hcju/isabelleps huggingface dataset:
sphere_trivial: fixes x :: "'a" shows "sphere x 0 = {x}"
path_image_rectpath: fixes a1 :: "complex"   and a3 :: "complex" assumes "Re a1 \<le> Re a3"   and "Im a1 \<le> Im a3" shows "path_image (rectpath a1 a3) = {z. Re z \<in> {Re a1, Re a3} \<and> Im z \<in> {Im a1..Im a3}} \<union> {z. Im z \<in> {Im a1, Im a3} \<and> Re z \<in> {Re a1..Re a3}}"
sel: fixes r :: "('d, 'e, 'f) Resumption.resumption" shows "the_gpv (lift_resumption r) = (case r of Resumption.resumption.Done None \<Rightarrow> return_pmf None | Resumption.resumption.Done (Some x') \<Rightarrow> return_spmf (Pure x') | Resumption.resumption.Pause out c \<Rightarrow> map_spmf (map_generat id id ((\<circ>) lift_resumption)) (return_spmf (IO out c)))"
cos_ge_zero: fixes x :: "real" assumes "- (pi / 2) \<le> x"   and "x \<le> pi / 2" shows "0 \<le> cos x"
tendsto_sin: fixes a :: "'b"   and F :: "'a filter" assumes "(f \<longlongrightarrow> a) F" shows "((\<lambda>x. sin (f x)) \<longlongrightarrow> sin a) F"
qSwapAbs_preserves_alphaAbs: fixes A :: "('f, 'g, 'e, 'd, 'h) qAbs"   and A' :: "('f, 'g, 'e, 'd, 'h) qAbs"   and zs :: "'e"   and z1 :: "'d"   and z2 :: "'d" assumes "FixVars TYPE('d) TYPE('e)"   and "FixVars.qGoodAbs A \<or> FixVars.qGoodAbs A'"   and "A $= A'" shows "A $[[z1 \<and> z2]]_zs $= A' $[[z1 \<and> z2]]_zs"
sound_inference_system_def: fixes \<Gamma> :: "'b inference set" shows "sound_inference_system \<Gamma> \<equiv> \<forall>CC D E I. Infer CC D E \<in> \<Gamma> \<longrightarrow> I \<Turnstile>m CC \<longrightarrow> I \<Turnstile> D \<longrightarrow> I \<Turnstile> E"
measure_measure_pmf_finite: fixes S :: "'a set"   and M :: "'a pmf" assumes "finite S" shows "measure_pmf.prob M S = sum (pmf M) S"
complex_Re_of_int: fixes z :: "int" shows "Re (of_int z) = real_of_int z"
inject_2: fixes x2 :: "jump"   and y2 :: "jump" shows "(Jump x2 = Jump y2) = (x2 = y2)"
rep_eq: fixes x :: "'a coreflexive"   and xa :: "'a coreflexive" shows "(x \<le> xa) = (Rep_coreflexive x \<le> Rep_coreflexive xa)"
object_ptr_kinds_eq_h3: shows "object_ptr_kinds h3 = object_ptr_kinds h'"
rel_symp: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "symp R" shows "symp (rel_bset R)"
cases: fixes a1 :: "'a"   and a2 :: "[:'a:]" assumes "slistmem a1 a2" obtains x :: "'a" and xs :: "[:'a:]" where "a1 = x__" and "a2 = x__ :# xs__" and "x__ \<noteq> \<bottom>" and "xs__ \<noteq> \<bottom>"   | x :: "'a" and xs :: "[:'a:]" and y :: "'a" where "a1 = x" and "a2 = y :# xs" and "slistmem x xs" and "y \<noteq> \<bottom>"
homotopic_with_subset_right: fixes P :: "('b \<Rightarrow> 'c) \<Rightarrow> bool"   and X :: "'b set"   and Y :: "'c set"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c"   and Z :: "'c set" assumes "homotopic_with_canon P X Y f g"   and "Y \<subseteq> Z" shows "homotopic_with_canon P X Z f g"
dens'': shows "has_density (distr (distr M (count_space UNIV) extract_int_pair) (count_space UNIV) (\<lambda>(x, y). f x y)) (count_space UNIV) (g (\<delta> \<circ> IntPairVal))"
