hcju/isabelleps huggingface dataset:
eval_Pr_Suc_converg_2: fixes n :: "nat"   and f :: "recf"   and g :: "recf"   and xs :: "nat list"   and x :: "nat" assumes "recfn (Suc n) (Pr n f g)"   and "n = length xs"   and "eval (Pr n f g) (Suc x # xs) \<down>" shows "eval (Pr n f g) (Suc x # xs) = eval g (x # the (eval (Pr n f g) (x # xs)) # xs)"
wfp_onE_min: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and A :: "'c set"   and x :: "'c"   and Q :: "'c set" assumes "wfp_on r A"   and "x \<in> Q"   and "Q \<subseteq> A" obtains z :: "'c" where "z \<in> Q" and "\<And>y. r y z \<Longrightarrow> y \<notin> Q"
ord_dvd_group_order: fixes G :: "('c, 'd) monoid_scheme"   and a :: "'c" assumes "Group.group G"   and "a \<in> carrier G" shows "group.ord G a dvd order G"
Var: fixes \<sigma> :: "'f \<Rightarrow> ('d, 'e) Term.term option"   and x :: "'f"   and t :: "('d, 'e) Term.term"   and P :: "(('d, 'f) Term.term \<times> ('d, 'e) Term.term) multiset" assumes "\<sigma> x = None \<or> \<sigma> x = Some t" shows "MATCH1 (P + {#(Var x, t)#}, \<sigma>) (P, \<sigma>(x \<mapsto> t))"
monoI: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b" assumes "class.order less_eq less"   and "\<And>x y. less_eq x y \<Longrightarrow> f x \<le> f y" shows "order.mono less_eq f"
fps_shift_def: fixes n :: "nat"   and f :: "'a fps" shows "fps_shift n f = Abs_fps (\<lambda>i. fps_nth f (i + n))"
risk_neutral_iff: fixes N :: "bool stream measure"   and q :: "real" assumes "N = bernoulli_stream q"   and "0 < q"   and "q < 1" shows "risk_neutral_prob N = (q = (1 + r - d) / (u - d))"
sign_r_pos_smult: fixes c :: "real"   and p :: "real poly"   and x :: "real" assumes "c \<noteq> 0"   and "p \<noteq> 0" shows "sign_r_pos (smult c p) x = (if 0 < c then sign_r_pos p x else \<not> sign_r_pos p x)"
int_cases4: fixes m :: "int" obtains n :: "nat" where "m = int n__"   | n :: "nat" where "0 < n" and "m = - int n"
cfs_smult: fixes R :: "('c, 'd) ring_scheme"   and a :: "'c"   and p :: "nat \<Rightarrow> 'c"   and n :: "nat" assumes "UP_ring R"   and "a \<in> carrier R"   and "p \<in> carrier (UP R)" shows "(a \<odot>\<^bsub>UP R\<^esub> p) n = a \<otimes>\<^bsub>R\<^esub> p n"
wp_omega_fix: fixes x :: "'b"   and p :: "'b Assertion" shows "wp (x ^ \<omega>) p = wp x (wp (x ^ \<omega>) p) \<sqinter> p"
fw_impl'_def: fixes n :: "nat"   and M :: "nat \<times> nat \<Rightarrow> 'b" shows "fw_impl' n M = fw' M n n"
freshSets_1: fixes C :: "'d" shows "[] \<sharp>* C"
that: fixes \<pi> :: "nat \<Rightarrow> 'n"   and n :: "nat"   and k :: "nat" assumes "is_path \<pi>"   and "\<pi> 0 = x"   and "\<pi> n = local.return"   and "\<pi> k = y"   and "k \<le> n" shows "thesis"
master1_bigo: fixes p' :: "real" assumes "g \<in> O(\<lambda>x. real x powr p')"   and "1 < (\<Sum>i<k. as ! i * bs ! i powr p')" shows "f \<in> O(\<lambda>x. real x powr p)"
