hcju/isabelleps huggingface dataset:
deterministic: fixes a :: "'edge"   and a' :: "'edge" assumes "valid_edge a"   and "valid_edge a'"   and "sourcenode a = sourcenode a'"   and "targetnode a \<noteq> targetnode a'"   and "intra_kind (kind a)"   and "intra_kind (kind a')" shows "\<exists>Q Q'. kind a = (Q)\<^sub>\<surd> \<and> kind a' = (Q')\<^sub>\<surd> \<and> (\<forall>s. (Q s \<longrightarrow> \<not> Q' s) \<and> (Q' s \<longrightarrow> \<not> Q s))"
EquivS: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>" shows "[\<phi> \<^bold>\<equiv> \<psi> in v] = ([\<phi> in v] = [\<psi> in v])"
ln_add_one_self_le_self2: fixes x :: "real" assumes "- 1 < x" shows "ln (1 + x) \<le> x"
Spy_analz_private_Key: fixes evs :: "event list"   and b :: "bool"   and A :: "agent" assumes "evs \<in> set_mr" shows "(Key (invKey (publicKey b A)) \<in> analz (knows Spy evs)) = (A \<in> bad)"
smap_Union_union: fixes m1 :: "'a \<Rightarrow> 'b set"   and ms :: "('a \<Rightarrow> 'b set) list" shows "m1 \<union>. \<Union>.ms = \<Union>.m1 # ms"
ide_u: shows "ide (tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0)"
lift_orient_lit_inst: fixes L :: "'a literal"   and t :: "'a trm"   and s :: "'a trm"   and polarity :: "sign"   and \<theta> :: "('a \<times> 'a trm) list"   and \<sigma> :: "('a \<times> 'a trm) list"   and \<eta> :: "('a \<times> 'a trm) list" assumes "orient_lit_inst L t s polarity \<theta>"   and "\<theta> \<doteq> \<sigma> \<lozenge> \<eta>" shows "orient_lit_inst L t s polarity \<sigma>"
fixed_point_imp_\<omega>_limit_set: fixes x :: "'a" assumes "x \<in> X"   and "f x = (0::'a)" shows "\<omega>_limit_set x = {x}"
simps_5: fixes \<phi> :: "'a ltln"   and \<psi> :: "'a ltln" shows "is_\<nu>LTL (\<phi> and\<^sub>n \<psi>) = (is_\<nu>LTL \<phi> \<and> is_\<nu>LTL \<psi>)"
mod_by_0: 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" assumes "class.semidom_modulo divide plus minus zero times one modulo" shows "modulo a zero = a"
yinv_1: shows "y \<otimes> y' = \<one>"
RS_simps_18: shows "Maps.cod S\<^sub>1 = Maps.MkIde (trg s)"
assms_1: shows "rt1 \<sqsubset>\<^bsub>i\<^esub> rt2"
ennreal_less_top_iff: fixes x :: "ennreal" shows "(x < \<top>) = (x \<noteq> \<top>)"
inj': fixes x :: "'a star"   and y :: "'a star" assumes "(*f* f) x = (*f* f) y" shows "x = y"
less_eq_empty: fixes q :: "iface" shows "Iface [] \<le> q"
step: fixes u :: "real"   and d :: "real"   and v :: "real"   and n :: "nat" shows "geom_rand_walk u d v (Suc n) = (\<lambda>w. (case w !! n of True \<Rightarrow> u | False \<Rightarrow> d) * geom_rand_walk u d v n w)"
comp_assoc: fixes f :: "'e \<Rightarrow> 'c"   and g :: "'d \<Rightarrow> 'e"   and h :: "'b \<Rightarrow> 'd" shows "f \<circ> g \<circ> h = f \<circ> (g \<circ> h)"
update_with_def: fixes v :: "'c"   and k :: "'b"   and f :: "'c \<Rightarrow> 'c"   and al :: "('b, 'c) assoc_list" shows "Assoc_List.update_with v k f al = Assoc_List (AList.update_with_aux v k f (assoc_list.impl_of al))"
