hcju/isabelleps huggingface dataset:
gt_left_comm0: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Gt Bet Cong A B C D" shows "Tarski_neutral_dimensionless.Gt Bet Cong B A C D"
vector_inf_closed: fixes x :: "'a"   and y :: "'a" assumes "x \<sqinter> top = x"   and "y \<sqinter> top = y" shows "x \<sqinter> y \<sqinter> top = x \<sqinter> y"
complete_development_fvs: fixes A :: "'a trm"   and A' :: "'a trm" assumes "A >>> A'" shows "fvs A' \<subseteq> fvs A"
that: fixes ns :: "'node list" assumes "g \<turnstile> Entry g-ns\<rightarrow>n"   and "distinct ns"   and "length ns = shortestPath g n" shows "thesis"
d_mu_Ints: fixes l :: "nat"   and k :: "nat" assumes "l \<le> k"   and "k < length fs" shows "d (Suc l) * \<mu> k l \<in> \<int>"
SuccI2: fixes \<R> :: "('d \<Rightarrow> real) set set"   and X :: "'d set"   and k :: "'d \<Rightarrow> nat"   and v :: "'d \<Rightarrow> real"   and R :: "('d \<Rightarrow> real) set"   and t :: "real"   and R' :: "('d \<Rightarrow> real) set" assumes "\<R> = {region X I r |I r. valid_region X k I r}"   and "v \<in> R"   and "R \<in> \<R>"   and "0 \<le> t"   and "R' = [v \<oplus> t]\<^sub>\<R>" shows "R' \<in> Succ \<R> R"
inject_1: fixes x1 :: "'a"   and y1 :: "'a" shows "(Pos x1 = Pos y1) = (x1 = y1)"
accessible_onI: fixes x :: "'b"   and A :: "'b set"   and P :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "x \<in> A"   and "\<And>y. \<lbrakk>y \<in> A; P y x\<rbrakk> \<Longrightarrow> accessible_on P A y" shows "accessible_on P A x"
simps_1: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" shows "sum_le P Q (Inl x) (Inl y) = P x y"
image_iff: fixes z :: "'b::type"   and f :: "'c::type \<Rightarrow> 'b::type"   and A :: "'c::type set" shows "((z::'b::type) \<in> (f::'c::type \<Rightarrow> 'b::type) ` (A::'c::type set)) = (\<exists>x::'c::type\<in>A. z = f x)"
dagger_sum_unfold: fixes x :: "('a \<times> 'a) set"   and y :: "('a \<times> 'a) set" shows "x\<^sup>* \<union> x\<^sup>* ; y ; (x \<union> y)\<^sup>* = (x \<union> y)\<^sup>*"
SP: fixes x :: "'b" assumes "x \<in> S" shows "P x"
cases: fixes x :: "'b tree \<times> 'b tree" obtains r :: "nat" and x\<^sub>1 :: "'b" and ts\<^sub>1 :: "'b tree list" and r' :: "nat" and x\<^sub>2 :: "'b" and ts\<^sub>2 :: "'b tree list" where "x = (Node r x\<^sub>1 ts\<^sub>1, Node r' x\<^sub>2 ts\<^sub>2)"
assms: shows "lset P \<inter> deadends p \<noteq> {}"
countably_additive_def: fixes M :: "'b set set \<times> ('b set \<Rightarrow> real)" shows "countably_additive M = (\<forall>f. range f \<subseteq> measurable_sets M \<and> (\<forall>m n. m \<noteq> n \<longrightarrow> f m \<inter> f n = {}) \<and> \<Union> (range f) \<in> measurable_sets M \<longrightarrow> (\<lambda>n. Measure.measure M (f n)) sums Measure.measure M (\<Union> (range f)))"
Always_includes_reachable: fixes F :: "'b program"   and A :: "'b set" assumes "F \<in> Always A" shows "reachable F \<subseteq> A"
