hcju/isabelleps huggingface dataset:
assms: fixes n :: "nat" assumes "n < length ias" shows "interrupt_action_ok (redT_updIs is (take n ias)) (ias ! n)"
\<phi>_def: shows "\<phi> = lucas_lehmer_hom M"
Bk: fixes x :: "'a" assumes "x \<in> S"   and "norm (f x - f a) < k" shows "norm (x - a) \<le> B * norm (f x - f a)"
length_pivot_positions_dim_row: fixes A :: "'a mat" assumes "row_echelon_form A" shows "length (pivot_positions A) \<le> dim_row A"
cases: fixes ci :: "('addr, 'heap) check_instr"   and P :: "(nat \<times> nat \<times> 'addr instr list \<times> (nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list) prog"   and ins :: "'addr instr list"   and xt :: "(nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list"   and t :: "'thread_id"   and a1 :: "'heap"   and a2 :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option"   and a3 :: "('addr, 'thread_id, 'heap) jvm_thread_action"   and a4 :: "'heap"   and a5 :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option" assumes "exec_meth ci P ins xt t a1 a2 a3 a4 a5" obtains ta :: "('addr, 'thread_id, 'heap) jvm_thread_action" and xcp :: "'addr option" and h' :: "'heap" and stk' :: "'addr val list" and loc' :: "'addr val list" and pc' :: "nat" and pc :: "nat" and h :: "'heap" and stk :: "'addr val list" and loc :: "'addr val list" where "a1 = ha__" and "a2 = (stka__, loca__, pca__, None)" and "a3 = ta" and "a4 = h'" and "a5 = (stk', loc', pc'a__, xcpb__)" and "(ta, xcpb__, h', [(stk', loc', undefined, undefined, pc'a__)]) \<in> exec_instr (ins ! pca__) P t ha__ stka__ loca__ undefined undefined pca__ []" and "pca__ < length ins" and "ci_app ci (ins ! pca__) P ha__ stka__ loca__ undefined undefined pca__ []"   | h :: "'heap" and xcp :: "'addr" and pc :: "nat" and pc' :: "nat" and d :: "nat" and stk :: "'addr val list" and loc :: "'addr val list" where "a1 = h" and "a2 = (stk, loc, pc, \<lfloor>xcp\<rfloor>)" and "a3 = \<lbrace>\<rbrace>" and "a4 = h" and "a5 = (Addr xcp # drop (length stk - d) stk, loc, pc', None)" and "match_ex_table P (cname_of h xcp) pc xt = \<lfloor>(pc', d)\<rfloor>" and "d \<le> length stk"
cr_option_gpv_def: fixes x :: "'g option"   and gpv :: "('g, 'h, 'i) gpv" shows "cr_option_gpv x gpv = (gpv = (lift_spmf \<circ> return_pmf) x)"
at_left_to_top': fixes c :: "real" shows "at_left c = filtermap (\<lambda>x. c - inverse x) at_top"
strip_iter_widen: fixes f :: "'a acom \<Rightarrow> 'a acom"   and c :: "'a acom"   and c' :: "'a acom" assumes "\<forall>c. strip (f c) = strip c"   and "iter_widen f c = Some c'" shows "strip c' = strip c"
useNodes_of_def: fixes g :: "'g" shows "useNodes_of g = fold (addN g) (\<alpha>n g) Mapping.empty"
concurrentD1: fixes s1 :: "'a \<times> ('a, 'b) operation"   and s2 :: "'a \<times> ('a, 'b) operation" assumes "hb.concurrent s1 s2" shows "\<not> hb s1 s2"
