hcju/isabelleps huggingface dataset:
twice_power: fixes a :: "nat" shows "2 * 2 ^ a = Suc (Suc (2 * bl_bin (Oc \<up> a)))"
supt_imp_funas_term_subset: fixes s :: "('c, 'd) Term.term"   and t :: "('c, 'd) Term.term" assumes "s \<rhd> t" shows "funas_term t \<subseteq> funas_term s"
inj_map: fixes f1 :: "'h \<Rightarrow> 'i"   and f2 :: "'j \<Rightarrow> 'k" assumes "inj f1"   and "inj f2" shows "inj (map_sum f1 f2)"
dependent_insertD: fixes a :: "'b"   and S :: "'b set" assumes "a \<notin> span S"   and "dependent (insert a S)" shows "dependent S"
reachable_snapshot_fun_upd: fixes s :: "'g process_name \<Rightarrow> ('f, 'g, 'h, 'e) Model.local_state"   and fld :: "'g process_name"   and val :: "('f, 'g, 'h, 'e) Model.local_state"   and m :: "'g" assumes "roots ((s(fld := val)) (mutator m)) = roots (s (mutator m))"   and "ghost_honorary_root ((s(fld := val)) (mutator m)) = ghost_honorary_root (s (mutator m))"   and "\<And>x. (x \<in> (\<Union>p. WL p (s(fld := val)))) = (x \<in> (\<Union>p. WL p s))"   and "sys_fM (s(fld := val)) = sys_fM s"   and "\<And>x. \<Union> (ran ` obj_fields ` set_option (sys_heap (s(fld := val)) x)) = \<Union> (ran ` obj_fields ` set_option (sys_heap s x))"   and "\<And>x. map_option obj_mark (sys_heap (s(fld := val)) x) = map_option obj_mark (sys_heap s x)"   and "filter (\<lambda>s. is_mw_Mutate s \<or> is_mw_Mutate_Payload s) (sys_mem_store_buffers (mutator m) (s(fld := val))) = filter (\<lambda>s. is_mw_Mutate s \<or> is_mw_Mutate_Payload s) (sys_mem_store_buffers (mutator m) s)" shows "mut_m.reachable_snapshot_inv m (s(fld := val)) = mut_m.reachable_snapshot_inv m s"
hyps_3: shows "st \<noteq> sh \<and> (st, s') \<in> (R \<inter> UNIV \<times> - {sh})\<^sup>* \<or> (st, sh) \<in> R\<^sup>* \<and> (sh, s') \<in> (R \<inter> UNIV \<times> - {sh})\<^sup>*"
arity_sym_ne_infinity_if_\<delta>_gt_0: fixes ground_heads_var :: "'a \<Rightarrow> 'b set"   and gt_sym :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and arity_sym :: "'b \<Rightarrow> enat"   and arity_var :: "'a \<Rightarrow> enat"   and wt_sym :: "'b \<Rightarrow> 'c"   and \<epsilon> :: "nat"   and \<delta> :: "nat"   and extf :: "'b \<Rightarrow> (('b, 'a) tm \<Rightarrow> ('b, 'a) tm \<Rightarrow> bool) \<Rightarrow> ('b, 'a) tm list \<Rightarrow> ('b, 'a) tm list \<Rightarrow> bool"   and f :: "'b" assumes "kbo_std_basis ground_heads_var gt_sym arity_sym arity_var wt_sym \<epsilon> \<delta> extf"   and "0 < \<delta>" shows "arity_sym f \<noteq> \<infinity>"
pseudo_divmod_main_list_invar: fixes d :: "'b list"   and lc :: "'b"   and q :: "'b list"   and r :: "'b list"   and n :: "nat"   and q' :: "'b list"   and r' :: "'b list" assumes "last d \<noteq> (0::'b)"   and "last d = lc"   and "d \<noteq> []"   and "pseudo_divmod_main_list lc q (rev r) (rev d) n = (q', rev r')"   and "n = 1 + length r - length d" shows "pseudo_divmod_main lc (monom (1::'b) n * Poly q) (Poly r) (Poly d) (length r - 1) n = (Poly q', Poly r')"
