hcju/isabelleps huggingface dataset:
freshenLC_eq_iff: fixes a :: "nat"   and v :: "'var"   and a' :: "nat"   and v' :: "'var" shows "(freshenLC a v = freshenLC a' v') = (a = a' \<and> v = v')"
assoc_mult_vec_mat: fixes v :: "'c vec"   and n1 :: "nat"   and A :: "'c mat"   and n2 :: "nat"   and B :: "'c mat"   and n3 :: "nat" assumes "v \<in> carrier_vec n1"   and "A \<in> carrier_mat n1 n2"   and "B \<in> carrier_mat n2 n3" shows "v \<^sub>v* (A * B) = v \<^sub>v* A \<^sub>v* B"
restrict_spmf_def: fixes p :: "'c spmf"   and A :: "'c set" shows "p \<upharpoonleft> A = map_pmf (\<lambda>x. x \<bind> (\<lambda>y. if y \<in> A then Some y else None)) p"
has_bochner_integral_null_measure_iff: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> 'd" shows "has_bochner_integral (null_measure M) f (0::'d) = (f \<in> borel_measurable M)"
Not_Rall: fixes F :: "'b \<Rightarrow> 'a \<Rightarrow> bool" shows "\<turnstile> \<not> (\<forall>x. F x) = (\<exists>x. \<not> F x)"
Quotient3_rep_reflp: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and Abs :: "'c \<Rightarrow> 'd"   and Rep :: "'d \<Rightarrow> 'c"   and a :: "'d" assumes "Quotient3 R Abs Rep" shows "R (Rep a) (Rep a)"
abelian_group_hom_def: fixes G :: "('c, 'd) ring_scheme"   and H :: "('e, 'f) ring_scheme"   and h :: "'c \<Rightarrow> 'e" shows "abelian_group_hom G H h \<equiv> abelian_group G \<and> abelian_group H \<and> abelian_group_hom_axioms G H h"
cmult: fixes c :: "'a"   and F :: "'b filter"   and f :: "'b \<Rightarrow> 'a" assumes "c \<noteq> (0::'a)" shows "o[F](\<lambda>x. c * f x) = o[F](f)"
bij_betw_inv_into: fixes f :: "'e \<Rightarrow> 'f"   and A :: "'e set"   and B :: "'f set" assumes "bij_betw f A B" shows "bij_betw (inv_into A f) B A"
maxim_in: fixes r :: "('d \<times> 'd) set"   and A :: "'d set" assumes "wo_rel r"   and "finite A"   and "A \<noteq> {}"   and "A \<subseteq> Field r" shows "wo_rel.maxim r A \<in> A"
union_fold_insert: fixes A :: "'g set"   and B :: "'g set" assumes "finite A" shows "A \<union> B = Finite_Set.fold insert B A"
lgc_invar_def: fixes TNs :: "('f inference set \<times> ('f \<times> 'l) set) llist"   and i :: "enat" shows "lgc_invar TNs i = (Inf_from (active_subset (Liminf_upto_llist (lmap snd TNs) i)) \<subseteq> \<Union> (from_F ` Liminf_upto_llist (lmap fst TNs) i) \<union> Sup_upto_llist (lmap (Red_I_\<G> \<circ> snd) TNs) i)"
subgroup_generated_def: fixes G :: "('b, 'c) monoid_scheme"   and S :: "'b set" shows "subgroup_generated G S = G\<lparr>carrier := generate G (carrier G \<inter> S)\<rparr>"
comm_groupE_6: fixes G :: "('c, 'd) monoid_scheme"   and x :: "'c" assumes "comm_group G"   and "x \<in> carrier G" shows "\<exists>y\<in>carrier G. y \<otimes>\<^bsub>G\<^esub> x = \<one>\<^bsub>G\<^esub>"
rel_intros_1: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and x1 :: "'b"   and y1 :: "'c" assumes "R x1 y1" shows "rel_behaviour R (Terminates x1) (Terminates y1)"
