Nuprl Lemma : Veldman-Coquand
∀X:Type. ∀n:ℕ. ∀p,q:wfd-tree(X).
∀[A,B:n:ℕ ⟶ (ℕn ⟶ X) ⟶ ℙ]. ∀[R,S:n-aryRel(X)].
(tree-secures(X;λm,s. ((A m s) ∨ ([[R]] m s));p)
⇒ tree-secures(X;λm,s. ((B m s) ∨ ([[S]] m s));q)
⇒ tree-secures(X;λm,s. (((A m s) ∨ (B m s)) ∨ (([[R]] m s) ∧ ([[S]] m s)));tree-tensor(n;p;q)))
Proof
Definitions occuring in Statement :
tree-tensor: tree-tensor(n;p;q),
tree-secures: tree-secures(T;A;p),
nary-rel-predicate: [[R]],
nary-rel: n-aryRel(T),
wfd-tree: wfd-tree(T),
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
prop: ℙ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
or: P ∨ Q,
and: P ∧ Q,
apply: f a,
lambda: λx.A[x],
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
nat: ℕ,
prop: ℙ,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
implies: P ⇒ Q,
so_apply: x[s],
guard: {T},
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
wfd-tree: wfd-tree(T),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
bfalse: ff,
Wsup: Wsup(a;b),
tree-secures: tree-secures(T;A;p),
int_seg: {i..j-},
lelt: i ≤ j < k,
tree-tensor: tree-tensor(n;p;q),
eq_int: (i =z j),
subtract: n - m,
cand: A c∧ B,
nary-rel-predicate: [[R]],
nary-rel: n-aryRel(T),
ge: i ≥ j ,
predicate-or-shift: A[x],
predicate-shift: A_x,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
true: True,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
seq-append: seq-append(n;m;s1;s2),
less_than: a < b,
squash: ↓T
Lemmas referenced :
wfd-tree-induction,
all_wf,
wfd-tree_wf,
uall_wf,
nat_wf,
int_seg_wf,
nary-rel_wf,
false_wf,
le_wf,
tree-secures_wf,
or_wf,
nary-rel-predicate_wf,
tree-tensor_wf,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
subtype_rel_self,
set_wf,
less_than_wf,
primrec-wf2,
Wsup_wf,
bool_wf,
eqtt_to_assert,
equal_wf,
btrue_wf,
void_wf,
int_seg_properties,
trivial-tree-secures,
tree-secures_functionality,
nat_properties,
ifthenelse_wf,
bfalse_wf,
predicate-or-shift_wf,
seq-append_wf,
subtype_rel_function,
int_seg_subtype,
not-le-2,
condition-implies-le,
add-associates,
minus-add,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-mul-special,
zero-mul,
add-zero,
add-commutes,
le-add-cancel2,
add-is-int-iff,
set_subtype_base,
int_subtype_base,
itermAdd_wf,
int_term_value_add_lemma,
seq-single_wf,
eq_int_wf,
assert_of_eq_int,
intformeq_wf,
int_formula_prop_eq_lemma,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
W_wf,
lelt_wf,
top_wf,
decidable__lt,
predicate-shift_wf,
lt_int_wf,
assert_of_lt_int,
not_functionality_wrt_uiff,
assert_wf,
decidable__equal_int
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
thin,
instantiate,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
cumulativity,
hypothesisEquality,
dependent_functionElimination,
sqequalRule,
lambdaEquality,
hypothesis,
applyEquality,
because_Cache,
functionEquality,
natural_numberEquality,
setElimination,
rename,
universeEquality,
dependent_set_memberEquality,
independent_pairFormation,
functionExtensionality,
productEquality,
independent_functionElimination,
unionElimination,
independent_isectElimination,
approximateComputation,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
isect_memberFormation,
equalityElimination,
productElimination,
equalityTransitivity,
equalitySymmetry,
inlFormation,
inrFormation,
hyp_replacement,
addEquality,
minusEquality,
multiplyEquality,
baseApply,
closedConclusion,
baseClosed,
promote_hyp,
lessCases,
imageMemberEquality,
axiomSqEquality,
imageElimination
Latex:
\mforall{}X:Type. \mforall{}n:\mBbbN{}. \mforall{}p,q:wfd-tree(X).
\mforall{}[A,B:n:\mBbbN{} {}\mrightarrow{} (\mBbbN{}n {}\mrightarrow{} X) {}\mrightarrow{} \mBbbP{}]. \mforall{}[R,S:n-aryRel(X)].
(tree-secures(X;\mlambda{}m,s. ((A m s) \mvee{} ([[R]] m s));p)
{}\mRightarrow{} tree-secures(X;\mlambda{}m,s. ((B m s) \mvee{} ([[S]] m s));q)
{}\mRightarrow{} tree-secures(X;\mlambda{}m,s. (((A m s) \mvee{} (B m s)) \mvee{} (([[R]] m s) \mwedge{} ([[S]] m s)));tree-tensor(n;p;q)))
Date html generated:
2019_06_20-PM-02_45_53
Last ObjectModification:
2018_09_17-PM-11_02_31
Theory : fan-theorem
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