hcju/isabelleps huggingface dataset:
module_criteria: fixes R :: "('h, 'i) ring_scheme"   and M :: "('h, 'j, 'k) module_scheme" assumes "cring R"   and "\<zero>\<^bsub>M\<^esub> \<in> carrier M"   and "\<forall>v w. v \<in> carrier M \<and> w \<in> carrier M \<longrightarrow> v \<oplus>\<^bsub>M\<^esub> w \<in> carrier M"   and "\<forall>v\<in>carrier M. \<exists>neg_v\<in>carrier M. v \<oplus>\<^bsub>M\<^esub> neg_v = \<zero>\<^bsub>M\<^esub>"   and "\<forall>c v. c \<in> carrier R \<and> v \<in> carrier M \<longrightarrow> c \<odot>\<^bsub>M\<^esub> v \<in> carrier M"   and "\<forall>v w. v \<in> carrier M \<and> w \<in> carrier M \<longrightarrow> v \<oplus>\<^bsub>M\<^esub> w = w \<oplus>\<^bsub>M\<^esub> v"   and "\<forall>v w x. v \<in> carrier M \<and> w \<in> carrier M \<and> x \<in> carrier M \<longrightarrow> v \<oplus>\<^bsub>M\<^esub> w \<oplus>\<^bsub>M\<^esub> x = v \<oplus>\<^bsub>M\<^esub> (w \<oplus>\<^bsub>M\<^esub> x)"   and "\<forall>v\<in>carrier M. v \<oplus>\<^bsub>M\<^esub> \<zero>\<^bsub>M\<^esub> = v"   and "\<forall>a b v. a \<in> carrier R \<and> b \<in> carrier R \<and> v \<in> carrier M \<longrightarrow> a \<otimes>\<^bsub>R\<^esub> b \<odot>\<^bsub>M\<^esub> v = a \<odot>\<^bsub>M\<^esub> (b \<odot>\<^bsub>M\<^esub> v)"   and "\<forall>v\<in>carrier M. \<one>\<^bsub>R\<^esub> \<odot>\<^bsub>M\<^esub> v = v"   and "\<forall>a b v. a \<in> carrier R \<and> b \<in> carrier R \<and> v \<in> carrier M \<longrightarrow> (a \<oplus>\<^bsub>R\<^esub> b) \<odot>\<^bsub>M\<^esub> v = a \<odot>\<^bsub>M\<^esub> v \<oplus>\<^bsub>M\<^esub> b \<odot>\<^bsub>M\<^esub> v"   and "\<forall>a v w. a \<in> carrier R \<and> v \<in> carrier M \<and> w \<in> carrier M \<longrightarrow> a \<odot>\<^bsub>M\<^esub> (v \<oplus>\<^bsub>M\<^esub> w) = a \<odot>\<^bsub>M\<^esub> v \<oplus>\<^bsub>M\<^esub> a \<odot>\<^bsub>M\<^esub> w" shows "module R M"
simps_18: fixes x1 :: "'a"   and x2 :: "'b"   and y1 :: "'a"   and y2 :: "'b" shows "((x1, x2) = (y1, y2)) = (x1 = y1 \<and> x2 = y2)"
add_mset_eq_singleton_iff: fixes x :: "'b"   and M :: "'b multiset"   and y :: "'b" shows "(x, M = y, {#}) = (M = {#} \<and> x = y)"
not_eq_order_implies_strict: fixes a :: "('a, 'b) Term.term"   and b :: "('a, 'b) Term.term" assumes "a \<noteq> b"   and "a \<sqsubseteq> b" shows "a \<sqsubset> b"
semiring_1_ow_def: fixes U :: "'c set"   and one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c" shows "semiring_1_ow U one times plus zero \<equiv> zero_neq_one_ow U one zero \<and> semiring_0_ow U plus zero times \<and> monoid_mult_ow U one times"
thmD6_Left: fixes x :: "item"   and \<delta> :: "symbol list"   and \<omega> :: "symbol list"   and k :: "nat"   and D :: "(nat \<times> symbol \<times> symbol list) list" assumes "wellformed_item x"   and "item_\<beta> x = \<delta> @ \<omega>"   and "item_end x = k"   and "LeftDerivation \<delta> D []" shows "inc_dot (length \<delta>) x \<in> \<pi> k {} {x}"
