hcju/isabelleps huggingface dataset:
simps_67: fixes f1 :: "'b \<Rightarrow> 'a"   and f2 :: "nat set \<Rightarrow> 'b \<Rightarrow> 'a"   and f3 :: "nat \<Rightarrow> 'b \<Rightarrow> 'a"   and f4 :: "nat \<Rightarrow> 'a"   and f5 :: "'b \<Rightarrow> 'a"   and f6 :: "nat \<Rightarrow> 'a"   and f7 :: "'b \<Rightarrow> 'a"   and f8 :: "'a"   and x4 :: "nat" shows "(case \<not>unicast x4 of broadcast x \<Rightarrow> f1 x | groupcast x xa \<Rightarrow> f2 x xa | unicast x xa \<Rightarrow> f3 x xa | \<not>unicast x \<Rightarrow> f4 x | send x \<Rightarrow> f5 x | deliver x \<Rightarrow> f6 x | receive x \<Rightarrow> f7 x | \<tau>\<^sub>s \<Rightarrow> f8) = f4 x4"
nsgPr1: fixes N :: "'a set"   and a :: "'a"   and h :: "'a" assumes "G \<triangleright> N"   and "a \<in> carrier G"   and "h \<in> N" shows "a \<cdot> (h \<cdot> \<rho> a) \<in> N"
lf: shows "0 < \<bar>lead_coeff f\<bar>"
dom_subdist: fixes x :: "'a"   and y :: "'a" shows "d x \<le> d (x + y)"
classical_operator_mult: fixes \<pi> :: "'c \<Rightarrow> 'd option"   and \<rho> :: "'e \<Rightarrow> 'c option" assumes "classical_operator_exists \<pi>"   and "classical_operator_exists \<rho>" shows "classical_operator \<pi> o\<^sub>C\<^sub>L classical_operator \<rho> = classical_operator (\<pi> \<circ>\<^sub>m \<rho>)"
distinct_33: fixes x3 :: "nat"   and x71 :: "msg"   and x72 :: "msg" shows "Nonce x3 \<noteq> \<lbrace>x71, x72\<rbrace>"
hyps_2: fixes \<sigma> :: "'a OCL_Types.type" assumes "\<Gamma> \<turnstile>\<^sub>E expr : \<sigma>" shows "\<tau> = \<sigma>"
relation_of_CSP1: fixes x :: "('c, 'd) action" shows "action.relation_of x is CSP1 healthy"
a_inv_closed: fixes x :: "'a list" assumes "x \<in> carrier (K [X])" shows "\<ominus>\<^bsub>K [X]\<^esub> x \<in> carrier (K [X])"
limB: shows "(sum f \<longlongrightarrow> limB) (finite_subsets_at_top B)"
vconn_implies_hpar_1: fixes \<mu> :: "'a"   and f :: "'a"   and g :: "'a" assumes "\<guillemotleft>\<mu> : f \<rightarrow>\<^sub>B g\<guillemotright>" shows "src\<^sub>B \<mu> = src\<^sub>B f"
wf_MainC: shows "wf_cdecl tprg (Main, MainCl)"
remove_nth_def: fixes n :: "nat"   and xs :: "'c list" shows "remove_nth n xs \<equiv> take n xs @ drop (Suc n) xs"
add_lessD2: fixes k :: "nat"   and m :: "nat"   and n :: "nat" assumes "k + m < n" shows "m < n"
last_map: fixes xs :: "'g list"   and f :: "'g \<Rightarrow> 'h" assumes "xs \<noteq> []" shows "last (map f xs) = f (last xs)"
BOLZANO: fixes a :: "real"   and b :: "real"   and P :: "real \<Rightarrow> real \<Rightarrow> bool" assumes "a \<le> b"   and "\<And>a b c. \<lbrakk>P a b; P b c; a \<le> b; b \<le> c\<rbrakk> \<Longrightarrow> P a c"   and "\<And>x. \<exists>d>0. \<forall>a b. a \<le> x \<and> x \<le> b \<and> b - a < d \<longrightarrow> P a b" shows "P a b"
wt_jvm_progD: fixes \<Phi> :: "char list \<Rightarrow> char list \<Rightarrow> (ty list \<times> ty err list) option list"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> nat \<times> nat \<times> instr list \<times> (nat \<times> nat \<times> char list \<times> nat \<times> nat) list) list) list" assumes "wf_jvm_prog\<^bsub>\<Phi>\<^esub> P" shows "\<exists>wt. wf_prog wt P"
Dom_preserves_Diag: fixes t :: "'a local.term" assumes "Diag t" shows "Diag (Dom t)"
