hcju/isabelleps huggingface dataset:
nlt__le: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "\<not> A B Lt C D" shows "C D Le A B"
proots_unbounded_line_def: fixes p :: "complex poly"   and st :: "complex"   and tt :: "complex" shows "proots_unbounded_line p st tt = proots_count p (unbounded_line st tt)"
simps: fixes a1 :: "('a \<times> real) msllist"   and a2 :: "real \<Rightarrow> real"   and a3 :: "(real \<Rightarrow> real) list" shows "is_expansion_aux a1 a2 a3 = ((\<exists>f basis. a1 = MSLNil \<and> a2 = f \<and> a3 = basis \<and> (\<forall>\<^sub>F x in at_top. f x = 0) \<and> length basis = Suc (expansion_level TYPE('a))) \<or> (\<exists>xs f C b e basis. a1 = MSLCons (C, e) xs \<and> a2 = f \<and> a3 = b # basis \<and> is_expansion_aux xs (\<lambda>x. f x - eval C x * b x powr e) (b # basis) \<and> is_expansion C basis \<and> (\<forall>x>e. f \<in> o(\<lambda>xa. b xa powr x)) \<and> ms_exp_gt e (ms_aux_hd_exp xs)))"
sum_upto_inverse_bound_1: fixes x :: "real" shows "0 \<le> sum_upto (\<lambda>i. 1 / real i) x"
eval_vars_det\<^sub>B: fixes eval\<^sub>A :: "('a \<Rightarrow> 'b) \<Rightarrow> 'c \<Rightarrow> 'b"   and eval\<^sub>B :: "('a \<Rightarrow> 'b) \<Rightarrow> 'd \<Rightarrow> bool"   and aexp_vars :: "'c \<Rightarrow> 'a set"   and bexp_vars :: "'d \<Rightarrow> 'a set"   and b :: "'d"   and mem\<^sub>1 :: "'a \<Rightarrow> 'b"   and mem\<^sub>2 :: "'a \<Rightarrow> 'b" assumes "sifum_lang_no_dma eval\<^sub>A eval\<^sub>B aexp_vars bexp_vars"   and "\<forall>x\<in>bexp_vars b. mem\<^sub>1 x = mem\<^sub>2 x" shows "eval\<^sub>B mem\<^sub>1 b = eval\<^sub>B mem\<^sub>2 b"
assms_1: shows "flowpipe X0 hl hu CX X1"
simplify_def: fixes S :: "'at Literal set set" shows "simplify S = {C \<in> S. \<not> strictly_redundant C S}"
get_exp_def: fixes g :: "'c" shows "H.get_exp g = (\<lambda>a. SOME k. a = g [^]\<^bsub>H\<^esub> k)"
array_grow'_def: fixes a :: "'a array" shows "array_grow' a \<equiv> array_grow a \<circ> nat_of_integer"
polyfun_subset: fixes N :: "nat set"   and N' :: "nat set"   and f :: "(nat \<Rightarrow> 'b) \<Rightarrow> 'b" assumes "N \<subseteq> N'"   and "polyfun N f" shows "polyfun N' f"
R8_d: shows "pmf (sds R8) d = 1 - pmf (sds R8) a"
simps_1: fixes s :: "('a, 'b) fmap" shows "state_list s [] = [s]"
add_smult_distrib_left_mat: fixes A :: "'b mat"   and nr :: "nat"   and nc :: "nat"   and B :: "'b mat"   and k :: "'b" assumes "A \<in> carrier_mat nr nc"   and "B \<in> carrier_mat nr nc" shows "k \<cdot>\<^sub>m (A + B) = k \<cdot>\<^sub>m A + k \<cdot>\<^sub>m B"
subfieldE_1: fixes K :: "'c set"   and R :: "('c, 'd) ring_scheme" assumes "subfield K R" shows "subring K R"
comp_char: fixes g :: "'a \<times> 'a"   and f :: "'a \<times> 'a" shows "VxV.comp g f = (if seq (fst g) (fst f) \<and> seq (snd g) (snd f) then (fst g \<cdot> fst f, snd g \<cdot> snd f) else (null, null))"
degree0_coeffs: fixes p :: "'a poly" assumes "degree p = 0" shows "\<exists>a. p = [:a:]"
