hcju/isabelleps huggingface dataset:
symmetry_functor_def: fixes A1 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and A2 :: "'c \<Rightarrow> 'c \<Rightarrow> 'c" shows "symmetry_functor A1 A2 \<equiv> (category A1 \<and> category A2) \<and> product_category A1 A2 \<and> product_category A2 A1"
redundant_scc_def: fixes g :: "'g"   and P :: "'val set"   and scc :: "'val set" shows "redundant_scc g P scc \<equiv> redundant_set g scc \<and> is_scc (induced_phi_graph g P) scc"
simps_3: fixes "is" :: "'b set"   and t :: "'b"   and b :: "bool" shows "interrupt_action_ok' is (IsInterrupted t b) = (b \<or> t \<notin> is)"
simps_2: fixes e1 :: "exp"   and e2 :: "exp"   and S :: "stack_elem list" shows "upds (Alts e1 e2 # S) = upds S"
worec_fixpoint: fixes H :: "('a \<Rightarrow> 'c) \<Rightarrow> 'a \<Rightarrow> 'c" assumes "r.adm_wo H" shows "r.worec H = H (r.worec H)"
submonoid_axioms: shows "submonoid M (S \<rightarrow>\<^sub>E S) (compose S) (identity S)"
tfr\<^sub>s\<^sub>s\<^sub>t_def: fixes S :: "('fun, 'var) stateful_strand_step list" shows "tfr\<^sub>s\<^sub>s\<^sub>t S \<equiv> tfr\<^sub>s\<^sub>e\<^sub>t (trms\<^sub>s\<^sub>s\<^sub>t S \<union> pair ` setops\<^sub>s\<^sub>s\<^sub>t S) \<and> list_all tfr\<^sub>s\<^sub>s\<^sub>t\<^sub>p S"
set_iterator_rev_linord_foldli_conv: fixes iti :: "('c \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'c \<Rightarrow> 'c) \<Rightarrow> 'c \<Rightarrow> 'c"   and S :: "'a set" shows "set_iterator_rev_linord iti S = (\<exists>l0. distinct l0 \<and> S = set l0 \<and> sorted (rev l0) \<and> iti = foldli l0)"
matrix_to_iarray_fst_QR_decomposition: fixes A :: "((real, 'c) vec, 'd) vec" shows "matrix_to_iarray (fst (QR_decomposition A)) = fst (QR_decomposition_iarrays (matrix_to_iarray A))"
mult_zero_left: fixes times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and a :: "'c" assumes "class.mult_zero times zero" shows "times zero a = zero"
subspace_Berlekamp: fixes f :: "'a mod_ring poly" assumes "degree f \<noteq> 0" shows "subspace class_ring {v. [v ^ CARD('a) = v] (mod f) \<and> degree v < degree f} local.poly_abelian_monoid"
max2_def: fixes a :: "'a"   and b :: "'a" shows "local.max2 a b \<equiv> if (a, b) \<in> r then b else a"
subseqs_distinctD: fixes ys :: "'c list"   and xs :: "'c list" assumes "ys \<in> set (subseqs xs)"   and "distinct xs" shows "distinct ys"
conjuncts: fixes p :: "fm"   and bs :: "real list" shows "(\<forall>q\<in>set (conjuncts p). Ifm bs q) = Ifm bs p"
frameImpClosed: fixes F :: "'b frame"   and \<Phi> :: "'c"   and p :: "(name \<times> name) list" assumes "F \<turnstile>\<^sub>F \<Phi>" shows "p \<bullet> F \<turnstile>\<^sub>F p \<bullet> \<Phi>"
ring_hom_cringI: fixes R :: "('c, 'd) ring_scheme"   and S :: "('e, 'f) ring_scheme"   and h :: "'c \<Rightarrow> 'e" assumes "cring R"   and "cring S"   and "h \<in> ring_hom R S" shows "ring_hom_cring R S h"
t_Lam: fixes x :: "name"   and \<Gamma> :: "(name \<times> ty) list"   and T\<^sub>1 :: "ty"   and e :: "trm"   and T\<^sub>2 :: "ty" assumes "x \<sharp> \<Gamma>"   and "(x, T\<^sub>1) # \<Gamma> \<turnstile> e : T\<^sub>2" shows "\<Gamma> \<turnstile> Lam [x].e : T\<^sub>1\<rightarrow>T\<^sub>2"
