hcju/isabelleps huggingface dataset:
Un_upper2_approx: fixes B :: "((nat \<times> (nat \<Rightarrow> 'a option)) \<times> 'a \<aproc>) set"   and A :: "((nat \<times> (nat \<Rightarrow> 'a option)) \<times> 'a \<aproc>) set" shows "B \<lessapprox> A \<union> B"
res_fold_A_endo_A: shows "ChamberComplexEndomorphism A (restrict1 fold_A (\<Union> A))"
M_times_2: fixes x :: "int"   and y :: "int" shows "M (x * M y) = M (x * y)"
is_continuous: fixes S :: "'a set"   and is_open :: "'a set \<Rightarrow> bool"   and S' :: "'b set"   and is_open' :: "'b set \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b"   and U :: "'b set" assumes "continuous_map S is_open S' is_open' f"   and "is_open' U" shows "is_open (f \<^sup>\<inverse> S U)"
Gi: fixes i :: "nat" assumes "i < length (map gs1.gso [0..<m])" shows "map gs1.gso [0..<m] ! i \<in> Rn"
DecidedImpliesUniform: fixes c :: "('p, 'v, 's) configuration"   and v :: "bool" assumes "\<forall>cfg. reachable c cfg \<longrightarrow> agreement cfg"   and "\<And>cc Q. terminationPseudo 1 cc Q"   and "initReachable c \<and> <\<bottom>, outM v> \<in># msgs c" shows "vUniform v c"
insert_4: assumes "\<And>s. s \<in> ss \<Longrightarrow> \<exists>rs. s = s0 \<star> rs \<and> set rs \<subseteq> rset" shows "\<exists>rs. \<Sqinter> ss = s0 \<star> rs \<and> set rs \<subseteq> rset"
cross_ratio_abs: fixes u :: "(real, 3) vec"   and v :: "(real, 3) vec"   and j :: "real"   and l :: "real"   and i :: "real"   and k :: "real" assumes "u \<noteq> 0"   and "v \<noteq> 0"   and "proj2_abs u \<noteq> proj2_abs v"   and "j \<noteq> 0"   and "l \<noteq> 0" shows "cross_ratio (proj2_abs u) (proj2_abs v) (proj2_abs (i *\<^sub>R u + j *\<^sub>R v)) (proj2_abs (k *\<^sub>R u + l *\<^sub>R v)) = j * k / (i * l)"
elims: fixes x :: "'f \<Rightarrow> ('g \<times> 'h \<Rightarrow>f nat) option"   and xa :: "('f, 'g, 'i) new_thread_action"   and y :: "'f \<Rightarrow> ('g \<times> 'h \<Rightarrow>f nat) option" assumes "redT_updT x xa = y" obtains ts :: "'f \<Rightarrow> ('g \<times> 'h \<Rightarrow>f nat) option" and t' :: "'f" and xb :: "'g" and m :: "'i" where "x = tsa__" and "xa = NewThread t' xb m" and "y = tsa__(t' \<mapsto> (xb, no_wait_locks))"   | ts :: "'f \<Rightarrow> ('g \<times> 'h \<Rightarrow>f nat) option" and v :: "'f" and va :: "bool" where "x = ts" and "xa = ThreadExists v va" and "y = ts"
\<C>_vars_consistent: fixes x\<^sub>A :: "Eg1.var" shows "Eg2_var\<^sub>C_of_Eg1 ` \<C>_vars x\<^sub>A = \<C>_vars\<^sub>C (Eg2_var\<^sub>C_of_Eg1 x\<^sub>A)"
mgu_gives_MGU: fixes s :: "('d, 'c) Term.term"   and t :: "('d, 'c) Term.term"   and \<theta> :: "'c \<Rightarrow> ('d, 'c) Term.term" assumes "mgu s t = Some \<theta>" shows "MGU \<theta> s t"
