hcju/isabelleps huggingface dataset:
b_def: shows "b = real_of_float (lower bivl + upper bivl) / 2"
globally_consistent_writable: fixes stop :: "'a"   and eval :: "((('a \<times> (Mode \<Rightarrow> 'b set)) \<times> ('b \<Rightarrow> 'c)) \<times> ('a \<times> (Mode \<Rightarrow> 'b set)) \<times> ('b \<Rightarrow> 'c)) set"   and some_val :: "'c"   and some_val' :: "'c"   and x :: "'b"   and A :: "'b \<Rightarrow> ('c \<times> 'c) option"   and dma :: "'b \<Rightarrow> Sec"   and mds :: "Mode \<Rightarrow> 'b set" assumes "sifum_security stop eval some_val some_val'"   and "x \<in> dom A"   and "sifum_security.globally_consistent dma A mds" shows "x \<notin> mds AsmNoWrite"
signed_of_int_def: fixes i :: "int" shows "signed_of_int i \<equiv> word_of_int i"
LIMSEQ_incseq_SUP: fixes X :: "nat \<Rightarrow> 'e" assumes "bdd_above (range X)"   and "incseq X" shows "X \<longlonglongrightarrow> Sup (range X)"
all1_matrix_dim_2: fixes nr :: "nat"   and nc :: "nat" shows "dim_col (all1_matrix nr nc) = nc"
nop_v11_n: shows "n \<in>\<^sub>\<circ> \<omega>"
fin2: fixes K :: "'b \<Rightarrow> 'b DBMEntry" assumes "theLe \<circ> K = id" shows "finite ({u c' + d |c' d. 0 < v c' \<and> v c' \<le> n \<and> c \<noteq> c' \<and> M (v c) (v c') = K d} \<union> {d |d. M (v c) 0 = K d})"
N2_def: shows "N2 \<equiv> alpha2 \<or> beta2 \<or> gamma2"
list_set_tl: fixes x :: "'b"   and xs :: "'b list" assumes "x \<in> set (tl xs)" shows "x \<in> set xs"
ENC_Exists: fixes n :: "nat"   and \<phi> :: "'a formula" assumes "wf_formula n (FExists \<phi>)" shows "lang n (ENC n (FOV (FExists \<phi>))) - {[]} = map \<pi> ` lang (Suc n) (ENC (Suc n) (FOV \<phi>)) - {[]}"
eval_in_hom: fixes b :: "'a"   and c :: "'a" assumes "ide b"   and "ide c" shows "\<guillemotleft>eval b c : local.prod (exp b c) b \<rightarrow> c\<guillemotright>"
disjunctiveD: fixes x :: "'c \<Rightarrow> 'd"   and y :: "'c"   and z :: "'c" assumes "x \<in> Apply.disjunctive" shows "x (sup y z) = sup (x y) (x z)"
dim_hyp1: shows "M_mat_R signs subsets \<in> carrier_mat (length signs) (length signs)"
z1n2_def: shows "z1n2 = z1 * n2"
iteration_slide_var: fixes x :: "'a"   and y :: "'a" shows "x \<cdot> (y \<cdot> x)\<^sup>\<infinity> \<le> (x \<cdot> y)\<^sup>\<infinity> \<cdot> x"
natural_transformation_axioms: shows "natural_transformation J C \<chi>.A.map D \<chi>"
that: assumes "h' = h"   and "r = f h" shows "thesis"
fls_divide_convert_times_inverse: fixes f :: "'b fls"   and g :: "'b fls" shows "f / g = f * inverse g"
append_eq_append_conv: fixes xs :: "'c list"   and ys :: "'c list"   and us :: "'c list"   and vs :: "'c list" assumes "length xs = length ys \<or> length us = length vs" shows "(xs \<cdot> us = ys \<cdot> vs) = (xs = ys \<and> us = vs)"
intro: fixes M :: "'c pre_map"   and a :: "'c" assumes "pre_digraph_map.face_cycle_set M (edge_rev M a) \<noteq> pre_digraph_map.face_cycle_set M a" shows "bidel_arc_diff_face_axioms M a"
