hcju/isabelleps huggingface dataset:
ShiftP_Eats_Neg: fixes u :: "name"   and u' :: "name"   and v :: "name"   and f :: "tm"   and k :: "tm"   and del :: "tm"   and g :: "tm"   and c :: "tm" assumes "atom u \<sharp> (u', v, f, k, del, g, c)"   and "atom u' \<sharp> (v, f, k, del, g, c)"   and "atom v \<sharp> (f, k, del, g, c)" shows "{ShiftP f k del g, Neg (SyntaxN.Ex u (SyntaxN.Ex u' (SyntaxN.Ex v (c EQ HPair (Var u) (Var v) AND Var u IN k AND HaddP del (Var u) (Var u')))))} \<turnstile> ShiftP (Eats f c) k del g"
minim_UnderS_underS: fixes A :: "'a set" assumes "A \<noteq> {}"   and "A \<subseteq> Field r" shows "r.UnderS A = r.underS (r.minim A)"
inductive_invariant_is_execution_invariant: fixes s :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option"   and P :: "('r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option) \<Rightarrow> bool" assumes "reachable s"   and "inductive_invariant P" shows "P s"
hd_dropWhile: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b list" assumes "dropWhile P xs \<noteq> \<epsilon>" shows "\<not> P (hd (dropWhile P xs))"
l_null: fixes x :: "'a" assumes "x \<in> carrier R" shows "\<zero> \<otimes> x = \<zero>"
N_def: shows "N \<equiv> while (\<lambda>A. f A \<noteq> A) f {}"
dist_proj: fixes x :: "'e \<Rightarrow>\<^sub>F 'd"   and i :: "'e"   and y :: "'e \<Rightarrow>\<^sub>F 'd" shows "dist ((x)\<^sub>F i) ((y)\<^sub>F i) \<le> dist x y"
fg_1: shows "f piecewise_C1_differentiable_on S"
Un_iff: fixes c :: "'b"   and A :: "'b set"   and B :: "'b set" shows "(c \<in> A \<union> B) = (c \<in> A \<or> c \<in> B)"
gf_cong2: shows "fixespointwise gf (- \<Union> A)"
assms_2: fixes x :: "'b"   and y :: "'b" assumes "(x, y) \<in> r'" shows "(g x, g y) \<in> r"
ab_tendsto: shows "(\<lambda>n. real_of_int \<bar>b n\<bar> / real_of_int (a (n - 1) * a n)) \<longlonglongrightarrow> 0"
DOWN_mapsto: shows "DOWN \<in> Univ \<rightarrow> UNIV"
tree_insert_in_traverse: fixes t :: "interval_tree"   and x :: "nat idx_interval" assumes "tree_sorted t" shows "in_traverse (Interval_Tree.insert x t) = ordered_insert x (in_traverse t)"
list_of_aux_code_1: fixes xs :: "'a list" shows "list_of_aux xs LNil = rev xs"
postcond_left_unit: fixes p :: "('b, 'c) urel" assumes "in\<alpha> \<sharp> p" shows "true ;; p = p"
ge_trans: fixes less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and y :: "'d"   and x :: "'d"   and z :: "'d" assumes "class.non_strict_order less_eq"   and "less_eq y x"   and "less_eq z y" shows "less_eq z x"
islimpt_Un: fixes x :: "'b"   and S :: "'b set"   and T :: "'b set" shows "x islimpt S \<union> T = (x islimpt S \<or> x islimpt T)"
null_eqI: fixes n :: "'c \<times> 'c" assumes "\<And>f. C.VV.comp n f = n \<and> C.VV.comp f n = n" shows "n = C.VV.null"
ennreal_plus_if: fixes a :: "real"   and b :: "real" shows "ennreal a + ennreal b = ennreal (if 0 \<le> a then if 0 \<le> b then a + b else a else b)"