aext_and: fixes P :: "'b upred"   and Q :: "'b upred"   and x :: "'b \<Longrightarrow> 'a" shows "(P \<and> Q) \<oplus>\<^sub>p x = (P \<oplus>\<^sub>p x \<and> Q \<oplus>\<^sub>p x)"
plus_floatarith_def: fixes f :: "floatarith"   and g :: "floatarith" shows "f + g = Add f g"
IH: shows "\<exists>n. (derived G ^^ n) K = {\<one>}"
assms_5: shows "\<langle>v|v\<rangle> = 1"
finite_intersection_of_complement: fixes P :: "'b set \<Rightarrow> bool"   and S :: "'b set" shows "(finite intersection_of P) S = (finite union_of (\<lambda>S. P (- S))) (- S)"
balance_right_in_tree: fixes l :: "('a, 'b) rbt"   and r :: "('a, 'b) rbt"   and x :: "'a"   and y :: "'b"   and k :: "'a"   and v :: "'b" assumes "inv1 l"   and "inv1l r"   and "bheight l = bheight r + 1"   and "inv2 l"   and "inv2 r" shows "((x, y) \<in> set (entries (balance_right l k v r))) = ((x, y) \<in> set (entries l) \<or> x = k \<and> y = v \<or> (x, y) \<in> set (entries r))"
K_GDD_axioms: shows "K_GDD \<V> \<B> \<K> \<G>"
foldr_conv_foldl: fixes f :: "'h \<Rightarrow> 'g \<Rightarrow> 'g"   and xs :: "'h list"   and a :: "'g" shows "foldr f xs a = foldl (\<lambda>x y. f y x) a (rev xs)"
simps_23: fixes vc :: "'a acom"   and vd :: "'a acom"   and v :: "char list"   and va :: "aexp"   and vb :: "'a" shows "(vc;; vd \<sqsubseteq> v ::= va {vb}) = False"
local: fixes \<sigma> :: "'a"   and s :: "'b"   and a :: "'c"   and \<sigma>' :: "'a"   and s' :: "'b" assumes "(\<sigma>, s) \<in> oreachable A S U"   and "P (\<sigma>, s)"   and "Q ((\<sigma>, s), a, \<sigma>', s')"   and "S \<sigma> \<sigma>' a" shows "P (\<sigma>', s')"
gauss_jordan_main_i: fixes nr :: "nat"   and A' :: "'a mat"   and nc :: "nat"   and A :: "'i mat"   and i :: "nat"   and j :: "nat"   and B' :: "'a mat" assumes "nr = dim_row A'"   and "nc = dim_col A'"   and "mat_rel R A A'"   and "i \<le> nr"   and "j \<le> nc" shows "mat_rel R (gauss_jordan_main_i ops nr nc A i j) (fst (gauss_jordan_main A' B' i j))"
rep_eq: fixes x :: "'c \<Rightarrow>\<^sub>L 'b"   and xa :: "'a \<Rightarrow>\<^sub>L 'c" shows "(*\<^sub>v) (x o\<^sub>L xa) = (*\<^sub>v) x \<circ> (*\<^sub>v) xa"
elims: fixes n :: "nat"   and fs :: "'a vec list"   and x :: "nat"   and y :: "'a vec" assumes "gram_schmidt_fs.gso n fs x = y" obtains i :: "nat" where "x = i" and "y = fs ! i + abelian_monoid.sumlist (module_vec TYPE('a) n) (map (\<lambda>j. - gram_schmidt_fs.\<mu> n fs i j \<cdot>\<^sub>v gram_schmidt_fs.gso n fs j) [0..<i])"
hyps_2: fixes y :: "ty"   and \<Delta> :: "binding list"   and X :: "tyvrs"   and \<Gamma> :: "binding list"   and M :: "ty"   and N :: "ty"   and P :: "ty" assumes "size_ty y < size_ty Q"   and "(\<Delta> @ [TVarB X y] @ \<Gamma>)\<turnstile>M<:N"   and "\<Gamma>\<turnstile>P<:y" shows "(\<Delta> @ [TVarB X P] @ \<Gamma>)\<turnstile>M<:N"
interaction_bounded_by_DoneI: fixes "consider" :: "'d \<Rightarrow> bool"   and x :: "'e" shows "interaction_bounded_by consider (Generative_Probabilistic_Value.Done x) 0"
