hcju/isabelleps huggingface dataset:
MM0: fixes F :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and G :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" shows "\<turnstile> \<box>(F \<longrightarrow> G) \<longrightarrow> \<box>F \<longrightarrow> \<box>G"
inj_on_Inl: fixes A :: "V set" shows "inj_on ZFC_Cardinals.Inl A"
inj_on_the_inv_into: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set" assumes "inj_on f A" shows "inj_on (the_inv_into A f) (f ` A)"
trivialPhi_in_allVars: fixes g :: "'g"   and v :: "'val"   and v' :: "'val" assumes "isTrivialPhi g v v'"   and "v \<in> allVars g" shows "v' \<in> allVars g"
measurable_ident_sets: fixes M :: "'c measure"   and M' :: "'c measure" assumes "sets M = sets M'" shows "(\<lambda>x. x) \<in> M \<rightarrow>\<^sub>M M'"
map_update: fixes m :: "'q"   and a :: "'v"   and b :: "nat" assumes "Q_invar m" shows "Q_lookup (Q_update a b m) = Q_lookup m(a \<mapsto> b)"
is_cnf_encode_problem_with_operator_interference_exclusion: fixes \<Pi> :: "'b strips_problem"   and t :: "nat" assumes "is_valid_problem_strips \<Pi>" shows "is_cnf (\<Phi>\<^sub>\<forall> \<Pi> t)"
domA_map_ran: fixes f :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and m :: "('d \<times> 'e) list" shows "domA (map_ran f m) = domA m"
assms_2: shows "eval f (z # xs) \<down>= 0"
Games_mkinterp: fixes G :: "char \<Rightarrow> (variable \<Rightarrow> real) set \<Rightarrow> (variable \<Rightarrow> real) set"   and C :: "char \<Rightarrow> real"   and F :: "char \<Rightarrow> real \<Rightarrow> real"   and P :: "char \<Rightarrow> real \<Rightarrow> bool" assumes "\<And>a. mono (G a)" shows "Games (mkinterp (C, F, P, G)) = G"
power_mult_distrib: fixes a :: "'b"   and b :: "'b"   and n :: "nat" shows "(a * b) ^ n = a ^ n * b ^ n"
cadjoint_is_cadjoint: fixes G :: "'c \<Rightarrow> 'd" assumes "bounded_clinear G" shows "is_cadjoint (cadjoint G) G"
islimpt_insert: fixes x :: "'b"   and a :: "'b"   and s :: "'b set" shows "x islimpt insert a s = x islimpt s"
Rep_perm_inject: fixes x :: "perm"   and y :: "perm" shows "(Rep_perm x = Rep_perm y) = (x = y)"
abs_eq: fixes x :: "int extended \<times> int extended" shows "\<gamma>_ivl (abs_ivl x) = \<gamma>_rep x"
mvt: fixes a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> real"   and f' :: "real \<Rightarrow> real \<Rightarrow> real" assumes "a < b"   and "continuous_on {a..b} f"   and "\<And>x. \<lbrakk>a < x; x < b\<rbrakk> \<Longrightarrow> (f has_derivative f' x) (at x)" obtains \<xi> :: "real" where "a < \<xi>" and "\<xi> < b" and "f b - f a = f' \<xi> (b - a)"
is_proper_prefix_eq_prefix: fixes a :: "'a list"   and b :: "'a list" shows "is_proper_prefix a b = (a \<noteq> b \<and> is_prefix a b)"
induct: fixes P :: "'b list \<Rightarrow> 'b list \<Rightarrow> bool"   and a0 :: "'b list"   and a1 :: "'b list" assumes "\<And>x xs y ys. (x = y \<Longrightarrow> P xs ys) \<Longrightarrow> P (x # xs) (y # ys)"   and "\<And>uv_. P [] uv_"   and "\<And>uu_. P uu_ []" shows "P a0 a1"
