hcju/isabelleps huggingface dataset:
simps: fixes \<sigma> :: "'e \<Rightarrow> ('d, 'e, 'f) val option"   and \<tau> :: "'e \<Rightarrow> ('d, 'e, 'f) val option"   and e :: "('d, 'e, 'f) expr" shows "(\<sigma>, \<tau>, e)\<^sub>e = e"
map_nth_default: fixes x :: "'b"   and xs :: "'b list" shows "map (nth_default x xs) [0..<length xs] = xs"
DAimpliesMR_E: fixes p :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list"   and x :: "int \<times> (int \<times> int \<times> protocol) \<times> (int \<times> int \<times> protocol) \<times> DummyContent" assumes "DenyAll \<in> set p" shows "\<exists>r. applied_rule_rev Cp x p = \<lfloor>r\<rfloor>"
finite_Collect_not: fixes P :: "'b \<Rightarrow> bool" assumes "finite {x. P x}" shows "finite {x. \<not> P x} = finite UNIV"
reach3: fixes s :: "Store"   and k :: "Location"   and y :: "Value"   and x :: "Value"   and l :: "Location" assumes "\<not> s\<turnstile> k reachable_from y"   and "\<not> s\<turnstile> k reachable_from x" shows "\<not> s\<langle>l := y\<rangle>\<turnstile> k reachable_from x"
proots_count_0_imp_empty: fixes p :: "'b poly"   and s :: "'b set" assumes "proots_count p s = 0"   and "p \<noteq> 0" shows "proots_within p s = {}"
id_comp: fixes g :: "'g \<Rightarrow> 'h" shows "id \<circ> g = g"
simps_3: fixes I :: "interp"   and \<phi> :: "fml" shows "game_sem I (? \<phi>) = (\<inter>) (fml_sem I \<phi>)"
next_step_step: fixes s\<^sub>0 :: "'b"   and lstep :: "'d \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and s :: "nat \<Rightarrow> 'b"   and i :: "nat"   and l :: "'d" assumes "fair_run s\<^sub>0 lstep s" shows "i \<le> fair_run.next_step_of lstep s l i \<and> (\<not> system.can_step lstep l (s (fair_run.next_step_of lstep s l i)) \<or> df_system.rstep lstep l s (fair_run.next_step_of lstep s l i))"
primitive_iff_content_eq_1: fixes f :: "'a poly" shows "primitive f = (content f = (1::'a))"
tl_app_in: fixes l :: "'a list"   and f :: "nat \<Rightarrow> 'b" assumes "l \<noteq> []" shows "map f [0..<length l - Suc 0] @ [f (length l - Suc 0)] = map f [0..<length l]"
Abs_filter_inverse: fixes y :: "'c set" assumes "y \<in> filters" shows "Filters.filter.Rep_filter (Filters.filter.Abs_filter y) = y"
opt_MatchAny_match_expr_def: fixes m :: "'b match_expr" shows "opt_MatchAny_match_expr m \<equiv> repeat_stabilize 2 opt_MatchAny_match_expr_once m"
mem_zmset_of: fixes x :: "'a"   and M :: "'a multiset" shows "(x \<in>#\<^sub>z zmset_of M) = (x \<in># M)"
degree_pdev_upd_le: fixes X :: "'a pdevs"   and n :: "nat"   and x :: "'a" assumes "degree X \<le> n" shows "degree (pdev_upd X n x) \<le> Suc n"
graph_def: fixes X :: "'a set"   and f :: "'a \<Rightarrow> 'b" shows "graph X f = {(x, f x) |x. x \<in> X}"
squarefree_part_uminus: fixes x :: "'a" assumes "x \<noteq> (0::'a)" shows "squarefree_part (- x) = - squarefree_part x"