path_entry0_empty_conv: fixes E :: "(nat \<times> nat) set"   and v :: "nat" shows "path_entry E [] v = (v = 0)"
translation_unit_eq: shows "identity M = (\<one>)\<^sub>L"
gHom_comp_ginjec: fixes F :: "('e, 'f) Group_scheme"   and G :: "('g, 'h) Group_scheme"   and H :: "('i, 'j) Group_scheme"   and f :: "'e \<Rightarrow> 'g"   and g :: "'g \<Rightarrow> 'i" assumes "Group F"   and "Group G"   and "Group H"   and "ginj\<^bsub>F,G\<^esub> f"   and "ginj\<^bsub>G,H\<^esub> g" shows "ginj\<^bsub>F,H\<^esub> (g \<circ>\<^bsub>F\<^esub> f)"
hyps_4: shows "local.general_U p = measure_pmf.expectation p local.ocU"
inverse_le_imp_le: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less_eq (inverse a) (inverse b)"   and "less zero a" shows "less_eq b a"
path_emptyI: fixes g :: "'b ugraph"   and u :: "'b" shows "path g u [] u"
intro: fixes x :: "'e \<Longrightarrow> 'd" assumes "\<And>\<sigma> v. get\<^bsub>x\<^esub> (put\<^bsub>x\<^esub> \<sigma> v) = v" shows "weak_lens x"
recfun_aux_domD: fixes x :: "'a" assumes "recfun_aux_dom x" shows "local.dom x"
csum_def: fixes r1 :: "('c \<times> 'c) set"   and r2 :: "('d \<times> 'd) set" shows "r1 +c r2 \<equiv> |Field r1 <+> Field r2|"
c_graph_lm_4: fixes f :: "nat \<Rightarrow> nat" shows "c_graph f = ce_rel_to_set (graph f)"
subtensor_prod_with_unit_vec: fixes j :: "nat"   and n :: "nat"   and i :: "nat"   and A :: "'b tensor" assumes "j < n" shows "subtensor (unit_vec n i \<otimes> A) j = (if i = j then A else tensor0 (dims A))"
rel_funI: fixes A :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and B :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'f" assumes "\<And>x y. A x y \<Longrightarrow> B (f x) (g y)" shows "rel_fun A B f g"
cases: fixes a1 :: "'a list"   and a2 :: "('b, 'c) sequent" assumes "(a1, a2) \<in> modRules2" obtains ps :: "'a list" and c :: "('b, 'c) sequent" and M :: "'c" and Ms :: "('b, 'c) form list" where "a1 = ps" and "a2 = c" and "ps \<noteq> []" and "ModalSequents.mset c = \<LM> Modal M Ms  \<RM>"
AD_aux: fixes a :: "(int \<times> int) set set"   and b :: "(int \<times> int) set set"   and po :: "int"   and p :: "((int \<times> int) set set, int) Combinators list"   and c :: "(int \<times> int) set set"   and d :: "(int \<times> int) set set" assumes "AllowPortFromTo a b po \<in> set p"   and "DenyAllFromTo c d \<in> set p"   and "allNetsDistinct p"   and "singleCombinators p"   and "a \<noteq> c \<or> b \<noteq> d" shows "dom (C (AllowPortFromTo a b po)) \<inter> dom (C (DenyAllFromTo c d)) = {}"
insert_5: fixes s :: "'a" assumes "s \<in> insert s__ ss" shows "\<exists>rs. s = s0 \<star> rs \<and> set rs \<subseteq> rset"
strict_boundedE: fixes b :: "'a"   and c :: "'a"   and a :: "'a" assumes "max b c < a" obtains "b < a" and "c < a"
IVT_min: fixes f :: "real \<Rightarrow> 'a"   and a :: "real"   and y :: "'a"   and b :: "real" assumes "f a \<le> y"   and "y \<le> f b"   and "a \<le> b"   and "continuous_on {a..b} f" obtains x :: "real" where "a \<le> x" and "x \<le> b" and "f x = y" and "\<And>x'. \<lbrakk>a \<le> x'; x' < x\<rbrakk> \<Longrightarrow> f x' < y"
