hcju/isabelleps huggingface dataset:
plossless_exec_gpv: fixes callee :: "'b \<Rightarrow> 'd \<Rightarrow> ('e \<times> 'b) spmf"   and I :: "'b \<Rightarrow> bool"   and \<I> :: "('d, 'e) \<I>"   and gpv :: "('f, 'd, 'e) gpv"   and s :: "'b" assumes "callee_invariant_on callee I \<I>"   and "plossless_gpv \<I> gpv"   and "\<I> \<turnstile>g gpv \<surd>"   and "\<And>s x. \<lbrakk>x \<in> outs_\<I> \<I>; I s\<rbrakk> \<Longrightarrow> lossless_spmf (callee s x)"   and "I s" shows "lossless_spmf (exec_gpv callee gpv s)"
sprod_closed: fixes a :: "'b"   and m :: "'a" assumes "a \<in> carrier R"   and "m \<in> carrier M" shows "a \<cdot>\<^sub>s m \<in> carrier M"
edge_preserving_def: fixes h :: "('a \<times> 'b) set"   and e1 :: "('c \<times> 'a \<times> 'a) set"   and e2 :: "('c \<times> 'b \<times> 'b) set" shows "edge_preserving h e1 e2 \<equiv> \<forall>(k, v1, v2)\<in>e1. \<forall>v1' v2'. (v1, v1') \<in> h \<and> (v2, v2') \<in> h \<longrightarrow> (k, v1', v2') \<in> e2"
DF_all_tickfree_traces1: fixes A :: "'a set" shows "T (DF A) = {s. set s \<subseteq> ev ` A}"
refl_ax_def: shows "refl_ax = Var X1 EQ Var X1"
Bit_B1_2t: fixes k :: "int" shows "k BIT True = 2 * k + 1"
analz_subset_iff: fixes G :: "msg set"   and H :: "msg set" shows "(analz G \<subseteq> analz H) = (G \<subseteq> analz H)"
iIN_iFROM_subset_same: fixes n :: "nat"   and d :: "nat" shows "[n\<dots>,d] \<subseteq> [n\<dots>]"
5496: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "- - (x \<sqinter> y) \<sqinter> - (x \<squnion> z) = bot"
red_lA_incr: fixes e :: "char list exp"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and l :: "char list \<Rightarrow> val option"   and e' :: "char list exp"   and h' :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and l' :: "char list \<Rightarrow> val option"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list" assumes "P \<turnstile> \<langle>e,(h, l)\<rangle> \<rightarrow> \<langle>e',(h', l')\<rangle>" shows "\<lfloor>dom l\<rfloor> \<squnion> \<A> e \<sqsubseteq> \<lfloor>dom l'\<rfloor> \<squnion> \<A> e'"
hyps: fixes p :: "'a poly" assumes "degree p < degree pa__" shows "P p"
term_ok_mk_all: fixes \<Theta> :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set"   and B :: "Core.term"   and \<tau> :: "typ"   and x :: "variable" assumes "wf_theory \<Theta>"   and "term_ok' (sig \<Theta>) B"   and "typ_of B = Some propT"   and "typ_ok \<Theta> \<tau>" shows "term_ok \<Theta> (mk_all x \<tau> B)"
collapse_2: fixes process :: "'a process" assumes "isChoice process" shows "Choice (ch1Of process) (ch2Of process) = process"
IH: fixes p :: "('a, 'out, 'b) gpv" shows "interaction_bound' (p \<bind> f) \<le> ib1 p + (\<Squnion>x\<in>results'_gpv p. interaction_bound' (f x))"
inject_3: fixes x3 :: "Vars.var"   and y3 :: "Vars.var" shows "(Upd x3 = Upd y3) = (x3 = y3)"
differentiable_at_withinI: fixes f :: "'c \<Rightarrow> 'd"   and x :: "'c"   and s :: "'c set" assumes "f differentiable at x" shows "f differentiable at x within s"
