hcju/isabelleps huggingface dataset:
INF_absorb: fixes k :: "'b"   and I :: "'b set"   and A :: "'b \<Rightarrow> 'a" assumes "k \<in> I" shows "A k \<sqinter> Inf (A ` I) = Inf (A ` I)"
assms_3: shows "typ_of f = Some (\<tau> \<rightarrow> \<tau>')"
oprod_wf_Id: fixes r :: "('d \<times> 'd) set"   and r' :: "('e \<times> 'e) set" assumes "Total r"   and "Total r'"   and "wf (r - Id)"   and "wf (r' - Id)" shows "wf (r *o r' - Id)"
external_WT'_iff: fixes P :: "'b prog"   and h :: "'heap"   and a :: "'addr"   and M :: "String.literal"   and vs :: "'addr val list"   and U :: "ty" shows "P,h \<turnstile> a\<bullet>M(vs) : U = (\<exists>hT Ts Ts' D. typeof_addr h a = \<lfloor>hT\<rfloor> \<and> map typeof\<^bsub>h\<^esub> vs = map Some Ts \<and> P \<turnstile> class_type_of hT sees M: Ts'\<rightarrow>U = Native in D \<and> P \<turnstile> Ts [\<le>] Ts')"
psubst_Ax: fixes \<theta>_n :: "(name \<times> coname \<times> trm) list"   and x :: "name"   and c :: "coname"   and P :: "trm"   and \<theta>_c :: "(coname \<times> name \<times> trm) list"   and a :: "coname"   and y :: "name"   and N :: "trm" assumes "\<theta>_n nmaps x to Some (c, P)"   and "\<theta>_c cmaps a to Some (y, N)" shows "\<theta>_n,\<theta>_c<Ax x a> = Cut <c>.P y.N"
lift_ident: fixes inp :: "'a \<Rightarrow> 'b option" shows "lift (\<lambda>v. v) inp = inp"
iteration_double: fixes x :: "'a" shows "(x\<^sup>\<infinity>)\<^sup>\<infinity> = \<top>"
eSuc_eint: fixes n :: "int" shows "eint n + 1 = eint (n + 1)"
star_of_0_eq: fixes y :: "'a" shows "(0 = star_of y) = ((0::'a) = y)"
dagger_trans_eq: fixes x :: "'a" shows "x\<^sup>\<dagger> \<cdot> x\<^sup>\<dagger> = x\<^sup>\<dagger>"
sup_commut_type: fixes \<tau> :: "'a OCL_Types.type"   and \<sigma> :: "'a OCL_Types.type" shows "\<tau> \<squnion> \<sigma> = \<sigma> \<squnion> \<tau>"
asym: fixes a :: "'b"   and b :: "'b" assumes "c1.lt a b"   and "c1.lt b a" shows "False"
poly_deg_zero_imp_monomial: fixes p :: "('c \<Rightarrow>\<^sub>0 'b) \<Rightarrow>\<^sub>0 'd" assumes "poly_deg p = (0::'b)" shows "monomial (lookup p 0) 0 = p"
m_ac_2: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier G"   and "y \<in> carrier G" shows "x \<otimes> y = y \<otimes> x"
bisimSymmetric: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi" assumes "\<Psi> \<rhd> P \<sim> Q" shows "\<Psi> \<rhd> Q \<sim> P"
I_def: shows "I \<equiv> \<Union>i\<in>Basis. \<Union>j\<in>C. {(i, j * e / 2 / real DIM('a))}"
det_bezout_matrix: fixes bezout :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<times> 'd \<times> 'd \<times> 'd \<times> 'd"   and a :: "'e"   and b :: "'e"   and A :: "(('d, 'f) vec, 'e) vec"   and j :: "'f" assumes "is_bezout_ext bezout"   and "a < b"   and "A $ a $ j \<noteq> (0::'d)" shows "det (bezout_matrix A a b j bezout) = (1::'d)"
reduce_basis_inv: fixes fs :: "int vec list" assumes "local.reduce_basis = fs" shows "LLL_invariant True m fs"
While: fixes b :: "bexp"   and C :: "com" assumes "\<turnstile>\<^sub>3 {\<lambda>(s, n). I (s, n) \<and> lmaps_to_axpr b True s} C { I \<and>* $ 1}" shows "\<turnstile>\<^sub>3 {(\<lambda>(s, n). I (s, n) \<and> vars b \<subseteq> dom s) \<and>* $ 1} WHILE b DO C { \<lambda>(s, n). I (s, n) \<and> lmaps_to_axpr b False s}"
