hcju/isabelleps huggingface dataset:
a_coset_join2: fixes x :: "'a"   and H :: "'a set" assumes "x \<in> carrier G"   and "subgroup H (add_monoid G)"   and "x \<in> H" shows "H +> x = H"
lincomb_empty: fixes a :: "'c \<Rightarrow> 'a" shows "lincomb a {} = \<zero>\<^bsub>M\<^esub>"
free: shows "\<forall>(x, \<tau>)\<in>set vs. (x, \<tau>) \<notin> fv t \<union> FV \<Gamma>"
norm_bound_Basis_lt: fixes b :: "'a"   and x :: "'a"   and e :: "real" assumes "b \<in> Basis"   and "norm x < e" shows "\<bar>x \<bullet> b\<bar> < e"
le: fixes p :: "tm"   and vs :: "'b list"   and bs :: "'b list" assumes "allpolys isnpoly p" shows "Ifm vs bs (le p) = Ifm vs bs (Le p)"
union_list'_nodes_finite: fixes AA :: "('c, 'd) dca list" assumes "list_all (finite \<circ> DCA.nodes) AA" shows "finite (DGCA.nodes (union_list' AA))"
assms_3: shows "f \<triangleright> n \<noteq> g \<triangleright> n"
l_one: fixes x :: "nat \<Rightarrow> 'a" assumes "x \<in> carrier P" shows "\<one>\<^bsub>P\<^esub> \<otimes>\<^bsub>P\<^esub> x = x"
elims_2: fixes x :: "nat \<times> nat" assumes "invar x" obtains n :: "nat" and l :: "nat" where "x = (n, l)" and "l0 \<le> l \<and> (int l0 \<le> \<lfloor>real l / c\<rfloor> \<longrightarrow> f1 * real l \<le> real n) \<and> real n \<le> f2 * real l"
subbdt_ex_dag_def: fixes p :: "ref"   and low :: "ref \<Rightarrow> ref"   and high :: "ref \<Rightarrow> ref"   and t :: "dag"   and var :: "ref \<Rightarrow> nat"   and bdt1 :: "bdt"   and no :: "ref"   and not :: "dag" assumes "Dag p low high t"   and "bdt t var = Some bdt1"   and "Dag no low high not"   and "no \<in> set_of t" shows "\<exists>bdt2. bdt not var = Some bdt2"
\<alpha>_finite: shows "\<alpha> \<noteq> \<top>"
has_integral_nonpos: fixes f :: "'a \<Rightarrow> real"   and i :: "real"   and s :: "'a set" assumes "(f has_integral i) s"   and "\<forall>x\<in>s. f x \<le> 0" shows "i \<le> 0"
induct: fixes P :: "pol \<Rightarrow> pol \<Rightarrow> bool"   and a0 :: "pol"   and a1 :: "pol" assumes "\<And>a b. P (Pc a) (Pc b)"   and "\<And>c i P. P P (Pc c) \<Longrightarrow> P (Pc c) (Pinj i P)"   and "\<And>i P c. P P (Pc c) \<Longrightarrow> P (Pinj i P) (Pc c)"   and "\<And>c P i Q. P Q (Pc c) \<Longrightarrow> P (Pc c) (PX P i Q)"   and "\<And>P i Q c. P Q (Pc c) \<Longrightarrow> P (PX P i Q) (Pc c)"   and "\<And>x P y Q. \<lbrakk>x = y \<Longrightarrow> P P Q; \<lbrakk>x \<noteq> y; y < x\<rbrakk> \<Longrightarrow> P (Pinj (x - y) P) Q; \<lbrakk>x \<noteq> y; \<not> y < x\<rbrakk> \<Longrightarrow> P (Pinj (y - x) Q) P\<rbrakk> \<Longrightarrow> P (Pinj x P) (Pinj y Q)"   and "\<And>x P Q y R. \<lbrakk>x = 0 \<Longrightarrow> P P (PX Q y R); \<lbrakk>x \<noteq> 0; x = 1\<rbrakk> \<Longrightarrow> P R P; \<lbrakk>x \<noteq> 0; x \<noteq> 1\<rbrakk> \<Longrightarrow> P R (Pinj (x - 1) P)\<rbrakk> \<Longrightarrow> P (Pinj x P) (PX Q y R)"   and "\<And>P x R y Q. \<lbrakk>y = 0 \<Longrightarrow> P (PX P x R) Q; \<lbrakk>y \<noteq> 0; y = 