hcju/isabelleps huggingface dataset:
wls_rawInduct: fixes phi :: "'sort \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> bool"   and phiAbs :: "'varSort \<times> 'sort \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> bool"   and s :: "'sort"   and X :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and xs :: "'varSort"   and s' :: "'sort"   and A :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set" assumes "\<And>xs x. phi (asSort xs) (Var xs x)"   and "\<And>delta inp binp. \<lbrakk>wlsInp delta inp; wlsBinp delta binp; liftAll2 phi (arOf delta) inp; liftAll2 phiAbs (barOf delta) binp\<rbrakk> \<Longrightarrow> phi (stOf delta) (Op delta inp binp)"   and "\<And>s xs x X. \<lbrakk>isInBar (xs, s); wls s X; phi s X\<rbrakk> \<Longrightarrow> phiAbs (xs, s) (Abs xs x X)" shows "(wls s X \<longrightarrow> phi s X) \<and> (wlsAbs (xs, s') A \<longrightarrow> phiAbs (xs, s') A)"
full_Suc_Branch3_iff: fixes n :: "nat"   and l :: "'a tree23"   and p :: "int \<times> 'a"   and m :: "'a tree23"   and q :: "int \<times> 'a"   and r :: "'a tree23" shows "full (Suc n) (Branch3 l p m q r) = (full n l \<and> full n m \<and> full n r)"
sep_conj_commuteI: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and sep_disj :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool"   and h :: "'a" assumes "class.sep_algebra plus zero sep_disj"   and "sep_algebra.sep_conj plus sep_disj P Q h" shows "sep_algebra.sep_conj plus sep_disj Q P h"
uint16_msb_test_bit: fixes x :: "uint16" shows "msb x = bit x 15"
contour_integral_has_field_derivative: fixes A :: "complex set"   and a :: "complex"   and z :: "complex"   and f :: "complex \<Rightarrow> complex"   and B :: "complex set" assumes "open A"   and "convex A"   and "a \<in> A"   and "z \<in> A"   and "\<And>z. z \<in> A \<Longrightarrow> f contour_integrable_on linepath a z"   and "f holomorphic_on A" shows "((\<lambda>z. contour_integral (linepath a z) f) has_field_derivative f z) (at z within B)"
VWF_asym: fixes x :: "V"   and y :: "V" assumes "(x, y) \<in> VWF"   and "(y, x) \<in> VWF" shows "False"
pivotandupdate_valuation_nonlhs: fixes s :: "('i, 'a) state"   and x\<^sub>i :: "nat"   and x\<^sub>j :: "nat"   and x :: "nat"   and c :: "'a" assumes "\<triangle> (\<T> s)"   and "\<nabla> s"   and "x\<^sub>i \<in> lvars (\<T> s)"   and "x\<^sub>j \<in> rvars_eq (eq_for_lvar (\<T> s) x\<^sub>i)" shows "x \<notin> lvars (\<T> s) - {x\<^sub>i} \<union> {x\<^sub>j} \<longrightarrow> Mapping.lookup (\<V> (pivot_and_update x\<^sub>i x\<^sub>j c s)) x = (if x = x\<^sub>i then Some c else Mapping.lookup (\<V> s) x)"
some_src_comp: fixes \<tau> :: "'a"   and \<sigma> :: "'a" assumes "\<tau> \<star> \<sigma> \<noteq> null" shows "some_src (\<tau> \<star> \<sigma>) = some_src \<sigma>"
deleteMin_def: fixes q :: "('a, 'b) BinomialHeap" shows "BinomialHeap_loc.deleteMin q \<equiv> if q = BinomialHeap_loc.empty then BinomialHeap_loc.empty else Abs_BinomialHeap (BinomialHeapStruc.deleteMin (Rep_BinomialHeap q))"