normalized_prod_msetI: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and A :: "'a multiset" assumes "class.normalization_semidom_multiplicative divide plus minus zero times one unit_factor normalize"   and "\<And>a. a \<in># A \<Longrightarrow> normalize a = a" shows "normalize (comm_monoid_mult.prod_mset times one A) = comm_monoid_mult.prod_mset times one A"
Nil_eq_concat_conv: fixes xss :: "'d list list" shows "([] = concat xss) = (\<forall>xs\<in>set xss. xs = [])"
cf_z: fixes n :: "nat" shows "dist (z n) (z (Suc n)) \<le> c ^ n * d"
aux_lemma_lose_1: fixes d :: "nat"   and k :: "nat"   and lose :: "nat \<Rightarrow> 'a"   and t :: "nat"   and x :: "'a"   and ka :: "nat" assumes "\<forall>j\<le>2 * d + (4 + k). lose (t + j) = x"   and "ka \<le> Suc d" shows "lose (Suc (Suc (t + k + ka))) = x"
noNeg_mapNegPos_helper: fixes ls :: "'b negation_type list"   and C :: "'b \<Rightarrow> 'c" assumes "getNeg ls = []" shows "map (Pos \<circ> C) (getPos ls) = NegPos_map C ls"
supp_Act: fixes \<alpha> :: "'c"   and x :: "('a, 'b, 'c) formula" shows "supp (Act \<alpha> x) = supp \<alpha> \<union> supp x - bn \<alpha>"
cases: fixes a :: "'a"   and \<delta> :: "('a, 'b) ta_rule set" assumes "a \<in> b_accessible \<delta>" obtains q :: "'a" and l :: "'b" and qs :: "'a list" where "a = q" and "q \<rightarrow> l qs \<in> \<delta>" and "\<And>x. x \<in> set qs \<Longrightarrow> x \<in> b_accessible \<delta>"
bit_horner_sum_bit_iff: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and division_segment :: "'b \<Rightarrow> 'b"   and bit :: "'b \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and drop_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and take_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and bs :: "bool list"   and n :: "nat" assumes "class.unique_euclidean_semiring_with_bit_shifts divide plus minus zero times one modulo euclidean_size division_segment bit push_bit drop_bit take_bit" shows "bit (comm_semiring_0.horner_sum plus zero times (zero_neq_one.of_bool one zero) (numeral.numeral one plus (num.Bit0 num.One)) bs) n = (n < length bs \<and> bs ! n)"
phi_def: fixes \<alpha>n :: "'b \<Rightarrow> 'c list"   and defs :: "'b \<Rightarrow> 'c \<Rightarrow> 'a set"   and phis :: "'b \<Rightarrow> 'c \<times> 'a \<Rightarrow> 'a list option"   and g :: "'b"   and v :: "'a" shows "CFG_SSA_wf_base.phi \<alpha>n defs phis g v = phis g (CFG_SSA_wf_base.defNode \<alpha>n defs phis g v, v)"
card_of_Un_ordLess_infinite_Field: fixes r :: "('a \<times> 'a) set"   and A :: "'b set"   and B :: "'b set" assumes "infinite (Field r)"   and "Card_order r"   and "|A| <o r"   and "|B| <o r" shows "|A \<union> B| <o r"
subresultant_def: fixes J :: "nat"   and F :: "'b poly"   and G :: "'b poly" shows "subresultant J F G = det (subresultant_mat J F G)"
not_le: fixes y :: "'a"   and x :: "'a" shows "(\<not> y \<le> x) = (x < y)"
prems_1: shows "\<forall>a\<in>insert x A. a dvd c"
HFinite_mult: fixes x :: "'a star"   and y :: "'a star" assumes "x \<in> HFinite"   and "y \<in> HFinite" shows "x * y \<in> HFinite"