BNTr7: fixes f :: "'a \<Rightarrow> 'a"   and a :: "'a"   and W1 :: "'a set"   and W2 :: "'a set" assumes "f \<in> carrier D \<rightarrow> carrier D"   and "a \<in> carrier D"   and "\<forall>x\<in>carrier D. x \<preceq> f x"   and "W1 \<in> WWa D f a"   and "W2 \<in> WWa D f a" shows "W1 \<subseteq> W2 \<or> W2 \<subseteq> W1"
Cons_1: fixes t :: "pratt"   and p :: "nat"   and a :: "nat"   and x :: "nat" assumes "valid_cert ys"   and "t \<in> set ys" shows "(t = Prime p \<longrightarrow> prime p) \<and> (t = Triple p a x \<longrightarrow> (\<forall>q\<in>#prime_factorization x. [a ^ ((p - 1) div q) \<noteq> 1] (mod p)) \<and> 0 < x)"
authK_not_AKcryptSK: fixes A :: "agent"   and authK :: "nat"   and tk :: "msg"   and evs :: "event list"   and K :: "nat" assumes "Crypt (shrK Tgs) \<lbrace>Agent A, Agent Tgs, Key authK, tk\<rbrace> \<in> parts (knows Spy evs)"   and "evs \<in> kerbIV" shows "\<not> AKcryptSK K authK evs"
split_paired_Ex: fixes P :: "'b \<times> 'c \<Rightarrow> bool" shows "(\<exists>x. P x) = (\<exists>a b. P (a, b))"
sqn_nsqn: fixes rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option"   and dip :: "nat" shows "sqn rt dip - 1 \<le> nsqn rt dip"
the_Some_image: fixes A :: "'a set" shows "the ` Some ` A = A"
b: shows "AE x in MA. (\<lambda>n. birkhoff_sum f n x / real n) \<longlonglongrightarrow> real_cond_exp M Invariants f x"
in_set_butlast_appendI: fixes x :: "'a"   and xs :: "'a list"   and ys :: "'a list" assumes "x \<in> set (butlast xs) \<or> x \<in> set (butlast ys)" shows "x \<in> set (butlast (xs @ ys))"
nat_to_pr_srj: fixes f :: "nat \<Rightarrow> nat" assumes "f \<in> PrimRec1" shows "\<exists>n. f = nat_to_pr n"
abelian_monoidE_2: fixes R :: "('a, 'b) ring_scheme" assumes "abelian_monoid R" shows "\<zero>\<^bsub>R\<^esub> \<in> carrier R"
cong_left_commutativity: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Cong A B C D" shows "Cong B A C D"
between_def: fixes p :: "alt \<Rightarrow> real"   and a :: "alt"   and b :: "alt"   and c :: "alt" shows "between p a b c \<equiv> p(b := (p a + p c) / 2)"
llscale: shows "matrix_seq.lowner_lub (\<lambda>n. complex_of_real c \<cdot>\<^sub>m matrix_sum d (\<lambda>n. denote_while_n (x1 0) (x1 1) (denote x2) n \<rho>) n) = complex_of_real c \<cdot>\<^sub>m matrix_seq.lowner_lub (matrix_sum d (\<lambda>n. denote_while_n (x1 0) (x1 1) (denote x2) n \<rho>))"
set_less_eq_aux'_def: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set"   and B :: "'a set" shows "ord.set_less_eq_aux' less_eq A B = (ord.set_less_eq_aux less_eq A B \<or> A = {} \<and> B \<in> infinite_complement_partition)"
Subterm: fixes t :: "('fun, 'var) Term.term"   and M :: "('fun, 'var) Term.term set"   and t' :: "('fun, 'var) Term.term" assumes "t \<in> SMP M"   and "t' \<sqsubseteq> t" shows "t' \<in> SMP M"
