hcju/isabelleps huggingface dataset:
put\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_ptr_in_heap: fixes shadow_root_ptr :: "(_) shadow_root_ptr"   and shadow_root :: "('o, 'm, 'n, 'v option) RShadowRoot_scheme"   and h :: "(_) heap"   and h' :: "(_) heap" assumes "put shadow_root_ptr shadow_root h = h'" shows "shadow_root_ptr |\<in>| shadow_root_ptr_kinds h'"
game_def: fixes \<A> :: "(unit, 'x + 'x \<times> 'hash, 'hash + unit) gpv" shows "game \<A> = key_gen \<bind> (\<lambda>key. exec_gpv (oracle key) \<A> (False, {}) \<bind> (\<lambda>(uu_, flag', L'). return_spmf flag'))"
lcp_eq: fixes u :: "'b list"   and v :: "'b list" assumes "\<^bold>|u\<^bold>| = \<^bold>|u \<and>\<^sub>p v\<^bold>|" shows "u = u \<and>\<^sub>p v"
vdomain_VLambda: fixes A :: "V"   and f :: "V \<Rightarrow> V" shows "\<D>\<^sub>\<circ> (VLambda A f) = A"
scalar_prod_right_zero: fixes v :: "'c vec"   and n :: "nat" assumes "v \<in> carrier_vec n" shows "v \<bullet> 0\<^sub>v n = (0::'c)"
symmetric_bibdI: assumes "\<b> = \<v>" shows "symmetric_bibd \<V> \<B> \<k> \<Lambda>"
ArrMap_def: shows "ArrMap = 1\<^sub>\<nat>"
simps_2: fixes var :: "nat"   and p :: "real mpoly"   and xs :: "real list" shows "convert_atom var (Eq p) xs = map_option EqUni (convert_poly var p xs)"
c2_2: fixes x :: "real" assumes "1 \<le> x" shows "sum_upto (\<lambda>n. - f n) x + g x \<le> c2 * h x"
pseudonatural_transformation_axioms: shows "pseudonatural_transformation (\<cdot>\<^sub>C) (\<star>\<^sub>C) \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C (\<cdot>\<^sub>D) (\<star>\<^sub>D) \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F F \<Phi>\<^sub>F F.map\<^sub>0 (\<lambda>f. \<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<cdot>\<^sub>D \<r>\<^sub>D[F f])"
cross3_simps_29: fixes a :: "'a"   and b :: "'a"   and n :: "nat" shows "(a * b) ^ n = a ^ n * b ^ n"
countable_Un: fixes A :: "'a set"   and B :: "'a set" assumes "countable A"   and "countable B" shows "countable (A \<union> B)"
paste_sub_Un: fixes P :: "('a \<times> 'b) set"   and Q :: "('a \<times> 'b) set" shows "P +< Q \<subseteq> P \<union> Q"
bind_empty: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> 'd measure" assumes "space M = {}" shows "M \<bind> f = count_space {}"
weakly_preferred_by_menI: fixes \<mu> :: "('m \<times> 'w) set"   and \<mu>' :: "('m \<times> 'w) set" assumes "\<And>m w. (m, w) \<in> \<mu> \<Longrightarrow> \<exists>w'. (m, w') \<in> \<mu>' \<and> w' \<in> m_weakly_prefers m \<mu>" shows "weakly_preferred_by_men \<mu> \<mu>'"
surjective: shows "additive.kernel.Class ` R = additive.kernel.Partition"
X: fixes s :: "'s" shows "{P s |P. P \<in> range (\<lambda>P s. 0)} = {0}"
FWI'_def: shows "FWI' = uncurry \<circ>\<circ>\<circ> FWI \<circ> curry"
domain_subset_valid_pub: shows "Domain Rel1 \<subseteq> valid_pub1"
colossless_restrict_gpvD: fixes gpv :: "('b, 'out, 'in) gpv" assumes "colossless_gpv \<I> (local.restrict_gpv gpv)"   and "\<I> \<turnstile>g gpv \<surd>" shows "colossless_gpv \<I> gpv"
fin_c: shows "\<forall>C\<in>Cs. finite C"
conv_isotone: fixes x :: "'a"   and y :: "'a" assumes "x \<le> y" shows "id x \<le> id y"
