Nuprl Lemma : fun-path-induction
∀[T:Type]
∀f:T ⟶ T
∀[R:T ⟶ T ⟶ (T List) ⟶ ℙ]
((∀x:T. R[x;x;[x]])
⇒ (∀L:T List. ∀x,y,z:T. (R[y;z;[y / L]] ⇒ R[x;z;[x; [y / L]]]) supposing ((¬(x = y ∈ T)) and (x = (f y) ∈ T)))
⇒ {∀L:T List. ∀x,y:T. R[x;y;L] supposing x=f*(y) via L})
Proof
Definitions occuring in Statement :
fun-path: y=f*(x) via L,
cons: [a / b],
nil: [],
list: T List,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
guard: {T},
so_apply: x[s1;s2;s3],
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
guard: {T},
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
so_lambda: λ2x.t[x],
uimplies: b supposing a,
prop: ℙ,
so_apply: x[s1;s2;s3],
so_apply: x[s],
fun-path: y=f*(x) via L,
and: P ∧ Q,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
length: ||as||,
list_ind: list_ind,
nil: [],
it: ⋅,
false: False,
not: ¬A,
select: L[n],
so_lambda: λ2x y.t[x; y],
top: Top,
so_apply: x[s1;s2],
subtract: n - m,
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
uiff: uiff(P;Q),
subtype_rel: A ⊆r B,
cons: [a / b],
ge: i ≥ j ,
le: A ≤ B
Lemmas referenced :
list_induction,
all_wf,
isect_wf,
fun-path_wf,
list_wf,
member-less_than,
length_of_nil_lemma,
stuck-spread,
base_wf,
nil_wf,
length_wf,
cons_wf,
equal_wf,
select_wf,
length_of_cons_lemma,
int_seg_properties,
subtract_wf,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
decidable__lt,
add-is-int-iff,
intformless_wf,
itermSubtract_wf,
int_formula_prop_less_lemma,
int_term_value_subtract_lemma,
false_wf,
int_seg_wf,
fun-path-cons,
less_than_wf,
not_wf,
list-cases,
product_subtype_list,
reduce_hd_cons_lemma,
non_neg_length,
and_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
lambdaFormation,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
lambdaEquality,
cumulativity,
functionExtensionality,
applyEquality,
hypothesis,
independent_functionElimination,
productElimination,
independent_pairEquality,
imageElimination,
voidElimination,
independent_isectElimination,
axiomEquality,
dependent_functionElimination,
rename,
baseClosed,
isect_memberEquality,
voidEquality,
because_Cache,
natural_numberEquality,
equalityTransitivity,
equalitySymmetry,
addEquality,
setElimination,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
independent_pairFormation,
computeAll,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
hyp_replacement,
applyLambdaEquality,
productEquality,
functionEquality,
universeEquality,
hypothesis_subsumption,
dependent_set_memberEquality
Latex:
\mforall{}[T:Type]
\mforall{}f:T {}\mrightarrow{} T
\mforall{}[R:T {}\mrightarrow{} T {}\mrightarrow{} (T List) {}\mrightarrow{} \mBbbP{}]
((\mforall{}x:T. R[x;x;[x]])
{}\mRightarrow{} (\mforall{}L:T List. \mforall{}x,y,z:T.
(R[y;z;[y / L]] {}\mRightarrow{} R[x;z;[x; [y / L]]]) supposing ((\mneg{}(x = y)) and (x = (f y))))
{}\mRightarrow{} \{\mforall{}L:T List. \mforall{}x,y:T. R[x;y;L] supposing x=f*(y) via L\})
Date html generated:
2018_05_21-PM-07_44_34
Last ObjectModification:
2017_07_26-PM-05_22_07
Theory : general
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