1\<rbrakk> \<Longrightarrow> P R Q; \<lbrakk>y \<noteq> 0; y \<noteq> 1\<rbrakk> \<Longrightarrow> P R (Pinj (y - 1) Q)\<rbrakk> \<Longrightarrow> P (PX P x R) (Pinj y Q)"   and "\<And>P1 x P2 Q1 y Q2. \<lbrakk>x = y \<Longrightarrow> P P1 Q1; x = y \<Longrightarrow> P P2 Q2; \<lbrakk>x \<noteq> y; y < x\<rbrakk> \<Longrightarrow> P (PX P1 (x - y) (Pc 0)) Q1; \<lbrakk>x \<noteq> y; y < x\<rbrakk> \<Longrightarrow> P P2 Q2; \<lbrakk>x \<noteq> y; \<not> y < x\<rbrakk> \<Longrightarrow> P (PX Q1 (y - x) (Pc 0)) P1; \<lbrakk>x \<noteq> y; \<not> y < x\<rbrakk> \<Longrightarrow> P P2 Q2\<rbrakk> \<Longrightarrow> P (PX P1 x P2) (PX Q1 y Q2)" shows "P a0 a1"
Rel_def: fixes r :: "'e \<Rightarrow> 'f \<Rightarrow> bool" shows "Transfer.Rel r \<equiv> r"
snoc_eq_iff_butlast: fixes xs :: "'c list"   and x :: "'c"   and ys :: "'c list" shows "(xs @ [x] = ys) = (ys \<noteq> [] \<and> butlast ys = xs \<and> last ys = x)"
is_processT1: fixes P :: "'a process" shows "([], {}) \<in> F P"
nba_g_run_ipath: fixes A :: "('c, 'd) nba"   and w :: "'c stream"   and r :: "'d stream"   and p :: "'d" assumes "run A (w ||| r) p" shows "ipath (g_E (nba_g A)) ((!!) (p ## r))"
inline1_Pause: fixes out :: "'call"   and c :: "('b, 'call, 'ret) rpv"   and s :: "'s" shows "local.inline1 (Generative_Probabilistic_Value.Pause out c) s = the_gpv (callee s out) \<bind> (\<lambda>react. case react of Pure (x, s') \<Rightarrow> local.inline1 (c x) s' | IO out' c' \<Rightarrow> return_spmf (Inr (out', c', c)))"
indRelR_iff_exists_source_target_relation: fixes Pred :: "('procS, 'procT) Proc \<times> ('procS, 'procT) Proc \<Rightarrow> bool" shows "(\<forall>(P, Q)\<in>indRelR. Pred (P, Q)) = (\<exists>Rel. (\<forall>S. (SourceTerm S, TargetTerm (\<lbrakk>S\<rbrakk>)) \<in> Rel) \<and> (\<forall>(P, Q)\<in>Rel. Pred (P, Q)))"
Swap: fixes X :: "('d, 'e, 'f, 'g, 'h) qTerm"   and Y :: "('d, 'e, 'f, 'g, 'h) qTerm"   and zs :: "'f"   and x :: "'g"   and y :: "'g" assumes "(X, Y) \<in> qSwapped" shows "(X, Y #[[x \<and> y]]_zs) \<in> qSwapped"
xml_of_typ_def: fixes uu :: "typerep" shows "xml_of_typ uu = [XML_Tree]"
simps: fixes w :: "'a list"   and I :: "(nat + nat set) list"   and \<phi> :: "'a formula" shows "wf_interp_for_formula (w, I) \<phi> = (wf_interp w I \<and> (\<forall>n\<in>FOV \<phi>. case I ! n of Inl x \<Rightarrow> True | Inr b \<Rightarrow> False) \<and> (\<forall>n\<in>SOV \<phi>. case I ! n of Inl x \<Rightarrow> False | Inr x \<Rightarrow> True))"
graph_map_def: fixes g :: "('c \<times> 'd) set" shows "graph_map g = (\<lambda>x. if x \<in> fst ` g then Some (SOME y. (x, y) \<in> g) else None)"
path_Append: fixes sourcenode :: "'f \<Rightarrow> 'g"   and targetnode :: "'f \<Rightarrow> 'g"   and valid_edge :: "'f \<Rightarrow> bool"   and Entry :: "'g"   and n :: "'g"   and as :: "'f list"   and n'' :: "'g"   and as' :: "'f list"   and n' :: "'g" assumes "CFG sourcenode targetnode valid_edge Entry"   and "CFG.path sourcenode targetnode valid_edge n as n''"   and "CFG.path sourcenode targetnode valid_edge n'' as' n'" shows "CFG.path sourcenode targetnode valid_edge n (as @ as') n'"
