Nuprl Lemma : p-fun-exp-add
∀[T:Type]. ∀[n,m:ℕ]. ∀[f:T ⟶ (T + Top)]. (f^n + m = f^n o f^m ∈ (T ⟶ (T + Top)))
Proof
Definitions occuring in Statement :
p-fun-exp: f^n,
p-compose: f o g,
nat: ℕ,
uall: ∀[x:A]. B[x],
top: Top,
function: x:A ⟶ B[x],
union: left + right,
add: n + m,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
squash: ↓T,
prop: ℙ,
nat: ℕ,
ge: i ≥ j ,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
implies: P ⇒ Q,
not: ¬A,
top: Top,
and: P ∧ Q,
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
p-fun-exp: f^n
Lemmas referenced :
equal_wf,
squash_wf,
true_wf,
p-fun-exp_wf,
nat_properties,
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,
le_wf,
p-fun-exp-compose,
iff_weakening_equal,
p-id_wf,
primrec_add,
p-compose_wf,
top_wf,
int_seg_wf,
primrec_wf,
nat_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
applyEquality,
thin,
lambdaEquality,
sqequalHypSubstitution,
imageElimination,
extract_by_obid,
isectElimination,
hypothesisEquality,
equalityTransitivity,
hypothesis,
equalitySymmetry,
because_Cache,
cumulativity,
functionExtensionality,
dependent_set_memberEquality,
addEquality,
setElimination,
rename,
dependent_functionElimination,
natural_numberEquality,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
imageMemberEquality,
baseClosed,
universeEquality,
productElimination,
independent_functionElimination,
lambdaFormation,
functionEquality,
unionEquality,
axiomEquality
Latex:
\mforall{}[T:Type]. \mforall{}[n,m:\mBbbN{}]. \mforall{}[f:T {}\mrightarrow{} (T + Top)]. (f\^{}n + m = f\^{}n o f\^{}m)
Date html generated:
2017_10_01-AM-09_14_39
Last ObjectModification:
2017_07_26-PM-04_49_38
Theory : general
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