hcju/isabelleps huggingface dataset:
gauss_sum_int_cong: fixes a :: "int"   and b :: "int" assumes "[a = b] (mod int n)" shows "gauss_sum_int a = gauss_sum_int b"
lookup_HSem_heap: fixes y :: "'d"   and h :: "('d \<times> 'e) list"   and ESem :: "'e \<Rightarrow> ('d \<Rightarrow> 'f) \<rightarrow> 'f"   and \<rho> :: "'d \<Rightarrow> 'f" assumes "y \<in> domA h" shows "(has_ESem.HSem ESem h\<cdot>\<rho>) y = ESem (the (map_of h y))\<cdot>(has_ESem.HSem ESem h\<cdot>\<rho>)"
nn_integral_measure_pmf_support: fixes A :: "'a set"   and f :: "'a \<Rightarrow> ennreal"   and M :: "'a pmf" assumes "finite A"   and "\<And>x. x \<in> A \<Longrightarrow> 0 \<le> f x"   and "\<And>x. \<lbrakk>x \<in> set_pmf M; x \<notin> A\<rbrakk> \<Longrightarrow> f x = 0" shows "integral\<^sup>N (measure_pmf M) f = (\<Sum>x\<in>A. f x * ennreal (pmf M x))"
dg_Cod_app_in_Obj: fixes f :: "V" assumes "f \<in>\<^sub>\<circ> \<AA>\<lparr>Arr\<rparr>" shows "\<AA>\<lparr>Cod\<rparr>\<lparr>f\<rparr> \<in>\<^sub>\<circ> \<AA>\<lparr>Obj\<rparr>"
length_take_prefixes: fixes xs :: "'a list"   and n :: "nat"   and ys :: "'a list" assumes "xs \<in> set (take n (prefixes ys))" shows "length xs < n"
hyps_2: assumes "\<forall>p. ps c p \<noteq> None"   and "Marker \<notin> set (msgs c i)" shows "Marker \<notin> set (msgs c' i)"
convergence_prop: fixes g :: "'g"   and v :: "'val"   and n :: "'node"   and ns :: "'node list"   and m :: "'node" assumes "necessaryPhi g (var g v) n"   and "g \<turnstile> n-ns\<rightarrow>m"   and "v \<in> allUses g m"   and "\<And>x. x \<in> set (tl ns) \<Longrightarrow> v \<notin> allDefs g x"   and "v \<notin> defs g n" shows "phis g (n, v) \<noteq> None"
comp_PRE_def: fixes R :: "('h \<times> 'g) set"   and P :: "'g \<Rightarrow> bool"   and Q :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and S :: "'g \<Rightarrow> bool" shows "comp_PRE R P Q S \<equiv> \<lambda>x. S x \<longrightarrow> P x \<and> (\<forall>y. (y, x) \<in> R \<longrightarrow> Q x y)"
DN_simps_3: fixes \<mu> :: "('a E.term, 'a) arr" assumes "arr \<mu>" shows "trg\<^sub>B (DN \<mu>) = DN (trg \<mu>)"
vars_subst_in_left_only_diff_iff: fixes x :: "nat"   and p :: "int mpoly" assumes "x \<notin> vars p" shows "vars (p - Var x) = insert x (vars p)"
flat_interpretation: fixes b :: "'g" shows "partial_function_definitions (flat_ord b) (flat_lub b)"
derangements_drop_minimal_cycle: fixes d :: "nat \<Rightarrow> nat"   and S :: "nat set"   and x :: "nat" assumes "d \<in> derangements S"   and "d (d x) = x" shows "Fun.swap x (d x) d \<in> derangements (S - {x, d x})"
split: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "guest \<Rightarrow> room \<Rightarrow> key \<times> key \<Rightarrow> 'a"   and f2 :: "guest \<Rightarrow> room \<Rightarrow> key \<times> key \<Rightarrow> 'a"   and f3 :: "guest \<Rightarrow> room \<Rightarrow> 'a"   and event :: "event" shows "P (case event of Check_in x xa xb \<Rightarrow> f1 x xa xb | Enter x xa xb \<Rightarrow> f2 x xa xb | Exit x xa \<Rightarrow> f3 x xa) = ((\<forall>x11 x12 x13. event = Check_in x11 x12 x13 \<longrightarrow> P (f1 x11 x12 x13)) \<and> (\<forall>x21 x22 x23. event = Enter x21 x22 x23 \<longrightarrow> P (f2 x21 x22 x23)) \<and> (\<forall>x31 x32. event = Exit x31 x32 \<longrightarrow> P (f3 x31 x32)))"
