hcju/isabelleps huggingface dataset:
simps_2: fixes x :: "('b, 'c) Combinators"   and xs :: "('b, 'c) Combinators list" shows "wellformed_policy1 (x # xs) = (DenyAll \<notin> set xs)"
one_less_of_natD: fixes n :: "nat" assumes "(1::'b) < of_nat n" shows "1 < n"
diagonalize_is_idempotent: shows "\<F>C.D \<circ> \<F>C.D = \<F>C.D"
B1: shows "\<^bold>\<forall>x xa. (dom x \<cong> cod xa \<^bold>\<leftarrow> E (x \<cdot> xa)) \<^bold>\<and> (E (x \<cdot> xa) \<^bold>\<leftarrow> dom x \<cong> cod xa)"
reds_avoids_live: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and e :: "exp"   and L :: "Vars.var list"   and \<Delta> :: "(Vars.var \<times> exp) list"   and z :: "exp"   and x :: "Vars.var" assumes "\<Gamma> : e \<Down>\<^bsub>L\<^esub> \<Delta> : z"   and "x \<in> set L"   and "x \<notin> domA \<Gamma>" shows "x \<notin> domA \<Delta>"
real_alg_2_2: fixes rai :: "int poly \<times> rat \<times> rat" assumes "invariant_1 rai" shows "real_of_2 (real_alg_2 rai) = real_of_1 rai"
padd_closed: fixes a :: "int"   and b :: "int"   and p\<^sub>1 :: "int \<times> int \<times> int"   and p\<^sub>2 :: "int \<times> int \<times> int" assumes "a \<in> carrier R"   and "b \<in> carrier R"   and "on_curvep a b p\<^sub>1"   and "on_curvep a b p\<^sub>2" shows "on_curvep a b (padd a p\<^sub>1 p\<^sub>2)"
extensionalD_def: fixes d :: "'d"   and A :: "'c set" shows "extensionalD d A = {f. \<forall>x. x \<notin> A \<longrightarrow> f x = d}"
valid_plan_valid_head: fixes h :: "'a"   and as :: "'a list"   and PROB :: "'a set" assumes "h # as \<in> valid_plans PROB" shows "h \<in> PROB"
vsv_vcomp: fixes r :: "V"   and s :: "V" assumes "vsv r"   and "vsv s" shows "vsv (r \<circ>\<^sub>\<circ> s)"
iwf_it: fixes c :: "'form" assumes "c \<in> set conclusions" shows "plain_iwf (it' c) (fst (tree.root (ts c)))"
uint_arith_simps_1: fixes a :: "'b word"   and b :: "'b word" shows "(a \<le> b) = (uint a \<le> uint b)"
cases: fixes a1 :: "'c"   and a2 :: "'c"   and succs :: "'c \<Rightarrow> 'c list" assumes "a1 [succs]\<rightarrow>* a2" obtains g :: "'c" where "a1 = g__" and "a2 = g__"   | g :: "'c" and g' :: "'c" and g'' :: "'c" where "a1 = g" and "a2 = g''" and "g' \<in> set (succs g)" and "g' [succs]\<rightarrow>* g''"
multiple_unfold_1: fixes n :: "nat" shows "multiple_blocks (Suc n) = multiple_blocks n + \<B>"
edge_ugraph_def: fixes es :: "nat set set" shows "edge_ugraph es \<equiv> (S_verts, es \<inter> S_edges)"
Spy_not_see_encrypted_key: fixes A :: "agent"   and nb :: "msg"   and B :: "agent"   and K :: "nat"   and na :: "msg"   and evs :: "event list" assumes "Says Server A \<lbrace>nb, Crypt (shrK A) \<lbrace>Agent B, Key K, na\<rbrace>, Crypt (shrK B) \<lbrace>Agent A, Agent B, Key K, nb\<rbrace>\<rbrace> \<in> set evs"   and "Notes Spy \<lbrace>na, nb, Key K\<rbrace> \<notin> set evs"   and "A \<notin> bad"   and "B \<notin> bad"   and "evs \<in> yahalom" shows "Key K \<notin> analz (knows Spy evs)"
