Nuprl Lemma : can-apply-fun-exp
∀[A:Type]. ∀[f:A ⟶ (A + Top)]. ∀[n:ℕ]. ∀[y:A]. ∀[m:ℕ]. ↑can-apply(f^m;y) supposing m ≤ n supposing ↑can-apply(f^n;y)
Proof
Definitions occuring in Statement :
p-fun-exp: f^n,
can-apply: can-apply(f;x),
nat: ℕ,
assert: ↑b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
top: Top,
le: A ≤ B,
function: x:A ⟶ B[x],
union: left + right,
universe: Type
Definitions unfolded in proof :
can-apply: can-apply(f;x),
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,
decidable: Dec(P),
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
p-fun-exp: f^n,
p-id: p-id(),
isl: isl(x),
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
true: True,
le: A ≤ B,
less_than': less_than'(a;b),
squash: ↓T,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
p-compose: f o g,
do-apply: do-apply(f;x),
outl: outl(x),
bfalse: ff
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,
assert_witness,
isl_wf,
top_wf,
p-fun-exp_wf,
le_wf,
assert_wf,
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
primrec0_lemma,
intformnot_wf,
intformeq_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
false_wf,
decidable__le,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
subtract-add-cancel,
assert_functionality_wrt_uiff,
p-compose_wf,
squash_wf,
true_wf,
p-fun-exp-add,
nat_wf,
equal_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
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,
independent_pairFormation,
computeAll,
independent_functionElimination,
cumulativity,
applyEquality,
functionExtensionality,
equalityTransitivity,
equalitySymmetry,
because_Cache,
unionElimination,
instantiate,
dependent_set_memberEquality,
addEquality,
imageElimination,
unionEquality,
universeEquality,
imageMemberEquality,
baseClosed,
productElimination,
addLevel,
impliesFunctionality,
functionEquality
Latex:
\mforall{}[A:Type]. \mforall{}[f:A {}\mrightarrow{} (A + Top)]. \mforall{}[n:\mBbbN{}]. \mforall{}[y:A].
\mforall{}[m:\mBbbN{}]. \muparrow{}can-apply(f\^{}m;y) supposing m \mleq{} n supposing \muparrow{}can-apply(f\^{}n;y)
Date html generated:
2017_10_01-AM-09_15_05
Last ObjectModification:
2017_07_26-PM-04_49_53
Theory : general
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