ahm_update_def: fixes k :: "'a"   and v :: "'b"   and hm :: "('a, 'b) hashmap" shows "ahm_update k v hm = HashMap (ArrayHashMap_Impl.ahm_update k v (impl_of hm))"
conf_ClassD: fixes P :: "'b prog"   and h :: "heap"   and v :: "val"   and C :: "char list" assumes "P,h \<turnstile> v :\<le> Class C" shows "v = Null \<or> (\<exists>a obj T. v = Addr a \<and> h a = \<lfloor>obj\<rfloor> \<and> obj_ty obj = T \<and> P \<turnstile> T \<le> Class C)"
simps_2: fixes n :: "nat" shows "zsplit0 (Bound n) = (if n = 0 then (1, C 0) else (0, Bound n))"
associative: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<in> A"   and "b \<in> A"   and "c \<in> A" shows "multA (multA a b) c = multA a (multA b c)"
65585: fixes x :: "'a"   and y :: "'a" shows "- (x \<squnion> - y) = - - y \<sqinter> - x"
intros_2: fixes \<phi> :: "'b stream \<Rightarrow> bool"   and xs :: "'b stream" assumes "ev \<phi> (stl xs)" shows "ev \<phi> xs"
prY_def: shows "prY = countable_preimages (A \<inter> range Y) Y"
simRec: shows "similar_mat (diag_block_mat (map fst Ms)) (diag_block_mat (map snd Ms))"
right_cancel: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<in> carrier R"   and "y \<in> carrier R"   and "z \<in> carrier R" shows "(y \<oplus> x = z \<oplus> x) = (y = z)"
tendsto_Lambert_W: fixes f :: "'a \<Rightarrow> real"   and L :: "real"   and F :: "'a filter" assumes "(f \<longlongrightarrow> L) F"   and "(\<forall>\<^sub>F x in F. - exp (- 1) \<le> f x) \<or> - exp (- 1) < L" shows "((\<lambda>x. Lambert_W (f x)) \<longlongrightarrow> Lambert_W L) F"
increasing_tendsto: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and "open" :: "'b set \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'b"   and l :: "'b"   and F :: "'d filter" assumes "class.order_topology less_eq less open"   and "\<forall>\<^sub>F n in F. less_eq (f n) l"   and "\<And>x. less x l \<Longrightarrow> \<forall>\<^sub>F n in F. less x (f n)" shows "filterlim f (topological_space.nhds open l) F"
control_within_topI: fixes \<sigma> :: "('e \<times> ('a, 'b, 'c, 'd) seqp) set"   and \<Gamma> :: "'c \<Rightarrow> ('a, 'b, 'c, 'd) seqp" assumes "\<And>p. p \<in> Range \<sigma> \<Longrightarrow> \<exists>pn. p = \<Gamma> pn" shows "control_within \<Gamma> \<sigma>"
weak_reduction_correspondence_simulation_impl_coupled_simulation_1: fixes Rel :: "('b \<times> 'b) set"   and Cal :: "'b processCalculus" assumes "weak_reduction_correspondence_simulation Rel Cal" shows "weak_reduction_coupled_simulation (cSim_cs Rel Cal) Cal"
map_\<I>_id: fixes \<I> :: "('i, 'j) \<I>" shows "map_\<I> id id \<I> = \<I>"
normal_inv_iff: fixes G :: "('c, 'd) monoid_scheme"   and N :: "'c set" assumes "Group.group G" shows "N \<lhd> G = (subgroup N G \<and> (\<forall>x\<in>carrier G. \<forall>h\<in>N. x \<otimes>\<^bsub>G\<^esub> h \<otimes>\<^bsub>G\<^esub> inv\<^bsub>G\<^esub> x \<in> N))"
