Nuprl Lemma : p-fun-exp-add1-sq
∀[A:Type]. ∀[f:A ⟶ (A + Top)]. ∀[x:A]. ∀[n:ℕ]. f^n + 1 x ~ f^n do-apply(f;x) supposing ↑can-apply(f;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,
natural_number: $n,
universe: Type,
sqequal: s ~ t
Definitions unfolded in proof :
do-apply: do-apply(f;x),
can-apply: can-apply(f;x),
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
implies: P ⇒ Q,
isl: isl(x),
p-fun-exp: f^n,
top: Top,
nat: ℕ,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
not: ¬A,
false: False,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
prop: ℙ,
p-id: p-id(),
p-compose: f o g,
outl: outl(x),
ifthenelse: if b then t else f fi ,
btrue: tt,
assert: ↑b,
bfalse: ff,
subtype_rel: A ⊆r B,
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
istype-assert,
btrue_wf,
bfalse_wf,
istype-nat,
istype-top,
istype-universe,
simple-primrec-add,
istype-void,
istype-le,
primrec1_lemma,
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
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,
istype-less_than,
primrec0_lemma,
subtract-1-ge-0,
istype-true,
not_wf,
bnot_wf,
assert_wf,
int_subtype_base,
equal-wf-base,
bool_wf,
eq_int_wf,
primrec-unroll,
uiff_transitivity,
eqtt_to_assert,
assert_of_eq_int,
iff_transitivity,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot,
equal_wf,
int_formula_prop_eq_lemma,
intformeq_wf,
satisfiable-full-omega-tt
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation_alt,
introduction,
cut,
hypothesis,
axiomSqEquality,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
applyEquality,
hypothesisEquality,
inhabitedIsType,
lambdaFormation_alt,
unionElimination,
equalityIstype,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
independent_functionElimination,
isect_memberEquality_alt,
isectIsTypeImplies,
universeIsType,
functionIsType,
unionIsType,
because_Cache,
instantiate,
universeEquality,
voidElimination,
dependent_set_memberEquality_alt,
natural_numberEquality,
independent_pairFormation,
promote_hyp,
setElimination,
rename,
intWeakElimination,
independent_isectElimination,
approximateComputation,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
functionIsTypeImplies,
intEquality,
baseClosed,
closedConclusion,
baseApply,
voidEquality,
isect_memberEquality,
lambdaFormation,
equalityElimination,
productElimination,
impliesFunctionality,
computeAll,
lambdaEquality,
dependent_pairFormation
Latex:
\mforall{}[A:Type]. \mforall{}[f:A {}\mrightarrow{} (A + Top)]. \mforall{}[x:A]. \mforall{}[n:\mBbbN{}].
f\^{}n + 1 x \msim{} f\^{}n do-apply(f;x) supposing \muparrow{}can-apply(f;x)
Date html generated:
2019_10_15-AM-11_07_44
Last ObjectModification:
2019_06_26-PM-04_19_11
Theory : general
Home
Index