Nuprl Lemma : cons_iseg
∀[T:Type]. ∀a,b:T. ∀l1,l2:T List. ([a / l1] ≤ [b / l2] ⇐⇒ (a = b ∈ T) ∧ l1 ≤ l2)
Proof
Definitions occuring in Statement :
iseg: l1 ≤ l2,
cons: [a / b],
list: T List,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
true: True,
listp: A List+,
decidable: Dec(P),
or: P ∨ Q,
false: False,
le: A ≤ B,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
top: Top,
so_apply: x[s1;s2;s3],
squash: ↓T,
uimplies: b supposing a,
ge: i ≥ j ,
prop: ℙ,
subtype_rel: A ⊆r B,
iseg: l1 ≤ l2,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
exists: ∃x:A. B[x],
member: t ∈ T,
rev_implies: P ⇐ Q
Lemmas referenced :
istype-void,
true_wf,
tl_wf,
reduce_tl_cons_lemma,
equal_wf,
length_of_cons_lemma,
non_neg_length,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
intformle_wf,
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,
less_than_wf,
list_ind_cons_lemma,
reduce_hd_cons_lemma,
hd_wf,
squash_wf,
ge_wf,
length_wf,
length_cons_ge_one,
subtype_rel_list,
top_wf,
cons_wf,
append_wf,
list_wf
Rules used in proof :
Error :isect_memberEquality_alt,
Error :lambdaEquality_alt,
Error :dependent_pairFormation_alt,
hyp_replacement,
applyLambdaEquality,
setElimination,
rename,
dependent_set_memberEquality,
addEquality,
unionElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
applyEquality,
lambdaEquality,
imageElimination,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
cumulativity,
because_Cache,
imageMemberEquality,
baseClosed,
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
Error :isect_memberFormation_alt,
Error :lambdaFormation_alt,
independent_pairFormation,
cut,
sqequalHypSubstitution,
productElimination,
thin,
hypothesis,
Error :productIsType,
Error :inhabitedIsType,
hypothesisEquality,
Error :equalityIsType1,
introduction,
extract_by_obid,
isectElimination,
Error :universeIsType,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}a,b:T. \mforall{}l1,l2:T List. ([a / l1] \mleq{} [b / l2] \mLeftarrow{}{}\mRightarrow{} (a = b) \mwedge{} l1 \mleq{} l2)
Date html generated:
2019_06_20-PM-02_12_50
Last ObjectModification:
2019_06_20-PM-02_08_37
Theory : list_1
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