Nuprl Lemma : fun-connected-step-back
∀[T:Type]. ∀f:T ⟶ T. ∀x,y:T. (x is f*(y) ⇒ x is f*(f y) supposing ¬(x = y ∈ T))
Proof
Definitions occuring in Statement :
fun-connected: y is f*(x),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
uimplies: b supposing a,
member: t ∈ T,
not: ¬A,
false: False,
fun-connected: y is f*(x),
exists: ∃x:A. B[x],
or: P ∨ Q,
fun-path: y=f*(x) via L,
select: L[n],
nil: [],
it: ⋅,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
subtract: n - m,
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
and: P ∧ Q,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
cons: [a / b],
bfalse: ff,
prop: ℙ,
append: as @ bs,
so_lambda: so_lambda3,
so_apply: x[s1;s2;s3],
ge: i ≥ j ,
true: True,
guard: {T},
nat: ℕ,
le: A ≤ B,
cand: A c∧ B,
decidable: Dec(P),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
listp: A List+,
subtype_rel: A ⊆r B,
satisfiable_int_formula: satisfiable_int_formula(fmla),
int_seg: {i..j-},
lelt: i ≤ j < k,
last: last(L),
sq_type: SQType(T)
Lemmas referenced :
list-cases,
length_of_nil_lemma,
stuck-spread,
istype-base,
null_nil_lemma,
product_subtype_list,
length_of_cons_lemma,
reduce_hd_cons_lemma,
null_cons_lemma,
istype-void,
fun-connected_wf,
istype-universe,
last_lemma,
fun-path_wf,
list_ind_nil_lemma,
last_wf,
hd_wf,
squash_wf,
ge_wf,
length_wf,
list_wf,
listp_properties,
length_wf_nat,
decidable__lt,
istype-false,
not-lt-2,
condition-implies-le,
minus-add,
minus-one-mul,
zero-add,
minus-one-mul-top,
add-commutes,
add_functionality_wrt_le,
add-associates,
add-zero,
le-add-cancel,
istype-less_than,
cons_wf,
nil_wf,
member-less_than,
append_wf,
length_nil,
non_neg_length,
length_cons,
length_append,
subtype_rel_list,
top_wf,
length-append,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
intformle_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
hd-append-sq,
cons-listp,
int_seg_wf,
subtract_wf,
list_ind_cons_lemma,
decidable__le,
add-is-int-iff,
itermSubtract_wf,
int_term_value_subtract_lemma,
false_wf,
istype-le,
equal_wf,
true_wf,
select_wf,
select_append_front,
subtype_rel_self,
iff_weakening_equal,
subtype_base_sq,
int_subtype_base,
general_arith_equation1,
length-singleton,
int_seg_properties,
select-nthtl0,
int_seg_subtype_nat
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
cut,
introduction,
sqequalRule,
sqequalHypSubstitution,
lambdaEquality_alt,
dependent_functionElimination,
thin,
hypothesisEquality,
voidElimination,
functionIsTypeImplies,
inhabitedIsType,
rename,
productElimination,
extract_by_obid,
isectElimination,
hypothesis,
unionElimination,
baseClosed,
independent_isectElimination,
Error :memTop,
imageElimination,
promote_hyp,
hypothesis_subsumption,
functionIsType,
equalityIstype,
universeIsType,
because_Cache,
instantiate,
universeEquality,
dependent_pairFormation_alt,
applyEquality,
independent_functionElimination,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
setElimination,
independent_pairFormation,
addEquality,
minusEquality,
dependent_set_memberEquality_alt,
imageMemberEquality,
independent_pairEquality,
axiomEquality,
hyp_replacement,
applyLambdaEquality,
voidEquality,
approximateComputation,
int_eqEquality,
pointwiseFunctionality,
baseApply,
closedConclusion,
productIsType,
cumulativity,
intEquality
Latex:
\mforall{}[T:Type]. \mforall{}f:T {}\mrightarrow{} T. \mforall{}x,y:T. (x is f*(y) {}\mRightarrow{} x is f*(f y) supposing \mneg{}(x = y))
Date html generated:
2020_05_20-AM-08_10_38
Last ObjectModification:
2019_12_31-PM-06_30_08
Theory : general
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