hcju/isabelleps huggingface dataset:
OclIncluding_valid_args_valid: fixes \<tau> :: "'a state \<times> 'a state"   and X :: "'a state \<times> 'a state \<Rightarrow> Set('b)"   and x :: "'a state \<times> 'a state \<Rightarrow> 'b" shows "(\<tau> \<Turnstile> \<upsilon> X->including\<^sub>S\<^sub>e\<^sub>t(x)) = (\<tau> \<Turnstile> \<delta> X \<and> \<tau> \<Turnstile> \<upsilon> x)"
sum_indep_erlang: fixes X :: "'a \<Rightarrow> real"   and Y :: "'a \<Rightarrow> real"   and l :: "real"   and k\<^sub>1 :: "nat"   and k\<^sub>2 :: "nat" assumes "indep_var borel X borel Y"   and "0 < l"   and "distributed M lborel X (\<lambda>x. ennreal (erlang_density k\<^sub>1 l x))"   and "distributed M lborel Y (\<lambda>x. ennreal (erlang_density k\<^sub>2 l x))" shows "distributed M lborel (\<lambda>x. X x + Y x) (\<lambda>x. ennreal (erlang_density (Suc k\<^sub>1 + Suc k\<^sub>2 - 1) l x))"
S: shows "set p \<subseteq> Xs \<times> Ys"
Node_Push_I: fixes p :: "(nat \<Rightarrow> 'a + nat) \<times> ('b + nat)"   and i :: "'a + nat" assumes "p \<in> Node" shows "apfst (Push i) p \<in> Node"
obj: shows "obj\<^bsub>CanonicalCat' T\<^esub> = Ty\<^bsub>aS\<^bsub>T\<^esub>\<^esub>"
vid_on_def: fixes A :: "V" shows "vid_on A = set {\<langle>a, a\<rangle> |a. a \<in>\<^sub>\<circ> A}"
badE: fixes P :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "nat \<Rightarrow> 'c" assumes "bad P f" obtains "\<And>i j. i < j \<Longrightarrow> \<not> P (f i) (f j)"
simps_3: fixes x21 :: "dag"   and x22 :: "ref"   and x23 :: "dag" shows "Node x21 x22 x23 \<noteq> Tip"
greaterThanLessThan_iff: fixes i :: "'e word"   and l :: "'e word"   and u :: "'e word" shows "(i \<in> signed.greaterThanLessThan l u) = (l <s i \<and> i <s u)"
distinct_mset_mono: fixes ys :: "'b list"   and xs :: "'b list" assumes "distinct ys"   and "mset xs \<subseteq># mset ys" shows "distinct xs"
fold_plus_sum_list_rev: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and xs :: "'a list" assumes "class.monoid_add plus zero" shows "fold plus xs = plus (monoid_add.sum_list plus zero (rev xs))"
top_neq_ennreal: fixes r :: "real" shows "\<top> \<noteq> ennreal r"
cone_ordLeq_Cnotzero: fixes r :: "('c \<times> 'c) set" assumes "(r, czero) \<notin> ordIso \<and> Card_order r" shows "cone \<le>o r"
sorted_wrt_mergesort_by_rel_merge: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a list"   and ys :: "'a list" assumes "\<And>x y. R x y \<or> R y x"   and "\<And>x y z. \<lbrakk>R x y; R y z\<rbrakk> \<Longrightarrow> R x z" shows "sorted_wrt R (mergesort_by_rel_merge R xs ys) = (sorted_wrt R xs \<and> sorted_wrt R ys)"
continuous_map_from_subtopology: fixes X :: "'d topology"   and X' :: "'e topology"   and f :: "'d \<Rightarrow> 'e"   and S :: "'d set" assumes "continuous_map X X' f" shows "continuous_map (subtopology X S) X' f"
borel_measurable_exp: shows "exp \<in> borel_measurable borel"
node\<^sub>i_bal_simp: fixes k :: "nat"   and ts :: "('a btree \<times> 'a) list"   and t :: "'a btree" shows "bal_up\<^sub>i (node\<^sub>i k ts t) = bal (Node ts t)"