7: shows "bprv (eqv \<psi> (subst \<phi> \<langle>\<psi>\<rangle> xx))"
is_pot_ordD2: fixes u :: "'t"   and v :: "'t" assumes "is_pot_ord"   and "u \<preceq>\<^sub>t v" shows "component_of_term u \<le> component_of_term v"
is_projection_on_iff_orthog: fixes M :: "'b set"   and \<pi> :: "'b \<Rightarrow> 'b" assumes "closed_csubspace M" shows "is_projection_on \<pi> M = (\<forall>h. h - \<pi> h \<in> orthogonal_complement M \<and> \<pi> h \<in> M)"
not_const_vs_to_es_list: fixes e :: "e"   and vs1 :: "e list"   and vs2 :: "e list"   and vs :: "v list" assumes "\<not> is_const e" shows "vs1 @ [e] @ vs2 \<noteq> $$* vs"
lcm_C: fixes a :: "'a"   and b :: "'a" shows "(a \<le> b) = (a l\<rightarrow> b = (1::'a))"
perm_2: fixes pi :: "(name \<times> name) list"   and x3 :: "name"   and x2 :: "name"   and x1 :: "pi" shows "pi \<bullet> x3{x2}.x1 = (pi \<bullet> x3){(pi \<bullet> x2)}.(pi \<bullet> x1)"
OUTfromChCorrect_data6: shows "OUTfromChCorrect data6"
int_zle_neg: fixes n :: "nat"   and m :: "nat" shows "(int n \<le> - int m) = (n = 0 \<and> m = 0)"
Suc: shows "ack i (Suc j) \<le> ack (Suc i) j"
ETP_base_axioms: fixes n :: "nat" shows "ETP_base (I n) local.domain local.range F F\<^sub>i\<^sub>n\<^sub>v"
len_bv_n: fixes n :: "nat" shows "\<forall>xs\<in>set_pmf (bv n). length xs = n"
x: shows "x \<le> - Itm vs (x # bs) s / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>"
fresh_ssubst_fm: fixes i :: "'a"   and V :: "name set"   and A :: "fm" assumes "atom i \<sharp> p \<bullet> V"   and "V \<subseteq> Va__" shows "atom i \<sharp> ssubst \<lfloor>A\<rfloor>V V F"
Abs_sat_inverse: fixes y :: "nat" assumes "y \<in> {..LENGTH('b)}" shows "nat_of (Abs_sat y) = y"
group_eq_pow_eq_mod: fixes G :: "('a, 'b) cyclic_group_scheme"   and a :: "nat"   and b :: "nat" assumes "Cyclic_Group.cyclic_group G"   and "\<^bold>g\<^bsub>G\<^esub> [^]\<^bsub>G\<^esub> a = \<^bold>g\<^bsub>G\<^esub> [^]\<^bsub>G\<^esub> b"   and "0 < order G" shows "[a = b] (mod order G)"
list_rel_list_rel_order_iff: fixes a :: "String.literal list"   and b :: "char list list"   and a' :: "String.literal list"   and b' :: "char list list" assumes "(a, b) \<in> monom_rel"   and "(a', b') \<in> monom_rel" shows "(a < a') = (b < b')"
hash_blind_view: fixes view :: "view" shows "hash_view (blind_view view) = hash_view (embed_view view)"
b': shows "b \<in> sets (Pi\<^sub>F (Collect finite) (\<lambda>_. borel))"
x'_def: shows "x' = geodesic_segment_param {px--x} px M"
Rel2Mon_def: fixes S :: "('a \<times> 'a) set" shows "Rel2Mon S = (\<lambda>\<sigma>. if \<exists>\<sigma>'. (\<sigma>, \<sigma>') \<in> S then Some ((), SOME \<sigma>'. (\<sigma>, \<sigma>') \<in> S) else None)"
float_minus: fixes a :: "int"   and b :: "int" shows "- float (a, b) = float (- a, b)"