VecHom_GMap_on_fbasis_is_FGModuleHom: fixes G :: "'a set"   and smult :: "('b, 'a) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and V :: "'c set"   and vs :: "'c list"   and smult' :: "('b, 'a) aezfun \<Rightarrow> 'd \<Rightarrow> 'd"   and T :: "'c \<Rightarrow> 'd"   and W :: "'d set" assumes "FGModule G smult V"   and "set vs \<subseteq> V \<and> V = R_scalar_mult.RSpan UNIV (aezfun_scalar_mult.fsmult smult) vs \<and> R_scalar_mult.R_lin_independent UNIV (aezfun_scalar_mult.fsmult smult) vs"   and "VectorSpaceHom (aezfun_scalar_mult.fsmult smult) V (aezfun_scalar_mult.fsmult smult') T"   and "FGModule G smult' W"   and "T ` V \<subseteq> W"   and "\<And>g v. \<lbrakk>g \<in> G; v \<in> set vs\<rbrakk> \<Longrightarrow> T (aezfun_scalar_mult.Gmult smult g v) = aezfun_scalar_mult.Gmult smult' g (T v)" shows "FGModuleHom G smult V smult' T"
SeqSkip: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and c\<^sub>2 :: "('b, 'a, 'c) com"   and s :: "('b, 'c) xstate" shows "\<Gamma>\<turnstile> (Seq Skip c\<^sub>2, s) \<rightarrow> (c\<^sub>2, s)"
fv\<^sub>s\<^sub>s\<^sub>t_unlabel_Cons: fixes l :: "'c"   and b :: "('b, 'a) stateful_strand_step"   and A :: "('c \<times> ('b, 'a) stateful_strand_step) list" shows "fv\<^sub>l\<^sub>s\<^sub>s\<^sub>t ((l, b) # A) = fv\<^sub>s\<^sub>s\<^sub>t\<^sub>p b \<union> fv\<^sub>l\<^sub>s\<^sub>s\<^sub>t A"
induct: fixes x :: "'c set"   and sp :: "'c set"   and A :: "'c set set"   and P :: "'c set \<Rightarrow> bool" assumes "x \<in> sigma_sets sp A"   and "\<And>a. a \<in> A \<Longrightarrow> P a"   and "P {}"   and "\<And>a. \<lbrakk>a \<in> sigma_sets sp A; P a\<rbrakk> \<Longrightarrow> P (sp - a)"   and "\<And>a. \<lbrakk>\<And>i. a i \<in> sigma_sets sp A; \<And>i. P (a i)\<rbrakk> \<Longrightarrow> P (\<Union> (range a))" shows "P x"
rw2: shows "foldl Gauss_Jordan_column_k (0, A) [0..<Suc k] = (if \<forall>m. is_zero_row_upt_k m (Suc k) (Gauss_Jordan_upt_k A k) then 0 else mod_type_class.to_nat (GREATEST n. \<not> is_zero_row_upt_k n (Suc k) (Gauss_Jordan_upt_k A k)) + 1, Gauss_Jordan_upt_k A k)"
q: fixes x :: "'call"   and s :: "'s" assumes "consider' x" shows "interaction_bounded_by consider (callee s x) q"
filter_single_mset: fixes P :: "'a \<Rightarrow> bool"   and x :: "'a" shows "filter_mset P {#x#} = (if P x then {#x#} else {#})"
0: shows "{} \<turnstile> Robinson_Arithmetic.neg (zer EQ suc (Var xx))"
B_def: shows "B = {#[:x:]. x \<in># prime_factorization (content p)#}"
compare_expansions_GT_flip: fixes F :: "'a"   and G :: "'a"   and c :: "real \<times> real" assumes "length (snd (dominant_term F)) = length (snd (dominant_term G))" shows "(compare_expansions F G = (GT, c)) = (compare_expansions G F = (LT, c))"
Inf_eq_MInfty: fixes S :: "ereal set" assumes "- \<infinity> \<in> S" shows "Inf S = - \<infinity>"
RefVar_def: fixes vf :: "state \<Rightarrow> vvar \<times> state"   and P :: "'b assn" shows "(vf ..; P) = (\<lambda>Y s. let (v, s') = vf s in P (In2 v) s')"
distinct_3: fixes x11 :: "'a"   and x12 :: "bool"   and x31 :: "'a"   and x32 :: "'b" shows "<x11, inM x12> \<noteq> <x31, x32>"