pseudominimal_inI: fixes x :: "'a"   and P :: "'a set" assumes "x \<in> P"   and "x \<noteq> bottom P"   and "\<And>z. \<lbrakk>z \<in> P; x \<succ> z\<rbrakk> \<Longrightarrow> z = bottom P" shows "pseudominimal_in P x"
input_sizes_shallow_model: fixes Y :: "nat"   and Z :: "nat"   and M :: "nat"   and N :: "nat" shows "input_sizes (shallow_model Y Z M N) = replicate (Suc N) M"
simps_4: fixes c :: "poly"   and n :: "nat"   and p :: "poly"   and c' :: "poly"   and n' :: "nat"   and p' :: "poly" shows "CN c n p +\<^sub>p CN c' n' p' = (if n < n' then CN (c +\<^sub>p CN c' n' p') n p else if n' < n then CN (CN c n p +\<^sub>p c') n' p' else let cc' = c +\<^sub>p c'; pp' = p +\<^sub>p p' in if pp' = 0\<^sub>p then cc' else CN cc' n pp')"
cases: fixes x :: "'a \<times> 'a list \<times> 'a list" obtains a :: "'a" where "x = (a__, [], [])"   | a :: "'a" and l :: "'a" where "x = (aa__, [], [l__])"   | a :: "'a" and l :: "'a" and v :: "'a" and va :: "'a list" where "x = (ab__, [], la__ # v # va)"   | a :: "'a" and f :: "'a" and fs :: "'a list" and l :: "'a" and ls :: "'a list" where "x = (ac__, f__ # fs__, l # ls)"   | a :: "'a" and f :: "'a" and fs :: "'a list" where "x = (a, f # fs, [])"
initialization_list_def: shows "initialization_list = Thread # sys_xcpts_list"
flow_split: fixes f :: "'v \<times> 'v \<Rightarrow> ennreal" assumes "flow \<Delta> f" shows "flow \<Delta>'' (split f)"
absorb2: fixes b :: "'e"   and a :: "'e" assumes "b \<le> a" shows "inf_class.inf a b = b"
c3: assumes "ip \<in> kD rt"   and "sqn rt ip = \<pi>\<^sub>2 r"   and "\<pi>\<^sub>5 r < the (dhops rt ip)" shows "P (rt(ip \<mapsto> r))"
ackloop_dom_longer: fixes m :: "nat"   and n :: "nat"   and l :: "nat list" assumes "ackloop_dom (ack m n # l)" shows "ackloop_dom (n # m # l)"
arr_cod_iff_arr: fixes f :: "'a" shows "A.arr (A.cod f) = A.arr f"
contains_propagates_fconj: fixes n :: "nat"   and g :: "form"   and h :: "form" assumes "infinite (deriv s)"   and "contains f n (0, FConj g h)" shows "\<exists>y. contains f (n + y) (0, g) \<or> contains f (n + y) (0, h)"
c_tr_singleton: fixes step :: "'c \<Rightarrow> 'a \<Rightarrow> 'c"   and out :: "'c \<Rightarrow> 'a \<Rightarrow> 'b"   and s :: "'c"   and x :: "'a" shows "c_tr step out s [x] = [(x, out s x)]"
inf_Int_eq: fixes R :: "'k set"   and S :: "'k set" shows "(\<lambda>x. x \<in> R) \<sqinter> (\<lambda>x. x \<in> S) = (\<lambda>x. x \<in> R \<inter> S)"
gamma1_def: shows "gamma1 \<equiv> $pc1 = #g \<and> pc1$ = #a \<and> sem$ = Suc<$sem> \<and> unchanged (x, y, pc2)"
proj6_invalidate: fixes rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option"   and dests :: "nat \<Rightarrow> nat option"   and dip :: "nat" shows "\<pi>\<^sub>6 (the (invalidate rt dests dip)) = \<pi>\<^sub>6 (the (rt dip))"
real_interval_of: fixes b :: "float" shows "real_interval (interval_of b) = interval_of (real_of_float b)"