axioms_1: fixes \<Omega> :: "'c set"   and M :: "'c set set" assumes "semiring_of_sets \<Omega> M" shows "subset_class \<Omega> M"
borel_measurableI: fixes f :: "'c \<Rightarrow> 'b"   and M :: "'c measure" assumes "\<And>S. open S \<Longrightarrow> f -` S \<inter> space M \<in> sets M" shows "f \<in> borel_measurable M"
local: fixes s :: "'a \<times> 'b"   and a :: "'c"   and s' :: "'a \<times> 'b" assumes "s \<in> oreachable A S U"   and "P s"   and "(s, a, s') \<in> automaton.trans A"   and "S (fst s) (fst s') a" shows "P s'"
assms_1: shows "P \<simeq>\<^sup>s Q"
iso_hcomp: fixes \<mu> :: "'d"   and \<nu> :: "'d" assumes "D.iso \<mu>"   and "D.iso \<nu>"   and "src\<^sub>D \<nu> = trg\<^sub>D \<mu>" shows "D.iso (\<nu> \<star>\<^sub>D \<mu>)"
get_child_nodes_new_document: fixes ptr' :: "(_) object_ptr"   and new_document_ptr :: "(_) document_ptr"   and h :: "(_) heap"   and h' :: "(_) heap"   and r :: "(_) heap \<Rightarrow> (_) heap \<Rightarrow> bool" assumes "ptr' \<noteq> cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r new_document_ptr"   and "h \<turnstile> new_document \<rightarrow>\<^sub>r new_document_ptr"   and "h \<turnstile> new_document \<rightarrow>\<^sub>h h'"   and "r \<in> Shadow_DOM.get_child_nodes_locs ptr'" shows "r h h'"
sorted_distinct_pred_def: fixes h :: "'b \<Rightarrow> int"   and a :: "'b"   and b :: "'b"   and t :: "'b Tree" shows "sorted_distinct_pred h a b t \<equiv> sortedTree h t \<and> a \<in> setOf t \<and> b \<in> setOf t \<and> h a = h b \<longrightarrow> a = b"
closed_segment_subset: fixes x :: "'b"   and S :: "'b set"   and y :: "'b" assumes "x \<in> S"   and "y \<in> S"   and "convex S" shows "{x--y} \<subseteq> S"
inj_decode: fixes e :: "nat" shows "inj_on (local.decode e) {M. local.well_base\<^sub>h M \<and> local.aligned\<^sub>d e M}"
S\<^sub>r_nS2: fixes s :: "'s" assumes "s \<in> S\<^sub>r" shows "s \<notin> S2"
path_def: fixes \<alpha>n :: "'a \<Rightarrow> 'b list"   and invar :: "'a \<Rightarrow> bool"   and inEdges' :: "'a \<Rightarrow> 'b \<Rightarrow> ('b \<times> 'c) list" shows "graph_path_base.path \<alpha>n invar inEdges' \<equiv> \<lambda>g. lfp (\<lambda>p x. (\<exists>n. x = [n] \<and> n \<in> set (\<alpha>n g) \<and> invar g) \<or> (\<exists>ns n'. x = n' # ns \<and> p ns \<and> n' \<in> set (graph_path_base.predecessors inEdges' g (hd ns))))"
set_bit_beyond: fixes x :: "'a word"   and n :: "nat"   and b :: "bool" assumes "size x \<le> n" shows "set_bit_class.set_bit x n b = x"
deformation_retraction_imp_homotopy_equivalent_space: fixes X :: "'a topology"   and s :: "'b \<Rightarrow> 'a"   and r :: "'a \<Rightarrow> 'b"   and Y :: "'b topology" assumes "homotopic_with (\<lambda>x. True) X X (s \<circ> r) id"   and "retraction_maps X Y r s" shows "X homotopy_equivalent_space Y"
independence2: fixes x :: "'a"   and y :: "'a" assumes "x ; y = 0" shows "x ; y\<^sup>\<star> = x"
pm: shows "pseudo_mod (F 1) (F 2) = ffp (pseudo_mod G1 G2)"