rep_eq: fixes x :: "nat"   and xa :: "float interval" shows "bounds_of_interval (arctan_float_interval x xa) = (case bounds_of_interval xa of (lx, ux) \<Rightarrow> (lb_arctan x lx, ub_arctan x ux))"
IH: shows "cnf (foldr (\<^bold>\<and>) fs (\<^bold>\<not> \<bottom>)) = \<Union> (cnf ` set fs)"
below_def: fixes L :: "(alt \<times> alt) set"   and a :: "alt"   and b :: "alt" shows "below L a b \<equiv> rem L a \<union> {(a, b)} \<union> {(x, a) |x. (x, b) \<in> L \<and> x \<noteq> a} \<union> {(a, y) |y. (b, y) \<in> L \<and> y \<noteq> a}"
RETURN_rule: fixes \<Phi> :: "'c \<Rightarrow> bool"   and x :: "'c" assumes "\<Phi> x" shows "RETURN x \<le> SPEC \<Phi>"
distinct_9: shows "T_i64 \<noteq> T_f64"
inverse_irred_power_poly_def: fixes d :: "'a"   and n :: "nat" shows "inverse_irred_power_poly d n = Poly (map (\<lambda>k. d * of_nat (stirling n (k + 1)) / fact (n - 1)) [0..<n])"
small_step_stuck: fixes s :: "'h option \<times> 'j"   and \<alpha> :: "('g, 'i) seq_label"   and c' :: "('g, 'h, 'i, 'j) com list \<times> 'h option \<times> 'j" shows "\<not> ([], s) \<rightarrow>\<^bsub>\<alpha>\<^esub> c'"
bal_imp_full: fixes t :: "'b tree234" assumes "bal t" shows "full (height t) t"
components_def: fixes S :: "'a set" shows "components S \<equiv> connected_component_set S ` S"
not_empty_eq_Ici_eq_empty: fixes l :: "'a multiset" shows "{} \<noteq> subset_mset.atLeast l"
infs_szip_snd: fixes P :: "'b \<Rightarrow> bool"   and u :: "'a stream"   and v :: "'b stream" shows "infs (P \<circ> snd) (u ||| v) = infs P v"
simps_2: fixes var :: "nat"   and p :: "real mpoly"   and as :: "atom list" shows "find_eq var (Less p # as) = (let (A, B) = find_eq var as in (A, Less p # B))"
assms: shows "of_int (fls_subdegree f) * f $$ fls_subdegree f \<noteq> (0::'a)"
zimage_def: fixes f :: "'b \<Rightarrow> 'a"   and A :: "'b zet" shows "zimage f A \<equiv> Abs_zet (f ` Rep_zet A)"
boundedIncLevel: fixes A :: "(nat \<times> 'b) set" shows "bounded (incLevel ` A) = bounded A"
semiring_normalization_rules_16: fixes lx :: "'a"   and ly :: "'a"   and rx :: "'a" shows "lx * ly * rx = lx * rx * ly"
that: fixes x :: "'a" assumes "x \<in> A"   and "disjnt C (B x)" shows "thesis"
shift1_head: fixes p :: "poly"   and n0 :: "nat" assumes "isnpolyh p n0" shows "head (shift1 p) = head p"
assms: shows "ops \<in> set (List_Supplement.embed \<psi>)"
prems: shows "ir_valid P (WHILE x1 DO c) c' Q"
limI: fixes X :: "nat \<Rightarrow> 'e"   and L :: "'e" assumes "X \<longlonglongrightarrow> L" shows "lim X = L"
arctan_le_mult: fixes x :: "real"   and y :: "real"   and z :: "real" assumes "0 < x"   and "x \<le> y"   and "arctan x \<le> x * z" shows "arctan y \<le> y * z"
card_idx_perm: shows "|UNIV| <o |UNIV|"
bernoulli_denom_0: shows "bernoulli_denom 0 = 1"
blocks_def: fixes m :: "'m"   and w :: "'w"   and \<mu> :: "('m \<times> 'w) set" shows "blocks m w \<mu> = (w \<in> m_strictly_prefers m \<mu> \<and> m \<in> w_strictly_prefers w \<mu>)"