ran_dom_the: fixes y :: "'d"   and G :: "'e \<Rightarrow> 'd set option"   and x :: "'e" assumes "y \<notin> \<Union> (ran G)"   and "x \<in> dom G" shows "y \<notin> the (G x)"
aezfun_setspan_proj_add: fixes A :: "'b set"   and x :: "('a, 'b) aezfun"   and y :: "('a, 'b) aezfun" shows "aezfun_setspan_proj A (x + y) = aezfun_setspan_proj A x + aezfun_setspan_proj A y"
Assoc_in_assoc: fixes f :: "'c local.term set"   and g :: "'c local.term set"   and h :: "'c local.term set"   and t :: "'c local.term"   and u :: "'c local.term"   and v :: "'c local.term" assumes "arr f"   and "arr g"   and "arr h"   and "t \<in> f"   and "u \<in> g"   and "v \<in> h" shows "\<^bold>\<a>\<^bold>[t, u, v\<^bold>] \<in> \<a>[f, g, h]"
iT_Plus_neg_card_le: fixes I :: "nat set"   and k :: "nat" shows "card (I \<oplus>- k) \<le> card I"
IH: shows "(\<sigma>, s) \<in> oreachable (opnet onp p\<^sub>1) (\<lambda>\<sigma> _. oarrivemsg I \<sigma>) (other U (net_tree_ips p\<^sub>1)) \<and> (\<sigma>, t) \<in> oreachable (opnet onp p\<^sub>2) (\<lambda>\<sigma> _. oarrivemsg I \<sigma>) (other U (net_tree_ips p\<^sub>2)) \<and> net_tree_ips p\<^sub>1 \<inter> net_tree_ips p\<^sub>2 = {}"
isomorphic_def: fixes a :: "'a"   and a' :: "'a" shows "B.isomorphic a a' = (\<exists>f. \<guillemotleft>f : a \<rightarrow>\<^sub>B a'\<guillemotright> \<and> B.iso f)"
wf_subset: fixes r :: "('a \<times> 'a) set"   and p :: "('a \<times> 'a) set" assumes "wf r"   and "p \<subseteq> r" shows "wf p"
handle_purity2: fixes \<Phi> :: "('d \<times> 'b) set \<Rightarrow> bool"   and A :: "'b \<Rightarrow> 'd \<Rightarrow> assn" assumes "CONSTRAINT (IS_PURE \<Phi>) A" shows "CONSTRAINT is_pure A"
invar_delete: fixes m :: "'b"   and a :: "'v" assumes "Q_invar m" shows "Q_invar (Q_delete a m)"
beta_step_imp_beta_reducible: fixes t :: "Core.term"   and t' :: "Core.term" assumes "t \<rightarrow>\<^sub>\<beta> t'" shows "beta_reducible t"
f_im_the_inv_into_f_im: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and x :: "'c set" assumes "inj_on f A"   and "order.greater_eq (f ` A) x" shows "f ` the_inv_into A f ` x = x"
cosh_complex: fixes z :: "complex" shows "(exp z + inverse (exp z)) / 2 = cos (\<i> * z)"
A3: shows "\<forall>S. (SourceTerm S, TargetTerm (\<lbrakk>S\<rbrakk>)) \<in> RelB"
simps_2: fixes x :: "'b" shows "extB.Bot + extB.Val x = extB.Val x"
not_summable_infsetsum_eq: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set" assumes "\<not> f abs_summable_on A" shows "infsetsum f A = (0::'d)"
elementaryCompDD_def: fixes C :: "CSet" shows "elementaryCompDD C \<equiv> (\<exists>x. OUT C = {x}) \<or> (\<forall>x\<in>OUT C. \<forall>y\<in>OUT C. outSetCorelated x \<inter> outSetCorelated y \<noteq> {})"
Abs_inverse: fixes y :: "'a" assumes "y \<in> A" shows "Rep (Abs y) = y"