\<I>'_1: shows "\<forall>X F. \<forall>X\<langle>\<or>\<noteq>: F\<rangle>\<^sub>s\<^sub>t \<in> set S \<longrightarrow> ineq_model \<I>' X F"
hrealpow_hyperpow_Infinitesimal_iff: fixes x :: "'a star"   and n :: "nat" shows "(x ^ n \<in> Infinitesimal) = (x pow hypnat_of_nat n \<in> Infinitesimal)"
append_is_Nil_conv: fixes xs :: "'a list"   and ys :: "'a list" shows "(xs @ ys = []) = (xs = [] \<and> ys = [])"
of_nat_floor: fixes r :: "'b" assumes "(0::'b) \<le> r" shows "of_nat (nat \<lfloor>r\<rfloor>) \<le> r"
is_Hcomp_HcompNml: fixes t :: "'a local.term"   and u :: "'a local.term" assumes "Nml t"   and "Nml u"   and "Src t = Trg u"   and "\<not> is_Prim\<^sub>0 t"   and "\<not> is_Prim\<^sub>0 u" shows "is_Hcomp (t \<^bold>\<lfloor>\<^bold>\<star>\<^bold>\<rfloor> u)"
step_def: fixes u :: "'g \<Rightarrow> 'node \<Rightarrow> 'val set"   and p :: "'g \<Rightarrow> 'node \<times> 'val \<Rightarrow> 'val list option"   and g :: "'g" shows "uninst.step u p g \<equiv> \<lambda>(ua, pa). (uninst.uses' (u(g := ua)) (p(g := pa)) g, uninst.phis' (u(g := ua)) (p(g := pa)) g)"
locations_def: fixes A :: "('a \<times> ('b, 'd) cconstraint \<times> ('b set \<times> 'a) pmf) set \<times> ('a \<Rightarrow> ('b, 'd) cconstraint)" shows "locations A \<equiv> fst ` Edges A \<union> (snd \<circ> snd \<circ> snd \<circ> snd) ` Edges A"
Inl_inject: fixes x :: "'ag"   and y :: "'ag" assumes "Inl x = Inl y" shows "x = y"
left_unique_conversep: fixes A :: "'e \<Rightarrow> 'd \<Rightarrow> bool" shows "left_unique A\<inverse>\<inverse> = right_unique A"
is_free_def: fixes t :: "'a" shows "is_free t = (unfree t \<noteq> None)"
mod_eq_diff_mod_0_conv: fixes a :: "nat"   and b :: "nat"   and m :: "nat" assumes "a \<le> b" shows "(a mod m = b mod m) = ((b - a) mod m = 0)"
elim_2: fixes P :: "pi"   and Rel :: "(pi \<times> pi) set"   and Q :: "pi"   and \<alpha> :: "freeRes"   and Q' :: "pi" assumes "P \<leadsto>[Rel] Q"   and "Q \<longmapsto> \<alpha> \<prec> Q'" shows "\<exists>P'. P \<longmapsto> \<alpha> \<prec> P' \<and> (P', Q') \<in> Rel"
sym_sum_mset_inter2_sets_size: fixes blv :: "'a set" assumes "blv \<in># \<B>" shows "int (size (\<Sum>b2\<in>#remove1_mset blv \<B>. mset_set {y. y \<subseteq> blv \<inter> b2 \<and> card y = 2})) = int (nat \<k> choose 2) * (\<Lambda> - 1)"
phi_inf_principal: fixes x :: "'a"   and y :: "'b" shows "\<exists>z. up_filter z = phi x \<sqinter> up_filter y"
0: fixes \<pi> :: "('state \<times> 'aprop set) stream"   and \<pi>' :: "('state \<times> 'aprop set) stream" shows "wfp AP' \<pi> \<and> wfp AP' \<pi>' \<longrightarrow> \<phi> [] [\<pi>, \<pi>'] = f (stateOf \<pi>) (stateOf \<pi>') \<and> \<psi> [] [\<pi>, \<pi>'] = g (stateOf \<pi>) (stateOf \<pi>')"