ext_inject: fixes precondition_of :: "'c list"   and add_effects_of :: "'c list"   and delete_effects_of :: "'c list"   and more :: "'d"   and precondition_of' :: "'c list"   and add_effects_of' :: "'c list"   and delete_effects_of' :: "'c list"   and more' :: "'d" shows "(\<lparr>strips_operator.precondition_of = precondition_of, add_effects_of = add_effects_of, delete_effects_of = delete_effects_of, \<dots> = more\<rparr> = \<lparr>strips_operator.precondition_of = precondition_of', add_effects_of = add_effects_of', delete_effects_of = delete_effects_of', \<dots> = more'\<rparr>) = (precondition_of = precondition_of' \<and> add_effects_of = add_effects_of' \<and> delete_effects_of = delete_effects_of' \<and> more = more')"
is_extensional: fixes f :: "'d" assumes "\<not> D.arr f" shows "trg\<^sub>D f = D.null"
add_mono_thms_linordered_semiring_3: fixes i :: "'c"   and j :: "'c"   and k :: "'c"   and l :: "'c" assumes "i \<le> j \<and> k = l" shows "i + k \<le> j + l"
post_meas0_is_state: fixes n :: "nat"   and v :: "complex Matrix.mat"   and i :: "nat" assumes "state n v"   and "prob0 n v i \<noteq> 0" shows "state n (post_meas0 n v i)"
restrict_Pi_cancel: fixes x :: "'c \<Rightarrow> 'd"   and I :: "'c set"   and A :: "'c \<Rightarrow> 'd set" shows "(restrict x I \<in> Pi I A) = (x \<in> Pi I A)"
complete_if_size_min_height: fixes kdt :: "'b kdt" assumes "KD_Tree.size_kdt kdt = 2 ^ min_height kdt" shows "KD_Tree.complete kdt"
derivation_shift_left_right_cancel: fixes D :: "(nat \<times> symbol \<times> symbol list) list"   and r :: "nat" assumes "derivation_ge D r" shows "derivation_shift (derivation_shift D r 0) 0 r = D"
tc: fixes x :: "'a" assumes "x \<cdot> x \<le> x" shows "x \<cdot> x\<^sup>\<star> = x"
hv_diff: fixes x :: "(real, 7) vec"   and y :: "(real, 7) vec" shows "Hv (x - y) = Hv x - Hv y"
IH: assumes "valid_reqs M"   and "wf_graph \<lparr>nodes = V, edges = E\<rparr>"   and "E' \<subseteq> E"   and "\<Union> (get_offending_flows M \<lparr>nodes = V, edges = E\<rparr>) \<subseteq> X" shows "\<Union> (get_offending_flows M \<lparr>nodes = V, edges = E - E'\<rparr>) \<subseteq> X - E'"
\<phi>_def: shows "\<phi> \<equiv> k.trnl\<^sub>\<epsilon> (h \<star> p\<^sub>0) (\<a>[k\<^sup>*, h, p\<^sub>0] \<cdot> tab)"
Y_1: shows "Complete_Partial_Order.chain (ord_spmf (=)) Y"
fiter_seq_choice: fixes c :: "'a::refinement_lattice" shows "(c::'a::refinement_lattice)\<^sup>\<star> = \<Sqinter> range ((\<^sup>;^) c)"
MapsToMorDomCod_3: fixes f :: "'b"   and g :: "'b" assumes "f \<approx>> g" shows "cod (f ;; g) = cod g"
obj_is_weak_unit: fixes a :: "'a" assumes "obj a" shows "weak_unit a"
f5: shows "adj_mat B \<in> carrier_mat n n \<and> B * adj_mat B = det B \<cdot>\<^sub>m 1\<^sub>m n"
rim: shows "r ` V \<subseteq> S \<inter> T"
all_consts_def: shows "all_consts = fst |`| fset_of_list rs |\<union>| C"
trms_list\<^sub>s\<^sub>t_is_trms\<^sub>s\<^sub>t: fixes S :: "('a, 'b) strand_step list" shows "trms\<^sub>s\<^sub>t S = set (trms_list\<^sub>s\<^sub>t S)"