sel_7: fixes x1 :: "real"   and x2 :: "real"   and x3 :: "real"   and x4 :: "real"   and x5 :: "real"   and x6 :: "real"   and x7 :: "real"   and x8 :: "real" shows "Im6 (Octo x1 x2 x3 x4 x5 x6 x7 x8) = x7"
double_def: fixes x :: "'b" shows "double x = Introduction.plus_class.plus x x"
arrE: fixes f :: "'c local.term set \<times> 'c local.term set" assumes "\<F>C.CC.arr f" obtains "\<F>C.arr (fst f) \<and> \<F>C.arr (snd f)"
simps_3: fixes vb :: "'b"   and vc :: "'b"   and va :: "('b, 'c) Combinators list" shows "noDenyAll1 (DenyAllFromTo vb vc # va) = noDenyAll (DenyAllFromTo vb vc # va)"
P: fixes n :: "nat" assumes "n < length cl" shows "part {..<brn (cl ! n)} (P n)"
degree_smult_le: fixes a :: "'c"   and p :: "'c poly" shows "degree (smult a p) \<le> degree p"
lift_cm_inv_cm_all': fixes P :: "('loc, 't) Propagate.configuration \<Rightarrow> bool"   and \<Delta> :: "('loc \<times> 't) zmultiset"   and c0 :: "('loc, 't) Propagate.configuration" assumes "\<And>c0 c1 loc t n. \<lbrakk>P c0; next_change_multiplicity' c0 c1 loc t n\<rbrakk> \<Longrightarrow> P c1"   and "\<forall>(loc, t)\<in>#\<^sub>z\<Delta>. \<exists>t'. t' \<in>\<^sub>A frontier (c_imp c0 loc) \<and> t' \<le> t"   and "P c0" shows "P (cm_all' c0 \<Delta>)"
to_fun_diff: fixes p :: "nat \<Rightarrow> nat \<Rightarrow> int"   and q :: "nat \<Rightarrow> nat \<Rightarrow> int"   and a :: "nat \<Rightarrow> int" assumes "p \<in> carrier (UP Zp)"   and "q \<in> carrier (UP Zp)"   and "a \<in> carrier Zp" shows "(p \<ominus>\<^bsub>UP Zp\<^esub> q) \<bullet> a = p \<bullet> a \<ominus> q \<bullet> a"
preserves_arr: fixes f :: "'a arrow_of_spans_data" assumes "V.arr f" shows "V.arr (trg f)"
RHS0: shows "Cring_Poly.compose R (Cring_Poly.compose R (monom P (f (deg R f)) (deg R f)) q) r = Cring_Poly.compose R (a \<odot>\<^bsub>P\<^esub> q [^]\<^bsub>P\<^esub> n) r"
int_word_sint: fixes x :: "int" shows "sint (word_of_int x) = (x + 2 ^ (LENGTH('b) - 1)) mod 2 ^ LENGTH('b) - 2 ^ (LENGTH('b) - 1)"
simps_1: fixes T :: "POPLmark.type" shows "type_ofB (VarB T) = T"
bal_length: fixes n :: "nat"   and xs :: "'a list"   and t :: "'a tree"   and zs :: "'a list" assumes "n \<le> length xs"   and "bal n xs = (t, zs)" shows "length zs = length xs - n"
proj_Cons_1: fixes n :: "'a"   and a :: "'b"   and A :: "('a strand_label \<times> 'b) list" shows "proj n ((ln n, a) # A) = (ln n, a) # proj n A"
R_def: shows "R = {(ct(s := D), ct) |ct s D. ct \<in> Pi\<^sub>E S K \<and> proper ct \<and> s \<in> S\<^sub>r \<and> D \<in> K s \<and> v (simple ct s) < \<integral>\<^sup>+ t. v (simple ct t) \<partial>measure_pmf D}"
rel_spmf_bindI: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'c spmf"   and q :: "'d spmf"   and P :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'e spmf"   and g :: "'d \<Rightarrow> 'f spmf" assumes "rel_spmf R p q"   and "\<And>x y. R x y \<Longrightarrow> rel_spmf P (f x) (g y)" shows "rel_spmf P (p \<bind> f) (q \<bind> g)"
simps_3: fixes f1 :: "'a"   and f2 :: "'a" shows "(case Leq_Rel of Leq_Rel \<Rightarrow> f1 | Lt_Rel \<Rightarrow> f2) = f1"
minim_Under_under: fixes r :: "('a \<times> 'a) set"   and A :: "'a set" assumes "wo_rel r"   and "A \<noteq> {}"   and "A \<subseteq> Field r" shows "Under r A = under r (minim r A)"
sphere_retract_of_punctured_universe: fixes r :: "real"   and a :: "'a" assumes "0 < r" shows "sphere a r retract_of - {a}"