config'_rand_snoc: fixes A :: "('b list \<Rightarrow> 'c pmf) \<times> ('b list \<times> 'c \<Rightarrow> 'b \<Rightarrow> ((nat \<times> nat list) \<times> 'c) pmf)"   and s :: "('b list \<times> 'c) pmf"   and rs :: "'b list"   and r :: "'b" shows "Partial_Cost_Model.config'_rand A s (rs @ [r]) = Partial_Cost_Model.config'_rand A s rs \<bind> (\<lambda>s. snd A s r \<bind> (\<lambda>(a, is'). return_pmf (step (fst s) r a, is')))"
carrier_dim_vec: fixes v :: "'a vec"   and n :: "nat" shows "(v \<in> carrier_vec n) = (dim_vec v = n)"
conseq: shows "P \<subseteq> {s. \<exists>Z. s \<in> P' Z \<and> (\<forall>t. t \<in> Q' Z \<longrightarrow> t \<in> Q)}"
approx_unique_real: fixes r :: "real star"   and s :: "real star"   and x :: "real star" assumes "r \<in> \<real>"   and "s \<in> \<real>"   and "r \<approx> x"   and "s \<approx> x" shows "r = s"
subst_cls_mset_single: fixes C :: "'a literal multiset"   and \<sigma> :: "'s" shows "{#C#} \<cdot>cm \<sigma> = {#C \<cdot> \<sigma>#}"
tj_stack_ss_stack_finished: shows "set (tj_stack s) \<subseteq> set (stack s) \<union> dom (finished s)"
lemma2_6_6_b_one: fixes r :: "('a \<times> 'a) set"   and q :: "'a"   and M :: "'a multiset"   and N :: "'a multiset" assumes "trans r"   and "irrefl r"   and "(add_mset q M, add_mset q N) \<in> mul_eq r" shows "(M, N) \<in> mul_eq r"
power_rep_eq: fixes P :: "('d, 'c) uexpr"   and n :: "nat" shows "\<lbrakk>P ^ n\<rbrakk>\<^sub>e = (\<lambda>b. \<lbrakk>P\<rbrakk>\<^sub>e b ^ n)"
gso_carrier': fixes j :: "nat" assumes "\<And>i. i \<le> j \<Longrightarrow> fs ! i \<in> carrier_vec n" shows "gso j \<in> carrier_vec n"
induct: fixes P1 :: "('sf, 'sc, 'sz) hp \<Rightarrow> bool"   and P2 :: "('sf, 'sc, 'sz) formula \<Rightarrow> bool"   and x1 :: "('sf, 'sc, 'sz) hp"   and x2 :: "('sf, 'sc, 'sz) formula" assumes "\<And>x \<theta>. P1 (x := \<theta>)"   and "\<And>x \<theta>. P1 (DiffAssign x \<theta>)"   and "\<And>\<phi>. \<lbrakk>FRadmit \<phi>; P2 \<phi>\<rbrakk> \<Longrightarrow> P1 (? \<phi>)"   and "\<And>ODE \<phi>. \<lbrakk>ORadmit ODE; FRadmit \<phi>; P2 \<phi>\<rbrakk> \<Longrightarrow> P1 (EvolveODE ODE \<phi>)"   and "\<And>a b. \<lbrakk>PRadmit a; P1 a; PRadmit b; P1 b\<rbrakk> \<Longrightarrow> P1 (a \<union>\<union> b)"   and "\<And>a b. \<lbrakk>PRadmit a; P1 a; PRadmit b; P1 b\<rbrakk> \<Longrightarrow> P1 (a ;; b)"   and "\<And>a. \<lbrakk>PRadmit a; P1 a\<rbrakk> \<Longrightarrow> P1 a**"   and "\<And>\<theta>1 \<theta>2. P2 (Geq \<theta>1 \<theta>2)"   and "\<And>p args. P2 ($\<phi> p args)"   and "\<And>\<phi>. \<lbrakk>FRadmit \<phi>; P2 \<phi>\<rbrakk> \<Longrightarrow> P2 (! \<phi>)"   and "\<And>\<phi> \<psi>. \<lbrakk>FRadmit \<phi>; P2 \<phi>; FRadmit \<psi>; P2 \<psi>\<rbrakk> \<Longrightarrow> P2 (\<phi> && \<psi>)"   and "\<And>\<phi> x. \<lbrakk>FRadmit \<phi>; P2 \<phi>\<rbrakk> \<Longrightarrow> P2 (Exists x \<phi>)"   and "\<And>\<alpha> \<phi>. \<lbrakk>PRadmit \<alpha>; P1 \<alpha>; FRadmit \<phi>; P2 \<phi>\<rbrakk> \<Longrightarrow> P2 (\<langle> \<alpha> \<rangle> \<phi>)" shows "(PRadmit x1 \<longrightarrow> P1 x1) \<and> (FRadmit x2 \<longrightarrow> P2 x2)"