take2_def: fixes xs :: "int list" shows "take2 xs \<equiv> let l = Challenge1A.take xs in (l, drop (length l) xs)"
erlang_CDF_nonneg: fixes l :: "real"   and k :: "nat"   and x :: "real" assumes "0 < l" shows "0 \<le> erlang_CDF k l x"
dg_Arr_vsubset_Vset: shows "\<AA>\<lparr>Arr\<rparr> \<subseteq>\<^sub>\<circ> Vset \<alpha>"
hb: shows "\<forall>\<Gamma>'' s' t'.  \<Gamma> \<subseteq> \<Gamma>''  \<longrightarrow> valid \<Gamma>'' \<longrightarrow> \<Gamma>'' \<turnstile> s' is t' : T\<^sub>1 \<longrightarrow> \<Gamma>'' \<turnstile> App s s' is App t t' : T\<^sub>2"
idom_hom_def: fixes hom :: "'a \<Rightarrow> 'b" shows "idom_hom hom \<equiv> comm_ring_hom hom"
assms_4: shows "integrable M (\<lambda>x. \<bar>f x\<bar> powr p)"
alpha_trans: fixes X :: "('f, 'g, 'e, 'd, 'h) qTerm"   and X' :: "('f, 'g, 'e, 'd, 'h) qTerm"   and X'' :: "('f, 'g, 'e, 'd, 'h) qTerm" assumes "FixVars TYPE('d) TYPE('e)"   and "FixVars.qGood X \<or> FixVars.qGood X' \<or> FixVars.qGood X''"   and "X #= X'"   and "X' #= X''" shows "X #= X''"
interp_ins_def: fixes ops :: "('c \<times> 'c option) list" shows "interp_ins ops \<equiv> foldl insert_spec [] ops"
approximating_bigstep_induct: fixes \<gamma> :: "('c \<Rightarrow> 'd \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'd \<Rightarrow> bool)"   and p :: "'d"   and rs :: "'c rule list"   and s :: "state"   and t :: "state"   and P :: "'c rule list \<Rightarrow> state \<Rightarrow> state \<Rightarrow> bool" assumes "\<gamma>,p\<turnstile> \<langle>rs, s\<rangle> \<Rightarrow>\<^sub>\<alpha> t"   and "\<And>t. P [] t t"   and "\<And>m a. \<lbrakk>Matching_Ternary.matches \<gamma> m a p; a = action.Accept\<rbrakk> \<Longrightarrow> P [Rule m a] Undecided (Decision FinalAllow)"   and "\<And>m a. \<lbrakk>Matching_Ternary.matches \<gamma> m a p; a = action.Drop \<or> a = Reject\<rbrakk> \<Longrightarrow> P [Rule m a] Undecided (Decision FinalDeny)"   and "\<And>m a. \<lbrakk>Matching_Ternary.matches \<gamma> m a p; a = Log \<or> a = Empty\<rbrakk> \<Longrightarrow> P [Rule m a] Undecided Undecided"   and "\<And>m a. \<not> Matching_Ternary.matches \<gamma> m a p \<Longrightarrow> P [Rule m a] Undecided Undecided"   and "\<And>rs X. P rs (Decision X) (Decision X)"   and "\<And>rs rs\<^sub>1 rs\<^sub>2 t t'. \<lbrakk>rs = rs\<^sub>1 @ rs\<^sub>2; \<gamma>,p\<turnstile> \<langle>rs\<^sub>1, Undecided\<rangle> \<Rightarrow>\<^sub>\<alpha> t; P rs\<^sub>1 Undecided t; \<gamma>,p\<turnstile> \<langle>rs\<^sub>2, t\<rangle> \<Rightarrow>\<^sub>\<alpha> t'; P rs\<^sub>2 t t'\<rbrakk> \<Longrightarrow> P rs Undecided t'" shows "P rs s t"