igFreshIGOpSTR_def: fixes MOD :: "('c, 'd, 'e, 'f, 'g, 'h, 'i, 'j, 'k) model_scheme" shows "igFreshIGOpSTR MOD \<equiv> \<forall>ys y delta inp binp. igFreshInp MOD ys y inp \<and> igFreshBinp MOD ys y binp \<longrightarrow> igFresh MOD ys y (igOp MOD delta inp binp)"
Rep_boolean_subset_inverse: fixes x :: "'b boolean_subset" shows "Abs_boolean_subset (Rep_boolean_subset x) = x"
out_degree_def: fixes G :: "('c, 'd) pre_digraph"   and v :: "'c" shows "out_degree G v \<equiv> card (out_arcs G v)"
phiarg_set: shows "{\<phi>'. \<exists>\<phi>. phiArg g \<phi> \<phi>'} = \<Union> (set ` {b. \<exists>a. phi g a = Some b})"
distinct_15: fixes x31 :: "com"   and x32 :: "com"   and x41 :: "bexp"   and x42 :: "com"   and x43 :: "com" shows "x31;; x32 \<noteq> IF x41 THEN x42 ELSE x43"
statelist'_TAKE: fixes s :: "('c, 'd) fmap"   and as :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) list"   and p :: "('c, 'd) fmap list" assumes "statelist' s as = p" shows "\<forall>n\<le>length as. exec_plan s (take n as) = p ! n"
plus_assoc: fixes a :: "'ag"   and b :: "'ag"   and c :: "'ag" assumes "a \<in> U"   and "b \<in> U"   and "c \<in> U" shows "a +\<^sub>o\<^sub>w b +\<^sub>o\<^sub>w c = a +\<^sub>o\<^sub>w (b +\<^sub>o\<^sub>w c)"
coeff_linear_power: fixes a :: "'b"   and n :: "nat" shows "coeff ([:a, 1::'b:] ^ n) n = (1::'b)"
chain_subset_def: fixes C :: "'b set set" shows "chain\<^sub>\<subseteq> C = set_chain C"
up_add_closed: fixes p :: "nat \<Rightarrow> 'a"   and q :: "nat \<Rightarrow> 'a" assumes "p \<in> up R"   and "q \<in> up R" shows "(\<lambda>i. p i \<oplus> q i) \<in> up R"
interv_diff_def: shows "interv_diff \<equiv> \<lambda>K i. interval_upperbound K \<bullet> i - interval_lowerbound K \<bullet> i"
subseteq_eqvt_1: fixes pi :: "(tyvrs \<times> tyvrs) list"   and X :: "'a set"   and Y :: "'a set" shows "pi \<bullet> (X \<subseteq> Y) = (pi \<bullet> X \<subseteq> pi \<bullet> Y)"
simps_1: fixes N :: "nat" shows "freeright (NONCE N) = NONCE N"
assms: shows "GEN_OP eq (=) (A \<rightarrow> A \<rightarrow> bool_rel)"
deadlock_free_dnorm: fixes Tr :: "'a \<Rightarrow> 'b set"   and Up :: "'a \<Rightarrow> 'b \<Rightarrow> 'a"   and s :: "'a" assumes "\<And>sa. sa \<in> \<RR> Tr Up s \<Longrightarrow> Tr sa \<noteq> {}" shows "deadlock_free_v2 ((P\<^sub>n\<^sub>o\<^sub>r\<^sub>m\<lbrakk>Tr,Up\<rbrakk>) s)"
perm_simps_2: fixes p :: "perm"   and tm1 :: "tm"   and tm2 :: "tm" shows "p \<bullet> tm1 EQ tm2 = (p \<bullet> tm1) EQ (p \<bullet> tm2)"
elims: fixes x :: "dynamic"   and y :: "bool option" assumes "unbox_bool x = y" obtains b :: "bool" where "x = DBool b" and "y = Some b"   | "x = DNil" and "y = None"   | v :: "nat" where "x = DNum v" and "y = None"
prj_simps_6: shows "C.cod \<mu>\<nu>_\<pi>.prj\<^sub>0 = \<pi>.apex"
assms: shows "\<S>\<bullet>\<C> \<turnstile> [$C v] : [] _> [t]"