that: fixes f :: "nat \<Rightarrow> nat" assumes "f permutes {..<length ls}"   and "permute_list f ls = map ((!) ls) (filter (\<lambda>i. i \<notin> set ss) [0..<length ls]) @ map ((!) ls) ss" shows "thesis"
e0'\<phi>e: shows "E0' \<cdot> \<phi> = E"
distinct_19: fixes x21 :: "floatarith"   and x22 :: "floatarith"   and x23 :: "form"   and x61 :: "form"   and x62 :: "form" shows "Assign x21 x22 x23 \<noteq> Conj x61 x62"
proves_beta_steps: fixes \<Theta> :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set"   and \<Gamma> :: "Core.term set"   and t :: "Core.term"   and u :: "Core.term" assumes "wf_theory \<Theta>"   and "finite \<Gamma>"   and "term_ok \<Theta> t"   and "t \<rightarrow>\<^sub>\<beta>\<^sup>* u"   and "\<forall>A\<in>\<Gamma>. term_ok \<Theta> A"   and "\<forall>A\<in>\<Gamma>. typ_of A = Some propT" shows "\<Theta>,\<Gamma> \<turnstile> mk_eq t u"
dvd_power_same: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and x :: "'a"   and y :: "'a"   and n :: "nat" assumes "class.comm_semiring_1 times one plus zero"   and "dvd.dvd times x y" shows "dvd.dvd times (power.power one times x n) (power.power one times y n)"
coreflexive_idempotent: fixes x :: "'a" assumes "x \<le> top" shows "x \<sqinter> x = x"
distinct_6: fixes x41 :: "expr"   and x42 :: "cmd"   and x43 :: "cmd" shows "if (x41) x42 else x43 \<noteq> Skip"
rev_induct: fixes P :: "'d list \<Rightarrow> bool"   and xs :: "'d list" assumes "P []"   and "\<And>x xs. P xs \<Longrightarrow> P (xs @ [x])" shows "P xs"
measurable_cong_simp: fixes M :: "'c measure"   and N :: "'c measure"   and M' :: "'d measure"   and N' :: "'d measure"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "M = N"   and "M' = N'"   and "\<And>w. w \<in> space M \<Longrightarrow> f w = g w" shows "(f \<in> M \<rightarrow>\<^sub>M M') = (g \<in> N \<rightarrow>\<^sub>M N')"
rev_is_Nil_conv: fixes xs :: "'c list" shows "(rev xs = []) = (xs = [])"
Resp_def: fixes \<delta> :: "'e \<Rightarrow> 'f \<times> 'g \<Rightarrow> 'e"   and \<gamma> :: "'e \<Rightarrow> 'f \<times> 'g \<Rightarrow> 'h"   and bot :: "'e"   and p :: "'f"   and ou :: "'h"   and s :: "('e, 'f, 'g) SLin_state"   and s' :: "('e, 'f, 'g) SLin_state" assumes "SLin \<delta> \<gamma> bot" shows "SLin.Resp \<delta> \<gamma> bot p ou s s' \<equiv> status s p = Pending \<and> initialized s \<and> pre_RDR.contains \<delta> bot (dstate s) (pending s p) \<and> ou = \<gamma> (dstate s) (pending s p) \<and> s' = s\<lparr>status := (status s)(p := Ready)\<rparr>"
ProcRec: fixes Procs :: "'d set"   and \<Gamma> :: "'d \<Rightarrow> ('f, 'd, 'g) com option"   and \<Theta> :: "('f set \<times> 'd \<times> 'f set \<times> 'f set) set"   and P :: "'d \<Rightarrow> 'e \<Rightarrow> 'f set"   and Q :: "'d \<Rightarrow> 'e \<Rightarrow> 'f set"   and A :: "'d \<Rightarrow> 'e \<Rightarrow> 'f set"   and F :: "'g set" assumes "\<forall>p\<in>Procs. \<forall>Z. \<Gamma>,\<Theta> \<union> (\<Union>p\<in>Procs. \<Union>Z. {(P p Z, p, Q p Z, A p Z)})\<turnstile>\<^bsub>/F \<^esub>(P p Z) the (\<Gamma> p) (Q p Z),(A p Z)"   and "Procs \<subseteq> dom \<Gamma>" shows "\<forall>p\<in>Procs. \<forall>Z. \<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>(P p Z) Call p (Q p Z),(A p Z)"