ordinal_le_div: fixes y :: "ordinal"   and z :: "ordinal"   and x :: "ordinal" assumes "0 < y"   and "y * z \<le> x" shows "z \<le> x div y"
ln5: shows "ln 5 \<in> {4 / 3..76 / 45}"
WW_eq_range_into: shows "WW = range (into_WW 0)"
add_diff_inverse: fixes a :: "'a multiset"   and b :: "'a multiset" assumes "a \<subseteq># b" shows "a + (b - a) = b"
prj_simps_7: shows "Maps.cod SPN_fgh.Prj\<^sub>1\<^sub>1 = SPN_fgh.\<mu>.apex"
\<sigma>': shows "\<sigma>' = (None, h(oref \<mapsto> blank P X), (Addr oref # stk, loc, C, M, pc + 1, No_ics) # frs, sh)"
l_coset_subset_G: fixes G :: "('c, 'd) monoid_scheme"   and H :: "'c set"   and x :: "'c" assumes "Group.group G"   and "H \<subseteq> carrier G"   and "x \<in> carrier G" shows "x <#\<^bsub>G\<^esub> H \<subseteq> carrier G"
rows_and_cols_transpose: fixes M :: "'b mat" shows "rows M = cols M\<^sup>T"
assms_3: shows "r \<cdot> q \<le> q \<cdot> r"
dim_row_def: shows "dim_row \<equiv> (Rep_mat ---> id) fst"
set_eq: shows "{a [^] i |i. i \<in> UNIV} = ([^]) a ` {0..G.ord a - 1}"
measure_zero_top: fixes M :: "'b measure"   and A :: "'b set" assumes "emeasure M A = \<top>" shows "Sigma_Algebra.measure M A = 0"
assms: fixes x :: "'a" assumes "x \<in> A" shows "f x \<in> O[F](g)"
ideD_1: fixes a :: "'a \<times> 'a" assumes "VoV.ide a" shows "VoV.arr a"
adopt_node_children_subset: fixes h :: "(_) heap"   and owner_document :: "(_) document_ptr"   and node :: "(_) node_ptr"   and h' :: "(_) heap"   and ptr :: "(_) object_ptr"   and children :: "(_) node_ptr list"   and children' :: "(_) node_ptr list" assumes "h \<turnstile> adopt_node owner_document node \<rightarrow>\<^sub>h h'"   and "h \<turnstile> get_child_nodes ptr \<rightarrow>\<^sub>r children"   and "h' \<turnstile> get_child_nodes ptr \<rightarrow>\<^sub>r children'"   and "DocumentClass.known_ptrs h"   and "DocumentClass.type_wf h" shows "set children' \<subseteq> set children"
selE: fixes s :: "'s"   and x :: "'x"   and f :: "'x \<Rightarrow> 'r option"   and r :: "'r" assumes "invar s"   and "x \<in> \<alpha> s"   and "f x = Some r" obtains x :: "'x" and r :: "'r" where "sel s f = Some r" and "x \<in> \<alpha> s" and "f x = Some r"
Numb_not_Scons: fixes k :: "nat"   and M :: "('b, 'c) Old_Datatype.node set"   and N :: "('b, 'c) Old_Datatype.node set" shows "Numb k \<noteq> Scons M N"
par_not_col_strict: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p"   and P :: "'p" assumes "A B Par C D"   and "Col C D P"   and "\<not> Col A B P" shows "A B ParStrict C D"
nnf_cnf_form: fixes F :: "'b formula" assumes "is_nnf F" shows "is_nnf (cnf_form_of F)"
A: shows "A = prod_emb I M I (Pi\<^sub>E I X)"
distinct_35: fixes x31 :: "'b poly"   and x32 :: "'b poly"   and x81 :: "'b poly"   and x82 :: "nat"   and x83 :: "'b poly" shows "Add x31 x32 \<noteq> CN x81 x82 x83"