abs_eq: fixes xa :: "'a ltl"   and x :: "'b" shows "f_abs (Abs xa) x = Abs (f xa x)"
simps_3: fixes ty_ent :: "'a \<Rightarrow> PDDL_STRIPS_Semantics.type option" shows "wf_fmla_atom ty_ent \<bottom> = False"
Sup_fun_def: fixes A :: "('b \<Rightarrow> 'c) set" shows "\<Squnion> A = (\<lambda>x. \<Squnion>f\<in>A. f x)"
iso_iff: shows "(gn \<in> Group.iso (reduced_homology_group (int n - 1) (nsphere (n - Suc 0))) (relative_homology_group (int n) (subtopology (nsphere n) {x. 0 \<le> x n}) {x. x n = 0})) = (carrier (relative_homology_group (int n) (subtopology (nsphere n) {x. 0 \<le> x n}) {x. x n = 0}) \<subseteq> gn ` carrier (reduced_homology_group (int n - 1) (nsphere (n - Suc 0))) \<and> (\<forall>x\<in>carrier (reduced_homology_group (int n - 1) (nsphere (n - Suc 0))). gn x = \<one>\<^bsub>relative_homology_group (int n) (subtopology (nsphere n) {x. 0 \<le> x n}) {x. x n = 0}\<^esub> \<longrightarrow> x = \<one>\<^bsub>reduced_homology_group (int n - 1) (nsphere (n - Suc 0))\<^esub>))"
upper_bits_unset_is_l2p: fixes n :: "nat" assumes "n < LENGTH('a)" shows "(\<forall>n'\<ge>n. n' < LENGTH('a) \<longrightarrow> \<not> p !! n') = (p < 2 ^ n)"
sep_conj_empty: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and sep_disj :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and P :: "'c \<Rightarrow> bool" assumes "class.sep_algebra plus zero sep_disj" shows "sep_algebra.sep_conj plus sep_disj P (sep_algebra.sep_empty zero) = P"
prefix_match_semantics_ipset_from_netmask: fixes pfx :: "'a prefix_match"   and a :: "'a word" assumes "valid_prefix pfx" shows "prefix_match_semantics pfx a = (a \<in> ipset_from_netmask (pfxm_prefix pfx) (~~ pfxm_mask pfx))"
simps_2: fixes l :: "('c \<times> nat) option"   and t :: "'c" shows "lock_action_ok l t Unlock = has_lock l t"
univalent_O_converse: fixes R :: "('e \<times> 'd) set" assumes "univalent (R\<inverse>)" shows "R O R\<inverse> = Id_on (Domain R)"
euler_C_nonneg: shows "0 \<le> euler_C TYPE('a)"
set1: shows "set (map ((*) r) ss) \<subseteq> R"
effect_deterministic_1: fixes f :: "'b Heap"   and h :: "heap"   and h' :: "heap"   and a :: "'b"   and h'' :: "heap"   and b :: "'b" assumes "effect f h h' a"   and "effect f h h'' b" shows "a = b"
drop_bit_Suc: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and drop_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and take_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and n :: "nat"   and a :: "'a" assumes "class.semiring_bit_shifts plus minus zero times one divide modulo bit push_bit drop_bit take_bit" shows "drop_bit (Suc n) a = drop_bit n (divide a (numeral.numeral one plus (num.Bit0 num.One)))"
simps_3: fixes P :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and Qannot :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and uu :: "(char list \<Rightarrow> int) \<Rightarrow> nat"   and C :: "acom" shows "varacom ({P/Qannot/uu} CONSEQ C) = support P \<union> varacom C \<union> support Qannot"
l_neg: fixes x :: "'c \<Rightarrow> 'a" assumes "x \<in> carrier F" shows "\<ominus>\<^bsub>F\<^esub> x \<oplus>\<^bsub>F\<^esub> x = \<zero>\<^bsub>F\<^esub>"