SUP_le_iff: fixes f :: "'e \<Rightarrow> 'c \<times> 'd"   and A :: "'e set"   and u :: "'c \<times> 'd" shows "less_eq_prod (Sup_prod (\<P> f A)) u = (\<forall>i\<in>A. less_eq_prod (f i) u)"
hk: fixes y :: "'b \<Rightarrow>\<^sub>0 int" assumes "Poly_Mapping.keys y \<subseteq> T" shows "Poly_Mapping.keys y \<subseteq> T \<and> h (k y) = y"
disc_145: fixes x9 :: "formula list" shows "is_Ands (Ands x9)"
rtrancl_converse: fixes r :: "('b \<times> 'b) set" shows "(r\<inverse>)\<^sup>* = (r\<^sup>*)\<inverse>"
conj_one_left: fixes x :: "'a" shows "\<one> \<^bold>\<sqinter> x = x"
neq_Leaf_iff: fixes t :: "'b tree" shows "(t \<noteq> \<langle>\<rangle>) = (\<exists>l a r. t = \<langle>l, a, r\<rangle>)"
mono_Ndet2: fixes P :: "'a process"   and Q :: "'a process"   and S :: "'a process" assumes "P \<sqsubseteq> Q" shows "\<forall>s. s \<notin> D (P |-| S) \<longrightarrow> Ra (P |-| S) s = Ra (Q |-| S) s"
comp_def: fixes f :: "'f \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'f" shows "f \<circ> g = (\<lambda>x. f (g x))"
behaves_def: shows "(\<Down>) = load OO (\<down>)"
ZOretrT_def: fixes theta :: "(('test, 'atom) com \<times> ('test, 'atom) com) set" shows "ZOretrT theta \<equiv> {(c, d). matchC_ZOC theta c d \<and> matchT_T c d}"
row_add_iterate_eq_Gauss_Jordan_in_ij: fixes A :: "(('b, 'c) vec, 'd) vec"   and i :: "'d"   and j :: "'c" shows "row_add_iterate (mult_row (interchange_rows A i (LEAST n. A $ n $ j \<noteq> (0::'b) \<and> i \<le> n)) i ((1::'b) / interchange_rows A i (LEAST n. A $ n $ j \<noteq> (0::'b) \<and> i \<le> n) $ i $ j)) (nrows A - 1) i j = Gauss_Jordan_in_ij A i j"
comp_ide_arr: fixes b :: "(('c + 'd) set, 'c + 'd \<Rightarrow> 'c + 'd) concrete_category.arr"   and f :: "(('c + 'd) set, 'c + 'd \<Rightarrow> 'c + 'd) concrete_category.arr" assumes "S.ide b"   and "S.seq b f" shows "S.comp b f = f"
walk_tl: fixes xs :: "'a list" assumes "walk xs" shows "walk (tl xs)"
sub3_preserves_dimensions_1: fixes sel_fun :: "(nat \<times> 'a) list \<Rightarrow> nat"   and mf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a"   and d :: "nat"   and l :: "nat"   and r :: "'a"   and A :: "'a mat"   and r' :: "'a"   and A' :: "'a mat" assumes "sub3 sel_fun mf d l (r, A) = (r', A')" shows "dim_row A' = dim_row A"
return_time_def: fixes f :: "'b \<Rightarrow> 'b"   and X :: "'b set"   and P :: "'b set"   and x :: "'b" assumes "auto_ll_on_open f X" shows "auto_ll_on_open.return_time f X P x = (if auto_ll_on_open.returns_to f X P x then SOME t. 0 < t \<and> t \<in> auto_ll_on_open.existence_ivl0 f X x \<and> auto_ll_on_open.flow0 f X x t \<in> P \<and> (\<forall>s\<in>{0<..<t}. auto_ll_on_open.flow0 f X x s \<notin> P) else 0)"
trivial_limit_at_top_linorder: shows "at_top \<noteq> \<bottom>"