elims_3: fixes x :: "formula" assumes "\<not> future_bounded x" obtains p :: "String.literal" and \<phi> :: "formula" and \<psi> :: "formula" where "x = formula.Let p \<phi>'__ \<psi>__" and "\<not> (future_bounded \<phi>'__ \<and> future_bounded \<psi>__)"   | \<phi> :: "formula" where "x = Neg \<phi>''__" and "\<not> future_bounded \<phi>''__"   | \<phi> :: "formula" and \<psi> :: "formula" where "x = Or \<phi>'''__ \<psi>'__" and "\<not> (future_bounded \<phi>'''__ \<and> future_bounded \<psi>'__)"   | \<phi> :: "formula" and \<psi> :: "formula" where "x = And \<phi>''''__ \<psi>''__" and "\<not> (future_bounded \<phi>''''__ \<and> future_bounded \<psi>''__)"   | l :: "formula list" where "x = Ands l" and "\<not> list_all future_bounded l"   | \<phi> :: "formula" where "x = Exists \<phi>'''''__" and "\<not> future_bounded \<phi>'''''__"   | y :: "nat" and \<omega> :: "agg_type \<times> event_data" and b :: "nat" and f :: "trm" and \<phi> :: "formula" where "x = Agg y \<omega> b f \<phi>''''''__" and "\<not> future_bounded \<phi>''''''__"   | I :: "\<I>" and \<phi> :: "formula" where "x = Prev I__ \<phi>'''''''__" and "\<not> future_bounded \<phi>'''''''__"   | I :: "\<I>" and \<phi> :: "formula" where "x = Next Ia__ \<phi>''''''''__" and "\<not> future_bounded \<phi>''''''''__"   | \<phi> :: "formula" and I :: "\<I>" and \<psi> :: "formula" where "x = Since \<phi>'''''''''__ Ib__ \<psi>'''__" and "\<not> (future_bounded \<phi>'''''''''__ \<and> future_bounded \<psi>'''__)"   | \<phi> :: "formula" and I :: "\<I>" and \<psi> :: "formula" where "x = Until \<phi> Ic__ \<psi>" and "\<not> (future_bounded \<phi> \<and> future_bounded \<psi> \<and> right Ic__ \<noteq> \<infinity>)"   | I :: "\<I>" and r :: "formula Regex.regex" where "x = MatchP Id__ r__" and "\<not> regex.pred_regex future_bounded r__"   | I :: "\<I>" and r :: "formula Regex.regex" where "x = MatchF I r" and "\<not> (regex.pred_regex future_bounded r \<and> right I \<noteq> \<infinity>)"
simps_3: fixes \<psi>\<^sub>1 :: "'b ltln"   and \<psi>\<^sub>2 :: "'b ltln"   and X :: "'b ltln set" shows "(\<psi>\<^sub>1 or\<^sub>n \<psi>\<^sub>2)[X]\<^sub>\<nu> = \<psi>\<^sub>1[X]\<^sub>\<nu> or\<^sub>n \<psi>\<^sub>2[X]\<^sub>\<nu>"
bempty_eqvt: fixes p :: "perm" shows "p \<bullet> bempty = bempty"
yoneda_lemma_for_hom_axioms_def: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a' :: "'a" shows "yoneda_lemma_for_hom_axioms C a' \<equiv> partial_magma.ide C a'"
simps_4: fixes x2 :: "'m"   and x1 :: "nat" shows "shadow_root_ptr.Ext x2 \<noteq> shadow_root_ptr.Ref x1"
wf_Ord_ordertype: fixes r :: "('a \<times> 'a) set"   and A :: "'a set" assumes "wf r"   and "trans r" shows "Ord (ordertype A r)"
width_def: fixes G :: "('e, 'f) Graph_scheme"   and T :: "('g, 'h) Graph_scheme"   and bag :: "'g \<Rightarrow> 'e set" assumes "TreeDecomposition G T bag" shows "TreeDecomposition.width T bag \<equiv> if V\<^bsub>T\<^esub> = {} then 0 else TreeDecomposition.max_bag_card T bag - 1"