ext_inject: fixes cpu_reg :: "CPU_register \<Rightarrow> word32"   and user_reg :: "'b word \<Rightarrow> word5 \<Rightarrow> word32"   and sys_reg :: "sys_reg \<Rightarrow> word32"   and mem :: "word8 \<Rightarrow> word36 \<Rightarrow> word8 option"   and mmu :: "MMU_state"   and cache :: "cpu_cache"   and dwrite :: "(int \<times> word32 \<times> CPU_register) list"   and state_var :: "sparc_state_var"   and traps :: "Trap set"   and undef :: "bool"   and more :: "'c"   and cpu_reg' :: "CPU_register \<Rightarrow> word32"   and user_reg' :: "'b word \<Rightarrow> word5 \<Rightarrow> word32"   and sys_reg' :: "sys_reg \<Rightarrow> word32"   and mem' :: "word8 \<Rightarrow> word36 \<Rightarrow> word8 option"   and mmu' :: "MMU_state"   and cache' :: "cpu_cache"   and dwrite' :: "(int \<times> word32 \<times> CPU_register) list"   and state_var' :: "sparc_state_var"   and traps' :: "Trap set"   and undef' :: "bool"   and more' :: "'c" shows "(\<lparr>cpu_reg = cpu_reg, user_reg = user_reg, sys_reg = sys_reg, mem = mem, mmu = mmu, cache = cache, dwrite = dwrite, state_var = state_var, traps = traps, undef = undef, \<dots> = more\<rparr> = \<lparr>cpu_reg = cpu_reg', user_reg = user_reg', sys_reg = sys_reg', mem = mem', mmu = mmu', cache = cache', dwrite = dwrite', state_var = state_var', traps = traps', undef = undef', \<dots> = more'\<rparr>) = (cpu_reg = cpu_reg' \<and> user_reg = user_reg' \<and> sys_reg = sys_reg' \<and> mem = mem' \<and> mmu = mmu' \<and> cache = cache' \<and> dwrite = dwrite' \<and> state_var = state_var' \<and> traps = traps' \<and> undef = undef' \<and> more = more')"
Int_stableI: fixes A :: "'c set set" assumes "\<And>a b. \<lbrakk>a \<in> A; b \<in> A\<rbrakk> \<Longrightarrow> a \<inter> b \<in> A" shows "Int_stable A"
unat_of_nat_eq: fixes x :: "nat" assumes "x < 2 ^ LENGTH('b)" shows "unat (word_of_nat x) = x"
list_orderI: fixes ops :: "('a \<times> 'a option) list"   and xs :: "'a list"   and x :: "'a"   and ys :: "'a list"   and y :: "'a"   and zs :: "'a list" assumes "interp_ins ops = xs @ [x] @ ys @ [y] @ zs" shows "list_order ops x y"
f'_def: fixes b :: "'b set" shows "f' b = (b \<subseteq> U\<^sub>V\<^sub>S \<and> f b)"
rScope: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and M :: "'a"   and xvec :: "name list"   and N :: "'a"   and P' :: "('a, 'b, 'c) psi"   and x :: "name"   and C :: "'d" assumes "\<Psi> \<rhd> P \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> P'"   and "\<And>C. Prop C \<Psi> P M xvec N P'"   and "x \<sharp> \<Psi>"   and "x \<sharp> M"   and "x \<sharp> xvec"   and "x \<sharp> N"   and "xvec \<sharp>* \<Psi>"   and "xvec \<sharp>* P"   and "xvec \<sharp>* M"   and "x \<sharp> C"   and "xvec \<sharp>* C" shows "Prop C \<Psi> (\<lparr>\<nu>x\<rparr>P) M xvec N (\<lparr>\<nu>x\<rparr>P')"
E_linear_plus2: fixes A :: "'c pmf"   and f :: "'c \<Rightarrow> real"   and g :: "'c \<Rightarrow> real" assumes "finite (set_pmf A)" shows "E (map_pmf f A) + E (map_pmf g A) = E (map_pmf (\<lambda>x. f x + g x) A)"
assms_4: fixes U :: "'a set" assumes "closedin (top_of_set S) U" shows "closedin (top_of_set T) (f ` U)"
ex: fixes a :: "'action"   and b :: "'action"   and p :: "'state" assumes "ind a b"   and "en a p"   and "en b p" shows "ex b (ex a p) = ex a (ex b p)"