op_map_empty_def: shows "op_map_empty \<equiv> Map.empty"
stable_translation_length_eq_additive_strength_aux: fixes f :: "'b \<Rightarrow> 'b"   and xi :: "'b Gromov_completion" assumes "isometry f"   and "Gromov_extension f xi = xi"   and "0 \<le> additive_strength f xi" shows "stable_translation_length f = additive_strength f xi"
F_image_mset_product: fixes g :: "'e \<Rightarrow> 'd \<Rightarrow> 'a"   and x :: "'e"   and A :: "'e multiset"   and B :: "'d multiset" shows "F {#g x j \<^bold>* F {#g i j. i \<in># A#}. j \<in># B#} = F (image_mset (g x) B) \<^bold>* F {#F {#g i j. i \<in># A#}. j \<in># B#}"
select_true: fixes n :: "i32"   and v1 :: "v"   and v2 :: "v" assumes "\<not> int_eq n 0" shows "\<lparr>[$C v1, $C v2, $C ConstInt32 n, $Select]\<rparr> \<leadsto> \<lparr>[$C v1]\<rparr>"
dp_aux4: fixes z :: "'a"   and w :: "'a"   and x :: "'a"   and y :: "'a" assumes "is_p_fun z"   and "is_total w"   and "z\<^sup>\<smile> ; x = 1" shows "(y ; z\<^sup>\<smile> \<cdot> w ; x\<^sup>\<smile>) ; z = y"
executable_visible_sticky: shows "executable \<inter> visible \<subseteq> sticky"
preserves_ide: fixes a :: "'a" assumes "B.ide a" shows "B.ide (src\<^sub>B a)"
DERIV_mult': fixes f :: "'a \<Rightarrow> 'a"   and D :: "'a"   and x :: "'a"   and s :: "'a set"   and g :: "'a \<Rightarrow> 'a"   and E :: "'a" assumes "(f has_field_derivative D) (at x within s)"   and "(g has_field_derivative E) (at x within s)" shows "((\<lambda>x. f x * g x) has_field_derivative f x * E + D * g x) (at x within s)"
i_take_eq_Nil: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'b" shows "(f \<Down> n = []) = (n = 0)"
can_lunit: fixes a :: "'c local.term set" assumes "\<F>C.ide a" shows "\<F>C.can \<l>[a]"
has_integral_empty_eq: fixes f :: "'c \<Rightarrow> 'd"   and i :: "'d" shows "(f has_integral i) {} = (i = (0::'d))"
comp_id: fixes f :: "'n \<Rightarrow> 'o" shows "f \<circ> id = f"
iterates: fixes f :: "'b \<Rightarrow> 'b"   and x :: "'b" shows "iterates f x = x ## iterates f (f x)"
Red_I_def: fixes N :: "('f \<times> 'l) set" shows "Red_I_\<G> N = (\<Inter>q\<in>Q. Red_I_\<G>_q q N)"
arr_cod_iff_arr: fixes f :: "('a, 'a set) Maps.arr" shows "Maps.arr (Maps.cod f) = Maps.arr f"
power_eq_abs: fixes a :: "'a"   and k :: "nat"   and b :: "'a" assumes "a ^ k = b ^ k"   and "0 < k" shows "\<bar>a\<bar> = \<bar>b\<bar>"
WT2: shows "\<I>2, \<I>2' \<turnstile>\<^sub>C conv2 \<surd>"
la_ex'_def: fixes fs :: "local_state \<times> global_state"   and a :: "action" shows "la_ex' fs a \<equiv> case la_ex fs a of None \<Rightarrow> fs | Some fs' \<Rightarrow> fs'"
DiffD1: fixes c :: "'c"   and A :: "'c set"   and B :: "'c set" assumes "c \<in> A - B" shows "c \<in> A"
ctxt_subst_append: fixes \<Delta> :: "binding list"   and \<Gamma> :: "binding list"   and X :: "tyvrs"   and T :: "ty" shows "(\<Delta> @ \<Gamma>)[X \<mapsto> T]\<^sub>e = \<Delta>[X \<mapsto> T]\<^sub>e @ \<Gamma>[X \<mapsto> T]\<^sub>e"