seq': fixes rs :: "'a rule list"   and rs\<^sub>1 :: "'a rule list"   and rs\<^sub>2 :: "'a rule list"   and \<Gamma> :: "char list \<Rightarrow> 'a rule list option"   and \<gamma> :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and p :: "'b"   and s :: "state"   and t :: "state"   and t' :: "state" assumes "rs = rs\<^sub>1 @ rs\<^sub>2"   and "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs\<^sub>1, s\<rangle> \<Rightarrow> t"   and "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs\<^sub>2, t\<rangle> \<Rightarrow> t'" shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs, s\<rangle> \<Rightarrow> t'"
is_normal: fixes n :: "nat"   and v :: "complex Matrix.mat" assumes "state n v" shows "\<parallel>Matrix.col v 0\<parallel> = 1"
AB_edge: fixes y :: "'v"   and x :: "'v" assumes "edge \<Gamma> y x" shows "edge G x y"
fixp_induct: fixes P :: "(real \<Rightarrow> bool spmf) \<Rightarrow> bool" assumes "spmf.admissible P"   and "P (\<lambda>bernoulli. return_pmf None)"   and "\<And>p. P p \<Longrightarrow> P (\<lambda>pa. coin_spmf \<bind> (\<lambda>b. if b then return_spmf (1 / 2 \<le> pa) else if pa < 1 / 2 then p (2 * pa) else p (2 * pa - 1)))" shows "P bernoulli"
partner_runs_def: fixes R :: "fid_t"   and R' :: "fid_t" shows "partner_runs R R' \<equiv> role (guessed_runs R) = role_comp (role (guessed_runs R')) \<and> owner (guessed_runs R) = partner (guessed_runs R') \<and> owner (guessed_runs R') = partner (guessed_runs R) \<and> matching (guessed_frame R) (guessed_frame R')"
smap_siterate: fixes f :: "'b \<Rightarrow> 'b"   and x :: "'b" shows "smap f (siterate f x) = siterate f (f x)"
comp_ide_arr: fixes b :: "'b"   and f :: "'b" assumes "C'.ide b"   and "C'.seq b f" shows "C' b f = f"
CanonicalCat_def: fixes T :: "(ZF, ZF) Axioms" shows "CanonicalCat T \<equiv> MakeCat (CanonicalCat' T)"
enn2ereal_mono: fixes x :: "ennreal"   and y :: "ennreal" assumes "x \<le> y" shows "enn2ereal x \<le> enn2ereal y"
NextPropagate'_def: fixes c0 :: "('b, 't, 'loc) Combined.configuration"   and c1 :: "('b, 't, 'loc) Combined.configuration"   and p :: "'b" shows "NextPropagate' c0 c1 p = (exchange_config c1 = exchange_config c0 \<and> init c1 = (init c0)(p := True) \<and> (\<forall>p'. Some (prop_config c1 p') = (if p' = p then propagate_all (prop_config c0 p') else Some (prop_config c0 p'))))"
Inter_lower: fixes B :: "'d set"   and A :: "'d set set" assumes "B \<in> A" shows "\<Inter> A \<subseteq> B"
wens_single_finite_def: fixes act :: "('b \<times> 'b) set"   and B :: "'b set"   and k :: "nat" shows "wens_single_finite act B k \<equiv> \<Union>i\<le>k. (wp act ^^ i) B"
state_update_vs_allInstances_at_post_including': fixes \<sigma>' :: "nat \<Rightarrow> \<langle>'d\<rangle>\<^sub>\<bottom>"   and oid :: "nat"   and Object :: "'d"   and Type :: "'d \<Rightarrow> \<langle>'e\<rangle>\<^sub>\<bottom>"   and \<sigma> :: "'d state"   and A :: "nat \<Rightarrow> \<langle>nat list list list\<rangle>\<^sub>\<bottom>" assumes "\<And>x. \<sigma>' oid = \<lfloor>x\<rfloor> \<Longrightarrow> x = Object"   and "Type Object \<noteq> None" shows "Type .allInstances() (\<sigma>, \<lparr>heap = \<sigma>'(oid \<mapsto> Object), assocs = A\<rparr>) = Type .allInstances()->including\<^sub>S\<^sub>e\<^sub>t(\<lambda>_. \<lfloor>\<lfloor>\<lceil>Type Object\<rceil>\<rfloor>\<rfloor>) (\<sigma>, \<lparr>heap = \<sigma>', assocs = A\<rparr>)"