bangPres: fixes P :: "pi"   and Rel' :: "(pi \<times> pi) set"   and Q :: "pi"   and Rel :: "(pi \<times> pi) set" assumes "P \<leadsto><Rel'> Q"   and "(P, Q) \<in> Rel"   and "\<And>P Q. (P, Q) \<in> Rel \<Longrightarrow> P \<leadsto><Rel'> Q"   and "\<And>P Q. (P, Q) \<in> Rel \<Longrightarrow> (P, Q) \<in> Rel'"   and "eqvt Rel'" shows "!P \<leadsto><bangRel Rel'> !Q"
arr_MkArr: fixes A :: "'o"   and B :: "'o"   and f :: "'a" assumes "A \<in> Obj"   and "B \<in> Obj"   and "f \<in> Hom A B" shows "arr (MkArr A B f)"
flush_store_buffer_append: fixes i :: "nat"   and ts :: "('p \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> unit \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list"   and sb :: "'p memref list"   and sb' :: "'p memref list"   and is\<^sub>s\<^sub>b :: "instr list"   and "is" :: "instr list"   and p :: "'p"   and \<theta> :: "nat \<Rightarrow> nat option"   and x :: "unit"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option"   and p\<^sub>s\<^sub>b :: "'p"   and \<O>' :: "nat set"   and m :: "nat \<Rightarrow> nat"   and \<S> :: "nat \<Rightarrow> bool option" assumes "i < length ts"   and "instrs (sb @ sb') @ is\<^sub>s\<^sub>b = is @ prog_instrs (sb @ sb')"   and "causal_program_history is\<^sub>s\<^sub>b (sb @ sb')"   and "ts ! i = (p, is, \<theta> |` (dom \<theta> - read_tmps (sb @ sb')), x, \<D>, \<O>, \<R>)"   and "p = hd_prog p\<^sub>s\<^sub>b (sb @ sb')"   and "last_prog p\<^sub>s\<^sub>b (sb @ sb') = p\<^sub>s\<^sub>b"   and "reads_consistent True \<O>' m sb"   and "history_consistent \<theta> p (sb @ sb')"   and "\<forall>sop\<in>write_sops sb. valid_sop sop"   and "distinct_read_tmps (sb @ sb')"   and "volatile_reads_consistent m sb" shows "\<exists>is'. instrs sb' @ is\<^sub>s\<^sub>b = is' @ prog_instrs sb' \<and> (ts, m, \<S>) \<Rightarrow>\<^sub>d\<^sup>* (ts[i := (last_prog (hd_prog p\<^sub>s\<^sub>b sb') sb, is', \<theta> |` (dom \<theta> - read_tmps sb'), x, \<D> \<or> outstanding_refs is_volatile_Write\<^sub>s\<^sub>b sb \<noteq> {}, acquired True sb \<O>, release sb (dom \<S>) \<R>)], flush sb m, share sb \<S>)"
assms_1: shows "has_bottom (P .\<^bold>\<le> y)"
alpha_qVar_iff: fixes X :: "('f, 'g, 'h, 'i, 'j) qTerm"   and xs' :: "'h"   and x' :: "'i" shows "(X #= qVar xs' x') = (X = qVar xs' x')"
prod_unfold_prod_mset: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and f :: "'b \<Rightarrow> 'a"   and A :: "'b set" assumes "class.comm_monoid_mult times one" shows "comm_monoid_mult.prod times one f A = comm_monoid_mult.prod_mset times one (image_mset f (mset_set A))"
cdinit_def: fixes g :: "'G"   and v0 :: "'V" shows "cdinit g v0 \<equiv> FOREACH (nodes (g.\<alpha> g)) (\<lambda>v wl. RETURN (qw.insert wl v infty.Infty)) (qw.empty ()) \<bind> (\<lambda>wl. RETURN (qw.insert wl v0 (Num (0::'W)), mr.sng v0 ([], 0::'W)))"
contrapos_np: fixes Q :: "bool"   and P :: "bool" assumes "\<not> (Q::bool)"   and "\<not> (P::bool) \<Longrightarrow> Q::bool" shows "P::bool"