top_sort: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and s :: "'a set" assumes "acyclicP r" shows "finite s \<longrightarrow> (\<exists>l. set l = s \<and> sorted_wrt (to_ord r) l \<and> distinct l)"
diaG: fixes X :: "'a set" assumes "X \<in> \<G>" shows "diameter X < d"
is_applicable_then_enabled: fixes s :: "nat \<Rightarrow> nat option"   and ast_op :: "char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat" assumes "s \<in> valid_states"   and "ast_op \<in> set ast\<delta>"   and "is_operator_applicable_in s (abs_ast_operator ast_op)" shows "enabled ((fst \<circ> the \<circ> lookup_action \<circ> abs_ast_operator) ast_op) s"
assms: fixes m :: "nat"   and n :: "nat" assumes "m \<le> n" shows "f n \<le> f m"
card_of_empty5: fixes A :: "'a set"   and B :: "'b set" assumes "|A| <o |B|" shows "B \<noteq> {}"
lm08: fixes f :: "'c \<Rightarrow> 'd"   and l :: "'c list" shows "argmaxList f l = concat (map (\<lambda>n. if Max (f ` set l) \<le> f (l ! n) then [l ! n] else []) [0..<length l])"
two_ideal_points: fixes H :: "circline" assumes "is_poincare_line H" shows "card (ideal_points H) = 2"
inv_solve_right: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<in> carrier G"   and "b \<in> carrier G"   and "c \<in> carrier G" shows "(a = b \<oplus> \<ominus> c) = (b = a \<oplus> c)"
refl_NS_point: fixes S :: "('a \<times> 'a) set"   and NS :: "('a \<times> 'a) set"   and s :: "'a" assumes "pre_order_pair S NS" shows "(s, s) \<in> NS"
append_simpl3: fixes id :: "'b"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and y :: "'c list"   and i :: "nat"   and z :: "'d"   and zs :: "'d list" assumes "Matrix_Tensor.mult id f"   and "length y < i" shows "i < length (z # zs) * length y \<longrightarrow> i - length y < length zs * length y"
bounded_semilattice_inf_top_ow_def: fixes U :: "'a set"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and le :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and ls :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and top :: "'a" shows "bounded_semilattice_inf_top_ow U inf le ls top \<equiv> semilattice_inf_ow U inf le ls \<and> order_top_ow U le ls top"
even_iff_mod_2_eq_zero: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c" assumes "class.semiring_parity plus minus zero times one divide modulo" shows "dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) a = (modulo a (numeral.numeral one plus (num.Bit0 num.One)) = zero)"
unit_char_1: fixes a :: "'c" assumes "C.obj a" shows "\<guillemotleft>H.unit a : H.map\<^sub>0 a \<Rightarrow>\<^sub>D H a\<guillemotright>"
fib_body_strict: fixes r :: "nat discr Box \<rightarrow> nat discr Box" shows "fib_body\<cdot>r\<cdot>\<bottom> = \<bottom>"
assms_2: fixes x :: "'var"   and \<rho> :: "'var \<Rightarrow> 'value" assumes "x \<in> domA \<Gamma>" shows "\<lbrakk> the (map_of \<Gamma> x) \<rbrakk>\<^bsub>\<rho>\<^esub> = \<lbrakk> the (map_of \<Gamma> x) \<rbrakk>\<^bsub>\<rho> f|` fv (the (map_of \<Gamma> x))\<^esub>"