measurable_piecewise_restrict: fixes C :: "'c set set"   and M :: "'c measure"   and f :: "'c \<Rightarrow> 'd"   and N :: "'d measure" assumes "countable C"   and "\<And>\<Omega>. \<Omega> \<in> C \<Longrightarrow> \<Omega> \<inter> space M \<in> sets M"   and "space M \<subseteq> \<Union> C"   and "\<And>\<Omega>. \<Omega> \<in> C \<Longrightarrow> f \<in> restrict_space M \<Omega> \<rightarrow>\<^sub>M N" shows "f \<in> M \<rightarrow>\<^sub>M N"
k_2: shows "\<forall>x\<in>{a..b} - k. g differentiable at x within {a..b}"
cmod_add_squared: fixes r1 :: "real"   and r2 :: "real"   and \<theta>1 :: "real"   and \<theta>2 :: "real" assumes "0 \<le> r1"   and "0 \<le> r2" shows "(cmod (complex_of_real r1 * exp (\<i> * complex_of_real \<theta>1) + complex_of_real r2 * exp (\<i> * complex_of_real \<theta>2)))\<^sup>2 = r1\<^sup>2 + r2\<^sup>2 + 2 * r1 * r2 * cos (\<theta>1 - \<theta>2)"
f: fixes x :: "'c" assumes "x \<in> results_gpv \<I>' gpv'" shows "f x = g x"
range_of_var_def: fixes x :: "nat" shows "range_of_var x \<equiv> {0..<numVals x}"
wf_trms_subst_rm_vars': fixes \<delta> :: "'a \<Rightarrow> ('fun, 'a) Term.term"   and X :: "'a set" assumes "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<delta>)" shows "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range (rm_vars X \<delta>))"
wfp_onE_min: fixes r :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and A :: "'e set"   and x :: "'e"   and Q :: "'e set" assumes "wfp_on r A"   and "x \<in> Q"   and "Q \<subseteq> A" obtains z :: "'e" where "z \<in> Q" and "\<And>y. r y z \<Longrightarrow> y \<notin> Q"
simps_3: fixes t :: "'b tree" shows "right (mirror t) = mirror (left t)"
rep_in_arr: fixes f :: "'c local.term set" assumes "arr f" shows "rep f \<in> f"
normal_ordinal_rec: fixes s :: "ordinal \<Rightarrow> ordinal \<Rightarrow> ordinal"   and z :: "ordinal" assumes "\<forall>p x. x < s p x" shows "normal (ordinal_rec z s)"
image_sub0: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'d set" assumes "f \<in> A \<rightarrow> B" shows "f ` A \<subseteq> B"
map_of_filter: fixes P :: "'a \<Rightarrow> bool"   and xs :: "('a \<times> 'b) list" shows "map_of (filter (\<lambda>x. P (fst x)) xs) = map_of xs |` Collect P"
mod_sub_eq_mod_swap: fixes a :: "nat"   and n :: "nat"   and b :: "nat"   and m :: "nat" assumes "a \<le> n"   and "b \<le> n" shows "((n - a) mod m = b mod m) = ((n - b) mod m = a mod m)"
assms_6: shows "f2 x2 y2 z2 = (0::'a)"
exhaust_sel: fixes contT :: "('d, 'e) contT" obtains "contT = ContT (run_cont contT)"
reachable_step: fixes E :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and s\<^sub>0 :: "'c"   and a :: "'c"   and b :: "'c" assumes "Graph_Start_Defs.reachable E s\<^sub>0 a"   and "E a b" shows "Graph_Start_Defs.reachable E s\<^sub>0 b"
valid: shows "\<Gamma>\<Turnstile>\<^bsub>/F\<^esub> P c Q,A"
minGraphProps2: fixes g :: "graph"   and f :: "face" assumes "minGraphProps g"   and "f \<in> \<F> g" shows "2 < |vertices f|"
map_poly_add: fixes h :: "'c \<Rightarrow> 'b"   and p :: "'c poly"   and q :: "'c poly" assumes "h (0::'c) = (0::'b)"   and "\<forall>p q. h (p + q) = h p + h q" shows "map_poly h (p + q) = map_poly h p + map_poly h q"