decomps\<^sub>e\<^sub>s\<^sub>t_exist: fixes A :: "('fun, 'var) Term.term set"   and N :: "('fun, 'var) Term.term set"   and \<I> :: "'var \<Rightarrow> ('fun, 'var) Term.term" assumes "finite A"   and "finite N" shows "\<exists>D\<in>decomps\<^sub>e\<^sub>s\<^sub>t A N \<I>. \<forall>t. A \<turnstile> t \<longrightarrow> A \<union> ik\<^sub>e\<^sub>s\<^sub>t D \<turnstile>\<^sub>c t"
norm_triangle_eq: fixes x :: "'c"   and y :: "'c" shows "(norm (x + y) = norm x + norm y) = (norm x *\<^sub>R y = norm y *\<^sub>R x)"
Q0_def: shows "Q0 = {z \<in> Q. fst z = fst z0}"
abs_eq: fixes x :: "'b ltl" shows "\<up>Unf\<^sub>G (Abs x) = Abs (Unf\<^sub>G x)"
results_gpv_inline: fixes \<I> :: "('a, 'b) \<I>"   and \<I>' :: "('c, 'd) \<I>"   and callee :: "'e \<Rightarrow> 'a \<Rightarrow> ('b \<times> 'e, 'c, 'd) gpv"   and I :: "'e \<Rightarrow> bool"   and x :: "'f"   and s' :: "'e"   and gpv :: "('f, 'a, 'b) gpv"   and s :: "'e" assumes "raw_converter_invariant \<I> \<I>' callee I"   and "(x, s') \<in> results_gpv \<I>' (inline callee gpv s)"   and "\<I> \<turnstile>g gpv \<surd>"   and "I s" shows "x \<in> results_gpv \<I> gpv \<and> I s'"
beta: fixes T :: "typ"   and s :: "Core.term"   and t :: "Core.term" shows "Abs T s $ t \<rightarrow>\<^sub>\<beta> subst_bv2 s 0 t"
octo_of_real_inverse: fixes x :: "real" shows "octo_of_real (inverse x) = inverse (octo_of_real x)"
int_pow_def: fixes G :: "('c, 'd) monoid_scheme"   and a :: "'c"   and z :: "int" shows "a [^]\<^bsub>G\<^esub> z = (let p = rec_nat \<one>\<^bsub>G\<^esub> (\<lambda>u b. b \<otimes>\<^bsub>G\<^esub> a) in if z < 0 then inv\<^bsub>G\<^esub> p (nat (- z)) else p (nat z))"
declassified_subterms_proj_GSMP_subset: fixes I :: "'var \<Rightarrow> ('fun, 'var) Term.term"   and A :: "('lbl strand_label \<times> ('fun, 'var) strand_step) list"   and n :: "'lbl" assumes "wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t I"   and "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range I)"   and "subst_domain I = \<V> \<and> ground (subst_range I)" shows "subterms\<^sub>s\<^sub>e\<^sub>t (declassified\<^sub>l\<^sub>s\<^sub>t A I) \<subseteq> GSMP (trms\<^sub>l\<^sub>s\<^sub>t (proj n A))"
image_cong: fixes M :: "'c set"   and N :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "M = N"   and "\<And>x. x \<in> N \<Longrightarrow> f x = g x" shows "f ` M = g ` N"
mem_rel_interior_cball: fixes x :: "'b"   and S :: "'b set" shows "(x \<in> rel_interior S) = (x \<in> S \<and> (\<exists>e>0. cball x e \<inter> affine hull S \<subseteq> S))"
arc_part_circlepath: fixes r :: "real"   and s :: "real"   and t :: "real"   and z :: "complex" assumes "r \<noteq> 0"   and "s \<noteq> t"   and "\<bar>s - t\<bar> < 2 * pi" shows "arc (part_circlepath z r s t)"
invertible_Gauss_Jordan_column_k: fixes i :: "nat"   and A :: "(('a, 'c) vec, 'b) vec"   and k :: "nat" shows "\<exists>P. invertible P \<and> snd (Gauss_Jordan_column_k (i, A) k) = P ** A"