veq_structure_simps_3: fixes name\<^sub>1 :: "name"   and ts :: "value list"   and name\<^sub>2 :: "name"   and us :: "value list" shows "veq_structure (Vconstr name\<^sub>1 ts) (Vconstr name\<^sub>2 us) = (name\<^sub>1 = name\<^sub>2 \<and> list_all2 veq_structure ts us)"
alpha_def_5: shows "val_Zp (pderiv f \<bullet> \<alpha>) = val_Zp (pderiv f \<bullet> a)"
f_map: fixes v :: "'v"   and a :: "'f" assumes "v \<in> V" shows "T (a \<sharp>\<cdot> v) = a \<sharp>\<star> T v"
span_minimal: fixes S :: "'b set"   and T :: "'b set" assumes "S \<subseteq> T"   and "csubspace T" shows "cspan S \<subseteq> T"
simps_10: fixes bs :: "real list"   and i :: "int"   and b :: "MIR.num" shows "Ifm bs (NDvd i b) = (\<not> real_of_int i rdvd Inum bs b)"
CFn_project_strict_1: fixes v :: "C \<rightarrow> CValue" shows "\<bottom> \<down>CFn v = \<bottom>"
meval_sched_appD: fixes ns :: "nat list"   and ms :: "nat list"   and cms\<^sub>1 :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list"   and mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and cms\<^sub>1' :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list"   and mem\<^sub>1' :: "'Var \<Rightarrow> 'Val" assumes "(cms\<^sub>1, mem\<^sub>1) \<rightarrow>\<^bsub>ns @ ms\<^esub> (cms\<^sub>1', mem\<^sub>1')" shows "\<exists>cms\<^sub>1'''' mem\<^sub>1''''. (cms\<^sub>1, mem\<^sub>1) \<rightarrow>\<^bsub>ns\<^esub> (cms\<^sub>1'''', mem\<^sub>1'''') \<and> (cms\<^sub>1'''', mem\<^sub>1'''') \<rightarrow>\<^bsub>ms\<^esub> (cms\<^sub>1', mem\<^sub>1')"
greatest_def: fixes L :: "('a, 'b) gorder_scheme"   and g :: "'a"   and A :: "'a set" shows "greatest L g A = (A \<subseteq> carrier L \<and> g \<in> A \<and> (\<forall>x\<in>A. x \<sqsubseteq>\<^bsub>L\<^esub> g))"
seq_UINF_distr: fixes A :: "'e set"   and F :: "'e \<Rightarrow> ('b, 'd) urel"   and Q :: "('d, 'c) urel" shows "(\<Sqinter> P \<in> A \<bullet> F P) ;; Q = (\<Sqinter> P \<in> A \<bullet> F P ;; Q)"
fls_left_inverse': fixes x :: "'a"   and f :: "'a fls"   and y :: "'a" assumes "x * fls_nth f (fls_subdegree f) = (1::'a)"   and "fls_nth f (fls_subdegree f) * y = (1::'a)" shows "fls_right_inverse f y * f = 1"
ext_total_axioms_def: fixes ext :: "('b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b list \<Rightarrow> 'b list \<Rightarrow> bool" shows "ext_total_axioms ext \<equiv> \<forall>A gt ys xs. (\<forall>y\<in>A. \<forall>x\<in>A. gt y x \<or> gt x y \<or> y = x) \<longrightarrow> ys \<in> lists A \<longrightarrow> xs \<in> lists A \<longrightarrow> ext gt ys xs \<or> ext gt xs ys \<or> ys = xs"
UniquePre: fixes a :: "'a"   and a1 :: "'a" assumes "a \<in> carrier D"   and "ExPre D a"   and "a1 \<in> carrier D \<and> a1 \<prec> a \<and> \<not> (\<exists>y\<in>carrier D. a1 \<prec> y \<and> y \<prec> a)" shows "Pre D a = a1"
k_reachable: shows "simAbs ` KBPAlg.reachable (set (k_frontier a)) = sim_equiv_class a ` jkbpC"