assms: shows "[g, f]\<^sub>\<circ> \<in>\<^sub>\<circ> (\<AA> \<times>\<^sub>D\<^sub>G \<BB>)\<lparr>Arr\<rparr>"
aux1: assumes "set l \<subseteq> {(d, v). w (u, v) = enat d}" shows "relax_outgoing'' l du V Q v = (if v \<in> snd ` set l then relax_outgoing' u du V Q v else Q v)"
rep_eq: fixes x :: "\<Pi>\<^sub>3 \<Rightarrow> \<o>" shows "eval\<o> (forall\<^sub>3 x) = (\<lambda>s w. \<forall>xa. eval\<o> (x xa) s w)"
shr_start_state: fixes addr2thread_id :: "'b \<Rightarrow> 'c"   and empty_heap :: "'a"   and allocate :: "'a \<Rightarrow> htype \<Rightarrow> ('a \<times> 'b) set"   and f :: "String.literal \<Rightarrow> String.literal \<Rightarrow> ty list \<Rightarrow> ty \<Rightarrow> 'e \<Rightarrow> 'b val list \<Rightarrow> 'd"   and P :: "'e prog"   and C :: "String.literal"   and M :: "String.literal"   and vs :: "'b val list" shows "shr (heap_base.start_state addr2thread_id empty_heap allocate f P C M vs) = heap_base.start_heap empty_heap allocate"
alphas_3: fixes bs :: "atom list"   and x :: "'a"   and R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> atom set"   and p :: "perm"   and cs :: "atom list"   and y :: "'a" shows "(bs, x) \<approx>lst R f p (cs, y) = (f x - set bs = f y - set cs \<and> (f x - set bs) \<sharp>* p \<and> R (p \<bullet> x) y \<and> p \<bullet> bs = cs)"
mor_fold: fixes s1 :: "('d, 'e, 'f) F1 \<Rightarrow> 'e"   and s2 :: "('d, 'e, 'f) F2 \<Rightarrow> 'f" shows "mor UNIV UNIV ctor1 ctor2 UNIV UNIV s1 s2 (fold1 s1 s2) (fold2 s1 s2)"
branch_delete: fixes p :: "bool list"   and ds :: "bool list"   and T :: "tree" assumes "branch p (delete ds T)" shows "branch p T \<or> p = ds"
locally_finite_compactD: fixes X :: "'c set"   and I :: "'d set"   and U :: "'d \<Rightarrow> 'c set"   and V :: "'c set" assumes "locally_finite_on X I U"   and "compact V"   and "V \<subseteq> X" shows "finite {i \<in> I. U i \<inter> V \<noteq> {}}"
closed_fip_Heine_Borel: fixes S :: "'b set"   and T :: "'c set"   and \<F> :: "'c set set" assumes "closed S"   and "T \<in> \<F>"   and "bounded T"   and "\<And>T. T \<in> \<F> \<Longrightarrow> closed T"   and "\<And>\<F>'. \<lbrakk>finite \<F>'; \<F>' \<subseteq> \<F>\<rbrakk> \<Longrightarrow> \<Inter> \<F>' \<noteq> {}" shows "\<Inter> \<F> \<noteq> {}"
R_def: shows "R = R1 \<union> R2 \<union> {([], [])}"
Image_mono: fixes r' :: "('c \<times> 'd) set"   and r :: "('c \<times> 'd) set"   and A' :: "'c set"   and A :: "'c set" assumes "r' \<subseteq> r"   and "A' \<subseteq> A" shows "r' `` A' \<subseteq> r `` A"
is_group: shows "Group.group (DirProds (\<lambda>p. Z (p ^ multiplicity p n)) (prime_factors n))"
balE: fixes c :: "(Vars.var \<times> exp) list \<times> exp \<times> stack_elem list"   and T :: "((Vars.var \<times> exp) list \<times> exp \<times> stack_elem list) list"   and c' :: "(Vars.var \<times> exp) list \<times> exp \<times> stack_elem list" assumes "c \<Rightarrow>\<^sup>b\<^sup>*\<^bsub>T\<^esub> c'" obtains "c \<Rightarrow>\<^sup>*\<^bsub>T\<^esub> c'" and "\<forall>c'\<in>set T. stack c \<lesssim> stack c'" and "stack c' = stack c"
