Nuprl Lemma : implies-vdf-eq
∀[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].
vdf-eq(A;f;L) supposing ∀i:ℕ||L|| + 1. ((∀j:ℕi. vdf-eq(A;f;firstn(j;L))) ⇒ vdf-eq(A;f;firstn(i;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),
firstn: firstn(n;as),
length: ||as||,
list: T List,
int_seg: {i..j-},
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s1;s2],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
product: x:A × B[x],
add: n + m,
natural_number: $n,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
sq_stable: SqStable(P),
implies: P ⇒ Q,
all: ∀x:A. B[x],
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
le: A ≤ B,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
prop: ℙ,
decidable: Dec(P),
or: P ∨ Q,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
guard: {T},
sq_type: SQType(T),
nat: ℕ,
less_than: a < b,
squash: ↓T,
ge: i ≥ j
Lemmas referenced :
sq_stable__vdf-eq,
int_seg_properties,
full-omega-unsat,
intformand_wf,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
int_seg_wf,
decidable__equal_int,
subtract_wf,
subtype_base_sq,
set_subtype_base,
lelt_wf,
int_subtype_base,
intformnot_wf,
intformeq_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_subtract_lemma,
decidable__le,
decidable__lt,
istype-le,
istype-less_than,
subtype_rel_self,
length_wf,
itermAdd_wf,
int_term_value_add_lemma,
nat_properties,
le_wf,
vdf-eq_wf,
firstn_wf,
primrec-wf2,
istype-nat,
length_wf_nat,
firstn_all,
subtype_rel_list,
top_wf,
list_wf,
very-dep-fun_wf,
istype-universe
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
lambdaEquality_alt,
applyEquality,
universeIsType,
hypothesis,
independent_functionElimination,
lambdaFormation_alt,
setElimination,
rename,
productElimination,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
dependent_functionElimination,
Error :memTop,
independent_pairFormation,
voidElimination,
unionElimination,
instantiate,
cumulativity,
intEquality,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
dependent_set_memberEquality_alt,
because_Cache,
productIsType,
promote_hyp,
hypothesis_subsumption,
addEquality,
productEquality,
imageElimination,
functionIsType,
functionEquality,
setIsType,
imageMemberEquality,
baseClosed,
universeEquality
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].
vdf-eq(A;f;L)
supposing \mforall{}i:\mBbbN{}||L|| + 1. ((\mforall{}j:\mBbbN{}i. vdf-eq(A;f;firstn(j;L))) {}\mRightarrow{} vdf-eq(A;f;firstn(i;L)))
Date html generated:
2020_05_19-PM-09_40_58
Last ObjectModification:
2020_03_06-PM-01_37_51
Theory : co-recursion-2
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