Q_step_def: shows "Q_step = make_mat n (Suc i) (snd (Smith_1x2 (Matrix.mat 1 2 (\<lambda>(a, b). if b = 0 then A $$ (0, 0) else A $$ (0, Suc i)))))"
lemma_termdiff2: fixes h :: "'a"   and z :: "'a"   and n :: "nat" assumes "h \<noteq> (0::'a)" shows "((z + h) ^ n - z ^ n) / h - of_nat n * z ^ (n - Suc 0) = h * (\<Sum>p<n - Suc 0. \<Sum>q<n - Suc 0 - p. (z + h) ^ q * z ^ (n - 2 - q))"
hyps: assumes "p \<noteq> 0" shows "card (proots_within p B) = card (proots_within (fcompose p q1 q2) A)"
succ_diff: fixes i :: "'a" assumes "local.aboveS i \<noteq> {}" shows "succ i \<noteq> i"
coeff_poly_cutoff: fixes n :: "nat"   and p :: "'b poly"   and k :: "nat" shows "coeff (poly_cutoff n p) k = (if k < n then coeff p k else (0::'b))"
subst_of_def: fixes ss :: "('c \<times> ('d, 'c) Term.term) list" shows "subst_of ss = foldr (\<lambda>(x, t) \<sigma>. \<sigma> \<circ>\<^sub>s subst x t) ss Var"
exprSec_Lo_eval_eq: fixes expr :: "exp"   and s :: "Concrete.var \<Rightarrow> nat"   and t :: "Concrete.var \<Rightarrow> nat" assumes "exprSec expr = Lo"   and "Example_PL.indisAbbrev s t" shows "eval expr s = eval expr t"
cs_sorted_list_of_cd': fixes prob :: "('a, 'b, 'c, 'd) ifc_problem"   and \<pi> :: "nat \<Rightarrow> 'a"   and k :: "nat" assumes "IFC prob" shows "IFC_def.cs prob \<pi> k = map \<pi> (sorted_list_of_set {i. IFC_def.is_cdi prob k \<pi> i}) @ [\<pi> k]"
sel_2: fixes x :: "complex"   and y :: "complex" shows "Im (x - y) = Im x - Im y"
map_transfer: fixes Rb :: "'a \<Rightarrow> 'c \<Rightarrow> bool"   and Sd :: "'b \<Rightarrow> 'd \<Rightarrow> bool" shows "rel_fun (rel_fun Rb Sd) (rel_fun (rel_pmf Rb) (rel_pmf Sd)) map_pmf map_pmf"
Nml_Nmlize_3: fixes t :: "'a local.term" assumes "Arr t" shows "Trg \<^bold>\<lfloor>t\<^bold>\<rfloor> = Trg t"
\<C>: fixes U :: "(nat \<Rightarrow> real) set" assumes "U \<in> \<C>" shows "openin (powertop_real UNIV) U"
scale_left_imp_eq: fixes a :: "'a"   and x :: "'b"   and y :: "'b" assumes "a \<noteq> (0::'a)"   and "a *s x = a *s y" shows "x = y"
l_set_mode_axioms: shows "l_set_mode ShadowRootClass.type_wf set_mode set_mode_locs"
frrestriction_fid_on: fixes A :: "V"   and B :: "V" shows "fid_on A \<restriction>\<^sup>r\<^sub>\<bullet> B = fid_on (A \<inter>\<^sub>\<circ> B)"
tat: shows "T\<^sub>p_on_rand' BIT (type1 init x y) (v @ r) = 75 / 10\<^sup>2 * real (length v) + T\<^sub>p_on_rand' BIT (type1 init x y) r \<and> Partial_Cost_Model.config'_rand BIT (type1 init x y) v = type1 init x y"
ars_r_def: fixes x :: "'a" shows "antirange_semiring.ars_r ar x = ar (ar x)"
da_elim_cases_52: fixes Env :: "env"   and B :: "lname set"   and vn :: "lname"   and A :: "assigned" assumes "Env\<turnstile> B \<guillemotright>In2 (LVar vn)\<guillemotright> A" obtains "A = \<lparr>nrm = B, brk = \<lambda>l. UNIV\<rparr>"
FI: shows "Field I = {r. Well_order r \<and> downset_on (Field r) p \<and> extension_on (Field r) r p}"
bell01_index_1: shows "|\<beta>\<^sub>0\<^sub>1\<rangle> $$ (0, 0) = 0"