hyps_4: shows "gen_contr_graph (with_proj H) (verts3 (with_proj H)) = G"
simps: fixes p :: "char list"   and ts :: "hterm list" shows "fterm_of_hterm (HFun p ts) = Fun p (map fterm_of_hterm ts)"
diff_chartsD: fixes d1 :: "('a, 'e) chart"   and d2 :: "('b, 'f) chart" assumes "d1 \<in> src.atlas"   and "d2 \<in> dest.atlas" shows "k-smooth_on (codomain d1 \<inter> inv_chart d1 -` (src.carrier \<inter> f -` domain d2)) (apply_chart d2 \<circ> f \<circ> inv_chart d1)"
lincomb_not_depend: fixes A :: "'c set"   and N :: "'c set"   and a :: "'c \<Rightarrow> 'a" assumes "finite A"   and "A \<subseteq> N"   and "LinearCombinations.submodule R N M"   and "a \<in> A \<rightarrow> carrier R" shows "lincomb a A = module.lincomb (md N) a A"
App: fixes ts :: "dB list"   and x :: "nat" assumes "listall NF ts" shows "NF (Var x \<degree>\<degree> ts)"
kingman_theorem_AE_aux2: fixes u :: "nat \<Rightarrow> 'a \<Rightarrow> real" assumes "subcocycle u" shows "\<exists>g. g \<in> borel_measurable Invariants \<and> (\<forall>x. g x < \<infinity>) \<and> (AE x in M. (\<lambda>xa. ereal (u xa x / real xa)) \<longlonglongrightarrow> g x)"
obt_3: shows "\<forall>x\<in>convex hull S. \<forall>y\<in>convex hull S. norm (x - y) \<le> norm (u - v)"
assms_4: fixes x :: "'call"   and s :: "'s" assumes "x \<in> outs_\<I> \<I>"   and "I s" shows "colossless_gpv \<I>' (callee s x)"
index_smult_vec_2: fixes a :: "'c"   and v :: "'c vec" shows "dim_vec (a \<cdot>\<^sub>v v) = dim_vec v"
choose_affine_subset: fixes S :: "'a set"   and d :: "int" assumes "affine S"   and "- 1 \<le> d"   and "d \<le> aff_dim S" obtains T :: "'a set" where "affine T" and "T \<subseteq> S" and "aff_dim T = d"
subdg_subdigraph_op_dg_op_dg: shows "op_dg \<BB> \<subseteq>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> op_dg \<CC>"
surjective_pairing: fixes t :: "'a \<times> 'c" shows "t = (fst t, snd t)"
add_0_iff: fixes plus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and minus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and zero :: "'k"   and times :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and one :: "'k"   and b :: "'k"   and a :: "'k" assumes "class.comm_semiring_1_cancel_crossproduct plus minus zero times one" shows "(b = plus b a) = (a = zero)"
ordertype_VWF_\<omega>: fixes A :: "V set" assumes "finite A" shows "ordertype A VWF \<in> elts \<omega>"
I: fixes Ps :: "('a, 'b) sequent list"   and c :: "('a, 'b) sequent"   and R :: "(('a, 'b) sequent list \<times> ('a, 'b) sequent) set"   and M :: "'b"   and \<Gamma> :: "('a, 'b) form multiset"   and N :: "'b"   and \<Delta> :: "('a, 'b) form multiset" assumes "(Ps, c) \<in> R"   and "R \<subseteq> modRules2" shows "extendRule ( M \<cdot> \<Gamma> \<Rightarrow>* N \<cdot> \<Delta>) (Ps, c) \<in> p_e R M N"