simps_1: fixes uu :: "'a" shows "is_exn (Some uu) = False"
moebius_over_square_sums: shows "(\<lambda>n. moebius_mu n / real (n\<^sup>2)) sums (6 / pi\<^sup>2)"
list_sequent_cost_def: fixes \<Gamma> :: "'b list"   and \<Delta> :: "'b list" shows "list_sequent_cost \<Gamma> \<Delta> = 2 * sum_list (map size (\<Gamma> @ \<Delta>)) + length (\<Gamma> @ \<Delta>)"
intra_SDG_path_valid_SDG_node_1: fixes n :: "'node SDG_node"   and ns :: "'node SDG_node list"   and n' :: "'node SDG_node" assumes "n i-ns\<rightarrow>\<^sub>d* n'" shows "valid_SDG_node n"
that: fixes A :: "'a set" assumes "open A"   and "x \<in> A"   and "\<And>y. \<lbrakk>y \<in> s; y \<in> A\<rbrakk> \<Longrightarrow> f y \<in> B" shows "thesis"
support_postList: fixes xs :: "(char list \<times> ((char list \<Rightarrow> int) \<Rightarrow> nat) \<times> 'a) list" shows "support (postList xs) \<subseteq> lesvars xs"
ring_hom_minus: fixes S :: "('c, 'd) ring_scheme"   and h :: "'a \<Rightarrow> 'c"   and a :: "'a"   and b :: "'a" assumes "ring S"   and "h \<in> ring_hom R S"   and "a \<in> carrier R"   and "b \<in> carrier R" shows "h (a \<ominus> b) = h a \<ominus>\<^bsub>S\<^esub> h b"
hmem_rel_induct: fixes P :: "hf \<Rightarrow> bool"   and x :: "hf" assumes "\<And>x. (\<And>y. (y, x) \<in> hmem_rel \<Longrightarrow> P y) \<Longrightarrow> P x" shows "P x"
lub_upper: fixes leq :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and lub :: "'f set \<Rightarrow> 'f"   and A :: "'f set"   and x :: "'f" assumes "partial_function_definitions leq lub"   and "Complete_Partial_Order.chain leq A"   and "x \<in> A" shows "leq x (lub A)"
norm_imp_pos_and_ge: fixes x :: "'c"   and n :: "real" assumes "norm x \<equiv> n" shows "0 \<le> norm x \<and> norm x \<le> n"
map_iterator_abs_genord_remove_abs2: fixes \<alpha> :: "'e \<Rightarrow> 'f"   and invar :: "'e \<Rightarrow> bool"   and iti :: "('g \<Rightarrow> bool) \<Rightarrow> ('h \<times> 'e \<Rightarrow> 'g \<Rightarrow> 'g) \<Rightarrow> 'g \<Rightarrow> 'g"   and m :: "'h \<Rightarrow> 'f option"   and R :: "'h \<times> 'f \<Rightarrow> 'h \<times> 'f \<Rightarrow> bool" assumes "map_iterator_abs_genord \<alpha> invar iti m R" obtains m' :: "'h \<Rightarrow> 'e option" where "map_iterator_genord iti m' (\<lambda>(k, v) (k', v'). R (k, \<alpha> v) (k', \<alpha> v'))" and "map_option \<alpha> \<circ> m' = m" and "\<And>k v. m' k = Some v \<Longrightarrow> invar v"
absorb_iff2: fixes b :: "'a::semilattice_inf"   and a :: "'a::semilattice_inf" shows "((b::'a::semilattice_inf) \<le> (a::'a::semilattice_inf)) = (a \<sqinter> b = b)"
ge: fixes x :: "real" assumes "0 < x"   and "x < 1" shows "e \<le> Im (vector_derivative \<gamma> (at x) * cnj (\<gamma> x - z))"
simps: fixes m :: "name"   and x :: "tm" assumes "atom m \<sharp> x" shows "IndP x = SyntaxN.Ex m (OrdP (Var m) AND x EQ Q_Ind (Var m))"