contract_extensionality: fixes Cn :: "'b set \<Rightarrow> 'b set"   and contraction :: "'b set \<Rightarrow> 'b \<Rightarrow> 'b set"   and K :: "'b set"   and A :: "'b set"   and \<phi> :: "'b"   and \<psi> :: "'b" assumes "AGM_Contraction Cn contraction"   and "K = Cn A"   and "Cn {\<phi>} = Cn {\<psi>}" shows "contraction K \<phi> = contraction K \<psi>"
L7_Controller_One_l0: fixes y :: "nat \<Rightarrow> nat list"   and l :: "nat \<Rightarrow> bit"   and z :: "nat \<Rightarrow> bit list" assumes "Controller_L y (fin_inf_append [Zero] l) l z"   and "y 0 = [500]" shows "l 0 = Zero"
local_var_approx_def: fixes prog :: "'a program_scheme" shows "local_var_approx prog \<equiv> \<Union> {set (local_vars p) |p. p \<in> set (program.processes prog)}"
Abs_swap2: fixes a :: "atom"   and x :: "'a"   and bs :: "atom list"   and b :: "atom" assumes "a \<notin> supp x - set bs"   and "b \<notin> supp x - set bs" shows "[bs]lst. x = [(a \<rightleftharpoons> b) \<bullet> bs]lst. (a \<rightleftharpoons> b) \<bullet> x"
simps_2: fixes v :: "val"   and r :: "proc \<times> val" shows "Some v \<bullet> r = Some v"
ex_b: shows "\<exists>b. A $ b \<noteq> (0::'b)"
card_seteq: fixes B :: "'g set"   and A :: "'g set" assumes "finite B"   and "A \<subseteq> B"   and "card B \<le> card A" shows "A = B"
a: shows "M[x\<turnstile>n>y] \<longrightarrow>\<^sub>c M'"
simps_3: fixes cmp :: "'b comparator"   and y :: "'b"   and x :: "'b"   and xs :: "'b list" shows "Sorting_Algorithms.sorted cmp (y # x # xs) = (compare cmp y x \<noteq> Greater \<and> Sorting_Algorithms.sorted cmp (x # xs))"
axioms_1: fixes TA :: "('a, 'b) tree_automaton_rec"   and A :: "'b \<Rightarrow> nat option" assumes "det_tree_automaton TA A" shows "ranked_tree_automaton TA A"
string_trans: fixes xa :: "char list"   and ya :: "char list"   and z :: "char list" assumes "(xa, ya) \<in> lexord {(x, y). x < y}"   and "(ya, z) \<in> lexord {(x, y). x < y}" shows "(xa, z) \<in> lexord {(x, y). x < y}"
part_fn_def: fixes r :: "nat"   and s :: "nat"   and Y :: "'a set"   and f :: "'a set \<Rightarrow> nat" shows "part_fn r s Y f = (f \<in> [Y]\<^bsup>r\<^esup> \<rightarrow> {..<s})"
in_oreturn: fixes x :: "'d"   and s :: "'e"   and v :: "'d" shows "(oreturn x s = Some v) = (v = x)"
comp_ide_self: fixes a :: "nat" assumes "J.ide a" shows "J.comp a a = a"
height_split_min: fixes t :: "'b tree"   and a :: "'b"   and t' :: "'b tree" assumes "Root_Balanced_Tree.split_min t = (a, t')"   and "t \<noteq> \<langle>\<rangle>" shows "height t' \<le> height t"
CONSTANT_def: fixes k :: "nat"   and l :: "nat list" shows "CONSTANT k l = k"
Mutator: shows "\<turnstile> Mutator  \<lbrace>False\<rbrace>"
assms: fixes h :: "heap \<times> nat set" assumes "h \<Turnstile> P" shows "h \<Turnstile> Q"
assms_6: shows "[a ^ ((p - 1) div q) \<noteq> 1] (mod p)"
fps_def: fixes L :: "('a, 'b) eq_object_scheme"   and f :: "'a \<Rightarrow> 'a" shows "fps L f = {x \<in> carrier L. f x .=\<^bsub>L\<^esub> x}"
