hcju/isabelleps huggingface dataset:
distinct_59: fixes x41 :: "'d"   and x42 :: "'a \<Rightarrow> nat"   and x43 :: "'a \<Rightarrow> 'b"   and x44 :: "('a, 'b, 'c, 'd) seqp"   and x45 :: "('a, 'b, 'c, 'd) seqp"   and x10 :: "'c" shows "{x41}unicast(x42, x43) . x44 \<triangleright> x45 \<noteq> call(x10)"
assms: shows "(E, {#}) \<in> unif\<^sup>*"
trans: fixes P :: "('a, 'b) Proc"   and Q :: "('a, 'b) Proc"   and Enc :: "'a \<Rightarrow> 'b"   and R :: "('a, 'b) Proc" assumes "(P, Q) \<in> encoding.indRelRPO Enc"   and "(Q, R) \<in> encoding.indRelRPO Enc" shows "(P, R) \<in> encoding.indRelRPO Enc"
"1_1": shows "\<forall>(r, c)\<in>set C. 0 < r \<and> c \<in> cs"
sin_cos_squared_add3: fixes x :: "'b" shows "cos x * cos x + sin x * sin x = (1::'b)"
dom_dom_restrict: fixes S :: "'b set"   and p :: "'b \<Rightarrow> 'c decision option" shows "dom (S \<triangleleft> p) = S \<inter> dom p"
wpo_ns_refl: fixes s :: "('f, 'v) Term.term" shows "s \<succeq> s"
subspace_add: fixes Sa :: "('b \<Rightarrow> real) set"   and x :: "'b \<Rightarrow> real"   and y :: "'b \<Rightarrow> real" assumes "Sa \<subseteq> dest.diff_fun_space"   and "dest.diff_fun_space.m1.subspace Sa"   and "x \<in> Sa"   and "y \<in> Sa" shows "x + y \<in> Sa"
edom_env: fixes S :: "'b set"   and m :: "'b \<Rightarrow> 'c" shows "edom (m f|` S) = edom m \<inter> S"
bottom_right_bottom: fixes a :: "'a" shows "gcd a (1::'a) = (1::'a)"
SeqWRP_HPair_Zero_Zero: fixes s :: "tm"   and k :: "tm"   and y :: "tm" shows "{SeqWRP s k y} \<turnstile> HPair Zero Zero IN s"
inext_nth_card_Max: fixes I :: "nat set"   and n :: "nat" assumes "finite I"   and "I \<noteq> {}"   and "card I \<le> Suc n" shows "I \<rightarrow> n = Max I"
linear: fixes x :: "'d DBMEntry"   and y :: "'d DBMEntry" shows "x \<preceq> y \<or> y \<preceq> x"
integrable_on_lebesgue_on: fixes S :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "integrable (lebesgue_on S) f"   and "S \<in> sets lebesgue" shows "f integrable_on S"
face_distance_le: fixes C :: "'a set"   and F :: "'a set"   and D :: "'a set" assumes "domain.chamber C"   and "order.greater_eq C F" shows "order.greater_eq (domain.chamber_distance C D) (domain.face_distance F D)"
dom_kildall_is_strict: fixes j :: "nat"   and res :: "nat list" assumes "j < length start"   and "dom_kildall start ! j = res" shows "j \<notin> set res"
equality: fixes r :: "('b, 'c) arrow_of_spans_data_scheme"   and r' :: "('b, 'c) arrow_of_spans_data_scheme" assumes "Chn r = Chn r'"   and "Dom r = Dom r'"   and "Cod r = Cod r'"   and "arrow_of_spans_data.more r = arrow_of_spans_data.more r'" shows "r = r'"
differentiable_bound_linearization: fixes a :: "'a"   and b :: "'a"   and S :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and f' :: "'a \<Rightarrow> 'a \<Rightarrow> 'b"   and x0 :: "'a"   and B :: "real" assumes "\<And>t. t \<in> {0..1} \<Longrightarrow> a + t *\<^sub>R (b - a) \<in> S"   and "\<And>x. x \<in> S \<Longrightarrow> (f has_derivative f' x) (at x within S)"   and "\<And>x. x \<in> S \<Longrightarrow> onorm (f' x - f' x0) \<le> B"   and "x0 \<in> S" shows "norm (f b - f a - f' x0 (b - a)) \<le> norm (b - a) * B"