nempty_sl_in_state_set: fixes sl :: "'a list" assumes "sl \<noteq> []" shows "sl \<in> state_set sl"
total_comp: fixes x :: "'a"   and y :: "'a" assumes "is_total x"   and "is_total y" shows "is_total (x ; y)"
no_races_imp_no_write_overlap: fixes T :: "nat set"   and \<gamma> :: "nat \<Rightarrow> thread_state option" assumes "\<not> group_race T \<gamma>" shows "\<forall>i\<in>T. \<forall>j\<in>T. i \<noteq> j \<longrightarrow> W (the (\<gamma> i)) \<inter> W (the (\<gamma> j)) = {}"
\<W>_bounded_by_\<X>: fixes P :: "(symbol \<times> character list) list set"   and k :: "nat" shows "\<W> P k \<subseteq> \<X> k"
unit_simps_3: shows "cod \<sigma> = k \<star> h"
proj_plus: fixes pair_of_term :: "'a \<Rightarrow> 'b \<times> 'c"   and term_of_pair :: "'b \<times> 'c \<Rightarrow> 'a"   and k :: "'c"   and p :: "'a \<Rightarrow>\<^sub>0 'd"   and q :: "'a \<Rightarrow>\<^sub>0 'd" assumes "term_powerprod pair_of_term term_of_pair" shows "term_powerprod.proj_poly term_of_pair k (p + q) = term_powerprod.proj_poly term_of_pair k p + term_powerprod.proj_poly term_of_pair k q"
next_in_basics_2: fixes A :: "'b set"   and a :: "'b"   and b :: "'b"   and u :: "'b" assumes "finite A"   and "A \<subseteq> {a..b}"   and "a \<in> A"   and "b \<in> A"   and "a < b"   and "u \<in> {a..<b}" shows "u < next_in A u"
normalize_simps_5: fixes b :: "'d set"   and c\<^sub>1 :: "('d, 'e, 'f) com"   and c\<^sub>2 :: "('d, 'e, 'f) com" shows "Language.normalize (Cond b c\<^sub>1 c\<^sub>2) = Cond b (Language.normalize c\<^sub>1) (Language.normalize c\<^sub>2)"
simps_2: fixes uu :: "'b \<Rightarrow> bool" shows "(uu::'b \<Rightarrow> bool) \<Turnstile> \<bottom> = False"
cs_single_pd_intermed: fixes \<pi> :: "nat \<Rightarrow> 'n"   and n :: "nat"   and k :: "nat" assumes "is_path \<pi>"   and "cs\<^bsup>\<pi>\<^esup> n = [\<pi> n]"   and "k \<le> n" shows "\<pi> n pd\<rightarrow> \<pi> k"
is_nan_def: fixes a :: "('a, 'b) IEEE.float" shows "is_nan a = (IEEE.exponent a = emax TYPE(('a, 'b) IEEE.float) \<and> fraction a \<noteq> 0)"
1: shows "finite {k. (T ^^ k) ((T ^^ n) x) \<in> A}"
assms: shows "f \<in>\<^sub>\<circ> vproduct I A"
fmdrop_set_fmdom: fixes A :: "('c, 'd) fmap" shows "fmdrop_set (fmdom' A) A = fmempty"
g0: fixes x :: "real" assumes "0 < x"   and "x \<le> a" shows "(g has_real_derivative g' x) (at x)"
in3: shows "dim_row A \<le> dim_row (two_block_non_interfering A B) - n + i"
elims_2: fixes x :: "'e \<times> ('a, 'f, 'g, 'h) SLin_state_scheme" assumes "P16 x" obtains s1 :: "'e" and s2 :: "('a, 'f, 'g, 'h) SLin_state_scheme" where "x = (s1, s2)" and "dstate s2 \<noteq> \<bottom> \<longrightarrow> initialized s2"
urel64_plus: fixes x :: "uint64"   and y :: "int"   and x' :: "uint64"   and y' :: "int" assumes "urel64 x y"   and "urel64 x' y'" shows "urel64 (plus_p64 pp x x') (plus_p p y y')"
can_exec_nil: fixes root :: "(att \<times> char list, att, nat) env" shows "can_exec root []"
content_field_poly: fixes f :: "'a poly" shows "content f = (if f = 0 then 0::'a else (1::'a))"