monad_right_unit: fixes m :: "'c\<cdot>'d" shows "m \<bind> return = m"
subst_idem_iff_self_support: fixes \<theta> :: "'a \<Rightarrow> ('b, 'a) Term.term" shows "subst_idem \<theta> = (\<theta> supports \<theta>)"
WF_leadsto: fixes N :: "state \<times> state \<Rightarrow> bool"   and P :: "state \<Rightarrow> bool"   and A :: "state \<times> state \<Rightarrow> bool"   and v :: "state \<Rightarrow> 'a"   and B :: "state \<times> state \<Rightarrow> bool" assumes "\<turnstile> N \<and> $P \<longrightarrow> $Enabled (<A>_v)"   and "\<turnstile> N \<and> <A>_v \<longrightarrow> B"   and "\<turnstile> \<box>(N \<and> [\<not> A]_v) \<longrightarrow> stable P" shows "\<turnstile> \<box>N \<and> WF(A)_v \<longrightarrow> (P \<leadsto> B)"
spmf_scale_spmf': fixes r :: "real"   and p :: "'c spmf"   and x :: "'c" assumes "r \<le> 1" shows "spmf (scale_spmf r p) x = max 0 r * spmf p x"
Cons_in_shuffles_leftI: fixes zs :: "'a list"   and xs :: "'a list"   and ys :: "'a list"   and z :: "'a" assumes "zs \<in> shuffles xs ys" shows "z # zs \<in> shuffles (z # xs) ys"
lock_thread_okD: fixes ls :: "('a, 'b) locks"   and ts :: "'b \<Rightarrow> ('c \<times> 'a \<Rightarrow>f nat) option"   and l :: "'a"   and t :: "'b" assumes "lock_thread_ok ls ts"   and "has_lock (ls $ l) t" shows "\<exists>xw. ts t = \<lfloor>xw\<rfloor>"
left_edge_explicit: shows "(\<lambda>y. twoC (0, y)) = (\<lambda>x. (a, g2 a + x *\<^sub>R (g1 a - g2 a)))"
bounded_byD2: fixes b :: "real"   and P :: "'a \<Rightarrow> real" assumes "bounded_by b P" shows "P \<tturnstile> \<lambda>s. b"
simps_6: fixes bs :: "real list"   and a :: "Ferrack.num"   and b :: "Ferrack.num" shows "Inum bs (Sub a b) = Inum bs a - Inum bs b"
Sup_union_distrib: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<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 A :: "'c set"   and B :: "'c set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "Sup (A \<union> B) = sup (Sup A) (Sup B)"
CELam_no_restr: fixes x :: "Vars.var"   and e :: "exp"   and \<rho> :: "Vars.var \<Rightarrow> C \<rightarrow> CValue"   and r :: "C" shows "(\<N>\<lbrakk> Lam [x]. e \<rbrakk>\<^bsub>\<rho>\<^esub>)\<cdot>r \<sqsubseteq> CFn\<cdot>(\<Lambda> v. \<N>\<lbrakk> e \<rbrakk>\<^bsub>\<rho>(x := v)\<^esub>)"
IH: assumes "set Vs \<subseteq> carrier_vec n" shows "lincomb_list (\<lambda>j. if i = j then 1::'a else (0::'a)) Vs = (if i < length Vs then Vs ! i else 0\<^sub>v n)"
hyps: shows "Q \<oplus> R \<longmapsto> \<alpha> \<prec> Q'"
Says_K6: fixes servK :: "nat"   and T3 :: "nat"   and evs :: "event list"   and A :: "agent"   and authK :: "nat"   and B :: "agent"   and Ts :: "nat"   and servTicket :: "msg" assumes "Crypt servK (Number T3) \<in> parts (knows Spy evs)"   and "Says Tgs A \<lbrace>Crypt authK \<lbrace>Key servK, Agent B, Number Ts\<rbrace>, servTicket\<rbrace> \<in> set evs"   and "Key servK \<notin> analz (knows Spy evs)"   and "A \<notin> bad"   and "B \<notin> bad"   and "evs \<in> kerbV" shows "Says B A (Crypt servK (Number T3)) \<in> set evs"