intro: fixes R :: "('e, 'f) Ring_scheme"   and M :: "('g, 'e, 'h) Module_scheme" assumes "Ring R"   and "\<And>a m. \<lbrakk>a \<in> carrier R; m \<in> carrier M\<rbrakk> \<Longrightarrow> a \<cdot>\<^sub>s\<^bsub>M\<^esub> m \<in> carrier M"   and "\<And>a b m. \<lbrakk>a \<in> carrier R; b \<in> carrier R; m \<in> carrier M\<rbrakk> \<Longrightarrow> (a \<plusminus>\<^bsub>R\<^esub> b) \<cdot>\<^sub>s\<^bsub>M\<^esub> m = a \<cdot>\<^sub>s\<^bsub>M\<^esub> m \<plusminus>\<^bsub>M\<^esub> b \<cdot>\<^sub>s\<^bsub>M\<^esub> m"   and "\<And>a m n. \<lbrakk>a \<in> carrier R; m \<in> carrier M; n \<in> carrier M\<rbrakk> \<Longrightarrow> a \<cdot>\<^sub>s\<^bsub>M\<^esub> (m \<plusminus>\<^bsub>M\<^esub> n) = a \<cdot>\<^sub>s\<^bsub>M\<^esub> m \<plusminus>\<^bsub>M\<^esub> a \<cdot>\<^sub>s\<^bsub>M\<^esub> n"   and "\<And>a b m. \<lbrakk>a \<in> carrier R; b \<in> carrier R; m \<in> carrier M\<rbrakk> \<Longrightarrow> (a \<cdot>\<^sub>r\<^bsub>R\<^esub> b) \<cdot>\<^sub>s\<^bsub>M\<^esub> m = a \<cdot>\<^sub>s\<^bsub>M\<^esub> (b \<cdot>\<^sub>s\<^bsub>M\<^esub> m)"   and "\<And>m. m \<in> carrier M \<Longrightarrow> 1\<^sub>r\<^bsub>R\<^esub> \<cdot>\<^sub>s\<^bsub>M\<^esub> m = m" shows "Module_axioms M R"
statA_prop: shows "\<forall>T. statT = ArrayT T \<longrightarrow> mode \<noteq> SuperM"
in_MPLS_charles_leq_2_pow_n: fixes PROB :: "(('c, bool) fmap \<times> ('c, bool) fmap) set"   and x :: "nat" assumes "finite PROB"   and "x \<in> MPLS_charles PROB" shows "x \<le> 2 ^ card (prob_dom PROB) - 1"
simps_1: fixes G :: "'b graph"   and M :: "'b SecurityInvariant_configured list"   and accu :: "('b \<times> 'b) list" shows "filter_IFS_no_violations_accu G M accu [] = accu"
CACH_VP: fixes G :: "((Class \<times> Method \<times> int) \<times> ((Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool) \<times> ((Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<times> Val \<Rightarrow> bool) \<times> ((Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool)) list" assumes "\<forall>c m par code l0. mbody_is c m (par, code, l0) \<longrightarrow> (\<exists>n Anno. MST\<down>(c, m) = Some (mkSPEC (Cachera n) Anno) \<and> deriv G c m l0 (Cachera n))"   and "\<forall>c m l A B I. G\<down>(c, m, l) = Some (A, B, I) \<longrightarrow> (\<exists>n. (A, B, I) = Cachera n \<and> deriv G c m l (Cachera n))" shows "VP"
ntrunc_equality: fixes M :: "('a, 'b) node set"   and N :: "('a, 'b) node set" assumes "\<And>k. ntrunc k M = ntrunc k N" shows "M = N"
Ball_image_comp: fixes f :: "'b \<Rightarrow> 'a"   and A :: "'b set"   and g :: "'a \<Rightarrow> bool" shows "Ball (f ` A) g = Ball A (g \<circ> f)"
that: fixes p' :: "'a poly" assumes "p = smult (content p) p'"   and "content p' = (1::'a)" shows "thesis"
simps_2: fixes uu :: "'b tree"   and uv :: "'b"   and r :: "'b tree" shows "len \<langle>uu, uv, r\<rangle> = 1 + len r"
