Nuprl Lemma : fun-path-append
∀[T:Type]. ∀[f:T ⟶ T]. ∀[L1,L2:T List]. ∀[x,y,z:T].
uiff(z=f*(x) via L1 @ [y / L2];{y=f*(x) via [y / L2] ∧ z=f*(y) via L1 @ [y]})
Proof
Definitions occuring in Statement :
fun-path: y=f*(x) via L,
append: as @ bs,
cons: [a / b],
nil: [],
list: T List,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
guard: {T},
and: P ∧ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
subtype_rel: A ⊆r B,
decidable: Dec(P),
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
squash: ↓T,
less_than: a < b,
sq_type: SQType(T),
so_apply: x[s],
so_lambda: λ2x.t[x],
it: ⋅,
nil: [],
colength: colength(L),
less_than': less_than'(a;b),
le: A ≤ B,
cons: [a / b],
or: P ∨ Q,
fun-path: y=f*(x) via L,
guard: {T},
uiff: uiff(P;Q),
prop: ℙ,
and: P ∧ Q,
top: Top,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
uimplies: b supposing a,
ge: i ≥ j ,
false: False,
implies: P ⇒ Q,
nat: ℕ,
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
so_apply: x[s1;s2;s3],
so_lambda: so_lambda3,
append: as @ bs,
subtract: n - m,
last: last(L),
select: L[n],
cand: A c∧ B,
true: True,
int_seg: {i..j-},
lelt: i ≤ j < k,
nat_plus: ℕ+,
listp: A List+,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q
Lemmas referenced :
list_wf,
nat_wf,
decidable__le,
int_term_value_add_lemma,
int_term_value_subtract_lemma,
int_formula_prop_not_lemma,
itermAdd_wf,
itermSubtract_wf,
intformnot_wf,
subtract_wf,
decidable__equal_int,
spread_cons_lemma,
int_subtype_base,
set_subtype_base,
subtype_base_sq,
subtract-1-ge-0,
istype-universe,
fun-path_wf,
nil_wf,
append_wf,
cons_wf,
length_wf,
le_wf,
istype-false,
colength_wf_list,
colength-cons-not-zero,
product_subtype_list,
list-cases,
length_of_nil_lemma,
length-append,
int_formula_prop_eq_lemma,
intformeq_wf,
length_of_cons_lemma,
member-less_than,
less_than_wf,
ge_wf,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
istype-void,
int_formula_prop_and_lemma,
istype-int,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformand_wf,
full-omega-unsat,
nat_properties,
reduce_hd_cons_lemma,
list_ind_nil_lemma,
int_seg_properties,
int_seg_wf,
list_ind_cons_lemma,
fun-path-cons,
false_wf,
add-is-int-iff,
decidable__lt,
nat_plus_properties,
length_wf_nat,
add_nat_plus,
listp_properties,
hd_wf,
top_wf,
subtype_rel_list,
length_append,
length_cons,
non_neg_length,
length_nil,
not_wf,
equal_wf,
iff_weakening_uiff
Rules used in proof :
universeEquality,
functionIsType,
intEquality,
applyEquality,
baseClosed,
closedConclusion,
baseApply,
equalityIsType4,
imageElimination,
instantiate,
productEquality,
dependent_set_memberEquality_alt,
because_Cache,
equalityIsType1,
hypothesis_subsumption,
promote_hyp,
unionElimination,
inhabitedIsType,
functionIsTypeImplies,
axiomEquality,
applyLambdaEquality,
equalitySymmetry,
equalityTransitivity,
independent_pairEquality,
productElimination,
universeIsType,
independent_pairFormation,
sqequalRule,
voidElimination,
isect_memberEquality_alt,
dependent_functionElimination,
int_eqEquality,
lambdaEquality_alt,
dependent_pairFormation_alt,
independent_functionElimination,
approximateComputation,
independent_isectElimination,
natural_numberEquality,
intWeakElimination,
rename,
setElimination,
hypothesis,
hypothesisEquality,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
lambdaFormation_alt,
thin,
cut,
introduction,
isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
productIsType,
hyp_replacement,
Error :memTop,
imageMemberEquality,
pointwiseFunctionality,
voidEquality,
isectEquality,
addEquality
Latex:
\mforall{}[T:Type]. \mforall{}[f:T {}\mrightarrow{} T]. \mforall{}[L1,L2:T List]. \mforall{}[x,y,z:T].
uiff(z=f*(x) via L1 @ [y / L2];\{y=f*(x) via [y / L2] \mwedge{} z=f*(y) via L1 @ [y]\})
Date html generated:
2020_05_20-AM-08_10_23
Last ObjectModification:
2020_01_17-AM-11_35_29
Theory : general
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