Nuprl Lemma : vdf-eq-witness
∀[A,B:Type]. ∀[C:A ⟶ B ⟶ Type]. ∀[f:very-dep-fun(A;B;a,b.C[a;b])]. ∀[L:(a:A × b:B × C[a;b]) List].
Ax ∈ vdf-eq(A;f;L) supposing vdf-eq(A;f;L)
Proof
Definitions occuring in Statement :
very-dep-fun: very-dep-fun(A;B;a,b.C[a; b]),
vdf-eq: vdf-eq(A;f;L),
list: T List,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s1;s2],
member: t ∈ T,
function: x:A ⟶ B[x],
product: x:A × B[x],
universe: Type,
axiom: Ax
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
all: ∀x:A. B[x],
prop: ℙ,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
and: P ∧ Q,
subtype_rel: A ⊆r B,
vdf-eq: vdf-eq(A;f;L),
select: L[n],
nil: [],
it: ⋅,
dep-all: dep-all(n;i.P[i]),
true: True,
decidable: Dec(P),
or: P ∨ Q,
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
let: let
Lemmas referenced :
vdf-eq_wf,
list_wf,
very-dep-fun_wf,
istype-universe,
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_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,
ge_wf,
istype-less_than,
first0,
subtype_rel_list,
top_wf,
length_of_nil_lemma,
stuck-spread,
istype-base,
firstn_wf,
istype-le,
length_wf,
subtract-1-ge-0,
istype-nat,
length_wf_nat,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
firstn_all,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
decidable__lt,
subtract-add-cancel
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalHypSubstitution,
hypothesis,
sqequalRule,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
universeIsType,
extract_by_obid,
isectElimination,
thin,
hypothesisEquality,
lambdaEquality_alt,
applyEquality,
dependent_functionElimination,
isect_memberEquality_alt,
isectIsTypeImplies,
inhabitedIsType,
productEquality,
functionIsType,
instantiate,
universeEquality,
lambdaFormation_alt,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
int_eqEquality,
Error :memTop,
independent_pairFormation,
voidElimination,
functionIsTypeImplies,
productIsType,
because_Cache,
baseClosed,
closedConclusion,
unionElimination,
dependent_set_memberEquality_alt,
productElimination,
dependentIntersectionElimination,
dependentIntersection_memberEquality
Latex:
\mforall{}[A,B:Type]. \mforall{}[C:A {}\mrightarrow{} B {}\mrightarrow{} Type]. \mforall{}[f:very-dep-fun(A;B;a,b.C[a;b])]. \mforall{}[L:(a:A \mtimes{} b:B \mtimes{} C[a;b]) List].
Ax \mmember{} vdf-eq(A;f;L) supposing vdf-eq(A;f;L)
Date html generated:
2020_05_19-PM-09_40_44
Last ObjectModification:
2020_03_06-PM-03_21_08
Theory : co-recursion-2
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