eq: shows "{..<x} = space (sigma UNIV (range atLeast)) - {x..}"
pair_by_index_inj1: fixes x1 :: "nat"   and y1 :: "nat"   and x2 :: "nat"   and y2 :: "nat" assumes "pair_by_index x1 y1 = pair_by_index x2 y2" shows "x1 = x2"
irreducibility_hom_axioms: shows "irreducibility_hom poly_lift"
unitary_def: fixes P :: "'a \<Rightarrow> real" shows "unitary P = (sound P \<and> bounded_by 1 P)"
mult_cancel_add_mset: fixes s :: "('a \<times> 'a) set"   and uu :: "'a"   and X :: "'a multiset"   and Y :: "'a multiset" assumes "trans s"   and "irrefl s" shows "((add_mset uu X, add_mset uu Y) \<in> mult s) = ((X, Y) \<in> mult s)"
subset_trans: fixes A :: "'g set"   and B :: "'g set"   and C :: "'g set" assumes "A \<subseteq> B"   and "B \<subseteq> C" shows "A \<subseteq> C"
unfold1_o_dtor1: shows "unfold1 (F1map id dtor1 dtor2) (F2map id dtor1 dtor2) \<circ> dtor1 = id"
extremum_unique: fixes a :: "'b" shows "(a \<le> \<bottom>) = (a = \<bottom>)"
same_upto_refl: fixes n :: "nat"   and j :: "nat"   and A :: "'b mat" shows "same_upto n j A A"
h0: fixes i :: "nat"   and j :: "nat" assumes "i < length subsets"   and "j < length signs" shows "map (\<lambda>index_list. map (z_R index_list) signs) subsets ! i ! j = z_R (subsets ! i) (signs ! j)"
lens_put_irr2: fixes X :: "'d \<Longrightarrow> 'e"   and Y :: "'f \<Longrightarrow> 'e"   and \<sigma> :: "'e"   and u :: "'d" assumes "X \<bowtie> Y" shows "get\<^bsub>Y\<^esub> (put\<^bsub>X\<^esub> \<sigma> u) = get\<^bsub>Y\<^esub> \<sigma>"
precond: shows "\<forall>A\<in>f -` {a} \<inter> F. finite A \<and> card A \<le> k"
surj_of_nat_mod_ring: fixes x :: "'b mod_ring" shows "\<exists>i<CARD('b). x = of_nat i"
impl_sem: fixes \<nu> :: "(real, 'a) vec \<times> (real, 'a) vec"   and I :: "('b, 'c, 'a) interp"   and A :: "('b, 'c, 'a) formula"   and B :: "('b, 'c, 'a) formula" shows "(\<nu> \<in> fml_sem I (A \<rightarrow> B)) = (\<nu> \<in> fml_sem I A \<longrightarrow> \<nu> \<in> fml_sem I B)"
labeled_red_inf_eq_red_inf: fixes \<iota> :: "('f \<times> 'l) inference"   and N :: "('f \<times> 'l) set" assumes "\<iota> \<in> Inf_FL" shows "(\<iota> \<in> Red_I_\<G> N) = (to_F \<iota> \<in> no_labels.Red_I_\<G> (fst ` N))"
fold_insert: fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and A :: "'a set"   and x :: "'a"   and z :: "'b" assumes "comp_fun_commute f"   and "finite A"   and "x \<notin> A" shows "Finite_Set.fold f z (insert x A) = f x (Finite_Set.fold f z A)"
nop_onto_vrange: shows "\<R>\<^sub>\<circ> vint_mult = \<int>\<^sub>\<circ>"
restrict_map_subset_eq: fixes m :: "'a \<Rightarrow> 'b option"   and R :: "'a set"   and m' :: "'a \<Rightarrow> 'b option"   and R' :: "'a set" assumes "m |` R = m'"   and "R' \<subseteq> R" shows "m |` R' = m' |` R'"
ospec: fixes A :: "'a option"   and P :: "'a \<Rightarrow> bool"   and x :: "'a" assumes "\<forall>x\<in>set_option A. P x"   and "A = Some x" shows "P x"