Macaulay_list_def: fixes ps :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) list" shows "pp_pm.punit.Macaulay_list ps = filter (\<lambda>p. p \<noteq> 0) (mat_to_polys (pp_pm.punit.Keys_to_list ps) (row_echelon (pp_pm.punit.Macaulay_mat ps)))"
iff_def: fixes P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool" shows "P \<longleftrightarrow> Q \<equiv> \<lambda>A. P A = Q A"
padic_set_memI': fixes f :: "nat \<Rightarrow> int"   and p :: "int" assumes "\<And>m. f m \<in> {0..<p ^ m}"   and "\<And>m n. m < n \<Longrightarrow> f n mod p ^ m = f m" shows "f \<in> padic_set p"
PREC: shows "\<forall>x y. h \<Turnstile> P x * F1 \<and>\<^sub>A P y * F2 \<longrightarrow> x = y"
hyps_3: assumes "s \<noteq> {}" shows "\<exists>a. is_closest v s x a"
degenerate: fixes x :: "'a" assumes "x \<in> A" shows "P (return_pmf x)"
mult_less_cancel_left1: fixes 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 times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and c :: "'b"   and b :: "'b" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less c (times c b) = ((less_eq zero c \<longrightarrow> less one b) \<and> (less_eq c zero \<longrightarrow> less b one))"
match_le: fixes test :: "nat \<Rightarrow> 'b \<Rightarrow> bool"   and r :: "'b regex"   and i :: "nat"   and j :: "nat" assumes "match test r i j" shows "i \<le> j"
axioms_3: fixes return :: "'a \<Rightarrow> 'b"   and bind :: "'b \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b"   and fail :: "'b"   and catch :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and get :: "('c \<Rightarrow> 'b) \<Rightarrow> 'b"   and put :: "'c \<Rightarrow> 'b \<Rightarrow> 'b" assumes "monad_catch_state return bind fail catch get put" shows "monad_catch_state_axioms catch get put"
dom_locals_assign_mono: fixes s :: "state"   and f :: "'a \<Rightarrow> state \<Rightarrow> state"   and n :: "'a" assumes "dom (locals (snd s)) \<subseteq> dom (locals (snd (f n s)))" shows "dom (locals (snd s)) \<subseteq> dom (locals (snd (assign f n s)))"
Seq_imp_eq_app: fixes s :: "hf"   and d :: "hf"   and x :: "hf"   and y :: "hf" assumes "Seq s d"   and "\<langle>x, y\<rangle> \<^bold>\<in> s" shows "app s x = y"
norm_triangle_mono: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and a :: "'a"   and r :: "real"   and b :: "'a"   and s :: "real" assumes "class.real_normed_vector minus dist norm plus zero uminus scaleR sgn uniformity open"   and "norm a \<le> r"   and "norm b \<le> s" shows "norm (plus a b) \<le> r + s"
free_vars_marg_dens2_cexpr: fixes vs :: "nat list"   and vs' :: "nat list"   and \<Gamma> :: "nat \<Rightarrow> pdf_type"   and \<delta> :: "cexpr"   and x :: "nat"   and y :: "nat" assumes "cdens_ctxt_invar vs vs' \<Gamma> \<delta>" shows "free_vars (marg_dens2_cexpr \<Gamma> vs x y \<delta>) \<subseteq> shift_var_set (set vs')"
Un_upper2: fixes B :: "'b set"   and A :: "'b set" shows "(B::'b set) \<subseteq> (A::'b set) \<union> B"
intro: fixes G :: "('e, 'f) monoid_scheme"   and A :: "'e set"   and B :: "'e set" assumes "Group.group G"   and "subgroup A G"   and "subgroup B G" shows "group_disjoint_sum G A B"
arccos_less_arccos: fixes x :: "real"   and y :: "real" assumes "- 1 \<le> x"   and "x < y"   and "y \<le> 1" shows "arccos y < arccos x"