coeff_normalize: fixes p :: "'b poly"   and n :: "nat" shows "coeff (normalize p) n = coeff p n div unit_factor (lead_coeff p)"
com_sifum_secure_def: fixes cmd :: "'Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var\<^sub>C set)" shows "conc.com_sifum_secure cmd \<equiv> case cmd of (c, mds\<^sub>s) \<Rightarrow> conc.low_indistinguishable mds\<^sub>s c c"
distinct_89: fixes x31 :: "floatarith"   and x32 :: "floatarith"   and x151 :: "floatarith"   and x152 :: "nat" shows "Mult x31 x32 \<noteq> x151 ^\<^sub>e x152"
cSup_upper: fixes x :: "'b multiset"   and X :: "'b multiset set" assumes "x \<in> X"   and "subset_mset.bdd_above X" shows "x \<subseteq># Sup X"
Qg_unit: fixes G :: "('c, 'd) Group_scheme"   and N :: "'c set" assumes "Group G"   and "G \<triangleright> N" shows "\<forall>x\<in>set_rcs G N. c_top G N N x = x"
labeling_consistent: fixes a :: "'a"   and b :: "'b" assumes "(a, b) \<in> R" shows "a.L a = b.L b"
is_processT5_S7: fixes t :: "'a event list"   and P :: "'a process"   and A :: "'a event set" assumes "t \<in> T P"   and "(t, A) \<notin> F P" shows "\<exists>x. x \<in> A \<and> t @ [x] \<in> T P"
simps_2: fixes x :: "'c"   and xs :: "'c list" shows "prefixes (x # xs) = [] # map ((#) x) (prefixes xs)"
foldl_ins_treap_of: fixes ys :: "('c \<times> 'd) list" assumes "distinct ys"   and "inj_on fst (set ys)"   and "inj_on snd (set ys)" shows "foldl (\<lambda>t' (x, p). ins x p t') \<langle>\<rangle> ys = treap_of (set ys)"
cond_action_oks_conv_set: fixes s :: "('d, 'e) locks \<times> (('e \<Rightarrow> ('x2 \<times> 'd \<Rightarrow>f nat) option) \<times> 'f) \<times> ('e \<Rightarrow> 'g wait_set_status option) \<times> 'e set"   and t :: "'e"   and cts :: "'e conditional_action list" shows "r2.cond_action_oks s t cts = (\<forall>ct\<in>set cts. r2.cond_action_ok s t ct)"
card_iff_exists: fixes X :: "'a set" shows "(0 < card X) = (finite X \<and> (\<exists>x. x \<in> X))"
SELECT_def: fixes P :: "'b \<Rightarrow> bool" shows "SELECT P \<equiv> if \<exists>x. P x then SPEC (\<lambda>uu_. \<exists>x. uu_ = Some x \<and> P x) else RETURN None"
p2: fixes p :: "int" assumes "mod_ring_locale TYPE('b) p" shows "2 \<le> p"
I'_def: shows "I' = (\<lambda>i. if i = Suc n then I j else if i = j then I (Suc n) else I i)"
node_ptr_no_shadow_root_ptr_cast: fixes shadow_root_ptr :: "(_) shadow_root_ptr" shows "\<not> is_node_ptr_kind (cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r shadow_root_ptr)"
braun_height_r_le: fixes l :: "'b tree"   and x :: "'b"   and r :: "'b tree" assumes "braun \<langle>l, x, r\<rangle>" shows "height r \<le> height l"
lift_Suc_mono_le: fixes f :: "nat \<Rightarrow> nat list"   and n :: "nat"   and n' :: "nat" assumes "\<And>n. f n \<le>\<^sub>v f (Suc n)"   and "n \<le> n'" shows "f n \<le>\<^sub>v f n'"