simps_8: fixes v :: "Cooper.num" shows "not (Eq v) = NOT (Eq v)"
span_li_not_depend_1: fixes S :: "'c set"   and N :: "'c set" assumes "S \<subseteq> N"   and "LinearCombinations.submodule K N V" shows "LinearCombinations.module.span K (V.vs N) S = V.module.span S"
Int_Un_eq_3: fixes S :: "'a set"   and T :: "'a set" shows "S \<union> S \<inter> T = S"
lookup_default_neq: fixes x :: "'b"   and y :: "'b"   and d :: "'c"   and m :: "('b, 'c) mapping" assumes "x \<noteq> y" shows "lookup (Mapping.default x d m) y = lookup m y"
jointgd_tool4: fixes n :: "nat"   and i :: "nat" assumes "n < i" shows "i - n = Suc (i - Suc n)"
consistent_pres_op_def: fixes op :: "'a \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> 'b" shows "consistent_pres_op op \<equiv> case op of (name, pres, effs, cost) \<Rightarrow> distinct (map fst (pres @ implicit_pres effs)) \<and> consistent_map_lists pres (implicit_pres effs)"
assoc_mult_mat: fixes A :: "'c mat"   and n\<^sub>1 :: "nat"   and n\<^sub>2 :: "nat"   and B :: "'c mat"   and n\<^sub>3 :: "nat"   and C :: "'c mat"   and n\<^sub>4 :: "nat" assumes "A \<in> carrier_mat n\<^sub>1 n\<^sub>2"   and "B \<in> carrier_mat n\<^sub>2 n\<^sub>3"   and "C \<in> carrier_mat n\<^sub>3 n\<^sub>4" shows "A * B * C = A * (B * C)"
scaleC_conv_of_complex: fixes r :: "complex"   and x :: "'b" shows "r *\<^sub>C x = of_complex r * x"
UNPROTECT_def: fixes x :: "'a" shows "UNPROTECT x \<equiv> x"
bijective_closed_imp_homeomorphic_map: fixes X :: "'c topology"   and Y :: "'d topology"   and f :: "'c \<Rightarrow> 'd" assumes "continuous_map X Y f"   and "closed_map X Y f"   and "f ` topspace X = topspace Y"   and "inj_on f (topspace X)" shows "homeomorphic_map X Y f"
pfa_butlast_def: shows "pfa_butlast \<equiv> \<lambda>(a, n). return (a, n - 1)"
ereal_x_minus_x: fixes x :: "ereal" shows "x - x = (if \<bar>x\<bar> = \<infinity> then \<infinity> else 0)"
cross_alt_poly_commute: fixes p :: "real poly"   and q :: "real poly"   and a :: "real"   and b :: "real" shows "cross_alt p q a b = cross_alt q p a b"
assms_2: shows "r = (SOME r'. recfn 1 r' \<and> Partial_Recursive.total r' \<and> f = (\<lambda>x. eval r' [x]))"
soundness: fixes p :: "'b fm"   and M :: "('b, 'c) kripke"   and s :: "'c" assumes "\<turnstile> p" shows "M, s \<Turnstile> p"
deriv_g: fixes s :: "complex"   and B :: "complex set" assumes "s0 < ereal (Re s)" shows "(g has_field_derivative eval_fds (fds_deriv f) s / eval_fds f s) (at s within B)"
seqI': fixes f :: "'a \<times> 'a"   and a :: "'a \<times> 'a"   and b :: "'a \<times> 'a"   and g :: "'a \<times> 'a"   and c :: "'a \<times> 'a" assumes "VV.in_hom f a b"   and "VV.in_hom g b c" shows "VV.seq g f"