partition_factors_def: fixes xs :: "('a \<times> nat) list" shows "partition_factors xs = (let n = sum_list (map snd xs) div 2 in case partition_factors_main n xs of ([], []) \<Rightarrow> ([], []) | ([], [x]) \<Rightarrow> ([], [x]) | ([], x # y # ys) \<Rightarrow> ([x], y # ys) | ([x], b) \<Rightarrow> ([x], b) | (x # y # ys, []) \<Rightarrow> ([x], y # ys) | (x # y # ys, ad # listb) \<Rightarrow> (x # y # ys, ad # listb))"
Cons_in_nlists_Suc: fixes x :: "'c"   and xs :: "'c list"   and A :: "'c set"   and n :: "nat" shows "(x # xs \<in> nlists A (Suc n)) = (x \<in> A \<and> xs \<in> nlists A n)"
simps: fixes v :: "'a" shows "pwalk v = (if v = s \<or> dist v = \<infinity> \<or> v \<notin> verts G then [] else pwalk (tail G (the (parent_edge v))) @ [the (parent_edge v)])"
assms_2: shows "qf = (\<lambda>x y. A1 * x\<^sup>2 + A1 * y\<^sup>2 + b1 * x + b2 * y + c)"
subgame_V': fixes G :: "('c, 'd) ParityGame_scheme"   and V' :: "'c set" assumes "ParityGame G"   and "V' \<subseteq> V\<^bsub>G\<^esub>" shows "V\<^bsub>ParityGame.subgame G V'\<^esub> = V'"
verit_la_disequality: fixes a :: "'a"   and b :: "'a" shows "a = b \<or> \<not> a \<le> b \<or> \<not> b \<le> a"
SUP_image: fixes Sup :: "'a set \<Rightarrow> 'a"   and g :: "'b \<Rightarrow> 'a"   and f :: "'c \<Rightarrow> 'b"   and A :: "'c set" shows "Sup (g ` f ` A) = Sup ((g \<circ> f) ` A)"
simps_10: fixes \<xi> :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltlc" shows "\<xi> \<Turnstile>\<^sub>c G\<^sub>c \<phi> = (\<forall>i. suffix i \<xi> \<Turnstile>\<^sub>c \<phi>)"
Bounded_vec_rows_Bounded_mat: fixes A :: "'b mat"   and B :: "'b" shows "(set (rows A) \<subseteq> Bounded_vec B) = (A \<in> Bounded_mat B)"
assms: shows "\<R>\<^sub>\<circ> xs \<subseteq>\<^sub>\<circ> A"
sin_zero_iff: fixes x :: "real" shows "(sin x = 0) = ((\<exists>n. even n \<and> x = real n * (pi / 2)) \<or> (\<exists>n. even n \<and> x = - (real n * (pi / 2))))"
vector_sneg_minus1: fixes x :: "('b, 'c) vec" shows "- x = - (1::'b) *s x"
lemma_2_5_13_b: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<le> b" shows "b l\<rightarrow> c \<le> a l\<rightarrow> c"
basis_completion_def: fixes v :: "'b vec" shows "basis_completion v \<equiv> let n = dim_vec v; drop_index = hd (concat (map (\<lambda>i. if v $ i \<noteq> (0::'b) then [i] else []) [0..<n])) in Let (concat (map (\<lambda>i. if i \<noteq> drop_index then [unit_vec n i] else []) [0..<n])) ((#) v)"
fmfilter_comp: fixes P :: "'d \<Rightarrow> bool"   and Q :: "'d \<Rightarrow> bool"   and m :: "('d, 'e) fmap" shows "fmfilter P (fmfilter Q m) = fmfilter (\<lambda>x. P x \<and> Q x) m"
distinct_47: fixes x21 :: "char list"   and x22 :: "expr"   and x121 :: "expr"   and x122 :: "expr" shows "Cast x21 x22 \<noteq> x121;; x122"