simps_2: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b" shows "successively P [x] = True"
qe_FM_def: shows "qe_FM = R\<^sub>e.lift_dnfeq_qe qe_FM\<^sub>1"
FUA: shows "FUadmitFO \<sigma> \<phi> UNIV"
disj_not_empty: shows "disjoint_family_on F {i. F i \<noteq> {}}"
intro: fixes Rep :: "'a \<Rightarrow> 'b"   and A :: "'b set"   and Abs :: "'b \<Rightarrow> 'a" assumes "\<And>x. Rep x \<in> A"   and "\<And>x. Abs (Rep x) = x"   and "\<And>y. y \<in> A \<Longrightarrow> Rep (Abs y) = y" shows "type_definition Rep Abs A"
intros_1: fixes v1 :: "'b val"   and r1 :: "'b rexp"   and v2 :: "'b val"   and r2 :: "'b rexp" assumes "\<turnstile> v1 : r1"   and "\<turnstile> v2 : r2" shows "\<turnstile> Seq v1 v2 : Times r1 r2"
coeff_pderiv: fixes p :: "'a poly"   and n :: "nat" shows "poly.coeff (pderiv p) n = of_nat (Suc n) * poly.coeff p (Suc n)"
hyps: assumes "is_regular_sequence fs" shows "is_regular_sequence (rev (remdups (rev fs)))"
realistic_executions_ind_def: fixes execs :: "'dom_t \<Rightarrow> 'action_t list list" shows "realistic_executions_ind execs \<equiv> \<forall>d. case execs d of [] \<Rightarrow> True | aseq # aseqs \<Rightarrow> realistic_AS_partial aseq \<and> set aseqs \<subseteq> AS_set"
mu_exchange_1: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and l :: "'b \<Rightarrow> 'b"   and u :: "'b \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'b"   and h :: "'b \<Rightarrow> 'b" assumes "class.order less_eq less"   and "ord.galois less_eq l u"   and "ord.isotone less_eq g"   and "ord.isotone less_eq h"   and "order.has_least_prefixpoint less_eq (l \<circ> h)"   and "order.has_least_prefixpoint less_eq (h \<circ> g)"   and "order.has_least_fixpoint less_eq (g \<circ> h)"   and "ord.lifted_less_eq less_eq (l \<circ> h \<circ> g) (g \<circ> h \<circ> l)" shows "less_eq (order.the_least_fixpoint less_eq (l \<circ> h)) (order.the_least_fixpoint less_eq (g \<circ> h))"
left_adjoint_preserved_by_iso: fixes f :: "'a"   and \<phi> :: "'a"   and f' :: "'a" assumes "is_left_adjoint f"   and "\<guillemotleft>\<phi> : f \<Rightarrow> f'\<guillemotright>"   and "local.iso \<phi>" shows "is_left_adjoint f'"
assms_1: shows "x \<rightarrow>\<^sub>i y"
SubstTerm_imp_subst_dbtm': fixes v :: "hf"   and y :: "hf"   and x :: "hf"   and x' :: "hf"   and e :: "(name, hf) finfun" assumes "SubstTerm v y x x'" obtains t :: "dbtm" and u :: "dbtm" where "y = \<lbrakk>quot_dbtm u\<rbrakk>e" and "x = \<lbrakk>quot_dbtm t\<rbrakk>e" and "x' = \<lbrakk>quot_dbtm (subst_dbtm u (decode_Var v) t)\<rbrakk>e"
sym_def: fixes r :: "('b \<times> 'b) set" shows "sym r = (\<forall>x y. (x, y) \<in> r \<longrightarrow> (y, x) \<in> r)"
induct: fixes P :: "char list set \<Rightarrow> char list \<Rightarrow> bool"   and a0 :: "char list set"   and a1 :: "char list" assumes "\<And>Y Xs. \<lbrakk>Y \<in> Xs; finite Xs; P (Xs - {Y}) (upChar Y)\<rbrakk> \<Longrightarrow> P Xs Y"   and "\<And>Y Xs. Y \<notin> Xs \<or> infinite Xs \<Longrightarrow> P Xs Y" shows "P a0 a1"