shows_list_vec_def: fixes vs :: "'a vec list" shows "shows_list vs \<equiv> shows_sep shows_vec (shows '', '') vs"
val_Zp_of_a_inv: fixes p :: "int"   and a :: "nat \<Rightarrow> int" assumes "padic_integers p"   and "a \<in> carrier (padic_int p)" shows "padic_integers.val_Zp p a = padic_integers.val_Zp p (\<ominus>\<^bsub>padic_int p\<^esub> a)"
arith_extra_simps_12: fixes a :: "'a" shows "(0::'a) - a = - a"
index_of_lookup_2: fixes p :: "'a"   and ls :: "'a list" assumes "p \<in> set ls" shows "ls ! index_of ls p = p"
length_rotate: fixes n :: "nat"   and xs :: "'c list" shows "length (rotate n xs) = length xs"
independent_explicit_module: fixes V :: "'b set" assumes "V \<subseteq> U\<^sub>V\<^sub>S" shows "(\<not> local.dependent V) = (\<forall>U\<subseteq>U\<^sub>V\<^sub>S. \<forall>f. \<forall>v\<in>U\<^sub>V\<^sub>S. finite U \<longrightarrow> U \<subseteq> V \<longrightarrow> (\<Sum>u\<in>U. f u *s u) = (0::'b) \<longrightarrow> v \<in> U \<longrightarrow> f v = (0::'a))"
symp_conv_conversep_eq: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "symp r = (r\<inverse>\<inverse> = r)"
assms_2: shows "degree f - J + (degree g - J) \<noteq> 0"
fresh_zero_perm: fixes a :: "atom" shows "a \<sharp> 0"
pathstart_compose: fixes f :: "'d \<Rightarrow> 'c"   and p :: "real \<Rightarrow> 'd" shows "pathstart (f \<circ> p) = f (pathstart p)"
terminal_tllist_of_llist: fixes y :: "'f"   and xs :: "'g llist" shows "terminal (tllist_of_llist y xs) = (if lfinite xs then y else undefined)"
measure_measure_spmf_conv_measure_pmf: fixes p :: "'h spmf"   and A :: "'h set" shows "Sigma_Algebra.measure (measure_spmf p) A = measure_pmf.prob p (Some ` A)"
inj_split_Cons: fixes X :: "('g list \<times> 'g) set" shows "inj_on (\<lambda>(xs, n). n # xs) X"
DERIV_cot: fixes x :: "'a" assumes "sin x \<noteq> (0::'a)" shows "(cot has_field_derivative - inverse ((sin x)\<^sup>2)) (at x)"
globally_sound_mode_use_def: fixes gc :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list \<times> ('Var \<Rightarrow> 'Val)" shows "globally_sound_mode_use gc = (\<forall>mdss. mdss \<in> reachable_mode_states gc \<longrightarrow> compatible_modes mdss)"
intros: fixes p :: "'b \<times> 'c" shows "sndsp p (snd p)"
order_on_defs_2: fixes A :: "'b set"   and r :: "('b \<times> 'b) set" shows "partial_order_on A r \<equiv> preorder_on A r \<and> antisym r"
F1set3_bd': fixes x :: "('e, 'f, 'd) F1"   and r2 :: "(bd_type_F2 \<times> bd_type_F2) set" shows "|F1set3 x| \<le>o dir_image (bd_F1 +c r2) Abs_bd_type_F"
append_replicate_commute: fixes n :: "nat"   and x :: "'c"   and k :: "nat" shows "replicate n x @ replicate k x = replicate k x @ replicate n x"
trnl\<^sub>\<eta>_def: fixes v :: "'a"   and \<mu> :: "'a" shows "trnl\<^sub>\<eta> v \<mu> \<equiv> (g \<star> \<mu>) \<cdot> \<a>[g, f, v] \<cdot> (\<eta> \<star> v) \<cdot> \<l>\<^sup>-\<^sup>1[v]"