intro: fixes ext :: "('b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b list \<Rightarrow> 'b list \<Rightarrow> bool" assumes "\<And>y x gt xs xs'. \<lbrakk>y \<noteq> x; gt y x\<rbrakk> \<Longrightarrow> ext gt (xs @ y # xs') (xs @ x # xs')" shows "ext_compat_list_axioms ext"
B: fixes x :: "'a" assumes "x \<in> S" shows "onorm (f' x) \<le> B"
negligible_cmultI: fixes c :: "real"   and f :: "nat \<Rightarrow> real" assumes "c \<noteq> 0 \<Longrightarrow> Negligible.negligible f" shows "Negligible.negligible (\<lambda>\<eta>. c * f \<eta>)"
is_character_data_ptr_kind_none: fixes node_ptr :: "(_) node_ptr" assumes "\<not> is_character_data_ptr_kind node_ptr" shows "cast node_ptr = None"
mkCone_def: fixes e :: "'c" shows "mkCone e \<equiv> \<lambda>j. if J.arr j then if j = J.Zero then e else f0 \<cdot> e else C.null"
bounded_Un: fixes S :: "'b set"   and T :: "'b set" shows "bounded (S \<union> T) = (bounded S \<and> bounded T)"
iterSTR_termFSwImorph_errMOD: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model" assumes "iwlsFSw MOD" shows "termFSwImorph (iterSTR (errMOD MOD)) (iterAbsSTR (errMOD MOD)) (errMOD MOD)"
ennreal_plus: fixes a :: "real"   and b :: "real" assumes "0 \<le> a"   and "0 \<le> b" shows "ennreal (a + b) = ennreal a + ennreal b"
fut_default_def: shows "fut_default = (\<psi>, sorted_list_of_set {\<langle>fold rderiv (replicate k (zero idx)) \<langle>\<psi>\<rangle>\<rangle> |k. True})"
uniqueness_of_interpolation_point_list: fixes xs_ys :: "('a \<times> 'a) list"   and p :: "'a poly"   and q :: "'a poly" assumes "distinct (map fst xs_ys)"   and "\<And>x y. (x, y) \<in> set xs_ys \<Longrightarrow> poly p x = y"   and "degree p < length xs_ys"   and "\<And>x y. (x, y) \<in> set xs_ys \<Longrightarrow> poly q x = y"   and "degree q < length xs_ys" shows "p = q"
HDomain_Incl_Subset_lemma: fixes r :: "tm"   and k :: "tm"   and k' :: "tm" shows "{HDomain_Incl r k, k' SUBS k} \<turnstile> HDomain_Incl r k'"
equivalence_pseudofunctor_def: fixes V\<^sub>C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and H\<^sub>C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<a>\<^sub>C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<i>\<^sub>C :: "'c \<Rightarrow> 'c"   and src\<^sub>C :: "'c \<Rightarrow> 'c"   and trg\<^sub>C :: "'c \<Rightarrow> 'c"   and V\<^sub>D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and H\<^sub>D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and \<a>\<^sub>D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and \<i>\<^sub>D :: "'d \<Rightarrow> 'd"   and src\<^sub>D :: "'d \<Rightarrow> 'd"   and trg\<^sub>D :: "'d \<Rightarrow> 'd"   and F :: "'c \<Rightarrow> 'd"   and \<Phi> :: "'c \<times> 'c \<Rightarrow> 'd" shows "equivalence_pseudofunctor V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi> \<equiv> pseudofunctor V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi> \<and> faithful_functor V\<^sub>C V\<^sub>D F \<and> equivalence_pseudofunctor_axioms V\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F"