cases: fixes a1 :: "'a list"   and a2 :: "'b sequent" assumes "(a1, a2) \<in> Ax" obtains i :: "nat" where "a1 = []" and "a2 = ( \<LM> At i  \<RM> \<Rightarrow>* \<LM> At i  \<RM>)"   | "a1 = []" and "a2 = ( \<LM> ff  \<RM> \<Rightarrow>* \<Empt>)"
Sup: fixes M :: "'a set"   and f :: "'a \<Rightarrow> 'a" assumes "Complete_Partial_Order.chain (\<le>) M"   and "\<forall>x\<in>M. x \<in> ccpo_class.iterates f" shows "\<Squnion> M \<in> ccpo_class.iterates f"
nilTrans_5: fixes \<Psi> :: "'b"   and M :: "'a"   and N :: "'a"   and P :: "('a, 'b, 'c) psi"   and P' :: "('a, 'b, 'c) psi" assumes "\<Psi> \<rhd> M\<langle>N\<rangle>.P \<longmapsto> \<tau> \<prec> P'" shows "False"
small_UN_V: fixes X :: "'c set"   and B :: "'c \<Rightarrow> V set" assumes "small X"   and "\<And>x. x \<in> X \<Longrightarrow> small (B x)" shows "small (\<Union> (B ` X))"
finite_FV: fixes S :: "Core.term set" assumes "finite S" shows "finite (FV S)"
circ_square: fixes x :: "'a" shows "(x * x)\<^sup>\<star> \<le> x\<^sup>\<star>"
fpr_same_suffix: fixes P :: "cld list"   and ctx :: "ctx"   and cl :: "cl"   and prefix :: "(ctx \<times> cld) list"   and path :: "(ctx \<times> cld) list" shows "find_path_rec_f P ctx cl prefix = Some path \<longrightarrow> (\<forall>suffix prefix' suffix'. path = prefix @ suffix \<and> find_path_rec_f P ctx cl prefix' = Some (prefix' @ suffix') \<longrightarrow> suffix = suffix')"
onorm_def: fixes f :: "'d \<Rightarrow> 'e" shows "onorm f = (SUP x. norm (f x) / norm x)"
intros_19: fixes i :: "nat"   and \<C> :: "t_context"   and ts :: "t list" assumes "i < length (label \<C>)"   and "label \<C> ! i = ts" shows "\<C> \<turnstile> [Br_if i] : ts @ [T_i32] _> ts"
more_arith_simps_6: fixes a :: "'d" shows "a * (1::'d) = a"
attributes_def: fixes "file" :: "('a \<times> 'b, 'a, 'c) env" shows "attributes file = (case file of Val (att, text) \<Rightarrow> att | Env att dir \<Rightarrow> att)"
IH: fixes t :: "'a trm" assumes "subterm t p' s" shows "x \<in> vars_of t"
hyps: fixes ll :: "('p, 'ba) label list"   and p :: "'p"   and v :: "'n"   and c' :: "'n list multiset"   and P :: "'p multiset" assumes "length ll \<le> length w"   and "(([entry fg p], {#}), ll, [v], c') \<in> trcl (trss fg)"   and "size P \<le> k"   and "{#[entry fg p]. p \<in># P#} \<subseteq># c'" shows "(v, mon_w fg ll, P) \<in> S_cs fg k"
uminus_INF: fixes B :: "'b \<Rightarrow> 'a"   and A :: "'b set" shows "- Inf (B ` A) = (SUP x\<in>A. - B x)"
Throw: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and cs :: "('e, 'd, 'f) com list"   and css :: "(('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list"   and s :: "'e" shows "\<Gamma>\<turnstile> (Throw # cs, css, Normal s) \<rightarrow> (cs, css, Abrupt s)"