checked_nsc_def: shows "checked_nsc = Incremental_Atom_Ops_For_NS_Constraint_Ops.checked_nsc checked_s preprocess"
inext_nth_UNIV: fixes a :: "nat" shows "UNIV \<rightarrow> a = a"
neutral: fixes A :: "'a set"   and g :: "'a \<Rightarrow> 'b" assumes "\<forall>x\<in>A. g x = (0::'b)" shows "sum g A = (0::'b)"
supset_pseudominimal_inD1: fixes P :: "'b set set"   and x :: "'b set" assumes "dual_order.pseudominimal_in P x" shows "x \<in> P"
ipset_from_cidr_def: fixes addr :: "'b word"   and pflength :: "nat" shows "ipset_from_cidr addr pflength \<equiv> ipset_from_netmask addr (mask pflength << LENGTH('b) - pflength)"
fm_in_K': fixes nb :: "nat"   and m :: "nat" assumes "1 \<le> nb"   and "nb \<le> m" shows "fm nb (y m) \<in> K' nb"
arden_r: fixes y :: "'a list set"   and x :: "'a list set"   and z :: "'a list set" assumes "\<not> l_ewp y"   and "x = x \<cdot> y + z" shows "x = z \<cdot> y\<^sup>\<star>"
poly_rel_one: shows "poly_rel (one_poly_i ops) 1"
dist_fst_le: fixes x :: "'b \<times> 'c"   and y :: "'b \<times> 'c" shows "dist (fst x) (fst y) \<le> dist x y"
Nfunc_LeastTr2_5: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a set"   and g :: "nat \<Rightarrow> 'a set" assumes "\<forall>l\<le>n. G \<guillemotright> f l "   and "\<forall>l\<le>n. G \<guillemotright> g l "   and "\<exists>i\<le>n. f i \<noteq> g i" shows "f (LEAST k. f k \<noteq> g k) \<noteq> g (LEAST k. f k \<noteq> g k)"
map_single: fixes c :: "'b"   and f :: "'b \<Rightarrow> 'c"   and x :: "'d" assumes "c = (0::'b) \<Longrightarrow> f (0::'b) = (0::'c)" shows "Poly_Mapping.map f (monomial c x) = monomial (f c) x"
rSeq: fixes \<alpha> :: "prog"   and \<nu> :: "fin_string \<Rightarrow> real"   and \<mu> :: "fin_string \<Rightarrow> real"   and \<beta> :: "prog"   and \<omega> :: "fin_string \<Rightarrow> real" assumes "[\<alpha>]\<nu> \<downharpoonleft> \<mu>"   and "[\<beta>]\<mu> \<downharpoonleft> \<omega>" shows "[\<alpha> ;; \<beta>]\<nu> \<downharpoonleft> \<omega>"
elims: fixes x :: "'a extB"   and y :: "'a extB" assumes "- x = y" obtains "x = extB.Bot" and "y = extB.Val top"   | uu_ :: "'a" where "x = extB.Val uu_" and "y = extB.Bot"
rat_of_vrat_vrat_of_rat: fixes a :: "rat" shows "rat_of_vrat (a\<^sub>\<rat>) = a"
dg_Par_is_arr_dg_Rel_is_arr: fixes \<alpha> :: "V"   and a :: "V"   and b :: "V"   and r :: "V" assumes "r : a \<mapsto>\<^bsub>dg_Par \<alpha>\<^esub> b" shows "r : a \<mapsto>\<^bsub>dg_Rel \<alpha>\<^esub> b"
atLeastLessThan_def: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and l :: "'a"   and u :: "'a" shows "ord.atLeastLessThan less_eq less l u \<equiv> ord.atLeast less_eq l \<inter> ord.lessThan less u"
diag_four_block_mat: fixes A :: "'c mat"   and n1 :: "nat"   and D :: "'c mat"   and n2 :: "nat"   and B :: "'c mat"   and C :: "'c mat" assumes "A \<in> carrier_mat n1 n1"   and "D \<in> carrier_mat n2 n2" shows "diag_mat (four_block_mat A B C D) = diag_mat A @ diag_mat D"
