hcju/isabelleps huggingface dataset:
ide_right: shows "ide (tab\<^sub>0 g \<star> Tfg.p\<^sub>0)\<^sup>*"
union_frontiers_nonneg: fixes summary :: "'a \<Rightarrow> 'a \<Rightarrow> 'b antichain"   and results_in :: "'c \<Rightarrow> 'b \<Rightarrow> 'c"   and c :: "('a, 'c) configuration"   and loc :: "'a"   and t :: "'c" assumes "dataflow_topology summary results_in" shows "0 \<le> zcount (\<Sum>loc'\<in>UNIV. dataflow_topology.after_summary results_in (zmset_of (mset_set (set_antichain (frontier (c_imp c loc'))))) (summary loc' loc)) t"
tdghm_dghm_comp_NTMap_app: fixes x :: "V"   and \<NN> :: "V" assumes "x \<in>\<^sub>\<circ> \<BB>\<lparr>Obj\<rparr>" shows "(\<NN> \<circ>\<^sub>T\<^sub>D\<^sub>G\<^sub>H\<^sub>M\<^sub>-\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<GG>)\<lparr>NTMap\<rparr>\<lparr>x\<rparr> = \<NN>\<lparr>NTMap\<rparr>\<lparr>\<GG>\<lparr>ObjMap\<rparr>\<lparr>x\<rparr>\<rparr>"
WHILET_unfold: fixes b :: "'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'c nres"   and s :: "'c" shows "WHILE\<^sub>T b f s = (if b s then f s \<bind> WHILE\<^sub>T b f else RETURN s)"
intro: fixes final1 :: "'a \<Rightarrow> bool"   and r1 :: "('d, 'b, 'a, 'c, 'e, 'f) semantics"   and final2 :: "'g \<Rightarrow> bool"   and r2 :: "('d, 'b, 'g, 'h, 'e, 'f) semantics"   and bisim :: "'b \<Rightarrow> 'a \<times> 'c \<Rightarrow> 'g \<times> 'h \<Rightarrow> bool"   and bisim_wait :: "'a \<Rightarrow> 'g \<Rightarrow> bool"   and \<tau>move1 :: "('d, 'b, 'a, 'c, 'e, 'f) \<tau>moves"   and \<tau>move2 :: "('d, 'b, 'g, 'h, 'e, 'f) \<tau>moves" assumes "FWdelay_bisimulation_obs final1 r1 final2 r2 bisim bisim_wait \<tau>move1 \<tau>move2"   and "FWdelay_bisimulation_diverge_axioms r1 r2 bisim \<tau>move1 \<tau>move2" shows "FWdelay_bisimulation_diverge final1 r1 final2 r2 bisim bisim_wait \<tau>move1 \<tau>move2"
ras_raw_invar_def: fixes s :: "'c array \<times> nat" shows "ras_raw_invar s \<equiv> snd s \<le> array_length (fst s)"
is_extensional: fixes f :: "'a" assumes "\<not> Right_a.arr f" shows "Right_a.R f = Right_a.null"
triangle_G: shows "Adj.G\<epsilon>o\<eta>G.map = G"
a_4: fixes x :: "'a"   and y :: "'a" shows "ad x \<le> ad (x \<cdot> y)"
agree_Nil: fixes s :: "'a \<Rightarrow> bool" shows "agree s Mapping.empty"
vp_nonzero: fixes v :: "'a \<Rightarrow> ant" assumes "valuation K v" shows "vp K v \<noteq> {\<zero>\<^bsub>Vr K v\<^esub>}"
mark_guards_simps_1: fixes f :: "'f" shows "mark_guards f Skip = Skip"
measurable_distr2: fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'd"   and L :: "'a measure"   and M :: "'b measure"   and N :: "'d measure"   and g :: "'a \<Rightarrow> 'b measure" assumes "(\<lambda>(x, y). f x y) \<in> L \<Otimes>\<^sub>M M \<rightarrow>\<^sub>M N"   and "g \<in> L \<rightarrow>\<^sub>M subprob_algebra M" shows "(\<lambda>x. distr (g x) N (f x)) \<in> L \<rightarrow>\<^sub>M subprob_algebra N"
add: fixes x :: "real"   and y :: "real"   and xa :: "'a" shows "(x + y) *\<^sub>R xa = x *\<^sub>R xa + y *\<^sub>R xa"