I_lift_dnf_qe: fixes qe :: "'a list \<Rightarrow> 'a fm"   and \<phi> :: "'a fm"   and xs :: "'b list" assumes "\<And>as. \<forall>a\<in>set as. depends\<^sub>0 a \<Longrightarrow> qfree (qe as)"   and "\<And>as. \<forall>a\<in>set as. depends\<^sub>0 a \<Longrightarrow> is_dnf_qe qe as" shows "I (lift_dnf_qe qe \<phi>) xs = I \<phi> xs"
eta_norm_preserves_proves: fixes \<Theta> :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set"   and \<Gamma> :: "Core.term set"   and t :: "Core.term"   and u :: "Core.term" assumes "wf_theory \<Theta>"   and "finite \<Gamma>"   and "\<Theta>,\<Gamma> \<turnstile> t"   and "eta_norm t = u"   and "\<forall>A\<in>\<Gamma>. term_ok \<Theta> A"   and "\<forall>A\<in>\<Gamma>. typ_of A = Some propT" shows "\<Theta>,\<Gamma> \<turnstile> u"
map_sorted_distinct_set_unique: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'b"   and xs :: "'c list"   and ys :: "'c list" assumes "class.linorder less_eq less"   and "inj_on f (set xs \<union> set ys)"   and "linorder.sorted less_eq (map f xs)"   and "distinct (map f xs)"   and "linorder.sorted less_eq (map f ys)"   and "distinct (map f ys)"   and "set xs = set ys" shows "xs = ys"
assms: shows "r = memory_write_asi asi addr byte_mask data s \<and> r = Some s' \<and> get_S (cpu_reg_val PSR s) = 0"
ref_two_step: fixes A :: "'e nres"   and R :: "('e \<times> 'f) set"   and B :: "'f nres"   and C :: "'f nres" assumes "A \<le> \<Down> R B"   and "B \<le> C" shows "A \<le> \<Down> R C"
supp_is_subset: fixes S :: "atom set"   and x :: "'b" assumes "S supports x"   and "finite S" shows "supp x \<subseteq> S"
finite_less_Inf_iff: fixes X :: "'b set"   and a :: "'b" assumes "finite X"   and "X \<noteq> {}" shows "(a < Inf X) = (\<forall>x\<in>X. a < x)"
inf_glb: fixes \<delta> :: "'e \<Rightarrow> 'f \<times> 'g \<Rightarrow> 'e"   and \<gamma> :: "'e \<Rightarrow> 'f \<times> 'g \<Rightarrow> 'h"   and bot :: "'e"   and s1 :: "'e"   and s2 :: "'e" assumes "RDR \<delta> \<gamma> bot" shows "pre_RDR.is_glb \<delta> (pre_RDR.inf \<delta> s1 s2) s1 s2"
well_def_subsets1: fixes l :: "nat list \<times> nat list"   and i :: "nat" assumes "l \<in> set (get_subsets_R (calculate_data_R p qs1))" shows "(i \<in> set (fst l) \<longrightarrow> i < length qs1) \<and> (i \<in> set (snd l) \<longrightarrow> i < length qs1)"
preserves_inverse_arrows: fixes f :: "'c"   and g :: "'c" assumes "C.inverse_arrows f g" shows "D.inverse_arrows (G f) (G g)"
le_minus_iff: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'d"   and b :: "'d" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less" shows "less_eq a (uminus b) = less_eq b (uminus a)"
hyperpow_not_zero: fixes r :: "'a star"   and n :: "nat star" assumes "r \<noteq> 0" shows "r pow n \<noteq> 0"
ts_inv_ok_inv_ext_upd_inv: fixes ts :: "'f \<Rightarrow> ('g \<times> 'h \<Rightarrow>f nat) option"   and I :: "'f \<Rightarrow> 'i option"   and ta :: "('f, 'j, 'k) new_thread_action"   and P :: "'i \<Rightarrow> 'f \<Rightarrow> 'j \<Rightarrow> 'k \<Rightarrow> bool" assumes "ts_inv_ok ts I"   and "thread_ok ts ta" shows "I \<subseteq>\<^sub>m upd_inv I P ta"
deltas_below: fixes d :: "'b \<Rightarrow> nat \<Rightarrow> nat"   and k :: "nat"   and i :: "nat"   and w :: "'b list" assumes "bounded d k"   and "i < k" shows "deltas d w i < k"