secure_refinement_simpler: fixes \<R>\<^sub>A :: "((((Eg1.var, aexp, bexp) Stmt \<times> (Mode \<Rightarrow> Eg1.var set)) \<times> (Eg1.var \<Rightarrow> nat)) \<times> ((Eg1.var, aexp, bexp) Stmt \<times> (Mode \<Rightarrow> Eg1.var set)) \<times> (Eg1.var \<Rightarrow> nat)) set"   and \<R> :: "((((Eg1.var, aexp, bexp) Stmt \<times> (Mode \<Rightarrow> Eg1.var set)) \<times> (Eg1.var \<Rightarrow> nat)) \<times> ((var\<^sub>C, aexp\<^sub>C, bexp\<^sub>C) Stmt \<times> (Mode \<Rightarrow> var\<^sub>C set)) \<times> (var\<^sub>C \<Rightarrow> nat)) set"   and P :: "((((var\<^sub>C, aexp\<^sub>C, bexp\<^sub>C) Stmt \<times> (Mode \<Rightarrow> var\<^sub>C set)) \<times> (var\<^sub>C \<Rightarrow> nat)) \<times> ((var\<^sub>C, aexp\<^sub>C, bexp\<^sub>C) Stmt \<times> (Mode \<Rightarrow> var\<^sub>C set)) \<times> (var\<^sub>C \<Rightarrow> nat)) set"   and abs_steps :: "((Eg1.var, aexp, bexp) Stmt \<times> (Mode \<Rightarrow> Eg1.var set)) \<times> (Eg1.var \<Rightarrow> nat) \<Rightarrow> ((var\<^sub>C, aexp\<^sub>C, bexp\<^sub>C) Stmt \<times> (Mode \<Rightarrow> var\<^sub>C set)) \<times> (var\<^sub>C \<Rightarrow> nat) \<Rightarrow> nat" assumes "secure_refinement_simpler \<R>\<^sub>A \<R> P abs_steps" shows "secure_refinement \<R>\<^sub>A \<R> P"
simps_1: fixes f :: "'d \<Rightarrow> 'c option" shows "None \<bind> f = None"
open_contains_cball_eq: fixes S :: "'a set" assumes "open S" shows "\<forall>x. (x \<in> S) = (\<exists>e>0. cball x e \<subseteq> S)"
map_add_comm: fixes m1 :: "'c \<Rightarrow> 'd option"   and m2 :: "'c \<Rightarrow> 'd option" assumes "dom m1 \<inter> dom m2 = {}" shows "m1 ++ m2 = m2 ++ m1"
MKT_3: shows "is_ord (MKT n l (MKT rn rl rr rh) h)"
degree_prod_eq: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b poly" assumes "\<And>x. x \<in> A \<Longrightarrow> f x \<noteq> 0" shows "Polynomial.degree (prod f A) = (\<Sum>x\<in>A. Polynomial.degree (f x))"
take_bit_of_exp: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and 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 modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and division_segment :: "'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 m :: "nat"   and n :: "nat" assumes "class.unique_euclidean_semiring_with_bit_shifts divide plus minus zero times one modulo euclidean_size division_segment bit push_bit drop_bit take_bit" shows "take_bit m (power.power one times (numeral.numeral one plus (num.Bit0 num.One)) n) = times (zero_neq_one.of_bool one zero (n < m)) (power.power one times (numeral.numeral one plus (num.Bit0 num.One)) n)"
induct: fixes x :: "fm"   and P :: "fm \<Rightarrow> bool" assumes "x \<in> boolean_axioms"   and "\<And>A. P (A IMP A)"   and "\<And>A B. P (A IMP A OR B)"   and "\<And>A. P (A OR A IMP A)"   and "\<And>A B C. P (A OR B OR C IMP (A OR B) OR C)"   and "\<And>C A B. P (C OR A IMP (C IMP B) IMP A OR B)" shows "P x"
