hcju/isabelleps huggingface dataset:
disj: shows "bvars\<^sub>s\<^sub>t (S @ send\<langle>Fun f X\<rangle>\<^sub>s\<^sub>t # S') \<inter> (subst_domain \<delta> \<union> range_vars \<delta>) = {}"
oneposs: fixes xs :: "'a list"   and x :: "'a"   and y :: "'a" assumes "set xs = {x, y}"   and "x \<noteq> y"   and "distinct xs"   and "x < y in xs" shows "xs = [x, y]"
P_eq: fixes x :: "'a"   and x' :: "'b" shows "P x x' = \<up> (P' x x')"
tyenv_wellformed_mode_update: fixes mds :: "Mode \<Rightarrow> 'Var set"   and \<Gamma> :: "'Var \<Rightarrow> 'BExp set option"   and \<S> :: "'Var set \<times> 'Var set"   and P :: "'BExp set"   and upd :: "'Var ModeUpd" assumes "tyenv_wellformed mds \<Gamma> \<S> P"   and "anno_type_stable \<Gamma> \<S> upd" shows "tyenv_wellformed (update_modes upd mds) \<Gamma> \<oplus>\<^sub>\<S> upd (add_anno_stable \<S> upd) P |` {v. stable (add_anno_stable \<S> upd) v}"
nevaln: shows "lcn \<leadsto>\<^bsup>m\<^esup> lcn'"
plus_ms_aux_MSLCons_eval: fixes c1 :: "'a"   and e1 :: "real"   and xs :: "('a \<times> real) msllist"   and c2 :: "'a"   and e2 :: "real"   and ys :: "('a \<times> real) msllist" shows "plus_ms_aux (MSLCons (c1, e1) xs) (MSLCons (c2, e2) ys) = CMP_BRANCH (COMPARE e1 e2) (MSLCons (c2, e2) (plus_ms_aux (MSLCons (c1, e1) xs) ys)) (MSLCons (c1 + c2, e1) (plus_ms_aux xs ys)) (MSLCons (c1, e1) (plus_ms_aux xs (MSLCons (c2, e2) ys)))"
cont_Sup: fixes lub :: "'c set \<Rightarrow> 'c"   and ord :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'a set" assumes "cont lub ord \<Union> (\<subseteq>) f" shows "cont lub ord Sup (\<le>) (\<lambda>x. \<Squnion> f x)"
add_diff_inverse: fixes a :: "'a"   and b :: "'a" assumes "a \<le> b" shows "a + (b - a) = b"
Cod_f'': shows "\<CC>\<lparr>Cod\<rparr>\<lparr>f''\<rparr> = b''"
hoare_sound: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> bool"   and c :: "com"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> bool" assumes "\<turnstile> {P} c {Q}" shows "\<Turnstile> {P} c {Q}"
Gets_imp_knows_Spy: fixes B :: "agent"   and X :: "msg"   and evs :: "event list" assumes "Gets B X \<in> set evs"   and "evs \<in> set_pur" shows "X \<in> knows Spy evs"
invmfun_def: fixes R :: "('g, 'h) Ring_scheme"   and M :: "('f, 'g, 'i) Module_scheme"   and N :: "('e, 'g, 'j) Module_scheme"   and f :: "'f \<Rightarrow> 'e" shows "invmfun R M N f = (\<lambda>y\<in>carrier N. SOME x. x \<in> carrier M \<and> f x = y)"
winding_number_zero_outside: fixes \<gamma> :: "real \<Rightarrow> complex"   and s :: "complex set"   and z :: "complex" assumes "path \<gamma>"   and "convex s"   and "pathfinish \<gamma> = pathstart \<gamma>"   and "z \<notin> s"   and "path_image \<gamma> \<subseteq> s" shows "winding_number \<gamma> z = 0"
stake_add: fixes m :: "nat"   and s :: "'b stream"   and n :: "nat" shows "stake m s @ stake n (sdrop m s) = stake (m + n) s"
cross_basis_3: shows "cross3 (axis 2 1) (axis 3 1) = axis 1 1"