comp_def: fixes g :: "'a \<times> 'a"   and f :: "'a \<times> 'a" shows "B.VV.comp g f = (if (B.arr (fst f) \<and> B.arr (snd f) \<and> src (fst f) = trg (snd f)) \<and> (B.arr (fst g) \<and> B.arr (snd g) \<and> src (fst g) = trg (snd g)) \<and> B.VxV.cod f = B.VxV.dom g then B.VxV.comp g f else B.VxV.null)"
bind_game_alt_def: fixes \<A> :: "'pub_input \<Rightarrow> ('msg \<times> 'challenge \<times> 'response \<times> 'challenge \<times> 'response) spmf" shows "abstract_com.bind_game \<A> = TRY key_gen \<bind> (\<lambda>(ck, vk). \<A> ck \<bind> (\<lambda>(c, m, d, m', d'). assert_spmf (m \<noteq> m' \<and> valid_msg m \<and> valid_msg m') \<bind> (\<lambda>_. let b = verify vk m c d; b' = verify vk m' c d' in assert_spmf (b \<and> b') \<bind> (\<lambda>_. return_spmf True)))) ELSE return_spmf False"
map\<^sub>0_simps_1: fixes a :: "'c" assumes "C.obj a" shows "D.obj (F.map\<^sub>0 a)"
tendsto_of_int_iff: fixes f :: "'a \<Rightarrow> int"   and c :: "int"   and F :: "'a filter" shows "((\<lambda>n. of_int (f n)) \<longlongrightarrow> of_int c) F = (\<forall>\<^sub>F n in F. f n = c)"
absorb_iff1: fixes b :: "'c"   and a :: "'c" shows "(b \<le> a) = (max a b = a)"
assms_2: shows "A \<equiv> {x \<in> space M. limsup (\<lambda>n. ereal (birkhoff_sum f n x)) = \<infinity>}"
strictly_subsumes_antisym: fixes C :: "'a literal multiset"   and D :: "'a literal multiset" assumes "strictly_subsumes C D" shows "\<not> strictly_subsumes D C"
fcomp_assoc: fixes r :: "V"   and s :: "V"   and t :: "V" shows "(r \<circ>\<^sub>\<bullet> s) \<circ>\<^sub>\<bullet> t = r \<circ>\<^sub>\<bullet> s \<circ>\<^sub>\<bullet> t"
mr: shows "to_mregex r = (mr, \<psi>s)"
next_symbol_not_complete: fixes x :: "item"   and t :: "symbol" assumes "next_symbol x = Some t" shows "\<not> is_complete x"
simps_3: fixes n :: "nat" shows "enumerate_aexp_inputs (V (R n)) = {}"
is_glb_def: fixes below :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and S :: "'c set"   and x :: "'c" assumes "class.po below" shows "po.is_glb below S x = (po.is_lb below S x \<and> (\<forall>u. po.is_lb below S u \<longrightarrow> below u x))"
id2: shows "[i..<k] = i # [Suc i..<k]"
l_comb_jointfun_jj1: fixes H :: "'a set"   and A :: "'b set"   and s :: "nat \<Rightarrow> 'b"   and n :: "nat"   and f :: "nat \<Rightarrow> 'a"   and t :: "nat \<Rightarrow> 'b"   and m :: "nat"   and g :: "nat \<Rightarrow> 'a" assumes "H \<subseteq> carrier M"   and "ideal R A"   and "s \<in> {j. j \<le> n} \<rightarrow> A"   and "f \<in> {j. j \<le> n} \<rightarrow> H"   and "t \<in> {j. j \<le> m} \<rightarrow> A"   and "g \<in> {j. j \<le> m} \<rightarrow> H" shows "l_comb R M n (jointfun n s m t) (jointfun n f m g) = l_comb R M n s f"
distinct_11: fixes x3 :: "'b" shows "Fault x3 \<noteq> Stuck"
SUP_le_iff: fixes f :: "'e \<Rightarrow> 'c \<times> 'd"   and A :: "'e set"   and u :: "'c \<times> 'd" shows "less_eq_prod (Sup_prod (f ` A)) u = (\<forall>i\<in>A. less_eq_prod (f i) u)"