card_order_on_Card_order: fixes A :: "'a set"   and r :: "('a \<times> 'a) set" assumes "card_order_on A r" shows "A = Field r \<and> Card_order r"
g_impl_rel_ext_internal_def: fixes Rm :: "('f \<times> 'h) set"   and Rv :: "('e \<times> 'g) set" shows "g_impl_rel_ext Rm Rv \<equiv> \<langle>Rm, \<langle>Rv\<rangle>fun_set_rel, \<langle>Rv\<rangle>slg_rel, \<langle>Rv\<rangle>list_set_rel\<rangle>gen_g_impl_rel_ext"
intros_1: fixes p :: "'d"   and A :: "('e, 'f, 'd, 'g) FSM_scheme" assumes "p = initial A" shows "p \<in> nodes A"
T_def: shows "T \<equiv> dirichlet_prod' a (\<lambda>x. real_of_int \<lfloor>x\<rfloor>)"
hyps: assumes "Suc 0 \<le> n" shows "of_nat (stirling (Suc n) 2) = fact n * (\<Sum>k = 1..n. (1::'a) / of_nat k)"
ospecD: fixes x :: "'d option"   and P :: "'d \<Rightarrow> bool"   and y :: "'d" assumes "ospec x P"   and "x = Some y" shows "P y"
inf_mono: fixes a :: "'d upred"   and c :: "'d upred"   and b :: "'d upred"   and d :: "'d upred" assumes "c \<sqsubseteq> a"   and "d \<sqsubseteq> b" shows "(c \<and> d) \<sqsubseteq> (a \<and> b)"
local_lipschitz_def: fixes T :: "'a set"   and X :: "'b set"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" shows "local_lipschitz T X f \<equiv> \<forall>x\<in>X. \<forall>t\<in>T. \<exists>u>0. \<exists>L. \<forall>t\<in>cball t u \<inter> T. L-lipschitz_on (cball x u \<inter> X) (f t)"
set_to_map_insert: fixes kv :: "'a \<times> 'b"   and S :: "('a \<times> 'b) set" assumes "fst kv \<notin> fst ` S" shows "set_to_map (insert kv S) = set_to_map S(fst kv \<mapsto> snd kv)"
f4: shows "4 \<le> rk {A, B, P, a}"
lq_suf_suf: fixes r :: "'b list"   and s :: "'b list" assumes "r \<le>p s" shows "r\<inverse>\<^sup>>s \<le>s s"
Limsup_const: fixes F :: "'b filter"   and c :: "'c" assumes "F \<noteq> bot" shows "Limsup F (\<lambda>x. c) = c"
HARootStates_notmem_Range_ChiRel: fixes S :: "'a"   and A :: "('a, 'b, 'c) hierauto" assumes "S \<in> States (HARoot A)" shows "S \<notin> Range (ChiRel A)"
rotate_toppos_kauff_mat: shows "kauff_mat (basic [vert, over] \<circ> basic [cap, vert]) = kauff_mat (basic [brick.under, vert] \<circ> basic [vert, cap])"
diff_inf_eq_sup: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "class.lattice_ab_group_add plus zero minus uminus less_eq less inf sup" shows "minus a (inf b c) = plus a (sup (uminus b) (uminus c))"
InfCard_ge_ord_of_nat: fixes \<kappa> :: "V"   and n :: "nat" assumes "InfCard \<kappa>" shows "ord_of_nat n \<le> \<kappa>"
unitary_bound: fixes P :: "'c \<Rightarrow> real" assumes "unitary P" shows "bounded_by 1 P"
Inl_Inr_False: fixes x :: "'g"   and y :: "'h" shows "(Inl x = Inr y) = False"
prv_fls: fixes \<phi> :: "'fmla" shows "prv (imp fls \<phi>)"
vpairsI: fixes x :: "V"   and r :: "V"   and a :: "V"   and b :: "V" assumes "x \<in>\<^sub>\<circ> r"   and "x = \<langle>a, b\<rangle>" shows "x \<in>\<^sub>\<circ> vpairs r"