suma_assoc_1: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p"   and E :: "'p"   and F :: "'p"   and G :: "'p"   and H :: "'p"   and I :: "'p"   and A' :: "'p"   and B' :: "'p"   and C' :: "'p"   and D' :: "'p"   and E' :: "'p"   and F' :: "'p"   and K :: "'p"   and L :: "'p"   and M :: "'p" assumes "SAMS A B C D E F"   and "SAMS D E F G H I"   and "A B C D E F SumA A' B' C'"   and "D E F G H I SumA D' E' F'"   and "A' B' C' G H I SumA K L M" shows "A B C D' E' F' SumA K L M"
r0: shows "keys r0 \<subseteq> set (take j fs)"
bisim1BinOp2: fixes P :: "'a prog"   and h :: "'heap"   and e2 :: "'addr expr1"   and e' :: "'addr expr1"   and xs :: "'addr val list"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and xcp :: "'addr option"   and e1 :: "'addr expr1"   and bop :: "bop"   and v1 :: "'addr val" assumes "P,e2,h \<turnstile> (e', xs) \<leftrightarrow> (stk, loc, pc, xcp)" shows "P,e1 \<guillemotleft>bop\<guillemotright> e2,h \<turnstile> (Val v1 \<guillemotleft>bop\<guillemotright> e', xs) \<leftrightarrow> (stk @ [v1], loc, length (compE2 e1) + pc, xcp)"
insertI_2: fixes s :: "real"   and a :: "real"   and b :: "real"   and P :: "real \<Rightarrow> bool" assumes "s \<in> {a..<b}"   and "s = a \<or> P s"   and "\<forall>t\<in>{s<..<b}. \<not> P t"   and "finite_Psegments P a s" shows "finite_Psegments P a b"
assms: shows "\<Gamma> \<turnstile> Fn x T A : \<theta>"
add_left_one: fixes q :: "'a"   and N :: "'a multiset"   and M :: "'a multiset"   and r :: "('a \<times> 'a) set" assumes "\<exists>I J K. add_mset q N = I + J \<and> add_mset q M = I + K \<and> J \<inter># K = {#} \<and> set_mset K \<subseteq> r \<down>m J" shows "\<exists>I2 J K. N = I2 + J \<and> M = I2 + K \<and> set_mset K \<subseteq> r \<down>m J"
continuous_within_topological: fixes x :: "'b"   and s :: "'b set"   and f :: "'b \<Rightarrow> 'c" shows "continuous (at x within s) f = (\<forall>B. open B \<longrightarrow> f x \<in> B \<longrightarrow> (\<exists>A. open A \<and> x \<in> A \<and> (\<forall>y\<in>s. y \<in> A \<longrightarrow> f y \<in> B)))"
id_is_mor_pr_rngs: shows "morphism_presheaves_of_rings Spec is_zariski_open local.im_sheaf im_sheaf_morphisms \<O>b (\<lambda>V. add_sheaf_spec (identity Spec \<^sup>\<inverse> Spec V)) (\<lambda>V. mult_sheaf_spec (identity Spec \<^sup>\<inverse> Spec V)) (\<lambda>V. zero_sheaf_spec (identity Spec \<^sup>\<inverse> Spec V)) (\<lambda>V. one_sheaf_spec (identity Spec \<^sup>\<inverse> Spec V)) local.im_sheaf im_sheaf_morphisms \<O>b (\<lambda>V. add_sheaf_spec (identity Spec \<^sup>\<inverse> Spec V)) (\<lambda>V. mult_sheaf_spec (identity Spec \<^sup>\<inverse> Spec V)) (\<lambda>V. zero_sheaf_spec (identity Spec \<^sup>\<inverse> Spec V)) (\<lambda>V. one_sheaf_spec (identity Spec \<^sup>\<inverse> Spec V)) (\<lambda>U. identity (local.im_sheaf U))"
state: shows "Partial_Cost_Model.config' (rTS h0) ([x, y], x # y # h) v = ([x, y], rev qs @ h)"