iT_Mult_mem_iff: fixes k :: "nat"   and x :: "nat"   and I :: "nat set" assumes "0 < k" shows "(x \<in> I \<otimes> k) = (x mod k = 0 \<and> x div k \<in> I)"
pref_ext: fixes xs :: "'b list"   and ys :: "'b list"   and zs :: "'b list" assumes "xs \<le>p ys" shows "xs \<le>p ys \<cdot> zs"
diff_lem: fixes A :: "char list set option"   and A' :: "char list set option"   and b :: "char list" assumes "A \<sqsubseteq> A'" shows "A \<ominus> b \<sqsubseteq> A' \<ominus> b"
roots_of_real_alg_2: fixes p :: "int poly" shows "distinct (map real_of (roots_of_real_alg p))"
nonempty_card_pos: fixes A :: "'c set" assumes "finite A"   and "A \<noteq> {}" shows "0 < card A"
pderiv_def: fixes p :: "nat \<Rightarrow> 'a" shows "pderiv p = poly_shift (n_mult p)"
Rep_cfun_inverse: fixes x :: "'a \<rightarrow> 'b" shows "Abs_cfun (Rep_cfun x) = x"
finite_ranking_induct: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and S :: "'c set"   and P :: "'c set \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'b" assumes "class.linorder less_eq less"   and "finite S"   and "P {}"   and "\<And>x S. \<lbrakk>finite S; \<And>y. y \<in> S \<Longrightarrow> less_eq (f y) (f x); P S\<rbrakk> \<Longrightarrow> P (insert x S)" shows "P S"
det_col_mul: fixes M :: "('c, 'd) sq_matrix"   and i :: "'d"   and c :: "'c"   and a :: "('c, 'd) vec" shows "Square_Matrix.det (upd_col M i (c *s a)) = c * Square_Matrix.det (upd_col M i a)"
cod_object: fixes f :: "'a" assumes "f \<in> Ar" shows "Cod f \<in> Ob"
WT_gpvD_1: fixes \<Gamma> :: "('h, 'i) \<I>"   and gpv :: "('j, 'h, 'i) gpv"   and out :: "'h"   and c :: "('j, 'h, 'i) rpv" assumes "\<Gamma> \<turnstile>g gpv \<surd>"   and "IO out c \<in> set_spmf (the_gpv gpv)" shows "out \<in> outs_\<I> \<Gamma>"
prv_scnj_imp: fixes \<chi> :: "'fmla"   and F :: "'fmla set"   and \<phi> :: "'fmla" assumes "\<chi> \<in> fmla"   and "F \<subseteq> fmla"   and "finite F"   and "\<phi> \<in> F"   and "prv (imp \<phi> \<chi>)" shows "prv (imp (scnj F) \<chi>)"
ltl_prop_equiv_implies_ltl_lang_equiv: fixes \<phi> :: "'b ltln"   and \<psi> :: "'b ltln" assumes "\<phi> \<sim>\<^sub>P \<psi>" shows "\<phi> \<sim>\<^sub>L \<psi>"
sym_block_intersections_index: fixes b1 :: "'a set"   and b2 :: "'a set" assumes "b1 \<in># \<B>"   and "b2 \<in># remove1_mset b1 \<B>" shows "b1 |\<inter>| b2 = \<Lambda>"
lub_below_iff: fixes below :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and S :: "nat \<Rightarrow> 'b"   and x :: "'b" assumes "class.cpo below"   and "po.chain below S" shows "below (po.lub below (range S)) x = (\<forall>i. below (S i) x)"
distinct_1: fixes x1 :: "'a"   and x21 :: "nat set"   and x22 :: "'a" shows "broadcast x1 \<noteq> groupcast x21 x22"