unification': fixes ts :: "fterm set"   and \<sigma> :: "char list \<Rightarrow> fterm" assumes "finite ts"   and "unifier\<^sub>t\<^sub>s \<sigma> ts" shows "\<exists>\<theta>. mgu\<^sub>t\<^sub>s \<theta> ts"
simps_1: fixes P :: "'c \<Rightarrow> bool" shows "takeWhile P [] = []"
virtual_simulates_direct_step: fixes ts :: "('p \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> unit \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list"   and m :: "nat \<Rightarrow> nat"   and \<S> :: "nat \<Rightarrow> bool option"   and ts' :: "('p \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> unit \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list"   and m' :: "nat \<Rightarrow> nat"   and \<S>' :: "nat \<Rightarrow> bool option" assumes "(ts, m, \<S>) \<Rightarrow>\<^sub>d (ts', m', \<S>')" shows "(remove_rels ts, m, \<S>) \<Rightarrow>\<^sub>v (remove_rels ts', m', \<S>')"
pt_all_eqvt: fixes pi :: "('b \<times> 'b) list"   and P :: "'a \<Rightarrow> bool" assumes "pt TYPE('a) TYPE('b)"   and "at TYPE('b)" shows "pi \<bullet> (\<forall>x. P x) = (\<forall>x. pi \<bullet> P (rev pi \<bullet> x))"
finsum_over_indexes_to_vectors: fixes vs :: "'a vec list"   and n :: "nat"   and l :: "nat"   and g :: "nat \<Rightarrow> int" assumes "set vs \<subseteq> carrier_vec n"   and "l = length vs" shows "\<exists>c. (\<Oplus>\<^bsub>module_vec TYPE('a) n\<^esub>x\<in>{0..<l}. of_int (g x) \<cdot>\<^sub>v vs ! x) = (\<Oplus>\<^bsub>module_vec TYPE('a) n\<^esub>v\<in>set vs. of_int (c v) \<cdot>\<^sub>v v)"
nth_rows: fixes i :: "nat"   and A :: "'c mat" assumes "i < dim_row A" shows "rows A ! i = row A i"
return_time0: fixes A :: "'a set" shows "A.return_time_function A -` {0} = UNIV - A.recurrent_subset A"
Skip: fixes P :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool" shows "\<turnstile>\<^sub>t {P} SKIP {P}"
filter_compliant_stateful_ACS_complies: fixes M :: "('b SecurityInvariant \<times> 'b SecurityInvariant_configured) list"   and G :: "'b list_graph" assumes "\<forall>(m_impl, m_spec)\<in>set M. SecurityInvariant_complies_formal_def m_impl m_spec"   and "wf_list_graph G" shows "TopoS_Stateful_Policy_impl.filter_compliant_stateful_ACS G (get_impl M) = TopoS_Stateful_Policy_Algorithm.filter_compliant_stateful_ACS (list_graph_to_graph G) (get_spec M) (edgesL G)"
partial_correctness: fixes V :: "nat set"   and I :: "nat set"   and J :: "nat set"   and Q_I_pos :: "(nat set \<times> 'a option list set) set"   and Q :: "(nat set \<times> 'a option list set) set"   and Q_J_pos :: "(nat set \<times> 'a option list set) set"   and Q_I_neg :: "(nat set \<times> 'a option list set) set"   and Qn :: "(nat set \<times> 'a option list set) set"   and Q_J_neg :: "(nat set \<times> 'a option list set) set"   and NQ_pos :: "(nat set \<times> 'a option list set) set"   and t :: "'a option list"   and NQ_neg :: "(nat set \<times> 'a option list set) set"   and R_NQ :: "'a option list set"   and R_I :: "'a option list set"   and n :: "nat"   and z :: "'a option list"   and xx :: "'a option