Nuprl Lemma : mccarthy91_wf1
∀x:ℤ. (mccarthy91(x) ∈ {m:ℤ| m = if x ≤z 101 then 91 else x - 10 fi ∈ ℤ} )
Proof
Definitions occuring in Statement :
mccarthy91: mccarthy91(x),
le_int: i ≤z j,
ifthenelse: if b then t else f fi ,
all: ∀x:A. B[x],
member: t ∈ T,
set: {x:A| B[x]} ,
subtract: n - m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
assert: ↑b,
ifthenelse: if b then t else f fi ,
bnot: ¬bb,
guard: {T},
sq_type: SQType(T),
bfalse: ff,
or: P ∨ Q,
decidable: Dec(P),
squash: ↓T,
true: True,
less_than': less_than'(a;b),
less_than: a < b,
uiff: uiff(P;Q),
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
mccarthy91: mccarthy91(x),
prop: ℙ,
and: P ∧ Q,
top: Top,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
uimplies: b supposing a,
ge: i ≥ j ,
false: False,
implies: P ⇒ Q,
nat: ℕ,
member: t ∈ T,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
has-value: (a)↓,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
subtype_rel: A ⊆r B,
subtract: n - m,
lelt: i ≤ j < k,
int_seg: {i..j-},
le: A ≤ B,
label: ...$L... t,
so_apply: x[s],
so_lambda: λ2x.t[x]
Lemmas referenced :
nat_wf,
decidable__le,
assert-bnot,
bool_subtype_base,
subtype_base_sq,
bool_cases_sqequal,
equal_wf,
eqff_to_assert,
int_term_value_subtract_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_not_lemma,
itermSubtract_wf,
intformeq_wf,
intformnot_wf,
decidable__equal_int,
top_wf,
assert_of_lt_int,
eqtt_to_assert,
bool_wf,
lt_int_wf,
subtract_wf,
le_wf,
less_than_wf,
ge_wf,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_and_lemma,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformand_wf,
full-omega-unsat,
nat_properties,
int-value-type,
value-type-has-value,
iff_weakening_equal,
int_term_value_add_lemma,
itermAdd_wf,
true_wf,
squash_wf,
int_subtype_base,
int_seg_wf,
lelt_wf,
decidable__lt,
int_seg_cases,
false_wf,
int_seg_subtype,
int_seg_properties,
assert_of_le_int,
le_int_wf,
equal-wf-base,
set_wf,
member_wf
Rules used in proof :
cut,
dependent_set_memberEquality,
cumulativity,
instantiate,
promote_hyp,
imageElimination,
baseClosed,
imageMemberEquality,
because_Cache,
sqequalAxiom,
isect_memberFormation,
lessCases,
productElimination,
equalitySymmetry,
equalityTransitivity,
equalityElimination,
unionElimination,
axiomEquality,
independent_pairFormation,
sqequalRule,
voidEquality,
voidElimination,
isect_memberEquality,
dependent_functionElimination,
intEquality,
int_eqEquality,
lambdaEquality,
dependent_pairFormation,
independent_functionElimination,
approximateComputation,
independent_isectElimination,
natural_numberEquality,
intWeakElimination,
rename,
setElimination,
hypothesis,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
addEquality,
callbyvalueReduce,
universeEquality,
applyEquality,
hypothesis_subsumption,
closedConclusion,
baseApply
Latex:
\mforall{}x:\mBbbZ{}. (mccarthy91(x) \mmember{} \{m:\mBbbZ{}| m = if x \mleq{}z 101 then 91 else x - 10 fi \} )
Date html generated:
2018_05_21-PM-00_30_40
Last ObjectModification:
2017_12_27-PM-06_46_58
Theory : int_2
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