diff_mod_0_imp_mod_eq: fixes b :: "nat"   and a :: "nat"   and m :: "nat" assumes "(b - a) mod m = 0"   and "a \<le> b" shows "a mod m = b mod m"
unfold_congs_8: fixes r :: "('b, 'c) sparc_state_scheme"   and r' :: "('b, 'c) sparc_state_scheme"   and v' :: "sparc_state_var"   and f :: "sparc_state_var \<Rightarrow> sparc_state_var"   and f' :: "sparc_state_var \<Rightarrow> sparc_state_var" assumes "r = r'"   and "state_var r' = v'"   and "\<And>v. v = v' \<Longrightarrow> f v = f' v" shows "state_var_update f r = state_var_update f' r'"
dfs2_invariant: fixes g :: "(node \<times> node) list"   and xs :: "node list"   and ys :: "node list" assumes "dfs2_dom (g, xs, ys)" shows "set ys \<subseteq> set (dfs2 g xs ys)"
a1: fixes r :: "'d rexp"   and s :: "'d rexp"   and t :: "'d rexp" shows "Alt (Alt r s) t ~ Alt r (Alt s t)"
eqTrueI: fixes P :: "bool" assumes "P" shows "P = True"
mkVarChannel_cl_inv: fixes g :: "'a gState_scheme"   and p :: "pState"   and v :: "varDecl"   and upd :: "((String.literal, variable) assoc_list \<Rightarrow> (String.literal, variable) assoc_list) \<Rightarrow> 'a gState_scheme \<times> pState \<Rightarrow> 'a gState_scheme \<times> pState" assumes "cl_inv (g, p)"   and "\<And>k v' cs. toVariable g p v = (k, v', cs) \<Longrightarrow> cl_inv (upd (lm.update k v') (g, p))" shows "cl_inv (mkVarChannel v upd g p)"
intros_2: fixes ck :: "bool"   and s1 :: "v sem_env"   and env :: "'b state"   and e :: "exp0"   and s2 :: "'b state"   and v1 :: "v" assumes "evaluate ck s1 env e (s2, Rval v1)" shows "evaluate ck s1 env (Raise e) (s2, Rerr (Rraise v1))"
AbGroup_subset_sum_list: fixes Gs :: "'a set list"   and H :: "'a set" assumes "\<forall>G\<in>set Gs. AbGroup G"   and "H \<in> set Gs" shows "H \<subseteq> (\<Sum>G\<leftarrow>Gs. G)"
fw_temp_def: shows "first_world \<equiv> \<lambda>sigma. sigma"
simps_1: fixes uu :: "'a option" shows "None + uu = None"
f1: shows "Broadcast (r, Expunge e mo r) \<in> set (history (oid (r, Expunge e mo r)))"
\<psi>_5: fixes p :: "'a" assumes "p \<in> carrier" shows "apply_chart (\<psi> p) p = (0::'b)"
simps_3: fixes indis :: "('a \<times> 'a) set"   and tval :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'a \<Rightarrow> 'a"   and tst :: "'b"   and c1 :: "('b, 'c) com"   and c2 :: "('b, 'c) com" assumes "PL_Indis indis" shows "PL_Indis.SC_siso tval aval indis (if tst then c1 else c2) = (PL_Indis.compatTst tval indis tst \<and> PL_Indis.SC_siso tval aval indis c1 \<and> PL_Indis.SC_siso tval aval indis c2)"
COND_REF: fixes x :: "'b"   and x' :: "'a" assumes "(x, x') \<in> R"   and "I x"   and "I' x'" shows "b x = b' x'"
sel_1: fixes x21 :: "'a"   and x22 :: "'a msllist" shows "un_MSLCons1 (MSLCons x21 x22) = x21"
word_reverse_eq_of_bl_rev_to_bl: fixes w :: "'b word" shows "word_reverse w = of_bl (rev (to_bl w))"