new_shadow_root_ptr: shows "h \<turnstile> new\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M \<rightarrow>\<^sub>r new_shadow_root_ptr"
smallomegaI_filterlim_at_infinity: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b"   and F :: "'a filter" assumes "LIM x F. f x / g x :> at_infinity" shows "f \<in> \<omega>[F](g)"
key_map_axioms: shows "Comm_Ring.key_map R (+) (\<cdot>) \<zero> \<one> \<pp>"
Match_1: fixes P :: "pi"   and b :: "name"   and x :: "name"   and P' :: "pi"   and a :: "name" assumes "P \<Longrightarrow>b<\<nu>x> \<prec> P'" shows "[a\<frown>a]P \<Longrightarrow>b<\<nu>x> \<prec> P'"
fls_X_power_base_factor_to_fps: fixes n :: "nat" shows "fls_base_factor_to_fps (fls_X ^ n) = 1"
iprev_in_imp_in: fixes n :: "nat"   and I :: "nat set" assumes "iprev n I \<in> I" shows "n \<in> I"
delete_invalid_block_eq: fixes b :: "'a set"   and bl :: "'a set" assumes "b \<notin># add_point_sys.add_block bl" shows "add_block_sys.del_block bl b = add_point_sys.add_block bl"
subgroup_mult_equality: fixes H :: "'a set"   and h1 :: "'a"   and h2 :: "'a" assumes "subgroup H G"   and "h1 \<in> H"   and "h2 \<in> H" shows "h1 \<otimes>\<^bsub>G\<lparr>carrier := H\<rparr>\<^esub> h2 = h1 \<otimes> h2"
norm_poly_mapping_def: shows "norm \<equiv> \<lambda>x. dist x 0"
set_map_of_compr: fixes xs :: "('b \<times> 'c) list" assumes "distinct (list.map fst xs)" shows "set xs = {(k, v). map_of xs k = Some v}"
intro: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and set_shadow_root :: "(_) element_ptr \<Rightarrow> (_) shadow_root_ptr option \<Rightarrow> ((_) heap, exception, unit) prog"   and set_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and set_mode :: "(_) shadow_root_ptr \<Rightarrow> shadow_root_mode \<Rightarrow> ((_) heap, exception, unit) prog"   and set_mode_locs :: "(_) shadow_root_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_tag_name :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, char list) prog"   and get_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_shadow_root :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, (_) shadow_root_ptr option) prog"   and get_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and attach_shadow_root :: "(_) element_ptr \<Rightarrow> shadow_root_mode \<Rightarrow> ((_) heap, exception, (_) shadow_root_ptr) prog" assumes "l_set_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf set_shadow_root set_shadow_root_locs"   and "l_set_mode type_wf set_mode set_mode_locs"   and "l_get_tag_name type_wf get_tag_name get_tag_name_locs"   and "l_get_shadow_root type_wf get_shadow_root get_shadow_root_locs"   and "l_attach_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms known_ptr set_shadow_root set_mode attach_shadow_root get_tag_name get_shadow_root" shows "l_attach_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr set_shadow_root set_shadow_root_locs set_mode set_mode_locs attach_shadow_root type_wf get_tag_name get_tag_name_locs get_shadow_root get_shadow_root_locs"