simps_3: fixes \<phi> :: "'a ltl" shows "\<^bold>G (F \<phi>) = \<^bold>G \<phi>"
swap_def: fixes p :: "'s \<times> 'r" shows "prod.swap p = (snd p, fst p)"
const_unconst: fixes u :: "'a"   and name :: "name" assumes "unconst u = Some name" shows "u = const name"
prefixes_neq_nil: fixes i :: "nat"   and x :: "nat" shows "0 < length (prefixes i x)"
legs_form_cospan_2: shows "C.cospan f.cod.leg0 g.cod.leg1"
sel_1: fixes x21 :: "'e"   and x22 :: "'e list" shows "hd (x21 # x22) = x21"
a0: shows "r (Suc k) (b $ 0) = a $ 0"
Base: fixes G :: "('c, 'd) ParityGame_scheme"   and v :: "'c"   and W :: "'c set"   and p :: "Player" assumes "ParityGame G"   and "v \<in> W" shows "v \<in> ParityGame.attractor_inductive G p W"
of_nat_diff: fixes plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and one :: "'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and n :: "nat"   and m :: "nat" assumes "class.semiring_1_cancel plus minus zero one times"   and "n \<le> m" shows "semiring_1.of_nat one plus zero (m - n) = minus (semiring_1.of_nat one plus zero m) (semiring_1.of_nat one plus zero n)"
abs_eq: fixes xa :: "circline_mat"   and x :: "complex_homo_coords" shows "on_circline (abs_circline xa) (abs_complex_homo x) = on_circline_clmat_hcoords xa x"
basis_in_ex: fixes \<E> :: "'a set" assumes "\<E> \<subseteq> \<E>" shows "\<exists>B. \<E>.basis_in \<E> B"
order_less_le_subst2: fixes a :: "'b"   and b :: "'b"   and f :: "'b \<Rightarrow> 'c"   and c :: "'c" assumes "a < b"   and "f b \<le> c"   and "\<And>x y. x < y \<Longrightarrow> f x < f y" shows "f a < c"
flincomb_Horbit_induced_veclist_reduce: fixes H :: "'a set"   and G :: "'a set"   and smult :: "('b, 'a) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and V :: "'c set"   and rrsmult :: "('b, 'a) aezfun \<Rightarrow> (('b, 'a) aezfun \<Rightarrow> 'c) \<Rightarrow> ('b, 'a) aezfun \<Rightarrow> 'c"   and vs :: "'c list"   and i :: "nat"   and hs :: "'a list"   and css :: "'b list list" assumes "InducedFinGroupRepresentation H G smult V rrsmult"   and "set vs \<subseteq> V"   and "i < length hs"   and "list_all2 (\<lambda>rs ms. length rs = length ms) css (aezfun_scalar_mult.negGorbit_list rrsmult hs (InducedFHModule.induced_vector H G smult V) vs)"   and "Group.is_rcoset_replist H G hs" shows "scalar_mult.lincomb (aezfun_scalar_mult.fsmult rrsmult) (concat css) (concat (aezfun_scalar_mult.negGorbit_list rrsmult hs (InducedFHModule.induced_vector H G smult V) vs)) ((1::'b) \<delta>\<delta> hs ! i) = scalar_mult.lincomb (aezfun_scalar_mult.fsmult smult) (css ! i) vs"
funpow_decreasing: fixes m :: "nat"   and n :: "nat"   and f :: "'c \<Rightarrow> 'c" assumes "m \<le> n"   and "mono f" shows "(f ^^ m) bot \<le> (f ^^ n) bot"
interval_integral_reflect: fixes a :: "ereal"   and b :: "ereal"   and f :: "real \<Rightarrow> 'b" shows "interval_lebesgue_integral lborel a b f = LBINT x=- b..- a. f (- x)"