converseD: fixes a :: "'a"   and b :: "'b"   and r :: "('b \<times> 'a) set" assumes "(a, b) \<in> r\<inverse>" shows "(b, a) \<in> r"
control_withinI: fixes \<sigma> :: "('i \<times> ('e, 'f, 'g, 'h) seqp) set"   and \<Gamma> :: "'g \<Rightarrow> ('e, 'f, 'g, 'h) seqp" assumes "\<And>p. p \<in> Range \<sigma> \<Longrightarrow> \<exists>pn. p \<in> subterms (\<Gamma> pn)" shows "control_within \<Gamma> \<sigma>"
quarter_discrim_self_zero: fixes a :: "(real, 3) vec"   and b :: "(real, 3) vec" assumes "proj2_abs a = proj2_abs b" shows "quarter_discrim a b = 0"
isCont_rabs: fixes a :: "'a"   and f :: "'a \<Rightarrow> real" assumes "isCont f a" shows "isCont (\<lambda>x. \<bar>f x\<bar>) a"
subgroup_inverse_equality: fixes u :: "'a \<Rightarrow> 'a" assumes "u \<in> G" shows "symmetric.inverse u = symmetric.sub.inverse u"
le_def: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and e\<^sub>1 :: "'b err"   and e\<^sub>2 :: "'b err" shows "le r e\<^sub>1 e\<^sub>2 = (case e\<^sub>2 of Err \<Rightarrow> True | OK y \<Rightarrow> case e\<^sub>1 of Err \<Rightarrow> False | OK x \<Rightarrow> x \<sqsubseteq>\<^bsub>r\<^esub> y)"
kruskal_injective_inv_2: fixes e :: "'b"   and f :: "'b" assumes "stone_algebra_class.comp_inf.arc e"   and "stone_algebra_class.comp_inf.acyclic f" shows "bounded_distrib_lattice_class.comp_inf.tc (top \<sqinter> e) \<sqinter> id f \<le> - e"
ct_suffix_cons_it: fixes xs :: "ct list"   and xs' :: "ct list" shows "ct_suffix xs (xs' @ xs)"
dim_span: fixes S :: "'b set" shows "dim (span S) = dim S"
ik\<^sub>s\<^sub>t_is_rcv_set: fixes A :: "('a, 'b) strand_step list" shows "ik\<^sub>s\<^sub>t A = {t. receive\<langle>t\<rangle>\<^sub>s\<^sub>t \<in> set A}"
rPar2: fixes \<Psi> :: "'b"   and \<Psi>\<^sub>P :: "'b"   and Q :: "('a, 'b, 'c) psi"   and M :: "'a"   and N :: "'a"   and Q' :: "('a, 'b, 'c) psi"   and P :: "('a, 'b, 'c) psi"   and A\<^sub>P :: "name list"   and C :: "'d" assumes "\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto> M\<lparr>N\<rparr> \<prec> Q'"   and "extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>"   and "distinct A\<^sub>P"   and "\<And>C. Prop C (\<Psi> \<otimes> \<Psi>\<^sub>P) Q M N Q'"   and "distinct A\<^sub>P"   and "A\<^sub>P \<sharp>* P"   and "A\<^sub>P \<sharp>* Q"   and "A\<^sub>P \<sharp>* \<Psi>"   and "A\<^sub>P \<sharp>* M"   and "A\<^sub>P \<sharp>* N"   and "A\<^sub>P \<sharp>* Q'"   and "A\<^sub>P \<sharp>* C" shows "Prop C \<Psi> (P \<parallel> Q) M N (P \<parallel> Q')"
sig: fixes i :: "nat" shows "(fst (r i), length (a i)) \<in> F"
branch_nominals_def: fixes branch :: "(('d, 'c) fm list \<times> 'c) list" shows "branch_nominals branch \<equiv> \<Union> (block_nominals ` set branch)"
HomFtorInMor': fixes C :: "('d, 'e, 'f) LSCategory_scheme"   and X :: "'d"   and f :: "'e" assumes "LSCategory C"   and "X \<in> obj\<^bsub>C\<^esub>"   and "f \<in> mor\<^bsub>C\<^esub>" shows "Hom\<^bsub>C\<^esub>[X,f] \<in> mor\<^bsub>SET'\<^esub>"