hn_refine_cons_pre: fixes P :: "assn"   and P' :: "assn"   and c :: "'c Heap"   and Q :: "assn"   and R :: "'d \<Rightarrow> 'c \<Rightarrow> assn"   and m :: "'d nres" assumes "P \<Longrightarrow>\<^sub>t P'"   and "hn_refine P' c Q R m" shows "hn_refine P c Q R m"
saturated_ordered_sel_res_empty_sel: fixes Sel :: "'at set"   and S :: "'at Literal set set" assumes "saturated_binary_rule (ordered_sel_resolvent Sel) S" shows "saturated_binary_rule ordered_resolvent (empty_selected_part Sel S)"
dom_mkIde: fixes A :: "'s set" assumes "A \<subseteq> Univ"   and "|A| <o \<AA>" shows "local.dom (mkIde A) = mkIde A"
rel_compp: fixes R1 :: "'b \<Rightarrow> 'f \<Rightarrow> bool"   and S1 :: "'f \<Rightarrow> 'd \<Rightarrow> bool"   and R2 :: "'c \<Rightarrow> 'g \<Rightarrow> bool"   and S2 :: "'g \<Rightarrow> 'e \<Rightarrow> bool" shows "rel_sp\<^sub>\<mu> (R1 OO S1) (R2 OO S2) = rel_sp\<^sub>\<mu> R1 R2 OO rel_sp\<^sub>\<mu> S1 S2"
lmeasurable_compact: fixes S :: "'a set" assumes "compact S" shows "S \<in> lmeasurable"
Op_def: fixes delta :: "'e"   and inp :: "'c \<Rightarrow> ('c, 'd, 'varSort, 'var, 'e) qTerm set option"   and binp :: "'d \<Rightarrow> ('c, 'd, 'varSort, 'var, 'e) qAbs set option" shows "Op delta inp binp \<equiv> asTerm (qOp delta (pickInp inp) (pickBinp binp))"
s_space: fixes p :: "nat"   and n :: "nat"   and base :: "nat \<Rightarrow> nat"   and upd :: "nat \<Rightarrow> nat"   and s :: "(nat \<Rightarrow> nat) set" assumes "kuhn_simplex p n base upd s" shows "s \<subseteq> {..<n} \<rightarrow> {..p}"
simps_15: fixes f1 :: "'c \<Rightarrow> 'a"   and f2 :: "'d \<Rightarrow> 'b"   and x21 :: "'c"   and x22 :: "('c, 'd) tllist" shows "tmap f1 f2 (TCons x21 x22) = TCons (f1 x21) (tmap f1 f2 x22)"
dim_Macaulay_mat_1: fixes ps :: "('t \<Rightarrow>\<^sub>0 'c) list" shows "dim_row (Macaulay_mat ps) = length ps"
conversep_converse_eq: fixes r :: "('b \<times> 'a) set" shows "(\<lambda>x xa. (x, xa) \<in> r)\<inverse>\<inverse> = (\<lambda>x xa. (x, xa) \<in> r\<inverse>)"
simps_1: fixes f :: "'c \<Rightarrow> 'a" shows "psubst f FF = FF"
IH: assumes "j < e_length (block i t)"   and "j \<noteq> gap i t" shows "\<tau> i j \<down>= e_nth (block i t) j"
axioms_2: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<mu> :: "'b arrow_of_spans_data" assumes "arrow_of_spans C \<mu>" shows "span_in_category C (Dom \<mu>)"
composition_closed: fixes M :: "'b set"   and composition :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit :: "'b"   and a :: "'b"   and b :: "'b" assumes "Group_Theory.monoid M composition unit"   and "a \<in> M"   and "b \<in> M" shows "composition a b \<in> M"
set_renaming_perm: fixes bs :: "atom set"   and p :: "perm" assumes "finite bs" shows "\<exists>q. (\<forall>b\<in>bs. q \<bullet> b = p \<bullet> b) \<and> supp q \<subseteq> bs \<union> p \<bullet> bs"