rewrite_negated_dst_ports_def: fixes m :: "'b common_primitive match_expr" shows "rewrite_negated_dst_ports m \<equiv> rewrite_negated_primitives (is_Dst_Ports, dst_ports_sel) Dst_Ports l4_ports_negate_one m"
sorted_Consb: fixes x :: "('b, 'c) Combinators"   and xs :: "('b, 'c) Combinators list"   and l :: "'b set list" assumes "all_in_list (x # xs) l"   and "singleCombinators (x # xs)"   and "FWNormalisationCore.sorted xs l \<and> (\<forall>y\<in>set xs. smaller x y l)" shows "FWNormalisationCore.sorted (x # xs) l"
subgroup_of_subgroup: fixes U :: "'a set" assumes "Group.group G"   and "subgroup U (G\<lparr>carrier := H\<rparr>)" shows "subgroup U G"
fg: shows "\<forall>z\<in>ball 0 1. g z \<in> h ` S \<and> f (g z) = z"
Hash_imp_body: fixes A :: "agent"   and X :: "msg"   and evs :: "event list" assumes "Hash \<lbrace>Key (shrK A), X\<rbrace> \<in> parts (knows Spy evs)"   and "evs \<in> recur"   and "A \<notin> bad" shows "X \<in> parts (knows Spy evs)"
split_asm: fixes P :: "'c \<Rightarrow> bool"   and f1 :: "nat \<Rightarrow> 'c"   and f2 :: "'c"   and x :: "enat" shows "P (case x of enat x \<Rightarrow> f1 x | \<infinity> \<Rightarrow> f2) = (\<not> ((\<exists>nat. x = enat nat \<and> \<not> P (f1 nat)) \<or> x = \<infinity> \<and> \<not> P f2))"
extf_ext_insert_arg: fixes f :: "'s"   and gt :: "('s, 'v) tm \<Rightarrow> ('s, 'v) tm \<Rightarrow> bool"   and xs :: "('s, 'v) tm list"   and z :: "('s, 'v) tm"   and ys :: "('s, 'v) tm list" shows "extf f gt (xs @ z # ys) (xs @ ys)"
Rep_SkewBinomialHeap_invar: fixes x :: "('a, 'b) SkewBinomialHeap" shows "BsSkewBinomialHeapStruc.bs_invar (Rep_SkewBinomialHeap x)"
maxdeg_def: fixes A :: "(('d \<Rightarrow>\<^sub>0 'c) \<Rightarrow>\<^sub>0 'e) set" shows "maxdeg A = Max (poly_deg ` A)"
FactGroup_def: fixes G :: "('e, 'f) monoid_scheme"   and H :: "'e set" shows "G Mod H = \<lparr>carrier = rcosets\<^bsub>G\<^esub> H, monoid.mult = (<#>\<^bsub>G\<^esub>), one = H\<rparr>"
finlsts_empty: shows "{}\<^sup>\<star> = {LNil}"
BExpr_lemma: fixes b :: "BExpr"   and X :: "Var set" assumes "BVars b \<subseteq> X"   and "\<forall>x. x \<in> X \<longrightarrow> CONTEXT x = low"   and "BExpr_noLoc b" shows "BExpr_low b"
sum_list_mono_complex_lex: fixes xs :: "complex list"   and ys :: "complex list" assumes "list_all2 (\<le>\<^sub>\<complex>) xs ys" shows "sum_list xs \<le>\<^sub>\<complex> sum_list ys"
code: fixes acc :: "real"   and n :: "real" shows "exp_series_stream_aux acc n = MSSCons acc (exp_series_stream_aux (acc / n) (n + 1))"
flowsto_from_empty_iff: fixes t :: "real set"   and b :: "('d \<times> 'e \<Rightarrow>\<^sub>L 'd) set"   and c :: "('d \<times> 'e \<Rightarrow>\<^sub>L 'd) set" shows "flowsto {} t b c"
poincare_between_transitivity: fixes a :: "complex_homo"   and x :: "complex_homo"   and b :: "complex_homo"   and y :: "complex_homo" assumes "a \<in> unit_disc"   and "x \<in> unit_disc"   and "b \<in> unit_disc"   and "y \<in> unit_disc"   and "poincare_between a x b"   and "poincare_between a b y" shows "poincare_between x b y"