ih: fixes B :: "'a multiset" assumes "\<forall>A. A \<subset># B \<longrightarrow> P A" shows "P B"
deg_def: fixes R :: "('d, 'e) Ring_scheme"   and S :: "('d, 'f) Ring_scheme"   and X :: "'d"   and p :: "'d" shows "deg R S X p = (if p = \<zero>\<^bsub>R\<^esub> then - \<infinity> else an (deg_n R S X p))"
completeness: fixes Cs :: "fterm literal set set" assumes "unification"   and "finite Cs"   and "\<forall>C\<in>Cs. finite C"   and "\<forall>F G. \<not> eval\<^sub>c\<^sub>s F G Cs" shows "\<exists>Cs'. resolution_deriv Cs Cs' \<and> {} \<in> Cs'"
approx_map_le: fixes t2 :: "'a \<Rightarrow> 'b option"   and s :: "'a \<Rightarrow> 'b"   and t1 :: "'a \<Rightarrow> 'b option" assumes "approx t2 s"   and "t1 \<subseteq>\<^sub>m t2" shows "approx t1 s"
X_poly_def: fixes R :: "('a, 'b) ring_scheme" shows "X_poly R = monom (UP R) \<one>\<^bsub>R\<^esub> 1"
for_comb_def: fixes f :: "'b \<Rightarrow> nat \<Rightarrow> 'b nres"   and a0 :: "'b"   and n :: "nat" shows "for_comb f a0 n = nfoldli [0..<n + 1] (\<lambda>x. True) (\<lambda>k a. f a k) a0"
aprimedivisor_gt_Suc_0_nat: fixes d :: "nat" assumes "Suc 0 < d" shows "Suc 0 < aprimedivisor d"
store_barrier_pending_mod_low_equal: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and t1 :: "'b sparc_state"   and t2 :: "'b sparc_state" assumes "low_equal s1 s2 \<and> t1 = store_barrier_pending_mod False s1 \<and> t2 = store_barrier_pending_mod False s2" shows "low_equal t1 t2"
recognises_single_possible_step: fixes e :: "((nat \<times> nat) \<times> transition) fset"   and s :: "nat"   and r :: "nat \<Rightarrow>f value option"   and l :: "String.literal"   and i :: "value list"   and s' :: "nat"   and t :: "transition"   and trace :: "(String.literal \<times> value list) list" assumes "possible_steps e s r l i = {|(s', t)|}"   and "recognises_execution e s' (evaluate_updates t i r) trace" shows "recognises_execution e s r ((l, i) # trace)"
simps_5: fixes jmps :: "jump set"   and e :: "expr"   and c1 :: "stmt"   and c2 :: "stmt" shows "jumpNestingOkS jmps (If(e) c1 Else c2) = (jumpNestingOkS jmps c1 \<and> jumpNestingOkS jmps c2)"
rs_total: fixes x :: "nat" shows "eval r_s [x] \<down>"
ownership_distinct: fixes i :: "nat"   and j :: "nat"   and p\<^sub>i :: "'p"   and is\<^sub>i :: "instr list"   and \<theta>\<^sub>i :: "nat \<Rightarrow> nat option"   and sb\<^sub>i :: "'p memref list"   and \<D>\<^sub>i :: "bool"   and \<O>\<^sub>i :: "nat set"   and \<R>\<^sub>i :: "nat \<Rightarrow> bool option"   and p\<^sub>j :: "'p"   and is\<^sub>j :: "instr list"   and \<theta>\<^sub>j :: "nat \<Rightarrow> nat option"   and sb\<^sub>j :: "'p memref list"   and \<D>\<^sub>j :: "bool"   and \<O>\<^sub>j :: "nat set"   and \<R>\<^sub>j :: "nat \<Rightarrow> bool option" assumes "i < length ts"   and "j < length ts"   and "i \<noteq> j"   and "ts ! i = (p\<^sub>i, is\<^sub>i, \<theta>\<^sub>i, sb\<^sub>i, \<D>\<^sub>i, \<O>\<^sub>i, \<R>\<^sub>i)"   and "ts ! j = (p\<^sub>j, is\<^sub>j, \<theta>\<^sub>j, sb\<^sub>j, \<D>\<^sub>j, \<O>\<^sub>j, \<R>\<^sub>j)" shows "(\<O>\<^sub>i \<union> all_acquired sb\<^sub>i) \<inter> (\<O>\<^sub>j \<union> all_acquired sb\<^sub>j) = {}"