orthogonal_to_span: fixes a :: "'b"   and S :: "'b set"   and x :: "'b" assumes "a \<in> span S"   and "\<And>y. y \<in> S \<Longrightarrow> orthogonal x y" shows "orthogonal x a"
is_bezout_ext_def: fixes bezout :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<times> 'd \<times> 'd \<times> 'd \<times> 'd" shows "is_bezout_ext bezout = (\<forall>a b. let (p, q, u, v, gcd_a_b) = bezout a b in p * a + q * b = gcd_a_b \<and> gcd_a_b dvd a \<and> gcd_a_b dvd b \<and> (\<forall>d'. d' dvd a \<and> d' dvd b \<longrightarrow> d' dvd gcd_a_b) \<and> gcd_a_b * u = - b \<and> gcd_a_b * v = a)"
divide_le_eq_numeral1_1: fixes b :: "'b"   and w :: "num"   and a :: "'b" shows "(b / numeral w \<le> a) = (b \<le> a * numeral w)"
IMP_PRE_CUSTOM_def: shows "IMP_PRE_CUSTOM \<equiv> IMP_PRE"
rel_restrict_sub: fixes R :: "('b \<times> 'b) set"   and A :: "'b set" shows "rel_restrict R A \<subseteq> R"
clinear_continuous_within: fixes f :: "'a \<Rightarrow> 'b"   and x :: "'a"   and s :: "'a set" assumes "bounded_clinear f" shows "continuous (at x within s) f"
rot_diamond_cube_vertical_boundary_explicit: shows "vertical_boundary rot_diamond_cube = {diamond_left_edges, diamond_right_edges}"
convex_linear_image_eq: fixes f :: "'b \<Rightarrow> 'c"   and s :: "'b set" assumes "linear f"   and "inj f" shows "convex (f ` s) = convex s"
vwb_lens_wb: fixes x :: "'a \<Longrightarrow> 'b" assumes "vwb_lens x" shows "wb_lens x"
ccw'_sortedP_polychain_of3: fixes u :: "real \<times> real"   and v :: "real \<times> real"   and w :: "real \<times> real"   and xs :: "(real \<times> real) list"   and x0 :: "real \<times> real" assumes "ccw'.sortedP 0 (u # v # w # xs)" shows "ccw'.sortedP x0 (map snd (polychain_of x0 (u # v # w # xs)))"
assms_3: shows "ipresIGFresh h' MOD' MOD''"
assms_3: shows "\<forall>x\<in>X. x \<noteq> mx \<longrightarrow> f x < f mx"
map_poly_pCons: fixes c :: "'b"   and p :: "'b poly"   and f :: "'b \<Rightarrow> 'c" assumes "c \<noteq> (0::'b) \<or> p \<noteq> 0" shows "map_poly f (pCons c p) = pCons (f c) (map_poly f p)"
landau_flip_4: fixes g :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and f :: "'b \<Rightarrow> 'c" shows "(g \<in> \<omega>[F](f)) = (f \<in> o[F](g))"
perp_in_distinct: fixes X :: "'p"   and A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "X PerpAt A B C D " shows "A \<noteq> B \<and> C \<noteq> D"
ll_def: shows "ll \<equiv> if degree q < degree p then at 0 else if degree q = degree p then nhds (lead_coeff q / lead_coeff p) else if 0 < sgn_pos_inf q * sgn_pos_inf p then at_top else at_bot"
srsd_def: fixes x :: "'a" shows "srsd x = (x \<cdot> \<top> = x)"
theI': fixes P :: "'c \<Rightarrow> bool" assumes "\<exists>!x. P x" shows "P (THE x. P x)"
lub: fixes xs' :: "('a, 'b) tllist" assumes "xs' \<in> A" shows "local.tllist_ord xs' xs"
pred_rel: fixes P :: "'b \<Rightarrow> bool"   and x :: "'b stream" shows "pred_stream P x = stream_all2 (eq_onp P) x x"