IH_2: assumes "poly q x\<^sub>0 \<noteq> 0"   and "poly (hd (q # r # ps'')) x\<^sub>0 \<noteq> 0"   and "ps' \<in> set (split_sign_changes (q # r # ps'') x\<^sub>0)" shows "\<forall>\<^sub>F x in at x\<^sub>0. sign_changes ps' x = sign_changes ps' x\<^sub>0"
distr_id2: fixes M :: "'c measure"   and N :: "'c measure" assumes "sets M = sets N" shows "distr N M (\<lambda>x. x) = N"
inv_in_homI: fixes f :: "'c"   and a :: "'c"   and b :: "'c" assumes "C.iso f"   and "\<guillemotleft>f : a \<Rightarrow>\<^sub>C b\<guillemotright>" shows "\<guillemotleft>C.inv f : b \<Rightarrow>\<^sub>C a\<guillemotright>"
distinct_29: fixes x31 :: "POPLmarkRecord.type"   and x32 :: "trm"   and x51 :: "trm"   and x52 :: "POPLmarkRecord.type" shows "(\<lambda><:x31. x32) \<noteq> x51 \<bullet>\<^sub>\<tau> x52"
f_of_newton_seq_closed: fixes m :: "nat" shows "f \<bullet> ns m \<in> carrier Zp"
prems_2: shows "suffix n w \<Turnstile>\<^sub>n af (\<phi> R\<^sub>n \<psi>) (w [Suc i \<rightarrow> n])[X]\<^sub>\<nu>"
to_tree_order_parent_child_rel: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and to :: "(_) object_ptr list"   and child :: "(_) object_ptr" assumes "Shadow_DOM.heap_is_wellformed h"   and "ShadowRootClass.type_wf h"   and "ShadowRootClass.known_ptrs h"   and "h \<turnstile> Shadow_DOM.to_tree_order ptr \<rightarrow>\<^sub>r to" shows "((ptr, child) \<in> (Shadow_DOM.parent_child_rel h)\<^sup>*) = (child \<in> set to)"
wuntil_coinduct: fixes S :: "'a set"   and s0 :: "'a"   and \<delta> :: "'a \<Rightarrow> 'a set"   and AP :: "'b set"   and L :: "'a \<Rightarrow> 'b set"   and AP' :: "'b set"   and \<chi> :: "('a \<times> 'b set) stream list \<Rightarrow> bool"   and \<pi>l :: "('a \<times> 'b set) stream list"   and \<psi> :: "('a \<times> 'b set) stream list \<Rightarrow> bool"   and \<phi> :: "('a \<times> 'b set) stream list \<Rightarrow> bool" assumes "Shallow S s0 \<delta> AP L AP'"   and "\<chi> \<pi>l"   and "\<And>\<pi>l. \<chi> \<pi>l \<Longrightarrow> \<psi> \<pi>l \<or> \<phi> \<pi>l \<and> \<chi> (map stl \<pi>l)" shows "Shallow.wuntil \<phi> \<psi> \<pi>l"
C: fixes c :: "('test, 'atom) com"   and d :: "('test, 'atom) com"   and s :: "'state"   and t :: "'state"   and c' :: "('test, 'atom) com"   and s' :: "'state" assumes "\<phi> c d"   and "s \<approx> t"   and "(c, s) \<rightarrow>c (c', s')" shows "\<exists>d' t'. (d, t) \<rightarrow>*c (d', t') \<and> s' \<approx> t' \<and> (\<phi> c' d' \<or> c' \<approx>wT d')"
small_trans: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b"   and h :: "'a \<Rightarrow> 'b" assumes "f \<in> o[F](g)"   and "g \<in> o[F](h)" shows "f \<in> o[F](h)"
cgenideal_prod: fixes a :: "'a"   and b :: "'a" assumes "a \<in> carrier R"   and "b \<in> carrier R" shows "PIdl a <#> PIdl b = PIdl (a \<otimes> b)"
sup_idem: fixes sup :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d" assumes "class.semilattice_sup sup less_eq less" shows "sup x x = x"