lemma_conf_temporal: fixes A :: "agent"   and B :: "agent"   and evs :: "event list"   and Tk :: "nat"   and K :: "nat" assumes "A \<notin> bad"   and "B \<notin> bad"   and "evs \<in> bankerb_gets"   and "Says Server A (Crypt (shrK A) \<lbrace>Number Tk, Agent B, Key K, Crypt (shrK B) \<lbrace>Number Tk, Agent A, Key K\<rbrace>\<rbrace>) \<in> set evs"   and "Key K \<in> analz (knows Spy evs)" shows "expiredK Tk evs"
FreshExt': fixes x :: "name"   and U :: "pi"   and S :: "pi"   and T :: "pi" assumes "x \<sharp> U" shows "(<\<nu>x>(S \<parallel> T) \<parallel> U, <\<nu>x>(S \<parallel> T \<parallel> U)) \<in> Rel"
eventually_at_right_to_top: fixes P :: "real \<Rightarrow> bool" shows "eventually P (at_right 0) = (\<forall>\<^sub>F x in at_top. P (inverse x))"
roots: shows "set (rroots2 p) = (if bac < 0 then {} else {- b2a + sqrt bac, - b2a - sqrt bac})"
fmdip_cons_falseI: fixes cld :: "cld"   and meth_defs :: "meth_def list"   and meth :: "char list"   and ctxcld_list :: "(ctx \<times> cld) list"   and ctxmeth_def_opt :: "(ctx \<times> meth_def) option"   and ctx :: "ctx" assumes "class_methods cld meth_defs"   and "find_meth_def_in_list meth_defs meth None"   and "find_meth_def_in_path ctxcld_list meth ctxmeth_def_opt" shows "find_meth_def_in_path ([(ctx, cld)] @ ctxcld_list) meth ctxmeth_def_opt"
bij_betw_nth: fixes xs :: "'a list"   and A :: "nat set"   and B :: "'a set" assumes "distinct xs"   and "A = {..<length xs}"   and "B = set xs" shows "bij_betw ((!) xs) A B"
numeral_code_1: fixes plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and one :: "'e" assumes "class.numeral plus" shows "numeral.numeral one plus num.One = one"
rel_FGf_def: fixes L1 :: "'k \<Rightarrow> 'u \<Rightarrow> bool"   and L2 :: "'l \<Rightarrow> 'v \<Rightarrow> bool"   and Co1 :: "'n \<Rightarrow> 'w \<Rightarrow> bool"   and Co2 :: "'o \<Rightarrow> 'x \<Rightarrow> bool"   and Contra1 :: "'q \<Rightarrow> 'y \<Rightarrow> bool"   and Contra2 :: "'r \<Rightarrow> 'z \<Rightarrow> bool" shows "rel_FGf L1 L2 Co1 Co2 Contra1 Contra2 = rel_F L1 L2 (=) Co1 Co2 (=) Contra1 Contra2 (=)"
unit_SE_def: fixes e :: "'f" shows "unit\<^sub>S\<^sub>E e = (\<lambda>\<sigma>. Some (e, \<sigma>))"
lCdom2: fixes a :: "('b, 'c) Combinators list"   and b :: "('b, 'c) Combinators list"   and c :: "('b, 'c) Combinators list" shows "list2FWpolicy (a @ b @ c) = list2FWpolicy ((a @ b) @ c)"
preserves_arr: fixes f :: "'a" assumes "L'.C.arr f" shows "L'.C.arr (H\<^sub>L a' f)"