order_axioms_def: fixes less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool" shows "class.order_axioms less_eq \<equiv> \<forall>x y. less_eq x y \<longrightarrow> less_eq y x \<longrightarrow> x = y"
comp_ide_arr: fixes b :: "'c"   and f :: "'c" assumes "C.ide b"   and "C.seq b f" shows "b \<cdot> f = f"
ordinal_numberTr4: fixes D :: "'c Order"   and X :: "'c Order set" assumes "Worder D"   and "X \<in> carrier ODnods"   and "D \<in> X" shows "X = ordinal_number D"
distinct_7: fixes x21 :: "'b"   and x22 :: "'b"   and x31 :: "'b"   and x32 :: "'b"   and x33 :: "'c" shows "DenyAllFromTo x21 x22 \<noteq> AllowPortFromTo x31 x32 x33"
code_simple: fixes \<C> :: "'b list set"   and c :: "'b list" assumes "code \<C>"   and "c \<in> \<C>" shows " c \<in>B \<C> "
assms_8: shows "InvariantWatchListsUniq (getWatchList state)"
type_equiv_def: fixes ev\<^sub>A :: "('a \<Rightarrow> 'b) \<Rightarrow> 'c \<Rightarrow> 'b"   and ev\<^sub>B :: "('a \<Rightarrow> 'b) \<Rightarrow> 'd \<Rightarrow> bool"   and aexp_vars :: "'c \<Rightarrow> 'a set"   and bexp_vars :: "'d \<Rightarrow> 'a set"   and dma :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> Sec"   and \<C>_vars :: "'a \<Rightarrow> 'a set"   and bexp_neg :: "'d \<Rightarrow> 'd"   and assign_post :: "'d set \<Rightarrow> 'a \<Rightarrow> 'c \<Rightarrow> 'd set"   and dma_type :: "'a \<Rightarrow> 'd set"   and pred_False :: "'d"   and t :: "'d set"   and P :: "'d set"   and t' :: "'d set" assumes "sifum_types ev\<^sub>A ev\<^sub>B aexp_vars bexp_vars dma \<C>_vars bexp_neg assign_post dma_type pred_False" shows "sifum_types.type_equiv ev\<^sub>B t P t' \<equiv> sifum_types.subtype ev\<^sub>B t P t' \<and> sifum_types.subtype ev\<^sub>B t' P t"
cases: fixes x :: "'v node \<times> 'v node" obtains x :: "'v" where "xa__ = (SOURCE, Inner xb__)"   | x :: "'v" and y :: "'v" where "xa__ = (Inner x, Inner y)"   | vb :: "'v" where "xa__ = (Inner vb, SOURCE)"   | v :: "'v node" where "xa__ = (v, SOURCE)"
sup_mono: fixes a :: "'c multiset"   and c :: "'c multiset"   and b :: "'c multiset"   and d :: "'c multiset" assumes "a \<subseteq># c"   and "b \<subseteq># d" shows "a \<union># b \<subseteq># c \<union># d"
bv_to_int_type: fixes w :: "bit list" shows "bv_to_int (norm_signed w) = bv_to_int w"
min_less_iff_disj: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "class.linorder less_eq less" shows "less (ord.min less_eq x y) z = (less x z \<or> less y z)"
imp_def: fixes p :: "fm"   and q :: "fm" shows "imp p q \<equiv> if p = F \<or> q = T \<or> p = q then T else if p = T then q else if q = F then not p else Imp p q"
\<X>_token_length: fixes t :: "symbol \<times> character list"   and k :: "nat" assumes "t \<in> \<X> k" shows "k + length (chars_of_token t) \<le> length Doc"
edge_obs_intra_subset: fixes a :: "'edge"   and S :: "'node set" assumes "valid_edge a"   and "intra_kind (kind a)"   and "sourcenode a \<notin> S" shows "obs_intra (targetnode a) S \<subseteq> obs_intra (sourcenode a) S"
assms: shows "n -as @ a # as'\<rightarrow>* n'"
vwb_lens_wb: fixes x :: "'e \<Longrightarrow> 'f" assumes "vwb_lens x" shows "wb_lens x"
mem_Collect_eq: fixes a :: "'e"   and P :: "'e \<Rightarrow> bool" shows "(a \<in> Collect P) = P a"