fib_conv_gen_fib: fixes n :: "nat" shows "fib n = gen_fib 0 1 n"
model_to_dimacs_model_append': fixes M :: "nat \<Rightarrow> bool"   and vs :: "nat list"   and vs' :: "nat list" shows "model_to_dimacs_model M (vs @ vs') = model_to_dimacs_model M vs @ model_to_dimacs_model M vs'"
continuous_on_def: fixes s :: "'d set"   and f :: "'d \<Rightarrow> 'e" shows "continuous_on s f = (\<forall>x\<in>s. (f \<longlongrightarrow> f x) (at x within s))"
Arg_unique: fixes r :: "real"   and a :: "real"   and z :: "complex" assumes "complex_of_real r * exp (\<i> * complex_of_real a) = z"   and "0 < r"   and "- pi < a"   and "a \<le> pi" shows "Arg z = a"
new_shadow_root_get_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t: fixes h :: "(_) heap"   and h' :: "(_) heap"   and new_shadow_root_ptr :: "(_) shadow_root_ptr"   and ptr :: "(_) document_ptr"   and getter :: "('z, 'x, 'y, (\<lparr>RShadowRoot.nothing :: unit, mode :: shadow_root_mode, child_nodes :: (_) node_ptr list, \<dots> :: 'ag option\<rparr> + 'ah) option) RDocument_scheme \<Rightarrow> 'aj" assumes "h \<turnstile> new\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M \<rightarrow>\<^sub>h h'"   and "h \<turnstile> new\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M \<rightarrow>\<^sub>r new_shadow_root_ptr"   and "ptr \<noteq> cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r new_shadow_root_ptr" shows "preserved (get_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr getter) h h'"
eval_fps_binomial: fixes z :: "complex"   and c :: "complex" assumes "cmod z < 1" shows "eval_fps (fps_binomial c) z = (1 + z) powr c"
lookup_raw_filter_raw: fixes xs :: "('a \<times> 'c) list \<times> 'o"   and P :: "'a \<times> 'c \<Rightarrow> bool"   and k :: "'a" assumes "oalist_inv xs" shows "lookup_raw (filter_raw P xs) k = (let v = lookup_raw xs k in if P (k, v) then v else (0::'c))"
DisP: fixes A :: "'b set"   and H :: "(('c, 'b) fm list \<times> 'b) set"   and i :: "'b"   and p :: "('c, 'b) fm"   and q :: "('c, 'b) fm" assumes "Hintikka A H"   and "(p \<^bold>\<or> q) at i in' H" shows "p at i in' H \<or> q at i in' H"
secure_while_aux: fixes n :: "nat"   and k :: "nat"   and m :: "nat"   and s :: "(Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list"   and b :: "BExpr"   and C :: "OBJ"   and ss :: "(Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list"   and t :: "(Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list"   and tt :: "(Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list"   and \<beta> :: "(Location \<times> Location) set" assumes "n \<le> k"   and "m \<le> k"   and " s , While b C \<rightarrow>\<^sub>n  ss "   and " t , While b C \<rightarrow>\<^sub>m  tt "   and "secure C"   and "\<And>s ss \<beta>. s \<equiv>\<^sub>\<beta> ss \<Longrightarrow> evalB b (fst s) = evalB b (fst ss)"   and "s \<equiv>\<^sub>\<beta> t" shows "\<exists>\<gamma>. ss \<equiv>\<^sub>\<gamma> tt \<and> Pbij_extends \<gamma> \<beta>"