cases: fixes a1 :: "sequent list \<times> sequent"   and a2 :: "form" assumes "rightPrincipal a1 a2" obtains C :: "sequent" and A :: "form" and Ps :: "sequent list" where "a1 = (Ps, C)" and "a2 = A" and "C = ( \<Empt> \<Rightarrow>* \<LM> A  \<RM>)"
integrable_on_cmult_left: fixes f :: "'a \<Rightarrow> 'b"   and S :: "'a set"   and c :: "real" assumes "f integrable_on S" shows "(\<lambda>x. of_real c * f x) integrable_on S"
dom_restrict_def: fixes S :: "'e set"   and p :: "'e \<Rightarrow> 'f decision option" shows "S \<triangleleft> p \<equiv> \<lambda>x. if x \<in> S then p x else \<bottom>"
assms_2: shows "\<S>\<bullet>\<C> \<turnstile> [Local n i vs es] : ts _> ts'"
IH: shows "(\<Sum>x\<in>#A. x) = (\<Sum>x\<in>set_mset A. x * count A x)"
comp_arr_inv: fixes f :: "'a"   and g :: "'a" assumes "C.inverse_arrows f g" shows "f \<cdot> g = C.dom g"
Aik0: shows "A $ from_nat i $ from_nat k = (0::'a)"
restrict_compose_right: fixes g :: "'g \<Rightarrow> 'f"   and f :: "'e \<Rightarrow> 'g"   and S :: "'e set" shows "restrict (g \<circ> restrict f S) S = restrict (g \<circ> f) S"
intro: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and id :: "'a" assumes "\<And>a b. f a b = f b a"   and "\<And>a b c. f (f a b) c = f a (f b c)"   and "\<And>x. f id x = x"   and "\<And>x. f x id = x" shows "Matrix_Tensor.mult id f"
q: shows "q = smult b (\<Prod>(c, l)\<in>set cs. c ^ Suc l)"
eventually_def: fixes P :: "'c \<Rightarrow> bool"   and F :: "'c filter" shows "eventually P F = Rep_filter F P"
sterms_rel': shows "sterms_rel = (\<lambda>gq gp. (gq, gp) \<in> {((\<Gamma>, q), \<Gamma>', p). \<Gamma> = \<Gamma>' \<and> p \<leadsto>\<^bsub>\<Gamma>\<^esub> q})"
esl_def: shows "esl = (\<lambda>(A, r, f). (A, r, \<lambda>x y. OK (f x y)))"
simps_4: fixes g :: "char list \<Rightarrow> bool"   and h :: "'a pfm \<Rightarrow> bool"   and p :: "'a pfm"   and q :: "'a pfm" shows "peval g h (p \<^bold>\<and>\<^sub>! q) = (peval g h p \<and> peval g h q)"
final_is_s_t_len_t: fixes channel :: "nat \<Rightarrow> ('d \<times> 'd) option"   and trans :: "'d \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and init :: "('d, 'e, 'f) configuration"   and final :: "('d, 'e, 'f) configuration"   and t :: "('d, 'e, 'f) event list" assumes "computation channel trans send recv init final"   and "distributed_system.trace channel trans send recv init t final" shows "final = distributed_system.s channel trans send recv init t (length t)"
root_main'_pos: fixes x :: "int"   and n :: "int"   and y :: "int"   and b :: "bool" assumes "0 \<le> x"   and "0 \<le> n"   and "root_main' x n = (y, b)" shows "0 \<le> y"
simps_1: fixes zs :: "nat list"   and i :: "nat"   and T :: "bool list list" shows "mk_eqcl [] zs i T = ([], zs)"
indRelRPO_def: shows "indRelRPO \<equiv> _Collect (x, y) (indRelRPOp x y)"