props_2: shows "chain_reparam_chain' (one_chain2 - ps2) subdiv"
word_mod_def: fixes a :: "'a word"   and b :: "'a word" shows "a mod b = word_of_int (uint a mod uint b)"
length_update: fixes l :: "('a \<times> 'b) list list"   and k :: "'a"   and v :: "'b" assumes "1 < length l" shows "1 < length (abs_update k v l)"
i_Exec_Comp_Stream_Init_def: shows "i_Exec_Comp_Stream_Init \<equiv> \<lambda>trans_fun input c n. f_Exec_Comp trans_fun (input \<Down> n) c"
monofun_cfun: fixes f :: "'c \<rightarrow> 'd"   and g :: "'c \<rightarrow> 'd"   and x :: "'c"   and y :: "'c" assumes "f \<sqsubseteq> g"   and "x \<sqsubseteq> y" shows "f\<cdot>x \<sqsubseteq> g\<cdot>y"
smallest_accepting_rank_properties_2: fixes \<Sigma> :: "'c set"   and \<delta> :: "'d \<Rightarrow> 'c \<Rightarrow> 'd"   and q\<^sub>0 :: "'d"   and w :: "nat \<Rightarrow> 'c"   and F :: "'d set"   and i :: "nat" assumes "mojmir \<Sigma> \<delta> q\<^sub>0 w F"   and "mojmir_def.smallest_accepting_rank \<Sigma> \<delta> q\<^sub>0 w F = Some i" shows "finite (mojmir_def.fail \<Sigma> \<delta> q\<^sub>0 w F)"
prems_4: shows "a \<in> read_only_reads \<O> (x # sb)"
intros_1: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and s :: "('e, 'f) xstate" shows "\<Gamma>\<turnstile>[],[] \<Down> s"
precedesTransitivity: fixes a :: "'b"   and b :: "'b"   and l :: "'b list"   and c :: "'b" assumes "precedes a b l"   and "precedes b c l" shows "precedes a c l"
h3: assumes "\<exists>(a', b', c')\<in>set eq. a' \<noteq> 0 \<and> - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and> (\<forall>(d, e, f)\<in>set eq. d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 + e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) + f = 0) \<and> (\<forall>(d, e, f)\<in>set les. d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 + e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) + f < 0)" shows "(\<exists>(a', b', c')\<in>set eq. (a' = 0 \<and> b' \<noteq> 0) \<and> (\<forall>(d, e, f)\<in>set eq. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f = 0) \<and> (\<forall>(d, e, f)\<in>set les. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f < 0) \<or> a' \<noteq> 0 \<and> - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and> ((\<forall>(d, e, f)\<in>set eq. d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 + e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) + f = 0) \<and> (\<forall>(d, e, f)\<in>set les. d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 + e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) + f < 0) \<or> (\<forall>(d, e, f)\<in>set eq. d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 + e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) + f = 0) \<and> (\<forall>(d, e, f)\<in>set les. d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 + e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) + f < 0))) \<or> (\<forall>(d, e, f)\<in>set eq. d = 0 \<and> e = 0 \<and> f = 0) \<and> (\<exists>x. \<forall>(a, b, c)\<in>set les. a * x\<^sup>2 + b * x + c < 0)"