hyps: assumes "a $ 0 = (1::'a)" shows "a ^ n $ 1 = of_nat n * a $ 1"
closed_Inter: fixes "open" :: "'c set \<Rightarrow> bool"   and K :: "'c set set" assumes "class.topological_space open"   and "\<forall>S\<in>K. topological_space.closed open S" shows "topological_space.closed open (\<Inter> K)"
elims: fixes x :: "'b option \<times> 'b option"   and y :: "'b option" assumes "New_max.merge_option x = y" obtains "x = (None, None)" and "y = None"   | xa :: "'b" where "x = (Some xa__, None)" and "y = Some xa__"   | xa :: "'b" where "x = (None, Some xa)" and "y = Some xa"   | a :: "'b" and b :: "'b" where "x = (Some a, Some b)" and "y = Some a"
linorder_of_list_permute: fixes \<pi> :: "nat \<Rightarrow> nat"   and xs :: "'b list" assumes "\<pi> permutes {..<length xs}"   and "distinct xs" shows "linorder_of_list (permute_list \<pi> xs) = map_relation (set xs) ((!) xs \<circ> inv \<pi> \<circ> index xs) (linorder_of_list xs)"
simps_2: fixes \<alpha> :: "V"   and \<alpha>s :: "V list" shows "\<omega>_sum (\<alpha> # \<alpha>s) = \<omega> \<up> \<alpha> + \<omega>_sum \<alpha>s"
Lcm_set_eq_fold: fixes xs :: "'a list" shows "normalization_euclidean_semiring_class.Lcm (set xs) = fold normalization_euclidean_semiring_class.lcm xs (1::'a)"
that: fixes d :: "real"   and t :: "'a \<Rightarrow> real" assumes "0 < d"   and "continuous_on (ball x d) t"   and "\<And>y. y \<in> ball x d \<Longrightarrow> flow0 y (t y) \<in> {a<--<b}"   and "\<And>y. y \<in> ball x d \<Longrightarrow> \<bar>t y\<bar> < e"   and "continuous_on (ball x d) t"   and "t x = 0" shows "thesis"
simps_2: fixes G :: "('a, 'b) graph"   and v :: "'a"   and u :: "'a"   and us :: "'a list" assumes "valid_unSimpGraph G" shows "valid_unSimpGraph.adj_path G v (u # us) = (valid_unMultigraph.adjacent G v u \<and> valid_unSimpGraph.adj_path G u us)"
realfun_mon_conv_le: fixes x :: "nat \<Rightarrow> 'b \<Rightarrow> real"   and y :: "'b \<Rightarrow> real"   and i :: "nat" assumes "x\<up>y" shows "x i \<le> y"
hyps: assumes "s \<le> n" shows "sum ((^) 2) {s..n} < 2 ^ (n + 1)"
det_col_uminus: fixes M :: "('c, 'd) sq_matrix"   and i :: "'d"   and a :: "('c, 'd) vec" shows "Square_Matrix.det (upd_col M i (- a)) = - Square_Matrix.det (upd_col M i a)"
learn_cpE_3: fixes \<psi> :: "nat \<Rightarrow> nat \<Rightarrow> nat option"   and U :: "(nat \<Rightarrow> nat option) set"   and s :: "nat \<Rightarrow> nat option"   and f :: "nat \<Rightarrow> nat option"   and n :: "nat" assumes "learn_cp \<psi> U s"   and "f \<in> U" shows "\<psi> (the (s (f \<triangleright> n))) \<in> U"
T_eq_bind: fixes s :: "'s" shows "T s = measure_pmf (K s) \<bind> (\<lambda>t. distr (T t) (stream_space (count_space UNIV)) ((##) t))"
connected_Icc: fixes a :: "'b"   and b :: "'b" shows "connected {a..b}"
Postdomination_axioms: shows "Postdomination 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"