measure_negligible_symdiff: fixes S :: "'b set"   and T :: "'b set" assumes "S \<in> lmeasurable"   and "negligible (S - T \<union> (T - S))" shows "Sigma_Algebra.measure lebesgue T = Sigma_Algebra.measure lebesgue S"
gets_def: fixes v :: "char list"   and e :: "(char list \<Rightarrow> 'a) \<Rightarrow> 'a" shows "VC_KAD.gets v e = {(s, s(v := e s)) |s. True}"
dim_gen_is_basis: fixes A :: "'c set" assumes "finite A"   and "A \<subseteq> carrier V"   and "V.module.gen_set A"   and "card A \<le> V.dim" shows "V.basis A"
reads_def: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem" shows "IFC_def.reads prob = ifc_problem.reads prob"
eq_iff: fixes f :: "'a \<Rightarrow> 'b"   and x :: "'a"   and y :: "'a" assumes "injective f" shows "(f x = f y) = (x = y)"
\<theta>fun_id_1: shows "t \<cdot> \<theta> = t"
t1: shows "local.iso \<lbrace>t1\<rbrace>"
iterateoi_correct: fixes S :: "'s" assumes "invar S" shows "map_iterator_linord (iterateoi S) (\<alpha> S)"
subcomplex_image: shows "order.greater_eq X (g \<turnstile> X) \<and> SimplicialComplex (g \<turnstile> X)"
H_rcoset_reps_H: shows "set H_rcoset_reps \<subseteq> H"
lcs_alt_2: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and lcs :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c"   and x :: "'c"   and b :: "'c"   and y :: "'c" assumes "class.ulcs_powerprod plus minus zero lcs"   and "plus a x = plus b y" shows "lcs x y = minus (plus b y) (lcs_powerprod.gcs plus minus lcs a b)"
int_sdiv_simps_3: fixes a :: "int" shows "a sdiv - 1 = - a"
reachable_target: fixes q :: "'d"   and A :: "('e, 'f, 'd, 'g) FSM_scheme"   and p :: "'d"   and r :: "(('e \<times> 'f) \<times> 'd) list" assumes "q \<in> reachable A p"   and "path A r q" shows "target r q \<in> reachable A p"
length_listset: fixes xi :: "'c list"   and xs :: "'c set list" assumes "xi \<in> listset xs" shows "length xi = length xs"
sublinearI: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "\<And>a b c P Q s. \<lbrakk>sound P; sound Q; 0 \<le> a; 0 \<le> b; 0 \<le> c\<rbrakk> \<Longrightarrow> a * t P s + b * t Q s \<ominus> c \<le> t (\<lambda>s'. a * P s' + b * Q s' \<ominus> c) s" shows "sublinear t"
VC_disjoint_def: fixes v :: "'b V_rec" shows "VC_disjoint v \<equiv> V\<^bsub>v\<^esub> \<inter> C\<^bsub>v\<^esub> = {}"
fg: shows "\<forall>y\<in>topspace Y. f (g y) = y"
simps: fixes f :: "'b \<Rightarrow> 'a option"   and g :: "'b \<Rightarrow> 'a option"   and x :: "'b" shows "(f;;g) x = (if g x = None then f x else g x)"
sort_leq_refl: fixes s :: "String.literal set" assumes "sort_ex cs s" shows "sort_leq cs s s"
\<beta>_boundedness_le: fixes m :: "int"   and x :: "'a" assumes "m \<le> int (k x)"   and "x \<in> X" shows "\<exists>U\<subseteq>\<R>. \<Union> U = {u \<in> V. u x \<le> real_of_int m}"
distinct_50: fixes x51 :: "'c"   and x52 :: "('a, 'b, 'c) Stmt"   and x53 :: "('a, 'b, 'c) Stmt" shows "Stop \<noteq> Stmt.If x51 x52 x53"
last_replicate: fixes n :: "nat"   and x :: "'a" assumes "n \<noteq> 0" shows "last (replicate n x) = x"