integrable_abs_iff: fixes f :: "'a \<Rightarrow> real"   and M :: "'a measure" assumes "f \<in> borel_measurable M" shows "integrable M (\<lambda>x. \<bar>f x\<bar>) = integrable M f"
sgn_uexpr_def: fixes u :: "('a, 'b) uexpr" shows "sgn u = uop sgn u"
edom_empty_iff_bot: fixes m :: "'a \<Rightarrow> 'b" shows "(edom m = {}) = (m = \<bottom>)"
oalist_inv_raw_map_pair: fixes xs :: "('a \<times> 'd) list"   and f :: "'a \<times> 'd \<Rightarrow> 'a \<times> 'e" assumes "oalist_inv_raw xs"   and "\<And>a b. comp (fst (f a)) (fst (f b)) = comp (fst a) (fst b)" shows "oalist_inv_raw (map_pair f xs)"
tllist_all2_ttlI: fixes P :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and Q :: "'i \<Rightarrow> 'j \<Rightarrow> bool"   and xs :: "('g, 'i) tllist"   and ys :: "('h, 'j) tllist" assumes "tllist_all2 P Q xs ys"   and "\<not> is_TNil xs \<or> \<not> is_TNil ys" shows "tllist_all2 P Q (ttl xs) (ttl ys)"
wf_1: shows "wf\<^sub>s\<^sub>t\<^sub>s' \<S> (decomp_rm\<^sub>e\<^sub>s\<^sub>t \<A>\<^sub>d)"
circle_right_edge_def: shows "circle_right_edge = (1, \<lambda>y. (d / 2, 0::'b))"
unifiable_def: fixes E :: "(('c, 'd) Term.term \<times> ('c, 'd) Term.term) set" shows "unifiable E = (\<exists>\<sigma>. \<sigma> \<in> unifiers E)"
N_comp_top: fixes x :: "'a" shows "N x * top = - (- x * top)"
infdist_le: fixes a :: "'b"   and A :: "'b set"   and x :: "'b" assumes "a \<in> A" shows "infdist x A \<le> dist x a"
mset_single_iff: fixes xs :: "'a list"   and x :: "'a" shows "(mset xs = {#x#}) = (xs = [x])"
convergent_liminf_cl: fixes X :: "nat \<Rightarrow> 'a" assumes "convergent X" shows "liminf X = lim X"
dtor2_def: fixes x :: "'d JF2" shows "dtor2 x = F2map id Abs_JF1 Abs_JF2 (str_final2 (Rep_JF2 x))"
trans_frame_rule: fixes \<Gamma> :: "assn"   and \<Gamma>' :: "assn"   and c :: "'a Heap"   and \<Gamma>'' :: "assn"   and R :: "'b \<Rightarrow> 'a \<Rightarrow> assn"   and a :: "'b nres"   and F :: "assn" assumes "RECOVER_PURE \<Gamma> \<Gamma>'"   and "vassn_tag \<Gamma>' \<Longrightarrow> hn_refine \<Gamma>' c \<Gamma>'' R a" shows "hn_refine (F * \<Gamma>) c (F * \<Gamma>'') R a"
abs_eq: shows "\<top> = [-\<infinity>, \<infinity>]"
perm_append_swap: fixes xs :: "'a list"   and ys :: "'a list" shows "xs @ ys <~~> ys @ xs"
le_add_diff_inverse: fixes plus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and minus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and zero :: "'g"   and times :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and one :: "'g"   and b :: "'g"   and a :: "'g" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "less_eq b a" shows "plus b (minus a b) = a"
less_Store_def: fixes s :: "Store"   and t :: "Store" shows "(s < t) = (s \<le> t \<and> \<not> t \<le> s)"
card_option: shows "CARD('a option) = Suc CARD('a)"
simulation_axioms_def: fixes R :: "('a \<times> 'b) set"   and A :: "('a, 'c) graph_rec_scheme"   and B :: "('b, 'd) graph_rec_scheme" shows "simulation_axioms R A B \<equiv> (\<forall>a b. a \<in> g_V A \<longrightarrow> (a, b) \<in> R \<longrightarrow> b \<in> g_V B) \<and> (\<forall>a0. a0 \<in> g_V0 A \<longrightarrow> (\<exists>b0. b0 \<in> g_V0 B \<and> (a0, b0) \<in> R)) \<and> (\<forall>a a' b. (a, a') \<in> g_E A \<longrightarrow> (a, b) \<in> R \<longrightarrow> (\<exists>b'. (b, b') \<in> g_E B \<and> (a', b') \<in> R))"