simps: fixes f :: "'b poly \<times> 'b \<Rightarrow> 'a"   and Gi_1 :: "'b poly"   and Gi :: "'b poly"   and ni_1 :: "nat"   and d1_1 :: "nat"   and hi_2 :: "'b" shows "subresultant_prs_main_impl f Gi_1 Gi ni_1 d1_1 hi_2 = (let gi_1 = lead_coeff Gi_1; ni = degree Gi; hi_1 = if d1_1 = 1 then gi_1 else dichotomous_Lazard gi_1 hi_2 d1_1; d1 = ni_1 - ni; pmod = pseudo_mod Gi_1 Gi in if pmod = 0 then f (Gi, if d1 = 1 then lead_coeff Gi else dichotomous_Lazard (lead_coeff Gi) hi_1 d1) else let gi = lead_coeff Gi; divisor = (- (1::'b)) ^ (d1 + 1) * gi_1 * hi_1 ^ d1; Gi_p1 = sdiv_poly pmod divisor in subresultant_prs_main_impl f Gi Gi_p1 ni d1 hi_1)"
fst_netgmap_netlift: fixes sr :: "'e \<Rightarrow> 'c \<times> 'd"   and s :: "'e net_state" shows "fst (netgmap sr s) = netlift sr s"
QuadRes_def: fixes p :: "int"   and a :: "int" shows "QuadRes p a = (\<exists>y. [y\<^sup>2 = a] (mod p))"
ee_trans: fixes G :: "('c, 'd) monoid_scheme"   and as :: "'c list"   and bs :: "'c list"   and cs :: "'c list" assumes "Group.monoid G"   and "essentially_equal G as bs"   and "essentially_equal G bs cs"   and "set as \<subseteq> carrier G"   and "set bs \<subseteq> carrier G"   and "set cs \<subseteq> carrier G" shows "essentially_equal G as cs"
strictly_subsumes_variants: fixes E :: "'a literal multiset"   and D :: "'a literal multiset"   and D' :: "'a literal multiset" assumes "strictly_subsumes E D"   and "variants D D'" shows "strictly_subsumes E D'"
maximal_atlas': fixes c :: "('a, 'b) chart" assumes "\<And>c'. c' \<in> charts \<Longrightarrow> k-smooth_compat c c'"   and "domain c \<subseteq> carrier" shows "c \<in> atlas"
i: shows "\<forall>x. e_length x = n \<longrightarrow> eval r_normal_form [i, x] = eval f (Partial_Recursive.list_decode x)"
assms_1: shows "(\<lambda>x. ereal (u x)) \<longlonglongrightarrow> - \<infinity>"
is_extensional: fixes f :: "'c" assumes "\<not> C.arr f" shows "G f = D.null"
e_hd_nth0: fixes e :: "nat" shows "e_hd e = e_nth e 0"
shortest_is_augmenting: fixes p :: "(nat \<times> nat) list" assumes "cf.isShortestPath s p t" shows "isAugmentingPath p"
in_monomials_iff: fixes x :: "nat \<Rightarrow>\<^sub>0 nat"   and m :: "'a mpoly" shows "(x \<in> monomials m) = (MPoly_Type.coeff m x \<noteq> (0::'a))"
eSubstInp_simp1: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model"   and ys :: "'varSort"   and Y :: "'c"   and delta :: "'opSym"   and inp :: "'index \<Rightarrow> 'c option"   and y :: "'var" assumes "igWls MOD (asSort ys) Y"   and "igWlsInp MOD delta inp" shows "eSubstInp MOD ys (OK Y) y (OKI inp) = OKI (igSubstInp MOD ys Y y inp)"
exec1_only1_aux: fixes ccs :: "com list"   and s :: "state"   and cs' :: "com list"   and t :: "state" assumes "(ccs, s) \<rightarrow> (cs', t)" shows "\<forall>c cs. ccs = c # cs \<longrightarrow> (\<exists>cs1. cs' = cs1 @ cs)"
finite_measure_finite_Union: fixes S :: "'c set"   and A :: "'c \<Rightarrow> 'a set" assumes "finite S"   and "A ` S \<subseteq> sets M"   and "disjoint_family_on A S" shows "Sigma_Algebra.measure M (\<Union> (A ` S)) = (\<Sum>i\<in>S. Sigma_Algebra.measure M (A i))"
Diff_Int_distrib2: fixes A :: "'d set"   and B :: "'d set"   and C :: "'d set" shows "(A - B) \<inter> C = A \<inter> C - B \<inter> C"
