Nuprl Lemma : p-fun-exp-compose
∀[T:Type]. ∀[n:ℕ]. ∀[h,f:T ⟶ (T + Top)]. (f^n o h = primrec(n;h;λi,g. f o g) ∈ (T ⟶ (T + Top)))
Proof
Definitions occuring in Statement :
p-fun-exp: f^n,
p-compose: f o g,
primrec: primrec(n;b;c),
nat: ℕ,
uall: ∀[x:A]. B[x],
top: Top,
lambda: λx.A[x],
function: x:A ⟶ B[x],
union: left + right,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
all: ∀x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
p-fun-exp: f^n,
decidable: Dec(P),
or: P ∨ Q,
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
sq_type: SQType(T),
le: A ≤ B,
less_than': less_than'(a;b)
Lemmas referenced :
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
less_than_wf,
top_wf,
primrec0_lemma,
decidable__le,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
nat_wf,
equal_wf,
squash_wf,
true_wf,
p-id-compose,
iff_weakening_equal,
subtype_base_sq,
int_subtype_base,
decidable__equal_int,
intformeq_wf,
itermAdd_wf,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
primrec_add,
false_wf,
le_wf,
p-id_wf,
p-compose_wf,
int_seg_wf,
primrec1_lemma,
primrec_wf,
p-fun-exp_wf,
p-compose-associative
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
lambdaFormation,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
independent_functionElimination,
axiomEquality,
functionEquality,
cumulativity,
unionEquality,
unionElimination,
because_Cache,
universeEquality,
applyEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
imageMemberEquality,
baseClosed,
productElimination,
instantiate,
dependent_set_memberEquality,
functionExtensionality,
addEquality,
hyp_replacement,
applyLambdaEquality
Latex:
\mforall{}[T:Type]. \mforall{}[n:\mBbbN{}]. \mforall{}[h,f:T {}\mrightarrow{} (T + Top)]. (f\^{}n o h = primrec(n;h;\mlambda{}i,g. f o g))
Date html generated:
2017_10_01-AM-09_14_33
Last ObjectModification:
2017_07_26-PM-04_49_36
Theory : general
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