find_works: fixes a :: "'c"   and xs :: "('c, 'd) bintree option list" assumes "Some a \<in> set (map (map_option priority) xs)" shows "\<exists>t. Binomial_Queue.find a xs = Some t \<and> priority t = a"
sqmatrix_add_0_left: fixes A :: "('c, 'd) sqmatrix" shows "0 + A = A"
snapshot_algorithm_must_terminate: fixes t :: "('a, 'b, 'c) event list" assumes "trace init t final" shows "\<exists>phi. (\<forall>p. ps (S t phi) p \<noteq> None) \<and> (\<forall>cid. Marker \<notin> set (msgs (S t phi) cid))"
1: shows "length (filter P l) = length (filter P ps) + length (filter P l') + length (filter P ss)"
uminus_ivl_def: shows "uminus \<equiv> map_fun rep_ivl abs_ivl uminus_rep"
weight_spmf_of_pmf: fixes p :: "'c pmf" shows "weight_spmf (spmf_of_pmf p) = 1"
f2: assumes "get_trap_set s \<noteq> {}" shows "reset_trap_val s \<noteq> False \<or> get_ET (cpu_reg_val PSR s) \<noteq> 0"
nr: shows "length (col (mat_multI ze pl ti nr m1 m2) j) = nr"
fold_Cons: fixes f :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'c"   and xs :: "'c list" shows "fold f (x # xs) = fold f xs \<circ> f x"
WTrtFAssNT: fixes 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"   and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and e\<^sub>1 :: "expr"   and e\<^sub>2 :: "expr"   and T' :: "ty"   and T :: "ty"   and F :: "char list"   and Cs :: "char list list" assumes "P,E,h \<turnstile> e\<^sub>1 : NT"   and "P,E,h \<turnstile> e\<^sub>2 : T'"   and "P \<turnstile> T' \<le> T" shows "P,E,h \<turnstile> e\<^sub>1\<bullet>F{Cs} := e\<^sub>2 : T"
simps_1: fixes cert :: "'c list"   and f :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and T :: "'c"   and pc :: "nat"   and x :: "'c" shows "merge cert f r T pc [] x = x"
Spy_not_see_encrypted_key: fixes A :: "agent"   and K' :: "nat"   and NA :: "msg"   and B :: "agent"   and K :: "nat"   and X :: "msg"   and evs :: "event list" assumes "Says Server A (Crypt K' \<lbrace>NA, Agent B, Key K, X\<rbrace>) \<in> set evs"   and "\<forall>NB. Notes Spy \<lbrace>NA, NB, Key K\<rbrace> \<notin> set evs"   and "A \<notin> bad"   and "B \<notin> bad"   and "evs \<in> ns_shared" shows "Key K \<notin> analz (knows Spy evs)"
\<Phi>_ide: 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"   and a :: "'b" assumes "two_set_categories_bij_betw_Univ S S' \<AA> \<phi>"   and "partial_magma.ide S a" shows "two_set_categories_bij_betw_Univ.\<Phi> S S' \<AA> \<phi> a = two_set_categories_bij_betw_Univ.\<Phi>o S S' \<AA> \<phi> a"
extract_var_non_zero_coeff: fixes p :: "'a mpoly"   and v :: "nat" shows "extract_var p v = (\<Sum>m\<in>{m'. MPoly_Type.coeff p m' \<noteq> (0::'a)}. MPoly_Type.monom (remove_key v m) (MPoly_Type.monom (Poly_Mapping.single v (lookup m v)) (MPoly_Type.coeff p m)))"
distinct_keys: fixes t :: "('a, 'b) RBT.rbt" shows "distinct (RBT.keys t)"
inject_2: fixes x2 :: "String.literal"   and y2 :: "String.literal" shows "(Str x2 = Str y2) = (x2 = y2)"
square_free_map_poly: fixes hom :: "'a \<Rightarrow> 'b"   and f :: "'a poly" assumes "field_hom_0' hom" shows "square_free (map_poly hom f) = square_free f"
sum_ennreal: fixes I :: "'a set"   and f :: "'a \<Rightarrow> real" assumes "\<And>i. i \<in> I \<Longrightarrow> 0 \<le> f i" shows "(\<Sum>i\<in>I. ennreal (f i)) = ennreal (sum f I)"
assms_3: shows "z\<^sup>\<smile> ; x = 1"
prems: shows "\<Theta>,\<Gamma> \<turnstile> B"