iso_incl_imp_same_eval: fixes A :: "'c ring"   and B :: "'c ring"   and a :: "'c"   and p :: "'c list" assumes "ring A"   and "ring B"   and "A \<lesssim> B"   and "a \<in> carrier A"   and "set p \<subseteq> carrier A" shows "ring.eval A p a = ring.eval B p a"
mat_tensor_prod_2_bis: fixes v :: "complex Matrix.mat"   and w :: "complex Matrix.mat" assumes "state 1 v"   and "state 1 w" shows "v \<Otimes> w = |Matrix.vec 4 (\<lambda>i. if i = 0 then v $$ (0, 0) * w $$ (0, 0) else if i = 3 then v $$ (1, 0) * w $$ (1, 0) else if i = 1 then v $$ (0, 0) * w $$ (1, 0) else v $$ (1, 0) * w $$ (0, 0))\<rangle>"
IH: fixes S :: "stack_elem list" assumes "fv (as @ \<Gamma>, body, S) \<subseteq> set L \<union> domA (as @ \<Gamma>)" shows "(as @ \<Gamma>, body, S) \<Rightarrow>\<^sup>* (\<Delta>, z, S)"
M_UNIV: shows "UNIV = set [M1, M2, M3]"
V_simps_3: fixes trn :: "('state, 'act, 'out) trans"   and tr :: "('state, 'act, 'out) trans list" assumes "\<phi> trn" shows "V (trn # tr) = f trn # V tr"
correctCompositionDiffLevelsA7: shows "correctCompositionDiffLevels sA7"
measure_notin_sets: fixes A :: "'h set"   and M :: "'h measure" assumes "A \<notin> sets M" shows "Sigma_Algebra.measure M A = 0"
mbind_nofailure: fixes S :: "'d list"   and f :: "'d \<Rightarrow> 'c \<Rightarrow> ('b \<times> 'c) option"   and \<sigma> :: "'c" shows "mbind S f \<sigma> \<noteq> None"
linear_linear: fixes f :: "'b \<Rightarrow> 'c" shows "bounded_linear f = linear f"
Col_cases: fixes A :: "'p"   and B :: "'p"   and C :: "'p" assumes "Col A B C \<or> Col A C B \<or> Col B A C \<or> Col B C A \<or> Col C A B \<or> Col C B A" shows "Col A B C"
Cons_eq_appendI: fixes x :: "'g"   and xs1 :: "'g list"   and ys :: "'g list"   and xs :: "'g list"   and zs :: "'g list" assumes "x # xs1 = ys"   and "xs = xs1 @ zs" shows "x # xs = ys @ zs"
cofinite_not_part: fixes X :: "nat cofinite set" shows "\<not> part X"
min_eqL: fixes a :: "'b"   and b :: "'b" assumes "a \<le> b" shows "min a b = a"
a_elemrcos_carrier: fixes a :: "'a"   and a' :: "'a" assumes "a \<in> carrier R"   and "a' \<in> a_kernel R S h +> a" shows "a' \<in> carrier R"
general_qe'': fixes S :: "(real \<Rightarrow> real \<Rightarrow> bool) set"   and X :: "(real \<Rightarrow> real \<Rightarrow> bool) \<Rightarrow> (real \<times> real \<times> real) set"   and F :: "real \<Rightarrow> bool"   and F\<epsilon> :: "real \<Rightarrow> bool"   and F\<^sub>i\<^sub>n\<^sub>f :: "bool"   and roots :: "real \<times> real \<times> real \<Rightarrow> real set"   and A :: "real set"   and B :: "real set"   and C :: "real set"   and D :: "real set" assumes "S = {(=), (<), (\<le>), (\<noteq>)}"   and "finite (X (=))"   and "finite (X (<))"   and "finite (X (\<le>))"   and "finite (X (\<noteq>))"   and "F = (\<lambda>x. \<forall>rel\<in>S. \<forall>(a, b, c)\<in>X rel. rel (a * x\<^sup>2 + b * x + c) 0)"   and "F\<epsilon> = (\<lambda>r. \<forall>rel\<in>S. \<forall>(a, b, c)\<in>X rel. \<exists>y>r. \<forall>x\<in>{r<..y}. rel (a * x\<^sup>2 + b * x + c) 0)"   and "F\<^sub>i\<^sub>n\<^sub>f = (\<forall>rel\<in>S. \<forall>(a, b, c)\<in>X