eq_int: (i =z j),
btrue: tt,
isr: isr(x),
assert: ↑b,
bfalse: ff,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
subtype_rel: A ⊆r B,
guard: {T},
mkW: mkW(a;f),
MoveChoice: MoveChoice(Pos;a.Mv[a]),
exposed-bfalse: exposed-bfalse,
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
eqof: eqof(d),
sq_type: SQType(T),
bnot: ¬bb,
le: A ≤ B,
less_than': less_than'(a;b),
subtract: n - m,
seq-adjoin: s++t,
seq-append: seq-append(n;m;s1;s2),
deq: EqDecider(T),
nequal: a ≠ b ∈ T ,
less_than: a < b,
int_seg: {i..j-},
lelt: i ≤ j < k,
isl: isl(x),
WfdSpread: WfdSpread(Pos;a.Mv[a]),
resigned: resigned(x)
Lemmas referenced :
basic_bar_induction,
MoveChoice_wf,
assert_wf,
isr_wf,
WfdSpread_wf,
unit_wf2,
W_sel_wf,
int_seg_wf,
nat_wf,
true_wf,
equal_wf,
decidable__assert,
all_wf,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
int_formula_prop_and_lemma,
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,
seq-adjoin_wf,
mkW_wf,
decidable_wf,
false_wf,
deq-exists,
WfdSpread-ext,
subtype_rel_weakening,
bool_wf,
eqtt_to_assert,
safe-assert-deq,
subtype_rel-equal,
and_wf,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
intformless_wf,
int_formula_prop_less_lemma,
ge_wf,
less_than_wf,
decidable-equal-deq,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
member_wf,
eq_int_wf,
assert_of_eq_int,
intformeq_wf,
int_formula_prop_eq_lemma,
neg_assert_of_eq_int,
bnot_wf,
not_wf,
equal-wf-base,
int_subtype_base,
top_wf,
lelt_wf,
bool_cases,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot,
add-member-int_seg2,
squash_wf,
subtract-add-cancel,
add-subtract-cancel,
decidable__equal_int,
lt_int_wf,
assert_of_lt_int,
int_seg_properties,
btrue_wf,
isl_wf,
bfalse_wf,
btrue_neq_bfalse
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
lambdaEquality,
applyEquality,
hypothesis,
independent_isectElimination,
functionEquality,
natural_numberEquality,
setElimination,
rename,
unionEquality,
equalityTransitivity,
equalitySymmetry,
unionElimination,
dependent_functionElimination,
independent_functionElimination,
because_Cache,
cumulativity,
functionExtensionality,
dependent_set_memberEquality,
addEquality,
approximateComputation,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
imageElimination,
imageMemberEquality,
baseClosed,
universeEquality,
productElimination,
promote_hyp,
productEquality,
inlEquality,
equalityElimination,
applyLambdaEquality,
instantiate,
inrEquality,
hyp_replacement,
intWeakElimination,
axiomEquality,
lessCases,
axiomSqEquality,
impliesFunctionality,
equalityUniverse,
levelHypothesis
Latex:
\mforall{}[Pos:Type]
((\mforall{}a,b:Pos. Dec(a = b))
{}\mRightarrow{} (\mforall{}[Mv:Pos {}\mrightarrow{} Type]. \mforall{}[P:WfdSpread(Pos;a.Mv[a]) {}\mrightarrow{} \mBbbP{}].
((\mforall{}a:Pos. \mforall{}f:Mv[a] {}\mrightarrow{} WfdSpread(Pos;a.Mv[a]). ((\mforall{}m:Mv[a]. P[f m]) {}\mRightarrow{} P[mkW(a;f)]))
{}\mRightarrow{} (\mforall{}w:WfdSpread(Pos;a.Mv[a]). P[w]))))
Date html generated:
2019_06_20-PM-02_03_04
Last ObjectModification:
2018_08_21-PM-01_57_51
Theory : spread
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