prems: shows "all_roots_real ([:- a, 1:] * p)"
update: assumes "alive (ref l) s"   and "alive v s"   and "\<tau> v \<le> ltype l" shows "P (Abs_Store (Rep_Store s\<lparr>vals := (vals (Rep_Store s))(l := v)\<rparr>))"
derives_def: fixes g :: "('c, 'd) rule set \<times> 'e" shows "derives g = {(lsl, rsl). \<exists>s1 s2 lhs rhs. s1 @ [NTS lhs] @ s2 = lsl \<and> s1 @ rhs @ s2 = rsl \<and> rule lhs rhs \<in> fst g}"
lsl_rpc: fixes x :: "'b"   and y :: "'b" shows "\<nu> (rpc x y) = rpc x y"
compact_Euclidean_space: fixes n :: "nat" shows "compact_space (Euclidean_space n) = (n = 0)"
mult_cancel_left1: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and c :: "'b"   and b :: "'b" assumes "class.ring_1_no_zero_divisors one times plus zero minus uminus" shows "(c = times c b) = (c = zero \<or> b = one)"
pap: shows "contour_integral p (\<lambda>w. 1 / (w - z)) = complex_of_real (2 * pi) * \<i> * winding_number g z"
simps_9: fixes v :: "'b formula"   and va :: "'b formula" shows "is_lit_plus (v \<^bold>\<and> va) = False"
assms_1: shows "send\<langle>t\<rangle>\<^sub>s\<^sub>t # S \<in> \<S>"
assms_2: fixes a1 :: "('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and a2 :: "('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" assumes "a1 \<in> A"   and "a2 \<in> A" shows "a1 + a2 \<in> A"
simps_2: fixes s :: "'a FOL_Fitting.term"   and k :: "nat" shows "TT[s/k] = TT"
pderivs_snoc: fixes s :: "'a list"   and c :: "'a"   and r :: "'a rexp" shows "pderivs (s @ [c]) r = pderiv_set c (pderivs s r)"
wsst_TYs_3: fixes w :: "'b word" shows "size w = LENGTH('b)"
zero_real_def: shows "0 \<equiv> Real (\<lambda>n. 0)"
map_ext: fixes xs :: "'d list"   and f :: "'d \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'e" assumes "\<And>x. x \<in> set xs \<longrightarrow> f x = g x" shows "map f xs = map g xs"
exponent_def: shows "IEEE.exponent \<equiv> map_fun Rep_float id (\<lambda>(uu_, e, uu_). unat e)"
simps_28: fixes f1 :: "'b \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'a"   and f2 :: "nat \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> 'a"   and f3 :: "nat \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> 'a"   and f4 :: "'b \<Rightarrow> 'a"   and f5 :: "nat \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'a"   and x31 :: "nat"   and x32 :: "'b"   and x33 :: "'b"   and x34 :: "'c"   and x35 :: "'c"   and x36 :: "'d" shows "(case Recv x31 x32 x33 x34 x35 x36 of Trans x xa xb \<Rightarrow> f1 x xa xb | Send x xa xb xc xd xe \<Rightarrow> f2 x xa xb xc xd xe | Recv x xa xb xc xd xe \<Rightarrow> f3 x xa xb xc xd xe | Snapshot x \<Rightarrow> f4 x | RecvMarker x xa xb \<Rightarrow> f5 x xa xb) = f3 x31 x32 x33 x34 x35 x36"
prj_simps_2: shows "C.dom f_src.prj\<^sub>1 = f.leg0 \<down>\<down> src.leg1"
cases: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and p :: "'b pmf"   and q :: "'c pmf" assumes "rel_pmf R p q" obtains pq :: "('b \<times> 'c) pmf" where "\<And>x y. (x, y) \<in> set_pmf pq \<Longrightarrow> R x y" and "map_pmf fst pq = p" and "map_pmf snd pq = q"