blue_dfs_correct: fixes G :: "('a, 'b) b_graph_rec_scheme" assumes "b_graph G"   and "finite ((g_E G)\<^sup>* `` g_V0 G)" shows "blue_dfs G \<le> SPEC (\<lambda>r. case r of None \<Rightarrow> \<forall>L. \<not> b_graph.is_lasso_prpl G L | Some L \<Rightarrow> b_graph.is_lasso_prpl G L)"
assms_2: shows "(stateA, stateB) \<in> transitionRelation F0 decisionVars"
One_Redex: fixes x :: "name"   and t :: "lam"   and s :: "lam"   and r :: "lam" assumes " App (Lam [x].t) s \<longrightarrow>\<^sub>1 r"   and "x \<sharp> (s, r)" shows "(\<exists>t' s'. r = App (Lam [x].t') s' \<and>  t \<longrightarrow>\<^sub>1 t' \<and>  s \<longrightarrow>\<^sub>1 s') \<or> (\<exists>t' s'. r = t'[x::=s'] \<and>  t \<longrightarrow>\<^sub>1 t' \<and>  s \<longrightarrow>\<^sub>1 s')"
pmf_of_mapping_mapping_of_pmf: fixes p :: "'b pmf" shows "pmf_of_mapping (mapping_of_pmf p) = p"
inner_diff_right: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and inner :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "class.real_inner minus dist norm plus zero uminus scaleR sgn uniformity open inner" shows "inner x (minus y z) = inner x y - inner x z"
toSink: fixes s :: "'state" assumes "s \<in> S" shows "toSink s \<in> \<delta> s \<inter> Sink"
et_let: fixes \<Gamma> :: "nat \<Rightarrow> pdf_type"   and e1 :: "expr"   and t1 :: "pdf_type"   and e2 :: "expr"   and t2 :: "pdf_type" assumes "\<Gamma> \<turnstile> e1 : t1"   and "case_nat t1 \<Gamma> \<turnstile> e2 : t2" shows "\<Gamma> \<turnstile> LET e1 IN e2 : t2"
fls_prpart_base_factor: fixes f :: "'a fls" shows "fls_prpart (fls_base_factor f) = 0"
sbelow_sctn_def: fixes X :: "'b set"   and sctn :: "'b sctn" shows "sbelow_sctn X sctn = SPEC (\<lambda>b. b \<longrightarrow> X \<subseteq> sbelow_halfspace sctn)"
strictly_maximal_literal_def: fixes S :: "'at Literal set"   and A :: "'at Literal" shows "strictly_maximal_literal S A \<equiv> A \<in> S \<and> (\<forall>B. B \<in> S \<and> A \<noteq> B \<longrightarrow> literal_ordering B A)"
psubst_image: fixes f :: "'c \<Rightarrow> 'a"   and p :: "('c, 'b) form" shows "params (psubst f p) = f ` params p"
IH: shows "lookup (\<lambda>x. x = k) al = lookup (\<lambda>x. x = k) (AList.clearjunk al)"
sup_formula_def: fixes x :: "'a formula"   and y :: "'a formula" shows "x \<squnion> y = Abs_formula (Rep_formula x \<union> Rep_formula y)"
bin_lcp_def: shows "symcode.\<alpha> = u\<^sub>0 \<cdot> u\<^sub>1 \<and>\<^sub>p u\<^sub>1 \<cdot> u\<^sub>0"
Domainp_apply2I: fixes A :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and B :: "'g \<Rightarrow> 'i \<Rightarrow> bool"   and C :: "'h \<Rightarrow> 'j \<Rightarrow> bool"   and f :: "'e \<Rightarrow> 'g \<Rightarrow> 'h"   and g :: "'f \<Rightarrow> 'i \<Rightarrow> 'j"   and x :: "'e"   and y :: "'f"   and x' :: "'g"   and y' :: "'i" assumes "(A ===> B ===> C) f g"   and "A x y"   and "B x' y'" shows "Domainp C (f x x')"
