Nuprl Lemma : vdf-eq_wf1
∀[A,B:Type]. ∀[C:A ⟶ B ⟶ Type]. ∀[L:(a:A × b:B × C[a;b]) List]. ∀[m:ℤ]. ∀[f:vdf(A;B;a,b.C[a;b];m - 1)].
vdf-eq(A;f;L) ∈ Type supposing ||L|| ≤ m
Proof
Definitions occuring in Statement :
vdf: vdf(A;B;a,b.C[a; b];n),
vdf-eq: vdf-eq(A;f;L),
length: ||as||,
list: T List,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s1;s2],
le: A ≤ B,
member: t ∈ T,
function: x:A ⟶ B[x],
product: x:A × B[x],
subtract: n - m,
natural_number: $n,
int: ℤ,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
nat: ℕ,
less_than: a < b,
squash: ↓T,
and: P ∧ Q,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
prop: ℙ,
le: A ≤ B,
vdf-eq: vdf-eq(A;f;L),
dep-all: dep-all(n;i.P[i]),
less_than': less_than'(a;b),
true: True
Lemmas referenced :
vdf-wf,
istype-le,
length_wf,
vdf_wf,
subtract_wf,
istype-int,
list_wf,
istype-universe,
decidable__lt,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermSubtract_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_subtract_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
subtract-add-cancel,
istype-top,
true_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
productEquality,
applyEquality,
universeIsType,
sqequalRule,
lambdaEquality_alt,
natural_numberEquality,
functionIsType,
inhabitedIsType,
instantiate,
universeEquality,
dependent_functionElimination,
unionElimination,
dependent_set_memberEquality_alt,
imageElimination,
productElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
int_eqEquality,
Error :memTop,
independent_pairFormation,
voidElimination,
because_Cache,
lessCases,
axiomSqEquality,
isect_memberEquality_alt,
isectIsTypeImplies,
imageMemberEquality,
baseClosed,
lambdaFormation_alt
Latex:
\mforall{}[A,B:Type]. \mforall{}[C:A {}\mrightarrow{} B {}\mrightarrow{} Type]. \mforall{}[L:(a:A \mtimes{} b:B \mtimes{} C[a;b]) List]. \mforall{}[m:\mBbbZ{}].
\mforall{}[f:vdf(A;B;a,b.C[a;b];m - 1)].
vdf-eq(A;f;L) \mmember{} Type supposing ||L|| \mleq{} m
Date html generated:
2020_05_19-PM-09_40_27
Last ObjectModification:
2020_03_05-PM-01_01_37
Theory : co-recursion-2
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