steps'_append: fixes qs :: "'c list"   and as :: "(nat \<times> nat list) list"   and n :: "nat"   and s :: "'c list"   and q :: "'c"   and a :: "nat \<times> nat list" assumes "length qs = length as"   and "length qs = n" shows "steps' s (qs @ [q]) (as @ [a]) (Suc n) = step (steps' s qs as n) q a"
extend_set_Diff_distrib: fixes A :: "'a set"   and B :: "'a set" shows "extend_set h (A - B) = extend_set h A - extend_set h B"
concurrent_store_expunge_independent: fixes i :: "'a"   and e1 :: "'b"   and mo :: "'a"   and r :: "'a"   and e2 :: "'b"   and mo2 :: "'a"   and xs :: "('a \<times> ('a, 'b) operation) event list"   and j :: "nat" assumes "\<not> hb (i, Store e1 mo i) (r, Expunge e2 mo2 r)"   and "\<not> hb (r, Expunge e2 mo2 r) (i, Store e1 mo i)"   and "xs prefix of j"   and "(i, Store e1 mo i) \<in> set (node_deliver_messages xs)"   and "(r, Expunge e2 mo2 r) \<in> set (node_deliver_messages xs)" shows "i \<noteq> mo2 \<and> r \<noteq> mo"
keys_phis'_codem': fixes g :: "'g"   and nodes_of_phis :: "('val, ('node \<times> 'val) set) mapping"   and "next" :: "'node \<times> 'val"   and next' :: "'val" assumes "g \<turnstile> nodes_of_phis \<approx>\<^sub>\<phi> uninst_code.ssa.phiNodes_of phis g"   and "next \<in> Mapping.keys (phis g)" shows "Mapping.keys (phis'_codem g next next' nodes_of_phis) = Mapping.keys (phis g) - {next}"
n: fixes j :: "nat" assumes "n < j" shows "b j = (0::'a)"
\<Delta>_def: fixes x :: "'a" shows "\<Delta> x \<equiv> l2 x - l1 x"
q_def: shows "q = idx_pm_of_pm (take j fs) r0"
assms: shows "x \<in> {- exp (- 1)..<0}"
simps_1: fixes eff :: "'b \<Rightarrow> 'a \<Rightarrow> 'a fset \<Rightarrow> bool"   and rs :: "'b stream"   and s :: "'a" shows "tree.root (RuleSystem_Defs.mkTree eff rs s) = (s, shd (RuleSystem_Defs.trim eff rs s))"
vsingleton_insert_inj_eq': fixes x :: "V"   and A :: "V"   and y :: "V" shows "(vinsert x A = set {y}) = (x = y \<and> A \<subseteq>\<^sub>\<circ> set {y})"
Sec1_deriv: fixes p :: "L"   and G :: "Var \<Rightarrow> L"   and c :: "IMP"   and H :: "Var \<Rightarrow> L"   and X :: "((Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool) set" assumes "(p, G, c, H) \<in> HS" shows "X \<rhd> c : Q p H"
node_deliver_messages_distinct: fixes xs :: "('a \<times> operation) event list"   and i :: "nat" assumes "xs prefix of i" shows "distinct (node_deliver_messages xs)"
subst_atm_mset_lists_empty_iff: fixes AAs :: "'a multiset list"   and \<eta>s :: "'s list" shows "(AAs \<cdot>\<cdot>aml \<eta>s = []) = (AAs = [] \<or> \<eta>s = [])"
rangeB'_def: shows "rangeB' = range ((*\<^sub>V) B)"
set_trie_delete: fixes xs :: "bool list"   and t :: "trie" shows "set_trie (delete xs t) = set_trie t - {xs}"
E_def: fixes M :: "real pmf" shows "E M = measure_pmf.expectation M (\<lambda>x. x)"
assms_2: shows "filterlim g (at (0::'a)) F"
not_gr_zero: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and n :: "'b" assumes "class.canonically_ordered_monoid_add plus zero less_eq less" shows "(\<not> less zero n) = (n = zero)"