subst_overrides: fixes \<sigma> :: "'a \<Rightarrow> 'b option"   and \<tau> :: "'a \<Rightarrow> 'b option"   and mem :: "'a \<Rightarrow> 'b" assumes "dom \<sigma> = dom \<tau>" shows "subst \<sigma> (subst \<tau> mem) = subst \<sigma> mem"
2: shows "((\<lambda>x. (\<Sum>b\<in>basis1. F (\<gamma> x) \<bullet> b * (vector_derivative \<gamma> (at x within {0..1}) \<bullet> b)) + (\<Sum>b\<in>basis2. F (\<gamma> x) \<bullet> b * (vector_derivative \<gamma> (at x within {0..1}) \<bullet> b))) has_integral integral {0..1} (\<lambda>x. \<Sum>b\<in>basis1. F (\<gamma> x) \<bullet> b * (vector_derivative \<gamma> (at x within {0..1}) \<bullet> b)) + integral {0..1} (\<lambda>x. \<Sum>b\<in>basis2. F (\<gamma> x) \<bullet> b * (vector_derivative \<gamma> (at x within {0..1}) \<bullet> b))) {0..1}"
P\<^sub>c_def: shows "P\<^sub>c \<equiv> ts_process T\<^sub>c"
neg_closed: fixes g :: "'r" assumes "g \<in> R" shows "- g \<in> R"
binomial_absorb_comp: fixes n :: "nat"   and k :: "nat" shows "(n - k) * (n choose k) = n * (n - 1 choose k)"
prems: shows "wf\<^sub>t\<^sub>r\<^sub>m (Fun g T)"
cnorm_le: fixes x :: "'a"   and y :: "'b" shows "(norm x \<le> norm y) = (cinner x x \<le> cinner y y)"
inner_minus_right: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and inner :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and x :: "'b"   and y :: "'b" assumes "class.real_inner minus dist norm plus zero uminus scaleR sgn uniformity open inner" shows "inner x (uminus y) = - inner x y"
self_finance_portfolio: fixes pf :: "'b \<Rightarrow> nat \<Rightarrow> 'c \<Rightarrow> real"   and Mkt :: "('c, 'b) discrete_market"   and v :: "real"   and asset :: "'b" assumes "portfolio pf" shows "portfolio (self_finance Mkt v pf asset)"
decision: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and rs :: "'c rule list"   and X :: "final_decision" shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs, Decision X\<rangle> \<Rightarrow> Decision X"
inv_solve_right: fixes a :: "'c"   and b :: "'c"   and c :: "'c" assumes "a \<in> carrier S"   and "b \<in> carrier S"   and "c \<in> carrier S" shows "(a = b \<oplus>\<^bsub>S\<^esub> \<ominus>\<^bsub>S\<^esub> c) = (b = a \<oplus>\<^bsub>S\<^esub> c)"
midpoint_eq_endpoint_2: fixes a :: "'c"   and b :: "'c" shows "(midpoint a b = b) = (a = b)"
g: fixes i :: "nat" assumes "m \<le> i"   and "i < k" shows "g i = f i"
filterlim_at_bot_mirror: fixes f :: "real \<Rightarrow> 'b"   and F :: "'b filter" shows "filterlim f F at_bot = (LIM x at_top. f (- x) :> F)"
S: fixes s :: "'a" assumes "rwof m0 cond step s"   and "I s"   and "cond s" shows "step s \<le>\<^sub>n SPEC I"
snd_gram_schmidt_int: fixes n :: "nat"   and us :: "int vec list" shows "snd (gram_schmidt_int n us) = gram_schmidt n (map of_int_hom.vec_hom us)"
mult: fixes f1 :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and g1 :: "'b \<Rightarrow> 'c"   and f2 :: "'b \<Rightarrow> 'c"   and g2 :: "'b \<Rightarrow> 'c" assumes "f1 \<in> \<Theta>[F](g1)"   and "f2 \<in> \<Theta>[F](g2)" shows "(\<lambda>x. f1 x * f2 x) \<in> \<Theta>[F](\<lambda>x. g1 x * g2 x)"
simps: fixes r :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and a :: "'e" shows "Domainp r a = (\<exists>aa b. a = aa \<and> r aa b)"
ccField_ccSquare: fixes S :: "Vars.var set" shows "ccField (S\<^sup>2) = S"