compute_tm_by_comp_eq_Some_iff: fixes prec :: "nat"   and ord :: "nat"   and I :: "float interval list"   and a :: "float interval list"   and f :: "floatarith"   and t0 :: "taylor_model option"   and i :: "float interval \<Rightarrow> bool"   and t :: "taylor_model" shows "(compute_tm_by_comp prec ord I a f t0 i = Some t) = (\<exists>z x2. t0 = Some x2 \<and> tm_floatarith prec ord [compute_bound_tm prec I a x2] [mid (compute_bound_tm prec a a x2)] f = Some z \<and> tm_comp prec ord I a (mid (compute_bound_tm prec a a x2)) z x2 = t \<and> i (compute_bound_tm prec I a x2))"
u_3: shows "w = (1 - u) *\<^sub>R x + u *\<^sub>R y"
inf_left_idem: fixes inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.semilattice_inf inf less_eq less" shows "inf x (inf x y) = inf x y"
trivial_derived_seq_imp_solvable: fixes H :: "'c set"   and n :: "nat" assumes "subgroup H H"   and "(derived H ^^ n) H = {\<one>\<^bsub>H\<^esub>}" shows "solvable_seq H H"
no_local_consts_in_consequences': fixes n :: "('form, 'rule) graph_node"   and p :: "'form" assumes "n \<in> sset nodes"   and "Reg p |\<in>| outPorts n" shows "lconsts p = {}"
iface_conjunct_Some: fixes i1 :: "iface"   and i2 :: "iface"   and x :: "iface"   and p_i :: "char list" assumes "iface_conjunct i1 i2 = Some x" shows "match_iface x p_i = (match_iface i1 p_i \<and> match_iface i2 p_i)"
JNF: shows "\<forall>A\<in>carrier_mat CARD('n) CARD('n). admits_SNF_JNF A"
nn_integral_mono2mono: fixes M :: "'a measure"   and ord :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and F :: "'b \<Rightarrow> 'a \<Rightarrow> ennreal" assumes "\<And>x. x \<in> space M \<Longrightarrow> monotone ord (\<le>) (\<lambda>f. F f x)" shows "monotone ord (\<le>) (\<lambda>f. integral\<^sup>N M (F f))"
lfp_unfold: fixes f :: "'b \<Rightarrow> 'b" assumes "mono f" shows "\<nu> f = f (\<nu> f)"
simps_2: fixes p :: "pat"   and q :: "pat" shows "pat_type \<langle>\<langle>p, q\<rangle>\<rangle> = pat_type p \<otimes> pat_type q"
atLeast_def: fixes l :: "'a Word.word" shows "signed.atLeast l \<equiv> {x. l \<le>s x}"
assms: shows "Q \<Longrightarrow>\<^sub>l\<^sup>^\<alpha> \<prec> Q'"
fact_dvd_poly_higher_pderiv_aux: fixes n :: "nat"   and p :: "int poly"   and x :: "int" shows "fact n dvd poly ((pderiv ^^ n) p) x"
Domainp_eq: shows "Domainp (=) = (\<lambda>x. x \<in> UNIV)"
eq_divide_eq: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "class.division_ring divide inverse one times plus zero minus uminus" shows "(a = divide b c) = (if c \<noteq> zero then times a c = b else a = zero)"
rt_strictly_fresher_not_fresh_asD: fixes dip :: "nat"   and rt1 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and rt2 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option" assumes "rt1 \<sqsubset>\<^bsub>dip\<^esub> rt2" shows "\<not> rt1 \<approx>\<^bsub>dip\<^esub> rt2"
finite_field_factorization_i: shows "(poly_rel ===> rel_prod R (list_all2 poly_rel)) (finite_field_factorization_i p ff_ops) finite_field_factorization"