step_pop: fixes F :: "'fenv_inca"   and f :: "'fun"   and fd :: "('dyn, 'var, 'fun, 'op, 'opinl) Inca.instr fundef"   and pc :: "nat"   and H :: "'henv"   and d :: "'dyn"   and \<Sigma> :: "'dyn list"   and st :: "('fun, 'dyn) frame list" assumes "Finca_get F f = Some fd"   and "pc < length (body fd)"   and "body fd ! pc = Inca.instr.IPop" shows "Sinca.step (State F H (Frame f pc (d # \<Sigma>) # st)) (State F H (Frame f (Suc pc) \<Sigma> # st))"
reindex_bij_witness: fixes S :: "'e set"   and i :: "'f \<Rightarrow> 'e"   and j :: "'e \<Rightarrow> 'f"   and T :: "'f set"   and h :: "'f \<Rightarrow> 'g"   and g :: "'e \<Rightarrow> 'g" assumes "\<And>a. a \<in> S \<Longrightarrow> i (j a) = a"   and "\<And>a. a \<in> S \<Longrightarrow> j a \<in> T"   and "\<And>b. b \<in> T \<Longrightarrow> j (i b) = b"   and "\<And>b. b \<in> T \<Longrightarrow> i b \<in> S"   and "\<And>a. a \<in> S \<Longrightarrow> h (j a) = g a" shows "prod g S = prod h T"
strategyproof_sds_axioms: shows "strategyproof_sds agents alts RD"
tl_obtain_elem: fixes xs :: "'a list" assumes "xs \<noteq> []"   and "tl xs = []" obtains e :: "'a" where "xs = [e]"
drlex_fun_def: shows "drlex_fun \<equiv> dord_fun rlex_fun"
bvars\<^sub>s\<^sub>s\<^sub>t_cons_subset: fixes A :: "('d, 'c) stateful_strand_step list"   and a :: "('d, 'c) stateful_strand_step" shows "bvars\<^sub>s\<^sub>s\<^sub>t A \<subseteq> bvars\<^sub>s\<^sub>s\<^sub>t (a # A)"
IH: assumes "\<And>x. x \<in> A \<Longrightarrow> finite (B x)" shows "(\<Prod>x\<in>A. sum (f x) (B x)) = (\<Sum>g\<in>Pi\<^sub>E A B. \<Prod>x\<in>A. f x (g x))"
dependent_live_varsp_dependent_live_vars_eq: fixes n' :: "'node" shows "dependent_live_varsp n' = (\<lambda>x xa xb. (x, xa, xb) \<in> dependent_live_vars n')"
apath_def: fixes u :: "'a"   and p :: "'b list"   and v :: "'a" shows "G.apath u p v \<equiv> G.awalk u p v \<and> distinct (G.awalk_verts u p)"
strip_while_filter: fixes lst :: "'a list" shows "filter ((\<noteq>) (0::'a)) (strip_while ((=) (0::'a)) lst) = filter ((\<noteq>) (0::'a)) lst"
thetaSeqTT_RetrT: fixes indis :: "('a \<times> 'a) set"   and tval :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'a \<Rightarrow> 'a" assumes "PL_Indis indis" shows "PL_Indis.thetaSeqTT tval aval indis \<subseteq> PL_Indis.RetrT tval aval indis (PL_Indis.thetaSeqTT tval aval indis \<union> PL_Indis.BisT tval aval indis)"
bernoulli_conv_bernoulli': fixes n :: "nat" assumes "n \<noteq> 1" shows "bernoulli n = bernoulli' n"
simps_14: fixes v :: "bexp"   and va :: "'a acom"   and vb :: "'a acom"   and vc :: "'a"   and vd :: "'a" shows "(IF v THEN va ELSE vb {vc} \<sqsubseteq> SKIP {vd}) = False"
psubstAbs_substAbs_compose_freshEnv: fixes A :: "('a, 'b, 'varSort, 'var, 'c) qAbs set"   and rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option"   and Y :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and ys :: "'varSort"   and y :: "'var" assumes "goodAbs A"   and "goodEnv rho"   and "good Y"   and "freshEnv ys y rho" shows "A $[Y / y]_ys $[rho] = A $[rho] $[Y #[rho] / y]_ys"