unique_session_keys: fixes A :: "agent"   and Tk :: "nat"   and B :: "agent"   and K :: "nat"   and X :: "msg"   and evs :: "event list"   and A' :: "agent"   and Tk' :: "nat"   and B' :: "agent"   and X' :: "msg" assumes "Says Server A (Crypt (shrK A) \<lbrace>Number Tk, Agent B, Key K, X\<rbrace>) \<in> set evs"   and "Says Server A' (Crypt (shrK A') \<lbrace>Number Tk', Agent B', Key K, X'\<rbrace>) \<in> set evs"   and "evs \<in> bankerb_gets" shows "A = A' \<and> Tk = Tk' \<and> B = B' \<and> X = X'"
subreachable_def: fixes A :: "(('a \<Rightarrow> 'b) \<times> 'c, 'd) automaton"   and U :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> bool"   and J :: "'a set" shows "subreachable A U J \<equiv> \<forall>I. \<forall>s\<in>oreachable A (\<lambda>s s'. I) U. \<exists>\<sigma>. (\<forall>j\<in>J. \<sigma> j = fst s j) \<and> (\<sigma>, snd s) \<in> reachable A I"
antisymI: fixes r :: "('a \<times> 'a) set" assumes "\<And>x y. \<lbrakk>(x, y) \<in> r; (y, x) \<in> r\<rbrakk> \<Longrightarrow> x = y" shows "antisym r"
precise_def: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and sep_disj :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and P :: "'a \<Rightarrow> bool" assumes "class.cancellative_sep_algebra plus zero sep_disj" shows "cancellative_sep_algebra.precise plus sep_disj P = (\<forall>h hp hp'. sep_algebra.sep_substate plus sep_disj hp h \<and> P hp \<and> sep_algebra.sep_substate plus sep_disj hp' h \<and> P hp' \<longrightarrow> hp = hp')"
d_state: fixes ii :: "nat" assumes "ii \<le> m" shows "d_state state ii = d fs ii"
le_type_scheme_refl: fixes sch :: "type_scheme" shows "sch \<le> sch"
intros_6: fixes X :: "msg"   and H :: "msg set"   and K :: "nat" assumes "X \<in> synth H"   and "Key K \<in> H" shows "Crypt K X \<in> synth H"
max_density_is_max: fixes G :: "nat set \<times> nat set set"   and G' :: "nat set \<times> nat set set" assumes "finite_graph G"   and "finite_graph G'"   and "nonempty_graph G'"   and "uwellformed G'"   and "subgraph G' G" shows "Ugraph_Lemmas.density G' \<le> max_density G"
l_set_disconnected_nodes_get_ancestors\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: fixes known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and set_disconnected_nodes :: "(_) document_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog"   and get_parent_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptrs :: "(_) heap \<Rightarrow> bool"   and get_ancestors :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and get_ancestors_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_root_node :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr) prog"   and get_root_node_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_set_disconnected_nodes_get_ancestors\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr set_disconnected_nodes set_disconnected_nodes_locs get_child_nodes get_child_nodes_locs get_parent get_parent_locs type_wf known_ptrs get_ancestors get_ancestors_locs get_root_node get_root_node_locs \<equiv> l_set_disconnected_nodes_get_parent set_disconnected_nodes_locs get_parent_locs \<and> l_get_root_node\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf known_ptr known_ptrs get_parent get_parent_locs get_child_nodes get_child_nodes_locs get_ancestors get_ancestors_locs get_root_node get_root_node_locs \<and> l_set_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf set_disconnected_nodes set_disconnected_nodes_locs"