drop_suffix_css_step: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and cs :: "('e, 'd, 'f) com list"   and pcss :: "(('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list"   and css :: "(('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list"   and s :: "('e, 'f) xstate"   and cs' :: "('e, 'd, 'f) com list"   and pcss' :: "(('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list"   and t :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> (cs, pcss @ css, s) \<rightarrow> (cs', pcss' @ css, t)" shows "\<Gamma>\<turnstile> (cs, pcss, s) \<rightarrow> (cs', pcss', t)"
not_le_imp_less: fixes less_eq :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and less :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and y :: "'f"   and x :: "'f" assumes "class.linorder less_eq less"   and "\<not> less_eq y x" shows "less x y"
assms_2: shows "\<forall>s\<in>set (subtrees (ls @ (sub, sep) # rs)). order k s"
eql: fixes t :: "trm"   and u :: "trm"   and i :: "name"   and x :: "trm" shows "(t EQ u)(i::=x) = subst i x t EQ subst i x u"
Max_wrt_map_relation_bij: fixes f :: "'alt \<Rightarrow> 'a" assumes "bij f" shows "f ` Max_wrt (R i) = Max_wrt (map_relation (inv f) (R i))"
coeff_pCons: fixes x :: "'b"   and xa :: "'b poly" shows "coeff (pCons x xa) = case_nat x (coeff xa)"
produces_imp_neg_notin_lits: fixes N :: "'b literal multiset set"   and C :: "'b literal multiset"   and A :: "'b" assumes "production N C = {A}" shows "Neg A \<notin># C"
simps: fixes A :: "('b \<times> nat) tree"   and a :: "'b"   and B1 :: "('b \<times> nat) tree"   and b :: "'b"   and uu :: "nat"   and B2 :: "('b \<times> nat) tree"   and c :: "'b"   and C :: "('b \<times> nat) tree" shows "rot2 A a \<langle>B1, (b, uu), B2\<rangle> c C = N (N A a B1) b (N B2 c C)"
disc_2: fixes x2 :: "'d" shows "is_Skip (Skip x2)"
cprank_maxE: fixes n :: "nat"   and A :: "'b tensor" assumes "cprank_max n A" obtains BS :: "'b tensor list" where "\<And>B. B \<in> set BS \<Longrightarrow> cprank_max1 B" and "\<And>B. B \<in> set BS \<Longrightarrow> dims A = dims B" and "listsum (dims A) BS = A" and "length BS = n"
StepT: fixes P :: "jvm_prog"   and C :: "char list"   and M :: "char list"   and pc :: "nat"   and ics :: "init_call_status"   and Cs :: "char list list"   and a :: "nat" assumes "exec_step_input P C M pc ics = StepT Cs a" shows "ics = Throwing Cs a"
landau_flip_1: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'c" shows "(f \<in> O[F](g)) = (g \<in> \<Omega>[F](f))"
to_tree_order_node_ptrs: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and nodes :: "(_) object_ptr list"   and ptr' :: "(_) object_ptr" assumes "heap_is_wellformed h"   and "type_wf h"   and "known_ptrs h"   and "h \<turnstile> to_tree_order ptr \<rightarrow>\<^sub>r nodes"   and "ptr' \<noteq> ptr"   and "ptr' \<in> set nodes" shows "is_node_ptr_kind ptr'"
read_only: fixes p :: "bool Heap"   and h :: "heap" assumes "ro_cond p"   and "success p h" shows "snd (the (execute p h)) = h"
iLast_True: fixes t0 :: "nat"   and I :: "nat set" shows "\<ominus> t t0 I. True"