enabled_repcsI: fixes cfg :: "('s \<times> ('c \<Rightarrow> real) set) cfg"   and st :: "'s \<times> ('c \<Rightarrow> real)"   and xs :: "('s \<times> ('c \<Rightarrow> real)) cfg stream" assumes "cfg \<in> R_G.valid_cfg"   and "abss st = state cfg"   and "MDP.MC.enabled (repcs st cfg) xs" shows "enabled_repcs (repcs st cfg) xs"
closest_pair_bf_c0_ne_c1: fixes ps :: "(int \<times> int) list"   and c\<^sub>0 :: "int \<times> int"   and c\<^sub>1 :: "int \<times> int" assumes "1 < length ps"   and "distinct ps"   and "(c\<^sub>0, c\<^sub>1) = closest_pair_bf ps" shows "c\<^sub>0 \<noteq> c\<^sub>1"
prog: shows "{X. G' \<in> stable X} \<in> progress_set F UNIV B"
inner_zero_right: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and inner :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and x :: "'b" assumes "class.real_inner minus dist norm plus zero uminus scaleR sgn uniformity open inner" shows "inner x zero = 0"
assms_1: shows "\<exists>i\<ge>n. \<parallel>c\<parallel>\<^bsub>t i\<^esub>"
intro: fixes bind :: "'d \<Rightarrow> ('e \<Rightarrow> 'd) \<Rightarrow> 'd"   and return :: "'e \<Rightarrow> 'd" assumes "\<And>x f g. bind (bind x f) g = bind x (\<lambda>y. bind (f y) g)"   and "\<And>x f. bind (return x) f = f x"   and "\<And>x. bind x return = x" shows "monad return bind"
assms_1: shows "\<not> is_Tensor (t \<^bold>\<lfloor>\<^bold>\<otimes>\<^bold>\<rfloor> u)"
gcd_mult_right_left_cancel: fixes one :: "'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and gcd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and lcm :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and a :: "'b"   and c :: "'b"   and b :: "'b" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize"   and "algebraic_semidom.coprime times one a c" shows "gcd a (times c b) = gcd a b"
abs_power2: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sgn :: "'c \<Rightarrow> 'c"   and abs :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and a :: "'c" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "abs (power.power one times a 2) = power.power one times a 2"
Com_def: fixes x :: "'b com \<times> 'b set \<times> ('b \<times> 'b) set \<times> ('b \<times> 'b) set \<times> 'b set" shows "Com x \<equiv> fst x"
el_gt0_imp_prod_gt0: fixes xs :: "'a list" assumes "\<And>y. y \<in> set xs \<Longrightarrow> (0::'a) < y" shows "(0::'a) < prod_list xs"
roundup_positiv: fixes emBits :: "nat" assumes "0 < emBits" shows "0 < roundup emBits 160"
assms_2: shows "P (nsqn\<^sub>r (dsn, dsk, Aodv_Basic.inv, hops, nhip))"
comm_monoid_add_ow_Ball_def: fixes S :: "'c set"   and pls :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and z :: "'c" shows "comm_monoid_add_ow S pls z = (ab_semigroup_add_ow S pls \<and> (\<forall>a\<in>S. pls z a = a) \<and> z \<in> S)"
foldr_def: shows "foldr \<equiv> \<lambda>uu uua. rec_list (\<lambda>f. id) (\<lambda>x xs xsa f. (f x \<circ>\<circ> xsa) f) uua uu"