comps_fixpointwise_imp_bij_betw: fixes f :: "'c \<Rightarrow> 'b"   and X :: "'c set"   and Y :: "'b set"   and g :: "'b \<Rightarrow> 'c" assumes "order.greater_eq Y (f ` X)"   and "order.greater_eq X (g ` Y)"   and "fixespointwise (g \<circ> f) X"   and "fixespointwise (f \<circ> g) Y" shows "bij_betw f X Y"
ccw'_sortedP_translate_comp: fixes x0 :: "real \<times> real"   and x :: "real \<times> real"   and g :: "'a \<Rightarrow> real \<times> real"   and xs :: "'a list" shows "ccw'.sortedP x0 (map ((+) x \<circ> g) xs) = ccw'.sortedP (x0 - x) (map g xs)"
collect_pseudominimals_below_in_poset: fixes P :: "'a set"   and x :: "'a"   and f :: "'a \<Rightarrow> 'c set"   and A :: "'c set"   and a :: "'a set" assumes "OrderingSetIso (\<^bold>\<le>) (\<^bold><) dual_order.greater_eq dual_order.greater (P .\<^bold>\<le> x) f"   and "f ` (P .\<^bold>\<le> x) = Pow A"   and "order.greater_eq {y. pseudominimal_in (P .\<^bold>\<le> x) y} a" shows "the_inv_into (P .\<^bold>\<le> x) f (\<Union> (f ` a)) \<in> P"
preserve_tranclp: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and S :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b"   and x :: "'a"   and y :: "'a" assumes "\<And>x y. R x y \<Longrightarrow> S (f x) (f y)"   and "R\<^sup>+\<^sup>+ x y" shows "S\<^sup>+\<^sup>+ (f x) (f y)"
assms_3: shows "\<not> (a\<^sub>e < a\<^sub>o \<and> other.s' \<delta> < v\<^sub>e \<and> v\<^sub>e - a\<^sub>e / a\<^sub>o * other.s' \<delta> < 0)"
get_offending_flows_union_mono: fixes M :: "'b SecurityInvariant_configured list"   and V :: "'b set"   and E :: "('b \<times> 'b) set"   and E' :: "('b \<times> 'b) set" assumes "valid_reqs M"   and "wf_graph \<lparr>nodes = V, edges = E\<rparr>"   and "E' \<subseteq> E" shows "\<Union> (get_offending_flows M \<lparr>nodes = V, edges = E'\<rparr>) \<subseteq> \<Union> (get_offending_flows M \<lparr>nodes = V, edges = E\<rparr>)"
is_shadow_root_ptr_kind_none: fixes document_ptr :: "(_) document_ptr" assumes "\<not> is_shadow_root_ptr_kind document_ptr" shows "cast document_ptr = None"
trg_trg: fixes \<mu> :: "'a arrow_of_spans_data" assumes "arr \<mu>" shows "trg (trg \<mu>) = trg \<mu>"
coerce_prod: shows "COERCE('e \<times> 'f, 'g \<times> 'h) = prod_map\<cdot>COERCE('e, 'g)\<cdot>COERCE('f, 'h)"
discriminant_cubic_depressed: fixes x :: "'a"   and y :: "'a"   and z :: "'a"   and f :: "'a"   and e :: "'a" assumes "[:- x, 1::'a:] * [:- y, 1::'a:] * [:- z, 1::'a:] = [:f, e, 0::'a, 1::'a:]" shows "discriminant_cubic_depressed e f = (x - y)\<^sup>2 * (x - z)\<^sup>2 * (y - z)\<^sup>2"
mat_of_rows_carrier_1: fixes n :: "nat"   and vs :: "'c vec list" shows "mat_of_rows n vs \<in> carrier_mat (length vs) n"
simps: fixes A :: "('a list \<Rightarrow> 'b) \<times> ('a list \<times> 'b \<Rightarrow> 'a \<Rightarrow> (nat \<times> nat list) \<times> 'b)" shows "Partial_Cost_Model.embed A = (\<lambda>s. return_pmf (fst A s), \<lambda>s r. return_pmf (snd A s r))"