hyps_3: shows "(v, M, P) \<in> S_cs fg 1"
ZObis_matchC_ZO_rev: fixes s :: "('test, 'atom) com"   and t :: "('test, 'atom) com" assumes "s \<approx>01 t" shows "matchC_ZO ZObis t s"
Zero_not_eq_j1: shows "quiver.Zero \<noteq> parallel_pair.j1"
calc_ideal_point1_mix: fixes z1 :: "complex"   and z2 :: "complex"   and A :: "complex"   and B :: "complex" assumes "(z1, z2) = calc_ideal_point1_cvec A B"   and "is_real A"   and "(cmod A)\<^sup>2 < (cmod B)\<^sup>2" shows "B * cnj z1 * z2 + cnj B * z1 * cnj z2 = - 2 * A * (B * cnj B)\<^sup>2"
4: shows "rat_poly_plus (rat_poly_times A (rat_poly_times (B - rat_poly_times (rat_poly_times A A) A) A)) (rat_poly_times (B - rat_poly_times (rat_poly_times A A) A) (rat_poly_times B (B - rat_poly_times (rat_poly_times A A) A))) = B ^ 3 - A"
godel_code_in: fixes i :: "nat"   and nl :: "nat list" assumes "i < length nl" shows "nl ! i \<in> {u. Pi (Suc i) ^ u dvd godel_code' nl (Suc 0)}"
l_set_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_set_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M ShadowRootClass.type_wf set_shadow_root set_shadow_root_locs"
step'_Com: fixes S :: "'av Abs_State.st option"   and X :: "char list set"   and c :: "'av Abs_State.st option acom" assumes "domo S \<subseteq> X"   and "vars (strip c) \<subseteq> X"   and "c \<in> Com X" shows "local.step' S c \<in> Com X"
tllist_of_llist_inject: fixes b :: "'b"   and xs :: "'a llist"   and c :: "'b"   and ys :: "'a llist" shows "(tllist_of_llist b xs = tllist_of_llist c ys) = (xs = ys \<and> (lfinite ys \<longrightarrow> b = c))"
is_sublist_distinct: fixes as :: "'a list"   and vs :: "'a list" assumes "is_sublist as vs"   and "distinct vs" shows "distinct as"
intro: fixes x :: "'b \<Longrightarrow> 'c" assumes "weak_lens x"   and "ief_lens_axioms x" shows "ief_lens x"
decomp: shows "{0..n + 1} = {0} \<union> {n + 1} \<union> {1..n}"
HCollect_hempty: fixes P :: "hf \<Rightarrow> bool" shows "HCollect P 0 = 0"
Units_l_inv: fixes x :: "'a" assumes "x \<in> Units R" shows "inv x \<otimes> x = \<one>"
fixp_induct: fixes P :: "('d \<Rightarrow> (real \<times> 'd) stream \<Rightarrow> real \<Rightarrow> 'd) \<Rightarrow> bool" assumes "tailrec.admissible (\<lambda>trace_at. P (curry (curry trace_at)))"   and "P (\<lambda>trace_at s \<omega>. undefined)"   and "\<And>j. P j \<Longrightarrow> P (\<lambda>a aa aaa. case aa of (t', s') ## \<omega> \<Rightarrow> if t' \<le> aaa then j s' \<omega> aaa else a)" shows "P trace_at"