list" assumes "V = I \<union> J"   and "Set.is_empty (I \<inter> J)"   and "1 \<le> card I"   and "1 \<le> card J"   and "Q_I_pos = projectQuery I (filterQuery I Q)"   and "Q_J_pos = filterQuery J Q"   and "Q_I_neg = filterQueryNeg I Qn"   and "Q_J_neg = filterQuery J (Qn - Q_I_neg)"   and "NQ_pos = newQuery J Q_J_pos (I, t)"   and "NQ_neg = newQuery J Q_J_neg (I, t)"   and "R_NQ = genericJoin J NQ_pos NQ_neg"   and "\<forall>x. (x \<in> R_I) = (wf_tuple n I x \<and> (\<forall>(A, X)\<in>Q_I_pos. restrict A x \<in> X) \<and> (\<forall>(A, X)\<in>Q_I_neg. restrict A x \<notin> X))"   and "\<forall>y. (y \<in> R_NQ) = (wf_tuple n J y \<and> (\<forall>(A, X)\<in>NQ_pos. restrict A y \<in> X) \<and> (\<forall>(A, X)\<in>NQ_neg. restrict A y \<notin> X))"   and "z = merge xx t"   and "t \<in> R_I"   and "xx \<in> R_NQ"   and "rwf_query n V Q Qn" shows "wf_tuple n V z \<and> (\<forall>(A, X)\<in>Q. restrict A z \<in> X) \<and> (\<forall>(A, X)\<in>Qn. restrict A z \<notin> X)"
Q: shows "Q \<in> carrier_mat CARD('nc mod_ring) CARD('nc mod_ring)"
c_sum_of_c_pair: fixes x :: "nat"   and y :: "nat" shows "c_sum (c_pair x y) = x + y"
axioms: fixes smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and M :: "'b set"   and smult' :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and T :: "'b \<Rightarrow> 'c" assumes "ModuleHom smult M smult' T" shows "RModuleHom UNIV smult M smult' T"
SecurityInvariant_def: fixes sinvar :: "'b graph \<Rightarrow> ('b \<Rightarrow> 'c) \<Rightarrow> bool"   and default_node_properties :: "'c"   and receiver_violation :: "bool" shows "SecurityInvariant sinvar default_node_properties receiver_violation \<equiv> SecurityInvariant_preliminaries sinvar \<and> SecurityInvariant_axioms sinvar default_node_properties receiver_violation"
filterlim_weak_subseq: fixes u :: "nat \<Rightarrow> nat" assumes "(\<lambda>x. ereal (real (u x))) \<longlonglongrightarrow> \<infinity>" shows "filterlim u sequentially sequentially"
axioms_1: fixes M :: "'b measure"   and F :: "'b measure" assumes "finite_measure_subalgebra M F" shows "finite_measure M"
Vv_is_Vv1_union_Vv2: shows "V\<^bsub>\<V>\<^esub> = V\<^bsub>\<V>1\<^esub> \<union> V\<^bsub>\<V>2\<^esub>"
poly_zero_imp_not_unit: fixes p :: "'a poly"   and x :: "'a" assumes "poly p x = (0::'a)" shows "\<not> p dvd 1"
supp_2: fixes fmla1 :: "fmla"   and fmla2 :: "fmla" shows "supp (fmla1 OR fmla2) = supp fmla1 \<union> supp fmla2"
False: shows "\<not> (\<exists>j<Suc n. 0 < P j)"
distinct_11: fixes x31 :: "'b"   and x32 :: "('a, 'b) form list" shows "Modal x31 x32 \<noteq> ff"
\<F>\<G>_semantics: fixes \<phi> :: "'a ltln"   and w :: "nat \<Rightarrow> 'a set" shows "\<F>\<G> \<phi> w = {\<psi> \<in> subformulas\<^sub>\<nu> \<phi>. w \<Turnstile>\<^sub>n F\<^sub>n (G\<^sub>n \<psi>)}"
seq_sentences_none: fixes P :: "'c option process"   and xs :: "'c option list"   and y :: "'c option"   and ys :: "'c option list" assumes "sequential P"   and "xs \<in> sentences P"   and "xs @ y # ys \<in> traces P" shows "y = None"