eval_Neg_imp_Neg: fixes e :: "(name, hf) finfun"   and \<alpha> :: "fm"   and x :: "hf" assumes "\<lbrakk>\<guillemotleft>\<alpha>\<guillemotright>\<rbrakk>e = q_Neg x" shows "\<exists>A. \<alpha> = Neg A \<and> \<lbrakk>\<guillemotleft>A\<guillemotright>\<rbrakk>e = x"
normalize_left_idem: fixes a :: "'a"   and b :: "'a" shows "lcm (normalize a) b = lcm a b"
qfN_div_prime_weak: fixes p :: "int"   and N :: "int"   and q :: "int"   and a :: "int"   and b :: "int" assumes "prime (p\<^sup>2 + N * q\<^sup>2)"   and "p\<^sup>2 + N * q\<^sup>2 dvd a\<^sup>2 + N * b\<^sup>2" shows "\<exists>u v. a\<^sup>2 + N * b\<^sup>2 = (u\<^sup>2 + N * v\<^sup>2) * (p\<^sup>2 + N * q\<^sup>2)"
V_iff_Nil: fixes trn :: "('state, 'act, 'out) trans"   and tr :: "('state, 'act, 'out) trans list" shows "(V (trn # tr) = []) = (\<not> \<phi> trn \<and> V tr = [])"
l2_lkr_after_def: fixes A :: "agent" shows "l2_lkr_after A \<equiv> {(s, s'). test_ended s \<and> s' = s\<lparr>l2_state.bad := {A} \<union> l2_state.bad s\<rparr>}"
HInv4a_def: fixes s :: "state"   and p :: "Proc" shows "HInv4a s p = (phase s p \<noteq> 0 \<longrightarrow> HInv4a1 s p \<and> HInv4a2 s p)"
U_2: fixes n :: "nat"   and x :: "'a" assumes "N1 \<le> n"   and "x \<in> U" shows "L * f x < (\<Sum>i<n. (nn_transfer_operator ^^ i) f x)"
valid_eval: fixes env :: "v sem_env"   and s :: "'a state"   and e :: "exp0" shows "evaluate True env s e (eval env e s)"
3962: fixes x :: "'a"   and y :: "'a" shows "- - (x \<squnion> y) \<sqinter> - - x = - - x"
eq_fold: fixes x :: "'c"   and A :: "'c fset" shows "fMax (finsert x A) = ffold max x A"
directional_derivative_def: fixes k :: "enat"   and a :: "'b"   and v :: "'b" shows "directional_derivative k a v = restrict0 (manifold_eucl.diff_fun_space k) (\<lambda>f. frechet_derivative f (at a) v)"
finite_Union: fixes X :: "'a set set" assumes "finite X"   and "X \<subseteq> generated_ring" shows "\<Union> X \<in> generated_ring"
alt_consistency_def: fixes C :: "('c, 'd) form set set" shows "alt_consistency C = (\<forall>S. S \<in> C \<longrightarrow> (\<forall>p ts. \<not> (Pred p ts \<in> S \<and> Neg (Pred p ts) \<in> S)) \<and> FF \<notin> S \<and> Neg TT \<notin> S \<and> (\<forall>Z. Neg (Neg Z) \<in> S \<longrightarrow> S \<union> {Z} \<in> C) \<and> (\<forall>A B. And A B \<in> S \<longrightarrow> S \<union> {A, B} \<in> C) \<and> (\<forall>A B. Neg (Or A B) \<in> S \<longrightarrow> S \<union> {Neg A, Neg B} \<in> C) \<and> (\<forall>A B. Or A B \<in> S \<longrightarrow> S \<union> {A} \<in> C \<or> S \<union> {B} \<in> C) \<and> (\<forall>A B. Neg (And A B) \<in> S \<longrightarrow> S \<union> {Neg A} \<in> C \<or> S \<union> {Neg B} \<in> C) \<and> (\<forall>A B. Impl A B \<in> S \<longrightarrow> S \<union> {Neg A} \<in> C \<or> S \<union> {B} \<in> C) \<and> (\<forall>A B. Neg (Impl A B) \<in> S \<longrightarrow> S \<union> {A, Neg B} \<in> C) \<and> (\<forall>P t. closedt 0 t \<longrightarrow> Forall P \<in> S \<longrightarrow> S \<union> {P[t/0]} \<in> C) \<and> (\<forall>P t. closedt 0 t \<longrightarrow> Neg (Exists P) \<in> S \<longrightarrow> S \<union> {Neg (P[t/0])} \<in> C) \<and> (\<forall>P x. (\<forall>a\<in>S. x \<notin> params a) \<longrightarrow> Exists P \<in> S \<longrightarrow> S \<union> {P[App x []/0]} \<in> C) \<and> (\<forall>P x. (\<forall>a\<in>S. x \<notin> params a) \<longrightarrow> Neg (Forall P) \<in> S \<longrightarrow> S \<union> {Neg (P[App x []/0])} \<in> C))"
relpow_Suc_E: fixes x :: "'b"   and z :: "'b"   and n :: "nat"   and R :: "('b \<times> 'b) set" assumes "(x, z) \<in> R ^^ Suc n" obtains y :: "'b" where "(x, y) \<in> R ^^ n" and "(y, z) \<in> R"
case_eq_if: fixes f1 :: "'b"   and f2 :: "nat \<Rightarrow> nat \<Rightarrow> 'c option \<Rightarrow> 'c tree \<Rightarrow> 'c tree \<Rightarrow> 'b"   and tree :: "'c tree" shows "(case tree of Leaf \<Rightarrow> f1 | Node x xa xb xc xd \<Rightarrow> f2 x xa xb xc xd) = (if tree = Leaf then f1 else f2 (l tree) (r tree) (val tree) (lchild tree) (rchild tree))"
max_factor_degree_mono: fixes ws :: "'a poly list"   and vs :: "'a poly list" shows "max_factor_degree (map degree (fold remove1 ws vs)) \<le> max_factor_degree (map degree vs)"
res_def: shows "MAC.INSEC.res \<equiv> RES (MAC.INSEC.insec_oracle \<oplus>\<^sub>O A_CHAN.send_oracle \<oplus>\<^sub>O A_CHAN.recv_oracle) Void"
ws_def: shows "ws = (SOME ws. set ws = W \<and> distinct ws)"
adjust_paras: fixes xs :: "nat list"   and n :: "nat"   and x :: "nat"   and y :: "nat"   and anything :: "nat list" assumes "length xs = n" shows "{\<lambda>nl. nl = xs @ x # y # anything} [Inc n, Dec (Suc n) 3, Goto (Suc 0)] {\<lambda>nl. nl = xs @ Suc x # 0 # anything}"
T: shows "T \<in> {s \<in> S. wellorder_class.enumerate S n < s}"
get_document_ptr_simp4: fixes document_ptr :: "(_) document_ptr"   and element_ptr :: "(_) element_ptr"   and f :: "('m, 'n, 'o, 'q, 't option) RElement_scheme"   and h :: "(_) heap" shows "get document_ptr (put element_ptr f h) = get document_ptr h"
A0_2: shows "(\<lambda>n. PS.prob (space M \<inter> (T ^^ n) -` K n) * indicat_real A0 n) \<longlonglongrightarrow> 0"
E_App2: fixes v :: "trm"   and t :: "trm"   and t' :: "trm" assumes "v \<in> value"   and "t \<longmapsto> t'" shows "v \<bullet> t \<longmapsto> v \<bullet> t'"
is_Class_conv: fixes T :: "ty" shows "is_Class T = (\<exists>C. T = Class C)"
mem_remove: fixes cld :: "'a"   and P :: "'a list" assumes "cld \<in> set P" shows "length (remove1 cld P) < length P"
Brouwer_degree2_homotopy_equivalence: fixes p :: "nat"   and f :: "(nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real"   and g :: "(nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real" assumes "continuous_map (nsphere p) (nsphere p) f"   and "continuous_map (nsphere p) (nsphere p) g"   and "homotopic_with (\<lambda>x. True) (nsphere p) (nsphere p) (f \<circ> g) id" obtains "\<bar>Brouwer_degree2 p f\<bar> = 1" and "\<bar>Brouwer_degree2 p g\<bar> = 1" and "Brouwer_degree2 p g = Brouwer_degree2 p f"