frontier_of_def: fixes X :: "'c topology"   and S :: "'c set" shows "X frontier_of S \<equiv> X closure_of S - X interior_of S"
union_diff_inter_eq_sup: fixes A :: "'b multiset"   and B :: "'b multiset" shows "A + B - A \<inter># B = A \<union># B"
diff_add_mset_swap: fixes b :: "'k"   and A :: "'k multiset"   and M :: "'k multiset" assumes "b \<notin># A" shows "add_mset b M - A = add_mset b (M - A)"
Mp_rel_i_def: fixes f :: "'i list"   and F :: "int poly" shows "Mp_rel_i f F = (map (arith_ops_record.to_int ops) f = coeffs (Mp F))"
point_eqI: fixes M :: "point"   and N :: "point" assumes "abscissa M = abscissa N"   and "ordinate M = ordinate N" shows "M = N"
val_def: fixes BinomialTree :: "('c, 'd) BinomialTree" shows "val BinomialTree = (case BinomialTree of Node x1 x2 x3 x4 \<Rightarrow> x1)"
spmf_rel_conversep: fixes R :: "'d \<Rightarrow> 'c \<Rightarrow> bool" shows "rel_spmf R\<inverse>\<inverse> = (rel_spmf R)\<inverse>\<inverse>"
vdiff_vunion_left: fixes A :: "V"   and B :: "V"   and C :: "V" shows "A \<union>\<^sub>\<circ> B -\<^sub>\<circ> C = A -\<^sub>\<circ> C \<union>\<^sub>\<circ> (B -\<^sub>\<circ> C)"
pred_list_it_correct: fixes g :: "'G"   and v :: "'V" assumes "invar g" shows "set_iterator (pred_list_it g v) (pred (\<alpha> g) v)"
assms: shows "lossless_gpv (\<I>_full \<oplus>\<^sub>\<I> \<I>_full) \<A>"
fpxs_root_order_pos: fixes f :: "'b fpxs" shows "0 < fpxs_root_order f"
get_child_nodes_reads: fixes ptr :: "(_) object_ptr"   and h :: "(_) heap"   and h' :: "(_) heap" shows "reads (Shadow_DOM.get_child_nodes_locs ptr) (Shadow_DOM.get_child_nodes ptr) h h'"
bounded_cbox: fixes a :: "'a"   and b :: "'a" shows "bounded (cbox a b)"
linord_of_list_def: fixes xs :: "'c list" shows "linord_of_list xs \<equiv> {(x, y). linord_of_listP x y xs}"
that_2: assumes "\<bar>u\<bar> = - u" shows "thesis"
hom_prod: fixes f :: "'b \<Rightarrow> int poly"   and X :: "'b set" shows "Ring_Hom_Poly.of_int_poly (prod f X) = (\<Prod>x\<in>X. Ring_Hom_Poly.of_int_poly (f x))"
semClosed: fixes S :: "'a set"   and s0 :: "'a"   and \<delta> :: "'a \<Rightarrow> 'a set"   and AP :: "'b set"   and L :: "'a \<Rightarrow> 'b set"   and AP' :: "'b set"   and \<phi> :: "'b dfmla"   and \<pi>l :: "('a \<times> 'b set) stream list"   and env :: "pvar \<Rightarrow> nat" assumes "Shallow S s0 \<delta> AP L AP'"   and "wff \<phi>"   and "FV \<phi> = {}"   and "Shallow.pointerOf s0 \<pi>l = s0" shows "Shallow.semClosed S s0 \<delta> AP L AP' \<phi> = Shallow.sem S s0 \<delta> AP L AP' \<phi> env \<pi>l"
domI: fixes h' :: "'b \<Rightarrow> 'b"   and b :: "'b"   and Q :: "'c \<Rightarrow> bool"   and x :: "'c"   and h :: "'c \<Rightarrow> 'c" assumes "recursive h' b"   and "\<not> Q x \<Longrightarrow> tailrec.dom Q h (h x)" shows "tailrec.dom Q h x"
greatest_Lower_cong_l: fixes x :: "'a"   and x' :: "'a"   and A :: "'a set" assumes "x .= x'"   and "x \<in> carrier L"   and "x' \<in> carrier L" shows "is_glb L x A = is_glb L x' A"