satL_e: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ"   and l :: "('fsym, 'psym) lit" shows "U.satL \<xi> l = M.satL \<xi> l"
f_compat: fixes t :: "dB"   and t' :: "dB" assumes "t \<rightarrow>\<^sub>\<beta>\<^sup>* t'" shows "f t \<rightarrow>\<^sub>\<beta>\<^sup>* f t'"
indmhom_someTr1: fixes N :: "('g, 'b, 'h) Module_scheme"   and f :: "'a \<Rightarrow> 'g"   and m :: "'a" assumes "R module N"   and "f \<in> mHom R M N"   and "m \<in> carrier M" shows "f (SOME xa. xa \<in> m \<uplus>\<^bsub>M\<^esub> ker\<^bsub>M,N\<^esub> f) = f m"
WT_calleeI: fixes \<I> :: "('d, 'e) \<I>"   and callee :: "'d \<Rightarrow> ('e \<times> 'f) spmf" assumes "\<And>call ret s. \<lbrakk>call \<in> outs_\<I> \<I>; (ret, s) \<in> set_spmf (callee call)\<rbrakk> \<Longrightarrow> ret \<in> responses_\<I> \<I> call" shows "\<I> \<turnstile>c callee \<surd>"
assms_7: shows "finite X \<longrightarrow> c < card X"
emptyLeftNonSync: fixes A :: "'c event set"   and t :: "'c event list" shows "\<forall>s. s setinterleaves (([], t), A) \<longrightarrow> (\<forall>t1. t1 \<in> set t \<longrightarrow> t1 \<notin> A)"
infinite_descent0_measure: fixes V :: "'a \<Rightarrow> nat"   and P :: "'a \<Rightarrow> bool"   and x :: "'a" assumes "\<And>x. V x = 0 \<Longrightarrow> P x"   and "\<And>x. \<lbrakk>0 < V x; \<not> P x\<rbrakk> \<Longrightarrow> \<exists>y. V y < V x \<and> \<not> P y" shows "P x"
bind_maybe_simps_1: fixes f :: "'d \<rightarrow> 'c\<cdot>maybe" shows "\<bottom> \<bind> f = \<bottom>"
16: shows "r\<^sup>T \<le> r\<^sup>T * (v * v\<^sup>T \<sqinter> g)\<^sup>\<star> * ((1::'a) \<squnion> e)"
Rep_ty_inverse: fixes x :: "ty" shows "Abs_ty (Rep_ty x) = x"
IH: assumes "t \<in> \<Union> (trms\<^sub>e\<^sub>s\<^sub>t\<^sub>p ` set A)" shows "t \<in> trms\<^sub>s\<^sub>t (to_st A)"
closest_point_refl: fixes S :: "'a set"   and x :: "'a" assumes "closed S"   and "S \<noteq> {}" shows "(closest_point S x = x) = (x \<in> S)"
finite_extensional_prod_nat: fixes A :: "'b set"   and b :: "nat" assumes "finite A"   and "0 < b" shows "finite {d \<in> extensional A. prod d A = b}"
poly_add_is_polynomial: fixes K :: "'a set"   and p1 :: "'a list"   and p2 :: "'a list" assumes "subring K R"   and "set p1 \<subseteq> K"   and "set p2 \<subseteq> K" shows "polynomial K (poly_add p1 p2)"
halt_steps: fixes A :: "abc_inst list"   and lm :: "nat list"   and n :: "nat" shows "abc_steps_l (length A, lm) A n = (length A, lm)"
\<phi>_in_hom: fixes a :: "'c"   and b :: "'c" assumes "C.ide a"   and "C.ide b" shows "\<guillemotleft>\<phi> (a, b) : F a \<otimes>\<^sub>D F b \<rightarrow>\<^sub>D F (a \<otimes>\<^sub>C b)\<guillemotright>"