list_seq_length_mono: fixes seq :: "nat \<Rightarrow> 'b list"   and i :: "nat"   and j :: "nat" assumes "\<And>i. \<exists>x. seq (Suc i) = x # seq i"   and "i < j" shows "length (seq i) < length (seq j)"
RBT_Impl_fold1_def: fixes f :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and rbt :: "('c, unit) rbt" shows "RBT_Impl_fold1 f rbt = fold f (tl (RBT_Impl.keys rbt)) (hd (RBT_Impl.keys rbt))"
intro: fixes G :: "('c, 'd) monoid_scheme" assumes "Group.monoid G"   and "Group.group_axioms G" shows "Group.group G"
cyclesI: fixes execute :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and enabled :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and w :: "'a list"   and p :: "'b" assumes "transition_system.path execute enabled w p"   and "fold execute w p = p" shows "w \<in> transition_system_initial.cycles execute enabled p"
mem_fTo_Chain: fixes X :: "'a set" assumes "X \<in> carrier fTo D" shows "Chain D X"
exhaust: fixes y :: "X4" obtains "y = Xd1"   | "y = Xd1'"   | "y = Xd2"   | "y = Xd2'"
assms: shows "d z +  |x\<rangle> \<nabla>(x + y) = \<nabla>(x + y)"
continuous_at_Sup_mono: fixes f :: "'c \<Rightarrow> 'd"   and S :: "'c set" assumes "mono f"   and "continuous (at_left (\<Squnion> S)) f"   and "S \<noteq> {}"   and "bdd_above S" shows "f (\<Squnion> S) = \<Squnion> (f ` S)"
Int: fixes a :: "'a set"   and i :: "nat"   and b :: "'a set" assumes "a \<in> A i"   and "b \<in> A i" shows "a \<inter> b \<in> A i"
ring_iso_def: fixes R :: "('e, 'g) ring_scheme"   and S :: "('f, 'h) ring_scheme" shows "ring_iso R S = {h \<in> ring_hom R S. bij_betw h (carrier R) (carrier S)}"
Abs_Rep_well_defined: fixes x :: "wall" assumes "is_tangle_diagram x" shows "Rep_Tangle_Diagram (Abs_Tangle_Diagram x) = x"
optimize_no_cond_correct: fixes x :: "rat vec" assumes "optimize_no_cond A b c = Inr x" shows "x \<in> max_lp A b c"
iso_cancel_right: fixes g :: "'c"   and f :: "'c"   and f' :: "'c" assumes "C.iso g"   and "f \<cdot>\<^sub>C g = f' \<cdot>\<^sub>C g"   and "C.seq f g"   and "C.iso g" shows "f = f'"
that: fixes y :: "'a"   and ys :: "'a stream" assumes "xs = y ## ys"   and "P y"   and "pred_stream P ys" shows "thesis"
init_spec: shows "gds_init gds \<le>\<^sub>n SPEC (\<lambda>s. stack s = [] \<and> pending s = {})"
outside_frontier_misses_closure: fixes S :: "'a set" assumes "bounded S" shows "outside (frontier S) \<subseteq> - closure S"
P10: fixes a :: "'a"   and b :: "'a" shows "(a l\<rightarrow> b = (1::'a)) = (a r\<rightarrow> b = (1::'a))"
frac_eq_eq: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and y :: "'a"   and z :: "'a"   and x :: "'a"   and w :: "'a" assumes "class.field divide inverse times one plus zero minus uminus"   and "y \<noteq> zero"   and "z \<noteq> zero" shows "(divide x y = divide w z) = (times x z = times w y)"
tSplice_tFirst: fixes t :: "('a, 'ePubAct, 'es, 'pPubAct, 'ps) BEState Trace"   and t' :: "('a, 'ePubAct, 'es, 'pPubAct, 'ps) BEState Trace"   and a :: "'a" assumes "tLength t = tLength t'"   and "envObs a (tFirst t) = envObs a (tFirst t')" shows "tFirst (t \<^bsub>\<^esub>\<bowtie>\<^bsub>a\<^esub> t') = tFirst t"