cod_char: fixes f :: "'a" shows "hom.cod f = (if hom.arr f then cod f else null)"
path_vertices_shift: fixes v :: "'d"   and v' :: "'d"   and pth :: "(('d \<times> 'c) \<times> 'd \<times> 'b) list" assumes "path v v' pth" shows "map fst (map fst pth) @ [v'] = v # map fst (map snd pth)"
det_echelon_form_of_euclidean_iarrays: fixes A :: "(('c, 'd) vec, 'd) vec" shows "det A = (let A' = echelon_form_of_det_iarrays (matrix_to_iarray A) euclid_ext2 in (1::'c) div fst A' * (\<Prod>i\<leftarrow>[0..<nrows_iarray (matrix_to_iarray A)]. snd A' !! i !! i))"
abs: fixes X :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and xs :: "'varSort"   and x :: "'var" assumes "good X"   and "\<And>Y. (X, Y) \<in> rel \<Longrightarrow> phi Y" shows "phiAbs (Abs xs x X)"
Poincare_recurrence_thm_6: fixes A :: "'a set" assumes "A \<in> sets M" shows "emeasure M (recurrent_subset_infty A) = emeasure M A"
poincare_line_circline_set_1: fixes z :: "complex_homo"   and w :: "complex_homo" assumes "z \<noteq> w" shows "z \<in> circline_set (poincare_line z w)"
finite_complement_partition: fixes A :: "'a set" assumes "finite A" shows "A \<in> infinite_complement_partition"
simps_2: fixes a :: "'b" shows "cycle [a] = 1"
Icarr: fixes I :: "'e set"   and R :: "('e, 'f) ring_scheme"   and i :: "'e" assumes "ideal I R"   and "i \<in> I" shows "i \<in> carrier R"
pref_drop: fixes u :: "'b list"   and v :: "'b list"   and p :: "nat" assumes "u \<le>p v" shows "drop p u \<le>p drop p v"
fmrel_to_rtrancl: fixes r :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and xm :: "'f \<rightharpoonup>\<^sub>f 'e"   and ym :: "'f \<rightharpoonup>\<^sub>f 'e" assumes "reflp r"   and "fmrel r\<^sup>*\<^sup>* xm ym" shows "(fmrel r)\<^sup>*\<^sup>* xm ym"
algebraic_exp_complex_iff: fixes x :: "complex" assumes "algebraic x" shows "algebraic (exp x) = (x = 0)"
simps: fixes ires :: "cell list"   and rs :: "nat"   and l :: "cell list"   and r :: "cell list" shows "wcode_on_left_moving_1_O ires rs (l, r) = (\<exists>ln rn. l = Oc # ires \<and> r = Oc # Bk \<up> ln @ Bk # Bk # Oc \<up> Suc rs @ Bk \<up> rn)"
intros_8: fixes e :: "expr"   and \<theta> :: "nat \<Rightarrow> nat option"   and s\<^sub>1 :: "stmt"   and s\<^sub>2 :: "stmt"   and t :: "nat" assumes "\<forall>sop. e \<noteq> Tmp sop" shows "\<theta>\<turnstile> (Cond e s\<^sub>1 s\<^sub>2, t) \<rightarrow>\<^sub>s ((Cond (Tmp (eval_expr t e)) s\<^sub>1 s\<^sub>2, t + used_tmps e), issue_expr t e)"
RedNewArray: fixes i :: "32 word"   and h' :: "'heap"   and a :: "'addr"   and h :: "'heap"   and T :: "ty"   and extTA :: "('addr, 'thread_id, 'heap) external_thread_action \<Rightarrow> ('addr, 'thread_id, 'b, 'heap) Jinja_thread_action"   and P :: "'addr J_prog"   and t :: "'thread_id"   and l :: "String.literal \<Rightarrow> 'addr val option" assumes "0 \<le>s i"   and "(h', a) \<in> allocate h (Array_type T (nat (sint i)))" shows "extTA,P,t \<turnstile> \<langle>newA T\<lfloor>Val (Intg i)\<rceil>,(h, l)\<rangle> -\<lbrace>NewHeapElem a (Array_type T (nat (sint i)))\<rbrace>\<rightarrow> \<langle>addr a,(h', l)\<rangle>"