status_rel_merge_status: fixes a :: "status"   and b :: "status" shows "((merge_status a b, SUCCESS) \<notin> status_rel) = (a = FAILED \<or> b = FAILED \<or> a = FOUND \<or> b = FOUND)"
assoc: fixes a :: "'g"   and b :: "'g"   and c :: "'g" shows "sup (sup a b) c = sup a (sup b c)"
inv1: shows "u1 \<cdot> u2 \<cdot> (inverse u2 \<cdot> inverse u1) = \<one>"
zny: shows "(SOME z. z \<notin> lesvars upds \<union> varacom ({A} WHILE b DO C)) \<notin> lesvars upds"
simps_2: fixes x :: "'a"   and xs :: "'a list"   and ys :: "'a list" shows "(x # xs) @ ys = x # xs @ ys"
the_trivialI: fixes vs :: "'a list"   and v :: "'a"   and v' :: "'a" assumes "set vs \<subseteq> {v, v'}"   and "v' \<noteq> v" shows "the_trivial v vs = (if set vs \<subseteq> {v} then None else Some v')"
Eq_gpv: shows "gen_lossless_gpv b \<I>_full gpv"
updates_simps_3: fixes k :: "'c"   and ks :: "'c list"   and v :: "'d"   and vs :: "'d list"   and ps :: "('c \<times> 'd) list" shows "updates (k # ks) (v # vs) ps = updates ks vs (update k v ps)"
hintikka_def: fixes H :: "('b, 'c) form set" shows "hintikka H = ((\<forall>p ts. \<not> (Pred p ts \<in> H \<and> Neg (Pred p ts) \<in> H)) \<and> FF \<notin> H \<and> Neg TT \<notin> H \<and> (\<forall>Z. Neg (Neg Z) \<in> H \<longrightarrow> Z \<in> H) \<and> (\<forall>A B. And A B \<in> H \<longrightarrow> A \<in> H \<and> B \<in> H) \<and> (\<forall>A B. Neg (Or A B) \<in> H \<longrightarrow> Neg A \<in> H \<and> Neg B \<in> H) \<and> (\<forall>A B. Or A B \<in> H \<longrightarrow> A \<in> H \<or> B \<in> H) \<and> (\<forall>A B. Neg (And A B) \<in> H \<longrightarrow> Neg A \<in> H \<or> Neg B \<in> H) \<and> (\<forall>A B. Impl A B \<in> H \<longrightarrow> Neg A \<in> H \<or> B \<in> H) \<and> (\<forall>A B. Neg (Impl A B) \<in> H \<longrightarrow> A \<in> H \<and> Neg B \<in> H) \<and> (\<forall>P t. closedt 0 t \<longrightarrow> Forall P \<in> H \<longrightarrow> P[t/0] \<in> H) \<and> (\<forall>P t. closedt 0 t \<longrightarrow> Neg (Exists P) \<in> H \<longrightarrow> Neg (P[t/0]) \<in> H) \<and> (\<forall>P. Exists P \<in> H \<longrightarrow> (\<exists>t. closedt 0 t \<and> P[t/0] \<in> H)) \<and> (\<forall>P. Neg (Forall P) \<in> H \<longrightarrow> (\<exists>t. closedt 0 t \<and> Neg (P[t/0]) \<in> H)))"
Linear_order_in_diff_Id: fixes r :: "('d \<times> 'd) set"   and a :: "'d"   and b :: "'d" assumes "Linear_order r"   and "a \<in> Field r"   and "b \<in> Field r" shows "((a, b) \<in> r) = ((b, a) \<notin> r - Id)"
linear_diff: fixes f :: "'a \<Rightarrow> 'b"   and x :: "'a"   and y :: "'a" assumes "linear f" shows "f (x - y) = f x - f y"
dvd_imp_subdegree_le: fixes f :: "'a fps"   and g :: "'a fps" assumes "f dvd g"   and "g \<noteq> 0" shows "subdegree f \<le> subdegree g"
is_cupcake_all_envD_2: fixes env :: "v sem_env" assumes "is_cupcake_all_env env" shows "cupcake_c_ns (c env)"