amin_gt: fixes z :: "ant"   and x :: "ant"   and y :: "ant" assumes "z < x"   and "z < y" shows "z < amin x y"
shows_prec_append: fixes shows_prec :: "nat \<Rightarrow> 'b \<Rightarrow> char list \<Rightarrow> char list"   and shows_list :: "'b list \<Rightarrow> char list \<Rightarrow> char list"   and p :: "nat"   and x :: "'b"   and r :: "char list"   and s :: "char list" assumes "class.show shows_prec shows_list" shows "shows_prec p x (r @ s) = shows_prec p x r @ s"
boundOutputInduct: fixes P :: "pi"   and a :: "name"   and x :: "name"   and P' :: "pi"   and F :: "'a \<Rightarrow> pi \<Rightarrow> name \<Rightarrow> name \<Rightarrow> pi \<Rightarrow> bool"   and C :: "'a" assumes "P \<longmapsto> a<\<nu>x> \<prec> P'"   and "x \<sharp> P"   and "\<And>P a x P' b C. \<lbrakk>P \<longmapsto> a<\<nu>x> \<prec> P'; \<And>C. F C P a x P'\<rbrakk> \<Longrightarrow> F C ([b\<frown>b]P) a x P'"   and "\<And>P a x P' b c C. \<lbrakk>P \<longmapsto> a<\<nu>x> \<prec> P'; \<And>C. F C P a x P'; b \<noteq> c\<rbrakk> \<Longrightarrow> F C ([b\<noteq>c]P) a x P'"   and "\<And>P a x P' C. \<lbrakk>P \<longmapsto> a[x] \<prec> P'; a \<noteq> x\<rbrakk> \<Longrightarrow> F C (<\<nu>x>P) a x P'"   and "\<And>P Q a x P' C. \<lbrakk>P \<longmapsto> a<\<nu>x> \<prec> P'; \<And>C. F C P a x P'\<rbrakk> \<Longrightarrow> F C (P \<oplus> Q) a x P'"   and "\<And>P Q a x Q' C. \<lbrakk>Q \<longmapsto> a<\<nu>x> \<prec> Q'; \<And>C. F C Q a x Q'\<rbrakk> \<Longrightarrow> F C (P \<oplus> Q) a x Q'"   and "\<And>P P' Q a x C. \<lbrakk>P \<longmapsto> a<\<nu>x> \<prec> P'; x \<sharp> Q; \<And>C. F C P a x P'\<rbrakk> \<Longrightarrow> F C (P \<parallel> Q) a x (P' \<parallel> Q)"   and "\<And>P Q Q' a x C. \<lbrakk>Q \<longmapsto> a<\<nu>x> \<prec> Q'; x \<sharp> P; \<And>C. F C Q a x Q'\<rbrakk> \<Longrightarrow> F C (P \<parallel> Q) a x (P \<parallel> Q')"   and "\<And>P P' a x y C. \<lbrakk>P \<longmapsto> a<\<nu>x> \<prec> P'; y \<noteq> a; y \<noteq> x; y \<sharp> C; \<And>C. F C P a x P'\<rbrakk> \<Longrightarrow> F C (<\<nu>y>P) a x (<\<nu>y>P')"   and "\<And>P a x P' C. \<lbrakk>P \<parallel> !P \<longmapsto> a<\<nu>x> \<prec> P'; \<And>C. F C (P \<parallel> !P) a x P'\<rbrakk> \<Longrightarrow> F C (!P) a x P'" shows "F C P a x P'"
initial_invar: shows "dwa_initial WA \<in> dwa_invar WA"
a_one: shows "ad (1::'a) = (zero_class.zero::'a)"
jprop: shows "j \<sqsubset> length cms\<^sub>1 \<and> x \<notin> differing_vars_lists mem\<^sub>1 mem\<^sub>2 mems j"
valid_split1: fixes fg :: "('a, 'b, 'c, 'd, 'e) flowgraph_rec_scheme"   and c :: "'a list multiset"   and c' :: "'a list multiset" assumes "valid fg (c + c')" shows "valid fg c \<and> valid fg c' \<and> mon_c fg c \<inter> mon_c fg c' = {}"
assms_2: shows "q2 \<in> reachable M q1"
m: shows "(deriv ^^ m) f \<xi> / fact m \<noteq> 0"