nat_filtration_def: fixes n :: "nat" shows "nat_filtration n = fct_gen_subalgebra M M (pseudo_proj_True n)"
l4_14R1: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and A' :: "'p"   and B' :: "'p" assumes "Bet A B C"   and "Cong A B A' B'" shows "\<exists>C'. A B C Cong3 A' B' C'"
A: shows "ipurge_tr I D (D x) (x # xs) = ipurge_tr I D (D x) xs"
elims: fixes getIJ :: "(nat set \<times> 'b option list set) set \<Rightarrow> (nat set \<times> 'b option list set) set \<Rightarrow> nat set \<Rightarrow> nat set \<times> nat set"   and x :: "nat set"   and xa :: "nat set \<times> 'b option list set"   and y :: "nat set \<times> 'b option list set" assumes "getIJ getIJ"   and "New_max.projectTable x xa = y" obtains V :: "nat set" and s :: "nat set" and t :: "'b option list set" where "x = V" and "xa = (s, t)" and "y = (s \<inter> V, restrict V ` t)"
disjE3: fixes A :: "bool"   and B :: "bool"   and C :: "bool" assumes "A \<or> B \<or> C" obtains "A"   | "B"   | "C"
wf_norm: fixes idx :: "'n"   and \<phi> :: "('a, 'k) aformula" assumes "local.wf idx \<phi>" shows "local.wf idx (norm \<phi>)"
nb\<^sub>1: fixes op' :: "('a \<times> 'b) strips_operator" assumes "op' \<in> set ops'" shows "\<exists>op\<in>set (map (strips_op_to_sasp \<Psi>) ops'). op \<in> set (map (strips_op_to_sasp \<Psi>) ops') \<and> op = \<phi>\<^sub>O\<inverse> \<Psi> op' \<and> is_valid_operator_sas_plus \<Psi> op"
inverses: fixes g :: "'a" assumes "g \<in> G" shows "\<exists>h\<in>G. binop g h = e \<and> binop h g = e"
prod_filtermap2: fixes F :: "'d filter"   and g :: "'f \<Rightarrow> 'e"   and G :: "'f filter" shows "F \<times>\<^sub>F filtermap g G = filtermap (apsnd g) (F \<times>\<^sub>F G)"
M_def: fixes x :: "int" shows "pl.M x = x mod p ^ l"
min_dist_z: fixes v :: "nat" shows "min_dist v v = 0"
entropy_distr: fixes MX :: "'d measure"   and X :: "'a \<Rightarrow> 'd"   and f :: "'d \<Rightarrow> real" assumes "distributed M MX X (\<lambda>x. ennreal (f x))"   and "\<And>x. x \<in> space MX \<Longrightarrow> 0 \<le> f x" shows "entropy b MX X = - (LINT x|MX. f x * log b (f x))"
\<K>_def: fixes kripke :: "('a, 'b) kripke" shows "\<K> kripke = (case kripke of Kripke x1 x2 \<Rightarrow> x2)"
irreducible\<^sub>d_factorization_exists_1: fixes p :: "'a poly" assumes "0 < Polynomial.degree p" shows "\<exists>fs. fs \<noteq> [] \<and> (\<forall>f\<in>set fs. irreducible\<^sub>d f \<and> Polynomial.degree f \<le> Polynomial.degree p) \<and> p = prod_list fs"
any_OK_conv: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list"   and X :: "ty err"   and T' :: "ty" shows "P \<turnstile> X \<le>\<^sub>\<top> OK T' = (\<exists>T. X = OK T \<and> subtype P T T')"
saturated_complete: fixes N :: "'a literal multiset set" assumes "saturated N"   and "\<not> satisfiable N" shows "{#} \<in> N"