exhaust: fixes y :: "'b iarray" obtains x :: "'b list" where "y = IArray x"
acquire_all_def: fixes ls :: "('c, 'd) locks"   and t :: "'d"   and ln :: "'c \<Rightarrow>f nat" shows "acquire_all ls t ln \<equiv> (\<lambda>(l, la). acquire_locks l t la) \<circ>$ ($ls, ln$)"
redundant_def: fixes \<alpha>n :: "'c \<Rightarrow> 'd list"   and inEdges' :: "'c \<Rightarrow> 'd \<Rightarrow> ('d \<times> 'e) list"   and defs :: "'c \<Rightarrow> 'd \<Rightarrow> 'f set"   and "uses" :: "'c \<Rightarrow> 'd \<Rightarrow> 'f set"   and phis :: "'c \<Rightarrow> 'd \<times> 'f \<Rightarrow> 'f list option"   and g :: "'c" shows "CFG_SSA_wf_base.redundant \<alpha>n inEdges' defs uses phis g \<equiv> \<exists>v\<in>CFG_SSA_base.allVars \<alpha>n inEdges' defs uses phis g. CFG_SSA_wf_base.trivial \<alpha>n inEdges' defs uses phis g v"
axioms_1: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and u_block_size :: "int"   and grouping :: "int"   and min_index :: "int" assumes "t_packing_design point_set block_collection u_block_size grouping min_index" shows "block_design point_set block_collection u_block_size"
sylow_conjugate: fixes G :: "('c, 'd) monoid_scheme"   and p :: "nat"   and a :: "nat"   and m :: "nat"   and P :: "'c set"   and Q :: "'c set" assumes "snd_sylow G p a m"   and "P \<in> group.subgroups_of_size G (p ^ a)"   and "Q \<in> group.subgroups_of_size G (p ^ a)" obtains g :: "'c" where "g \<in> carrier G" and "Q = g <#\<^bsub>G\<^esub> (P #>\<^bsub>G\<^esub> inv\<^bsub>G\<^esub> g)"
ireturn_rule: fixes \<Phi> :: "'c \<Rightarrow> bool"   and x :: "'c" assumes "\<Phi> x" shows "RETURN x \<le> SPEC \<Phi>"
trcl_rev_uncons: fixes c :: "'a"   and w :: "'b list"   and e :: "'b"   and c' :: "'a"   and T :: "('a \<times> 'b \<times> 'a) set" assumes "(c, w @ [e], c') \<in> trcl T" shows "\<exists>ch. (c, w, ch) \<in> trcl T \<and> (ch, e, c') \<in> T"
allAllocationsUniverse: fixes G :: "'d set"   and N :: "'c set" shows "allAllocations N G \<subseteq> allocationsUniverse"
bipartite_V: shows "\<^bold>V\<^bsub>\<Gamma>'\<^esub> \<subseteq> A \<Gamma>' \<union> B \<Gamma>'"
min_absorb2: fixes y :: "'a"   and x :: "'a" assumes "y \<le> x" shows "min x y = y"
trace_measurement: fixes d :: "nat"   and n :: "nat"   and M :: "nat \<Rightarrow> complex mat"   and A :: "complex mat" assumes "measurement d n M"   and "A \<in> carrier_mat d d" shows "trace (matrix_sum d (\<lambda>k. M k * A * adjoint (M k)) n) = trace A"
G: fixes y :: "'e" shows "monotone (fun_ord ordb) ordb (\<lambda>g. U2 (G (C2 g)) y)"
subset_singleton_iff_Uniq: fixes A :: "'b set" shows "(\<exists>a. A \<subseteq> {a}) = (\<exists>\<^sub>\<le>\<^sub>1 x. x \<in> A)"
point: fixes v :: "'a vec" assumes "v \<in> X" shows "(a v \<cdot>\<^sub>v v) $ i = a v * v $ i"
vbrelation: shows "vpairs (\<FF>\<lparr>ArrMap\<rparr>) = \<FF>\<lparr>ArrMap\<rparr>"