wt_Unifier_if_Unifier: fixes arity :: "'a \<Rightarrow> nat"   and public :: "'a \<Rightarrow> bool"   and Ana :: "('a, 'b) Term.term \<Rightarrow> ('a, 'b) Term.term list \<times> ('a, 'b) Term.term list"   and \<Gamma> :: "('a, 'b) Term.term \<Rightarrow> ('a, 'c) Term.term"   and s :: "('a, 'b) Term.term"   and t :: "('a, 'b) Term.term"   and \<delta> :: "'b \<Rightarrow> ('a, 'b) Term.term" assumes "typed_model arity public Ana \<Gamma>"   and "intruder_model.wf\<^sub>t\<^sub>r\<^sub>m arity s"   and "intruder_model.wf\<^sub>t\<^sub>r\<^sub>m arity t"   and "\<Gamma> s = \<Gamma> t"   and "Unifier \<delta> s t" shows "\<exists>\<theta>. Unifier \<theta> s t \<and> typed_model.wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<Gamma> \<theta> \<and> (\<forall>t\<in>subst_range \<theta>. intruder_model.wf\<^sub>t\<^sub>r\<^sub>m arity t)"
mark_guards_simps_3: fixes f :: "'f"   and r :: "('d \<times> 'd) set" shows "mark_guards f (Spec r) = Spec r"
Well_Scoped_Graph_axioms: shows "Well_Scoped_Graph nodes inPorts outPorts vertices nodeOf edges hyps"
analz_NI_I_K_synth_analz_NI_E: fixes K :: "msg set"   and Bad :: "agent set"   and NI :: "msg set"   and I :: "msg set" assumes "Keys_bad K Bad"   and "NI \<subseteq> payload"   and "K \<subseteq> range LtK"   and "I \<subseteq> valid" shows "analz (NI \<union> I \<union> K \<union> Tags) \<subseteq> synth (analz (extr Bad NI (local.abs I))) \<union> - payload"
godel_code_eq_1: fixes nl :: "nat list" shows "(godel_code nl = 1) = (nl = [])"
flip_altsublist_adjacent_sum_list: fixes ss :: "'w list"   and ts :: "'w list" assumes "ss \<in> lists S"   and "flip_altsublist_adjacent ss ts" shows "sum_list ts = sum_list ss"
simulation_axioms: shows "simulation bs\<^sub>0 blstep cs\<^sub>0 clstep sim_rel2"
wf_config_def: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and E :: "char list \<Rightarrow> ty option"   and s :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and e :: "char list exp"   and T :: "ty" shows "P,E,s \<turnstile> e : T \<surd> \<equiv> P,E \<turnstile> s \<surd> \<and> P,E,hp s \<turnstile> e : T"
monoid_homomorphism_axioms: shows "monoid_homomorphism Class M (\<cdot>) \<one> Partition ([\<cdot>]) (Class \<one>)"
gen_def: fixes P :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and cs :: "'e list"   and w :: "'f" shows "gen P cs w \<equiv> \<forall>c\<in>set cs. P c w"
SN_on_weakening: fixes R1 :: "('a \<times> 'a) set"   and A :: "'a set"   and R2 :: "('a \<times> 'a) set" assumes "SN_on R1 A" shows "SN_on (R1 \<inter> R2) A"
symp_conv_conversep_eq: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "symp r = (r\<inverse>\<inverse> = r)"
domain_lsc_hull_between: fixes f :: "'b \<Rightarrow> ereal" shows "domain f \<subseteq> domain (lsc_hull f) \<and> domain (lsc_hull f) \<subseteq> closure (domain f)"
int_and_numerals_2: fixes x :: "num"   and y :: "num" shows "numeral (num.Bit0 x) AND numeral (num.Bit1 y) = 2 * (numeral x AND numeral y)"
axioms_1: fixes point_set :: "'c set"   and block_collection :: "'c set multiset" assumes "finite_incidence_system point_set block_collection" shows "incidence_system point_set block_collection"
map\<^sub>0_def: fixes a :: "'a" shows "F.map\<^sub>0 a \<equiv> src\<^sub>D (F a)"