interior_ic_ci_union: fixes a :: "real"   and b :: "real"   and c :: "real" assumes "a < b"   and "b < c" shows "b \<notin> interior ({a..<b} \<union> {b<..c})"
set_cases_2: fixes e :: "'a"   and a :: "('b, 'a) xstate" assumes "e \<in> set2_xstate a" obtains "a = Fault e"
igConsIPresIGWls_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'e, 'f, 'g, 'h) model_scheme" shows "igConsIPresIGWls MOD \<equiv> igVarIPresIGWls MOD \<and> igAbsIPresIGWls MOD \<and> igOpIPresIGWls MOD"
tauChainCases: fixes P :: "('a, 'b, 'c) psi"   and P' :: "('a, 'b, 'c) psi"   and \<Psi> :: "'b" assumes "\<Psi> \<rhd> P \<Longrightarrow>\<^sup>^\<^sub>\<tau> P'" obtains "P = P'"   | "\<Psi> \<rhd> P \<Longrightarrow>\<^sub>\<tau> P'"
is_ideal: shows "ideal (\<Inter>i\<le>n. I (Suc i)) R"
axioms_1: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and s :: "nat"   and t :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'a"   and l :: "nat \<Rightarrow> nat" assumes "Labeling c s t f l" shows "NPreflow c s t f"
h_bounds: fixes x :: "real"   and h :: "real \<Rightarrow> real" assumes "x\<^sub>1 \<le> x"   and "h \<in> set hs" shows "\<bar>h x\<bar> \<le> hb * x / ln x powr (1 + e)"
dom_prob_proj: fixes PROB :: "(('d, 'e) fmap \<times> ('d, 'f) fmap) set"   and vs :: "'d set" shows "prob_dom (prob_proj PROB vs) \<subseteq> vs"
equiv_selfloop_per_union_id: fixes S :: "'b set"   and F :: "('b \<times> 'b) set"   and x :: "'b" assumes "equiv S F"   and "x \<in> S" shows "per_union F x x = F"
evaluate_list_mono_strong: fixes R :: "'a state \<Rightarrow> exp0 \<Rightarrow> 'a state \<times> (v, v) result \<Rightarrow> bool"   and s :: "'a state"   and es :: "exp0 list"   and r :: "'a state \<times> (v list, v) result"   and Q :: "'a state \<Rightarrow> exp0 \<Rightarrow> 'a state \<times> (v, v) result \<Rightarrow> bool" assumes "Big_Step_Unclocked_Single.evaluate_list R s es r"   and "\<And>s e r. \<lbrakk>e \<in> set es; R s e r\<rbrakk> \<Longrightarrow> Q s e r" shows "Big_Step_Unclocked_Single.evaluate_list Q s es r"
mod_eq_0_iff_dvd: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.semidom_modulo divide plus minus zero times one modulo" shows "(modulo a b = zero) = dvd.dvd times b a"
vs: shows "set vs = {v \<in> pverts (list_digraph G_list). out_arcs (with_proj (list_digraph G_list)) v \<noteq> {}}"
eval_r_s: fixes s :: "nat \<Rightarrow> nat option" assumes "cons_lim s" shows "s = (\<lambda>x. eval (cons_lim.r_s s) [x])"
tiTable_SampleT_sending_data: fixes req :: "nat \<Rightarrow> reqType list"   and a1 :: "nat \<Rightarrow> aType list"   and stop :: "nat \<Rightarrow> stopType list"   and lose :: "nat \<Rightarrow> bool list"   and st :: "nat \<Rightarrow> GatewayStatus"   and buffer :: "nat \<Rightarrow> ECall_Info list"   and ack :: "nat \<Rightarrow> GatewayStatus list"   and i1 :: "nat \<Rightarrow> ECall_Info list"   and vc :: "nat \<Rightarrow> vcType list"   and d :: "nat"   and t :: "nat"   and x :: "nat" assumes "tiTable_SampleT req a1 stop lose (fin_inf_append [init_state] st) (fin_inf_append [[]] buffer) ack i1 vc st"   and "\<forall>j\<le>2 * d. lose (t + j) = [False]"   and "\<forall>t4\<le>t + d + d. a1 t4 = []"   and "fin_inf_append [init_state] st (Suc (t + x)) = sending_data"   and "Suc (t + x) \<le> 2 * d + t" shows "ack (Suc (t + x)) = [sending_data]"