y2_sum: shows "(\<lambda>i. y i * y i) abs_summable_on A"
assertLiteralEffect_4: fixes state :: "State"   and l :: "Literal"   and d :: "bool" assumes "InvariantWatchListsContainOnlyClausesFromF (getWatchList state) (getF state)"   and "InvariantWatchesEl (getF state) (getWatch1 state) (getWatch2 state)" shows "isPrefix (getQ state) (getQ (assertLiteral l d state))"
imp_steps: fixes l\<^sub>1 :: "lit"   and l\<^sub>2 :: "lit" assumes "sat_lit \<sigma> l\<^sub>1"   and "(l\<^sub>1, l\<^sub>2) \<in> (imp_graph cnf)\<^sup>*" shows "sat_lit \<sigma> l\<^sub>2"
simps_1: fixes uu :: "'d"   and uv :: "'d"   and F :: "'c" shows "Map (MkArr uu uv F) = F"
bal_split_right_1: fixes ls :: "('b btree \<times> 'b) list"   and rs :: "('b btree \<times> 'b) list"   and t :: "'b btree" assumes "bal (Node (ls @ rs) t)" shows "bal (Node rs t)"
one: fixes R :: "('a, 'b) ring_scheme"   and H :: "'a set" shows "\<one>\<^bsub>R\<^esub> \<in> generate_field R H"
trace_decomp_head: fixes channel :: "nat \<Rightarrow> ('a \<times> 'a) option"   and trans :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and c :: "('a, 'b, 'c) configuration"   and ev :: "('a, 'b, 'c) event"   and t :: "('a, 'b, 'c) event list"   and c' :: "('a, 'b, 'c) configuration" assumes "distributed_system channel"   and "distributed_system.trace channel trans send recv c (ev # t) c'" shows "\<exists>c''. distributed_system.next channel trans send recv c ev c'' \<and> distributed_system.trace channel trans send recv c'' t c'"
Un_eq_UN: fixes A :: "'a set"   and B :: "'a set" shows "A \<union> B = (\<Union>b. if b then A else B)"
I0: shows "I 0 = finprod (ideals_set R) I {..Suc n} <+> I (Suc (Suc n)) \<cdot> I 0"
vangle_collinear: fixes u :: "'b"   and v :: "'b" assumes "vangle u v \<in> {0, pi}" shows "collinear {0::'b, u, v}"
dvd_times_left_cancel_iff: fixes a :: "'b"   and b :: "'b"   and c :: "'b" assumes "a \<noteq> (0::'b)" shows "(a * b dvd a * c) = (b dvd c)"
dim_2: shows "dim_col (char_matrix B ev ^\<^sub>m k) = n"
game_induct: fixes P :: "game \<Rightarrow> bool"   and \<alpha> :: "game" assumes "\<And>a. P (Game a)"   and "\<And>x \<theta>. P (x := \<theta>)"   and "\<And>x \<theta>. P (ODE x \<theta>)"   and "\<And>\<phi>. P (? \<phi>)"   and "\<And>\<alpha> \<beta>. \<lbrakk>P \<alpha>; P \<beta>\<rbrakk> \<Longrightarrow> P (\<alpha> \<union>\<union> \<beta>)"   and "\<And>\<alpha> \<beta>. \<lbrakk>P \<alpha>; P \<beta>\<rbrakk> \<Longrightarrow> P (\<alpha> ;; \<beta>)"   and "\<And>\<alpha>. P \<alpha> \<Longrightarrow> P \<alpha>**"   and "\<And>\<alpha>. P \<alpha> \<Longrightarrow> P \<alpha>^d" shows "P \<alpha>"