elims: fixes x :: "'a rexp"   and xa :: "'a rexp"   and y :: "'a rexp" assumes "pnTimes x xa = y" obtains r :: "'a rexp" where "x = Zero" and "xa = r__" and "y = Zero"   | r :: "'a rexp" and s :: "'a rexp" and t :: "'a rexp" where "x = Plus r s__" and "xa = t" and "y = pnPlus (pnTimes r t) (pnTimes s__ t)"   | s :: "'a rexp" where "x = One" and "xa = sa__" and "y = Times One sa__"   | v :: "'a" and s :: "'a rexp" where "x = Atom v__" and "xa = sb__" and "y = Times (Atom v__) sb__"   | v :: "'a rexp" and va :: "'a rexp" and s :: "'a rexp" where "x = Times va__ va" and "xa = sc__" and "y = Times (Times va__ va) sc__"   | v :: "'a rexp" and s :: "'a rexp" where "x = Star v" and "xa = s" and "y = Times (Star v) s"
zero: fixes x :: "'b" shows "0 *\<^sub>R x = (0::'b)"
drop_append: fixes n :: "nat"   and xs :: "'d list"   and ys :: "'d list" shows "drop n (xs @ ys) = drop n xs @ drop (n - length xs) ys"
FormP_Disj: fixes x :: "tm"   and y :: "tm" shows "{FormP x, FormP y} \<turnstile> FormP (Q_Disj x y)"
Abs_bit1_inject: fixes x :: "int"   and y :: "int" assumes "x \<in> {0..<1 + 2 * int CARD('a)}"   and "y \<in> {0..<1 + 2 * int CARD('a)}" shows "(Abs_bit1 x = Abs_bit1 y) = (x = y)"
pcp_def: fixes C :: "'b formula set set" shows "pcp (C::'b formula set set) \<equiv> \<forall>S::'b formula set\<in>C. \<bottom> \<notin> S \<and> (\<forall>k::'b. Atom k \<in> S \<longrightarrow> \<^bold>\<not> (Atom k) \<in> S \<longrightarrow> False) \<and> (\<forall>(F::'b formula) G::'b formula. F \<^bold>\<and> G \<in> S \<longrightarrow> F \<triangleright> G \<triangleright> S \<in> C) \<and> (\<forall>(F::'b formula) G::'b formula. F \<^bold>\<or> G \<in> S \<longrightarrow> F \<triangleright> S \<in> C \<or> G \<triangleright> S \<in> C) \<and> (\<forall>(F::'b formula) G::'b formula. F \<^bold>\<rightarrow> G \<in> S \<longrightarrow> \<^bold>\<not> F \<triangleright> S \<in> C \<or> G \<triangleright> S \<in> C) \<and> (\<forall>F::'b formula. \<^bold>\<not> (\<^bold>\<not> F) \<in> S \<longrightarrow> F \<triangleright> S \<in> C) \<and> (\<forall>(F::'b formula) G::'b formula. \<^bold>\<not> (F \<^bold>\<and> G) \<in> S \<longrightarrow> \<^bold>\<not> F \<triangleright> S \<in> C \<or> \<^bold>\<not> G \<triangleright> S \<in> C) \<and> (\<forall>(F::'b formula) G::'b formula. \<^bold>\<not> (F \<^bold>\<or> G) \<in> S \<longrightarrow> \<^bold>\<not> F \<triangleright> \<^bold>\<not> G \<triangleright> S \<in> C) \<and> (\<forall>(F::'b formula) G::'b formula. \<^bold>\<not> (F \<^bold>\<rightarrow> G) \<in> S \<longrightarrow> F \<triangleright> \<^bold>\<not> G \<triangleright> S \<in> C)"
hn_refine_def: fixes \<Gamma> :: "assn"   and c :: "'d Heap"   and \<Gamma>' :: "assn"   and R :: "'e \<Rightarrow> 'd \<Rightarrow> assn"   and m :: "'e nres" shows "hn_refine \<Gamma> c \<Gamma>' R m \<equiv> nofail m \<longrightarrow> <\<Gamma>> c <\<lambda>r. \<Gamma>' * (\<exists>\<^sub>Ax. R x r * \<up> (RETURN x \<le> m))>\<^sub>t"
