Nuprl Lemma : vdf-eq_wf
∀[A,B:Type]. ∀[C:A ⟶ B ⟶ Type]. ∀L:(a:A × b:B × C[a;b]) List. ∀[f:very-dep-fun(A;B;a,b.C[a;b])]. (vdf-eq(A;f;L) ∈ ℙ)
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,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
member: t ∈ T,
function: x:A ⟶ B[x],
product: x:A × B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
decidable: Dec(P),
or: P ∨ Q,
nat: ℕ,
less_than: a < b,
squash: ↓T,
and: P ∧ Q,
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
prop: ℙ,
subtype_rel: A ⊆r B,
very-dep-fun: very-dep-fun(A;B;a,b.C[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,
very-dep-fun_wf,
list_wf,
istype-universe,
decidable__lt,
length_wf,
subtract_wf,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermSubtract_wf,
itermVar_wf,
intformless_wf,
istype-int,
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,
istype-le,
vdf_wf,
itermAdd_wf,
int_term_value_add_lemma,
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,
lambdaFormation_alt,
universeIsType,
sqequalRule,
lambdaEquality_alt,
applyEquality,
productEquality,
functionIsType,
inhabitedIsType,
instantiate,
universeEquality,
dependent_functionElimination,
natural_numberEquality,
unionElimination,
dependent_set_memberEquality_alt,
imageElimination,
productElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
int_eqEquality,
Error :memTop,
independent_pairFormation,
voidElimination,
equalityTransitivity,
equalitySymmetry,
isectIsType,
addEquality,
because_Cache,
lessCases,
axiomSqEquality,
isect_memberEquality_alt,
isectIsTypeImplies,
imageMemberEquality,
baseClosed
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{}[f:very-dep-fun(A;B;a,b.C[a;b])]. (vdf-eq(A;f;L) \mmember{} \mBbbP{})
Date html generated:
2020_05_19-PM-09_40_30
Last ObjectModification:
2020_03_05-AM-11_17_05
Theory : co-recursion-2
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