euclidean_inner: fixes x :: "'a"   and y :: "'a" shows "x \<bullet> y = (\<Sum>b\<in>Basis. x \<bullet> b * (y \<bullet> b))"
subset_add_mset_notin_subset_mset: fixes A :: "'b multiset"   and b :: "'b"   and B :: "'b multiset" assumes "A \<subseteq># add_mset b B"   and "b \<notin># A" shows "A \<subseteq># B"
real_resource_wiring: fixes \<eta> :: "nat" shows "CIPHER.res \<eta> = RES (\<dagger>CIPHER.AUTH.auth_oracle \<oplus>\<^sub>O key_channel_send \<eta> \<oplus>\<^sub>O key_channel_recv \<eta>) (None, Void)"
sup_compl_top: fixes x :: "'a" shows "x + - x = 1"
simps_2: fixes x :: "'b"   and xs :: "'b list" shows "rotate1 (x # xs) = xs @ [x]"
gauss: shows "gauss_jordan A (0\<^sub>m nr 0) = (C, D)"
fresh_context: fixes a :: "name"   and \<Gamma> :: "(name \<times> ty) list" assumes "a \<sharp> \<Gamma>" shows "\<nexists>\<tau>. (a, \<tau>) \<in> set \<Gamma>"
mod_mult_cong_right: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and division_segment :: "'b \<Rightarrow> 'b"   and c :: "'b"   and a :: "'b"   and b :: "'b"   and d :: "'b" assumes "class.unique_euclidean_semiring divide plus minus zero times one modulo euclidean_size division_segment" shows "unique_euclidean_semiring.cong modulo (modulo c (times a b)) d a = unique_euclidean_semiring.cong modulo c d a"
neq_p: shows "[p - 1 \<noteq> 1] (mod p)"
listprepend_correct: fixes \<gamma> :: "'b \<Rightarrow> bool"   and as :: "'b negation_type list"   and ds :: "'b negation_type list list" shows "dnf_to_bool \<gamma> (listprepend as ds) = (\<exists>a\<in>set as. dnf_to_bool \<gamma> [[a]] \<and> dnf_to_bool \<gamma> ds)"
to_nat_less_ncols: fixes a :: "'b"   and A :: "(('c, 'b) Finite_Cartesian_Product.vec, 'd) Finite_Cartesian_Product.vec" shows "to_nat a < ncols A"
FGModuleIso_axioms_def: fixes V :: "'a set"   and T :: "'a \<Rightarrow> 'b"   and W :: "'b set" shows "FGModuleIso_axioms V T W \<equiv> bij_betw T V W"
op_assoc_empty_def: shows "op_assoc_empty \<equiv> op_map_empty"
uGraph_from_list_invar_subset: fixes L :: "(nat \<times> int \<times> nat) list"   and L' :: "(nat \<times> int \<times> nat) list" assumes "uGraph_from_list_invar L"   and "set L' \<subseteq> set L"   and "distinct L'" shows "uGraph_from_list_invar L'"
graph_join_def: fixes g\<^sub>1 :: "'b ugraph"   and g\<^sub>2 :: "'b ugraph" shows "graph_join g\<^sub>1 g\<^sub>2 \<equiv> graph (nodes g\<^sub>1 \<union> nodes g\<^sub>2) (edges g\<^sub>1 \<union> edges g\<^sub>2)"
fM_rel_inv_def: shows "fM_rel_inv = (\<lambda>s. (\<forall>x. sys_ghost_hs_in_sync x s, sys_ghost_hs_phase s, gc_fM s, sys_fM s, filter is_mw_fM (sys_mem_store_buffers gc s), sys_mem_lock s = Some gc) \<in> fM_rel)"
cod_eqI': fixes a :: "(bool, 'arr list) concrete_category.arr"   and f :: "(bool, 'arr list) concrete_category.arr" assumes "a \<in> codomains f" shows "a = cod f"