Call: fixes \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and p :: "'e"   and bdy :: "('d, 'e, 'f) com"   and s :: "'d" assumes "\<Gamma> p = Some bdy"   and "\<Gamma>\<turnstile>bdy \<down> Normal s" shows "\<Gamma>\<turnstile>Call p \<down> Normal s"
orderE: fixes b :: "'e multiset"   and a :: "'e multiset" assumes "b \<subseteq># a" obtains "a = a \<union># b"
elims_2: fixes x :: "'b val" assumes "is_Addr x" obtains a :: "'b" where "x = Addr a"
scc_root_transfer': fixes s' :: "('v, 'a) state_scheme"   and r :: "'v"   and scc :: "'v set" assumes "discovered s = discovered s'"   and "tree_edges s = tree_edges s'" shows "scc_root s r scc = scc_root s' r scc"
int_simps_27: fixes P :: "'a \<Rightarrow> bool" shows "\<turnstile> (P \<or> P) = P"
push_down_rank_senior: fixes x :: "nat"   and n :: "nat"   and y :: "nat"   and i :: "nat" assumes "rank x n = rank y n"   and "rank x n = Some i" shows "senior x n = senior y n"
seq_comp_rep: fixes P :: "'c option process"   and Q :: "'c option process" assumes "weakly_sequential P" shows "Rep_process (P ; Q) = (seq_comp_failures P Q, {})"
integrable_scaleR_left: fixes c :: "'b"   and M :: "'c measure"   and f :: "'c \<Rightarrow> real" assumes "c \<noteq> (0::'b) \<Longrightarrow> integrable M f" shows "integrable M (\<lambda>x. f x *\<^sub>R c)"
total_lenlex: fixes r :: "('b \<times> 'b) set" assumes "total r" shows "total (lenlex r)"
Quotient3_real_alg: shows "Quotient3 (\<lambda>x y. real_of_3 x = real_of_3 y) Real_Alg_Quotient rep_real_alg"
borel_measurable_LIMSEQ_metric: fixes f :: "nat \<Rightarrow> 'c \<Rightarrow> 'd"   and M :: "'c measure"   and g :: "'c \<Rightarrow> 'd" assumes "\<And>i. f i \<in> borel_measurable M"   and "\<And>x. x \<in> space M \<Longrightarrow> (\<lambda>i. f i x) \<longlonglongrightarrow> g x" shows "g \<in> borel_measurable M"
x_c_glb: fixes z :: "'a"   and x :: "'a"   and y :: "'a" assumes "z \<cdot> 1\<^sub>\<pi> \<le> x \<cdot> 1\<^sub>\<pi> \<and> z \<cdot> 1\<^sub>\<pi> \<le> y \<cdot> 1\<^sub>\<pi>" shows "z \<cdot> 1\<^sub>\<pi> \<le> x \<cdot> 1\<^sub>\<pi> \<parallel> (y \<cdot> 1\<^sub>\<pi>)"
monotone_rel_prodI: fixes ordb :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ordc :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and f :: "'b \<times> 'c \<Rightarrow> 'd"   and orda :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<And>a. monotone ordb ordc (\<lambda>b. f (a, b))"   and "\<And>b. monotone orda ordc (\<lambda>a. f (a, b))"   and "class.preorder orda (mk_less orda)"   and "class.preorder ordb (mk_less ordb)"   and "class.preorder ordc (mk_less ordc)" shows "monotone (rel_prod orda ordb) ordc f"
inv_into_funcset: fixes f :: "'d \<Rightarrow> 'c"   and A :: "'d set"   and B :: "'c set" assumes "f ` A = B" shows "restrict (inv_into A f) B \<in> B \<rightarrow> A"