from_nat_suc: fixes times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and uminus :: "'d \<Rightarrow> 'd"   and one :: "'d"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and Rep :: "'d \<Rightarrow> int"   and Abs :: "int \<Rightarrow> 'd"   and j :: "nat" assumes "class.mod_type times minus plus zero uminus one less_eq less Rep Abs" shows "mod_type.from_nat Abs (j + 1) = plus (mod_type.from_nat Abs j) one"
Boole: fixes p :: "('a, 'b) fm"   and ps :: "('a, 'b) fm list" assumes "(\<^bold>\<not> p) # ps \<turnstile> \<^bold>\<bottom>" shows "ps \<turnstile> p"
InvariantWatchCharacterizationInBackjumpPrefix: fixes state :: "State" assumes "InvariantWatchCharacterization (getF state) (getWatch1 state) (getWatch2 state) (getM state)" shows "let l = getCl state; level = getBackjumpLevel state; prefix = prefixToLevel level (getM state); state' = state\<lparr>getConflictFlag := False, getQ := [], getM := prefix\<rparr> in InvariantWatchCharacterization (getF state') (getWatch1 state') (getWatch2 state') (getM state')"
sel_1: fixes x :: "gauss_int"   and y :: "gauss_int" shows "ReZ (x * y) = ReZ x * ReZ y - ImZ x * ImZ y"
not_fst_snd_pref: shows "\<not> u\<^sub>0 \<cdot> u\<^sub>1 \<le>p \<alpha>"
nn_integral_has_integral_lebesgue: fixes \<Omega> :: "'a set"   and f :: "'a \<Rightarrow> real"   and I :: "real" assumes "\<And>x. x \<in> \<Omega> \<Longrightarrow> 0 \<le> f x"   and "(f has_integral I) \<Omega>" shows "\<integral>\<^sup>+ x. ennreal (indicat_real \<Omega> x * f x) \<partial>lborel = ennreal I"
gmw_asym_axioms: shows "gmw_asym S1_14_OT R1_14_OT S2_14_OT R2_14_OT protocol_14_OT adv_14_OT"
simps_1: fixes c :: "'b" shows "isnpolyh (c)\<^sub>p = (\<lambda>k. True)"
weakStatImpTransitive: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Rel :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set"   and Q :: "('a, 'b, 'c) psi"   and R :: "('a, 'b, 'c) psi"   and Rel' :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set"   and Rel'' :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set" assumes "\<Psi> \<rhd> P \<lessapprox><Rel> Q"   and "(\<Psi>, Q, R) \<in> Rel'"   and "{(\<Psi>', S, U) |\<Psi>' S U. \<exists>T. (\<Psi>', S, T) \<in> Rel \<and> (\<Psi>', T, U) \<in> Rel'} \<subseteq> Rel''"   and "\<And>\<Psi>' S T. (\<Psi>', S, T) \<in> Rel' \<Longrightarrow> \<Psi>' \<rhd> S \<lessapprox><Rel'> T"   and "\<And>\<Psi>' S T S'. \<lbrakk>(\<Psi>', S, T) \<in> Rel'; \<Psi>' \<rhd> S \<Longrightarrow>\<^sup>^\<^sub>\<tau> S'\<rbrakk> \<Longrightarrow> \<exists>T'. \<Psi>' \<rhd> T \<Longrightarrow>\<^sup>^\<^sub>\<tau> T' \<and> (\<Psi>', S', T') \<in> Rel'" shows "\<Psi> \<rhd> P \<lessapprox><Rel''> R"
minus_equality: fixes G :: "('b, 'c) ring_scheme"   and y :: "'b"   and x :: "'b" assumes "abelian_group G"   and "y \<oplus>\<^bsub>G\<^esub> x = \<zero>\<^bsub>G\<^esub>"   and "x \<in> carrier G"   and "y \<in> carrier G" shows "\<ominus>\<^bsub>G\<^esub> x = y"
blocking_on_def: fixes Xd :: "'d \<Rightarrow> 'e"   and Xh :: "'d \<Rightarrow> 'f"   and Pd :: "'e \<Rightarrow> ('d \<times> 'd) set"   and Ch :: "'f \<Rightarrow> 'd set \<Rightarrow> 'd set"   and ds :: "'e set"   and X :: "'d set"   and h :: "'f"   and X' :: "'d set" assumes "Contracts Xd Xh Pd Ch" shows "Contracts.blocking_on Pd Ch ds X h X' = (X' \<noteq> Ch h X \<and> X' = Ch h (X \<union> X') \<and> X' \<subseteq> Contracts.CD_on Pd ds (X \<union> X'))"