Un_set_offending_flows_bound_minus_subseteq: fixes V :: "'a set"   and E :: "('a \<times> 'a) set"   and nP :: "'a \<Rightarrow> 'b"   and X :: "('a \<times> 'a) set"   and E' :: "('a \<times> 'a) set" assumes "wf_graph \<lparr>nodes = V, edges = E\<rparr>"   and "\<Union> (set_offending_flows \<lparr>nodes = V, edges = E\<rparr> nP) \<subseteq> X" shows "\<Union> (set_offending_flows \<lparr>nodes = V, edges = E - E'\<rparr> nP) \<subseteq> X - E'"
ineq1: fixes x :: "real" assumes "x \<in> {0..1}" shows "x * (1 - x) \<le> 1 / 4"
remap_lang: fixes f :: "'Q2 \<Rightarrow> 'a" assumes "inj_on f a2.Q" shows "ta_lang (ta_remap f TA') = ta_lang TA'"
card_nth_roots: fixes c :: "complex"   and n :: "nat" assumes "c \<noteq> 0"   and "0 < n" shows "card {z. z ^ n = c} = n"
invoke_not: fixes Pred :: "(_) object_ptr \<Rightarrow> bool"   and ptr :: "(_) object_ptr"   and f :: "(_) object_ptr \<Rightarrow> 'o \<Rightarrow> ((_) heap, exception, 'n) prog"   and xs :: "(((_) object_ptr \<Rightarrow> bool) \<times> ((_) object_ptr \<Rightarrow> 'o \<Rightarrow> ((_) heap, exception, 'n) prog)) list"   and args :: "'o" assumes "\<not> Pred ptr" shows "invoke ((Pred, f) # xs) ptr args = invoke xs ptr args"
HistP_def: fixes rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState"   and p :: "PrIds" shows "HistP rmhist p = (Init HInit rmhist p \<and> \<box>[HNext rmhist p]_(c p, r p, m p, rmhist!p))"
gso_connect: fixes us :: "'a vec list" shows "snd (main us) = gram_schmidt n us"
neutral: fixes A :: "'a multiset" assumes "\<forall>x\<in>#A. x = (0::'a)" shows "\<Sum>\<^sub># A = (0::'a)"
col_mult2: fixes A :: "'b mat"   and nr :: "nat"   and n :: "nat"   and B :: "'b mat"   and nc :: "nat"   and j :: "nat" assumes "A \<in> carrier_mat nr n"   and "B \<in> carrier_mat n nc"   and "j < nc" shows "col (A * B) j = A *\<^sub>v col B j"
poincare_line_poincare_collinear3_general: fixes a :: "complex_homo"   and a1 :: "complex_homo"   and a2 :: "complex_homo" assumes "a \<in> circline_set (poincare_line a1 a2)"   and "a1 \<noteq> a2" shows "poincare_collinear {a, a1, a2}"
theI_unique: fixes P :: "'a \<Rightarrow> bool"   and x :: "'a" assumes "\<exists>!x. P x" shows "P x = (x = The P)"
Nil: fixes le :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "linorder_list0.sortedP le []"
llength_LCons: fixes x :: "'c"   and xs :: "'c llist" shows "llength (LCons x xs) = eSuc (llength xs)"
clo: fixes z :: "'a" assumes "z \<in> A"   and "\<forall>y\<in>S. (y, z) \<in> r" shows "(L, z) \<in> r"
CE_result_correct: shows "proof_result CEProof = ([], [], [CEReq])"
scaleQ_as_qprod: fixes a :: "'e"   and x :: "'e['f, 'g]" shows "a *\<^sub>Q x \<cong>\<^sub>Q (a *\<^sub>Q \<one>) \<^bold>\<cdot> x"