distinct_10: fixes x2 :: "'b" shows "ack \<noteq> right x2"
closed_segment_degen_complex_2: fixes lb :: "complex"   and ub :: "complex"   and x :: "complex" assumes "Im lb = Im ub"   and "Re lb \<le> Re ub" shows "(x \<in> closed_segment lb ub) = (Im x = Im lb \<and> Re lb \<le> Re x \<and> Re x \<le> Re ub)"
well_formed_same_procs: fixes procs :: "(char list \<times> char list list \<times> char list list \<times> cmd) list"   and p :: "char list"   and ins :: "char list list"   and outs :: "char list list"   and c :: "cmd"   and ins' :: "char list list"   and outs' :: "char list list"   and c' :: "cmd" assumes "well_formed procs"   and "(p, ins, outs, c) \<in> set procs"   and "(p, ins', outs', c') \<in> set procs" shows "ins = ins' \<and> outs = outs' \<and> c = c'"
irreducible_linear_poly: fixes b :: "'a"   and a :: "'a" assumes "b \<noteq> (0::'a)"   and "algebraic_semidom_class.coprime a b" shows "irreducible [:a, b:]"
simps_3: fixes x1 :: "'b"   and l1 :: "'b list"   and x2 :: "'b"   and l2 :: "'b list" shows "subset_sorted (x1 # l1) (x2 # l2) = (if x1 < x2 then False else if x1 = x2 then subset_sorted l1 l2 else subset_sorted (x1 # l1) l2)"
assms_3: shows "\<forall>r\<in>R. Card_order r"
mult_wfactors_fmset: fixes as :: "'a list"   and a :: "'a"   and bs :: "'a list"   and b :: "'a"   and cs :: "'a list" assumes "wfactors G as a"   and "wfactors G bs b"   and "wfactors G cs (a \<otimes> b)"   and "a \<in> carrier G"   and "b \<in> carrier G"   and "set as \<subseteq> carrier G"   and "set bs \<subseteq> carrier G"   and "set cs \<subseteq> carrier G" shows "fmset G cs = fmset G as + fmset G bs"
flow_has_vderiv_on_ex_ivl: fixes s :: "'a" assumes "s \<in> S" shows " D flow t\<^sub>0 s = (\<lambda>t. f t (flow t\<^sub>0 s t)) on ex_ivl s"
pseudo_proj_False_singleton: fixes w :: "bool stream"   and n :: "nat" assumes "w = pseudo_proj_False n w" shows "pseudo_proj_False n -` {w} \<inter> space (bernoulli_stream p) \<in> sets (bernoulli_stream p)"
hk_sub: shows "{x. \<not> (h x = x \<and> k x = x)} \<subseteq> S - y ` K"
localImpGlobalCons: fixes \<phi> :: "i \<Rightarrow> bool"   and \<xi> :: "i \<Rightarrow> bool" assumes "\<lfloor>\<phi> \<^bold>\<rightarrow> \<xi>\<rfloor>" shows "\<lfloor>\<phi>\<rfloor> \<longrightarrow> \<lfloor>\<xi>\<rfloor>"
is_acc_run_def: fixes r :: "nat \<Rightarrow> 'a node" shows "gba.is_acc_run r \<equiv> gba.is_run r \<and> gba.is_acc r"
take_Suc0_2: fixes l :: "'c list" assumes "0 < length l" shows "take (Suc 0) l = [l ! 0]"
bin_bl_bin': fixes n :: "nat"   and w :: "int"   and bs :: "bool list" shows "bl_to_bin (bin_to_bl_aux n w bs) = bl_to_bin_aux bs (take_bit n w)"
cmod_cis: fixes a :: "complex" assumes "a \<noteq> 0" shows "cor (cmod a) * cis (arg a) = a"
awhd_if_cas: fixes G :: "('c, 'd) pre_digraph"   and u :: "'c"   and p :: "'d list"   and v :: "'c" assumes "wf_digraph G"   and "pre_digraph.cas G u p v" shows "hd (pre_digraph.awalk_verts G u p) = u"