eventually_at_le: fixes P :: "'a \<Rightarrow> bool"   and a :: "'a"   and S :: "'a set" shows "eventually P (at a within S) = (\<exists>d>0. \<forall>x\<in>S. x \<noteq> a \<and> dist x a \<le> d \<longrightarrow> P x)"
recvmsg_simps_3: fixes P :: "'d \<Rightarrow> bool"   and ip :: "nat"   and m :: "'d" shows "recvmsg P (unicast ip m) = True"
Rel_match_def: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'c"   and y :: "'d" shows "Rel_match R x y = R x y"
l1_step3_refines_a0i_commit_skip_i: fixes A :: "agent"   and B :: "agent"   and gny :: "msg"   and Ra :: "fid_t" shows "{R01iai \<inter> UNIV \<times> l1_inv3} (a0i_commit A B (Exp gny (NonceF (Ra $ nx))))\<^sup>=, l1_step3 Ra A B gny {> R01iai}"
swapAlts_def: fixes a :: "'b"   and b :: "'b"   and u :: "'b" shows "swapAlts a b u \<equiv> if u = a then b else if u = b then a else u"
good_imp_qGood_pick: fixes X :: "('b, 'c, 'varSort, 'var, 'd) qTerm set" assumes "good X" shows "qGood (pick X)"
homeomorphic_imp_surjective_map: fixes X :: "'b topology"   and Y :: "'c topology"   and f :: "'b \<Rightarrow> 'c" assumes "homeomorphic_map X Y f" shows "f ` topspace X = topspace Y"
braun_indices1_le: fixes i :: "nat"   and t :: "'a tree" assumes "i \<in> braun_indices t" shows "Suc 0 \<le> i"
typing_alt_simps_18: fixes \<Gamma> :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type"   and src :: "'a expr"   and k :: "call_kind"   and role :: "String.literal option"   and "from" :: "String.literal"   and \<tau> :: "'a OCL_Types.type" shows "(\<Gamma> \<turnstile>\<^sub>E AssociationEndCall src k role from : \<tau>) = (k = DotCall \<and> (\<exists>\<C> \<D> a aa ab ac b. \<tau> = assoc_end_type (a, aa, ab, ac, b) \<and> \<Gamma> \<turnstile>\<^sub>E src : \<langle>\<C>\<rangle>\<^sub>\<T>[1] \<and> association_end \<C> role from \<D> (a, aa, ab, ac, b)))"
cod_dom: fixes f :: "'b" assumes "D.arr f" shows "D.cod (D.dom f) = D.dom f"
identity_\<Pi>\<^sub>1_def: shows "(\<^bold>=) \<equiv> (\<^bold>=\<^sub>1)"
body'_strict: shows "body'\<cdot>\<bottom> = \<bottom>"
map_option_bind: fixes f :: "'b \<Rightarrow> 'a"   and x :: "'c option"   and g :: "'c \<Rightarrow> 'b option" shows "map_option f (Option.bind x g) = Option.bind x (map_option f \<circ> g)"
rett_instr_result: fixes i :: "sparc_operation \<times> inst_operand list"   and s :: "'b sparc_state" assumes "fst i = ctrl_type RETT \<and> get_ET (cpu_reg_val PSR s) \<noteq> 1 \<and> get_S (cpu_reg_val PSR s) \<noteq> 0 \<and> get_WIM_bit (nat ((uint (get_CWP (cpu_reg_val PSR s)) + 1) mod NWINDOWS)) (cpu_reg_val WIM s) = 0 \<and> get_addr (snd i) s AND 3 = 0" shows "snd (rett_instr i s) = False"
finprod_cong: fixes A :: "'c set"   and B :: "'c set"   and f :: "'c \<Rightarrow> 'a list"   and g :: "'c \<Rightarrow> 'a list" assumes "A = B"   and "(f \<in> B \<rightarrow> carrier (K [X])) = True"   and "\<And>i. i \<in> B =simp=> f i = g i" shows "finsum (K [X]) f A = finsum (K [X]) g B"
select_convs_4: fixes states :: "hf set"   and init :: "hf set"   and final :: "hf set"   and nxt :: "hf \<Rightarrow> 'b \<Rightarrow> hf set"   and eps :: "(hf \<times> hf) set"   and more :: "'c" shows "nfa.nxt \<lparr>nfa.states = states, init = init, final = final, nxt = nxt, eps = eps, \<dots> = more\<rparr> = nxt"