con3: fixes env :: "v sem_env"   and cn :: "(char list, char list) id0 option"   and es :: "exp0 list"   and s1 :: "'a state"   and s2 :: "'a state"   and err :: "v error_result" assumes "do_con_check (c env) cn (length es)"   and "Big_Step_Unclocked_Single.evaluate_list (Big_Step_Unclocked_Single.evaluate env) s1 (rev es) (s2, Rerr err)" shows "Big_Step_Unclocked_Single.evaluate env s1 (Con cn es) (s2, Rerr err)"
simps_3: fixes x :: "'g"   and xs :: "'g list"   and y :: "'g"   and ys :: "'g list" shows "shuffles (x # xs) (y # ys) = (#) x ` shuffles xs (y # ys) \<union> (#) y ` shuffles (x # xs) ys"
a_coset_join2: fixes x :: "'c"   and H :: "'c set" assumes "x \<in> carrier S"   and "subgroup H (add_monoid S)"   and "x \<in> H" shows "H +>\<^bsub>S\<^esub> x = H"
simps_5: shows "witness_acyclic_events (PeerC, 0) = 4"
1: shows "distinct (filter (\<lambda>ia. ia = i \<or> ia = j) [0..<length ls])"
assms_3: shows "proj2_abs p \<noteq> proj2_abs q"
append_Cons_nth_left: fixes i :: "nat"   and xs :: "'a list"   and u :: "'a"   and ys :: "'a list" assumes "i < length xs" shows "(xs @ u # ys) ! i = xs ! i"
Units_def: fixes G :: "('a, 'b) monoid_scheme" shows "Units G = {y \<in> carrier G. \<exists>x\<in>carrier G. x \<otimes>\<^bsub>G\<^esub> y = \<one>\<^bsub>G\<^esub> \<and> y \<otimes>\<^bsub>G\<^esub> x = \<one>\<^bsub>G\<^esub>}"
m_lcomm: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<in> carrier G"   and "y \<in> carrier G"   and "z \<in> carrier G" shows "x \<otimes>\<^bsub>G\<^esub> (y \<otimes>\<^bsub>G\<^esub> z) = y \<otimes>\<^bsub>G\<^esub> (x \<otimes>\<^bsub>G\<^esub> z)"
to_mregex_ok: fixes r :: "Formula.formula Regex.regex"   and mr :: "mregex"   and ys :: "Formula.formula list" assumes "to_mregex r = (mr, ys)" shows "set ys = atms r \<and> ok (length ys) mr"
foldl_conv_foldr: fixes f :: "'e \<Rightarrow> 'f \<Rightarrow> 'e"   and a :: "'e"   and xs :: "'f list" shows "foldl f a xs = foldr (\<lambda>x y. f y x) (rev xs) a"
hb_antisym: fixes x :: "'msgid \<times> 'op"   and y :: "'msgid \<times> 'op" assumes "hb x y"   and "hb y x" shows "False"
assms_2: shows "\<forall>\<^sub>F x in F. 0 \<le> g x"
WT_plus_oracleI: fixes \<I>l :: "('a, 'b) \<I>"   and left :: "'c \<Rightarrow> 'a \<Rightarrow> ('b \<times> 'c) spmf"   and s :: "'c"   and \<I>r :: "('d, 'e) \<I>"   and right :: "'c \<Rightarrow> 'd \<Rightarrow> ('e \<times> 'c) spmf" assumes "\<I>l \<turnstile>c left s \<surd>"   and "\<I>r \<turnstile>c right s \<surd>" shows "\<I>l \<oplus>\<^sub>\<I> \<I>r \<turnstile>c (left \<oplus>\<^sub>O right) s \<surd>"
simps_5: fixes x2 :: "'d"   and x1 :: "'d" shows "Abrupt x2 \<noteq> Normal x1"
assms_5: shows "m1 \<le>\<^sub>L A"
coeff_lowerPoly: fixes b :: "nat"   and i :: "nat"   and mp :: "'b mpoly"   and x :: "nat \<Rightarrow>\<^sub>0 nat" shows "MPoly_Type.coeff (lowerPoly b i mp) x = MPoly_Type.coeff mp (lowerPowers b i x)"
hn_refine_def: fixes \<Gamma> :: "assn"   and c :: "'a Heap"   and \<Gamma>' :: "assn"   and R :: "'b \<Rightarrow> 'a \<Rightarrow> assn"   and m :: "'b nres" shows "hn_refine \<Gamma> c \<Gamma>' R m \<equiv> nofail m \<longrightarrow> <\<Gamma>> c <\<lambda>r. \<Gamma>' * (\<exists>\<^sub>Ax. R x r * \<up> (RETURN x \<le> m))>\<^sub>t"