supportE_preTz_Un: fixes C :: "acom"   and e :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> nat"   and x :: "char list" shows "supportE (\<lambda>l. preTz C (e l) (l x)) \<subseteq> insert x (\<Union>n. supportE (\<lambda>l. preTz C (e l) n))"
comp_vwb_lens: fixes x :: "'d \<Longrightarrow> 'e"   and y :: "'e \<Longrightarrow> 'f" assumes "vwb_lens x"   and "vwb_lens y" shows "vwb_lens (x ;\<^sub>L y)"
subpath_reversepath: fixes g :: "real \<Rightarrow> 'c" shows "subpath 1 0 g = reversepath g"
LIMSEQ_erf_remainder_integral: fixes x :: "real"   and n :: "nat" assumes "0 < x" shows "(\<lambda>m. integral {x..x + real m} (\<lambda>t. exp (- t\<^sup>2) / t ^ (2 * n))) \<longlonglongrightarrow> erf_remainder_integral n x"
iterateoi_def: fixes olist_it :: "'d \<Rightarrow> (('b \<times> 'c) list \<Rightarrow> bool) \<Rightarrow> ('b \<times> 'c \<Rightarrow> ('b \<times> 'c) list \<Rightarrow> ('b \<times> 'c) list) \<Rightarrow> ('b \<times> 'c) list \<Rightarrow> ('b \<times> 'c) list"   and S :: "'d" shows "poly_map_iterateoi_defs.iterateoi olist_it S \<equiv> it_to_it (olist_it S)"
\<omega>_limit_set_closed: fixes x :: "'a" shows "closed (\<omega>_limit_set x)"
fds_nth_minus: fixes f :: "'b fds"   and g :: "'b fds" shows "fds_nth (f - g) = (\<lambda>n. fds_nth f n - fds_nth g n)"
max_Var_floatariths_leI: fixes xs :: "floatarith list"   and F :: "nat" assumes "\<And>i. i < length xs \<Longrightarrow> max_Var_floatarith (xs ! i) \<le> F" shows "max_Var_floatariths xs \<le> F"
orthogonal_rotation_or_rotoinversion: fixes Q :: "(('a, 'b) vec, 'b) vec" shows "orthogonal_matrix Q = (rotation_matrix Q \<or> rotoinversion_matrix Q)"
compact_imp_closed: fixes "open" :: "'d set \<Rightarrow> bool"   and s :: "'d set" assumes "class.t2_space open"   and "topological_space.compact open s" shows "topological_space.closed open s"
typedef_lens_def: fixes Rep :: "'a \<Rightarrow> 'b"   and Abs :: "'b \<Rightarrow> 'a"   and A :: "'b set" assumes "type_definition Rep Abs A" shows "type_definition.typedef_lens Rep Abs = \<lparr>lens_get = Abs, lens_put = \<lambda>s. Rep\<rparr>"
winding_number_ivt_neg: fixes \<gamma> :: "real \<Rightarrow> complex"   and z :: "complex"   and w :: "real" assumes "valid_path \<gamma>"   and "z \<notin> path_image \<gamma>"   and "Re (winding_number \<gamma> z) \<le> w"   and "w \<le> 0" shows "\<exists>t\<in>{0..1}. Re (winding_number (subpath 0 t \<gamma>) z) = w"
Black: fixes l :: "('a, 'b) rbt"   and k :: "'a"   and v :: "'b"   and r :: "('a, 'b) rbt" assumes "t = Branch B l k v r" shows "thesis"