class_wf: fixes P :: "'a prog"   and C :: "String.literal"   and c :: "'a option class"   and wf_md :: "'a prog \<Rightarrow> String.literal \<Rightarrow> 'a mdecl \<Rightarrow> bool" assumes "class P C = \<lfloor>c\<rfloor>"   and "wf_prog wf_md P" shows "wf_cdecl wf_md P (C, c)"
bind_returns_result_E2: fixes h :: "'ap"   and f :: "('ap, 'aq, 'as) prog"   and g :: "'as \<Rightarrow> ('ap, 'aq, 'ar) prog"   and y :: "'ar" assumes "h \<turnstile> Heap_Error_Monad.bind f g \<rightarrow>\<^sub>r y"   and "pure f h" obtains x :: "'as" where "h \<turnstile> f \<rightarrow>\<^sub>r x" and "h \<turnstile> g x \<rightarrow>\<^sub>r y"
token_succeeds_run_merge: fixes \<Sigma> :: "'c set"   and \<delta> :: "'d \<Rightarrow> 'c \<Rightarrow> 'd"   and q\<^sub>0 :: "'d"   and w :: "nat \<Rightarrow> 'c"   and F :: "'d set"   and x :: "nat"   and n :: "nat"   and y :: "nat" assumes "mojmir \<Sigma> \<delta> q\<^sub>0 w F"   and "x \<le> n"   and "y \<le> n"   and "semi_mojmir_def.token_run \<delta> q\<^sub>0 w x n = semi_mojmir_def.token_run \<delta> q\<^sub>0 w y n"   and "mojmir_def.token_succeeds \<delta> q\<^sub>0 w F x" shows "mojmir_def.token_succeeds \<delta> q\<^sub>0 w F y"
execute_operator_def: fixes s :: "'a \<Rightarrow> bool option"   and op :: "'a strips_operator" shows "s \<then> op \<equiv> s ++ map_of (effect_to_assignments op)"
P_new: shows "v0 \<leadsto>P_new\<leadsto> last P_new"
inj_right_lang_imp_eq_app_right_index: assumes "inj_on right_lang (dfa.states M)" shows "bij_betw (\<lambda>q. eq_app_right language `` {path_to q}) accessible (UNIV // eq_app_right language)"
run_alt_writer: fixes alt :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and m :: "('e, 'f, 'd) writerT"   and m' :: "('e, 'f, 'd) writerT" shows "run_writer (alt_writer alt m m') = alt (run_writer m) (run_writer m')"
elims_2: fixes x :: "'b match_expr" assumes "normalized_nnf_match x" obtains "x = MatchAny"   | uu_ :: "'b" where "x = Match uu_"   | uv_ :: "'b" where "x = MatchNot (Match uv_)"   | m1 :: "'b match_expr" and m2 :: "'b match_expr" where "x = MatchAnd m1 m2" and "normalized_nnf_match m1 \<and> normalized_nnf_match m2"
checkpoint_simplex': fixes s :: "QDelta ns_constraint list \<times> (('a, ('a \<times> QDelta atom) list) mapping \<times> ((nat, QDelta) mapping \<Rightarrow> (nat, QDelta) mapping) \<times> 'a list) \<times> ('a, QDelta) state" shows "checkpoint_simplex (Simplex_State s) = Simplex_Checkpoint (checkpoint_simplex' s)"
is_sentence_symbol: fixes a :: "symbol list"   and i :: "nat" assumes "is_sentence a"   and "i < length a" shows "is_symbol (a ! i)"
that: fixes S' :: "ty" assumes "\<Gamma>\<turnstile>U<:S"   and "TVarB X U # \<Gamma> \<turnstile> s : S'"   and "(TVarB X U # \<Gamma>)\<turnstile>S'<:U'" shows "thesis"
length_butlast: fixes xs :: "'g list" shows "length (butlast xs) = length xs - 1"
e: shows "Esigma (g * N) = Esigma g * Esigma N"
absorb_iff2: fixes b :: "'e"   and a :: "'e" shows "(b \<sqsubseteq> a) = (a \<sqinter> b = b)"
substitutes_on_Rf_mono_on: fixes A :: "'a set"   and f :: "'a set \<Rightarrow> 'a set" assumes "substitutes_on A f"   and "finite A" shows "Basis.mono_on (Pow A) (Rf f)"