basic_trans_rules_23: fixes x :: "'c"   and y :: "'c"   and z :: "'c" assumes "x \<le> y"   and "y \<le> z" shows "x \<le> z"
rep_eq: fixes x :: "'c \<Rightarrow>\<^sub>0 'b" shows "Poly_Mapping.range x = range (lookup x) - {0::'b}"
case_resumption_Terminal: fixes t :: "'g \<Rightarrow> 'f"   and l :: "'h \<Rightarrow> ('g, 'h, 'i, 'j) resumption \<Rightarrow> 'f"   and br :: "'i \<Rightarrow> ('j \<Rightarrow> ('g, 'h, 'i, 'j) resumption) \<Rightarrow> 'f"   and a :: "'g" shows "(case Terminal a of Terminal x \<Rightarrow> t x | Linear x xa \<Rightarrow> l x xa | Branch x xa \<Rightarrow> br x xa) = t a"
scc_of_eq: fixes u :: "'a"   and v :: "'a" assumes "u \<in> scc_of v" shows "scc_of u = scc_of v"
iso_class_rep_def: fixes F :: "'b set" shows "VoV.iso_class_rep F \<equiv> SOME f. f \<in> F"
e: shows "\<exists>\<alpha>\<in>{0..1}. p \<approx>[\<R>] mix_pmf \<alpha> local.some_best local.some_worst"
get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_ptr_in_heap: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and h :: "(_) heap"   and ptr :: "(_) element_ptr"   and getter :: "('n, 'l, 'm, 'p, 'r option) RElement_scheme \<Rightarrow> 'u" assumes "l_get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_lemmas type_wf"   and "h \<turnstile> ok get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr getter" shows "ptr |\<in>| element_ptr_kinds h"
1: shows "r\<^sup>T \<le> r\<^sup>T * prim_W w v e\<^sup>\<star>"
pred_entailment_def: fixes ev\<^sub>A :: "('a \<Rightarrow> 'b) \<Rightarrow> 'c \<Rightarrow> 'b"   and ev\<^sub>B :: "('a \<Rightarrow> 'b) \<Rightarrow> 'd \<Rightarrow> bool"   and aexp_vars :: "'c \<Rightarrow> 'a set"   and bexp_vars :: "'d \<Rightarrow> 'a set"   and dma :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> Sec"   and \<C>_vars :: "'a \<Rightarrow> 'a set"   and bexp_neg :: "'d \<Rightarrow> 'd"   and assign_post :: "'d set \<Rightarrow> 'a \<Rightarrow> 'c \<Rightarrow> 'd set"   and dma_type :: "'a \<Rightarrow> 'd set"   and pred_False :: "'d"   and P :: "'d set"   and P' :: "'d set" assumes "sifum_types ev\<^sub>A ev\<^sub>B aexp_vars bexp_vars dma \<C>_vars bexp_neg assign_post dma_type pred_False" shows "sifum_types.pred_entailment ev\<^sub>B P P' \<equiv> \<forall>mem. sifum_lang_no_dma.pred ev\<^sub>B P mem \<longrightarrow> sifum_lang_no_dma.pred ev\<^sub>B P' mem"
symP_def: fixes \<phi> :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "symP \<phi> \<equiv> \<forall>x y. \<phi> x y \<longrightarrow> \<phi> y x"
alw_coinduct: fixes X :: "'c stream \<Rightarrow> bool"   and x :: "'c stream"   and \<phi> :: "'c stream \<Rightarrow> bool" assumes "X x"   and "\<And>x. X x \<Longrightarrow> \<phi> x"   and "\<And>x. \<lbrakk>X x; \<not> alw \<phi> (stl x)\<rbrakk> \<Longrightarrow> X (stl x)" shows "alw \<phi> x"