bnd: shows "real_of_int \<bar>number\<bar> \<le> real (N ^ (2 * m) * 2 ^ m * m)"
intro: fixes Xd :: "'d \<Rightarrow> 'e"   and Xh :: "'d \<Rightarrow> 'f"   and Pd :: "'e \<Rightarrow> ('d \<times> 'd) set"   and Ch :: "'f \<Rightarrow> 'd set \<Rightarrow> 'd set" assumes "Contracts Xd Xh Pd Ch"   and "ContractsWithLAD_axioms Ch" shows "ContractsWithLAD Xd Xh Pd Ch"
3: shows "coreflexive ((w \<sqinter> - prim_EP w v e) * e\<^sup>T)"
Main_no_call_target: fixes a :: "'edge"   and Q :: "('var \<Rightarrow> 'val option) \<times> 'ret \<Rightarrow> bool"   and r :: "'ret"   and f :: "(('var \<Rightarrow> 'val option) \<Rightarrow> 'val option) list" assumes "valid_edge a"   and "kind a = Q:r\<hookrightarrow>\<^bsub>Main\<^esub>f" shows "False"
remove_shadow_root_locs_def: shows "remove_shadow_root_locs = (\<lambda>element_ptr shadow_root_ptr. set_shadow_root_locs element_ptr \<union> {delete\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M shadow_root_ptr})"
preprocess_unsat: fixes cs :: "('i \<times> 'a ns_constraint) list"   and t :: "(nat \<times> linear_poly) list"   and as :: "('i \<times> 'a atom) list"   and trans_v :: "(nat, 'a) mapping \<Rightarrow> (nat, 'a) mapping"   and U :: "'i list"   and I :: "'i set"   and v :: "nat \<Rightarrow> 'a" assumes "preprocess cs = (t, as, trans_v, U)"   and "(I, v) \<Turnstile>\<^sub>i\<^sub>n\<^sub>s\<^sub>s  set cs" shows "\<exists>v'. (I, v') \<Turnstile>\<^sub>i\<^sub>a\<^sub>s set as \<and> v' \<Turnstile>\<^sub>t t"
get_elements_by_class_name_def: shows "get_elements_by_class_name = (\<lambda>ptr class_name. Heap_Error_Monad.bind (to_tree_order ptr) (map_filter_M (\<lambda>ptr. case cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr of None \<Rightarrow> return None | Some element_ptr \<Rightarrow> Heap_Error_Monad.bind (get_attribute element_ptr ''class'') (\<lambda>class_name_opt. if class_name_opt = Some class_name then return (Some element_ptr) else return None))))"
mirror_elem_inj_on: fixes I :: "nat set" assumes "finite I" shows "inj_on (\<lambda>x. mirror_elem x I) I"
inf_sup_aci_5: fixes x :: "'a"   and y :: "'a" shows "x \<sqinter> y = y \<sqinter> x"
distinct_conv_nth: fixes xs :: "'g list" shows "distinct xs = (\<forall>i<length xs. \<forall>j<length xs. i \<noteq> j \<longrightarrow> xs ! i \<noteq> xs ! j)"
gpv_stop_def: shows "gpv_stop \<equiv> corec_gpv (\<lambda>gpv. map_spmf (map_generat Some id (\<lambda>uu input. case input of None \<Rightarrow> Inl (Generative_Probabilistic_Value.Done None) | Some input' \<Rightarrow> Inr (uu input'))) (the_gpv gpv))"
adm_PE_base_lf_rep: fixes r :: "(nat \<Rightarrow> ValD) admS dual \<times> (nat \<Rightarrow> ValD) admS" shows "adm (\<lambda>x. x \<in> PE_base_lf_rep r)"
mult_imp_div_pos_less: fixes y :: "'a"   and x :: "'a"   and z :: "'a" assumes "(0::'a) < y"   and "x < z * y" shows "x / y < z"
closed_compact_sums: fixes S :: "'a set"   and T :: "'a set" assumes "closed S"   and "compact T" shows "closed (\<Union>x\<in>S. \<Union>y\<in>T. {x + y})"
random_dictatorship_unique_favorites: fixes R :: "'agent \<Rightarrow> 'alt \<Rightarrow> 'alt \<Rightarrow> bool" assumes "is_pref_profile R"   and "has_unique_favorites R" shows "RD R = map_pmf (favorite R) (pmf_of_set agents)"