admissible_dom_prog: fixes P :: "'l \<Rightarrow> 'm \<Rightarrow> 'm \<Rightarrow> 'n \<Rightarrow> bool" shows "dom_prog.admissible (\<lambda>f. \<forall>x h h' r. h \<turnstile> f x \<rightarrow>\<^sub>r r \<longrightarrow> h \<turnstile> f x \<rightarrow>\<^sub>h h' \<longrightarrow> P x h h' r)"
intros_2: fixes T :: "nat" shows "cNumber T \<in> cpayload"
M1: fixes p :: "'a"   and q :: "'a"   and s :: "'a"   and r :: "'a" assumes "p \<parallel> q"   and "p \<parallel> s"   and "r \<parallel> q" shows "r \<parallel> s"
assms_2: shows "a \<in> carrier R - {\<zero>}"
sum_indep_random_variable: fixes X :: "'a \<Rightarrow> 'b"   and Y :: "'a \<Rightarrow> 'b" assumes "indep_var borel X borel Y"   and "random_variable borel X"   and "random_variable borel Y" shows "distr M borel (\<lambda>x. X x + Y x) = (distr M borel X \<star> distr M borel Y)"
hm_update_impl: shows "imp_map_update is_hashmap hm_update"
map_upd_Some_unfold: fixes m :: "'f \<Rightarrow> 'e option"   and a :: "'f"   and b :: "'e"   and x :: "'f"   and y :: "'e" shows "((m(a \<mapsto> b)) x = Some y) = (x = a \<and> b = y \<or> x \<noteq> a \<and> m x = Some y)"
f: shows "\<guillemotleft>\<psi> (a', a) (\<Psi>.FUN a' (\<iota> \<chi>')) : a' \<rightarrow> a\<guillemotright>"
continuous_on_fst: fixes s :: "'a set"   and f :: "'a \<Rightarrow> 'b \<times> 'c" assumes "continuous_on s f" shows "continuous_on s (\<lambda>x. fst (f x))"
appl_methds_def: fixes G :: "prog"   and S :: "qtname"   and rt :: "ref_ty" shows "appl_methds G S rt = (\<lambda>sig. {(mh, pTs') |mh pTs'. mh \<in> mheads G S rt \<lparr>name = name sig, parTs = pTs'\<rparr> \<and> G\<turnstile>parTs sig[\<preceq>]pTs'})"
f_pre_additive: fixes f :: "'a \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "is_conjugation f g" shows "(f (x + y) \<le> z) = (f x + f y \<le> z)"
rotate_append: fixes l :: "'c list"   and q :: "'c list" shows "rotate (length l) (l @ q) = q @ l"
stop_no_eval: fixes mds :: "Mode \<Rightarrow> 'Var set"   and mem :: "'Var \<Rightarrow> 'Val"   and c' :: "('Var, 'AExp, 'BExp) Stmt"   and mds' :: "Mode \<Rightarrow> 'Var set"   and mem' :: "'Var \<Rightarrow> 'Val" shows "(\<langle>Stop, mds, mem\<rangle>\<^sub>w, \<langle>c', mds', mem'\<rangle>\<^sub>w) \<notin> eval\<^sub>w"
multpw_add: fixes X1 :: "'a multiset"   and Y1 :: "'a multiset"   and ns :: "('a \<times> 'a) set"   and X2 :: "'a multiset"   and Y2 :: "'a multiset" assumes "(X1, Y1) \<in> multpw ns"   and "(X2, Y2) \<in> multpw ns" shows "(X1 + X2, Y1 + Y2) \<in> multpw ns"
m_lcomm: 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 "x \<oplus> (y \<oplus> z) = y \<oplus> (x \<oplus> z)"
vsv_vintersection: fixes s :: "V" shows "vsv (r \<inter>\<^sub>\<circ> s)"
omega_fun_def: fixes f :: "'b \<Rightarrow> 'b" shows "omega_fun f = lfp (Omega_fun f id)"
l_set_disconnected_nodes_get_ancestors_si_axioms: shows "l_set_disconnected_nodes_get_ancestors_si set_disconnected_nodes_locs get_ancestors_si_locs"
abs_eq: fixes x :: "int" shows "Word.the_int (Word.Word x) = take_bit LENGTH('b) x"