rel. \<exists>x. \<forall>y<x. rel (a * y\<^sup>2 + b * y + c) 0)"   and "roots = (\<lambda>(a, b, c). if a = 0 \<and> b \<noteq> 0 then {- c / b} else if a \<noteq> 0 \<and> 0 \<le> b\<^sup>2 - 4 * a * c then {(- b + sqrt (b\<^sup>2 - 4 * a * c)) / (2 * a)} \<union> {(- b - sqrt (b\<^sup>2 - 4 * a * c)) / (2 * a)} else {})"   and "A = \<Union> (roots ` X (=))"   and "B = \<Union> (roots ` X (<))"   and "C = \<Union> (roots ` X (\<le>))"   and "D = \<Union> (roots ` X (\<noteq>))" shows "(\<exists>x. F x) = (F\<^sub>i\<^sub>n\<^sub>f \<or> (\<exists>r\<in>A. F r) \<or> (\<exists>r\<in>B. F\<epsilon> r) \<or> (\<exists>r\<in>C. F r) \<or> (\<exists>r\<in>D. F\<epsilon> r))"
intsum: shows "integrable M (\<lambda>w. \<Sum>i. X w * indpre i w)"
uminus_rel_def: fixes X :: "('a \<times> 'a) set" shows "uminus_rel X \<equiv> let x = SOME x. x \<in> X in comm.additive.inverse (fst x) / snd x"
subdegree_numeral: fixes n :: "num" shows "subdegree (numeral n) = 0"
g2_im1_Rn: shows "gs2.gso (i - 1) \<in> Rn"
hyps: shows "a \<noteq> m \<and> a \<in> set xs \<longrightarrow> length (list_neq (remove1 a xs) m) < length (list_neq xs m)"
convergent_bounded_family_nat: fixes f :: "nat \<Rightarrow> nat \<Rightarrow> real" assumes "\<And>x. bounded (range (\<lambda>n. f n x))" obtains k :: "nat \<Rightarrow> nat" where "strict_mono k" and "\<And>x. convergent (\<lambda>n. f (k n) x)"
smU: shows "(s, m) \<in> set (\<Upsilon> p)"
finite_is_class: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list" shows "finite {C. is_class P C}"
vars_clause_list_def: fixes Cs :: "('b, 'a) Term.term literal multiset list" shows "vars_clause_list Cs = \<Union> (vars_clause ` set Cs)"
brc_cond_def: shows "brc_cond \<equiv> \<lambda>(Q, W, rcm). W \<noteq> []"
updownI': fixes dm :: "nat"   and lm :: "nat"   and pm :: "(nat \<times> int) list \<Rightarrow> nat"   and d :: "nat"   and a :: "rat array"   and v :: "(nat \<times> int) list \<Rightarrow> real" assumes "linearization dm lm pm"   and "d \<le> dm" shows "<linearization.gridI dm lm pm a v> linearization.updownI' dm lm pm d a <\<lambda>r. linearization.gridI dm lm pm a (updown' dm lm d v)>\<^sub>t"
LCons_llist_of_stream: fixes x :: "'b"   and xs :: "'b stream" shows "LCons x (llist_of_stream xs) = llist_of_stream (x ## xs)"
product_member: fixes dxs1 :: "'b set_dlist"   and dxs2 :: "'c set_dlist" assumes "ID CEQ('b) \<noteq> None"   and "ID CEQ('c) \<noteq> None" shows "DList_Set.member (DList_Set.product dxs1 dxs2) = (\<lambda>(a, b). DList_Set.member dxs1 a \<and> DList_Set.member dxs2 b)"
hyps_4: shows "ncc (eucl_of_list ` take DIM('a) ` env)"
OnlyOccurenceDisables: fixes trans :: "'a \<Rightarrow> 'b \<Rightarrow> 'c messageValue \<Rightarrow> 'b"   and sends :: "'a \<Rightarrow> 'b \<Rightarrow> 'c messageValue \<Rightarrow> ('a, 'c) message \<Rightarrow> nat"   and cfg1 :: "('a, 'c, 'b) configuration"   and m :: "('a, 'c) message"   and cfg2 :: "('a, 'c, 'b) configuration"   and m' :: "('a, 'c) message" assumes "asynchronousSystem.steps trans sends cfg1 m cfg2"   and "enabled cfg1 m'"   and "\<not> enabled cfg2 m'" shows "m = m'"
