Nuprl Lemma : can-apply-fun-exp-add
∀[A:Type]. ∀[n,m:ℕ]. ∀[f:A ⟶ (A + Top)]. ∀[x:A].
{(↑can-apply(f^m;x)) ∧ (↑can-apply(f^n;do-apply(f^m;x))) ∧ (do-apply(f^n + m;x) = do-apply(f^n;do-apply(f^m;x)) ∈ A)}
supposing ↑can-apply(f^n + m;x)
Proof
Definitions occuring in Statement :
p-fun-exp: f^n,
do-apply: do-apply(f;x),
can-apply: can-apply(f;x),
nat: ℕ,
assert: ↑b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
top: Top,
guard: {T},
and: P ∧ Q,
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,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
squash: ↓T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
all: ∀x:A. B[x],
top: Top,
true: True,
guard: {T},
uiff: uiff(P;Q),
and: P ∧ Q,
implies: P ⇒ Q,
nat: ℕ,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
cand: A c∧ B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
assert_functionality_wrt_uiff,
can-apply_wf,
p-fun-exp_wf,
p-compose_wf,
top_wf,
squash_wf,
true_wf,
p-fun-exp-add,
subtype_rel_dep_function,
subtype_rel_union,
assert_witness,
do-apply_wf,
assert_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,
nat_wf,
can-apply-compose,
equal_wf,
iff_weakening_equal,
do-apply-compose
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
hypothesis,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
applyEquality,
sqequalRule,
cumulativity,
hypothesisEquality,
functionExtensionality,
independent_isectElimination,
lambdaEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
functionEquality,
unionEquality,
lambdaFormation,
isect_memberEquality,
voidElimination,
voidEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
productElimination,
independent_pairEquality,
independent_functionElimination,
axiomEquality,
dependent_set_memberEquality,
addEquality,
setElimination,
rename,
dependent_functionElimination,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
independent_pairFormation,
computeAll,
universeEquality
Latex:
\mforall{}[A:Type]. \mforall{}[n,m:\mBbbN{}]. \mforall{}[f:A {}\mrightarrow{} (A + Top)]. \mforall{}[x:A].
\{(\muparrow{}can-apply(f\^{}m;x))
\mwedge{} (\muparrow{}can-apply(f\^{}n;do-apply(f\^{}m;x)))
\mwedge{} (do-apply(f\^{}n + m;x) = do-apply(f\^{}n;do-apply(f\^{}m;x)))\}
supposing \muparrow{}can-apply(f\^{}n + m;x)
Date html generated:
2017_10_01-AM-09_14_45
Last ObjectModification:
2017_07_26-PM-04_49_42
Theory : general
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