KGnotempty: shows "map ((\<inter>) K) \<GG> \<noteq> []"
add_diff_cancel_right: fixes a :: "'b"   and c :: "'b"   and b :: "'b" shows "a + c - (b + c) = a - b"
moebius_similarity_def: fixes a :: "complex"   and b :: "complex" shows "moebius_similarity a b = mk_moebius a b 0 1"
fset_to_im: fixes f :: "'b \<Rightarrow> 'a"   and X :: "'b set" shows "{f x |x. x \<in> X} = f ` X"
linear_sum': fixes k :: "enat"   and f :: "('c \<Rightarrow> real) \<Rightarrow> real"   and S :: "'d set"   and g :: "'d \<Rightarrow> 'c \<Rightarrow> real" assumes "\<forall>x. x \<in> manifold_eucl.diff_fun_space k \<longrightarrow> f x \<in> UNIV"   and "\<forall>x. x \<in> S \<longrightarrow> g x \<in> manifold_eucl.diff_fun_space k"   and "linear_on (manifold_eucl.diff_fun_space k) UNIV (*\<^sub>R) (*\<^sub>R) f" shows "f (sum g S) = (\<Sum>a\<in>S. f (g a))"
val2FormAreSingleLiteralClauses: fixes clause :: "Literal list"   and valuation :: "Literal list" shows "clause el val2form valuation \<longrightarrow> (\<exists>literal. clause = [literal] \<and> literal el valuation)"
finfun_Diag_const_code_2: fixes b :: "'b"   and g :: "('a, 'c) finfun"   and a :: "'a"   and c :: "'c" shows "finfun_Diag (finfun_const b) (finfun_update_code g a c) = finfun_update_code (finfun_Diag (finfun_const b) g) a (b, c)"
Abs_viewres_inverse: fixes y :: "(('a \<times> 'b) \<times> ('a, 'b) FingerTreeStruc) option" assumes "y \<in> {r. case r of None \<Rightarrow> True | Some (a, t) \<Rightarrow> FingerTreeStruc.ft_invar t}" shows "Rep_viewres (Abs_viewres y) = y"
vec_lambda_inverse: fixes y :: "'a \<Rightarrow> 'b" assumes "y \<in> UNIV" shows "($h) (vec_lambda y) = y"
lformula_nFOr: fixes \<phi> :: "('a, 'k) aformula"   and \<psi> :: "('a, 'k) aformula" assumes "lformula (FOr \<phi> \<psi>)" shows "lformula (nFOr \<phi> \<psi>)"
MergeGuards_iff: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and F :: "'f set"   and P :: "'e set"   and c :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set" shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P merge_guards c Q,A = \<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P c Q,A"
LEX_cases: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "nat \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> 'a" assumes "LEX P f g" obtains "f = g"   | k :: "nat" where "\<forall>i<k. f i = g i" and "P (f k) (g k)"
add_pow_rdistr_int: fixes a :: "'a"   and b :: "'a"   and k :: "int" assumes "a \<in> carrier R"   and "b \<in> carrier R" shows "a \<otimes> [k] \<cdot> b = [k] \<cdot> (a \<otimes> b)"
a: fixes a :: "complex" shows "(cmod a)\<^sup>2 = Re (a * cnj a)"
simps_7: fixes f1 :: "'b"   and f2 :: "'b"   and f3 :: "'b" shows "(case Eq of Eq \<Rightarrow> f1 | Lt \<Rightarrow> f2 | Gt \<Rightarrow> f3) = f1"
lst_of_sp_is_vert: fixes g :: "('c \<times> nat, 'd) rgraph_scheme"   and v1 :: "'c \<times> nat"   and es :: "('c \<times> nat) edge list"   and v2 :: "'c \<times> nat"   and subs :: "(('c \<times> nat) \<times> 'c \<times> nat) set" assumes "SubRel.subpath g v1 es v2 subs" shows "v2 \<in> Graph.vertices g"