leq_path_asym2: fixes Cs :: "char list list"   and Cs' :: "char list list"   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 C :: "char list"   and wf_md :: "(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 \<Rightarrow> char list \<Rightarrow> char list \<times> ty list \<times> ty \<times> char list list \<times> expr \<Rightarrow> bool" assumes "P,C \<turnstile> Cs \<sqsubseteq> Cs'"   and "P,C \<turnstile> Cs' \<sqsubseteq> Cs"   and "wf_prog wf_md P" shows "Cs = Cs'"
length_subformulae: fixes F :: "'a formula" shows "length (subformulae F) = size F"
div_by_1: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and a :: "'d" assumes "class.semidom_divide divide plus minus zero times one" shows "divide a one = a"
comm_semiring_ow_axioms_def: fixes U :: "'b set"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" shows "comm_semiring_ow_axioms U plus times \<equiv> \<forall>a b c. a \<in> U \<longrightarrow> b \<in> U \<longrightarrow> c \<in> U \<longrightarrow> times (plus a b) c = plus (times a c) (times b c)"
takeWhile_nth: fixes j :: "nat"   and P :: "'a \<Rightarrow> bool"   and xs :: "'a list" assumes "j < length (takeWhile P xs)" shows "takeWhile P xs ! j = xs ! j"
x: shows "\<forall>x. f (0::'a) x = x"
Sample_def: fixes req :: "nat \<Rightarrow> reqType list"   and dt :: "nat \<Rightarrow> ECall_Info list"   and a1 :: "nat \<Rightarrow> aType list"   and stop :: "nat \<Rightarrow> stopType list"   and lose :: "nat \<Rightarrow> bool list"   and ack :: "nat \<Rightarrow> GatewayStatus list"   and i1 :: "nat \<Rightarrow> ECall_Info list"   and vc :: "nat \<Rightarrow> vcType list" shows "Sample req dt a1 stop lose ack i1 vc \<equiv> msg 1 req \<and> msg 1 a1 \<and> msg 1 stop \<longrightarrow> (\<exists>st buffer. Sample_L req dt a1 stop lose (fin_inf_append [init_state] st) (fin_inf_append [[]] buffer) ack i1 vc st buffer)"
sign_simps_34: fixes bm :: "'b"   and am :: "'b"   and ck :: "'b" shows "bm * (am * ck) = am * (bm * ck)"
symP_R2P: fixes r :: "('c \<times> 'c) set" shows "symP (R2P r) = sym r"
slpa_silent_moves_callstacks_eq: fixes cs :: "'edge list"   and as :: "'edge list"   and S :: "'node SDG_node set"   and f :: "'edge \<Rightarrow> ('var, 'val, 'ret, 'pname) edge_kind"   and m :: "'node"   and msx :: "'node list"   and ms :: "'node list"   and s :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and m' :: "'node"   and ms' :: "'node list"   and s' :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and rs :: "'edge list" assumes "same_level_path_aux cs as"   and "S,f \<turnstile> (m # msx @ ms,s) =as\<Rightarrow>\<^sub>\<tau> (m' # ms',s')"   and "length ms = length ms'"   and "valid_call_list cs m"   and "\<forall>i<length rs. rs ! i \<in> get_return_edges (cs ! i)"   and "valid_return_list rs m"   and "length rs = length cs"   and "msx = targetnodes rs" shows "ms = ms'"
LIMSEQ_inverse_real_of_nat_add_minus_mult: fixes r :: "real" shows "(\<lambda>n. r * (1 + - inverse (real (Suc n)))) \<longlonglongrightarrow> r"
ffold_finsert_fremove: fixes f :: "'e \<Rightarrow> 'f \<Rightarrow> 'f"   and z :: "'f"   and x :: "'e"   and A :: "'e fset" assumes "comp_fun_commute f" shows "ffold f z (finsert x A) = f x (ffold f z (A |-| {|x|}))"
squarefreeI: fixes n :: "'b" assumes "\<And>x. x\<^sup>2 dvd n \<Longrightarrow> x dvd (1::'b)" shows "squarefree n"
simps_2: fixes \<eta> :: "nat" shows "valid_mac_query \<eta> Look = True"