sums_stieltjes_gamma: fixes n :: "nat" shows "(\<lambda>k. ln (real (k + 1)) ^ n / real (k + 1) - (ln (real (k + 2)) ^ (n + 1) - ln (real (k + 1)) ^ (n + 1)) / (real n + 1)) sums stieltjes_gamma n"
cl_subset_carrier: fixes X :: "'a set" shows "X_insert.cl X \<subseteq> insert x X"
sameDom_lift_simp2: fixes f :: "'d \<Rightarrow> 'b"   and inp :: "'a \<Rightarrow> 'd option"   and inp' :: "'a \<Rightarrow> 'c option" shows "sameDom (lift f inp) inp' = sameDom inp inp'"
cod_char: fixes f :: "'a \<times> 'a \<times> 'a" shows "VVV.cod f = (if VVV.arr f then VxVxV.cod f else VxVxV.null)"
simps_2: fixes v :: "'a"   and va :: "'a list" shows "merge (v # va) [] = v # va"
exhaust_sel: fixes model :: "('b, 'c) model" obtains "model = Model (R model) (V model)"
infinitely_many1_axioms: shows "infinitely_many1 (\<lambda>i. i \<in> I)"
impl_of_RBT_HM: shows "impl_of_RBT_HM hm \<in> {hm. HashMap_Impl.invar hm}"
nat_induct2: fixes P :: "nat \<Rightarrow> bool"   and n :: "nat" assumes "P 0"   and "P 1"   and "\<And>n. P n \<Longrightarrow> P (n + 2)" shows "P n"
kim: shows "k ` ({0..1} \<times> U) \<subseteq> C"
focus_Polys_Compl: fixes p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and X :: "'b set" assumes "p \<in> P[- X]" shows "focus X p = monomial p 0"
get_disconnected_nodes_ok: fixes h :: "(_) heap"   and document_ptr :: "(_) document_ptr" assumes "ShadowRootClass.type_wf h"   and "document_ptr |\<in>| document_ptr_kinds h" shows "h \<turnstile> ok get_disconnected_nodes document_ptr"
embed_bool_def: fixes P :: "'a \<Rightarrow> bool" shows "\<guillemotleft> P \<guillemotright> \<equiv> \<lambda>s. if P s then 1 else 0"
oexp_def: fixes r :: "('d \<times> 'd) set"   and s :: "('e \<times> 'e) set" assumes "Well_order r"   and "Well_order s" shows "r ^o s = {(f, g). f \<in> FinFunc r s \<and> g \<in> FinFunc r s \<and> ((let m = wo_rel.max_fun_diff s f g in (f m, g m) \<in> r) \<or> f = g)}"
l_span_closed2Tr0: fixes A :: "'b set"   and H :: "'a set"   and s :: "'b"   and f :: "'a" assumes "ideal R A"   and "H \<subseteq> carrier M"   and "Ring R"   and "s \<in> A"   and "f \<in> linear_span R M (carrier R) H" shows "s \<cdot>\<^sub>s f \<in> linear_span R M A H"
A1: shows "A1 \<in> carrier_mat 1 (dim_col A)"
that: fixes e' :: "nat \<Rightarrow> real" assumes "e' \<in> UNIV \<rightarrow> I"   and "pdevs_val e (pdevs_of_list xs) = pdevs_val e' (pdevs_of_list ys)" shows "thesis"
ordinal_order_antisym: fixes x :: "ordinal"   and y :: "ordinal" assumes "x \<le> y"   and "y \<le> x" shows "x = y"
nextElem_nth: fixes xs :: "'a list"   and i :: "nat"   and z :: "'a" assumes "distinct xs"   and "i < |xs|" shows "nextElem xs z (xs ! i) = (if |xs| = i + 1 then z else xs ! (i + 1))"