valid_eval_mmuaux: fixes args :: "Monitor.args"   and cur :: "nat"   and aux :: "nat \<times> nat queue \<times> nat \<times> bool list \<times> bool list \<times> (event_data option list, nat) mapping \<times> (nat, (event_data option list, nat + nat) mapping) mapping \<times> event_data option list set list \<times> nat"   and auxlist :: "(nat \<times> event_data option list set \<times> event_data option list set) list"   and nt :: "nat"   and res :: "event_data option list set list"   and aux' :: "nat \<times> nat queue \<times> nat \<times> bool list \<times> bool list \<times> (event_data option list, nat) mapping \<times> (nat, (event_data option list, nat + nat) mapping) mapping \<times> event_data option list set list \<times> nat"   and res' :: "event_data option list set list"   and auxlist' :: "(nat \<times> event_data option list set \<times> event_data option list set) list" assumes "valid_mmuaux args cur aux auxlist"   and "cur \<le> nt"   and "eval_mmuaux args nt aux = (res, aux')"   and "eval_until (args_ivl args) nt auxlist = (res', auxlist')" shows "res = res' \<and> valid_mmuaux args cur aux' auxlist'"
dead: shows "t \<in> r1.deadlocked s1 \<or> r1.deadlock s1"
prems_1: assumes "length p \<le> length q" shows "take (length p) q \<noteq> p"
subst_0_lift_1: fixes t :: "'c FOL_Fitting.term"   and s :: "'c FOL_Fitting.term" shows "liftt t[s/0] = t"
cond_map_pmf: fixes p :: "'c pmf"   and f :: "'c \<Rightarrow> 'd"   and s :: "'d set" assumes "set_pmf p \<inter> f -` s \<noteq> {}" shows "cond_pmf (map_pmf f p) s = map_pmf f (cond_pmf p (f -` s))"
fps_mult_nth_eq0: fixes n :: "nat"   and f :: "'b fps"   and g :: "'b fps" assumes "n < subdegree f + subdegree g" shows "(f * g) $ n = (0::'b)"
elims: fixes x :: "nat"   and xa :: "'b vec list"   and xb :: "'b vec list"   and y :: "'b vec list" assumes "gram_schmidt_sub x xa xb = y" obtains n :: "nat" and us :: "'b vec list" where "x = na__" and "xa = us__" and "xb = []" and "y = us__"   | n :: "nat" and us :: "'b vec list" and w :: "'b vec" and ws :: "'b vec list" where "x = n" and "xa = us" and "xb = w # ws" and "y = gram_schmidt_sub n ((adjuster n w us + w) # us) ws"
simps_2: fixes x :: "complex" shows "Im (- x) = - Im x"
extractFrameEqvt_1: fixes p :: "(name \<times> name) list"   and P :: "('a, 'b, 'c) psi" shows "p \<bullet> extractFrame P = extractFrame (p \<bullet> P)"
dim_eq_card: fixes B :: "('a, 'b) vec set"   and V :: "('a, 'b) vec set" assumes "vec.span B = vec.span V"   and "vec.independent B" shows "vec.dim V = card B"
lift_nat_def: fixes f :: "nat \<Rightarrow> nat" shows "lift_nat f = int \<circ> f \<circ> nat"
eq_swap_range: fixes ind :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and w\<^sub>1 :: "'a list"   and w\<^sub>2 :: "'a list" assumes "traces ind"   and "traces.eq_swap ind w\<^sub>1 w\<^sub>2" shows "set w\<^sub>1 = set w\<^sub>2"
extend_lin_independent_to_basis: fixes us :: "'b list" assumes "set us \<subseteq> V"   and "R_lin_independent us" shows "\<exists>vs. set (vs @ us) \<subseteq> V \<and> V = RSpan (vs @ us) \<and> R_lin_independent (vs @ us)"
in_components_self: fixes s :: "'d set" shows "(s \<in> components s) = (connected s \<and> s \<noteq> {})"