indicator_simps_1: fixes x :: "'c"   and S :: "'c set" assumes "x \<in> S" shows "indicator S x = (1::'d)"
append_expunge_ids_imply_messages_same: fixes i :: "'a"   and mo :: "'a"   and xs :: "('a \<times> ('a, 'b) operation) event list"   and j :: "nat"   and e1 :: "'b"   and r :: "'a"   and e2 :: "'b" assumes "i = mo"   and "xs prefix of j"   and "(i, Append i e1) \<in> set (node_deliver_messages xs)"   and "(r, Expunge e2 mo r) \<in> set (node_deliver_messages xs)" shows "e1 = e2"
\<Phi>a_def: fixes S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and S' :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<AA> :: "('d \<times> 'd) set"   and \<phi> :: "'b \<Rightarrow> 'c" assumes "two_set_categories_bij_betw_Univ S S' \<AA> \<phi>" shows "two_set_categories_bij_betw_Univ.\<Phi>a S \<AA> \<phi> = (\<lambda>f. \<lambda>x'\<in>\<phi> ` set_category_data.set S (set_category.some_img S \<AA>) (partial_magma.dom S f). \<phi> (set_category_given_img.Fun S (set_category.some_img S \<AA>) f (inv_into (Collect (category.terminal S)) \<phi> x')))"
S2: shows "c_assoc_value ls (c_pair (c_snd (c_snd (c_fst key))) (c_snd key)) = univ_for_pr (c_pair (c_snd (c_snd (c_fst key))) (c_snd key))"
chain_singleton: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a" shows "chain R (LCons x LNil)"
triangle: fixes f :: "'d"   and g :: "'d" assumes "D.ide f"   and "D.ide g"   and "src\<^sub>D g = trg\<^sub>D f" shows "(g \<star>\<^sub>D \<l>\<^sub>D[f]) \<cdot>\<^sub>D \<a>\<^sub>D[g, src\<^sub>D g, f] = \<r>\<^sub>D[g] \<star>\<^sub>D f"
antisym: fixes b :: "nat list"   and a :: "nat list" assumes "b \<le>\<^sub>v a"   and "a \<le>\<^sub>v b" shows "a = b"
range_varsE: fixes x :: "'a"   and \<sigma> :: "'a \<Rightarrow> ('b, 'a) Term.term" assumes "x \<in> range_vars \<sigma>" obtains t :: "('b, 'a) Term.term" where "x \<in> fv t" and "t \<in> subst_range \<sigma>"
take_map: fixes n :: "nat"   and f :: "'e \<Rightarrow> 'd"   and xs :: "'e list" shows "take n (map f xs) = map f (take n xs)"
n_huntington2: fixes x :: "'a"   and y :: "'a" shows "!(!x + t y) + !(!x + !y) = t x"
assms: shows "R[a\<turnstile>c>b] = NotR x.N c"
strict_trans1: fixes b :: "'a word"   and a :: "'a word"   and c :: "'a word" assumes "b \<le>s a"   and "c <s b" shows "c <s a"
P_m: shows "homogeneous\<^sub>l\<^sub>s\<^sub>t m \<A> Sec \<and> m \<in> GSMP (trms\<^sub>l\<^sub>s\<^sub>t \<A>)"
AbstrLevels_S4opt_A32: fixes i :: "AbstrLevelsID" assumes "sS4opt \<in> AbstrLevel i" shows "sA32 \<notin> AbstrLevel i"
hensel_factor_id: shows "local.divide (f \<bullet> a) (f' \<bullet> a) \<otimes> (f' \<bullet> a) = f \<bullet> a"
subgroup_in_rcosets: fixes H :: "'c set"   and G :: "('c, 'd) monoid_scheme" assumes "subgroup H G"   and "Group.group G" shows "H \<in> rcosets\<^bsub>G\<^esub> H"
trivial_subset: fixes Y :: "'a set"   and X :: "'a set" assumes "trivial Y"   and "X \<subseteq> Y" shows "trivial X"