le_funI: fixes f :: "'d \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'e" assumes "\<And>x. f x \<le> g x" shows "f \<le> g"
sum_list_nonneg_eq_0_iff: fixes xs :: "'b multiset list" assumes "\<And>x. x \<in> set xs \<Longrightarrow> {#} \<subseteq># x" shows "(monoid_add.sum_list (+) {#} xs = {#}) = (\<forall>x\<in>set xs. x = {#})"
inv_into_f_eq: fixes f :: "'e \<Rightarrow> 'f"   and A :: "'e set"   and x :: "'e"   and y :: "'f" assumes "inj_on f A"   and "x \<in> A"   and "f x = y" shows "inv_into A f y = x"
rel_nondetT'_phantom: fixes A :: "'i \<Rightarrow> 'j \<Rightarrow> bool" shows "rel_nondetT' A = rel_nondetT' \<top>"
prefix_diff3: fixes l :: "'c list"   and m :: "'c list"   and t :: "'c list" shows "l @ m - (l @ t) = m - t"
mult_nonpos_nonneg: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.ordered_semiring_0 plus zero less_eq less times"   and "less_eq a zero"   and "less_eq zero b" shows "less_eq (times a b) zero"
fresh_fmmap_erase_fresh: fixes x :: "'b"   and \<Gamma> :: "('a, ty) fmap" assumes "atom x \<sharp> \<Gamma>" shows "atom x \<sharp> fmmap erase \<Gamma>"
swing_pos_def: fixes x :: "wall"   and y :: "wall" shows "swing_pos x y \<equiv> x = r_over_braid \<and> y = l_over_braid"
IH: assumes "wf\<^sub>s\<^sub>t (V \<union> fv u) (S @ S')"   and "fv t' \<subseteq> wfvarsoccs\<^sub>s\<^sub>t S \<union> (V \<union> fv u)" shows "wf\<^sub>s\<^sub>t (V \<union> fv u) (S @ [\<langle>a: t \<doteq> t'\<rangle>\<^sub>s\<^sub>t] @ S')"
assms_1: shows "(R1 ===> R2 ===> R3) C C"
intensional_simps_4: fixes y :: "'a \<Rightarrow> bool" shows "\<not> \<not> y \<equiv> y"
card_order_on_Card_order: fixes A :: "'c set"   and r :: "('c \<times> 'c) set" assumes "card_order_on A r" shows "A = Field r \<and> Card_order r"
interaction_bounded_by_mono: fixes "consider" :: "'d \<Rightarrow> bool"   and gpv :: "('e, 'd, 'f) gpv"   and n :: "enat"   and m :: "enat" assumes "interaction_bounded_by consider gpv n"   and "n \<le> m" shows "interaction_bounded_by consider gpv m"
set_build: fixes ps :: "(real, 'a) vec list"   and ks :: "'a list" assumes "0 < length ps"   and "\<forall>k. distinct (map (\<lambda>p. p $ k) ps)" shows "set ps = set_kdt (build ks ps)"
seq_comp_failures: fixes P :: "'c option process"   and Q :: "'c option process" assumes "weakly_sequential P" shows "failures (P ; Q) = seq_comp_failures P Q"
from_nat_mono: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and one :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Rep :: "'b \<Rightarrow> int"   and Abs :: "int \<Rightarrow> 'b"   and i :: "nat"   and j :: "nat" assumes "class.mod_type times minus plus zero uminus one less_eq less Rep Abs"   and "i < j"   and "j < CARD('b)" shows "less (mod_type.from_nat Abs i) (mod_type.from_nat Abs j)"
mono: fixes callee :: "'f \<Rightarrow> 'd \<Rightarrow> ('e \<times> 'f) spmf"   and x :: "('b, 'd, 'e) gpv \<times> 'f" shows "spmf.mono_body (\<lambda>exec_gpv. case x of (c, s) \<Rightarrow> the_gpv c \<bind> case_generat (\<lambda>x. return_spmf (x, s)) (\<lambda>out c. callee s out \<bind> (\<lambda>(x, y). curry exec_gpv (c x) y)))"