I': shows "Q \<Longrightarrow>\<^sub>t Q'"
proj_ltake: fixes t :: "cnf llist"   and n :: "nat"   and active :: "'a \<Rightarrow> cnf \<Rightarrow> bool"   and c :: "'a"   and tCMP :: "'a \<Rightarrow> cnf \<Rightarrow> 'b" assumes "\<forall>n'. enat n' \<le> llength t \<longrightarrow> n \<le> n' \<longrightarrow> \<not> active c (lnth t n')" shows "dynamic_component.proj tCMP active c t = dynamic_component.proj tCMP active c (ltake (enat n) t)"
gamma_smooth: shows "\<gamma> C1_differentiable_on {0..1}"
fv\<^sub>s\<^sub>s\<^sub>t_cons_subset: fixes A :: "('b, 'a) stateful_strand_step list"   and a :: "('b, 'a) stateful_strand_step" shows "fv\<^sub>s\<^sub>s\<^sub>t A \<subseteq> fv\<^sub>s\<^sub>s\<^sub>t (a # A)"
simps_1: fixes c :: "'a" shows "maxdegree (c)\<^sub>p = 0"
bij_ind_in_set_bound: fixes f :: "nat \<Rightarrow> nat"   and v0 :: "nat set"   and M :: "nat" assumes "\<And>x. f x = card {y \<in> v0. y < x}" shows "bij_betw f ({0..<M} \<inter> v0) {0..<card ({0..<M} \<inter> v0)}"
fds_zeta_altdef: shows "fds_zeta = fds (\<lambda>n. if n = 0 then 0::'a else (1::'a))"
abstract_perfect_hiding: fixes \<eta> :: "nat"   and \<A> :: "('grp \<Rightarrow> ((nat \<times> nat) \<times> 'b) spmf) \<times> ('grp \<Rightarrow> 'b \<Rightarrow> bool spmf)" shows "ped_commit.perfect_hiding_ind_cpa \<eta> \<A>"
aw: shows "adjuster n w us + w \<in> carrier_vec n"
simps: fixes trsys :: "'c \<Rightarrow> 'd \<Rightarrow> 'c \<Rightarrow> bool"   and \<tau>move :: "'c \<Rightarrow> 'd \<Rightarrow> 'c \<Rightarrow> bool"   and a1 :: "'c"   and a2 :: "'c" shows "\<tau>trsys.silent_move trsys \<tau>move a1 a2 = (\<exists>s s' tl. a1 = s \<and> a2 = s' \<and> trsys s tl s' \<and> \<tau>move s tl s')"
simps_10: fixes i :: "nat"   and T :: "ty"   and e :: "nat exp"   and pc :: "nat"   and d :: "nat" shows "compxE\<^sub>2 {i:T; e} pc d = compxE\<^sub>2 e pc d"
dim_vec_last: fixes v :: "'b vec"   and n :: "nat" shows "dim_vec (vec_last v n) = n"
induct: fixes P1 :: "POPLmarkRecord.type \<Rightarrow> bool"   and P3 :: "(char list \<times> POPLmarkRecord.type) list \<Rightarrow> bool"   and P2 :: "char list \<times> POPLmarkRecord.type \<Rightarrow> bool"   and type :: "POPLmarkRecord.type" assumes "\<And>x. P1 (TVar x)"   and "P1 Top"   and "\<And>x1 x2. \<lbrakk>P1 x1; P1 x2\<rbrakk> \<Longrightarrow> P1 (x1 \<rightarrow> x2)"   and "\<And>x1 x2. \<lbrakk>P1 x1; P1 x2\<rbrakk> \<Longrightarrow> P1 (\<forall><:x1. x2)"   and "\<And>x. P3 x \<Longrightarrow> P1 (RcdT x)"   and "\<And>x1 x2. P1 x2 \<Longrightarrow> P2 (x1, x2)"   and "P3 []"   and "\<And>x1 x2. \<lbrakk>P2 x1; P3 x2\<rbrakk> \<Longrightarrow> P3 (x1 \<Colon> x2)" shows "P1 type"