set_plus_mono2_b: fixes C :: "'c set"   and D :: "'c set"   and E :: "'c set"   and F :: "'c set"   and x :: "'c" assumes "C \<subseteq> D"   and "E \<subseteq> F"   and "x \<in> C + E" shows "x \<in> D + F"
pred_inject_3: fixes P1 :: "'d \<Rightarrow> bool"   and P2 :: "'e \<Rightarrow> bool"   and a :: "'e" shows "pred_xstate P1 P2 (Fault a) = P2 a"
pair_return_pmf1: fixes x :: "'d"   and y :: "'e pmf" shows "pair_pmf (return_pmf x) y = map_pmf (Pair x) y"
plus_ereal_0: fixes x :: "ereal" shows "x + ereal 0 = x"
a: shows "ideal_generated {a} = \<Union> {I n |n. n \<in> UNIV}"
check_unsat: fixes check :: "('a, 'b) state \<Rightarrow> ('a, 'b) state"   and s :: "('a, 'b) state" assumes "Check check"   and "\<not> \<U> s"   and "\<Turnstile>\<^sub>n\<^sub>o\<^sub>l\<^sub>h\<^sub>s s"   and "\<diamond> s"   and "\<triangle> (\<T> s)"   and "\<nabla> s"   and "\<U> (check s)" shows "minimal_unsat_state_core (check s)"
vertex_bipartite_web_2: fixes \<Gamma> :: "('a, 'b) web_scheme"   and x :: "'a" assumes "web \<Gamma>" shows "vertex (bipartite_web_of \<Gamma>) (Inr x) = (vertex \<Gamma> x \<and> x \<notin> A \<Gamma>)"
T_def: fixes x :: "real" shows "??.Stirling_Formula.T x = 1 / (2 * x) + 1 / (2 * (x + 1))"
max_fun_diff_commute: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" shows "s.max_fun_diff f g = s.max_fun_diff g f"
bind_return': fixes M :: "'a measure" shows "M \<bind> return M = M"
map_image_filter_correct_aux1: fixes m :: "'m1"   and f :: "'u1 \<times> 'v1 \<Rightarrow> ('u2 \<times> 'v2) option"   and k' :: "'u2"   and v' :: "'v2" assumes "invar1 m"   and "transforms_to_unique_keys (\<alpha>1 m) f" shows "invar2 (map_image_filter f m) \<and> (\<alpha>2 (map_image_filter f m) k' = Some v') = (\<exists>k v. \<alpha>1 m k = Some v \<and> f (k, v) = Some (k', v'))"
add_assoc: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "class.semigroup_add plus" shows "plus (plus a b) c = plus a (plus b c)"
measurable_product_coordinates: fixes i :: "'b" shows "(\<lambda>x. x i) \<in> borel_measurable borel"
discr_Par: fixes cl :: "('test, 'atom, 'choice) cmd list" assumes "properL cl"   and "discrL cl" shows "discr (Par cl)"
unit_in_vhom: shows "\<guillemotleft>\<eta>' : src f \<Rightarrow> g' \<star> f\<guillemotright>"
pw_Inf_nofail: shows "nofail (Inf (f ` C)) = (\<exists>x\<in>f ` C. nofail x)"
simps_1: fixes x :: "'a list"   and ya :: "'a list" shows "(IArray x = IArray ya) = (x = ya)"
simps_13: fixes v :: "'b rexp"   and va :: "'b rexp" shows "le_rexp (Plus v va) One = False"
prems_1: shows "n -as' @ [a']\<rightarrow>* n'"
cross_matrix_mult: fixes A :: "((real, 3) vec, 3) vec"   and x :: "(real, 3) vec"   and y :: "(real, 3) vec" shows "Finite_Cartesian_Product.transpose A *v (A *v x) \<times> (A *v y) = det A *\<^sub>R x \<times> y"
nmsort_def: fixes xs :: "'b list" shows "nmsort xs = merge_all (runs xs)"