F_def: shows "F \<equiv> {h. h holomorphic_on S \<and> h ` S \<subseteq> ball 0 1 \<and> h 0 = 0 \<and> inj_on h S}"
Conj_injective_iff: fixes A :: "fm"   and B :: "fm"   and A' :: "fm"   and B' :: "fm" shows "(A AND B = A' AND B') = (A = A' \<and> B = B')"
comp_null_1: fixes f :: "'a1" shows "C1.null \<cdot>\<^sub>1 f = C1.null"
greatest_plus_one_not_zero: shows "(GREATEST n. \<not> is_zero_row_upt_k n k A) + (1::'rows) \<noteq> (0::'rows)"
system_step_reflclp_def: fixes \<sigma> :: "nat \<Rightarrow> ('g, 'h, 'i, 'j, 'k) system_state" shows "system_step_reflclp \<sigma> = (\<lambda>sh sh'. \<exists>pls. system_step pls sh' sh)\<^sup>=\<^sup>= (\<sigma> 0) (\<sigma> 1)"
mset_le_not_sym: fixes M :: "'b multiset"   and N :: "'b multiset" assumes "M < N" shows "\<not> N < M"
lookup_vectorize_poly: fixes pair_of_term :: "'a \<Rightarrow> 'b \<times> 'c"   and term_of_pair :: "'b \<times> 'c \<Rightarrow> 'a"   and p :: "'a \<Rightarrow>\<^sub>0 'd"   and k :: "'c" assumes "term_powerprod pair_of_term term_of_pair" shows "lookup (term_powerprod.vectorize_poly term_of_pair p) k = term_powerprod.proj_poly term_of_pair k p"
Pow_set_2: fixes x :: "'b"   and xs :: "'b list" shows "Pow (set (x # xs)) = (let A = Pow (set xs) in A \<union> insert x ` A)"
span_zero: fixes U :: "'c set" shows "\<zero>\<^bsub>V\<^esub> \<in> span U"
RBT_inverse: fixes y :: "('b, 'c) RBT_Impl.rbt" assumes "y \<in> {t. is_rbt t}" shows "rbt.impl_of (RBT y) = y"
all_ipd_imp_ret: fixes \<pi> :: "nat \<Rightarrow> 'n" assumes "is_path \<pi>"   and "\<forall>i. \<pi> i \<noteq> local.return \<longrightarrow> (\<exists>j>i. \<pi> j = ipd (\<pi> i))" shows "\<exists>j. \<pi> j = local.return"
fr_in_2: shows "frontier (inside (path_image c2 \<union> path_image c)) = path_image c2 \<union> path_image c"
dgrad_p_set_def: fixes d :: "'a \<Rightarrow> nat"   and m :: "nat" shows "punit.dgrad_p_set d m = {p. local.punit.pp_of_term ` keys p \<subseteq> dgrad_set d m}"
fresh_psubst_ty: fixes X :: "W.tvar"   and \<theta> :: "(W.tvar \<times> ty) list"   and T :: "ty" assumes "X \<sharp> \<theta>"   and "X \<sharp> T" shows "X \<sharp> \<theta><T>"
to_nat_on_infinite: fixes S :: "'c set" assumes "countable S"   and "infinite S" shows "bij_betw (to_nat_on S) S UNIV"
except_except: fixes p :: "'a \<Rightarrow>\<^sub>0 'b"   and S :: "'a set"   and T :: "'a set" shows "except (except p S) T = except p (S \<union> T)"
beta_reds_into_beta_eta_reds: fixes s :: "dB"   and t :: "dB" assumes "s \<rightarrow>\<^sub>\<beta>\<^sup>* t" shows "s \<rightarrow>\<^sub>\<beta>\<^sub>\<eta>\<^sup>* t"
Collect_finite_subset_eq_lists: fixes T :: "'b set" shows "{A. finite A \<and> A \<subseteq> T} = set ` lists T"
pseudominimal_in_below_in_less_eq_glbound: fixes P :: "'a set"   and x :: "'a"   and w :: "'a"   and y :: "'a"   and b :: "'a" assumes "pseudominimal_in (P .\<^bold>\<le> x) w"   and "pseudominimal_in (P .\<^bold>\<le> y) w"   and "glbound_in_of P x y b" shows "b \<succeq> w"
lm020: fixes bids :: "'a \<times> 'b set \<Rightarrow> 'c"   and N :: "'a set"   and G :: "'b set"   and a :: "('a \<times> 'b set) set" shows "summedBidVectorRel bids N G || a = summedBid bids ` (N \<times> (Pow G - {{}}) \<inter> a)"