map_FGcontra_def: fixes l1 :: "'au \<Rightarrow> 'bh"   and co1 :: "'av \<Rightarrow> 'bi"   and co2 :: "'bd \<Rightarrow> 'bq"   and co3 :: "'aw \<Rightarrow> 'bj"   and co4 :: "'ax \<Rightarrow> 'bk"   and co5 :: "'ay \<Rightarrow> 'bl"   and contra1 :: "'bm \<Rightarrow> 'az"   and contra2 :: "'bn \<Rightarrow> 'ba"   and contra3 :: "'bo \<Rightarrow> 'bb"   and contra4 :: "'bp \<Rightarrow> 'bc"   and contra5 :: "'br \<Rightarrow> 'bf" shows "map_FGcontra l1 co1 co2 co3 co4 co5 contra1 contra2 contra3 contra4 contra5 = map_F l1 co1 co3 co1 co4 co5 (map_G contra1 contra2 contra3 contra4 co1 co2) contra1 contra5"
node_deliver_messages_Deliver: fixes history :: "nat \<Rightarrow> 'c event list"   and msg_id :: "'c \<Rightarrow> 'd"   and m :: "'c" assumes "network history msg_id" shows "trivial_network.node_deliver_messages [Deliver m] = [m]"
pr_tuple_2: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and pr0 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and pr1 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b"   and g :: "'b"   and a :: "'b"   and b :: "'b" assumes "elementary_category_with_binary_products C pr0 pr1"   and "partial_magma.arr C f \<and> partial_magma.arr C g \<and> partial_magma.dom C f = partial_magma.dom C g"   and "partial_magma.cod C f = a"   and "partial_magma.cod C g = b" shows "C (pr0 a b) (elementary_category_with_binary_products.tuple C pr0 pr1 f g) = g"
poly_mult_lead_coeff: fixes K :: "'a list set"   and p1 :: "'a list list"   and p2 :: "'a list list" assumes "subring K (K [X])"   and "polynomial\<^bsub>K [X]\<^esub> K p1"   and "polynomial\<^bsub>K [X]\<^esub> K p2"   and "p1 \<noteq> []"   and "p2 \<noteq> []" shows "lead_coeff (UP.poly_mult p1 p2) = lead_coeff p1 \<otimes>\<^bsub>K [X]\<^esub> lead_coeff p2"
succ_of_rel_def: fixes s :: "('c \<times> 'd) set" shows "succ_of_rel s \<equiv> \<lambda>a. {b. (a, b) \<in> s}"
Basis_le_infnorm: fixes b :: "'a"   and x :: "'a" assumes "b \<in> Basis" shows "\<bar>x \<bullet> b\<bar> \<le> infnorm x"
simps: fixes uu :: "'a ext" shows "uu\<^sup>\<star> = Top"
elims_2: fixes x :: "'b formula" assumes "is_disj x" obtains F :: "'b formula" and G :: "'b formula" where "x = F \<^bold>\<or> G" and "is_lit_plus F \<and> is_disj G"   | v :: "'b" where "x = Atom v__" and "is_lit_plus (Atom v__)"   | "x = \<bottom>" and "is_lit_plus \<bottom>"   | v :: "'b formula" where "x = \<^bold>\<not> va__" and "is_lit_plus (\<^bold>\<not> va__)"   | v :: "'b formula" and va :: "'b formula" where "x = vb__ \<^bold>\<and> vaa__" and "is_lit_plus (vb__ \<^bold>\<and> vaa__)"   | v :: "'b formula" and va :: "'b formula" where "x = v \<^bold>\<rightarrow> va" and "is_lit_plus (v \<^bold>\<rightarrow> va)"
inj_on_inverseI: fixes A :: "'b set"   and g :: "'c \<Rightarrow> 'b"   and f :: "'b \<Rightarrow> 'c" assumes "\<And>x. x \<in> A \<Longrightarrow> g (f x) = x" shows "inj_on f A"
S_relaxed: fixes v :: "'v"   and u :: "'v" assumes "v \<in> S" shows "D u \<le> \<delta> s v + w (v, u)"
mHom_lin: fixes N :: "('g, 'b, 'h) Module_scheme"   and m :: "'a"   and f :: "'a \<Rightarrow> 'g"   and a :: "'b" assumes "R module N"   and "m \<in> carrier M"   and "f \<in> mHom R M N"   and "a \<in> carrier R" shows "f (a \<cdot>\<^sub>s m) = a \<cdot>\<^sub>s\<^bsub>N\<^esub> f m"