oseq_unicastTE: fixes \<sigma> :: "nat \<Rightarrow> 'e"   and l :: "'h"   and s\<^sub>i\<^sub>p :: "'e \<Rightarrow> nat"   and s\<^sub>m\<^sub>s\<^sub>g :: "'e \<Rightarrow> 'f"   and p :: "('e, 'f, 'g, 'h) seqp"   and q :: "('e, 'f, 'g, 'h) seqp"   and a :: "'f seq_action"   and \<sigma>' :: "nat \<Rightarrow> 'e"   and r :: "('e, 'f, 'g, 'h) seqp"   and \<Gamma> :: "'g \<Rightarrow> ('e, 'f, 'g, 'h) seqp"   and i :: "nat" assumes "((\<sigma>, {l}unicast(s\<^sub>i\<^sub>p, s\<^sub>m\<^sub>s\<^sub>g) . p \<triangleright> q), a, \<sigma>', r) \<in> oseqp_sos \<Gamma> i" obtains "a = unicast (s\<^sub>i\<^sub>p (\<sigma> i)) (s\<^sub>m\<^sub>s\<^sub>g (\<sigma> i))" and "\<sigma>' i = \<sigma> i" and "r = p"   | "a = \<not>unicast (s\<^sub>i\<^sub>p (\<sigma> i))" and "\<sigma>' i = \<sigma> i" and "r = q"
run_definitions_1: fixes s\<^sub>0 :: "'b"   and lstep :: "'d \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and l :: "nat \<Rightarrow> 'd"   and s :: "nat \<Rightarrow> 'b" assumes "df_system s\<^sub>0 lstep" shows "df_system.is_lrun s\<^sub>0 lstep l s \<equiv> s 0 = s\<^sub>0 \<and> (\<forall>i. lstep (l i) (s i) (s (Suc i)))"
prefix_concat_minus: fixes xs :: "'b list"   and ys :: "'b list" assumes "prefix xs ys" shows "xs @ ys - xs = ys"
IH: fixes S :: "(nat \<Rightarrow> real) set"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real) \<Rightarrow>\<^sub>0 int" assumes "simplicial_chain p S c" shows "simplicial_chain p S (simplicial_subdivision p c)"
is_finite_zerosign: fixes s :: "nat"   and a :: "('a, 'b) IEEE.float" shows "is_finite (zerosign s a) = is_finite a"
lift_pref_profile_Pareto_iff: fixes x :: "'b"   and y :: "'b" shows "Pareto R' x y = (x \<in> alts' \<and> y \<in> alts' \<and> (x \<notin> alts \<or> Pareto R x y))"
cases: fixes a :: "Regions.intv" assumes "Regions.isIntv a" obtains uu_ :: "nat" where "a = Regions.intv.Intv uu_"
path_join_hd: fixes p :: "nat list"   and q :: "nat list" assumes "p \<noteq> []" shows "hd (path_join p q) = hd p"
prime_elem_not_unit': fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and x :: "'a" assumes "class.algebraic_semidom divide plus minus zero times one"   and "ASSUMPTION (comm_semiring_1.prime_elem times one zero x)" shows "\<not> dvd.dvd times x one"
zero_poly_def: shows "0 \<equiv> Abs_poly (\<lambda>_. 0::'b)"
flush_update_other': fixes a :: "nat"   and sb :: "'b memref list"   and m :: "nat \<Rightarrow> nat"   and v :: "nat" assumes "a \<notin> outstanding_refs is_non_volatile_Write\<^sub>s\<^sub>b sb"   and "outstanding_refs is_volatile_Write\<^sub>s\<^sub>b sb = {}" shows "flush sb (m(a := v)) = (flush sb m)(a := v)"
simps_2: fixes \<sigma> :: "(char list \<times> 'a list) trace"   and v :: "'a list"   and i :: "nat"   and t1 :: "'a MFOTL.trm"   and t2 :: "'a MFOTL.trm" shows "MFOTL.sat \<sigma> v i (formula.Eq t1 t2) = (MFOTL.eval_trm v t1 = MFOTL.eval_trm v t2)"
segmentation: fixes P :: "'a set"   and Q :: "'a set" assumes "P \<in> \<P>"   and "2 \<le> card Q"   and "Q \<subseteq> P" shows "\<exists>S P1 P2. P = \<Union> S \<union> P1 \<union> P2 \<union> Q \<and> (\<forall>a\<in>S \<union> {P1, P2}. \<forall>b\<in>S \<union> {P1, P2}. a \<noteq> b \<longrightarrow> a \<inter> b = {}) \<and> P1 \<noteq> P2 \<and> P1 \<notin> S \<and> P2 \<notin> S \<and> (\<forall>x\<in>S. is_segment x) \<and> is_prolongation P1 \<and> is_prolongation P2"