decr_simps: fixes minSoFar :: "'ord"   and js :: "'ord stream" shows "decr minSoFar js = (if \<not> ev (\<lambda>js. (shd js, minSoFar) \<in> Ord \<and> shd js \<noteq> minSoFar) js then undefined else if (shd js, minSoFar) \<in> Ord \<and> shd js \<noteq> minSoFar then shd js ## decr (shd js) js else decr minSoFar (stl js))"
mat_inverse_simps_2: fixes c :: "'d" assumes "c \<noteq> (0::'d)" shows "mat c ** mat (inverse c) = mat (1::'d)"
disj1': fixes x :: "Vars.var" shows "carrier ((FBinds \<Gamma>\<cdot>ae) x) \<subseteq> - domA \<Delta>"
assigns_expr_simp: fixes e :: "expr" shows "assigns (In1l e) = assignsE e"
pnormal_last_length: fixes p :: "'a list" assumes "0 < length p"   and "last p \<noteq> (0::'a)" shows "pnormal p"
onorm_sum_le: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and z :: "real" assumes "finite S"   and "\<And>s. s \<in> S \<Longrightarrow> bounded_linear (f s)"   and "(\<Sum>s\<in>S. onorm (f s)) \<le> z" shows "onorm (\<lambda>x. \<Sum>s\<in>S. f s x) \<le> z"
incl_in_img_cod: fixes f :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "arr f" shows "incl_in (img f) (cod f)"
Left_Coset_def: fixes composition :: "'d \<Rightarrow> 'e \<Rightarrow> 'c"   and x :: "'d"   and H :: "'e set" shows "coset_notation.Left_Coset composition x H = {composition x h |h. h \<in> H}"
full_to_part: fixes C :: "com"   and s :: "char list \<Rightarrow> int"   and m :: "nat"   and s' :: "char list \<Rightarrow> int" assumes "(C, s) \<Rightarrow> m \<Down> s'" shows "(C, part s) \<Rightarrow>\<^sub>A m \<Down> part s'"
simps_7: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a" shows "(case Cext of Cext \<Rightarrow> f1 | Text \<Rightarrow> f2 | Fext \<Rightarrow> f3) = f1"
igSubstAbsIPresIGWlsAbs_def: fixes MOD :: "('c, 'd, 'varSort, 'sort, 'e, 'f, 'g, 'h, 'i) Iteration.model_scheme" shows "igSubstAbsIPresIGWlsAbs MOD \<equiv> \<forall>ys Y y us s A. isInBar (us, s) \<and> igWls MOD (asSort ys) Y \<and> igWlsAbs MOD (us, s) A \<longrightarrow> igWlsAbs MOD (us, s) (igSubstAbs MOD ys Y y A)"
list_separated_toString_intersperse: fixes sep :: "char"   and f :: "'a \<Rightarrow> char list"   and xs :: "'a list" shows "intersperse sep (map f xs) = list_separated_toString [sep] f xs"
invar_init: fixes w :: "'a \<times> 'a \<Rightarrow> enat"   and s :: "'a" shows "Dijkstra_Invar w s (Dijkstra.initD s) D.initS"
N_prime_mono: fixes q :: "real"   and p :: "int"   and p' :: "int" assumes "fbnorm q"   and "prime p"   and "prime p'"   and "p \<le> p'" shows "fbnorm.N q p \<le> fbnorm.N q p'"
eq_\<I>_generat_reflI: fixes generat :: "('h, 'i, 'j \<Rightarrow> 'k) generat"   and A :: "'h \<Rightarrow> 'h \<Rightarrow> bool"   and \<I> :: "('i, 'j) \<I>"   and D :: "'k \<Rightarrow> 'k \<Rightarrow> bool" assumes "\<And>x. x \<in> generat_pures generat \<Longrightarrow> A x x"   and "\<And>out c. generat = IO out c \<Longrightarrow> out \<in> outs_\<I> \<I> \<and> (\<forall>input\<in>responses_\<I> \<I> out. D (c input) (c input))" shows "eq_\<I>_generat A \<I> D generat generat"