runit_in_hom: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b local.term set" assumes "free_monoidal_category C"   and "partial_magma.ide (free_monoidal_category.comp C) a" shows "partial_magma.in_hom (free_monoidal_category.comp C) (free_monoidal_category.runit\<^sub>F\<^sub>M\<^sub>C C a) (free_monoidal_category.tensor\<^sub>F\<^sub>M\<^sub>C C a (free_monoidal_category.unity\<^sub>F\<^sub>M\<^sub>C C)) a"
remainder_recovery: fixes K :: "'a set"   and A :: "'a set"   and \<psi> :: "'a"   and B :: "'a set"   and \<phi> :: "'a" assumes "K = Cn A"   and "K \<turnstile> \<psi>"   and "B \<in> K .\<bottom>. \<phi>" shows "B \<turnstile> \<phi> .\<longrightarrow>. \<psi>"
poly_card_roots_leq: fixes a :: "real"   and p :: "real poly" shows "card {x. x \<le> a \<and> poly p x = 0} = count_roots_below p a"
Subset_trans: fixes H :: "fm set"   and a :: "tm"   and b :: "tm"   and c :: "tm" assumes "H \<turnstile> a SUBS b"   and "H \<turnstile> b SUBS c" shows "H \<turnstile> a SUBS c"
simps: fixes P :: "'a prog"   and a1 :: "('b \<times> ('c, 'b) obs_event action) list"   and a2 :: "('b \<times> ('c, 'b) obs_event action) llist" shows "ta_hb_consistent P a1 a2 = ((\<exists>obs. a1 = obs \<and> a2 = LNil) \<or> (\<exists>obs ob obs'. a1 = obs \<and> a2 = LCons ob obs' \<and> ta_hb_consistent P (obs @ [ob]) obs' \<and> (case ob of (t, NormalAction (ReadMem ad al v)) \<Rightarrow> \<exists>w. w \<in> write_actions (llist_of (obs @ [ob])) \<and> (ad, al) \<in> action_loc P (llist_of (obs @ [ob])) w \<and> value_written P (llist_of (obs @ [ob])) w (ad, al) = v \<and> P,llist_of (obs @ [ob]) \<turnstile> w \<le>hb length obs \<and> (\<forall>w'\<in>write_actions (llist_of (obs @ [ob])). (ad, al) \<in> action_loc P (llist_of (obs @ [ob])) w' \<longrightarrow> P,llist_of (obs @ [ob]) \<turnstile> w \<le>hb w' \<and> P,llist_of (obs @ [ob]) \<turnstile> w' \<le>hb length obs \<or> is_volatile P al \<and> P,llist_of (obs @ [ob]) \<turnstile> w \<le>so w' \<and> P,llist_of (obs @ [ob]) \<turnstile> w' \<le>so length obs \<longrightarrow> w' = w) | (t, NormalAction _) \<Rightarrow> True | (t, _) \<Rightarrow> True)))"
prm_apply_injective: fixes \<pi> :: "'b prm" shows "inj (($) \<pi>)"
alphaAbs_imp_asAbs_equal: fixes qA :: "('c, 'd, 'varSort, 'var, 'e) qAbs"   and qB :: "('c, 'd, 'varSort, 'var, 'e) qAbs" assumes "qGoodAbs qA"   and "qA $= qB" shows "asAbs qA = asAbs qB"
flat_unfold: fixes ws :: "'d list stream" assumes "shd ws \<noteq> []" shows "flat ws = shd ws @- flat (stl ws)"
unit_pred_cases: fixes P :: "unit pred \<Rightarrow> bool"   and Q :: "unit pred" assumes "P \<bottom>"   and "P (single ())" shows "P Q"
select_convs_4: fixes ext :: "real_int"   and lan :: "nat_int"   and own :: "cars"   and more :: "'a" shows "more \<lparr>ext = ext, lan = lan, own = own, \<dots> = more\<rparr> = more"
poly_length_mult: fixes a :: "'a"   and q :: "'a list" shows "length ([- a, 1::'a] *** q) = Suc (length q)"
eq_ord_equivTr: fixes E :: "('c, 'd) Order_scheme"   and F :: "('c, 'd) Order_scheme" assumes "ord_equiv D E"   and "E = F" shows "ord_equiv D F"