cont_true_cont_2: fixes v :: "i"   and p :: "\<o>" shows "[ContingentlyFalse p \<^bold>\<rightarrow> Contingent p in v]"
UN_least: fixes A :: "'f set"   and B :: "'f \<Rightarrow> 'g set"   and C :: "'g set" assumes "\<And>x. x \<in> A \<Longrightarrow> B x \<subseteq> C" shows "\<Union> (B ` A) \<subseteq> C"
secure_refinements: fixes i :: "nat" assumes "i < length cms" shows "secure_refinement (\<R>\<^sub>A_rel (cms ! i)) (\<R>_rel (cms ! i)) (P_rel (cms ! i))"
add_closed: fixes a :: "'a"   and b :: "'a" assumes "a \<in> S"   and "b \<in> S" shows "a \<oplus>\<^sub>o\<^sub>w b \<in> S"
primitive_remainder_sequence_def: fixes F :: "nat \<Rightarrow> 'b poly"   and n :: "nat \<Rightarrow> nat"   and \<delta> :: "nat \<Rightarrow> nat"   and f :: "nat \<Rightarrow> 'b"   and k :: "nat"   and \<beta> :: "nat \<Rightarrow> 'b" shows "primitive_remainder_sequence F n \<delta> f k \<beta> \<equiv> (((\<forall>i. f i = lead_coeff (F i)) \<and> (\<forall>i. n i = degree (F i))) \<and> (\<forall>i. \<delta> i = n i - n (Suc i)) \<and> n 2 \<le> n 1) \<and> (F 1 \<noteq> 0 \<and> F 2 \<noteq> 0) \<and> (\<forall>i. i \<noteq> 0 \<longrightarrow> (F i = 0) = (k < i)) \<and> (\<forall>i. \<beta> i \<noteq> (0::'b)) \<and> (\<forall>i\<ge>3. i \<le> Suc k \<longrightarrow> Polynomial.smult (\<beta> i) (F i) = pseudo_mod (F (i - 2)) (F (i - 1)))"
subint1_def: fixes G :: "prog" shows "subint1 G = {(I, J). ? i:iface G I: J \<in> set (isuperIfs i)}"
in_set_conv_decomp_first: fixes x :: "'a"   and xs :: "'a list" shows "(x \<in> set xs) = (\<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set ys)"
lcs_eq: fixes H :: "'a set"   and a :: "'a"   and b :: "'a" assumes "G \<guillemotright> H "   and "a \<in> carrier G"   and "b \<in> carrier G" shows "(\<rho> a \<cdot> b \<in> H) = (a \<diamondsuit> H = b \<diamondsuit> H)"
tick_count_strict_def: fixes r :: "'b run"   and c :: "clock"   and n :: "nat" shows "tick_count_strict r c n = card {i. i < n \<and> hamlet (Rep_run r i c)}"
dvdI: fixes a :: "'a"   and b :: "'a"   and k :: "'a" assumes "a = b * k" shows "b dvd a"
trapped_sol_left: fixes K :: "'a set"   and x :: "'a" assumes "compact K"   and "K \<subseteq> X"   and "x \<in> X"   and "trapped_backward x K" shows "{..0} \<subseteq> existence_ivl0 x"
guha_matched_rule_inversion: fixes \<gamma> :: "'c set \<Rightarrow> 'd \<Rightarrow> bool"   and ft :: "('c, 'e) flow_entry_match list"   and p :: "'d"   and a :: "'e" assumes "guha_table_semantics \<gamma> ft p (Some a)" shows "\<exists>fe\<in>set ft. a = ofe_action fe \<and> \<gamma> (ofe_fields fe) p \<and> (\<forall>fe'\<in>set ft. ofe_prio fe < ofe_prio fe' \<longrightarrow> \<not> \<gamma> (ofe_fields fe') p)"
dom_map_option_comp: fixes g :: "'c \<Rightarrow> 'b"   and m :: "'a \<Rightarrow> 'c option" shows "dom (map_option g \<circ> m) = dom m"
simps_2: fixes e :: "'c match_expr" shows "MatchAnd_foldr [e] = e"
N1: shows "n1 = (replicate (length (fst M)) False)[q := True]"
inf_sup_aci_5: fixes inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "class.lattice inf less_eq less sup" shows "sup x y = sup y x"