adopt_node_is_l_adopt_node: shows "l_adopt_node DocumentClass.type_wf DocumentClass.known_ptr DocumentClass.known_ptrs get_parent adopt_node adopt_node_locs get_child_nodes get_owner_document"
prefix_to_wordset_ipset_from_cidr: fixes pfx :: "'b prefix_match" assumes "valid_prefix pfx" shows "prefix_to_wordset pfx = ipset_from_cidr (pfxm_prefix pfx) (pfxm_length pfx)"
assms_2: fixes n :: "nat" shows "S n \<noteq> {}"
fs: fixes fa :: "int poly" assumes "fa \<in> set fs" shows "Mp fa = fa \<and> monic fa"
arc_not_bot: fixes x :: "'a" assumes "arc x" shows "x \<noteq> bot"
rep_eq: fixes x :: "'c stone_phi_pair"   and xa :: "'c stone_phi_pair" shows "Rep_stone_phi_pair (x \<squnion> xa) = stone_phi.pairs_sup (Rep_stone_phi_pair x) (Rep_stone_phi_pair xa)"
IterI2: fixes Y :: "'b list set"   and yrhs :: "'b trm set"   and ES :: "('b list set \<times> 'b trm set) set"   and X :: "'b list set"   and Q :: "('b list set \<times> 'b trm set) set \<Rightarrow> bool" assumes "(Y, yrhs) \<in> ES"   and "X \<noteq> Y"   and "\<And>Y yrhs. \<lbrakk>(Y, yrhs) \<in> ES; X \<noteq> Y\<rbrakk> \<Longrightarrow> Q (Remove ES Y yrhs)" shows "Q (Iter X ES)"
simps_3: fixes p :: "'a pfm"   and q :: "'a pfm" shows "static (p \<^bold>\<or>\<^sub>! q) = (static p \<and> static q)"
tree_type: fixes t :: "'c bro"   and h :: "nat" assumes "t \<in> Bp h" shows "tree t \<in> B h \<union> B (Suc h)"
mplus_fail: fixes m :: "'r match" shows "Fixrec.fail +++ m = m"
SUP_mono: fixes A :: "'b set"   and B :: "'c set"   and f :: "'b \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "\<And>n. n \<in> A \<Longrightarrow> \<exists>m\<in>B. f n \<le> g m" shows "\<Squnion>(f ` A) \<le> \<Squnion>(g ` B)"
interleave_rev_cons_1: fixes l :: "'b list"   and a :: "'b list"   and b :: "'b list"   and x :: "'b" assumes "l \<in> a \<otimes> b" shows "l @ [x] \<in> a @ [x] \<otimes> b"
induct: fixes P :: "'d list \<Rightarrow> ('d \<Rightarrow> 'e set \<times> 'e set) \<Rightarrow> 'd list \<Rightarrow> bool"   and a0 :: "'d list"   and a1 :: "'d \<Rightarrow> 'e set \<times> 'e set"   and a2 :: "'d list" assumes "\<And>\<alpha> w. P [] \<alpha> w"   and "\<And>v va \<alpha>. P (v # va) \<alpha> []"   and "\<And>e1 w1 \<alpha> e2 w2. \<lbrakk>fst (\<alpha> e1) \<inter> mon_pl (map \<alpha> (e2 # w2)) = {} \<Longrightarrow> P w1 \<alpha> (e2 # w2); fst (\<alpha> e2) \<inter> mon_pl (map \<alpha> (e1 # w1)) = {} \<Longrightarrow> P (e1 # w1) \<alpha> w2\<rbrakk> \<Longrightarrow> P (e1 # w1) \<alpha> (e2 # w2)" shows "P a0 a1 a2"
elims: fixes x :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and xa :: "'a"   and xb :: "'a list"   and y :: "'a" assumes "rexp_of_list x xa xb = y" obtains OP :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" and N :: "'a" where "x = OP__" and "xa = N__" and "xb = []" and "y = N__"   | OP :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" and N :: "'a" and r :: "'a" where "x = OPa__" and "xa = Na__" and "xb = [r__]" and "y = r__"   | OP :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" and N :: "'a" and r :: "'a" and v :: "'a" and va :: "'a list" where "x = OP" and "xa = N" and "xb = r # v # va" and "y = OP r (rexp_of_list OP N (v # va))"