biimp_def: fixes F :: "'a formula"   and G :: "'a formula" shows "F \<^bold>\<leftrightarrow> G \<equiv> (F \<^bold>\<rightarrow> G) \<^bold>\<and> (G \<^bold>\<rightarrow> F)"
borel_Suc_expectation: fixes f :: "bool stream \<Rightarrow> real"   and n :: "nat"   and g :: "bool stream \<Rightarrow> 'a"   and N :: "'a measure"   and z :: "bool stream" assumes "f \<in> borel_measurable (F (Suc n))"   and "g \<in> F n \<rightarrow>\<^sub>M N"   and "set_discriminating n g N"   and "g -` {g z} \<in> sets (F n)"   and "\<forall>y z. g y = g z \<and> y !! n = z !! n \<longrightarrow> f y = f z" shows "expectation (\<lambda>x. f x * indicat_real (g -` {g z}) x) = prob (g -` {g z}) * (p * f (pseudo_proj_True n z) + (1 - p) * f (pseudo_proj_False n z))"
lunit_simps_5: fixes f :: "'b" assumes "D.ide f" shows "D.cod \<l>\<^sub>D[f] = f"
not_fic_crename_aux: fixes M :: "trm"   and c :: "coname"   and a :: "coname"   and b :: "coname" assumes "fic M c"   and "c \<sharp> (a, b)" shows "fic (M[a\<turnstile>c>b]) c"
AE_E_prob: fixes P :: "'a \<Rightarrow> bool" assumes "almost_everywhere M P" obtains S :: "'a set" where "S \<subseteq> {x \<in> space M. P x}" and "S \<in> events" and "prob S = 1"
prv_T: fixes \<phi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "Fvars \<phi> = {}" shows "prv (eqv (subst T \<langle>\<phi>\<rangle> xx) \<phi>)"
assms: shows "\<exists>A B. TarskiLen A B l1 \<and> TarskiLen A B l2"
\<delta>_states_mono: fixes \<delta> :: "('a, 'b) ta_rule set"   and \<delta>' :: "('a, 'b) ta_rule set" assumes "\<delta> \<subseteq> \<delta>'" shows "\<delta>_states \<delta> \<subseteq> \<delta>_states \<delta>'"
Step2: fixes g1 :: "'a"   and g2 :: "'a" assumes "g1 \<in> carrier G"   and "g2 \<in> carrier G" shows "(\<lambda>H\<in>{H. H \<subseteq> carrier G}. g1 \<otimes> g2 <# H #> inv (g1 \<otimes> g2)) = (\<lambda>H\<in>{H. H \<subseteq> carrier G}. g1 <# H #> inv g1) \<otimes>\<^bsub>BijGroup {H. H \<subseteq> carrier G}\<^esub> (\<lambda>H\<in>{H. H \<subseteq> carrier G}. g2 <# H #> inv g2)"
PositiveImplies_def: fixes v :: "'c zmultiset"   and w :: "'c zmultiset" shows "PositiveImplies v w = (\<forall>t. 0 < zcount v t \<longrightarrow> 0 < zcount w t)"
simps_3: fixes K :: "clock"   and n :: "nat" shows "\<lbrakk> K \<not>\<Up> n \<rbrakk>\<^sub>p\<^sub>r\<^sub>i\<^sub>m = {\<rho>. \<not> hamlet (Rep_run \<rho> n K)}"
addN_conv: fixes ga :: "'g"   and n :: "'node"   and m :: "('var, 'node set) mapping"   and va :: "'var" shows "Mapping.lookup (addN ga n m) va = (case Mapping.lookup m va of None \<Rightarrow> if va \<in> set (sorted_list_of_set (uses ga n)) then Some {n} else None | Some N \<Rightarrow> if va \<in> set (sorted_list_of_set (uses ga n)) then Some (insert n N) else Some N)"
F_UN: shows "\<Union> (F ` P) = F (\<Union> P)"