igFreshIGAbs1STR_def: fixes MOD :: "('c, 'd, 'e, 'f, 'g, 'h, 'i, 'j, 'k) model_scheme" shows "igFreshIGAbs1STR MOD \<equiv> \<forall>ys y X. igFreshAbs MOD ys y (igAbs MOD ys y X)"
Min_le: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and A :: "'c set"   and x :: "'c" assumes "class.linorder less_eq less"   and "finite A"   and "x \<in> A" shows "less_eq (linorder.Min less_eq A) x"
SourcesA82_L1: shows "Sources level1 sA82 = {sA6, sA72}"
ae2ce_def: fixes ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and x :: "Vars.var" shows "ae2ce ae x = a2c\<cdot>(ae x)"
take_strict: fixes n :: "nat" shows "\<psi>\<^sup>E\<^bsub>n\<^esub>\<cdot>\<bottom> = \<bottom>"
preserves_ide: fixes a :: "'a" assumes "Adj.C.ide a" shows "Adj.D.ide (G a)"
typed_no_annotated_stop: fixes \<Gamma> :: "'Var \<Rightarrow> Sec option"   and c :: "('Var, 'AExp, 'BExp) Stmt"   and \<Gamma>' :: "'Var \<Rightarrow> Sec option" assumes "\<turnstile> \<Gamma> {c} \<Gamma>'" shows "\<not> has_annotated_stop c"
\<omega>_limit_set_in_compact_subset: fixes K :: "'a set"   and x :: "'a" assumes "compact K"   and "K \<subseteq> X"   and "x \<in> X"   and "trapped_forward x K" shows "\<omega>_limit_set x \<subseteq> K"
X3_UNIV: shows "UNIV = set [X3.Xd1, X3.Xd1', X3.Xd2]"
subset_Compl_singleton: fixes A :: "'g set"   and b :: "'g" shows "(A \<subseteq> - {b}) = (b \<notin> A)"
face_cycle_succ_pred: fixes a :: "'b" shows "G.face_cycle_succ (G.face_cycle_pred a) = a"
card_minus_fMin: fixes s :: "'b fset" assumes "s \<noteq> {||}" shows "card (fset s - {fMin s}) < card (fset s)"
i_set_finite_as_iMODb: fixes I :: "nat set" assumes "I \<in> i_set"   and "finite I" shows "\<exists>r m c. I = [ r, mod m, c ]"
eval_fds_inverse: fixes f :: "'a fds"   and s :: "'a" assumes "fds_abs_converges f s"   and "fds_abs_converges (inverse f) s"   and "fds_nth f 1 \<noteq> (0::'a)" shows "eval_fds (inverse f) s = inverse (eval_fds f s)"
lfp_loop_greatest: fixes G :: "'a \<Rightarrow> bool"   and body :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and P :: "'a \<Rightarrow> real"   and Q :: "'a \<Rightarrow> real" assumes "\<And>R. \<lbrakk>\<lambda>s. \<guillemotleft> G \<guillemotright> s * wp body R s + \<guillemotleft> \<N> G \<guillemotright> s * P s \<tturnstile> R; sound R\<rbrakk> \<Longrightarrow> Q \<tturnstile> R"   and "healthy (wp body)"   and "sound P"   and "sound Q" shows "Q \<tturnstile> lfp_exp (\<lambda>Q s. \<guillemotleft> G \<guillemotright> s * wp body Q s + \<guillemotleft> \<N> G \<guillemotright> s * P s)"
1: shows "t = \<^bold>\<langle>un_Prim t\<^bold>\<rangle> \<and> C.arr (un_Prim t) \<and> Diag t \<and> Diag u \<and> u \<noteq> \<^bold>\<I>"
dghm_comp_ObjMap_app: fixes \<alpha> :: "V"   and \<BB> :: "V"   and \<CC> :: "V"   and \<GG> :: "V"   and \<AA> :: "V"   and \<FF> :: "V"   and a :: "V" assumes "\<GG> : \<BB> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<CC>"   and "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>"   and "a \<in>\<^sub>\<circ> \<AA>\<lparr>Obj\<rparr>" shows "(\<GG> \<circ>\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<FF>)\<lparr>ObjMap\<rparr>\<lparr>a\<rparr> = \<GG>\<lparr>ObjMap\<rparr>\<lparr>\<FF>\<lparr>ObjMap\<rparr>\<lparr>a\<rparr>\<rparr>"