image_restrict_eq: fixes f :: "'b \<Rightarrow> 'a"   and A :: "'b set" shows "restrict f A ` A = f ` A"
r_i: fixes G :: "('e, 'f) Group_scheme"   and a :: "'e" assumes "Group G"   and "a \<in> carrier G" shows "a \<cdot>\<^bsub>G\<^esub> \<rho>\<^bsub>G\<^esub> a = \<one>\<^bsub>G\<^esub>"
rel_funD: fixes A :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and B :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'e"   and x :: "'b"   and y :: "'c" assumes "rel_fun A B f g"   and "A x y" shows "B (f x) (g y)"
between_implies_scaled_diff: fixes S :: "'a"   and T :: "'a"   and X :: "'a"   and Y :: "'a" assumes "between (S, T) X"   and "between (S, T) Y"   and "S \<noteq> Y" obtains c :: "real" where "X - Y = c *\<^sub>R (S - Y)"
Collect_case_prodD: fixes x :: "'a \<times> 'b"   and A :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "x \<in> {(x, y). A x y}" shows "A (fst x) (snd x)"
vector_3_2: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "vector [x, y, z] $ 2 = y"
path_image_reversepath: fixes g :: "real \<Rightarrow> 'b" shows "path_image (reversepath g) = path_image g"
arb_actI: fixes K :: "'a \<Rightarrow> 'a pmf set"   and s :: "'a" assumes "Markov_Decision_Process K" shows "Markov_Decision_Process.arb_act K s \<in> K s"
redT_updW_Some_otherD: fixes t' :: "'b"   and ws :: "'b \<Rightarrow> 'c wait_set_status option"   and wa :: "('b, 'c) wait_set_action"   and ws' :: "'b \<Rightarrow> 'c wait_set_status option"   and t :: "'b"   and w :: "'c wait_set_status" assumes "redT_updW t' ws wa ws'"   and "ws' t = \<lfloor>w\<rfloor>"   and "t \<noteq> t'" shows "case w of InWS w' \<Rightarrow> ws t = \<lfloor>InWS w'\<rfloor> | PostWS wake_up_status \<Rightarrow> ws t = \<lfloor>w\<rfloor> \<or> (\<exists>w'. ws t = \<lfloor>InWS w'\<rfloor>)"
2: fixes g :: "'a \<Rightarrow> 'b" assumes "sup_continuous g" shows "sup_continuous (\<lambda>s. f s (g s))"
submodule_whole: fixes M :: "('e, 'f, 'g) Module_scheme"   and R :: "('f, 'h) Ring_scheme" assumes "R module M" shows "submodule R M (carrier M)"
Extend_CONS: fixes \<AA> :: "'i"   and x :: "'x"   and k :: "'k"   and P :: "'v" assumes "#\<^sub>V \<AA> = size x" shows "Extend k 0 (CONS x \<AA>) P = CONS (extend k (if eval P 0 then True else False) x) (Extend k 0 \<AA> (downshift P))"
assms_1: shows "compile_env \<A> \<O> F1 = Ok F2"
intros: fixes xs :: "'d list" shows "cyclic1 xs (rotate1 xs)"
snoc_mid_Cons: fixes xs :: "'c list"   and x :: "'c"   and w :: "'a"   and ys :: "'a list"   and y :: "'a"   and z :: "'b"   and zs :: "'b list" assumes "as = xs @ [x]"   and "bs = w # ys @ [y]"   and "cs = z # zs" shows "P"
ereal_less_divide_iff: fixes c :: "ereal"   and a :: "ereal"   and b :: "ereal" assumes "0 < c"   and "c < \<infinity>" shows "(a < b / c) = (a * c < b)"
id_ring_hom: fixes R :: "('a, 'b) ring_scheme" shows "id \<in> ring_hom R R"
kernel_from_trivial_group: assumes "trivial_group G" shows "kernel G H h = carrier G"
assms_2: shows "ipresIGFreshAbs hA MOD MOD'"