Prefix_is_action: fixes c :: "'d \<Rightarrow> 'c" shows "R (true \<turnstile> do c \<and> (\<lambda>(A, A'). alpha_rp.more A' = alpha_rp.more A)) \<in> {p. is_CSP_process p}"
mk_less_def: fixes R :: "'f \<Rightarrow> 'f \<Rightarrow> bool" shows "mk_less R = (\<lambda>x y. R x y \<and> \<not> R y x)"
confluent_beta_eta: shows "Commutation.confluent (\<rightarrow>\<^sub>\<beta>\<^sub>\<eta>)"
normalize_Units: fixes x :: "'a" assumes "x \<in> Units" shows "normalize x = (1::'a)"
abs_eq: fixes xa :: "'a set"   and x :: "'a \<Rightarrow> 'b option" shows "xa \<lhd>\<^sub>p Abs_pfun x = Abs_pfun (x |` xa)"
euler_err_fas_def: fixes ode_ops :: "ode_ops"   and X0 :: "'e list"   and h :: "floatarith"   and CX :: "floatarith list" shows "ode_operations.euler_err_fas ode_ops X0 h CX = map (ode_operations.euler_err_fas_nth ode_ops X0 h CX) [0..<length X0]"
E_wf_2: fixes G :: "'a graph" assumes "wf_graph G" shows "snd ` edges G \<subseteq> nodes G"
cert_def: shows "c \<equiv> make_cert step \<tau>s \<bottom>"
open_Collect_positive: fixes s :: "'c set"   and f :: "'c \<Rightarrow> real" assumes "continuous_on s f" shows "\<exists>A. open A \<and> A \<inter> s = {x \<in> s. 0 < f x}"
search_enum_nonempty_eq0: fixes s :: "nat \<Rightarrow> nat option"   and b :: "nat"   and i :: "nat"   and y :: "nat" assumes "total_cons s"   and "s b \<down>= i"   and "\<phi> i y \<down>"   and "e_length b \<le> y" shows "\<exists>x. eval (total_cons.r_search_enum s) [x, b] \<down>= 0"
rep_eq: fixes x :: "perm"   and xa :: "'a set['b]" shows "set_bset (x \<bullet> xa) = x \<bullet> set_bset xa"
in_net_ips_fst_init_missing: fixes np :: "nat \<Rightarrow> ('c, 'd seq_action) automaton"   and onp :: "nat \<Rightarrow> ((nat \<Rightarrow> 'e) \<times> 'f, 'd seq_action) automaton"   and sr :: "'c \<Rightarrow> 'e \<times> 'f"   and i :: "nat"   and s :: "'c net_state" assumes "openproc np onp sr"   and "i \<in> net_ips s" shows "fst (openproc.initmissing np sr (netgmap sr s)) i = the (fst (netgmap sr s) i)"
rtrancl_comp_trancl_conv: fixes r :: "('b \<times> 'b) set" shows "r\<^sup>* O r = r\<^sup>+"
WT_resource_of_oracle: fixes s :: "'s_core" assumes "IC1 s" shows "\<I>E \<oplus>\<^sub>\<I> (\<I>CA \<oplus>\<^sub>\<I> \<I>CU) \<turnstile>res RES (callee_of_core core1) s \<surd>"
degree_eqI: fixes x :: "'a pdevs"   and d :: "nat" assumes "pdevs_apply x d \<noteq> (0::'a)"   and "\<And>j. d < j \<Longrightarrow> pdevs_apply x j = (0::'a)" shows "degree x = Suc d"
nofail_simps_3: fixes x :: "'b" shows "nofail (RETURN x) = True"
empty_not_cyc_free: fixes v :: "'c \<Rightarrow> nat"   and n :: "nat"   and M :: "nat \<Rightarrow> nat \<Rightarrow> 'd DBMEntry" assumes "\<forall>c. v c \<le> n \<longrightarrow> 0 < v c"   and "[M]\<^bsub>v,n\<^esub> = {}" shows "\<not> (\<forall>i xs. i \<le> n \<and> set xs \<subseteq> {0..n} \<longrightarrow> \<one> \<le> len M i i xs)"