assms: fixes r :: "'a state \<times> (v, v) result" shows "evaluate True env st1 e1 r = evaluate True env st2 e2 r"
envObs_def: fixes a :: "'a" shows "envObs a \<equiv> \<lambda>s. (envObsC (es s), ps s a, pubActs s)"
frechet_correctness: fixes I :: "('a, 'b, 'd) interp"   and \<theta> :: "('a, 'd) trm"   and \<nu> :: "(real, 'd) vec" assumes "is_interp I"   and "dfree \<theta>" shows "(sterm_sem I \<theta> has_derivative frechet I \<theta> \<nu>) (at \<nu>)"
sup_aci_2: fixes x :: "'c multiset"   and y :: "'c multiset"   and z :: "'c multiset" shows "x \<union># y \<union># z = x \<union># (y \<union># z)"
pseudonatural_equivalence_axioms: shows "pseudonatural_equivalence V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C F \<Phi>\<^sub>F G \<Phi>\<^sub>G \<tau>\<^sub>0 \<tau>\<^sub>1"
dfs_isNode: fixes S :: "'b"   and xs :: "'a list" assumes "invariant S"   and "list_all isNode xs" shows "set_of (dfs S xs) \<subseteq> {x. isNode x}"
cases: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b list" assumes "class.linorder less_eq less" obtains "x = []"   | xa :: "'b" and ys :: "'b list" where "x = xa # ys"
OUT_k: fixes x :: "'v" shows "d_OUT k x = (if x \<in> RF\<^sup>\<circ> (TER f) then d_OUT h x else d_OUT g x)"
Greatest_le: fixes P :: "nat \<Rightarrow> bool"   and k :: "nat"   and l :: "nat" assumes "P k"   and "\<And>k. P k \<Longrightarrow> k \<le> l" shows "k \<le> (GREATEST k. P k)"
exec1_def: fixes P :: "instr list"   and cf :: "int \<times> (char list \<Rightarrow> int) \<times> int list"   and cf' :: "int \<times> (char list \<Rightarrow> int) \<times> int list" shows "P \<turnstile> cf \<rightarrow> cf' \<equiv> (P !! fst cf, cf) \<mapsto> cf' \<and> 0 \<le> fst cf \<and> fst cf < size P"
simps_2: fixes A :: "specID"   and m :: "Secrets" shows "know A (sKS m) = (ine A (sE m) \<or> sKS m \<in> LocalSecrets A)"
mHom_mem: fixes N :: "('g, 'b, 'h) Module_scheme"   and f :: "'a \<Rightarrow> 'g"   and m :: "'a" assumes "R module N"   and "f \<in> mHom R M N"   and "m \<in> carrier M" shows "f m \<in> carrier N"
conseq12: fixes G :: "'a triple set"   and P' :: "'a assn"   and c :: "com"   and Q' :: "'a assn"   and P :: "'a assn"   and Q :: "'a assn" assumes "G|-{P'}. c .{Q'}"   and "\<forall>Z s. P Z s \<longrightarrow> (\<forall>s'. (\<forall>Z'. P' Z' s \<longrightarrow> Q' Z' s') \<longrightarrow> Q Z s')" shows "G|-{P}. c .{Q}"
wf_\<Gamma>_append: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and rs1 :: "'c rule list"   and rs2 :: "'c rule list" shows "iptables_bigstep_fixedbackground.wf_\<Gamma> \<Gamma> (rs1 @ rs2) = (iptables_bigstep_fixedbackground.wf_\<Gamma> \<Gamma> rs1 \<and> iptables_bigstep_fixedbackground.wf_\<Gamma> \<Gamma> rs2)"
sc_sim_depth: fixes \<Gamma> :: "'b formula list"   and A :: "'b list"   and \<Delta> :: "'b formula list"   and B :: "'b list" assumes "sc \<Gamma> A \<Delta> B = {}" shows "image_mset Atom (mset A) + mset \<Gamma> \<Rightarrow> image_mset Atom (mset B) + mset \<Delta> \<down> sum_list (map size (\<Gamma> @ \<Delta>)) + (if set A \<inter> set B = {} then 0 else 1)"