c: fixes u :: "complex" assumes "cmod u < 1" shows "Arctan u - G u = c"
csa_list_copr_rel: fixes p :: "real poly"   and qs :: "real poly list" assumes "p \<noteq> 0"   and "\<forall>q. List.member qs q \<longrightarrow> algebraic_semidom_class.coprime p q" shows "characterize_consistent_signs_at_roots p qs = characterize_consistent_signs_at_roots_copr p qs"
sel_1: fixes x1 :: "'a" shows "un_Terminates (Terminates x1) = x1"
size_prod_def: shows "index_class.size \<equiv> prod_index_def.size index_class.size index_class.size"
cut_ge_def: fixes I :: "'a set"   and t :: "'a" shows "I \<down>\<ge> t \<equiv> {x \<in> I. t \<le> x}"
open_times_image: fixes S :: "'c set"   and c :: "'c" assumes "open S"   and "c \<noteq> (0::'c)" shows "open ((*) c ` S)"
fresh_star_insert_elim: fixes x :: "'a"   and S :: "'a set"   and c :: "'b"   and C :: "prop" shows "(insert x S \<sharp>* c \<Longrightarrow> PROP C) \<equiv> (\<lbrakk>x \<sharp> c; S \<sharp>* c\<rbrakk> \<Longrightarrow> PROP C)"
support_on_mono: fixes carrier :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'e" assumes "\<And>x. \<lbrakk>x \<in> carrier; f x \<noteq> (0::'d)\<rbrakk> \<Longrightarrow> g x \<noteq> (0::'e)" shows "support_on carrier f \<subseteq> support_on carrier g"
fstI: fixes x :: "'a \<times> 'b"   and y :: "'a"   and z :: "'b" assumes "x = (y, z)" shows "fst x = y"
nprt_add: fixes A :: "'a matrix"   and B :: "'a matrix" assumes "disj_matrices A B" shows "nprt (A + B) = nprt A + nprt B"
graeffe_poly_def: fixes c :: "'a"   and as :: "'a list"   and m :: "nat" shows "graeffe_poly c as m = Polynomial.smult (c ^ 2 ^ m) (\<Prod>a\<leftarrow>as. [:- (a ^ 2 ^ m), 1::'a:])"
distinct_tl: fixes xs :: "'g list" assumes "distinct xs" shows "distinct (tl xs)"
no_Trace_implies_no_Failure: fixes s :: "'b event list"   and P :: "'b process" assumes "s \<notin> T P" shows "(s, {}) \<notin> F P"
Abs_Node_inverse: fixes y :: "(nat \<Rightarrow> 'a + nat) \<times> ('b + nat)" assumes "y \<in> Node" shows "Rep_Node (Abs_Node y) = y"
stabiliser_subgroup: fixes G :: "('d, 'e) monoid_scheme"   and action :: "'d \<Rightarrow> 'f \<Rightarrow> 'f"   and x :: "'f" assumes "orbit_stabiliser G action" shows "subgroup (orbit_stabiliser.stabiliser G action x) G"
IH_2: fixes k :: "nat"   and sub :: "'a btree"   and sep :: "'a" assumes "split_max k tt = (sub, sep)"   and "nonempty_lasttreebal tt"   and "tt \<noteq> Leaf" shows "height sub = height tt"