HInv4b_def: fixes s :: "state"   and p :: "Proc" shows "HInv4b s p = (phase s p = 1 \<longrightarrow> (\<forall>bk\<in>blocksOf s p. bal bk < mbal (dblock s p)))"
eval_Pr_converg_le: fixes n :: "nat"   and f :: "recf"   and g :: "recf"   and xs :: "nat list"   and x :: "nat"   and y :: "nat" assumes "recfn (Suc n) (Pr n f g)"   and "n = length xs"   and "eval (Pr n f g) (x # xs) \<down>"   and "y \<le> x" shows "eval (Pr n f g) (y # xs) \<down>"
Max_eqI: 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 "\<And>y. y \<in> A \<Longrightarrow> less_eq y x"   and "x \<in> A" shows "linorder.Max less_eq A = x"
le_infI1: fixes inf :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and less_eq :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and less :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and a :: "'b::type"   and x :: "'b::type"   and b :: "'b::type" assumes "class.semilattice_inf (inf::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (less::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool)"   and "(less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (a::'b::type) (x::'b::type)" shows "(less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) ((inf::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (a::'b::type) (b::'b::type)) (x::'b::type)"
repFreePath_1: shows "path (product M2 M1) p (initial M2, initial M1) \<and> target p (initial M2, initial M1) = (target (butlast seq || tr2b) (initial M2), target (butlast seq || tr1b) (initial M1))"
Step1_4_3b: fixes p :: "PrIds"   and rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState" shows "\<turnstile> RPCFail crCh rmCh rst p \<and> $S3 rmhist p \<and> S6 rmhist p$ \<and> unchanged (e p, c p, m p) \<longrightarrow> MemFail memCh (resbar rmhist) p"
triangle_num_less: fixes m :: "nat"   and n :: "nat" assumes "m < n" shows "triangle_num m < triangle_num n"
E: shows "E = (\<lambda>s. 1 + 2 * (nat (s ''a'') + 1 - nat (s ''z'')))"
abs_conv_abscissa_def: fixes f :: "'a fds" shows "abs_conv_abscissa f = (INF s\<in>{s. fds_abs_converges f s}. ereal (s \<bullet> (1::'a)))"
assms: shows "(t, s) \<in> set (db'\<^sub>s\<^sub>s\<^sub>t A I D)"
Un_subset_iff: fixes A :: "'b set"   and B :: "'b set"   and C :: "'b set" shows "(A \<union> B \<subseteq> C) = (A \<subseteq> C \<and> B \<subseteq> C)"
front_trace_is_tickfree: fixes P :: "('a event list \<times> 'a event set) set \<times> 'a event list set"   and t :: "'a event list"   and X :: "'a event set" assumes "is_process P"   and "(t @ [tick], X) \<in> FAILURES P" shows "tickFree t"
ev_2: shows "\<forall>\<^sub>F x in F. 0 \<le> g x"
pmf_bind_pmf_of_set: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'd pmf"   and x :: "'d" assumes "A \<noteq> {}"   and "finite A" shows "pmf (local.pmf_of_set A \<bind> f) x = (\<Sum>xaa\<in>A. pmf (f xaa) x) / real (card A)"
smult_minus_1_back: fixes v :: "'a Matrix.vec" assumes "v \<in> carrier_vec n" shows "\<ominus>\<^bsub>NC.V\<^esub> v = - (1::'a) \<cdot>\<^sub>v v"