state_rank_eq_rank_SOME: fixes \<Sigma> :: "'c set"   and \<delta> :: "'d \<Rightarrow> 'c \<Rightarrow> 'd"   and q\<^sub>0 :: "'d"   and w :: "nat \<Rightarrow> 'c"   and q :: "'d"   and n :: "nat" assumes "semi_mojmir \<Sigma> \<delta> q\<^sub>0 w" shows "semi_mojmir_def.state_rank \<Sigma> \<delta> q\<^sub>0 w q n = (if semi_mojmir_def.configuration \<delta> q\<^sub>0 w q n \<noteq> {} then semi_mojmir_def.rank \<Sigma> \<delta> q\<^sub>0 w (SOME x. x \<in> semi_mojmir_def.configuration \<delta> q\<^sub>0 w q n) n else None)"
fps_inv_ginv: shows "fps_inv = fps_ginv fps_X"
preserves_subset_stable: fixes v :: "'d \<Rightarrow> 'e"   and P :: "'e \<Rightarrow> bool" shows "preserves v \<subseteq> stable {s. P (v s)}"
no_black_refs_def: shows "no_black_refs = (\<lambda>s. \<forall>x. \<not> black x s)"
sublocale_sas_plus_finite_domain_representation_ii_6: fixes \<Psi> :: "('c, 'd) sas_plus_problem" assumes "is_valid_problem_sas_plus \<Psi>" shows "\<forall>v\<in>dom (\<Psi>\<^sub>G\<^sub>+). the ((\<Psi>\<^sub>G\<^sub>+) v) \<in> \<R>\<^sub>+ \<Psi> v"
face: fixes X :: "'a set"   and Y :: "'a set" assumes "X \<in> \<F>'"   and "Y \<in> \<F>'" shows "X \<inter> Y face_of X"
frequentlyE: fixes P :: "'a \<Rightarrow> bool"   and F :: "'a filter" assumes "frequently P F" obtains x :: "'a" where "P x"
in_renamed_in_unlabelled_local_2: fixes \<alpha> :: "'d \<Rightarrow> 'd"   and \<beta> :: "'e \<Rightarrow> 'e"   and l :: "'e"   and ls :: "('e \<Rightarrow> ('d, 'e, 'f) val option) \<times> ('e \<Rightarrow> ('d, 'e, 'f) val option) \<times> ('d, 'e, 'f) expr" assumes "bij \<alpha>"   and "bij \<beta>" shows "(\<beta> l \<in> LID\<^sub>L (\<R>\<^sub>L \<alpha> \<beta> ls)) = (l \<in> LID\<^sub>L ls)"
not_trivial_limit_within: fixes x :: "'c"   and S :: "'c set" shows "(at x within S = bot) \<noteq> (x \<in> closure (S - {x}))"
succeed_t_alt_def: shows "succeed_t i = {n. rank n n = Some i \<and> token_run n n = q\<^sub>0}"
NegPos_set_1: fixes a :: "'b"   and x :: "'b negation_type list" shows "(Pos a \<in> set x) = (a \<in> set (getPos x))"
ode_d2_def: fixes ode_ops :: "ode_ops" shows "ode_operations.ode_d2 ode_ops \<equiv> map_fun id Blinfun (\<lambda>x. if x \<in> ode_operations.Csafe ode_ops then ode_operations.ode_d ode_ops 1 x else (\<lambda>_. 0))"
a_f_3_in_f: fixes e1 :: "'v"   and e2 :: "'v" assumes "(e1, e2) \<in> f" shows "\<not> P (nP e1) (nP e2)"
the_inv_signed_lconjaction: fixes s :: "'a" shows "the_inv (signed_lconjaction s) = signed_rconjaction s"
iso_in_hom_def: fixes f :: "'d"   and a :: "'d"   and b :: "'d" shows "\<guillemotleft>f : a \<cong>\<^sub>D b\<guillemotright> \<equiv> \<guillemotleft>f : a \<Rightarrow>\<^sub>D b\<guillemotright> \<and> D.iso f"
set_update: fixes s :: "'b"   and f :: "'b \<Rightarrow> 'b" shows "State_Monad.set s \<bind> (\<lambda>_. State_Monad.update f) = State_Monad.set (f s)"