select_convs_2: fixes Functions :: "'b \<Rightarrow> (real, 'a) vec \<Rightarrow> real"   and Predicates :: "'a \<Rightarrow> (real, 'a) vec \<Rightarrow> bool"   and Contexts :: "'c \<Rightarrow> ((real, 'a) vec \<times> (real, 'a) vec) set \<Rightarrow> ((real, 'a) vec \<times> (real, 'a) vec) set"   and Programs :: "'a \<Rightarrow> (((real, 'a) vec \<times> (real, 'a) vec) \<times> (real, 'a) vec \<times> (real, 'a) vec) set"   and ODEs :: "'a \<Rightarrow> (real, 'a) vec \<Rightarrow> (real, 'a) vec"   and ODEBV :: "'a \<Rightarrow> 'a set"   and more :: "'d" shows "Predicates \<lparr>Functions = Functions, Predicates = Predicates, Contexts = Contexts, Programs = Programs, ODEs = ODEs, ODEBV = ODEBV, \<dots> = more\<rparr> = Predicates"
comp_red_monic_basis_nonzero: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and xs :: "('t \<Rightarrow>\<^sub>0 'c) list" assumes "p \<in> set (comp_red_monic_basis xs)" shows "p \<noteq> 0"
Max_Suc_Suc_Max: fixes A :: "nat set" assumes "finite A"   and "A \<noteq> {}" shows "Max (insert 0 (Suc ` A)) = Suc (Max (insert 0 A))"
defined_def: fixes X :: "'a state \<times> 'a state \<Rightarrow> 'b" shows "\<delta> X \<equiv> \<lambda>\<tau>. if X \<tau> = UML_Types.bot_class.bot \<tau> \<or> X \<tau> = null \<tau> then false \<tau> else true \<tau>"
bots_def: shows "Sup_lattice.bots Inf = \<Sqinter> {}"
seq_Sup_distr: fixes A :: "('b, 'd) urel set"   and Q :: "('d, 'c) urel" shows "\<Or>A ;; Q = \<Or>((\<lambda>P. P ;; Q) ` A)"
pivot_fun_swaprows: fixes A :: "'c mat"   and f :: "nat \<Rightarrow> nat"   and jj :: "nat"   and nr :: "nat"   and nc :: "nat"   and l :: "nat"   and k :: "nat" assumes "pivot_fun A f jj"   and "dim_row A = nr"   and "dim_col A = nc"   and "f l = jj"   and "f k = jj"   and "l < nr"   and "k < nr"   and "jj \<le> nc" shows "pivot_fun (swaprows l k A) f jj"
inRS2: fixes x :: "('a, 'b) Combinators"   and p :: "('a, 'b) Combinators list" assumes "x \<notin> set p" shows "x \<notin> set (removeShadowRules2 p)"
fmadd_idempotent: fixes m :: "('d, 'e) fmap" shows "m ++\<^sub>f m = m"
subgraph_refl: shows "subgraph (app_iso hom G) (app_iso hom G)"
mult_assoc: fixes c :: "'a"   and b :: "'a"   and a :: "'a" shows "c * (b * a) = c * b * a"
is_CSP_process_def: fixes P :: "('c, 'd) alpha_rp_scheme \<times> ('c, 'd) alpha_rp_scheme \<Rightarrow> bool" shows "is_CSP_process P \<equiv> P is CSP1 healthy \<and> P is CSP2 healthy \<and> P is R healthy"
prems_1: shows "\<Gamma> \<Rightarrow> F \<^bold>\<rightarrow> G, \<Delta> \<down> n"
natural_transformation_axioms: shows "natural_transformation (\<cdot>\<^sub>J) (\<cdot>\<^sub>B) (at (A.dom a) D) (at (A.cod a') D) Da'oDa.map"
Inf_set_def: fixes A :: "'b set set" shows "\<Inter> A = {x. \<Sqinter> ((\<in>) x ` A)}"