ivl_disj_un_two_7: fixes l :: "'a"   and m :: "'a"   and u :: "'a" assumes "l \<le> m"   and "m \<le> u" shows "{l..<m} \<union> {m..u} = {l..u}"
word_and_notzeroD: fixes w :: "'b"   and w' :: "'b" assumes "w AND w' \<noteq> (0::'b)" shows "w \<noteq> (0::'b) \<and> w' \<noteq> (0::'b)"
B: shows "sinks_aux I D U ys \<subseteq> V"
assoc: fixes a :: "'b DBMEntry"   and b :: "'b DBMEntry"   and c :: "'b DBMEntry" shows "dbm_add (dbm_add a b) c = dbm_add a (dbm_add b c)"
in_inv_image: fixes x :: "'d"   and y :: "'d"   and r :: "('e \<times> 'e) set"   and f :: "'d \<Rightarrow> 'e" shows "((x, y) \<in> inv_image r f) = ((f x, f y) \<in> r)"
step'_def: fixes num' :: "int \<Rightarrow> 'a"   and plus' :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and \<gamma> :: "'a \<Rightarrow> int set" assumes "Abs_Int num' plus' \<gamma>" shows "Abs_Int.step' num' plus' = Step (\<lambda>x e S. case S of None \<Rightarrow> None | Some S \<Rightarrow> Some (update S x (Gamma_semilattice.aval' num' plus' e S))) (\<lambda>b S. S)"
Nonce_compromise: fixes evs :: "event list"   and KK :: "nat set"   and N :: "nat" assumes "evs \<in> set_cr"   and "\<forall>K\<in>KK. \<not> KeyCryptNonce K N evs" shows "(Nonce N \<in> analz (Key ` KK \<union> knows Spy evs)) = (Nonce N \<in> analz (knows Spy evs))"
CallNull\<^sub>1: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> nat exp) list) list"   and e :: "nat exp"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> val list"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> val list"   and es :: "nat exp list"   and vs :: "val list"   and s\<^sub>2 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> val list"   and M :: "char list" assumes "P \<turnstile>\<^sub>1 \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>null,s\<^sub>1\<rangle>"   and "P \<turnstile>\<^sub>1 \<langle>es,s\<^sub>1\<rangle> [\<Rightarrow>] \<langle>map Val vs,s\<^sub>2\<rangle>" shows "P \<turnstile>\<^sub>1 \<langle>e\<bullet>M(es),s\<^sub>0\<rangle> \<Rightarrow> \<langle>THROW NullPointer,s\<^sub>2\<rangle>"
x0_less_x1: shows "x\<^sub>0 < x\<^sub>1"
nprv_clear5_4: fixes \<phi>1 :: "'fmla"   and \<phi>2 :: "'fmla"   and \<phi>3 :: "'fmla"   and \<phi>5 :: "'fmla"   and \<psi> :: "'fmla"   and \<phi>4 :: "'fmla" assumes "nprv {\<phi>1, \<phi>2, \<phi>3, \<phi>5} \<psi>"   and "\<phi>1 \<in> fmla"   and "\<phi>2 \<in> fmla"   and "\<phi>3 \<in> fmla"   and "\<phi>4 \<in> fmla"   and "\<phi>5 \<in> fmla"   and "\<psi> \<in> fmla" shows "nprv {\<phi>1, \<phi>2, \<phi>3, \<phi>4, \<phi>5} \<psi>"
Pair_synth: fixes X :: "msg"   and Y :: "msg"   and H :: "msg set" assumes "\<langle>X, Y\<rangle> \<in> synth H" obtains "\<langle>X, Y\<rangle> \<in> H"   | "X \<in> synth H" and "Y \<in> synth H"