ds2_def: fixes R :: "('e, 'f) Ring_scheme"   and M :: "('g, 'e, 'h) Module_scheme"   and M1 :: "('g, 'e, 'h) Module_scheme"   and M2 :: "('g, 'e, 'h) Module_scheme" shows "M1 \<Oplus>\<^bsub>R,M\<^esub> M2 = (R module M \<and> msubmodule R M M1 \<and> msubmodule R M M2 \<and> (\<forall>x\<in>carrier M. \<exists>m1\<in>carrier M1. \<exists>m2\<in>carrier M2. x = m1 \<plusminus>\<^bsub>M\<^esub> m2) \<and> carrier M1 \<inter> carrier M2 = {\<zero>\<^bsub>M\<^esub>})"
padic_zero_simp_1: fixes p :: "int"   and n :: "nat" shows "padic_zero p n = \<zero>\<^bsub>residue_ring (p ^ n)\<^esub>"
minus_plus: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and s :: "'c"   and t :: "'c"   and u :: "'c" assumes "class.comm_powerprod plus minus zero"   and "plus.adds plus s t" shows "plus (minus t s) u = minus (plus t u) s"
\<mu>: shows "\<guillemotleft>\<mu> : UP f \<Rightarrow> UP g\<guillemotright>"
eq\<^sub>P: shows "equal_class.equal \<equiv> local.ltln\<^sub>P_eq_test"
edkac_rel_def: shows "edkac_rel \<equiv> {((f, brk, itc), f, brk) |f brk itc. itc + ek_analysis_defs.ekMeasure (cf_of f) s t < 2 * card V * card E + card V}"
atomic_ev_wait_one_preserves_invariants: fixes s :: "state_t"   and tid :: "thread_id_t" assumes "atomic_step_invariant s" shows "atomic_step_invariant (atomic_step_ev_wait_one tid s)"
rename_image_constrains: fixes h :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'c set" assumes "bij h" shows "rename h ` (A co B) = h ` A co h ` B"
taylor_closed: fixes f :: "nat \<Rightarrow> nat \<Rightarrow> int"   and a :: "nat \<Rightarrow> int" assumes "f \<in> carrier (UP Zp)"   and "a \<in> carrier Zp" shows "taylor a f \<in> carrier (UP Zp)"
simps_7: fixes t :: "'a local.term" shows "Can \<^bold>\<r>\<^bold>[t\<^bold>] = Can t"
exp_times_self_ge: fixes x :: "real" shows "- exp (- 1) \<le> x * exp x"
to_bl_ucast: fixes w :: "'c word" shows "to_bl (ucast w) = replicate (LENGTH('b) - LENGTH('c)) False @ drop (LENGTH('c) - LENGTH('b)) (to_bl w)"
approx_tse: fixes xs :: "real list"   and vs :: "float interval option list"   and x :: "nat"   and ivlx :: "float interval"   and c :: "float"   and prec :: "nat"   and s :: "nat"   and f :: "floatarith"   and ivl :: "float interval" assumes "bounded_by xs vs"   and "vs ! x = Some ivlx"   and "real_of_float c \<in>\<^sub>r ivlx"   and "x < length vs"   and "x < length xs"   and "approx_tse prec x s c 1 f vs = Some ivl" shows "interpret_floatarith f xs \<in>\<^sub>r ivl"
frrestriction_fcomp: fixes r :: "V"   and s :: "V"   and A :: "V" shows "(r \<circ>\<^sub>\<bullet> s) \<restriction>\<^sup>r\<^sub>\<bullet> A = r \<restriction>\<^sup>r\<^sub>\<bullet> A \<circ>\<^sub>\<bullet> s"
InvariantsVarsAfterApplyDecide: fixes Vbl :: "nat set"   and state :: "State"   and F0 :: "Literal list list" assumes "\<not> Vbl \<subseteq> vars (elements (getM state))"   and "InvariantConsistent (getM state)"   and "InvariantUniq (getM state)"   and "InvariantWatchListsContainOnlyClausesFromF (getWatchList state) (getF state)"   and "InvariantWatchListsUniq (getWatchList state)"   and "InvariantWatchListsCharacterization (getWatchList