start_remove_id: fixes x :: "'b"   and xs :: "'b list"   and ys :: "'b list" assumes "x \<notin> set xs" shows "start_remove xs x ys = rev ys @ xs"
subspace_def: fixes S :: "'c set" shows "ideal.subspace S = ((0::'c) \<in> S \<and> (\<forall>x\<in>S. \<forall>y\<in>S. x + y \<in> S) \<and> (\<forall>c. \<forall>x\<in>S. c * x \<in> S))"
rep_eq: shows "lookup 0 = (\<lambda>k. 0::'b)"
quotient_of_nonzero_2: fixes r :: "rat" shows "snd (quotient_of r) \<noteq> 0"
simps_24: fixes vc :: "bexp"   and vd :: "'b acom"   and ve :: "'b acom"   and vf :: "'b"   and v :: "char list"   and va :: "aexp"   and vb :: "'b" shows "(IF vc THEN vd ELSE ve {vf} \<sqsubseteq> v ::= va {vb}) = False"
goedel_numbering_phi: shows "goedel_numbering \<phi>"
paste_1: fixes V :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and H :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<nu> :: "'c"   and \<mu> :: "'c" assumes "weak_composition V H" shows "H \<nu> \<mu> = V (H (partial_magma.cod V \<nu>) \<mu>) (H \<nu> (partial_magma.dom V \<mu>))"
ds: shows "matrix_sum d (\<lambda>k. f (k + 1)) n \<in> carrier_mat d d"
inv_eq_imp_eq: fixes x :: "'a"   and y :: "'a" assumes "x \<in> Units G"   and "y \<in> Units G"   and "inv x = inv y" shows "x = y"
tag_name_eq2_h: fixes ptr' :: "(_) element_ptr" assumes "ptr' \<noteq> new_element_ptr" shows "|h \<turnstile> get_tag_name ptr'|\<^sub>r = |h2 \<turnstile> get_tag_name ptr'|\<^sub>r"
ag_inv_inv: fixes A :: "('e, 'f) aGroup_scheme"   and x :: "'e" assumes "aGroup A"   and "x \<in> carrier A" shows "-\<^sub>a\<^bsub>A\<^esub> (-\<^sub>a\<^bsub>A\<^esub> x) = x"
intro_elim_5: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>" assumes "[\<phi> \<^bold>\<rightarrow> \<psi> in v]"   and "[\<psi> \<^bold>\<rightarrow> \<phi> in v]" shows "[\<phi> \<^bold>\<equiv> \<psi> in v]"
isFaultE: fixes s :: "('e, 'f) xstate" assumes "isFault s" obtains f :: "'f" where "s = Fault f"
decl: shows "v \<in> statesDecls (states prog !! pState.idx p)"
S: fixes H :: "fm set"   and A :: "fm"   and B :: "fm"   and C :: "fm"   and H' :: "fm set" assumes "H \<turnstile> A IMP B IMP C"   and "H' \<turnstile> A IMP B" shows "H \<union> H' \<turnstile> A IMP C"
UNpart_cases: fixes II :: "nat set"   and cl :: "('test, 'atom, 'choice) cmd list"   and P :: "nat \<Rightarrow> nat set set" assumes "II \<in> UNpart cl P" obtains n :: "nat" and I :: "nat set" where "n < length cl" and "I \<in> P n" and "II = local.shift cl n I"
simps: fixes x :: "nat"   and l :: "cell list"   and r :: "cell list" shows "inv_end5_loop x (l, r) = (\<exists>i j. i + j = x \<and> 0 < x \<and> 0 < j \<and> l = Oc \<up> i @ [Bk] \<and> r = Oc \<up> j @ Bk # Oc \<up> x)"
min_gallery_in_subcomplex: fixes X :: "'b set set"   and Y :: "'b set set"   and Cs :: "'b set list" assumes "ChamberComplex X"   and "ChamberComplex.ChamberSubcomplex X Y"   and "SimplicialComplex.min_maxsimpchain X Cs"   and "order.greater_eq Y (set Cs)" shows "SimplicialComplex.min_maxsimpchain Y Cs"
inf_def: fixes x :: "'a"   and y :: "'a" shows "x \<sqinter> y = - (- x \<squnion> - y)"