rank_token_squats: fixes x :: "nat"   and n :: "nat" assumes "token_squats x"   and "x \<le> n" shows "\<exists>i. rank x n = Some i"
lunit'_simps_5: fixes f :: "'a" assumes "C.ide f" shows "C.cod \<l>\<^sub>C\<^sup>-\<^sup>1[f] = trg\<^sub>C f \<star>\<^sub>C f"
measure_spmf_return_spmf: fixes x :: "'h" shows "measure_spmf (return_spmf x) = return (count_space UNIV) x"
proj_arcs_H_simps_4: shows "proj_arcs_H (rev_H uw) = rev_G uv"
tell_state_def: fixes w :: "'w"   and m :: "('s, 'm) stateT" shows "local.tell_state w m = StateT (\<lambda>s. tell w (run_state m s))"
routing_port_ranges_superseted: fixes a1 :: "char list"   and b1 :: "'b wordinterval"   and tbl :: "'b routing_rule list" assumes "(a1, b1) \<in> set (routing_port_ranges tbl wordinterval_UNIV)" shows "\<exists>b2. (a1, b2) \<in> set (routing_ipassmt_wi tbl) \<and> wordinterval_to_set b1 \<subseteq> wordinterval_to_set b2"
intercept_upf_dec_def: shows "intercept_upf_dec = (\<lambda>(L, D) (x, c, t). if (x, c, t) \<in> L \<or> length x \<noteq> prf_dlen then Generative_Probabilistic_Value.Done (None, L, D) else Generative_Probabilistic_Value.Pause (Inr (x @ c, t)) Generative_Probabilistic_Value.Done \<bind> (\<lambda>_. Generative_Probabilistic_Value.Done (None, L, insert (x, c, t) D)))"
hs_get_roots_locs_def: shows "hs_get_roots_locs = prefixed ''mark_loop_get_roots'' - {mark_loop_get_roots_init_type}"
digraph_axioms_def: fixes \<alpha> :: "V"   and \<CC> :: "V" shows "digraph_axioms \<alpha> \<CC> \<equiv> (vcard \<CC> = 4\<^sub>\<nat> \<and> \<D>\<^sub>\<circ> (\<CC>\<lparr>Dom\<rparr>) = \<CC>\<lparr>Arr\<rparr> \<and> \<R>\<^sub>\<circ> (\<CC>\<lparr>Dom\<rparr>) \<subseteq>\<^sub>\<circ> \<CC>\<lparr>Obj\<rparr>) \<and> (\<D>\<^sub>\<circ> (\<CC>\<lparr>Cod\<rparr>) = \<CC>\<lparr>Arr\<rparr> \<and> \<R>\<^sub>\<circ> (\<CC>\<lparr>Cod\<rparr>) \<subseteq>\<^sub>\<circ> \<CC>\<lparr>Obj\<rparr>) \<and> \<CC>\<lparr>Obj\<rparr> \<subseteq>\<^sub>\<circ> Vset \<alpha> \<and> (\<forall>A B. A \<subseteq>\<^sub>\<circ> \<CC>\<lparr>Obj\<rparr> \<longrightarrow> B \<subseteq>\<^sub>\<circ> \<CC>\<lparr>Obj\<rparr> \<longrightarrow> A \<in>\<^sub>\<circ> Vset \<alpha> \<longrightarrow> B \<in>\<^sub>\<circ> Vset \<alpha> \<longrightarrow> \<Union>\<^sub>\<circ> ((\<lambda>a\<in>\<^sub>\<circ>A. \<Union>\<^sub>\<circ> (VLambda B (Hom \<CC> a) `\<^sub>\<circ> B)) `\<^sub>\<circ> A) \<in>\<^sub>\<circ> Vset \<alpha>)"
walk_array_foldl': fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and xs :: "'c list"   and x :: "'b" shows "walk_array f (Array xs) x = foldl f x xs"
fmset_def: fixes G :: "('c, 'd) monoid_scheme"   and as :: "'c list" shows "fmset G as = mset (map (assocs G) as)"
a: shows "(c::'a::refinement_lattice) \<sqsubseteq> c // (i::'a::refinement_lattice) \<parallel> i"