closedin_Times: fixes S :: "'a set"   and S' :: "'a set"   and T :: "'b set"   and T' :: "'b set" assumes "closedin (top_of_set S) S'"   and "closedin (top_of_set T) T'" shows "closedin (top_of_set (S \<times> T)) (S' \<times> T')"
m_def: shows "m = abs_Bonk_Schramm_extension M"
rep_eq: fixes x :: "'b ratfps"   and xa :: "'b ratfps" shows "Rep_ratfps (x + xa) = Rep_ratfps x + Rep_ratfps xa"
SUP_eq_top_iff: fixes f :: "'c \<Rightarrow> 'b"   and A :: "'c set" shows "(Sup (f ` A) = top) = (\<forall>x<top. \<exists>i\<in>A. x < f i)"
map_graph_preserves_restricted: fixes G :: "('b, 'c) labeled_graph"   and f :: "('c \<times> 'd) set" assumes "graph G" shows "graph (map_graph f G)"
closure_contains_Inf: fixes S :: "real set" assumes "S \<noteq> {}"   and "bdd_below S" shows "\<Sqinter> S \<in> closure S"
strong: assumes "set qs \<subseteq> set init" shows "map_pmf (\<lambda>(l, w, i). (Lxy l {x, y}, nths w {index init x, index init y}, Lxy init {x, y})) (Partial_Cost_Model.config'_rand BIT (fst BIT init \<bind> (\<lambda>is. return_pmf (init, is))) qs) = Partial_Cost_Model.config'_rand BIT (fst BIT (Lxy init {x, y}) \<bind> (\<lambda>is. return_pmf (Lxy init {x, y}, is))) (Lxy qs {x, y})"
gcd_proj2_if_dvd_nat: fixes y :: "nat"   and x :: "nat" assumes "y dvd x" shows "gcd x y = y"
low_eq: shows "subst \<sigma> (fst (mems ! j)) =\<^bsub>map snd cms\<^sub>1 ! j\<^esub>\<^sup>l subst \<sigma> (snd (mems ! j))"
pre_FaceDiv_between1: fixes g' :: "graph"   and ram1 :: "nat"   and ram2 :: "nat"   and f :: "face" assumes "pre_splitFace g' ram1 ram2 f []" shows "between (vertices f) ram1 ram2 \<noteq> []"
integral_shift: fixes a :: "real"   and c :: "real"   and b :: "real"   and f :: "real \<Rightarrow> 'a" assumes "continuous_on {a + c..b + c} f" shows "integral {a..b} (f \<circ> (\<lambda>x. x + c)) = integral {a + c..b + c} f"
WT_gpvI: fixes gpv :: "('f, 'd, 'e) gpv"   and \<Gamma> :: "('d, 'e) \<I>" assumes "\<And>out c. IO out c \<in> set_spmf (the_gpv gpv) \<Longrightarrow> out \<in> outs_\<I> \<Gamma> \<and> (\<forall>input\<in>responses_\<I> \<Gamma> out. \<Gamma> \<turnstile>g c input \<surd>)" shows "\<Gamma> \<turnstile>g gpv \<surd>"
tree_delete_in_traverse: fixes t :: "interval_tree"   and x :: "nat idx_interval" assumes "tree_sorted t" shows "in_traverse (delete x t) = remove_elt_list x (in_traverse t)"
integral_mono_AE: fixes M :: "'b measure"   and f :: "'b \<Rightarrow> real"   and g :: "'b \<Rightarrow> real" assumes "integrable M f"   and "integrable M g"   and "AE x in M. f x \<le> g x" shows "integral\<^sup>L M f \<le> integral\<^sup>L M g"
intros_10: fixes F :: "'b formula"   and \<Gamma> :: "'b formula multiset"   and G :: "'b formula"   and \<Delta> :: "'b formula multiset"   and n :: "nat" assumes "F, \<Gamma> \<Rightarrow> G, \<Delta> \<down> n" shows "\<Gamma> \<Rightarrow> F \<^bold>\<rightarrow> G, \<Delta> \<down> Suc n"