set_less_aux_conv_set_less_eq_aux: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set"   and B :: "'a set" shows "ord.set_less_aux less_eq A B = (ord.set_less_eq_aux less_eq A B \<and> \<not> ord.set_less_eq_aux less_eq B A)"
birkhoff_sum_add: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b"   and n :: "nat"   and x :: "'a" shows "birkhoff_sum (\<lambda>x. f x + g x) n x = birkhoff_sum f n x + birkhoff_sum g n x"
distinguishing_formula_supp_aux: fixes P :: "'state"   and Q :: "'state" assumes "\<not> P =\<cdot> Q" shows "supp (Abs_bset {- p \<bullet> (SOME x. supp x \<subseteq> supp (p \<bullet> P) \<and> x distinguishes (p \<bullet> P) from (p \<bullet> Q)) |p. True}) \<subseteq> supp P"
countable_basis_openI: fixes B :: "'c set set" assumes "countable B"   and "topological_basis B" shows "countable_basis open B"
valid_elim: fixes x :: "name"   and T :: "ty"   and \<Gamma> :: "(name \<times> ty) list" assumes "valid ((x, T) # \<Gamma>)" shows "x \<sharp> \<Gamma> \<and> valid \<Gamma>"
m_distinct: shows "distinct_mset (mset_set (\<V> - {x}))"
msb_word_def: fixes a :: "'a word" shows "msb a = (bin_sign (signed_take_bit (LENGTH('a) - 1) (uint a)) = - 1)"
triv_path: fixes w :: "'a \<times> 'a \<Rightarrow> enat"   and u :: "'a"   and v :: "'a" shows "WGraph.path w u [w (u, v)] v"
distinct_1: shows "T_i32 \<noteq> T_i64"
selection_closed_id: fixes uminus :: "'c \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and x :: "'c" assumes "class.p_algebra uminus inf less_eq less sup bot top" shows "x = inf (uminus (uminus x)) x"
AE_prob_1: fixes M :: "'b measure"   and A :: "'b set" assumes "prob_space M"   and "Sigma_Algebra.measure M A = 1" shows "AE x in M. x \<in> A"
Disjunctuous_def: fixes H :: "'a upred \<Rightarrow> 'a upred" shows "Disjunctuous H = (\<forall>P Q. H (P \<sqinter> Q) = H P \<sqinter> H Q)"
iso_sum_groupI: fixes I :: "'b set"   and G :: "'b \<Rightarrow> ('c, 'd) monoid_scheme"   and H :: "'b \<Rightarrow> ('e, 'f) monoid_scheme" assumes "\<And>i. i \<in> I \<Longrightarrow> G i \<cong> H i"   and "\<And>i. i \<in> I \<Longrightarrow> Group.group (G i)"   and "\<And>i. i \<in> I \<Longrightarrow> Group.group (H i)" shows "sum_group I G \<cong> sum_group I H"
det_addrow_mat: fixes k :: "nat"   and l :: "nat"   and n :: "nat"   and a :: "'a" assumes "k \<noteq> l" shows "Determinant.det (addrow_mat n a k l) = (1::'a)"
line_integral_exists_on_typeII_Cube_boundaries': fixes i :: "real \<times> real"   and j :: "real \<times> real"   and twoC :: "real \<times> real \<Rightarrow> real \<times> real"   and F :: "real \<times> real \<Rightarrow> real \<times> real"   and k :: "int"   and \<gamma> :: "real \<Rightarrow> real \<times> real" assumes "green_typeII_cube i j twoC F"   and "(k, \<gamma>) \<in> boundary twoC" shows "line_integral_exists F {j} \<gamma>"
produces_imp_Pos_in_lits: fixes C :: "'a literal multiset"   and A :: "'a" assumes "production C = {A}" shows "Pos A \<in># C"
pmdl_closed_monom_mult: fixes p :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and B :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set"   and c :: "'b"   and t :: "'x \<Rightarrow>\<^sub>0 nat" assumes "p \<in> local.punit.pmdl B" shows "local.punit.monom_mult c t p \<in> local.punit.pmdl B"