IH: fixes f :: "'d \<Rightarrow> 'a"   and g :: "'d \<Rightarrow> 'b" assumes "higher_differentiable_on S f n"   and "higher_differentiable_on S g n"   and "open S" shows "higher_differentiable_on S (\<lambda>x. f x ** g x) n"
lambda_system_Compl: fixes x :: "'a set"   and f :: "'a set \<Rightarrow> ennreal" assumes "x \<in> lambda_system \<Omega> M f" shows "\<Omega> - x \<in> lambda_system \<Omega> M f"
bilinear_r_lin: fixes M1 :: "('g, 'a, 'h) Module_scheme"   and M2 :: "('i, 'a, 'h) Module_scheme"   and N :: "('j, 'a, 'h) Module_scheme"   and m1 :: "'g"   and m2 :: "'i"   and r :: "'a"   and f :: "'g \<times> 'i \<Rightarrow> 'j" assumes "R module M1"   and "R module M2"   and "R module N"   and "m1 \<in> carrier M1"   and "m2 \<in> carrier M2"   and "r \<in> carrier R"   and "bilinear_map f R M1 M2 N" shows "f (m1, r \<cdot>\<^sub>s\<^bsub>M2\<^esub> m2) = r \<cdot>\<^sub>s\<^bsub>N\<^esub> f (m1, m2)"
correctCompositionDiffLevelsS4: shows "correctCompositionDiffLevels sS4"
triangle_G: shows "Adj'.G\<epsilon>o\<eta>G.map = G'"
at_eq_bot_iff: fixes a :: "'a" shows "(at a = bot) = open {a}"
lm119: fixes l :: "'a list"   and X :: "'a set" shows "(!) l ` set (filterpositions2 (\<lambda>x. x \<in> X) l) \<subseteq> X \<inter> set l"
List_cons: fixes p :: "ref"   and h :: "ref \<Rightarrow> ref"   and a :: "ref"   and ps :: "ref list" shows "List p h (a # ps) = (p = a \<and> p \<noteq> Null \<and> List (h p) h ps)"
sup_assoc: fixes x :: "'b"   and y :: "'b"   and z :: "'b" shows "x + y + z = x + (y + z)"
dirichlet_prod_altdef2: fixes f :: "nat \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> 'a" shows "dirichlet_prod f g = (\<lambda>n. \<Sum>(r, d)\<in>{(r, d). r * d = n}. f r * g d)"
Posix1a: fixes s :: "'b list"   and r :: "'b rexp"   and v :: "'b val" assumes "s \<in> r \<rightarrow> v" shows "\<turnstile> v : r"
set_s_A: fixes n :: "nat" shows "set (s_A n) = set s0"
lookup_eq_zero_in_keys_contradict: fixes f :: "'c \<Rightarrow>\<^sub>0 'b"   and k :: "'c" assumes "poly_mapping.lookup f k = (0::'b)" shows "k \<notin> Poly_Mapping.keys f"
abs_Gromov_completion_in_Gromov_boundary: fixes u :: "nat \<Rightarrow> 'c" assumes "Gromov_converging_at_boundary u" shows "abs_Gromov_completion u \<in> Gromov_boundary"
Keys_Un: fixes A :: "('b \<Rightarrow>\<^sub>0 'c) set"   and B :: "('b \<Rightarrow>\<^sub>0 'c) set" shows "Keys (A \<union> B) = Keys A \<union> Keys B"
unit_closed: shows "cq.one_rel \<in> cq.carrier_quotient_ring"
lt_in_keys: fixes p :: "'t \<Rightarrow>\<^sub>0 'e" assumes "p \<noteq> 0" shows "lt p \<in> keys p"
DiaP': fixes a :: "'c"   and ps :: "('b, 'c) fm list"   and branch :: "(('b, 'c) fm list \<times> 'c) list"   and p :: "('b, 'c) fm"   and i :: "'c"   and A :: "'c set"   and n :: "nat" assumes "(\<^bold>\<diamond> p) at a in' set ((ps, a) # branch)"   and "i \<notin> A \<union> branch_nominals ((ps, a) # branch)"   and "\<nexists>a. p = Nom a"   and "\<not> witnessed p a ((ps, a) # branch)"   and "A, n \<turnstile> ((\<^bold>@ i p) # (\<^bold>\<diamond> Nom i) # ps, a) # branch" shows "A, n \<turnstile> (ps, a) # branch"