lt_tail: fixes pair_of_term :: "'c \<Rightarrow> 'd \<times> 'e"   and term_of_pair :: "'d \<times> 'e \<Rightarrow> 'c"   and ord :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_strict :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_term :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ord_term_strict :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and p :: "'c \<Rightarrow>\<^sub>0 'f" assumes "ordered_term pair_of_term term_of_pair ord ord_strict ord_term ord_term_strict"   and "ordered_term.tail term_of_pair ord_term p \<noteq> 0" shows "ord_term_strict (ordered_term.lt term_of_pair ord_term (ordered_term.tail term_of_pair ord_term p)) (ordered_term.lt term_of_pair ord_term p)"
valid_rga_msg_def: fixes list :: "('c \<times> 'd \<times> bool) list"   and msg :: "'c \<times> ('c, 'd) operation" shows "valid_rga_msg list msg \<equiv> case msg of (i, Insert e None) \<Rightarrow> fst e = i | (i, Insert e (Some pos)) \<Rightarrow> fst e = i \<and> pos \<in> element_ids list | (i, Delete pos) \<Rightarrow> pos \<in> element_ids list"
suffix_plus_com: fixes \<sigma> :: "nat \<Rightarrow> 'b"   and m :: "nat"   and n :: "nat" shows "\<sigma> |\<^sub>s m |\<^sub>s n = \<sigma> |\<^sub>s m + n"
simps_4: fixes uv :: "pdf_operator"   and e :: "expr" shows "free_vars_expr_code (uv $$ e) = free_vars_expr_code e"
rel_resource_Grp: fixes f :: "'k \<Rightarrow> 'i"   and g :: "'j \<Rightarrow> 'l" shows "rel_resource (BNF_Def.Grp UNIV f)\<inverse>\<inverse> (BNF_Def.Grp UNIV g) = BNF_Def.Grp UNIV (map_resource f g)"
assms_4: fixes inputs :: "real vec list" assumes "input_sizes m1 = map dim_vec inputs" shows "evaluate_net m1 inputs = evaluate_net m2 inputs"
preserves_hseq: fixes \<mu> :: "'b"   and \<nu> :: "'b" assumes "B.arr \<mu> \<and> B.arr \<nu> \<and> src\<^sub>B \<mu> = trg\<^sub>B \<nu>" shows "C.arr (H \<mu>) \<and> C.arr (H \<nu>) \<and> src\<^sub>C (H \<mu>) = trg\<^sub>C (H \<nu>)"
to_fun_monic_monom: fixes b :: "'a"   and n :: "nat" assumes "b \<in> carrier R" shows "to_fun (monom P \<one> n) b = b [^] n"
open_image_vec_nth: fixes S :: "('a, 'b) vec set"   and i :: "'b" assumes "open S" shows "open ((\<lambda>x. x $ i) ` S)"
cardSuc_least: fixes r :: "('c \<times> 'c) set"   and r' :: "('d \<times> 'd) set" assumes "Card_order r"   and "Card_order r'"   and "r <o r'" shows "cardSuc r \<le>o r'"
listwordinterval_compress: fixes ss :: "('b word \<times> 'b word) list" shows "\<Union> (interval_of ` set (listwordinterval_compress ss)) = \<Union> (interval_of ` set ss)"
inj_on_F: shows "inj_on (\<lambda>f. f \<circ> \<pi>) F"
filterlim_at_infinity_imp_filterlim_at_top: fixes f :: "'b \<Rightarrow> real"   and F :: "'b filter" assumes "filterlim f at_infinity F"   and "\<forall>\<^sub>F x in F. 0 < f x" shows "filterlim f at_top F"