state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchesEl (getF state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchesDiffer (getF state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchCharacterization (getF state) (getWatch1 state) (getWatch2 state) (getM state)"   and "InvariantVarsM (getM state) F0 Vbl"   and "InvariantVarsF (getF state) F0 Vbl"   and "getQ state = []" shows "let state' = applyDecide state Vbl in InvariantVarsM (getM state') F0 Vbl \<and> InvariantVarsF (getF state') F0 Vbl \<and> InvariantVarsQ (getQ state') F0 Vbl"
F_is_functor: shows "functor (\<cdot>\<^sub>D) (\<cdot>\<^sub>C) F"
qring_zero: fixes I :: "'a set" assumes "ideal R I" shows "\<zero>\<^bsub>R /\<^sub>r I\<^esub> = I"
c: shows "\<forall>i\<in>{0..<n}. c \<noteq> - of_nat i"
know2knows_s: fixes A :: "specID"   and m :: "Secrets" assumes "know A (sKS m)" shows "knows A [sE m]"
y: shows "\<not> is_zero_iarray (Gauss_Jordan_iarrays (matrix_to_iarray A) !! mod_type_class.to_nat y)"
older_seniors_older: fixes y :: "nat"   and x :: "nat"   and n :: "nat" assumes "y \<in> older_seniors x n" shows "y < x"
C: shows "(\<upsilon> x) \<tau> = true \<tau>"
safe_satisfied: fixes c :: "('b, 't, 'loc) Combined.configuration" assumes "cri.InvGlobNonposImpRecordsNonpos (exchange_config c)"   and "inv_init_imp_prop_safe c"   and "InvGlobPointstampsEq c" shows "safe_combined c"
prod_cases6: fixes y :: "'g \<times> 'h \<times> 'i \<times> 'j \<times> 'k \<times> 'l" obtains a :: "'g" and b :: "'h" and c :: "'i" and d :: "'j" and e :: "'k" and f :: "'l" where "y = (a, b, c, d, e, f)"
induct: fixes P :: "'a \<Rightarrow> (nat \<times> 'a) list \<Rightarrow> (nat \<times> 'a) list \<Rightarrow> bool"   and a0 :: "'a"   and a1 :: "(nat \<times> 'a) list"   and a2 :: "(nat \<times> 'a) list" assumes "\<And>y arr. P y arr []"   and "\<And>y v va. P y [] (v # va)"   and "\<And>y i a arr j b brr. \<lbrakk>i < j \<Longrightarrow> P y arr ((j, b) # brr); \<lbrakk>\<not> i < j; j < i\<rbrakk> \<Longrightarrow> P y ((i, a) # arr) brr; \<lbrakk>\<not> i < j; \<not> j < i\<rbrakk> \<Longrightarrow> P y arr brr\<rbrakk> \<Longrightarrow> P y ((i, a) # arr) ((j, b) # brr)" shows "P a0 a1 a2"
pt_pi_fresh_fresh: fixes pi :: "('b \<times> 'b) list"   and x :: "'a" assumes "pt TYPE('a) TYPE('b)"   and "at TYPE('b)"   and "\<forall>(a, b)\<in>set pi. a \<sharp> x \<and> b \<sharp> x" shows "pi \<bullet> x = x"
hyps_2: shows "p[d := X] \<in> grid (b[d := X]) ds"
ParCL_MtransC: fixes c1 :: "('test, 'atom) com"   and s :: "'state"   and c1' :: "('test, 'atom) com"   and s' :: "'state"   and c2 :: "('test, 'atom) com" assumes "(c1, s) \<rightarrow>*c (c1', s')" shows "(c1 | c2, s) \<rightarrow>*c (c1' | c2, s')"
assms_2: fixes i :: "'b"   and j :: "'c"   and k :: "'d" assumes "(i, j, k) \<in> A" shows "P i j k is H"
abs_eq: fixes xa :: "nat"   and x :: "'a llist \<times> 'b" shows "tdropn xa (case x of (xs, a) \<Rightarrow> tllist_of_llist a xs) = (case apfst (ldropn xa) x of (xs, a) \<Rightarrow> tllist_of_llist a xs)"