span_span: fixes A :: "('t \<Rightarrow>\<^sub>0 'c) set" shows "pmdl (pmdl A) = pmdl A"
M_3: shows "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> f x \<le> f M"
cases: fixes s :: "('a, 'b) locks \<times> (('b \<Rightarrow> ('x2 \<times> 'a \<Rightarrow>f nat) option) \<times> 'c) \<times> ('b \<Rightarrow> 'd wait_set_status option) \<times> 'b set"   and t :: "'b" assumes "r2.not_final_thread s t" obtains x :: "'x2" and ln :: "'a \<Rightarrow>f nat" where "thr s t = \<lfloor>(x__, ln__)\<rfloor>" and "\<not> final2 x__"   | x :: "'x2" and ln :: "'a \<Rightarrow>f nat" where "thr s t = \<lfloor>(xa__, lna__)\<rfloor>" and "lna__ \<noteq> no_wait_locks"   | x :: "'x2" and w :: "'d wait_set_status" and ln :: "'a \<Rightarrow>f nat" where "thr s t = \<lfloor>(x, ln)\<rfloor>" and "wset s t = \<lfloor>w\<rfloor>"
hd_interact: fixes xs :: "'a list list"   and ys :: "'a list list" assumes "xs \<noteq> []"   and "hd xs \<noteq> []" shows "hd (interact xs ys) = hd (hd xs)"
3: shows "card (pseudoAllocation aa) = card G \<and> card (pseudoAllocation a) = card G"
partGen_def: fixes P :: "'c set set" shows "partGen P \<equiv> {gen P I |I. I \<in> P}"
ide_compE: fixes g :: "'b"   and f :: "'b" assumes "\<phi>.B.ide (g \<cdot>\<^sub>B f)" obtains "\<phi>.B.seq g f" and "\<phi>.B.seq f g" and "g \<cdot>\<^sub>B f = \<phi>.B.dom f" and "g \<cdot>\<^sub>B f = \<phi>.B.cod g"
lmap_eq_LNil: fixes f :: "'b \<Rightarrow> 'a"   and xs :: "'b llist" shows "(lmap f xs = []\<^sub>l) = (xs = []\<^sub>l)"
f1_def: fixes f :: "'sf"   and x :: "'sz" shows "f1 f x = $f f (singleton (Var x))"
lproject_llength: fixes A :: "'b set"   and w :: "'b llist" shows "llength (lproject A w) = esize (liset A w)"
dim_row: fixes n :: "nat"   and A :: "complex Matrix.mat" assumes "gate n A" shows "dim_row A = 2 ^ n"
ir_If': fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and b :: "bexp"   and b' :: "bexp"   and c\<^sub>1 :: "com"   and c\<^sub>1' :: "com"   and Q\<^sub>1 :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and c\<^sub>2 :: "com"   and c\<^sub>2' :: "com"   and Q\<^sub>2 :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool" assumes "ir_hoare (\<lambda>s s'. P s s' \<and> bval b s \<and> bval b' s') c\<^sub>1 c\<^sub>1' Q\<^sub>1"   and "ir_hoare (\<lambda>s s'. P s s' \<and> \<not> bval b s \<and> \<not> bval b' s') c\<^sub>2 c\<^sub>2' Q\<^sub>2" shows "ir_hoare P (IF b THEN c\<^sub>1 ELSE c\<^sub>2) (IF b' THEN c\<^sub>1' ELSE c\<^sub>2') (\<lambda>t t'. Q\<^sub>1 t t' \<or> Q\<^sub>2 t t')"
measure_space_closed: fixes \<Omega> :: "'c set"   and M :: "'c set set"   and \<mu> :: "'c set \<Rightarrow> ennreal" assumes "measure_space \<Omega> M \<mu>" shows "M \<subseteq> Pow \<Omega>"
lebesgue_measurable_imp_measurable_on_real: fixes u :: "'a \<Rightarrow> real"   and S :: "'a set" assumes "u \<in> borel_measurable lebesgue"   and "S \<in> sets lebesgue" shows "u measurable_on S"
midpoint_bet: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and B :: "'a"   and A :: "'a"   and C :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Midpoint Bet Cong B A C" shows "Bet A B C"
nth_mem: fixes n :: "nat"   and xs :: "'f list" assumes "n < length xs" shows "xs ! n \<in> set xs"