sigma_sets_eq: fixes \<Omega> :: "'c set"   and M :: "'c set set" assumes "sigma_algebra \<Omega> M" shows "sigma_sets \<Omega> M = M"
Collect_empty_eq: fixes P :: "'k \<Rightarrow> bool" shows "(Collect P = {}) = (\<forall>x. \<not> P x)"
extensions_are_safe_uminus: fixes x' :: "nat"   and p :: "int mpoly"   and \<V> :: "nat set"   and A :: "int mpoly multiset"   and q :: "int mpoly" assumes "x' \<in> vars p"   and "x' \<notin> \<V>"   and "\<Union> (vars ` set_mset A) \<subseteq> \<V>"   and "MPoly_Type.coeff p (monomial (Suc 0) x') = - 1"   and "vars q \<subseteq> \<V>"   and "q \<in> More_Modules.ideal (insert p (set_mset A \<union> polynomial_bool))"   and "x' \<notin> vars (p + Var x')"   and "(Var x' + p)\<^sup>2 - (Var x' + p) \<in> More_Modules.ideal polynomial_bool" shows "q \<in> More_Modules.ideal (set_mset A \<union> polynomial_bool)"
is_unit_div_mult2_eq: fixes b :: "'a"   and c :: "'a"   and a :: "'a" assumes "is_unit b"   and "is_unit c" shows "a div (b * c) = a div b div c"
prems_1: shows "insert p P \<subseteq> {a. irreducible a}"
pwd_spy_iii_rel: fixes n :: "nat" assumes "n \<in> bad_prikey \<inter> bad_id_shakey" shows "s\<^sub>0 \<Turnstile> fst (pwd_spy_iii n)"
bs_insert_def: fixes i :: "nat"   and s :: "integer" shows "bs_insert i s \<equiv> set_bit_class.set_bit s i True"
first_countable_topology_def: fixes "open" :: "'c set \<Rightarrow> bool" shows "class.first_countable_topology open \<equiv> class.topological_space open \<and> class.first_countable_topology_axioms open"
spair_comm: fixes p :: "'t \<Rightarrow>\<^sub>0 'b"   and q :: "'t \<Rightarrow>\<^sub>0 'b" shows "spair p q = - spair q p"
intersection_calculus_axioms: shows "intersection_calculus Bot Inf Q entails_q Red_I_q Red_F_q"
antisym: fixes b :: "'b"   and a :: "'b" assumes "b \<le> a"   and "a \<le> b" shows "a = b"
dnz: shows "(deriv ^^ n) f \<xi> \<noteq> 0"
lemma0: fixes i :: "'a"   and u :: "'a"   and v :: "'a" assumes "i \<in> Basis" shows "content (cbox u v) / (interval_upperbound (cbox u v) \<bullet> i - interval_lowerbound (cbox u v) \<bullet> i) = (if content (cbox u v) = 0 then 0 else \<Prod>j\<in>Basis - {i}. interval_upperbound (cbox u v) \<bullet> j - interval_lowerbound (cbox u v) \<bullet> j)"
arr_dom_iff_arr: fixes f :: "'a \<times> 'a" shows "H.VV.arr (H.VV.dom f) = H.VV.arr f"
nontriv: assumes "class.nontriv TYPE('a)" shows "1 < CARD('a)"
a_coset_add_inv1: fixes M :: "'a set"   and x :: "'a"   and y :: "'a" assumes "M +> x \<oplus> \<ominus> y = M"   and "x \<in> carrier G"   and "y \<in> carrier G"   and "M \<subseteq> carrier G" shows "M +> x = M +> y"
assms_2: shows "local.diagonalize A bezout = (P, S, Q)"
p6: shows "{OrdP (Var k)} \<turnstile> (SyntaxN.All sj (SyntaxN.All lj (SeqQuoteP (Var i) (Var i') (Var si) ki IMP SeqQuoteP (Var j) (Var j') (Var sj) (Var lj) IMP HaddP ki (Var lj) (Var k) IMP (Var i IN Var j IMP PfP (Q_Mem (Var i') (Var j'))) AND (Var i SUBS Var j IMP PfP (Q_Subset (Var i') (Var j'))))))(j'::=Var j')"
