Nuprl Lemma : p-fun-exp-add-sq
∀[A:Type]. ∀[f:A ⟶ (A + Top)]. ∀[x:A]. ∀[m,n:ℕ]. f^n + m x ~ f^n do-apply(f^m;x) supposing ↑can-apply(f^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,
apply: f a,
function: x:A ⟶ B[x],
union: left + right,
add: n + m,
universe: Type,
sqequal: s ~ 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,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
all: ∀x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
le: A ≤ B,
less_than': less_than'(a;b),
decidable: Dec(P),
or: P ∨ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
p-fun-exp: f^n,
do-apply: do-apply(f;x),
p-id: p-id(),
outl: outl(x),
sq_type: SQType(T),
guard: {T},
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
p-compose: f o g,
can-apply: can-apply(f;x),
squash: ↓T,
true: True
Lemmas referenced :
nat_properties,
full-omega-unsat,
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_wf,
can-apply_wf,
p-fun-exp_wf,
false_wf,
le_wf,
decidable__le,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
nat_wf,
top_wf,
subtype_rel_dep_function,
subtype_rel_union,
primrec0_lemma,
add-zero,
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
zero-add,
inl-do-apply,
lt_int_wf,
bool_wf,
equal-wf-T-base,
equal-wf-base,
intformeq_wf,
int_formula_prop_eq_lemma,
le_int_wf,
bnot_wf,
itermAdd_wf,
int_term_value_add_lemma,
primrec-unroll,
uiff_transitivity,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
assert_functionality_wrt_uiff,
bnot_of_lt_int,
assert_of_le_int,
equal_wf,
can-apply-fun-exp,
not_wf,
assert_of_bnot,
do-apply_wf,
p-fun-exp-add1-sq,
bool_subtype_base,
isl_wf,
squash_wf,
true_wf,
p-compose_wf,
subtract-add-cancel
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,
approximateComputation,
independent_functionElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
sqequalAxiom,
cumulativity,
because_Cache,
functionExtensionality,
applyEquality,
equalityTransitivity,
equalitySymmetry,
dependent_set_memberEquality,
unionElimination,
functionEquality,
unionEquality,
universeEquality,
instantiate,
addEquality,
baseClosed,
baseApply,
closedConclusion,
equalityElimination,
productElimination,
imageElimination,
imageMemberEquality,
hyp_replacement,
applyLambdaEquality
Latex:
\mforall{}[A:Type]. \mforall{}[f:A {}\mrightarrow{} (A + Top)]. \mforall{}[x:A]. \mforall{}[m,n:\mBbbN{}].
f\^{}n + m x \msim{} f\^{}n do-apply(f\^{}m;x) supposing \muparrow{}can-apply(f\^{}m;x)
Date html generated:
2018_05_21-PM-06_29_39
Last ObjectModification:
2018_05_19-PM-04_40_42
Theory : general
Home
Index