assms_3: shows "\<not> (\<exists>x\<in>s. f x = (0::'b))"
assms_3: shows "\<delta> ` fv t \<subseteq> range Var"
shortestPath_is_simple: fixes s :: "nat"   and p :: "(nat \<times> nat) list"   and t :: "nat" assumes "isShortestPath s p t" shows "isSimplePath s p t"
union_diff2: fixes A :: "'b set"   and B :: "'b set"   and g :: "'b \<Rightarrow> 'c" assumes "finite A"   and "finite B" shows "sum g (A \<union> B) = sum g (A - B) + sum g (B - A) + sum g (A \<inter> B)"
order_ow_axioms: shows "order_ow U\<^sub>2 le\<^sub>2 ls\<^sub>2"
pw: shows "pairwise (\<lambda>i j. x i \<noteq> x j \<and> y i \<noteq> y j) K"
next_free_eventually: fixes \<phi> :: "'b pltl" shows "next_free (F\<^sub>p \<phi>) = next_free \<phi>"
isSend_def: fixes event :: "('d, 'e, 'f) event" shows "isSend event = (\<exists>x21 x22 x23 x24 x25 x26. event = Send x21 x22 x23 x24 x25 x26)"
IH2': shows "frechet I t2 (fst \<nu>) (snd \<nu>) = frechet I t2 (fst \<nu>') (snd \<nu>')"
ord_adds_term: fixes u :: "'x \<Rightarrow>\<^sub>0 nat"   and v :: "'x \<Rightarrow>\<^sub>0 nat" assumes "local.punit.adds_term u v" shows "u \<preceq> v"
residue_primroot_modulus_mult_coprime_imp_Carmichael_coprime: fixes m :: "nat"   and n :: "nat"   and g :: "nat" assumes "coprime m n"   and "residue_primroot (m * n) g" shows "coprime (Carmichael m) (Carmichael n)"
simps_84: fixes f1 :: "char list \<Rightarrow> 'c trm list \<Rightarrow> 'b"   and f2 :: "'c trm \<Rightarrow> 'c trm \<Rightarrow> 'b"   and f3 :: "'c formula \<Rightarrow> 'b"   and f4 :: "'c formula \<Rightarrow> 'c formula \<Rightarrow> 'b"   and f5 :: "'c formula \<Rightarrow> 'b"   and f6 :: "\<I> \<Rightarrow> 'c formula \<Rightarrow> 'b"   and f7 :: "\<I> \<Rightarrow> 'c formula \<Rightarrow> 'b"   and f8 :: "'c formula \<Rightarrow> \<I> \<Rightarrow> 'c formula \<Rightarrow> 'b"   and f9 :: "'c formula \<Rightarrow> \<I> \<Rightarrow> 'c formula \<Rightarrow> 'b"   and x3 :: "'c formula" shows "(case Neg x3 of Pred x xa \<Rightarrow> f1 x xa | Eq x xa \<Rightarrow> f2 x xa | Neg x \<Rightarrow> f3 x | Or x xa \<Rightarrow> f4 x xa | Exists x \<Rightarrow> f5 x | Prev x xa \<Rightarrow> f6 x xa | Next x xa \<Rightarrow> f7 x xa | Since x xa xb \<Rightarrow> f8 x xa xb | Until x xa xb \<Rightarrow> f9 x xa xb) = f3 x3"
measure_preserving_finite_measure: fixes f :: "'b \<Rightarrow> 'c"   and M :: "'b measure"   and N :: "'c measure" assumes "f \<in> measure_preserving M N" shows "finite_measure M = finite_measure N"
reduce_weight_def: fixes \<Gamma> :: "('k, 'l) web_scheme"   and x :: "'k"   and r :: "real" shows "reduce_weight \<Gamma> x r = \<Gamma>\<lparr>weight := \<lambda>y. weight \<Gamma> y - ennreal (if x = y then r else 0)\<rparr>"
continuous_on_Pair: fixes s :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'd" assumes "continuous_on s f"   and "continuous_on s g" shows "continuous_on s (\<lambda>x. (f x, g x))"