bounded_iff: fixes b :: "'a"   and c :: "'a"   and a :: "'a" shows "(max b c \<le> a) = (b \<le> a \<and> c \<le> a)"
closed_sequential_limits: fixes S :: "'c set" shows "closed S = (\<forall>x l. (\<forall>n. x n \<in> S) \<and> x \<longlonglongrightarrow> l \<longrightarrow> l \<in> S)"
maximalideal_prime: fixes I :: "'a set" assumes "maximalideal I R" shows "primeideal I R"
retraction_is_epi: fixes g :: "'d" assumes "D.retraction g" shows "D.epi g"
inject_2: fixes x2 :: "freeRes"   and x1 :: "pi"   and y2 :: "freeRes"   and y1 :: "pi" shows "(x2 \<prec> x1 = y2 \<prec> y1) = (x2 = y2 \<and> x1 = y1)"
pcr_cr_eq: shows "pcr_tllist (=) (=) = cr_tllist"
len_gt_0: fixes len_of :: "'c itself \<Rightarrow> nat" assumes "class.len len_of" shows "0 < len_of TYPE('c)"
convex_hull_def: fixes Vs :: "'a vec set" shows "n.convex_hull Vs = {x. \<exists>Ws c. finite Ws \<and> Ws \<subseteq> Vs \<and> n.convex_lincomb c Ws x}"
sz: shows "\<forall>i\<le>n. s i = (0::'a)"
substE_substE: fixes i :: "nat"   and j :: "nat"   and \<Gamma> :: "binding list"   and V :: "POPLmarkRecord.type"   and U :: "POPLmarkRecord.type" assumes "i \<le> j" shows "\<Gamma>[Suc j \<mapsto>\<^sub>\<tau> V]\<^sub>e[i \<mapsto>\<^sub>\<tau> U[j - i \<mapsto>\<^sub>\<tau> V]\<^sub>\<tau>]\<^sub>e = \<Gamma>[i \<mapsto>\<^sub>\<tau> U]\<^sub>e[j \<mapsto>\<^sub>\<tau> V]\<^sub>e"
has_next_filter_generator_induct: fixes g :: "('a, 'c) generator"   and s :: "'c"   and Q :: "'c \<Rightarrow> bool" assumes "list.has_next (local.filter_generator g) s"   and "\<And>s. \<lbrakk>list.has_next g s; P (fst (list.next g s))\<rbrakk> \<Longrightarrow> Q s"   and "\<And>s. \<lbrakk>list.has_next g s; \<not> P (fst (list.next g s)); Q (snd (list.next g s))\<rbrakk> \<Longrightarrow> Q s" shows "Q s"
correct_def: fixes key_gen :: "('b \<times> 'c) spmf"   and commit :: "'b \<Rightarrow> 'd \<Rightarrow> ('e \<times> 'f) spmf"   and verify :: "'c \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and valid_msg :: "'d \<Rightarrow> bool" shows "abstract_commitment.correct key_gen commit verify valid_msg \<equiv> \<forall>m. valid_msg m \<longrightarrow> spmf (abstract_commitment.correct_game key_gen commit verify m) True = 1"
dist_fs_finite: fixes z1 :: "complex"   and z2 :: "complex" shows "dist_fs (of_complex z1) (of_complex z2) = 2 * cmod (z1 - z2) / (sqrt (1 + (cmod z1)\<^sup>2) * sqrt (1 + (cmod z2)\<^sup>2))"
has_def: fixes n :: "nat"   and S :: "'b set" shows "has n S \<equiv> \<exists>xs. hasw xs S \<and> length xs = n"
finfun_upd_apply_code: fixes f :: "'b \<Rightarrow>f 'a"   and a :: "'b"   and b :: "'a"   and a' :: "'b" shows "finfun_update_code f a b $ a' = (if a = a' then b else f $ a')"
notstutstep_nextsuffix1: fixes s :: "nat \<Rightarrow> 'a"   and n :: "nat" assumes "\<not> stutstep s n" shows "nextsuffix (s |\<^sub>s n) = s |\<^sub>s Suc n"