intro: fixes plus\<^sub>M\<^sub>_\<^sub>1 :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero\<^sub>M\<^sub>_\<^sub>1 :: "'d"   and minus\<^sub>M\<^sub>_\<^sub>1 :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus\<^sub>M\<^sub>_\<^sub>1 :: "'d \<Rightarrow> 'd"   and scale\<^sub>1 :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and plus\<^sub>M\<^sub>_\<^sub>2 :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and zero\<^sub>M\<^sub>_\<^sub>2 :: "'f"   and minus\<^sub>M\<^sub>_\<^sub>2 :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and uminus\<^sub>M\<^sub>_\<^sub>2 :: "'f \<Rightarrow> 'f"   and scale\<^sub>2 :: "'g \<Rightarrow> 'f \<Rightarrow> 'f" assumes "module_with plus\<^sub>M\<^sub>_\<^sub>1 zero\<^sub>M\<^sub>_\<^sub>1 minus\<^sub>M\<^sub>_\<^sub>1 uminus\<^sub>M\<^sub>_\<^sub>1 scale\<^sub>1"   and "module_with plus\<^sub>M\<^sub>_\<^sub>2 zero\<^sub>M\<^sub>_\<^sub>2 minus\<^sub>M\<^sub>_\<^sub>2 uminus\<^sub>M\<^sub>_\<^sub>2 scale\<^sub>2" shows "module_pair_with plus\<^sub>M\<^sub>_\<^sub>1 zero\<^sub>M\<^sub>_\<^sub>1 minus\<^sub>M\<^sub>_\<^sub>1 uminus\<^sub>M\<^sub>_\<^sub>1 scale\<^sub>1 plus\<^sub>M\<^sub>_\<^sub>2 zero\<^sub>M\<^sub>_\<^sub>2 minus\<^sub>M\<^sub>_\<^sub>2 uminus\<^sub>M\<^sub>_\<^sub>2 scale\<^sub>2"
upper: shows "\<forall>\<^sub>F F in finite_subsets_at_top S. sum f F \<le> B"
intro: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" assumes "category C" shows "dual_category C"
order_on_defs_5: fixes A :: "'a set"   and r :: "('a \<times> 'a) set" shows "well_order_on A r \<equiv> linear_order_on A r \<and> wf (r - Id)"
induct: fixes P :: "'b \<Rightarrow> 'b list \<Rightarrow> bool"   and a0 :: "'b"   and a1 :: "'b list" assumes "\<And>x. P x []"   and "\<And>x y. P x [y]"   and "\<And>x y v va. (\<And>xa ya. (xa, ya) = bezout_coefficients y (prod_list (v # va)) \<Longrightarrow> P (xa * x) (v # va)) \<Longrightarrow> P x (y # v # va)" shows "P a0 a1"
span_subset_cspan: fixes A :: "'d set" shows "span A \<subseteq> cspan A"
unstream_gen_length: fixes s :: "'s" shows "local.gen_length_cons 0 s = length (unstream g s)"
residues_closed: fixes a :: "nat \<Rightarrow> int"   and k :: "nat" assumes "a \<in> carrier Zp" shows "a k \<in> carrier (residue_ring (p ^ k))"
terminal_is_nil: fixes v :: "'form \<times> nat list" assumes "v |\<in>| vertices" shows "(outPorts (nodeOf v) = {||}) = (snd v = [])"
cases: fixes AP' :: "('U, 'C, 'Out) Noninterference.aprop set"   and a :: "(('St, 'U, 'C) state \<times> ('U, 'C, 'Out) Noninterference.aprop set) stream" assumes "wfp AP' a" obtains s :: "('St, 'U, 'C) state" and A :: "('U, 'C, 'Out) Noninterference.aprop set" and \<pi> :: "(('St, 'U, 'C) state \<times> ('U, 'C, 'Out) Noninterference.aprop set) stream" where "a = (s, A) ## \<pi>" and "s \<in> UNIV" and "A \<subseteq> AP'" and "A \<inter> UNIV = L s" and "stateOf \<pi> \<in> \<delta> s" and "wfp AP' \<pi>"
row_length_def: fixes id :: "'a"   and f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and xs :: "'a list list" assumes "Matrix_Tensor.mult id f" shows "mult.row_length xs \<equiv> if xs = [] then 0 else length (hd xs)"
Rep_cases: fixes y :: "int" assumes "y \<in> uints LENGTH('b)" obtains x :: "'b word" where "y = uint x"
