Nuprl Lemma : Coquand-fan-theorem
∀[T:Type]
(finite-type(T)
⇒ (∀p:wfd-tree(T). ∀A:n:ℕ ⟶ (ℕn ⟶ T) ⟶ ℙ.
((∀n:ℕ. ∀s:ℕn ⟶ T. ((A n s) ⇒ (∀m:{n...}. ∀t:ℕm ⟶ T. ((t = s ∈ (ℕn ⟶ T)) ⇒ (A m t)))))
⇒ (p|A)
⇒ (∃N:ℕ. ∀m:{N...}. ∀as:ℕm ⟶ T. (A m as)))))
Proof
Definitions occuring in Statement :
tree-bars: (p|A),
finite-type: finite-type(T),
wfd-tree: wfd-tree(T),
int_upper: {i...},
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
prop: ℙ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
wfd-tree: wfd-tree(T),
member: t ∈ T,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
nat: ℕ,
prop: ℙ,
int_upper: {i...},
uimplies: b supposing a,
and: P ∧ Q,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
guard: {T},
ge: i ≥ j ,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
bfalse: ff,
tree-bars: (p|A),
Wsup: Wsup(a;b),
int_seg: {i..j-},
lelt: i ≤ j < k,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
predicate-shift: A_x,
seq-append: seq-append(n;m;s1;s2),
less_than: a < b,
true: True,
squash: ↓T,
finite-type: finite-type(T),
pi1: fst(t),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
l_exists: (∃x∈L. P[x]),
nat_plus: ℕ+,
select: L[n],
cons: [a / b],
surject: Surj(A;B;f),
seq-single: seq-single(t),
subtract: n - m,
nequal: a ≠ b ∈ T ,
cand: A c∧ B
Lemmas referenced :
W-induction,
bool_wf,
ifthenelse_wf,
all_wf,
nat_wf,
int_seg_wf,
int_upper_wf,
equal_wf,
subtype_rel_dep_function,
int_seg_subtype,
false_wf,
int_upper_properties,
nat_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
int_upper_subtype_nat,
tree-bars_wf,
exists_wf,
W_wf,
eqtt_to_assert,
le_wf,
int_seg_properties,
intformless_wf,
itermConstant_wf,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
predicate-shift_wf,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
finite-type_wf,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
itermAdd_wf,
int_term_value_add_lemma,
seq-append_wf,
seq-single_wf,
lt_int_wf,
assert_of_lt_int,
top_wf,
less_than_wf,
lelt_wf,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
decidable__lt,
imax-list-ub,
cons_wf,
map_wf,
upto_wf,
length_of_cons_lemma,
non_neg_length,
map_length,
length_wf,
add_nat_plus,
length_wf_nat,
nat_plus_wf,
nat_plus_properties,
add-is-int-iff,
select_wf,
add-associates,
add-swap,
add-commutes,
zero-add,
not-le-2,
condition-implies-le,
minus-add,
minus-one-mul,
minus-one-mul-top,
add_functionality_wrt_le,
le-add-cancel,
eq_int_wf,
assert_of_eq_int,
int_subtype_base,
neg_assert_of_eq_int,
add-member-int_seg2,
l_exists_iff,
l_member_wf,
cons_member,
member_map,
equal-wf-T-base,
member_upto2,
int_seg_subtype_nat
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
thin,
instantiate,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
cumulativity,
hypothesis,
sqequalRule,
lambdaEquality,
hypothesisEquality,
universeEquality,
voidEquality,
functionEquality,
applyEquality,
natural_numberEquality,
setElimination,
rename,
because_Cache,
functionExtensionality,
independent_isectElimination,
independent_pairFormation,
dependent_functionElimination,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
computeAll,
equalityElimination,
productElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
dependent_set_memberEquality,
applyLambdaEquality,
promote_hyp,
addLevel,
hyp_replacement,
levelHypothesis,
addEquality,
lessCases,
sqequalAxiom,
imageMemberEquality,
baseClosed,
imageElimination,
pointwiseFunctionality,
baseApply,
closedConclusion,
minusEquality,
int_eqReduceTrueSq,
int_eqReduceFalseSq,
setEquality,
productEquality,
inrFormation
Latex:
\mforall{}[T:Type]
(finite-type(T)
{}\mRightarrow{} (\mforall{}p:wfd-tree(T). \mforall{}A:n:\mBbbN{} {}\mrightarrow{} (\mBbbN{}n {}\mrightarrow{} T) {}\mrightarrow{} \mBbbP{}.
((\mforall{}n:\mBbbN{}. \mforall{}s:\mBbbN{}n {}\mrightarrow{} T. ((A n s) {}\mRightarrow{} (\mforall{}m:\{n...\}. \mforall{}t:\mBbbN{}m {}\mrightarrow{} T. ((t = s) {}\mRightarrow{} (A m t)))))
{}\mRightarrow{} (p|A)
{}\mRightarrow{} (\mexists{}N:\mBbbN{}. \mforall{}m:\{N...\}. \mforall{}as:\mBbbN{}m {}\mrightarrow{} T. (A m as)))))
Date html generated:
2017_04_17-AM-09_39_01
Last ObjectModification:
2017_02_27-PM-05_36_19
Theory : fan-theorem
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