\<eta>s0'_def: shows "\<eta>s0' = map inv_renaming \<rho>s \<odot>s \<eta>s0"
subdegree_leI: fixes f :: "'a fps"   and n :: "nat" assumes "f $ n \<noteq> (0::'a)" shows "subdegree f \<le> n"
Re_complex_div_ge_0: fixes a :: "complex"   and b :: "complex" shows "(0 \<le> Re (a / b)) = (0 \<le> Re (a * cnj b))"
mkarr_Runit': fixes t :: "'c local.term" assumes "Arr t" shows "\<F>C.mkarr \<^bold>\<r>\<^sup>-\<^sup>1\<^bold>[t\<^bold>] = \<F>C.\<rho>' (\<F>C.mkarr t)"
Un_empty: fixes A :: "'b set"   and B :: "'b set" shows "(A \<union> B = {}) = (A = {} \<and> B = {})"
NotAResultNotRF: shows "NotAResult \<noteq> RPCFailure"
node_ptr_kinds_eq_h: shows "node_ptr_kinds h2 = node_ptr_kinds h |\<union>| {|cast\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r new_character_data_ptr|}"
gInduct_gConsIndif_recAll_isSurjAll: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) Recursion.model" assumes "presWlsAll (rec MOD) (recAbs MOD) MOD"   and "presCons (rec MOD) (recAbs MOD) MOD"   and "gConsIndif MOD"   and "gInduct MOD" shows "isSurjAll (rec MOD) (recAbs MOD) MOD"
exhaust: fixes y :: "'d action_Rep" obtains x11 :: "'d" and x12 :: "'d" where "y = In_Rep x11 x12"   | x21 :: "'d" and x22 :: "name list" and x23 :: "'d" where "y = Out_Rep x21 x22 x23"   | "y = Tau_Rep"
lQuots_simps_1: fixes A :: "'c list set" shows "lQuots [] A = A"
dnf_True: fixes f :: "'b \<Rightarrow> bool" shows "dnf_to_bool f dnf_True"
boundedE: fixes b :: "'c multiset"   and c :: "'c multiset"   and a :: "'c multiset" assumes "b \<union># c \<subseteq># a" obtains "b \<subseteq># a" and "c \<subseteq># a"
ATV_EF: fixes A :: "'a attree"   and I :: "'a set"   and s :: "'a set" assumes "\<turnstile>\<^sub>V A"   and "(I, s) = attack A" shows "Kripke {s. \<exists>i\<in>I. i \<rightarrow>\<^sub>i* s} I \<turnstile> EF s"
Red_I_Sup_in_Liminf: shows "Red_I (Sup_llist Ns) \<subseteq> Red_I (Liminf_llist Ns)"
line_integral_on_path: shows "one_chain_line_integral F {i} \<gamma> = one_chain_line_integral F {i} (two_chain_horizontal_boundary two_chain)"
big_subsetI: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'c" assumes "f \<in> \<Omega>[F](g)" shows "\<Omega>[F](f) \<subseteq> \<Omega>[F](g)"
ex_post_efficient_sds_axioms: shows "ex_post_efficient_sds agents alts RSD"
Sup_least: fixes A :: "'al set"   and z :: "'al" assumes "A \<subseteq> U"   and "z \<in> U"   and "\<And>x. x \<in> A \<Longrightarrow> x \<le>\<^sub>o\<^sub>w z" shows "\<Squnion>\<^sub>o\<^sub>w A \<le>\<^sub>o\<^sub>w z"
simCasesCont: fixes Rel :: "(pi \<times> pi) set"   and Q :: "pi"   and P :: "pi"   and C :: "'a" assumes "eqvt Rel"   and "\<And>a y Q'. \<lbrakk>Q \<longmapsto> a<\<nu>y> \<prec> Q'; y \<sharp> P; y \<sharp> Q; y \<sharp> C\<rbrakk> \<Longrightarrow> \<exists>P'. P \<longmapsto> a<\<nu>y> \<prec> P' \<and> (P', Q') \<in> Rel"   and "\<And>\<alpha> Q'. Q \<longmapsto> \<alpha> \<prec> Q' \<Longrightarrow> \<exists>P'. P \<longmapsto> \<alpha> \<prec> P' \<and> (P', Q') \<in> Rel" shows "P \<leadsto>[Rel] Q"