expandRight: fixes R :: "pi"   and P :: "pi"   and Q :: "pi"   and Rel :: "(pi \<times> pi) set" assumes "(R, expandSet P Q) \<in> sumComposeSet"   and "hnf P"   and "hnf Q"   and "Id \<subseteq> Rel" shows "R \<leadsto>[Rel] P \<parallel> Q"
ws_2: shows "set ws \<subseteq> RSpan vs \<and> RSpan vs = RSpan ws \<and> R_lin_independent ws"
induct: fixes P :: "Cooper.num \<Rightarrow> bool"   and a0 :: "Cooper.num" assumes "\<And>n. P (Bound n)"   and "\<And>a. P a \<Longrightarrow> P (Neg a)"   and "\<And>a b. \<lbrakk>P a; P b\<rbrakk> \<Longrightarrow> P (Add a b)"   and "\<And>a b. \<lbrakk>P a; P b\<rbrakk> \<Longrightarrow> P (Sub a b)"   and "\<And>c a. P a \<Longrightarrow> P (Mul c a)"   and "\<And>n i a. P a \<Longrightarrow> P (CN n i a)"   and "\<And>v. P (C v)" shows "P a0"
polytope_interval: fixes a :: "'a"   and b :: "'a" shows "polytope (cbox a b)"
unique_expression3_2: fixes H :: "'a set"   and f :: "nat \<Rightarrow> 'a"   and n :: "nat"   and s :: "nat \<Rightarrow> 'b"   and l :: "nat" assumes "H \<subseteq> carrier M"   and "f \<in> {k. k \<le> Suc n} \<rightarrow> H"   and "s \<in> {k. k \<le> Suc n} \<rightarrow> carrier R"   and "l \<le> Suc n"   and "f l \<notin> f ` ({k. k \<le> Suc n} - {l})"   and "l \<noteq> Suc n" shows "\<exists>g m t. g \<in> {l. l \<le> m} \<rightarrow> H \<and> inj_on g {l. l \<le> m} \<and> t \<in> {l. l \<le> m} \<rightarrow> carrier R \<and> l_comb R M (Suc n) s f = l_comb R M m t g \<and> t m = s l \<and> g m = f l"
of_rat_val_constraints: fixes cs :: "constraint set"   and v :: "nat \<Rightarrow> rat" shows "of_rat_val v \<Turnstile>\<^sub>r\<^sub>c\<^sub>s cs = v \<Turnstile>\<^sub>c\<^sub>s cs"
implies_itself_same: fixes x :: "'c"   and y :: "'c" shows "- x \<squnion> x = - y \<squnion> y"
err_le_unfold: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" shows "Err.le r (OK a) (OK b) = r a b"
represents_zero: fixes B :: "'b set" shows "represents B 0 (0::'b)"
assms_1: shows "convert_atom var a (xs' @ x # xs) = Some a'"
wf_interp_for_formula_FExists: fixes I :: "(nat + nat set) list"   and \<phi> :: "'a formula"   and w :: "'a list" assumes "wf_formula (length I) (FExists \<phi>)" shows "wf_interp_for_formula (w, I) (FExists \<phi>) = (\<forall>p. wf_interp_for_formula (w, Inl p # I) \<phi>)"
neg_divide_le_eq: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and c :: "'a"   and b :: "'a"   and a :: "'a" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less c zero" shows "less_eq (divide b c) a = less_eq (times a c) b"
must_syncE: fixes P :: "'addr J_prog"   and t :: "'thread_id"   and x :: "'addr expr \<times> (String.literal \<Rightarrow> 'addr val option)"   and m :: "'heap" assumes "red_mthr.must_sync P t x m" obtains taa :: "('addr, 'thread_id, 'heap) J_thread_action" and x' :: "'addr expr \<times> (String.literal \<Rightarrow> 'addr val option)" and m' :: "'heap" and s :: "('addr, 'thread_id) locks \<times> (('thread_id \<Rightarrow> (('addr expr \<times> (String.literal \<Rightarrow> 'addr val option)) \<times> 'addr \<Rightarrow>f nat) option) \<times> 'heap) \<times> ('thread_id \<Rightarrow> 'addr wait_set_status option) \<times> 'thread_id set" where "red_mthr.r_syntax P t x m taa x' m'" and "red_mthr.actions_ok s t taa" and "m = shr s"