simps_5: fixes n :: "nat"   and f :: "recf"   and g :: "recf"   and xs :: "nat list" shows "rec_exec (Pr n f g) xs = (if last xs = 0 then rec_exec f (butlast xs) else rec_exec g (butlast xs @ [last xs - 1, rec_exec (Pr n f g) (butlast xs @ [last xs - 1])]))"
UNIF_subst_domain_range_vars_Int: fixes ss :: "('c \<Rightarrow> ('d, 'c) Term.term) list"   and E :: "(('d, 'c) Term.term \<times> ('d, 'c) Term.term) multiset"   and E' :: "(('d, 'c) Term.term \<times> ('d, 'c) Term.term) multiset" assumes "UNIF ss E E'" shows "subst_domain (compose ss) \<inter> range_vars (compose ss) = {}"
getEnv_swapEnv2: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option"   and xs :: "'varSort"   and zs :: "'varSort"   and z1 :: "'var"   and z2 :: "'var"   and x :: "'var"   and X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set" assumes "rho xs (x @xs[z1 \<and> z2]_zs) = Some X" shows "(rho &[z1 \<and> z2]_zs) xs x = Some (X #[z1 \<and> z2]_zs)"
int_simps_23: fixes P :: "'b \<Rightarrow> bool" shows "\<turnstile> (P \<or> #True) = #True"
top_on_step': fixes C :: "(char list \<Rightarrow> 'av) option acom" assumes "top_on\<^sub>c C (- vars C)" shows "top_on\<^sub>c (step' \<top> C) (- vars C)"
decompose_prod_root_unity_def: fixes p :: "'c poly" shows "decompose_prod_root_unity p = decompose_prod_root_unity_main p (degree p)"
simps_1: fixes sp_impl_subj_subj :: "partition_id_t \<Rightarrow> partition_id_t \<Rightarrow> bool"   and sp_impl_subj_obj :: "partition_id_t \<Rightarrow> obj_id_t \<Rightarrow> mode_t \<Rightarrow> bool"   and current :: "thread_id_t"   and obj :: "obj_id_t \<Rightarrow> obj_t"   and thread :: "thread_id_t \<Rightarrow> thread_t"   and more :: "'a" shows "sp_impl_subj_subj \<lparr>sp_impl_subj_subj = sp_impl_subj_subj, sp_impl_subj_obj = sp_impl_subj_obj, current = current, obj = obj, thread = thread, \<dots> = more\<rparr> = sp_impl_subj_subj"
p'_props_2: shows "awalk_verts y p' = (y # ys) @ y # zs"
norm_infsetsum'_bound: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set" assumes "infsetsum'_converges (\<lambda>x. norm (f x)) A" shows "norm (infsetsum' f A) \<le> infsetsum' (\<lambda>x. norm (f x)) A"
post_th_0: fixes A :: "nat set" shows "computable A = total_recursive (chf A)"
markedElementIsMarkedTrue: fixes m :: "'a"   and M :: "('a \<times> bool) list" shows "(m \<in> set (markedElements M)) = ((m, True) \<in> set M)"
l_get_host_def: fixes get_host :: "(_) shadow_root_ptr \<Rightarrow> ((_) heap, exception, (_) element_ptr) prog"   and get_host_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_get_host get_host get_host_locs \<equiv> (\<forall>element_ptr h. pure (get_host element_ptr) h) \<and> (\<forall>node_ptr h h'. reads get_host_locs (get_host node_ptr) h h')"
bigo_const4: fixes c :: "'a" assumes "c \<noteq> (0::'a)" shows "O(\<lambda>x. 1::'a) \<subseteq> O(\<lambda>x. c)"
OpAddNumUbx: shows "typeof_opubx OpAddNumUbx = (map typeof xs, codomain)"