SD_efficient: fixes R :: "'agent \<Rightarrow> 'alt \<Rightarrow> 'alt \<Rightarrow> bool" assumes "is_pref_profile R" shows "SD_efficient R (sds R)"
assm: shows "v=v1\<parallel>v2 \<and> v1=v3\<parallel>v4"
DF\<^sub>S\<^sub>K\<^sub>I\<^sub>P_unfold: fixes A :: "'a set" shows "DF\<^sub>S\<^sub>K\<^sub>I\<^sub>P A = ((\<sqinter>z\<in>A \<rightarrow>  DF\<^sub>S\<^sub>K\<^sub>I\<^sub>P A) |-| SKIP)"
fps_deriv_power: fixes a :: "'a fps"   and n :: "nat" shows "fps_deriv (a ^ n) = fps_const (of_nat n) * fps_deriv a * a ^ (n - 1)"
simps_2: fixes x :: "'b"   and xs :: "'b list" shows "distinct (x # xs) = (x \<notin> ran\<^sub>l xs \<and> distinct xs)"
simps_9: fixes f1 :: "'h \<Rightarrow> 'e"   and f2 :: "'i \<Rightarrow> 'f"   and x1 :: "('h \<times> 'i) pmf" shows "map_var0 f1 f2 (PMF x1) = PMF (map_pmf (map_prod f1 f2) x1)"
assms: shows "i < card {i. i < length xs \<and> i \<in> I}"
A_def: shows "A \<equiv> rec_nat S (\<lambda>n a. if \<exists>U\<subseteq>a. closed U \<and> \<phi> U \<and> U \<inter> B n = {} then SOME U. U \<subseteq> a \<and> closed U \<and> \<phi> U \<and> U \<inter> B n = {} else a)"
wf_ciI: fixes ci :: "'a instr \<Rightarrow> (nat \<times> nat \<times> 'a instr list \<times> (nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list) prog \<Rightarrow> 'b \<Rightarrow> 'a val list \<Rightarrow> 'a val list \<Rightarrow> String.literal \<Rightarrow> String.literal \<Rightarrow> nat \<Rightarrow> 'a frame list \<Rightarrow> bool" assumes "ci_stk_offer ci"   and "\<And>ins P h stk loc C M pc frs. ci ins P h stk loc C M pc frs \<Longrightarrow> check_instr' ins P h stk loc C M pc frs"   and "\<And>ins P h stk loc C M pc pc' frs. \<lbrakk>ci ins P h stk loc C M pc frs; ins_jump_ok ins pc'\<rbrakk> \<Longrightarrow> ci ins P h stk loc C M pc' frs" shows "wf_ci ci"
assms: fixes i :: "nat" assumes "i < length xs" shows "xs ! i = (0::'a)"
cnorm_le_square: fixes x :: "'a"   and a :: "real" shows "(norm x \<le> a) = (0 \<le> a \<and> cinner x x \<le> complex_of_real (a\<^sup>2))"
freshOutputTransition: fixes P :: "pi"   and a :: "name"   and b :: "name"   and P' :: "pi"   and c :: "name" assumes "P \<Longrightarrow>a[b] \<prec> P'"   and "c \<sharp> P" shows "c \<sharp> P'"
to_tree_order_subset: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and nodes :: "(_) object_ptr list"   and node :: "(_) object_ptr"   and nodes' :: "(_) object_ptr list" assumes "heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M h"   and "h \<turnstile> Shadow_DOM.to_tree_order ptr \<rightarrow>\<^sub>r nodes"   and "node \<in> set nodes"   and "h \<turnstile> Shadow_DOM.to_tree_order node \<rightarrow>\<^sub>r nodes'"   and "ShadowRootClass.known_ptrs h"   and "ShadowRootClass.type_wf h" shows "set nodes' \<subseteq> set nodes"