BrnFT_Int_BrnNFT_2: fixes cl :: "('test, 'atom, 'choice) cmd list"   and dl :: "('d, 'e, 'f) cmd list" assumes "length cl = length dl" shows "BrnNFT cl dl \<inter> BrnFT cl dl = {}"
llist_inf_le: fixes s :: "'a llist"   and A :: "'a set"   and t :: "'a llist" assumes "s \<in> A\<^sup>\<omega>" shows "(s \<le> t) = (s = t)"
induct: fixes P1 :: "'b ann_com \<Rightarrow> bool"   and P2 :: "'b com \<Rightarrow> bool"   and ann_com :: "'b ann_com"   and com :: "'b com" assumes "\<And>x1 x2. P1 (AnnBasic x1 x2)"   and "\<And>x1 x2. \<lbrakk>P1 x1; P1 x2\<rbrakk> \<Longrightarrow> P1 (AnnSeq x1 x2)"   and "\<And>x1 x2 x3 x4. \<lbrakk>P1 x3; P1 x4\<rbrakk> \<Longrightarrow> P1 (AnnCond1 x1 x2 x3 x4)"   and "\<And>x1 x2 x3. P1 x3 \<Longrightarrow> P1 (AnnCond2 x1 x2 x3)"   and "\<And>x1 x2 x3 x4. P1 x4 \<Longrightarrow> P1 (AnnWhile x1 x2 x3 x4)"   and "\<And>x1 x2 x3. P2 x3 \<Longrightarrow> P1 (AnnAwait x1 x2 x3)"   and "\<And>x. (\<And>xa xaa xaaa. \<lbrakk>xa \<in> set x; xaa \<in> Basic_BNFs.fsts xa; xaaa \<in> set_option xaa\<rbrakk> \<Longrightarrow> P1 xaaa) \<Longrightarrow> P2 (Parallel x)"   and "\<And>x. P2 (Basic x)"   and "\<And>x1 x2. \<lbrakk>P2 x1; P2 x2\<rbrakk> \<Longrightarrow> P2 (Seq x1 x2)"   and "\<And>x1 x2 x3. \<lbrakk>P2 x2; P2 x3\<rbrakk> \<Longrightarrow> P2 (Cond x1 x2 x3)"   and "\<And>x1 x2 x3. P2 x3 \<Longrightarrow> P2 (While x1 x2 x3)" shows "P1 ann_com \<and> P2 com"
assms_2: fixes theta :: "('a \<times> 'a) set" assumes "theta \<in> Theta" shows "bisim Retr theta"
nth_list_update_eq: fixes i :: "nat"   and xs :: "'e list"   and x :: "'e" assumes "i < length xs" shows "xs[i := x] ! i = x"
map_sel_2: fixes a :: "'a literal"   and f :: "'a \<Rightarrow> 'b" assumes "is_neg a" shows "atm_of (map_literal f a) = f (atm_of a)"
exp_conj_assoc: fixes P :: "'c \<Rightarrow> real"   and Q :: "'c \<Rightarrow> real"   and R :: "'c \<Rightarrow> real" assumes "unitary P"   and "unitary Q"   and "unitary R" shows "P && (Q && R) = P && Q && R"
chamber_facet_is_chamber_facet: fixes X :: "'b set set"   and C :: "'b set"   and D :: "'b set"   and z :: "'b set" assumes "ChamberComplex X"   and "SimplicialComplex.maxsimp X C"   and "SimplicialComplex.maxsimp X D"   and "z \<lhd> C"   and "order.greater_eq D z" shows "z \<lhd> D"
sigma_sets_Inter: fixes A :: "'c set set"   and sp :: "'c set"   and a :: "nat \<Rightarrow> 'c set" assumes "A \<subseteq> Pow sp"   and "\<And>i. a i \<in> sigma_sets sp A" shows "\<Inter> (range a) \<in> sigma_sets sp A"
monotone_def: fixes orda :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ordb :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'e" shows "monotone orda ordb f = (\<forall>x y. orda x y \<longrightarrow> ordb (f x) (f y))"
integral_completion: fixes f :: "'a \<Rightarrow> 'b"   and M :: "'a measure" assumes "f \<in> borel_measurable M" shows "integral\<^sup>L (completion M) f = integral\<^sup>L M f"
comp_cod_arr: fixes f :: "'s"   and b :: "'s" assumes "S.arr f"   and "S.cod f = b" shows "b \<cdot> f = f"