set_shadow_root_locs_def: shows "set_shadow_root_locs \<equiv> l_set_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_set_shadow_root_locs"
ordinal_rec_oSuc: fixes z :: "ordinal"   and s :: "ordinal \<Rightarrow> ordinal \<Rightarrow> ordinal"   and x :: "ordinal" shows "ordinal_rec z s (oSuc x) = s x (ordinal_rec z s x)"
P: fixes m :: "nat" assumes "m dvd p * q" shows "m = 1 \<or> m = p * q"
unit_l_inv: fixes x :: "'a" assumes "unit x" shows "local.inv x ** x = one"
antisym_conv2: fixes x :: "'a"   and y :: "'a" assumes "le x y" shows "(\<not> lt x y) = (x = y)"
Abs_IF2_inverse: fixes y :: "'d IIT \<Rightarrow> ((bd_type_F1 + (bd_type_F1, bd_type_F1, bd_type_F1) F1) + bd_type_F2 + (bd_type_F2, bd_type_F2, bd_type_F2) F2) set \<Rightarrow> 'd + bool" assumes "y \<in> car_init2 undefined" shows "Rep_IF2 (Abs_IF2 y) = y"
assms_4: fixes x :: "'c" assumes "x \<in> A" shows "g x = h x"
vconn_implies_hpar_1: fixes \<mu> :: "'a"   and f :: "'a"   and g :: "'a" assumes "S.in_hom \<mu> f g" shows "S.src \<mu> = S.src f"
sympI: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<And>a b. r a b \<Longrightarrow> r b a" shows "symp r"
Df: fixes x :: "'a" shows "(f has_derivative Df x) (at x)"
assms_3: shows "\<not> i icd\<^bsup>\<pi>\<^esup>\<rightarrow> l"
i_set_cont_inext: fixes I :: "nat set"   and n :: "nat" assumes "I \<in> i_set_cont"   and "n \<in> I"   and "finite I \<Longrightarrow> n < Max I" shows "inext n I = Suc n"
assms_1: shows "\<langle>c \<and> t\<rangle> < n"
foldli_mono_dres_aux2: fixes f :: "'f \<Rightarrow> 'g dres \<Rightarrow> 'g dres"   and f' :: "'f \<Rightarrow> 'g dres \<Rightarrow> 'g dres"   and l :: "'f list"   and c :: "'g \<Rightarrow> bool"   and \<sigma> :: "'g dres" assumes "\<And>x. f x dSUCCEED = dSUCCEED"   and "\<And>x. f' x dFAIL = dFAIL"   and "\<And>a x x'. x \<le> x' \<Longrightarrow> f a x \<le> f' a x'" shows "foldli l (case_dres False False c) f \<sigma> \<le> foldli l (case_dres False False c) f' \<sigma>"
2: shows "class.nontriv TYPE('a) = (2 \<le> CARD('a))"
Carmichael_dvd_totient: fixes n :: "nat" shows "Carmichael n dvd totient n"
inv_partI: fixes m_j :: "nat"   and m_i :: "nat"   and p :: "'a mat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool"   and A :: "'a mat" assumes "\<And>i j. \<lbrakk>i < n; j < n; j < m_j \<or> j = m_j \<and> m_i \<le> i\<rbrakk> \<Longrightarrow> p A i j" shows "local.inv_part p A m_i m_j"
mapsTo_Op: fixes C :: "('d, 'e, 'f) Category_scheme"   and f :: "'e"   and Y :: "'d"   and X :: "'d" shows "f maps\<^bsub>Op C\<^esub> Y to X = f maps\<^bsub>C\<^esub> X to Y"
simps_2: fixes s :: "'a \<Rightarrow> bool option"   and ops :: "'a strips_operator list"   and opss :: "'a strips_operator list list" shows "trace_parallel_plan_strips s (ops # opss) = s # (if are_all_operators_applicable s ops \<and> are_all_operator_effects_consistent ops then trace_parallel_plan_strips (execute_parallel_operator s ops) opss else [])"