assms_2: assumes "\<psi> \<pi>l" shows "\<phi> \<pi>l"
init_nrec_pos: fixes s :: "('c, 'd) configuration stream" assumes "holds init_config s" shows "holds (\<lambda>c. \<forall>t. 0 \<le> zcount (c_records c) t) s"
strict_implies_not_eq: fixes a :: "'a word"   and b :: "'a word" assumes "a <s b" shows "a \<noteq> b"
rel_inject_3: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and x3 :: "'b formula"   and y3 :: "'c formula" shows "rel_formula R (\<^bold>\<not> x3) (\<^bold>\<not> y3) = rel_formula R x3 y3"
"3_5": shows "set (x # y # xs) \<subseteq> {0..n}"
Gauss_Jordan_PA_def: fixes A :: "(('d, 'f) vec, 'e) vec" shows "Gauss_Jordan_PA A = Gauss_Jordan_upt_k_PA A (ncols A - 1)"
add_nonpos_neg: fixes a :: "'b multiset"   and b :: "'b multiset" assumes "a \<subseteq># {#}"   and "b \<subset># {#}" shows "a + b \<subset># {#}"
openin_subtopology_self: fixes S :: "'b set" shows "openin (top_of_set S) S"
BlockRedNone: fixes e :: "char list exp"   and h :: "heap"   and l :: "sfields"   and V :: "char list"   and sh :: "sheap"   and b :: "bool"   and e' :: "char list exp"   and h' :: "heap"   and l' :: "sfields"   and sh' :: "sheap"   and b' :: "bool"   and P :: "(char list list \<times> char list exp) prog"   and T :: "ty" assumes "P \<turnstile> \<langle>e,(h, l(V := None), sh),b\<rangle> \<rightarrow> \<langle>e',(h', l', sh'),b'\<rangle>"   and "l' V = None"   and "\<not> assigned V e" shows "P \<turnstile> \<langle>{V:T; e},(h, l, sh),b\<rangle> \<rightarrow> \<langle>{V:T; e'},(h', l'(V := l V), sh'),b'\<rangle>"
properties_existence: fixes f :: "'b \<Rightarrow> 'b"   and x0 :: "'b" obtains lambda :: "nat" and mu :: "nat" where "fx0.properties f x0 lambda mu"
subtensor_combine_subtensor: fixes A :: "'b tensor" assumes "dims A \<noteq> []" shows "subtensor_combine (tl (dims A)) (map (subtensor A) [0..<hd (dims A)]) = A"
Pow_set_conv_fold: fixes xs :: "'b list"   and A :: "'b set" shows "Pow (set xs \<union> A) = fold (\<lambda>x A. A \<union> insert x ` A) xs (Pow A)"
set_disconnected_nodes_get_disconnected_nodes_different_pointers: fixes ptr :: "(_) document_ptr"   and ptr' :: "(_) document_ptr"   and w :: "((_) heap, exception, unit) prog"   and h :: "(_) heap"   and h' :: "(_) heap"   and r :: "(_) heap \<Rightarrow> (_) heap \<Rightarrow> bool" assumes "ptr \<noteq> ptr'"   and "w \<in> set_disconnected_nodes_locs ptr"   and "h \<turnstile> w \<rightarrow>\<^sub>h h'"   and "r \<in> get_disconnected_nodes_locs ptr'" shows "r h h'"
square_free_factorization_int'_def: fixes f :: "int poly" shows "square_free_factorization_int' f = (if degree f = 0 then (lead_coeff f, []) else let c = content f; d = sgn (lead_coeff f) * c; g = sdiv_poly f d in (d, square_free_factorization_int_main g))"