on_stack_ge_def: fixes I :: "'c \<Rightarrow> node_state option"   and l :: "nat"   and v :: "'c" shows "on_stack_ge I l v \<equiv> case I v of None \<Rightarrow> False | Some (STACK j) \<Rightarrow> l \<le> j | Some DONE \<Rightarrow> False"
inext_closed: fixes n :: "nat"   and I :: "nat set" assumes "n \<in> I" shows "inext n I \<in> I"
ii: shows "atom i \<sharp> (j, p \<bullet> j, s, k, p)"
set_of_variables_is_finite_cl: fixes C :: "'b literal set" assumes "finite C" shows "finite (vars_of_cl C)"
subst_dom_vars_in_subst: fixes v :: "'c"   and s :: "'c \<Rightarrow> ('d, 'c) Term.term" assumes "v \<in> subst_domain s" shows "s v \<noteq> Var v"
mul_eq_not_equal_imp_elt: fixes M :: "'a multiset"   and N :: "'a multiset"   and r :: "('a \<times> 'a) set"   and y :: "'a" assumes "(M, N) \<in> mul_eq r"   and "y \<in> set_mset M - set_mset N" shows "\<exists>z\<in>#N. (y, z) \<in> r"
Ord_not_less: fixes x :: "V"   and y :: "V" assumes "Ord x"   and "Ord y" shows "(\<not> x < y) = (y \<le> x)"
nat_pow_consistent: fixes x :: "'a"   and n :: "nat"   and H :: "'a set" shows "x [^] n = x [^]\<^bsub>G\<lparr>carrier := H\<rparr>\<^esub> n"
array_copy_ind: fixes n :: "nat"   and xs :: "'b list"   and xs' :: "'b list"   and k :: "nat" assumes "n \<le> length xs"   and "n \<le> length xs'"   and "k < n" shows "array_copy xs xs' n ! k = xs ! k"
assms_1: shows "|x\<acute>=f & G on U S @ t\<^sub>0] C = (\<lambda>s. True)"
that: fixes V' :: "'x set" assumes "(monomial (1::'b) t, V') \<in> set (snd (split t U S))"   and "card V \<le> card V'" shows "thesis"
directional_derivative_scaleR: fixes k :: "enat"   and a :: "'c"   and c :: "real"   and x :: "'c" assumes "k \<noteq> 0" shows "directional_derivative k a (c *\<^sub>R x) = c *\<^sub>R directional_derivative k a x"
AbstrLevels_A9_A93: fixes i :: "AbstrLevelsID" assumes "sA9 \<in> AbstrLevel i" shows "sA93 \<notin> AbstrLevel i"
berlekamp_basis_basis: fixes u :: "'b mod_ring poly" assumes "0 < degree u" shows "vectorspace.basis class_ring (W u) (set (berlekamp_basis u))"
abs_eq: fixes xa :: "'b prm"   and x :: "'b ptrm" shows "xa \<cdot> abs_trm x = abs_trm (xa \<bullet> x)"
adv_P1_game_def: fixes m1 :: "'msg1"   and m2 :: "'msg2"   and D :: "'view1 \<Rightarrow> bool spmf" shows "adv_P1_game m1 m2 D = \<bar>2 * spmf (P1_game_alt m1 m2 D) True - 1\<bar>"
scale_zero_right: fixes a :: "'a" shows "a *b (0::'c) = (0::'c)"
induct: fixes P :: "'b list \<Rightarrow> 'b list \<Rightarrow> bool"   and a0 :: "'b list"   and a1 :: "'b list" assumes "\<And>cn xis. P [cn] xis"   and "\<And>ci v va xi xis. P (v # va) xis \<Longrightarrow> P (ci # v # va) (xi # xis)"   and "\<And>uv_. P [] uv_"   and "\<And>v vb vc. P (v # vb # vc) []" shows "P a0 a1"
sub0: fixes g :: "real \<Rightarrow> complex"   and z :: "complex" shows "P (subpath 0 0 g) z"
rm_vars_dom_subset: fixes V :: "'c set"   and s :: "'c \<Rightarrow> ('d, 'c) Term.term" shows "subst_domain (rm_vars V s) \<subseteq> subst_domain s"