foundedD: fixes subs :: "'a \<Rightarrow> 'a set"   and P :: "'a \<Rightarrow> bool"   and delta :: "'a" assumes "founded subs P (tree subs delta)"   and "terminal subs delta" shows "P delta"
invol_dual_var: fixes x :: "'b" shows "- (- x) = x"
possible_steps_apply_guards: fixes e :: "((nat \<times> nat) \<times> transition) fset"   and s :: "nat"   and r :: "nat \<Rightarrow>f value option"   and l :: "String.literal"   and i :: "value list"   and s' :: "nat"   and t :: "transition" assumes "possible_steps e s r l i = {|(s', t)|}" shows "apply_guards (Guards t) (join_ir i r)"
surjective: fixes r :: "('a, 'b) eq_object_scheme" shows "r = \<lparr>carrier = carrier r, eq = (.=\<^bsub>r\<^esub>), \<dots> = eq_object.more r\<rparr>"
listset_mono: fixes xs :: "'a set list"   and ys :: "'a set list" assumes "length xs = length ys"   and "\<And>i. i < length ys \<Longrightarrow> xs ! i \<subseteq> ys ! i" shows "listset xs \<subseteq> listset ys"
wf_mlex: fixes R :: "('b \<times> 'b) set"   and f :: "'b \<Rightarrow> nat" assumes "wf R" shows "wf (f <*mlex*> R)"
minus_mult_distrib_mat_vec: fixes A :: "'b mat"   and nr :: "nat"   and nc :: "nat"   and B :: "'b mat"   and v :: "'b Matrix.vec" assumes "A \<in> carrier_mat nr nc"   and "B \<in> carrier_mat nr nc"   and "v \<in> carrier_vec nc" shows "(A - B) *\<^sub>v v = A *\<^sub>v v - B *\<^sub>v v"
irreflpI: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "\<And>a. \<not> R a a" shows "irreflp R"
qtm_in_set_subset: fixes t :: "('c, 'd) BinomialTree"   and q :: "('c, 'd) BinomialTree list" assumes "t \<in> set q" shows "tree_to_multiset t \<subseteq># queue_to_multiset q"
minus_Integer_def: fixes a :: "Integer"   and b :: "Integer" shows "a - b = (case a of MkI\<cdot>x \<Rightarrow> \<Lambda> (MkI\<cdot>y). MkI\<cdot>(x - y))\<cdot>b"
round_up_less1: fixes x :: "real"   and p :: "int" assumes "x < 1 / 2"   and "0 < p" shows "round_up p x < 1"
disj_com: fixes x :: "'a"   and y :: "'a" shows "x ## y = y ## x"
path_first_vertex: fixes x :: "'a"   and xs :: "'a list" assumes "H.path (x # xs)" shows "x \<notin> set xs"
1: fixes ars :: "String.literal \<Rightarrow> String.literal set list option"   and ss\<^sub>1 :: "String.literal set list"   and ss\<^sub>2 :: "String.literal set list" assumes "ars \<in> ran (map_of (map (apsnd map_of) arss))"   and "ss\<^sub>1 \<in> ran ars"   and "ss\<^sub>2 \<in> ran ars" shows "length ss\<^sub>1 = length ss\<^sub>2"
wp_assert: fixes p :: "'b Assertion"   and q :: "'b Assertion" shows "wp {\<cdot> p } q = p \<sqinter> q"
finprod_insert: fixes G :: "('c, 'd) monoid_scheme"   and F :: "'e set"   and a :: "'e"   and f :: "'e \<Rightarrow> 'c" assumes "Group.comm_monoid G"   and "finite F"   and "a \<notin> F"   and "f \<in> F \<rightarrow> carrier G"   and "f a \<in> carrier G" shows "finprod G f (insert a F) = f a \<otimes>\<^bsub>G\<^esub> finprod G f F"