small_steps_append: fixes \<pi> :: "char list \<Rightarrow> com option"   and cs\<^sub>1 :: "com \<times> (char list \<Rightarrow> int \<Rightarrow> int)"   and cs\<^sub>2 :: "com \<times> (char list \<Rightarrow> int \<Rightarrow> int)"   and cs\<^sub>3 :: "(com \<times> (char list \<Rightarrow> int \<Rightarrow> int)) option" assumes "small_steps \<pi> cs\<^sub>1 (Some cs\<^sub>2)"   and "small_steps \<pi> cs\<^sub>2 cs\<^sub>3" shows "small_steps \<pi> cs\<^sub>1 cs\<^sub>3"
assms_4: shows "s0 \<bullet> (1::'a) \<le> c"
le: fixes A :: "'a set" assumes "A \<subseteq> set_pmf p" shows "measure_pmf.prob p A \<le> measure_pmf.prob q {y \<in> set_pmf q. \<exists>x\<in>A. R x y}"
prod_le_def: fixes P1 :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and P2 :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "prod_le P1 P2 = (\<lambda>(p1, p2) (q1, q2). P1 p1 q1 \<and> P2 p2 q2)"
erh_rHom_coeff: fixes A :: "('g, 'h) Ring_scheme"   and B :: "('g, 'i) Ring_scheme"   and Y :: "'g"   and h :: "'a \<Rightarrow> 'g"   and c :: "nat \<times> (nat \<Rightarrow> 'a)" assumes "PolynRg A B Y"   and "h \<in> rHom S B"   and "pol_coeff S c" shows "erh R S X A B Y h 0 c = cmp h (snd c) 0"
gt_unary: fixes t :: "('s, 'v) tm"   and s :: "('s, 'v) tm" assumes "wt t \<ge>\<^sub>p wt s"   and "\<not> head t \<le>\<ge>\<^sub>h\<^sub>d head s"   and "num_args t = 1"   and "\<exists>f\<in>local.ground_heads (head t). arity_sym f = 1 \<and> wt_sym f = 0"   and "arg t \<ge>\<^sub>t s" shows "t >\<^sub>t s"
dMk: fixes k :: "nat" assumes "k < n" shows "M k \<in> carrier_mat d d"
c_d: shows "(c, d) \<in> thetaZOSeqI"
x_in_u_eq: fixes x :: "nat"   and u :: "nat" shows "(x \<in> nat_to_set u) = (c_in x u = 1)"
in_snapshot_valid_ref: fixes r :: "'e"   and s :: "'f process_name \<Rightarrow> ('g, 'f, 'h, 'e) Model.local_state" assumes "in_snapshot r s"   and "valid_refs_inv s" shows "obj_at (\<lambda>s. True) r s"
ball_divide_subset: fixes d :: "real"   and x :: "'a"   and e :: "real" assumes "1 \<le> d" shows "ball x (e / d) \<subseteq> ball x e"
iptables_bigstepD_1: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and r :: "'c rule list"   and s :: "state"   and t :: "state" assumes "\<Gamma>,\<gamma>,p\<turnstile> \<langle>r, s\<rangle> \<Rightarrow> t"   and "r = []" shows "s = t"
zero_le_divide_1_iff: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and inverse :: "'d \<Rightarrow> 'd"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sgn :: "'d \<Rightarrow> 'd"   and abs :: "'d \<Rightarrow> 'd"   and a :: "'d" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "less_eq zero (divide one a) = less_eq zero a"
X: fixes x :: "'a" assumes "x \<in> topspace X"   and "P (prod_topology X Y)" shows "P (subtopology (prod_topology X Y) ({x} \<times> topspace Y))"
op: fixes u :: "'a set" assumes "u \<in> U" shows "open u"
ib: shows "i + 1 \<le> j \<and> \<not> ps (S t (i + 1)) q \<noteq> None \<and> ps (S t (i + 1)) p \<noteq> None \<and> (\<forall>k. i + 1 \<le> k \<and> k < j \<longrightarrow> occurs_on (t ! k) \<noteq> q) \<and> j - (i + 1) = n \<and> Marker \<in> set (msgs (S t (i + 1)) cid) \<and> cs (S t i) cid = cs (S t (i + 1)) cid"
