Nuprl Lemma : assert-C_TYPE_eq
∀[a,b:C_TYPE()].  uiff(↑C_TYPE_eq(a;b);a = b ∈ C_TYPE())
Proof
Definitions occuring in Statement : 
C_TYPE_eq: C_TYPE_eq(a;b)
, 
C_TYPE: C_TYPE()
, 
assert: ↑b
, 
uiff: uiff(P;Q)
, 
uall: ∀[x:A]. B[x]
, 
equal: s = t ∈ T
Definitions unfolded in proof : 
uall: ∀[x:A]. B[x]
, 
member: t ∈ T
, 
C_TYPE_eq: C_TYPE_eq(a;b)
, 
so_lambda: λ2x.t[x]
, 
so_apply: x[s]
, 
implies: P 
⇒ Q
, 
C_TYPE_eq_fun: C_TYPE_eq_fun(a)
, 
C_Void: C_Void()
, 
C_TYPE_ind: C_TYPE_ind, 
select: L[n]
, 
uimplies: b supposing a
, 
all: ∀x:A. B[x]
, 
it: ⋅
, 
so_lambda: λ2x y.t[x; y]
, 
top: Top
, 
so_apply: x[s1;s2]
, 
C_Void?: C_Void?(v)
, 
pi1: fst(t)
, 
eq_atom: x =a y
, 
assert: ↑b
, 
ifthenelse: if b then t else f fi 
, 
btrue: tt
, 
uiff: uiff(P;Q)
, 
and: P ∧ Q
, 
prop: ℙ
, 
true: True
, 
C_Int: C_Int()
, 
bfalse: ff
, 
false: False
, 
C_Int?: C_Int?(v)
, 
not: ¬A
, 
C_Struct: C_Struct(fields)
, 
C_Struct?: C_Struct?(v)
, 
C_Array: C_Array(length;elems)
, 
C_Array?: C_Array?(v)
, 
C_Pointer: C_Pointer(to)
, 
C_Pointer?: C_Pointer?(v)
, 
guard: {T}
, 
bool: 𝔹
, 
unit: Unit
, 
band: p ∧b q
, 
int_seg: {i..j-}
, 
lelt: i ≤ j < k
, 
decidable: Dec(P)
, 
or: P ∨ Q
, 
satisfiable_int_formula: satisfiable_int_formula(fmla)
, 
exists: ∃x:A. B[x]
, 
subtype_rel: A ⊆r B
, 
C_Struct-fields: C_Struct-fields(v)
, 
pi2: snd(t)
, 
nat: ℕ
, 
ge: i ≥ j 
, 
squash: ↓T
, 
sq_type: SQType(T)
, 
bnot: ¬bb
, 
l_all: (∀x∈L.P[x])
, 
le: A ≤ B
, 
iff: P 
⇐⇒ Q
, 
rev_implies: P 
⇐ Q
, 
less_than: a < b
, 
cand: A c∧ B
, 
nequal: a ≠ b ∈ T 
, 
C_Array-length: C_Array-length(v)
, 
C_Array-elems: C_Array-elems(v)
, 
C_Pointer-to: C_Pointer-to(v)
Lemmas referenced : 
C_Pointer-to_wf, 
le_wf, 
decidable__equal_int, 
C_Array-elems_wf, 
C_Array-length_wf, 
equal-wf-base-T, 
C_TYPE_subtype_base, 
atom_subtype_base, 
product_subtype_base, 
list_subtype_base, 
assert_of_band, 
iff_weakening_uiff, 
iff_transitivity, 
band_wf, 
select-upto, 
lelt_wf, 
length_wf_nat, 
length_upto, 
assert-bl-all, 
less_than_wf, 
neg_assert_of_eq_int, 
assert-bnot, 
bool_subtype_base, 
subtype_base_sq, 
bool_cases_sqequal, 
eqff_to_assert, 
list_extensionality, 
squash_wf, 
nat_properties, 
pi2_wf, 
assert_of_eq_atom, 
int_formula_prop_eq_lemma, 
intformeq_wf, 
top_wf, 
subtype_rel_product, 
pi1_wf_top, 
int_formula_prop_less_lemma, 
intformless_wf, 
decidable__lt, 
int_formula_prop_wf, 
int_term_value_var_lemma, 
int_term_value_constant_lemma, 
int_formula_prop_le_lemma, 
int_formula_prop_not_lemma, 
int_formula_prop_and_lemma, 
itermVar_wf, 
itermConstant_wf, 
intformle_wf, 
intformnot_wf, 
intformand_wf, 
satisfiable-full-omega-tt, 
decidable__le, 
int_seg_properties, 
select_wf, 
eq_atom_wf, 
upto_wf, 
int_seg_wf, 
bl-all_wf, 
assert_of_eq_int, 
C_Struct-fields_wf, 
length_wf, 
eq_int_wf, 
eqtt_to_assert, 
bool_wf, 
C_TYPE_eq_wf, 
assert_witness, 
C_Pointer_wf, 
C_Pointer?_wf, 
nat_wf, 
C_Array_wf, 
C_Array?_wf, 
list_wf, 
l_member_wf, 
l_all_wf2, 
C_Struct_wf, 
C_Struct?_wf, 
C_Int_wf, 
btrue_neq_bfalse, 
bfalse_wf, 
C_Int?_wf, 
and_wf, 
btrue_wf, 
false_wf, 
true_wf, 
C_Void_wf, 
C_Void?_wf, 
base_wf, 
stuck-spread, 
equal_wf, 
C_TYPE_eq_fun_wf, 
assert_wf, 
uiff_wf, 
C_TYPE_wf, 
all_wf, 
C_TYPE-induction
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
isect_memberFormation, 
introduction, 
cut, 
thin, 
sqequalRule, 
lemma_by_obid, 
sqequalHypSubstitution, 
isectElimination, 
lambdaEquality, 
hypothesis, 
applyEquality, 
hypothesisEquality, 
independent_functionElimination, 
baseClosed, 
independent_isectElimination, 
lambdaFormation, 
isect_memberEquality, 
voidElimination, 
voidEquality, 
independent_pairFormation, 
natural_numberEquality, 
axiomEquality, 
equalityTransitivity, 
equalitySymmetry, 
dependent_set_memberEquality, 
setElimination, 
rename, 
productElimination, 
setEquality, 
productEquality, 
atomEquality, 
spreadEquality, 
because_Cache, 
dependent_functionElimination, 
independent_pairEquality, 
unionElimination, 
equalityElimination, 
dependent_pairFormation, 
int_eqEquality, 
intEquality, 
computeAll, 
equalityEquality, 
imageElimination, 
promote_hyp, 
instantiate, 
cumulativity, 
imageMemberEquality, 
functionEquality, 
addLevel, 
impliesFunctionality, 
substitution
Latex:
\mforall{}[a,b:C\_TYPE()].    uiff(\muparrow{}C\_TYPE\_eq(a;b);a  =  b)
Date html generated:
2016_05_16-AM-08_46_01
Last ObjectModification:
2016_01_17-AM-09_44_50
Theory : C-semantics
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