IP: fixes k :: "'d"   and it :: "'d set"   and v :: "'a"   and \<sigma> :: "'c" assumes "k \<in> it"   and "invar v"   and "m k = Some (\<alpha> v)"   and "it \<subseteq> dom m"   and "I it \<sigma>" shows "I (it - {k}) (f (k, v) \<sigma>)"
annul_mod_def: fixes b :: "bool"   and s :: "'b sparc_state" shows "annul_mod b s \<equiv> s\<lparr>state_var := write_annul b (state_var s)\<rparr>"
induct: fixes x1 :: "('s, 'v) tm"   and x2 :: "('s, 'v) tm"   and P :: "('s, 'v) tm \<Rightarrow> ('s, 'v) tm \<Rightarrow> bool" assumes "x1 >\<^sub>t x2"   and "\<And>t s. \<lbrakk>vars_mset s \<subseteq># vars_mset t; wt s < wt t\<rbrakk> \<Longrightarrow> P t s"   and "\<And>t s. \<lbrakk>wt t = wt s; \<not> head t \<le>\<ge>\<^sub>h\<^sub>d head s; num_args t = 1; \<exists>f\<in>local.ground_heads (head t). arity_sym f = 1 \<and> wt_sym f = 0; arg t >\<^sub>t s \<and> P (arg t) s \<or> arg t = s\<rbrakk> \<Longrightarrow> P t s"   and "\<And>t s. \<lbrakk>vars_mset s \<subseteq># vars_mset t; wt t = wt s; head t >\<^sub>h\<^sub>d head s\<rbrakk> \<Longrightarrow> P t s"   and "\<And>t s. \<lbrakk>vars_mset s \<subseteq># vars_mset t; wt t = wt s; head t = head s; \<forall>f\<in>local.ground_heads (head t). extf f (\<lambda>x1 x2. x1 >\<^sub>t x2 \<and> P x1 x2) (args t) (args s)\<rbrakk> \<Longrightarrow> P t s" shows "P x1 x2"
cod_comp: fixes g :: "'a1 \<times> 'a2"   and f :: "'a1 \<times> 'a2" assumes "A1xA2.seq g f" shows "A1xA2.cod (g \<cdot>\<^sub>A\<^sub>1\<^sub>x\<^sub>A\<^sub>2 f) = A1xA2.cod g"
semiattractive_subset: fixes A :: "'c set"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and B :: "'c set" assumes "semiattractive A less_eq"   and "B \<subseteq> A" shows "semiattractive B less_eq"
euclid_ext_i: fixes ops :: "'b arith_ops_record"   and R :: "'b \<Rightarrow> 'c \<Rightarrow> bool" assumes "euclidean_ring_ops ops R" shows "(R ===> R ===> rel_prod (rel_prod R R) R) (arith_ops.euclid_ext_aux_i ops (arith_ops_record.one ops) (arith_ops_record.zero ops) (arith_ops_record.zero ops) (arith_ops_record.one ops)) (euclid_ext_aux (1::'c) (0::'c) (0::'c) (1::'c))"
C: shows "C \<subseteq> {x. ennreal (1 / real (Suc n)) \<le> f x}"
a_assoc: fixes x :: "nat \<Rightarrow> int"   and y :: "nat \<Rightarrow> int"   and z :: "nat \<Rightarrow> int" assumes "x \<in> carrier (UP INTEG)"   and "y \<in> carrier (UP INTEG)"   and "z \<in> carrier (UP INTEG)" shows "x \<oplus>\<^bsub>UP INTEG\<^esub> y \<oplus>\<^bsub>UP INTEG\<^esub> z = x \<oplus>\<^bsub>UP INTEG\<^esub> (y \<oplus>\<^bsub>UP INTEG\<^esub> z)"
mod_add_right_eq: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size" shows "modulo (plus a (modulo b c)) c = modulo (plus a b) c"
lcoset_el_reduce: fixes a :: "'g" assumes "a \<in> G" shows "a +o G = G"
simps_2: fixes f :: "('fsym, 'tp) efsym"   and Tl :: "('fsym, 'tp) efsym trm list" shows "GE.wt (Fn f Tl) = (GE_wtFsym f \<and> list_all GE.wt Tl \<and> GE_arOf f = map GE.tpOf Tl)"