g_1: shows "uniformly_continuous_on (closure (range to_metric_completion)) g"
Pd_shuffle_to_top_simps_3: fixes d :: "'b"   and ds' :: "'b set"   and Y :: "'a set" assumes "d \<notin> ds'" shows "TruePref.Pd_shuffle_to_top ds' Y d = Pd d"
map_ide_simp: fixes a :: "'a" assumes "B.ide a" shows "B.\<ll>'.map a = B.inv (B.\<ll> a)"
comp_in_homI': fixes f :: "'a"   and g :: "'a"   and a :: "'a"   and c :: "'a" assumes "C.arr f"   and "C.arr g"   and "C.dom f = a"   and "C.cod g = c"   and "C.dom g = C.cod f" shows "\<guillemotleft>g \<cdot>\<^sub>C f : a \<Rightarrow>\<^sub>C c\<guillemotright>"
Z_def: shows "Z \<equiv> Y \<union> wfvarsoccs\<^sub>s\<^sub>s\<^sub>t T2"
lift_CFGExit_wf: fixes sourcenode :: "'g \<Rightarrow> 'h"   and targetnode :: "'g \<Rightarrow> 'h"   and kind :: "'g \<Rightarrow> ('i, 'j, 'k, 'l) edge_kind"   and valid_edge :: "'g \<Rightarrow> bool"   and Entry :: "'h"   and get_proc :: "'h \<Rightarrow> 'l"   and get_return_edges :: "'g \<Rightarrow> 'g set"   and procs :: "('l \<times> 'i list \<times> 'i list) list"   and Main :: "'l"   and Exit :: "'h"   and Def :: "'h \<Rightarrow> 'i set"   and Use :: "'h \<Rightarrow> 'i set"   and ParamDefs :: "'h \<Rightarrow> 'i list"   and ParamUses :: "'h \<Rightarrow> 'i set list"   and H :: "'i set"   and L :: "'i set" assumes "CFGExit_wf sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit Def Use ParamDefs ParamUses"   and "Postdomination sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit" shows "CFGExit_wf src trg knd (lift_valid_edge valid_edge sourcenode targetnode kind Entry Exit) NewEntry (lift_get_proc get_proc Main) (lift_get_return_edges get_return_edges valid_edge sourcenode targetnode kind) procs Main NewExit (lift_Def Def Entry Exit H L) (lift_Use Use Entry Exit H L) (lift_ParamDefs ParamDefs) (lift_ParamUses ParamUses)"
assms_2: shows "x \<in> X \<inter> fmdom' f \<inter> fmdom' g"
negchecks_model_db_subset: fixes D' :: "(('b, 'c) Term.term \<times> ('b, 'c) Term.term) set"   and D :: "(('b, 'c) Term.term \<times> ('b, 'c) Term.term) set"   and \<I> :: "'c \<Rightarrow> ('b, 'c) Term.term"   and X :: "'c list"   and F :: "(('b, 'c) Term.term \<times> ('b, 'c) Term.term) list"   and F' :: "(('b, 'c) Term.term \<times> ('b, 'c) Term.term) list" assumes "D' \<subseteq> D"   and "negchecks_model \<I> D X F F'" shows "negchecks_model \<I> D' X F F'"
distinct_1: fixes x1 :: "name"   and x2 :: "name" shows "Const x1 \<noteq> Free x2"
lsum_mono: fixes al :: "'a list"   and bl :: "'a list"   and f :: "'a \<Rightarrow> nat" assumes "al \<le> bl" shows "lsum f al \<le> lsum f bl"
card_set_of_vector: shows "card (set_of_vector X) = CARD('n)"
is_faithful: fixes f :: "'a"   and f' :: "'a" assumes "hom.par f f'"   and "e \<star> f \<star> src \<mu> = e \<star> f' \<star> src \<mu>" shows "f = f'"
monomial_0I: fixes c :: "'c"   and t :: "'d" assumes "c = (0::'c)" shows "monomial c t = 0"
