Nuprl Lemma : list_append_singleton_ind
∀[T:Type]. ∀[Q:(T List) ⟶ ℙ]. (Q[[]] ⇒ (∀ys:T List. ∀x:T. (Q[ys] ⇒ Q[ys @ [x]])) ⇒ {∀zs:T List. Q[zs]})
Proof
Definitions occuring in Statement :
append: as @ bs,
cons: [a / b],
nil: [],
list: T List,
uall: ∀[x:A]. B[x],
prop: ℙ,
guard: {T},
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
guard: {T},
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
nat: ℕ,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
squash: ↓T,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
le: A ≤ B
Lemmas referenced :
all_wf,
list_wf,
append_wf,
cons_wf,
nil_wf,
equal-wf-T-base,
nat_wf,
length_wf_nat,
int_subtype_base,
set_wf,
less_than_wf,
primrec-wf2,
equal_wf,
length_zero,
nat_properties,
decidable__equal_int,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformeq_wf,
itermVar_wf,
itermConstant_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
list_decomp_reverse,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
le_wf,
squash_wf,
true_wf,
length_append,
subtype_rel_list,
top_wf,
subtype_rel_self,
iff_weakening_equal,
length-singleton,
non_neg_length,
decidable__le,
intformle_wf,
itermAdd_wf,
int_formula_prop_le_lemma,
int_term_value_add_lemma,
length-append,
length_wf,
subtract_wf,
add-is-int-iff,
itermSubtract_wf,
int_term_value_subtract_lemma,
false_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
sqequalRule,
lambdaEquality,
functionEquality,
applyEquality,
Error :functionIsType,
Error :universeIsType,
universeEquality,
rename,
setElimination,
baseApply,
closedConclusion,
baseClosed,
intEquality,
natural_numberEquality,
productElimination,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
because_Cache,
dependent_functionElimination,
unionElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
hyp_replacement,
imageElimination,
cumulativity,
imageMemberEquality,
instantiate,
dependent_set_memberEquality,
addEquality,
pointwiseFunctionality,
promote_hyp
Latex:
\mforall{}[T:Type]. \mforall{}[Q:(T List) {}\mrightarrow{} \mBbbP{}].
(Q[[]] {}\mRightarrow{} (\mforall{}ys:T List. \mforall{}x:T. (Q[ys] {}\mRightarrow{} Q[ys @ [x]])) {}\mRightarrow{} \{\mforall{}zs:T List. Q[zs]\})
Date html generated:
2019_06_20-PM-01_45_38
Last ObjectModification:
2018_09_26-PM-02_50_58
Theory : list_1
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