assms: fixes \<gamma> :: "'a" shows "\<langle>\<gamma>, \<psi>\<rangle> = \<langle>\<gamma>, \<phi>\<rangle>"
schroeder_2: fixes y :: "'a"   and x :: "'a"   and z :: "'a" shows "(y ; x \<cdot> z = 0) = (y \<cdot> z ; x\<^sup>\<smile> = 0)"
that: fixes g :: "real \<Rightarrow> real" assumes "g \<in> o(\<lambda>x. ln (b * x) powr - (e / 2 + 1))"   and "\<forall>\<^sub>F x in at_top. 1 + ln (b * x) powr (- e / 2) + g x \<le> 1 + ln (b * x + hb * x / ln x powr (1 + e)) powr (- e / 2)"   and "\<forall>\<^sub>F x in at_top. 0 < 1 + ln (b * x) powr (- e / 2) + g x" shows "thesis"
MkArr_in_hom: fixes A :: "'a2 \<Rightarrow> 'b"   and B :: "'a2 \<Rightarrow> 'b"   and f :: "'a2 \<Rightarrow> 'b" assumes "A \<in> Collect (functor (\<cdot>\<^sub>A\<^sub>2) (\<cdot>\<^sub>B))"   and "B \<in> Collect (functor (\<cdot>\<^sub>A\<^sub>2) (\<cdot>\<^sub>B))"   and "f \<in> Collect (natural_transformation (\<cdot>\<^sub>A\<^sub>2) (\<cdot>\<^sub>B) A B)" shows "\<guillemotleft>A2_B.MkArr A B f : A2_B.MkIde A \<rightarrow>\<^sub>[\<^sub>A\<^sub>2\<^sub>,\<^sub>B\<^sub>] A2_B.MkIde B\<guillemotright>"
fds_commutes: fixes f :: "'a fds"   and g :: "'a fds" assumes "\<And>m n. \<lbrakk>0 < m; 0 < n\<rbrakk> \<Longrightarrow> fds_nth f m * fds_nth g n = fds_nth g n * fds_nth f m" shows "f * g = g * f"
in_segment_2: fixes x :: "'b"   and a :: "'b"   and b :: "'b" shows "(x \<in> {a<--<b}) = (a \<noteq> b \<and> (\<exists>u>0. u < 1 \<and> x = (1 - u) *\<^sub>R a + u *\<^sub>R b))"
P3: shows "Bet B A A0 \<and> Cong A A0 B' A'"
v1: fixes x :: "'a"   and b1 :: "'b"   and b2 :: "'b" shows "h x (b1 + b2) = h x b1 + h x b2"
diff_zero: fixes a :: "'b" shows "a - (0::'b) = a"
val_Pg: fixes v :: "'a \<Rightarrow> ant" assumes "valuation K v" shows "Pg K v \<in> carrier K - {\<zero>} \<and> v (Pg K v) = Lv K v"
insert_dom_less_eq: fixes x :: "'g"   and f :: "'g \<Rightarrow> 'h option"   and f' :: "'g \<Rightarrow> 'i option"   and y :: "'h"   and y' :: "'i" assumes "x \<notin> dom f"   and "x \<notin> dom f'"   and "dom (f(x \<mapsto> y)) = dom (f'(x \<mapsto> y'))" shows "dom f = dom f'"
power2_less_imp_less: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and x :: "'b"   and y :: "'b" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "less (power.power one times x 2) (power.power one times y 2)"   and "less_eq zero y" shows "less x y"
comp_fun_idem_axioms: shows "comp_fun_idem Set.remove"
induct: fixes nr :: "nat"   and nc :: "nat"   and A :: "'c mat"   and zero :: "'c"   and P :: "nat \<Rightarrow> nat \<Rightarrow> bool"   and a0 :: "nat"   and a1 :: "nat" assumes "\<And>i j. \<lbrakk>\<lbrakk>i < nr; j < nc; A $$ (i, j) = zero\<rbrakk> \<Longrightarrow> P i (Suc j); \<lbrakk>i < nr; j < nc; A $$ (i, j) \<noteq> zero\<rbrakk> \<Longrightarrow> P (Suc i) (Suc j)\<rbrakk> \<Longrightarrow> P i j" shows "P a0 a1"
except_def: fixes g :: "graph"   and v :: "nat" shows "except g v \<equiv> |filter (\<lambda>f. final f \<and> 5 \<le> |vertices f|) (facesAt g v)|"