fconverse_vempty: shows "[]\<^sub>\<circ>\<inverse>\<^sub>\<bullet> = []\<^sub>\<circ>"
prefixToLevel_auxPrefixToLevel_auxHigherLevel: fixes i :: "nat"   and j :: "nat"   and a :: "('b \<times> bool) list"   and k :: "nat" assumes "i \<le> j" shows "prefixToLevel_aux a i k = prefixToLevel_aux (prefixToLevel_aux a j k) i k"
nth_Cons': fixes x :: "'a"   and xs :: "'a list"   and n :: "nat" shows "(x # xs) ! n = (if n = 0 then x else xs ! (n - 1))"
exhaust: fixes y :: "Enum.finite_4" obtains "y = a\<^sub>1"   | "y = a\<^sub>2"   | "y = a\<^sub>3"   | "y = a\<^sub>4"
smult_l_distr: fixes x :: "'a vec"   and a :: "'a"   and b :: "'a" assumes "True"   and "True"   and "x \<in> carrier_vec n" shows "(a + b) \<cdot>\<^sub>v x = a \<cdot>\<^sub>v x + b \<cdot>\<^sub>v x"
less_cSup_iff: fixes X :: "'a set"   and y :: "'a" assumes "X \<noteq> {}"   and "bdd_above X" shows "(y < Sup X) = (\<exists>x\<in>X. y < x)"
k_dfs_gen_dfs_unfold: shows "k_dfs = KBPAlg.dfs k_empt (k_frontier a)"
xor_commute: fixes a :: "'a"   and b :: "'a" shows "a \<oplus> b = b \<oplus> a"
c_pair_of_c_fst_c_snd: fixes u :: "nat" shows "c_pair (c_fst u) (c_snd u) = u"
simps_2: fixes ts :: "('b btree \<times> 'b) list"   and t :: "'b btree" shows "bal (Node ts t) = ((\<forall>sub\<in>set (subtrees ts). height sub = height t) \<and> Ball (set (subtrees ts)) bal \<and> bal t)"
abs_eq: shows "0\<^sub>h = abs_complex_homo 0\<^sub>h\<^sub>c"
commute: fixes x :: "'a"   and y :: "'a" shows "f x y = f y x"
rank_lt: fixes a :: "V"   and b :: "V" assumes "a \<in> elts b" shows "rank a < rank b"
assms_1: shows "keys (p + q) = {lt p, tt q}"
inverse_unique: fixes f :: "'a \<times> 'a"   and g :: "'a \<times> 'a" assumes "B.VV.inverse_arrows f g" shows "B.VV.inv f = g"
dghm_proj_snd_HomDom: fixes \<AA> :: "V"   and \<BB> :: "V" shows "\<pi>\<^sub>D\<^sub>G\<^sub>.\<^sub>2 \<AA> \<BB>\<lparr>HomDom\<rparr> = \<AA> \<times>\<^sub>D\<^sub>G \<BB>"
double_compl: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and x :: "'a" assumes "class.boolean_algebra minus uminus inf less_eq less sup bot top" shows "uminus (uminus x) = x"
monom_times_poly_shift': fixes n :: "nat"   and p :: "'b poly" assumes "n \<le> poly_subdegree p" shows "monom (1::'b) n * poly_shift n p = p"
inject: fixes x2 :: "nat"   and y2 :: "nat" shows "(Suc (x2::nat) = Suc (y2::nat)) = (x2 = y2)"
Above_shuffle_to_top: fixes x :: "'a"   and Y :: "'a set"   and X :: "'a set"   and y :: "'a" assumes "x \<in> Above (TruePref.shuffle_to_top Y (Xd x)) X"   and "y \<in> Y"   and "inj_on Xd Y"   and "y \<in> X" shows "x = y"
sinks_interference_eq: fixes u :: "'b"   and D :: "'c \<Rightarrow> 'b"   and x :: "'c"   and I :: "('b \<times> 'b) set"   and xs :: "'c list" shows "((u, D x) \<in> I \<or> (\<exists>v\<in>sinks I D u xs. (v, D x) \<in> I)) = (D x \<in> sinks I D u (xs @ [x]))"
components_full_pmdl_subset: fixes K :: "'k set" shows "component_of_term ` Keys (full_pmdl K) \<subseteq> K"