cases: fixes x :: "('x \<Rightarrow>\<^sub>0 nat) \<times> 'x set \<times> ('x \<Rightarrow>\<^sub>0 nat) set" obtains t :: "'x \<Rightarrow>\<^sub>0 nat" and U :: "'x set" and S :: "('x \<Rightarrow>\<^sub>0 nat) set" where "x = (t, U, S)"
shift_snth_ge: fixes xs :: "'b list"   and i :: "nat"   and \<sigma> :: "nat \<Rightarrow> 'b" assumes "length xs \<le> i" shows "(xs @- \<sigma>) i = \<sigma> (i - length xs)"
abs_eq: fixes xa :: "'a \<Rightarrow> 'b"   and x :: "'a \<Rightarrow> 'b" assumes "eq_onp (\<lambda>x. x \<in> bcontfun) xa xa"   and "eq_onp (\<lambda>x. x \<in> bcontfun) x x" shows "dist (Bcontfun xa) (Bcontfun x) = (SUP x. dist (xa x) (x x))"
S1: shows "\<forall>y\<in>S. {x, y} \<notin> E"
subset: fixes r :: "('a \<times> 'a) set"   and B :: "'a set"   and A :: "'a set" assumes "MaxR r"   and "set_option (MaxR.MaxR_opt r B) \<subseteq> A"   and "A \<subseteq> B" shows "MaxR.MaxR_opt r B = MaxR.MaxR_opt r A"
entailsI: fixes P :: "'a \<Rightarrow> real"   and Q :: "'a \<Rightarrow> real" assumes "\<And>s. P s \<le> Q s" shows "P \<tturnstile> Q"
case_x_in_range: shows "\<forall>x\<in>cbox a b. integral (cbox (g1 x) (g2 x)) (\<lambda>y. g (x, y)) = integral UNIV (\<lambda>y. f (x, y))"
ln_aform_err_def: fixes p :: "nat"   and X :: "(real \<times> real pdevs) \<times> real" shows "ln_aform_err p X = (let ivl = ivl_of_aform_err p X; l = lower ivl in if 0 < l then min_range_mono p Ln inverse l (upper ivl) X else None)"
magQ_def: fixes x :: "'e['f, 'g]" shows "\<lbrakk>x\<rbrakk>\<^sub>Q = mag (fromQ x)"
rel_gpv_conv_rel_gpv'': fixes A :: "'g \<Rightarrow> 'j \<Rightarrow> bool"   and C :: "'h \<Rightarrow> 'k \<Rightarrow> bool" shows "rel_gpv A C = rel_gpv'' A C (=)"
the_inv_into_f_f: fixes f :: "'d \<Rightarrow> 'e"   and A :: "'d set"   and x :: "'d" assumes "inj_on f A"   and "x \<in> A" shows "the_inv_into A f (f x) = x"
tdev': fixes x :: "'a pdevs"   and p :: "nat" shows "tdev x \<le> tdev' p x"
atLeastLessThanPlusOne_atLeastAtMost_integer: fixes l :: "integer"   and u :: "integer" shows "{l..<u + 1} = {l..u}"
bl_not_bin: fixes n :: "nat"   and w :: "int" shows "map Not (bin_to_bl n w) = bin_to_bl n (NOT w)"
all_left_def: shows "local.all_left \<equiv> nfoldli [0..<length xs] (\<lambda>_. True) (\<lambda>i (stk, lf). ASSERT (set stk \<subseteq> {0..<length xs}) \<bind> (\<lambda>_. let stk = local.pop stk (xs ! i) in ASSERT (stk = local.der_stack i) \<bind> (\<lambda>_. ASSERT (i < length lf) \<bind> (\<lambda>_. if stk = [] then let lf = lf[i := None] in RETURN (i # stk, lf) else let lf = lf[i := Some (hd stk)] in RETURN (i # stk, lf))))) ([], replicate (length xs) None) \<bind> (\<lambda>(uu_, lf). RETURN lf)"
IH_3: assumes "\<langle>c: t \<doteq> s\<rangle> \<in> set A" shows "\<langle>c: (t \<cdot> \<theta>) \<doteq> s \<cdot> \<theta>\<rangle> \<in> set (A \<cdot>\<^sub>s\<^sub>s\<^sub>t \<theta>)"