dioph_rhs_bound: fixes l :: "int"   and ks :: "int list"   and xs :: "nat list" shows "\<bar>(l - eval_dioph ks (map (\<lambda>x. x mod 2) xs)) div 2\<bar> \<le> max \<bar>l\<bar> (sum_list (map abs ks))"
rel_funD: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and B :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'd"   and g :: "'b \<Rightarrow> 'e"   and x :: "'a"   and y :: "'b" assumes "rel_fun A B f g"   and "A x y" shows "B (f x) (g y)"
HEndPhase1_HInv5_q: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and q :: "Proc" assumes "HEndPhase1 s s' p"   and "HInv5 s"   and "Inv1 s"   and "Inv2a s'"   and "Inv2a s"   and "Inv2b s"   and "Inv2c s"   and "HInv3 s"   and "phase s' q = 2"   and "p \<noteq> q"   and "\<not> maxBalInp s' (bal (dblock s' q)) (inp (dblock s' q))" shows "\<exists>D\<in>MajoritySet. \<exists>qq. \<forall>d\<in>D. bal (dblock s' q) < mbal (disk s' d qq) \<and> \<not> hasRead s' q d qq"
nonneg_lincomb_list_def: fixes c :: "nat \<Rightarrow> 'a"   and Vs :: "'a vec list"   and b :: "'a vec" shows "m.nonneg_lincomb_list c Vs b = (m.lincomb_list c Vs = b \<and> (\<forall>i<length Vs. (0::'a) \<le> c i))"
UINF_ind_const: fixes P :: "('d, 'e) uexpr" shows "(\<Sqinter> i \<bullet> P) = P"
canonical_proj_ext_zero: fixes R :: "('a, 'b) ring_scheme"   and I :: "nat \<Rightarrow> 'a set" assumes "ring R" shows "ring.canonical_proj_ext R I 0 = (\<lambda>a. [I 0 +>\<^bsub>R\<^esub> a])"
iMOD2_set_iMOD_set_subset: shows "iMOD2_set \<subseteq> iMOD_set"
alloc_new_array_in_Store: fixes s :: "Store"   and T :: "Arraytype"   and n :: "nat" shows "Rep_Store s\<lparr>newAID := \<lambda>S. if T = S then Suc (newAID (Rep_Store s) T) else newAID (Rep_Store s) S, vals := (vals (Rep_Store s))(arrLenLoc T (newAID (Rep_Store s) T) := intgV (int n))\<rparr> \<in> Store"
range_binaryset_eq: fixes A :: "'c set"   and B :: "'c set" shows "range (binaryset A B) = {A, B, {}}"
BoxDmd2_simple: fixes F :: "'a \<Rightarrow> bool"   and G :: "'a \<Rightarrow> bool" shows "\<turnstile> \<box>F \<and> \<diamond>G \<longrightarrow> \<diamond>(G \<and> F)"
inner_prod_with_row_bra_vec_col_ket_vec: fixes u :: "complex Matrix.vec"   and v :: "complex Matrix.vec" assumes "dim_vec u = dim_vec v" shows "\<langle>u|v\<rangle> = Matrix.row \<langle>|u\<rangle>| 0 \<bullet> Matrix.col |v\<rangle> 0"
unsatReportExtensiveExplain: fixes F0 :: "Literal list list"   and F :: "Literal list list"   and conflictFlag :: "bool"   and C :: "Literal list" assumes "InvariantEquivalent F0 F"   and "InvariantCEntailed conflictFlag F C"   and "conflictFlag"   and "C = []" shows "\<not> satisfiable F0"
map_in_list_rel_conv: fixes l :: "'a list"   and \<alpha> :: "'a \<Rightarrow> 'b"   and I :: "'a \<Rightarrow> bool" shows "((l, map \<alpha> l) \<in> \<langle>br \<alpha> I\<rangle>list_rel) = (\<forall>x\<in>set l. I x)"
cases: fixes x :: "'a list" obtains "x = []"   | xa :: "'a" where "x = [xa__]"   | xa :: "'a" and y :: "'a" and ys :: "'a list" where "x = xa # y # ys"