drop_update_swap: fixes m :: "nat"   and n :: "nat"   and xs :: "'b list"   and x :: "'b" assumes "m \<le> n" shows "drop m (xs[n := x]) = (drop m xs)[n - m := x]"
simps_4: fixes f1 :: "'a"   and f2 :: "'b \<Rightarrow> 'b tree list \<Rightarrow> 'a" shows "(case Node of Node \<Rightarrow> f1 | Fork x xa \<Rightarrow> f2 x xa) = f1"
ev_at_HLD_imp_snth: fixes X :: "'a set"   and n :: "nat"   and \<omega> :: "'a stream" assumes "ev_at (HLD X) n \<omega>" shows "\<omega> !! n \<in> X"
coef_hd_gt_0: fixes ground_heads_var :: "'a \<Rightarrow> 'b set"   and gt_sym :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and \<epsilon> :: "nat"   and \<delta> :: "nat"   and extf :: "'b \<Rightarrow> (('b, 'a) tm \<Rightarrow> ('b, 'a) tm \<Rightarrow> bool) \<Rightarrow> ('b, 'a) tm list \<Rightarrow> ('b, 'a) tm list \<Rightarrow> bool"   and arity_sym :: "'b \<Rightarrow> enat"   and arity_var :: "'a \<Rightarrow> enat"   and wt_sym :: "'b \<Rightarrow> hmultiset"   and coef_sym :: "'b \<Rightarrow> nat \<Rightarrow> hmultiset"   and A :: "'a pvar \<Rightarrow> zhmultiset"   and \<zeta> :: "('b, 'a) hd"   and i :: "nat" assumes "tkbo_coefs ground_heads_var gt_sym \<epsilon> \<delta> extf arity_sym arity_var wt_sym coef_sym"   and "tkbo_coefs.legal_zpassign ground_heads_var \<delta> arity_sym wt_sym A" shows "0 < eval_tpoly A (map_tpoly (\<lambda>x. x) zhmset_of (tkbo_coefs.coef_hd coef_sym \<zeta> i))"
one_conv: fixes x :: "'a" shows "1' \<cdot> x ; 1 = 1' \<cdot> x ; x\<^sup>\<smile>"
lm099: fixes x :: "'b"   and X :: "'b set"   and Y :: "'b set" assumes "x \<in> X" shows "1 \<in> (X <|| Y) `` {x}"
not_irreducibleE: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and x :: "'a" assumes "class.comm_semiring_1 times one plus zero"   and "\<not> comm_semiring_1.irreducible times one zero x" obtains "x = zero"   | "dvd.dvd times x one"   | a :: "'a" and b :: "'a" where "x = times a b" and "\<not> dvd.dvd times a one" and "\<not> dvd.dvd times b one"
insertI: fixes A :: "'a::type set"   and a :: "'a::type" assumes "finite (A::'a::type set)" shows "finite (insert (a::'a::type) (A::'a::type set))"
Abs_inverse: fixes m :: "int" assumes "m \<in> {0..<int CARD('a bit0)}" shows "Rep_bit0 (Abs_bit0 m) = m"
enfoldli_def: fixes l :: "'f list"   and c :: "'e \<Rightarrow> bool"   and f :: "'f \<Rightarrow> 'e \<Rightarrow> ('d + 'e) nres"   and s :: "'e" shows "enfoldli l c f s \<equiv> nfoldli l (\<lambda>xa. case xa of Inl e \<Rightarrow> False | Inr x \<Rightarrow> c x) (\<lambda>x s. ASSERT (\<not> isl s) \<bind> (\<lambda>_. Let (projr s) (f x))) (Inr s)"
partition_filter1: fixes P :: "'c \<Rightarrow> bool"   and xs :: "'c list" shows "fst (partition P xs) = filter P xs"
graph_restrict_aimpl: fixes Vl :: "'a set"   and Vr :: "'a set"   and E :: "('a \<times> 'a) set" shows "op_graph_restrict Vl Vr E = E_of_succ (\<lambda>v. if v \<in> Vl then {x \<in> E `` {v}. x \<in> Vr} else {})"