zero_in_hom_deg_set: fixes A :: "(('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c) set"   and z :: "nat" assumes "0 \<in> A" shows "0 \<in> hom_deg_set z A"
asprod_div_eq: fixes n :: "int"   and x :: "ant"   and y :: "ant" assumes "n \<noteq> 0"   and "n *\<^sub>a x = n *\<^sub>a y" shows "x = y"
less_asym: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c"   and P :: "bool" assumes "class.preorder less_eq less"   and "less x y"   and "\<not> P \<Longrightarrow> less y x" shows "P"
minimal_good_strategy_def: fixes v :: "'a"   and \<sigma> :: "'a \<Rightarrow> 'a" shows "minimal_good_strategy v \<sigma> \<equiv> \<sigma> \<in> good v \<and> (\<forall>\<sigma>''. (\<sigma>'', \<sigma>) \<in> r - Id \<longrightarrow> \<sigma>'' \<notin> good v)"
set_integrable_mult_right_iff: fixes a :: "'b"   and M :: "'c measure"   and A :: "'c set"   and f :: "'c \<Rightarrow> 'b" assumes "a \<noteq> (0::'b)" shows "set_integrable M A (\<lambda>t. a * f t) = set_integrable M A f"
union_set_fold: fixes xs :: "'c list"   and A :: "'c set" shows "set xs \<union> A = fold insert xs A"
filter_def: shows "Mapping.filter \<equiv> map_fun id (map_fun rep Mapping) (\<lambda>P m k. case m k of None \<Rightarrow> None | Some v \<Rightarrow> if P k v then Some v else None)"
target_select: fixes a :: "'d"   and b :: "('d, 'e) expr \<times> ('f data \<Rightarrow> bool) \<times> 'e set \<times> 'f pupdate"   and c :: "'d" shows "target (a, b, c) = c"
path_image_rectpath_cbox_minus_box: fixes a :: "complex"   and b :: "complex" assumes "Re a \<le> Re b"   and "Im a \<le> Im b" shows "path_image (rectpath a b) = cbox a b - box a b"
incrementI2: fixes f :: "real \<Rightarrow> real"   and x :: "real"   and D :: "real"   and h :: "real star" assumes "NSDERIV f x :> D" shows "increment f x h = (*f* f) (hypreal_of_real x + h) - hypreal_of_real (f x)"
atLeast0AtMost: fixes n :: "nat" shows "{0..n} = {..n}"
map_comp: fixes g1 :: "'i \<Rightarrow> 'e"   and g2 :: "'j \<Rightarrow> 'f"   and g3 :: "'k \<Rightarrow> 'g"   and g4 :: "'l \<Rightarrow> 'h"   and f1 :: "'m \<Rightarrow> 'i"   and f2 :: "'n \<Rightarrow> 'j"   and f3 :: "'o \<Rightarrow> 'k"   and f4 :: "'p \<Rightarrow> 'l"   and v :: "('m, 'n, 'o, 'p) F\<^sub>m" shows "map_F\<^sub>m g1 g2 g3 g4 (map_F\<^sub>m f1 f2 f3 f4 v) = map_F\<^sub>m (g1 \<circ> f1) (g2 \<circ> f2) (g3 \<circ> f3) (g4 \<circ> f4) v"
base: fixes u :: "nat \<Rightarrow> 'b \<Rightarrow> real"   and f :: "'b \<Rightarrow> real"   and x :: "nat \<Rightarrow> real"   and M :: "'b set set \<times> ('b set \<Rightarrow> real)"   and y :: "real" assumes "u\<up>f"   and "\<And>n. x n \<in> sfis (u n) M"   and "x\<up>y" shows "y \<in> nnfis f M"
bound_compare_defs_10: fixes c :: "'a"   and b :: "'a option" shows "c >\<^sub>u\<^sub>b b = \<rhd>\<^sub>u\<^sub>b (<) c b"