partial_appd: fixes vs' :: "nat list"   and t :: "dB" assumes "length vs' \<le> n" shows "var_dist vs' ((Abs ^^ n) (var_dist vs (liftn n t 0))) \<leftrightarrow> (Abs ^^ (n - length vs')) (var_dist (vsubsts n vs' vs) (liftn (n - length vs') t 0))"
float_zero: fixes b :: "int" shows "float (0, b) = 0"
sinl_below: fixes x :: "'c"   and y :: "'c" shows "(sinl\<cdot>x \<sqsubseteq> sinl\<cdot>y) = (x \<sqsubseteq> y)"
fmap_resT_simps_1: fixes f :: "'g \<rightarrow> 'e" shows "fmap\<cdot>f\<cdot>\<bottom> = \<bottom>"
member_remove: fixes x :: "'d"   and y :: "'d"   and A :: "'d set" shows "(x \<in> Set.remove y A) = (x \<in> A \<and> x \<noteq> y)"
invariance_of_dimension_subspaces: fixes U :: "'a set"   and S :: "'a set"   and V :: "'b set"   and f :: "'a \<Rightarrow> 'b" assumes "openin (top_of_set U) S"   and "subspace U"   and "subspace V"   and "continuous_on S f"   and "f ` S \<subseteq> V"   and "inj_on f S"   and "S \<noteq> {}" shows "dim U \<le> dim V"
gf_U: fixes x :: "'s"   and y :: "'s" shows "(gf_U x y \<longlongrightarrow> U x y) (at_left 1)"
unit_simps_1: fixes a :: "'c" assumes "C.obj a" shows "D.arr (H.unit a)"
freshImEnvAt_def: fixes xs :: "'varSort"   and x :: "'var"   and rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option"   and ys :: "'varSort"   and y :: "'var" shows "freshImEnvAt xs x rho ys y \<equiv> rho ys y = None \<and> (ys \<noteq> xs \<or> y \<noteq> x) \<or> (\<exists>Y. rho ys y = Some Y \<and> fresh xs x Y)"
borel_measurable_dist: fixes f :: "'b \<Rightarrow> 'c"   and M :: "'b measure"   and g :: "'b \<Rightarrow> 'c" assumes "f \<in> borel_measurable M"   and "g \<in> borel_measurable M" shows "(\<lambda>x. dist (f x) (g x)) \<in> borel_measurable M"
semigroup_add_def: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c" shows "class.semigroup_add plus \<equiv> \<forall>a b c. plus (plus a b) c = plus a (plus b c)"
outgoing_def: fixes G :: "('k, 'l) graph_scheme"   and x :: "'k" shows "\<^bold>O\<^bold>U\<^bold>T\<^bsub>G\<^esub> x = {y. (x, y) \<in> \<^bold>E\<^bsub>G\<^esub>}"
Koszul_syz_sigs_auxE_lt_Koszul_syz: fixes v :: "'t" assumes "0 \<notin> set fs"   and "v \<in> set (Koszul_syz_sigs_aux fs 0)" obtains i :: "nat" and j :: "nat" where "i < j" and "j < length fs" and "v = lt (fs ! i \<odot> monomial (1::'b) (term_of_pair (0::'a, j)) - fs ! j \<odot> monomial (1::'b) (term_of_pair (0::'a, i)))"
assms_2: assumes "c"   and "c'" shows "(t, t') \<in> R"
Cd_XD_smallest: fixes d :: "'b" assumes "d \<in> ds" shows "Cd d (XD_smallest \<inter> Field (Pd d)) = Cd d (X \<inter> Field (Pd d))"
z: shows "\<forall>w. P w z = (P w x \<and> P w y)"
update_convs_1: fixes ext' :: "real_int \<Rightarrow> real_int"   and ext :: "real_int"   and lan :: "nat_int"   and own :: "cars"   and more :: "'a" shows "ext_update ext' \<lparr>ext = ext, lan = lan, own = own, \<dots> = more\<rparr> = \<lparr>ext = ext' ext, lan = lan, own = own, \<dots> = more\<rparr>"