stabilizer_contains_P: fixes P :: "'a set"   and p :: "nat" assumes "finite (carrier (G\<lparr>carrier := P\<rparr>))"   and "P \<in> conjP.subgroups_of_size p" shows "P \<subseteq> group_action.stabilizer (G\<lparr>carrier := P\<rparr>) (conjP.conjugation_action p) P"
\<O>_on_emptyset: shows "\<O> {} = {\<O>b}"
dord_pm_alt: fixes ord :: "('a \<Rightarrow>\<^sub>0 'c) \<Rightarrow> ('a \<Rightarrow>\<^sub>0 'c) \<Rightarrow> bool" shows "dord_pm ord = (\<lambda>x y. deg_pm x < deg_pm y \<or> deg_pm x = deg_pm y \<and> ord x y)"
EntryPath_butlast_less_last: fixes g :: "'g"   and ns :: "'node list"   and z :: "'node" assumes "EntryPath g ns"   and "z \<in> set (butlast ns)" shows "shortestPath g z < shortestPath g (last ns)"
vec_def: fixes n :: "nat"   and x :: "'a list" shows "Matrix_Legacy.vec n x = (length x = n)"
compact_lfiniteD: fixes xs :: "'c llist" assumes "ccpo.compact lSup (\<sqsubseteq>) xs" shows "lfinite xs"
eq_up_to_le: fixes k :: "nat"   and n :: "nat"   and \<pi> :: "nat \<Rightarrow> 'n"   and \<pi>' :: "nat \<Rightarrow> 'n" assumes "k \<le> n"   and "\<pi> =\<^bsub>n\<^esub> \<pi>'" shows "\<pi> =\<^bsub>k\<^esub> \<pi>'"
not_outside_2: fixes x :: "'b"   and y :: "'b" assumes "R i x y" shows "y \<in> carrier"
separatedin_sym: fixes X :: "'c topology"   and S :: "'c set"   and T :: "'c set" shows "separatedin X S T = separatedin X T S"
sums_append2: fixes xs :: "'a list"   and ys :: "'a list" shows "sums (xs @ ys) = butlast (sums xs) @ map ((+) (sum_list xs)) (sums ys)"
assms_2: fixes n :: "('a, 'b) insert_message" assumes "n \<in> M \<union> {m}" shows "I m \<notin> deps (Insert n)"
dijkstra_list_def: fixes l :: "('b \<times> 'b \<times> nat) list"   and s :: "'b" shows "dijkstra_list l s \<equiv> if valid_graph_rep l then Some (dijkstra (from_list l) s) else None"
fold_set_union: fixes A :: "'a set"   and B :: "'a set"   and x :: "'b" assumes "finite A"   and "finite B" shows "Finite_Set.fold f x (A \<union> B) = Finite_Set.fold f (Finite_Set.fold f x A) B"
MC_T_not_sconst: fixes f :: "'s \<Rightarrow> 'd"   and u :: "'d"   and X :: "'s cfg set"   and r :: "ennreal"   and cfg :: "'s cfg"   and x :: "'s" assumes "\<forall>l. \<forall>cfg\<in>\<Union> (cfg_on ` f -` {u}) \<inter> X. emeasure (measure_pmf (action cfg)) (f -` {u}) \<le> r"   and "r < 1"   and "cfg \<in> cfg_on x"   and "almost_everywhere (MC.T cfg) (pred_stream (\<lambda>x. x \<in> X))" shows "AE \<omega> in MC.T cfg. smap (f \<circ> state) \<omega> \<noteq> sconst u"
R5_4: shows "pmf (sds R5) d = 1 / 2"
pnTimes_flatten_PLUS: fixes X1 :: "'a rexp set"   and r :: "'a rexp" assumes "X1 \<noteq> {}"   and "finite X1"   and "\<forall>x\<in>X1. \<not> is_Zero x \<and> \<not> is_Plus x" shows "pnTimes (flatten PLUS X1) r = flatten PLUS (Timess X1 r)"
bind_def: fixes xs :: "'f list"   and f :: "'f \<Rightarrow> 'e list" shows "xs \<bind> f = concat (map f xs)"