max_atm_imp_Interp: fixes C :: "'a literal multiset"   and A :: "'a" assumes "C \<in> N"   and "Pos A \<in># C"   and "A = Max (atms_of C)"   and "S C = {#}" shows "Interp C \<Turnstile> C"
not_None_the_mem: fixes x :: "'a option"   and A :: "'a set" assumes "x \<noteq> None" shows "(the x \<in> A) = (x \<in> A\<^sub>\<bottom>)"
cont: fixes out :: "'e"   and c :: "('d, 'e, 'f) rpv"   and gpv :: "('d, 'e, 'f) gpv"   and input :: "'f"   and \<I> :: "('e, 'f) \<I>"   and x :: "('d, 'e, 'f) gpv" assumes "IO out c \<in> set_spmf (the_gpv gpv)"   and "input \<in> responses_\<I> \<I> out"   and "x \<in> sub_gpvs \<I> (c input)" shows "x \<in> sub_gpvs \<I> gpv"
is_digit_intro: fixes n :: "nat" shows "is_digit_char (digit_to_char n) = is_digit n"
free_vars: shows "free_vars e \<subseteq> set vs \<union> set vs'"
if\<^sub>2: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and b :: "bool"   and e1 :: "exp"   and e2 :: "exp"   and S :: "stack_elem list" shows "(\<Gamma>, Bool b, Alts e1 e2 # S) \<Rightarrow> (\<Gamma>, if b then e1 else e2, S)"
vconverse_vconst_on: fixes A :: "V"   and c :: "V" shows "(vconst_on A c)\<inverse>\<^sub>\<circ> = set {c} \<times>\<^sub>\<circ> A"
quorum_union: fixes Q :: "'a set"   and Q' :: "'a set" assumes "i1.orig.quorum Q"   and "i1.orig.quorum Q'" shows "i1.orig.quorum (Q \<union> Q')"
concat_map_cons_rev: fixes j :: "'b"   and xss :: "'b list list" shows "rev (concat (map ((#) j) xss)) = concat (map (\<lambda>xs. xs @ [j]) (rev (map rev xss)))"
null_eqI: fixes n :: "'a" assumes "\<And>f. n \<star> f = n \<and> f \<star> n = n" shows "n = H.null"
remove_max_val: fixes t :: "'b"   and m :: "'a"   and t' :: "'b" assumes "\<not> is_empty t"   and "(m, t') = remove_max t" shows "m = val (as_tree t)"
mismatchPres: fixes P :: "pi"   and Rel :: "(pi \<times> pi) set"   and Q :: "pi"   and Rel' :: "(pi \<times> pi) set"   and a :: "name"   and b :: "name" assumes "P \<leadsto><Rel> Q"   and "Rel \<subseteq> Rel'" shows "[a\<noteq>b]P \<leadsto><Rel'> [a\<noteq>b]Q"
mult_left_le: fixes one :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and c :: "'d"   and a :: "'d" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero"   and "less_eq c one"   and "less_eq zero a" shows "less_eq (times a c) a"
a1: shows "s' = snd (fst (load_sub2 addr asi r win w s)) \<and> get_S (cpu_reg_val PSR s) = 0"
intro: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and s :: "nat"   and t :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'a"   and k :: "nat set" assumes "NFlow c s t f"   and "NCut c s t k" shows "NFlowCut c s t f k"
matchId: fixes a :: "name"   and P :: "pi" shows "[a\<frown>a]P \<sim>\<^sup>s\<^sub>l P"
dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t_steps_iff_7: fixes l :: "'d strand_label"   and X :: "'f list"   and F :: "(('e, 'f) Term.term \<times> ('e, 'f) Term.term) list"   and G :: "(('e, 'f) Term.term \<times> ('e, 'f) Term.term) list"   and A :: "('d strand_label \<times> ('e, 'f) stateful_strand_step) list" shows "((l, \<forall>X\<langle>\<or>\<noteq>: F \<or>\<notin>: G\<rangle>) \<in> set A) = ((l, \<forall>X\<langle>\<or>\<noteq>: F \<or>\<notin>: G\<rangle>) \<in> set (dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t A))"