simps_1: fixes uu :: "'a list" shows "T_chop 0 uu = 1"
algorithmic_symmetry_1: fixes \<Gamma> :: "(name \<times> ty) list"   and s :: "trm"   and t :: "trm"   and T :: "ty" assumes "\<Gamma> \<turnstile> s \<Leftrightarrow> t : T" shows "\<Gamma> \<turnstile> t \<Leftrightarrow> s : T"
injectivity_subexpr1_2: fixes a :: "'b"   and A :: "'b set"   and X :: "'b set"   and P :: "'b set set"   and X' :: "'b set"   and P' :: "'b set set"   and k' :: "nat" assumes "a \<notin> A"   and "X \<in> P \<and> X' \<in> P'"   and "insert (insert a X) (P - {X}) = insert (insert a X') (P' - {X'})"   and "(partition_on A P \<and> card P = Suc k') \<and> partition_on A P' \<and> card P' = Suc k'" shows "X = X'"
reflclp_mulex_on_add_right: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set"   and M :: "'b multiset"   and N :: "'b multiset"   and a :: "'b" assumes "(mulex_on P A)\<^sup>=\<^sup>= M N"   and "M \<in> multisets A"   and "a \<in> A" shows "mulex_on P A M (N + {#a#})"
ord_eq_less_trans: fixes a :: "'c word"   and b :: "'c word"   and c :: "'c word" assumes "a = b"   and "b <s c" shows "a <s c"
max_lor: fixes i :: "nat"   and nl :: "nat list" assumes "i < length nl" shows "Max {u. loR [godel_code nl, Pi (Suc i), u]} = nl ! i"
inject_1: fixes x2 :: "bool"   and y2 :: "bool" shows "(Bool x2 = Bool y2) = (x2 = y2)"
upb2_def: shows "upb2 = loc_upb (c_snd (c_snd n)) x"
vars_if_subtermeq_set: fixes x :: "'d"   and M :: "('c, 'd) Term.term set" assumes "Var x \<sqsubseteq>\<^sub>s\<^sub>e\<^sub>t M" shows "x \<in> fv\<^sub>s\<^sub>e\<^sub>t M"
split: fixes P :: "'j \<Rightarrow> bool"   and f1 :: "'j"   and f2 :: "'k \<Rightarrow> 'k list \<Rightarrow> 'j"   and list :: "'k list" shows "P (case list of [] \<Rightarrow> f1 | x # xa \<Rightarrow> f2 x xa) = ((list = [] \<longrightarrow> P f1) \<and> (\<forall>x21 x22. list = x21 # x22 \<longrightarrow> P (f2 x21 x22)))"
dagger_ext: fixes x :: "'b \<times> 'c" shows "less_eq_prod x (prod.qstar x)"
inf_apply: fixes f :: "'b \<Rightarrow> 'a"   and g :: "'b \<Rightarrow> 'a"   and x :: "'b" shows "(f \<sqinter> g) x = f x \<sqinter> g x"
Gauss_Jordan_upt_k_PA_def: fixes A :: "(('a, 'c) vec, 'b) vec"   and k :: "nat" shows "Gauss_Jordan_upt_k_PA A k = (let foldl = foldl Gauss_Jordan_column_k_PA (mat (1::'a), 0, A) [0..<Suc k] in (fst foldl, snd (snd foldl)))"
nsp: assumes "is_leveln_ftree 0 s"   and "is_measured_ftree s"   and "\<And>a b. p a \<Longrightarrow> p (a + b)"   and "\<not> p i"   and "p (i + gmft s)"   and "nsplitTree p i s = (l1, nd, r1)" shows "toList s = toList l1 @ nodeToList nd @ toList r1 \<and> \<not> p (i + gmft l1) \<and> p (i + gmft l1 + gmn nd) \<and> is_leveln_ftree 0 l1 \<and> is_measured_ftree l1 \<and> is_leveln_ftree 0 r1 \<and> is_measured_ftree r1 \<and> is_leveln_node 0 nd \<and> is_measured_node nd"