i_shrink_nth_eq_i_last_message_hold_nth: fixes k :: "nat"   and f :: "nat \<Rightarrow> 'b message_af"   and n :: "nat" assumes "0 < k" shows "(f \<div> k) n = (f \<longmapsto> k) (n * k + k - Suc 0)"
card_braun_indices: fixes t :: "'a tree" shows "card (braun_indices t) = size t"
rel_inject_1: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and x1 :: "'b"   and y1 :: "'c" shows "rel_exp R (Var x1) (Var y1) = R x1 y1"
dist: fixes x :: "'a"   and y :: "'a" assumes "P x \<and> P y" shows "dist (f x) (f y) < e"
tllist_all2_TCons2: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and ts' :: "('a, 'c) tllist"   and x :: "'b"   and ts :: "('b, 'd) tllist" shows "tllist_all2 P Q ts' (TCons x ts) = (\<exists>x' ts''. ts' = TCons x' ts'' \<and> P x' x \<and> tllist_all2 P Q ts'' ts)"
below: fixes d :: "'a \<rightarrow> 'a"   and x :: "'a" assumes "deflation d" shows "d\<cdot>x \<sqsubseteq> x"
simps_15: fixes ip :: "nat"   and sn :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option"   and rreqs :: "(nat \<times> nat) set"   and store :: "nat \<Rightarrow> (p \<times> nat list) option"   and msg :: "msg"   and data :: "nat"   and dests :: "nat \<Rightarrow> nat option"   and dip :: "nat"   and oip :: "nat"   and hops :: "nat"   and dsn :: "nat"   and dsk :: "k"   and osn :: "nat"   and sip :: "nat"   and handled :: "bool"   and more :: "'a" shows "sip \<lparr>ip = ip, sn = sn, rt = rt, rreqs = rreqs, store = store, msg = msg, data = data, dests = dests, dip = dip, oip = oip, hops = hops, dsn = dsn, dsk = dsk, osn = osn, sip = sip, handled = handled, \<dots> = more\<rparr> = sip"
inject: fixes x1 :: "node fset"   and x2 :: "node \<Rightarrow> assertion_gadget"   and x3 :: "(node fset \<times> command_gadget \<times> node fset) list"   and y1 :: "node fset"   and y2 :: "node \<Rightarrow> assertion_gadget"   and y3 :: "(node fset \<times> command_gadget \<times> node fset) list" shows "(Graph x1 x2 x3 = Graph y1 y2 y3) = (x1 = y1 \<and> x2 = y2 \<and> x3 = y3)"
proj2_rep_def: fixes x :: "proj2" shows "proj2_rep x \<equiv> SOME v. v \<in> Rep_proj2 x"
llength_ltake': fixes n :: "nat"   and xs :: "'a llist" assumes "enat n < llength xs" shows "llength (ltake (enat n) xs) = enat n"
WHILEI_refine_genR: fixes I' :: "'c \<Rightarrow> bool"   and x' :: "'c"   and x :: "'d"   and R' :: "('d \<times> 'c) set"   and I :: "'d \<Rightarrow> bool"   and b :: "'d \<Rightarrow> bool"   and b' :: "'c \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'd nres"   and f' :: "'c \<Rightarrow> 'c nres"   and R :: "('d \<times> 'c) set" assumes "I' x' \<Longrightarrow> (x, x') \<in> R'"   and "\<And>x x'. \<lbrakk>(x, x') \<in> R'; I' x'\<rbrakk> \<Longrightarrow> I x"   and "\<And>x x'. \<lbrakk>(x, x') \<in> R'; I x; I' x'\<rbrakk> \<Longrightarrow> b x = b' x'"   and "\<And>x x'. \<lbrakk>(x, x') \<in> R'; b x; b' x'; I x; I' x'\<rbrakk> \<Longrightarrow> f x \<le> \<Down> R' (f' x')"   and "\<And>x x'. \<lbrakk>(x, x') \<in> R'; \<not> b x; \<not> b' x'; I x; I' x'\<rbrakk> \<Longrightarrow> (x, x') \<in> R" shows "WHILE\<^bsup>I\<^esup> b f x \<le> \<Down> R (WHILE\<^bsup>I'\<^esup> b' f' x')"