weak_paretoD: fixes scf :: "('a \<Rightarrow> ('b \<times> 'b) set) \<Rightarrow> ('b \<times> 'b) set"   and A :: "'b set"   and Is :: "'a set"   and Pcond :: "'b set \<Rightarrow> 'a set \<Rightarrow> ('a \<Rightarrow> ('b \<times> 'b) set) \<Rightarrow> bool"   and P :: "'a \<Rightarrow> ('b \<times> 'b) set"   and x :: "'b"   and y :: "'b" assumes "weak_pareto scf A Is Pcond"   and "Pcond A Is P"   and "x \<in> A"   and "y \<in> A"   and "\<And>i. i \<in> Is \<Longrightarrow> x \<^bsub>(P i)\<^esub>\<prec> y" shows "x \<^bsub>(scf P)\<^esub>\<prec> y"
of_radiant_def: fixes \<omega> :: "real" shows "of_radiant \<omega> = axis 1 (cos \<omega>) + axis 2 (sin \<omega>)"
not_ccw_eq: fixes p :: "real \<times> real"   and q :: "real \<times> real"   and r :: "real \<times> real"   and S :: "(real \<times> real) set" assumes "distinct3 p q r"   and "in3 S p q r" shows "(\<not> ccw p q r) = ccw p r q"
assert_def: fixes assert_bound :: "'a \<times> 'b atom \<Rightarrow> ('a, 'b) state \<Rightarrow> ('a, 'b) state"   and check :: "('a, 'b) state \<Rightarrow> ('a, 'b) state"   and a :: "'a \<times> 'b atom"   and s :: "('a, 'b) state" assumes "Assert' assert_bound check" shows "Assert'.assert assert_bound check a s \<equiv> check (assert_bound a s)"
dens'': shows "has_density (distr (distr M lborel extract_real) borel f) lborel (g (\<delta> \<circ> RealVal))"
atLeastatMost_empty_iff: fixes a :: "'c word"   and b :: "'c word" shows "(signed.atLeastAtMost a b = {}) = (\<not> a \<le>s b)"
filter_conv_filter_it': fixes acc :: "'b list"   and P :: "'b \<Rightarrow> bool"   and xs :: "'b list" shows "rev acc @ filter P xs = filter_it' acc P xs"
QQ2_a_def: shows "QQ2_a \<equiv> [: p, t, left, right, atom, mrk \<leadsto> p', t', left', right', atom', mrk' . root \<noteq> nil \<and> p' = root \<and> t' = nil \<and> mrk' = mrk \<union> {root} \<and> left' = left \<and> right' = right \<and> atom' = atom :]"
connected_component_sym: fixes S :: "'a set"   and x :: "'a"   and y :: "'a" assumes "connected_component S x y" shows "connected_component S y x"
flush_instr_cache_privilege: fixes s :: "'b sparc_state"   and s' :: "'b sparc_state" assumes "get_S (cpu_reg_val PSR s) = 0"   and "s' = flush_instr_cache s" shows "get_S (cpu_reg_val PSR s') = 0"
labels_multiset: fixes \<sigma> :: "'d \<times> ('c \<times> 'd) list"   and \<alpha> :: "'c"   and r :: "('c \<times> 'c) set" assumes "length (labels \<sigma>) \<le> 1"   and "set (labels \<sigma>) \<subseteq> {\<alpha>}" shows "(r|labels \<sigma>|, {#\<alpha>#}) \<in> mul_eq r"
subGB: shows "generate G (carrier G \<inter> B) \<subseteq> B"
starset_insert: fixes x :: "'c"   and A :: "'c set" shows "*s* insert x A = insert (star_of x) (*s* A)"
fctrs: shows "factorize_complex_main (lr_fps_denominator' cs) = Some fctrs"
assms_4: fixes g :: "'a" assumes "g \<in> G" shows "invertible g"
deriv: fixes y :: "real" assumes "y \<in> {0..1}" shows "(G has_real_derivative g y) (at y within {0..1})"
spanning_subset_independent: fixes B :: "'b set"   and A :: "'b set" assumes "B \<subseteq> A"   and "vs1.independent A"   and "A \<subseteq> vs1.span B" shows "A = B"