coprime_dvd_mult_nat2: fixes k :: "nat"   and n :: "nat"   and m :: "nat" assumes "coprime k n"   and "k dvd n * m" shows "k dvd m"
Red_preserves_defass: fixes P :: "(char list list \<times> char list exp) prog"   and e :: "char list exp"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   and e' :: "char list exp"   and s' :: "heap \<times> sfields \<times> sheap"   and b' :: "bool" assumes "wf_J_prog P"   and "P \<turnstile> \<langle>e,s,b\<rangle> \<rightarrow>* \<langle>e',s',b'\<rangle>"   and "\<D> e \<lfloor>dom (lcl s)\<rfloor>" shows "\<D> e' \<lfloor>dom (lcl s')\<rfloor>"
max_reg_V_I: fixes n :: "nat" shows "max_reg (V (I n)) = None"
fds_left_inverse_unique: fixes f :: "'b fds"   and g :: "'b fds" assumes "f * g = 1" shows "f = inverse g"
ideD_2: fixes a :: "'b" assumes "D.ide a" shows "D.dom a = a"
smult_l_distr: fixes a :: "'a"   and b :: "'a"   and x :: "nat \<Rightarrow> 'a" assumes "a \<in> carrier R"   and "b \<in> carrier R"   and "x \<in> carrier P" shows "(a \<oplus> b) \<odot>\<^bsub>P\<^esub> x = a \<odot>\<^bsub>P\<^esub> x \<oplus>\<^bsub>P\<^esub> b \<odot>\<^bsub>P\<^esub> x"
oLimit_induct: fixes P :: "ordinal \<Rightarrow> bool"   and a :: "ordinal" assumes "P 0"   and "\<And>x. P x \<Longrightarrow> P (oSuc x)"   and "\<And>f. \<lbrakk>OrdinalInduct.strict_mono f; \<forall>n. P (f n)\<rbrakk> \<Longrightarrow> P (oLimit f)" shows "P a"
image_iff: fixes z :: "'e"   and f :: "'f \<Rightarrow> 'e"   and A :: "'f set" shows "(z \<in> f ` A) = (\<exists>x\<in>A. z = f x)"
assms_5: shows "C \<in> \<Union> (\<Union>(k, v)\<in>{0..<length \<pi>} \<times> set (\<Pi>\<^sub>\<V>). {{{(State k (index (\<Pi>\<^sub>\<V>) v))\<inverse>, (State (Suc k) (index (\<Pi>\<^sub>\<V>) v))\<^sup>+} \<union> {(Operator k (index (\<Pi>\<^sub>\<O>) op))\<^sup>+ |op. op \<in> set (\<Pi>\<^sub>\<O>) \<and> v \<in> set (delete_effects_of op)}}})"
7: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<squnion> (y \<squnion> z) = x \<squnion> y \<squnion> z"
accs_exclusive_aux: fixes \<delta>n :: "('a, 'b) ta_rule set" assumes "accs \<delta>n n q"   and "\<delta>n = \<delta>' \<union> \<delta>"   and "\<delta>_states \<delta>' \<inter> \<delta>_states \<delta> = {}"   and "q \<in> \<delta>_states \<delta>'" shows "accs \<delta>' n q"
insert_not_empty: fixes a :: "'a"   and A :: "'a set" shows "insert a A \<noteq> {}"
feq: shows "Funcs (USubst.adjoint (dotsubstt \<theta>) I \<omega>) = Funcs (repc I dotid (term_sem I \<theta> \<omega>))"
derivation_shift_cons: fixes d :: "nat \<times> symbol \<times> symbol list"   and D :: "(nat \<times> symbol \<times> symbol list) list"   and left :: "nat"   and right :: "nat" shows "derivation_shift (d # D) left right = (fst d - left + right, snd d) # derivation_shift D left right"
in_diff1: fixes x :: "'d"   and A :: "'d set"   and a :: "'d" assumes "x \<in> A - {a}" shows "x \<noteq> a"