succ_notin_self: fixes i :: "V" shows "ZFC_in_HOL.succ i \<notin> elts i"
lt_minus_eqI: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" assumes "lt p \<prec>\<^sub>t lt q" shows "lt (p - q) = lt q"
vec_times_mat_eq_def: fixes y :: "'b vec"   and A :: "'b mat"   and c :: "'b vec" shows "[y\<^sub>v*A]=c = ((\<forall>i<dim_vec c. (A\<^sup>T *\<^sub>v y) $ i = c $ i) \<and> dim_col A\<^sup>T = dim_vec y \<and> dim_row A\<^sup>T = dim_vec c)"
connectedin_separation: fixes X :: "'c topology"   and S :: "'c set" shows "connectedin X S = (S \<subseteq> topspace X \<and> (\<nexists>C1 C2. C1 \<union> C2 = S \<and> C1 \<noteq> {} \<and> C2 \<noteq> {} \<and> C1 \<inter> X closure_of C2 = {} \<and> C2 \<inter> X closure_of C1 = {}))"
mset_le_distrib: fixes X :: "'a multiset"   and A :: "'a multiset"   and B :: "'a multiset" assumes "X \<subseteq># A + B" obtains Xa :: "'a multiset" and Xb :: "'a multiset" where "X = Xa + Xb" and "Xa \<subseteq># A" and "Xb \<subseteq># B"
IH: fixes b :: "bool"   and m :: "nat" shows "mssnth (sin_series_stream_aux b (fact m) (real m)) n = (if b then - 1 else 1) * (- 1) ^ n / fact (2 * n + m)"
HVZK1: fixes x1 :: "'a"   and w1 :: "'b"   and x0 :: "'g" assumes "(x1, w1) \<in> Rel1" shows "\<forall>c\<in>challenge_space. \<Sigma>_OR.R (x0, x1) (Inr w1) c = \<Sigma>_OR.S (x0, x1) c"
functor_Diag: shows "functor (\<cdot>) CC.comp Diag"
mono_read_FD: fixes P :: "'d \<Rightarrow> 'e process"   and Q :: "'d \<Rightarrow> 'e process"   and c :: "'d \<Rightarrow> 'e" assumes "\<And>x. P x \<le> Q x" shows "read c UNIV P \<le> read c UNIV Q"
cfg: shows "continuous_on {0..1} (\<lambda>x. f (g x))"
rGamma_series_def: fixes z :: "'b"   and n :: "nat" shows "rGamma_series z n = pochhammer z (n + 1) / (fact n * exp (z * of_real (ln (real n))))"
mkArr_restrict_eq': fixes A :: "'s set"   and B :: "'s set"   and F :: "'s \<Rightarrow> 's" assumes "arr (mkArr A B (restrict F A))" shows "mkArr A B (restrict F A) = mkArr A B F"
dither_def: shows "dither \<equiv> [(W0, 1), (R, 2), (L, 1), (L, 0)]"
ridmap_ind_bijec: fixes A :: "('c, 'd) Ring_scheme" assumes "Ring A" shows "bijec\<^bsub>A /\<^sub>r (ker\<^bsub>A,A\<^esub> ridmap A),A\<^esub> (ridmap A\<degree>\<^bsub>A,A\<^esub>)"
distinguishing_formula_def: fixes P :: "'state"   and Q :: "'state" shows "distinguishing_formula P Q \<equiv> Conj (Abs_bset {- p \<bullet> (SOME x. supp x \<subseteq> supp (p \<bullet> P) \<and> x distinguishes (p \<bullet> P) from (p \<bullet> Q)) |p. True})"
floor_less_iff: fixes x :: "'d"   and z :: "int" shows "(\<lfloor>x\<rfloor> < z) = (x < of_int z)"
map_index'_map_zip: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'b \<Rightarrow> 'a"   and xs :: "'b list" shows "map_index' n f xs = map2 f [n..<n + length xs] xs"
simps: fixes a1 :: "'action list"   and a2 :: "'state" shows "R.path a1 a2 = ((\<exists>p. a1 = [] \<and> a2 = p) \<or> (\<exists>a p r. a1 = a # r \<and> a2 = p \<and> ren a p \<and> R.path r (ex a p)))"