StaticDownDynCastReds: fixes e :: "expr"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and a :: "nat"   and Cs :: "char list list"   and C :: "char list"   and Cs' :: "char list list"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option" assumes "P,E \<turnstile> \<langle>e,s\<rangle> \<rightarrow>* \<langle>ref (a, Cs @ [C] @ Cs'),s'\<rangle>" shows "P,E \<turnstile> \<langle>Cast C e,s\<rangle> \<rightarrow>* \<langle>ref (a, Cs @ [C]),s'\<rangle>"
normalize_field_eq_1: fixes x :: "'a" assumes "x \<noteq> (0::'a)" shows "normalize x = (1::'a)"
assms_3: fixes x :: "'a"   and xs :: "'a list" assumes "x \<notin> set xs"   and "P xs" shows "P (x # xs)"
cblinfun_inv_left: fixes A :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b" assumes "iso_cblinfun A" shows "cblinfun_inv A o\<^sub>C\<^sub>L A = id_cblinfun"
LIMSEQ_inverse_zero: fixes X :: "nat \<Rightarrow> real" assumes "\<And>r. \<exists>N. \<forall>n\<ge>N. r < X n" shows "(\<lambda>n. inverse (X n)) \<longlonglongrightarrow> 0"
id: shows "smult c (\<Prod>a\<leftarrow>as. [:- a, 1:]) = of_int_poly p"
assms_1: shows "\<forall>t\<in>FP. t \<in> closure"
lincomb_is_mod_hom: fixes S :: "'c set" assumes "finite S"   and "S \<subseteq> carrier M" shows "mod_hom R (func_space S) M (\<lambda>a. lincomb a S)"
assms: shows "(\<lambda>j. cos (\<theta> j - \<Theta>)) \<longlonglongrightarrow> 1"
E'_def: fixes n :: "nat" shows "E' n = fst (SOME (E', ws'). E' \<in> \<E> \<and> P \<turnstile> (E', ws') \<surd> \<and> (enat n \<le> llength E \<longrightarrow> enat n \<le> llength E') \<and> ltake (enat (n - 1)) E = ltake (enat (n - 1)) E' \<and> (0 < n \<longrightarrow> action_tid E' (n - 1) = action_tid E (n - 1) \<and> (if n - 1 \<in> read_actions E then sim_action else (=)) (action_obs E' (n - 1)) (action_obs E (n - 1)) \<and> (\<forall>i<n - 1. i \<in> read_actions E \<longrightarrow> ws' i = ws i)) \<and> (\<forall>r\<in>read_actions E'. n - 1 \<le> r \<longrightarrow> P,E' \<turnstile> ws' r \<le>hb r))"
cases: fixes x :: "((('Var, 'AExp, 'BExp) Stmt \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)) \<times> (('Var, 'AExp, 'BExp) Stmt \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)" obtains c\<^sub>1 :: "('Var, 'AExp, 'BExp) Stmt" and mds :: "Mode \<Rightarrow> 'Var set" and mem\<^sub>1 :: "'Var \<Rightarrow> 'Val" and c\<^sub>2 :: "('Var, 'AExp, 'BExp) Stmt" and mds\<^sub>2 :: "Mode \<Rightarrow> 'Var set" and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val" where "x = (\<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle>, \<langle>c\<^sub>2, mds\<^sub>2, mem\<^sub>2\<rangle>)"
write_cpu_y_privilege: fixes s' :: "'b sparc_state"   and w :: "word32"   and s :: "'b sparc_state" assumes "s' = snd (fst (write_cpu w Y s)) \<and> get_S (cpu_reg_val PSR s) = 0" shows "get_S (cpu_reg_val PSR s') = 0"
effect_lookupI: fixes h' :: "heap"   and h :: "heap"   and x :: "'b"   and r :: "'b ref" assumes "h' = h"   and "x = Ref.get h r" shows "effect (!r) h h' x"
Agent_Sign: fixes X :: "agent"   and X' :: "msg"   and Y' :: "msg" shows "Messages.Agent X \<noteq> Sign X' Y'"