iTILL_iIN_subset_conv: fixes n' :: "nat"   and n :: "nat"   and d :: "nat" shows "([\<dots>n'] \<subseteq> [n\<dots>,d]) = (n = 0 \<and> n' \<le> d)"
meet_ord_def: fixes x :: "'a"   and y :: "'a" shows "(ar x \<le> ar y) = (ar y \<cdot> ar x = ar x)"
ideal_points_conjugate: fixes H :: "circline" assumes "is_poincare_line H" shows "ideal_points (conjugate_circline H) = conjugate ` ideal_points H"
nl: assumes "w1 = NEG_INF \<and> w2 \<noteq> NEG_INF \<and> w2 <s 0" shows "P w1 w2"
continuous_discrete_range_constant: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "connected S"   and "continuous_on S f"   and "\<And>x. x \<in> S \<Longrightarrow> \<exists>e>0. \<forall>y. y \<in> S \<and> f y \<noteq> f x \<longrightarrow> e \<le> norm (f y - f x)" shows "f constant_on S"
add_mset_multisets_iff: fixes a :: "'b"   and M :: "'b multiset"   and A :: "'b set" shows "(add_mset a M \<in> multisets A) = (a \<in> A \<and> M \<in> multisets A)"
transfer_rule_of_bool: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool" assumes "R (0::'b) (0::'c)"   and "R (1::'b) (1::'c)" shows "((=) ===> R) of_bool of_bool"
O_assoc: fixes R :: "('e \<times> 'h) set"   and S :: "('h \<times> 'g) set"   and T :: "('g \<times> 'f) set" shows "(R O S) O T = R O S O T"
T'_def: shows "T' = (\<lambda>n. n * (n - 1) div 2)"
absorb2: fixes b :: "'c DBMEntry"   and a :: "'c DBMEntry" assumes "b \<preceq> a" shows "linordered_monoid.min a b = b"
unfoldE_1: fixes P :: "pi"   and Q :: "pi" assumes "P \<simeq> Q" shows "P \<leadsto><weakBisim> Q"
divide_poly_list_def: fixes f :: "'b poly"   and g :: "'b poly" shows "divide_poly_list f g = (let cg = coeffs g in if cg = [] then g else let cf = coeffs f; cgr = rev cg in poly_of_list (divide_poly_main_list (hd cgr) [] (rev cf) cgr (1 + length cf - length cg)))"
rep_zero_iff: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and equal :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and rep_nat :: "'a \<Rightarrow> nat"   and abs_nat :: "nat \<Rightarrow> 'a"   and x :: "'a" assumes "class.nat minus plus zero equal less_eq less rep_nat abs_nat" shows "(rep_nat x = 0) = (x = zero)"
order_on_defs_3: fixes A :: "'b set"   and r :: "('b \<times> 'b) set" shows "linear_order_on A r \<equiv> partial_order_on A r \<and> total_on A r"
alg_dvd_mono: fixes a :: "'b"   and c :: "'b"   and b :: "'b"   and d :: "'b" assumes "a alg_dvd c"   and "b alg_dvd d" shows "a * b alg_dvd c * d"
pref_prolong': fixes u :: "'b list"   and w :: "'b list"   and z :: "'b list"   and v :: "'b list" assumes "u \<le>p w \<cdot> z"   and "v \<cdot> u \<le>p z" shows "u \<le>p w \<cdot> v \<cdot> u"
is_unit_gcd_iff: fixes a :: "'a"   and b :: "'a" shows "is_unit (gcd a b) = (gcd a b = (1::'a))"
exhaust: fixes y :: "('l, 'm, 'n) state" obtains x1 :: "'l" and x2 :: "'m" and x3 :: "'n list" where "y = State x1 x2 x3"
g_1: fixes x :: "('a \<Rightarrow> real) \<Rightarrow> real" assumes "x \<in> tangent_space p" shows "g x \<in> sub.tangent_space p"
LIMSEQ_offset: fixes f :: "nat \<Rightarrow> 'b"   and k :: "nat"   and a :: "'b" assumes "(\<lambda>n. f (n + k)) \<longlonglongrightarrow> a" shows "f \<longlonglongrightarrow> a"
poly_plus_rightE: fixes p :: "'c poly"   and x :: "'c"   and y :: "'c" obtains c :: "'c" where "poly p (x + y) = poly p x + c * y"
empty_type: shows "{} \<in> {S. \<exists>m n. {m..n} = S}"