upper_asymptotic_density_shift: fixes A :: "nat set"   and k :: "nat"   and l :: "int" shows "ereal (upper_asymptotic_density A) = limsup (\<lambda>x. ereal (real (card (A \<inter> {k..nat (int x + l)})) / real x))"
cr_list_def: shows "list_R1.cr_list \<equiv> \<lambda>x y. x = list_R1.list_to_list_R1 y"
atU_add: fixes U :: "'a set"   and c :: "'a list multiset"   and ce :: "'a list multiset" assumes "atU U c" shows "atU U (c + ce) \<and> atU U (ce + c)"
map_upds_list_update2_drop: fixes xs :: "'a list"   and i :: "nat"   and m :: "'a \<Rightarrow> 'b option"   and ys :: "'b list"   and y :: "'b" assumes "length xs \<le> i" shows "m(xs [\<mapsto>] ys[i := y]) = m(xs [\<mapsto>] ys)"
preserves_section_retraction: fixes e :: "'a"   and m :: "'a" assumes "A.ide (e \<cdot>\<^sub>A m)" shows "B.ide (F e \<cdot>\<^sub>B F m)"
iMin_insert_remove: fixes n :: "'a"   and I :: "'a set" shows "iMin (insert n I) = (if I - {n} = {} then n else min n (iMin (I - {n})))"
ThinChamberComplexFolding_g: shows "ThinChamberComplexFolding X g"
lex_half_segments_last: fixes c :: "real \<times> real"   and d :: "real \<times> real"   and X :: "(real \<times> real) \<times> (real \<times> real) pdevs" assumes "(c, d) \<in> set (half_segments_of_aform X)" shows "Counterclockwise_2D_Arbitrary.lex (snd (last (half_segments_of_aform X))) d"
igFreshIGVar_fromMOD: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) Recursion.model" assumes "gFreshGVar MOD" shows "igFreshIGVar (fromMOD MOD)"
min_term_min: fixes v :: "'x \<Rightarrow>\<^sub>0 nat" shows "lex_pm i.punit.min_term v"
wffp_purgeIdle_Nil: fixes sl :: "('St, 'U, 'C) state list" assumes "wffp sl"   and "purgeIdle sl = []" shows "\<exists>n st. 0 < n \<and> sl = replicate n (Idle st)"
shift_snth_less: fixes p :: "nat"   and xs :: "'c list"   and s :: "'c stream" assumes "p < length xs" shows "(xs @- s) !! p = xs ! p"
guha_unmatched: fixes ft :: "('a, 'b) flow_entry_match list"   and \<gamma> :: "'a set \<Rightarrow> 'c \<Rightarrow> bool"   and p :: "'c" assumes "\<forall>fe\<in>set ft. \<gamma> (ofe_fields fe) p = False" shows "guha_table_semantics \<gamma> ft p None"
reduction_doesnt_break_length_signs_R: fixes signs :: "rat list list"   and qs :: "real poly list"   and p :: "real poly"   and subsets :: "(nat list \<times> nat list) list" assumes "\<forall>x\<in>set signs. length x = length qs"   and "satisfy_equation_R p qs subsets signs"   and "invertible_mat (M_mat_R signs subsets)" shows "\<forall>x\<in>set (reduction_signs_R p qs signs subsets (M_mat_R signs subsets)). length x = length qs"
sum_in_spanI: fixes q :: "'b \<Rightarrow> 'a"   and B :: "'b set" shows "(\<Sum>b\<in>B. q b *s b) \<in> span B"
GF_advice_a1: fixes \<phi> :: "'b ltln"   and w :: "nat \<Rightarrow> 'b set"   and X :: "'b ltln set" assumes "\<F> \<phi> w \<subseteq> X"   and "w \<Turnstile>\<^sub>n \<phi>" shows "w \<Turnstile>\<^sub>n \<phi>[X]\<^sub>\<Pi>\<^sub>1"
lem2: fixes z :: "'b" assumes "norm (z - y) < d" shows "norm (g z - g y) \<le> B * norm (z - y)"