Nuprl Lemma : fun-path-before
∀[T:Type]. ∀f:T ⟶ T. ∀L:T List. ∀x,y,a,b:T. a before b ∈ L ⇒ a is f*(b) supposing x=f*(y) via L
Proof
Definitions occuring in Statement :
fun-connected: y is f*(x),
fun-path: y=f*(x) via L,
l_before: x before y ∈ l,
list: T List,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
uimplies: b supposing a,
prop: ℙ,
implies: P ⇒ Q,
so_apply: x[s],
fun-path: y=f*(x) via L,
select: L[n],
nil: [],
it: ⋅,
so_lambda: λ2x y.t[x; y],
top: Top,
so_apply: x[s1;s2],
subtract: n - m,
and: P ∧ Q,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
false: False,
not: ¬A,
int_seg: {i..j-},
guard: {T},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
uiff: uiff(P;Q),
iff: P ⇐⇒ Q,
cons: [a / b],
nat_plus: ℕ+,
true: True,
subtype_rel: A ⊆r B,
rev_implies: P ⇐ Q,
cand: A c∧ B,
nat: ℕ,
le: A ≤ B
Lemmas referenced :
list_induction,
all_wf,
isect_wf,
fun-path_wf,
l_before_wf,
fun-connected_wf,
list_wf,
length_of_nil_lemma,
stuck-spread,
base_wf,
member-less_than,
nil_wf,
less_than_wf,
equal-wf-T-base,
int_seg_wf,
equal-wf-base-T,
not_wf,
equal-wf-base,
length_wf,
cons_wf,
equal_wf,
select_wf,
length_of_cons_lemma,
int_seg_properties,
subtract_wf,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
decidable__lt,
add-is-int-iff,
intformless_wf,
itermSubtract_wf,
int_formula_prop_less_lemma,
int_term_value_subtract_lemma,
false_wf,
fun-path-cons,
cons_before,
list-cases,
null_nil_lemma,
btrue_wf,
member-implies-null-eq-bfalse,
btrue_neq_bfalse,
product_subtype_list,
add_nat_plus,
length_wf_nat,
nat_plus_wf,
nat_plus_properties,
intformeq_wf,
int_formula_prop_eq_lemma,
reduce_hd_cons_lemma,
fun-connected-step,
squash_wf,
true_wf,
iff_weakening_equal,
cons_member,
fun-connected_transitivity,
fun-connected_weakening_eq,
l_before_member,
nil_member,
nat_wf,
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
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
cumulativity,
because_Cache,
functionExtensionality,
applyEquality,
hypothesis,
functionEquality,
independent_functionElimination,
rename,
dependent_functionElimination,
universeEquality,
baseClosed,
independent_isectElimination,
isect_memberEquality,
voidElimination,
voidEquality,
productElimination,
independent_pairEquality,
imageElimination,
axiomEquality,
productEquality,
natural_numberEquality,
minusEquality,
equalityTransitivity,
equalitySymmetry,
addEquality,
setElimination,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
independent_pairFormation,
computeAll,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
hypothesis_subsumption,
dependent_set_memberEquality,
imageMemberEquality,
applyLambdaEquality,
inrFormation,
inlFormation
Latex:
\mforall{}[T:Type]. \mforall{}f:T {}\mrightarrow{} T. \mforall{}L:T List. \mforall{}x,y,a,b:T. a before b \mmember{} L {}\mRightarrow{} a is f*(b) supposing x=f*(y) via L
Date html generated:
2018_05_21-PM-07_46_23
Last ObjectModification:
2017_07_26-PM-05_23_51
Theory : general
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