snoc_eq_iff_butlast': fixes ys :: "'a list"   and xs :: "'a list"   and x :: "'a" shows "(ys = xs @ [x]) = (ys \<noteq> [] \<and> butlast ys = xs \<and> last ys = x)"
left_unique_cr_spmf_gpv: shows "left_unique spmf_to_gpv.cr_spmf_gpv"
Carmichael_number_not_prime: fixes n :: "nat" assumes "Carmichael_number n" shows "\<not> prime n"
these_insert_None: fixes A :: "'b option set" shows "Option.these (insert None A) = Option.these A"
fBall_funion: fixes A :: "'c fset"   and B :: "'c fset"   and P :: "'c \<Rightarrow> bool" shows "fBall (A |\<union>| B) P = (fBall A P \<and> fBall B P)"
redT_updWs_WokenUp_SuspendD: fixes t :: "'b"   and ws :: "'b \<Rightarrow> 'c wait_set_status option"   and was :: "('b, 'c) wait_set_action list"   and ws' :: "'b \<Rightarrow> 'c wait_set_status option"   and w :: "'c wait_set_status" assumes "redT_updWs t ws was ws'"   and "Notified \<in> set was \<or> WokenUp \<in> set was"   and "ws' t = \<lfloor>w\<rfloor>" shows "\<exists>w. Suspend w \<in> set was"
to_mini_mset_is: fixes \<Gamma> :: "'b formula multiset" shows "is_mini_mset (image_mset to_mini_formula \<Gamma>)"
bv_comp_bernoulli: fixes n :: "nat"   and l :: "nat" assumes "n < l" shows "map_pmf (\<lambda>y. y ! n) (bv l) = bernoulli_pmf (5 / 10)"
Diff_infinite_finite: fixes T :: "'d set"   and S :: "'d set" assumes "finite T"   and "infinite S" shows "infinite (S - T)"
snoc_eq_iff_butlast: fixes xs :: "'a list"   and x :: "'a"   and ys :: "'a list" shows "(append xs [x] = ys) = (ys \<noteq> [] \<and> butlast ys = xs \<and> last ys = x)"
m_zmax_gt_eachTr: fixes n :: "nat"   and f :: "nat \<Rightarrow> int" shows "(\<forall>j\<le>n. f j \<in> Zset) \<longrightarrow> (\<forall>j\<le>n. f j \<le> m_zmax n f)"
snapshot_step_cs_preservation_p: fixes c :: "('a, 'b, 'c) configuration"   and ev :: "('a, 'b, 'c) event"   and c' :: "('a, 'b, 'c) configuration"   and p :: "'a"   and cid :: "nat"   and q :: "'a" assumes "c \<turnstile> ev \<mapsto> c'"   and "\<not> regular_event ev"   and "occurs_on ev = p"   and "channel cid = Some (p, q)" shows "map Msg (fst (cs c cid)) @ takeWhile ((\<noteq>) Marker) (msgs c cid) = map Msg (fst (cs c' cid)) @ takeWhile ((\<noteq>) Marker) (msgs c' cid) \<and> snd (cs c cid) = snd (cs c' cid)"
card0: shows "card X = 0 \<longrightarrow> (\<exists>f. TernaryOrdering.ordering f ord X)"
largest_eq: shows "largest TYPE(('a, 'b) IEEE.float) = (2 ^ (LENGTH('b) + 1) - 1) * 2 powr real_of_int (2 ^ (LENGTH('a) - 1) - int LENGTH('b) - 1)"
foundation5: fixes \<tau> :: "'a state \<times> 'a state"   and P :: "'a state \<times> 'a state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>"   and Q :: "'a state \<times> 'a state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" assumes "\<tau> \<Turnstile> P and Q" shows "\<tau> \<Turnstile> P \<and> \<tau> \<Turnstile> Q"
K_not_neg_in_consistency: fixes p :: "'b fm" assumes "\<not> \<turnstile> (\<^bold>\<not> p)" shows "{p} \<in> {set G |G. \<not> \<turnstile> imply G \<^bold>\<bottom>}"