differing_vars_elim: fixes dma :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> Sec"   and \<C>_vars :: "'a \<Rightarrow> 'a set"   and eval :: "((('c \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)) \<times> ('c \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)) set"   and x :: "'a"   and mem\<^sub>1 :: "'a \<Rightarrow> 'b"   and mem\<^sub>2 :: "'a \<Rightarrow> 'b"   and mems :: "(('a \<Rightarrow> 'b) \<times> ('a \<Rightarrow> 'b)) list"   and i :: "nat" assumes "sifum_security_init dma \<C>_vars eval"   and "x \<in> sifum_security_init.differing_vars_lists mem\<^sub>1 mem\<^sub>2 mems i" shows "fst (mems ! i) x \<noteq> mem\<^sub>1 x \<or> snd (mems ! i) x \<noteq> mem\<^sub>2 x"
injective_scale: fixes c :: "'c" assumes "c \<noteq> (0::'c)" shows "inj ((*s) c)"
skelInp_def2: fixes inp :: "'a \<Rightarrow> ('b, 'c, 'varSort, 'var, 'd) qTerm set option" assumes "goodInp inp" shows "skelInp inp = lift skel inp"
COND_REF: fixes x :: "'b"   and x' :: "'a" assumes "(x, x') \<in> R'"   and "I x"   and "I' x'" shows "b x = b' x'"
monic_prod_list: fixes as :: "'b poly list" assumes "\<And>a. a \<in> set as \<Longrightarrow> monic a" shows "monic (prod_list as)"
b_e_type_loop: fixes \<C> :: "t_context"   and e :: "b_e"   and ts :: "t list"   and ts' :: "t list"   and tf :: "tf"   and es :: "b_e list" assumes "\<C> \<turnstile> [e] : ts _> ts'"   and "e = Loop tf es" shows "\<exists>ts'' tfn tfm. tf = tfn _> tfm \<and> ts = ts'' @ tfn \<and> ts' = ts'' @ tfm \<and> \<C>\<lparr>label := [tfn] @ label \<C>\<rparr> \<turnstile> es : tf"
ife_FAIL_to_ASSERT_cnv: fixes \<Phi> :: "bool"   and m :: "'a nres" shows "(if \<Phi> then m else FAIL) = op_nres_ASSERT_bnd \<Phi> m"
assoc'_naturality: fixes \<mu> :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and \<nu> :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and \<tau> :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.arr \<mu>"   and "Span.arr \<nu>"   and "Span.arr \<tau>"   and "Span.src \<mu> = Span.trg \<nu>"   and "Span.src \<nu> = Span.trg \<tau>" shows "Span.\<a>' (Span.cod \<mu>) (Span.cod \<nu>) (Span.cod \<tau>) \<bullet> (\<mu> \<circ> \<nu> \<circ> \<tau>) = ((\<mu> \<circ> \<nu>) \<circ> \<tau>) \<bullet> Span.\<a>' (Span.dom \<mu>) (Span.dom \<nu>) (Span.dom \<tau>)"
simple_rotations_cycle: fixes r :: "Vertex \<Rightarrow> Vertex"   and v :: "Vertex"   and w :: "Vertex" assumes "r \<in> simple_rotations"   and "r \<noteq> id"   and "r v = w"   and "v \<noteq> w" shows "r w \<noteq> v"
fin_1: fixes i :: "nat" shows "f i \<noteq> \<infinity>"
associated_iff_dvd: 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 unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize" shows "(normalize a = normalize b) = (dvd.dvd times a b \<and> dvd.dvd times b a)"
assms_4: shows "set (characterize_consistent_signs_at_roots_copr p qs) \<subseteq> set signs"
l_null: fixes R :: "('a, 'b) ring_scheme"   and x :: "'a" assumes "semiring R"   and "x \<in> carrier R" shows "\<zero>\<^bsub>R\<^esub> \<otimes>\<^bsub>R\<^esub> x = \<zero>\<^bsub>R\<^esub>"