Domainp_eq_Collect: fixes A :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and \<UU> :: "'d set" shows "(Domainp A = (\<lambda>x. x \<in> \<UU>)) = (\<UU> = Collect (Domainp A))"
L_split_sup: fixes x :: "'a"   and y :: "'a" shows "x * (y \<squnion> L) \<le> x * y \<squnion> L"
cont2cont_lambda: fixes f :: "'c \<Rightarrow> 'b \<Rightarrow> 'd" assumes "\<And>y. cont (\<lambda>x. f x y)" shows "cont f"
nonz_q: shows "\<forall>q\<in>set qs. q \<noteq> 0"
assoc'_simps_4: fixes f :: "'b"   and g :: "'b"   and h :: "'b" assumes "D.ide f"   and "D.ide g"   and "D.ide h"   and "src\<^sub>D f = trg\<^sub>D g"   and "src\<^sub>D g = trg\<^sub>D h" shows "D.dom \<a>\<^sub>D\<^sup>-\<^sup>1[f, g, h] = D.dom f \<star>\<^sub>D D.dom g \<star>\<^sub>D D.dom h"
leg1_composite_1: shows "H\<mu>H\<nu>\<pi>.leg1 = \<mu>.leg1 \<cdot> \<mu>\<nu>\<pi>.Prj\<^sub>1"
bisimStructCong: fixes P :: "ccs"   and Q :: "ccs" assumes "P \<equiv>\<^sub>s Q" shows "P \<sim> Q"
ts\<^sub>d_def: shows "ts\<^sub>d \<equiv> map (\<lambda>(p, is, \<theta>, sb, \<D>, \<O>, \<R>'). (p, is, \<theta>, sb, \<D>, \<O>, Map.empty)) ts"
S_def: shows "S = four_block_mat (Matrix.mat 1 1 (\<lambda>(a, b). H $$ (0, 0))) (0\<^sub>m 1 (dim_col A - 1)) (0\<^sub>m (dim_row A - 1) 1) S'"
pdevs_apply_trunc_pdevs: fixes p :: "nat"   and X :: "'a pdevs"   and n :: "nat" shows "pdevs_apply (trunc_pdevs p X) n = eucl_truncate_down p (pdevs_apply X n)"
IH: fixes res :: "('a, 'b) resource"   and res' :: "('a, 'b) resource"   and gpv :: "('c, 'a, 'b) gpv" assumes "outs_\<I> \<I> \<turnstile>\<^sub>R res \<sim> res'"   and "\<I> \<turnstile>g gpv \<surd>"   and "\<I> \<turnstile>res res \<surd>" shows "rel_spmf (rel_prod (=) (eq_resource_on (outs_\<I> \<I>))) (exec_gpv' gpv res) (exec_gpv' gpv res')"
apply_power_eq_iff: fixes f :: "'b perm"   and m :: "nat"   and a :: "'b"   and n :: "nat" shows "(apply (f ^ m) a = apply (f ^ n) a) = (m mod order f a = n mod order f a)"
less_sets_def: fixes A :: "'a set"   and B :: "'a set" shows "A \<lless> B \<equiv> \<forall>x\<in>A. \<forall>y\<in>B. x < y"
Ex: fixes j :: "name"   and i :: "name"   and x :: "tm"   and A :: "fm" assumes "atom j \<sharp> (i, x)" shows "(SyntaxN.Ex j A)(i::=x) = SyntaxN.Ex j (A(i::=x))"
rev_below_trans: fixes below :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and y :: "'b"   and z :: "'b"   and x :: "'b" assumes "class.po below"   and "below y z"   and "below x y" shows "below x z"
inf_greatest: fixes A :: "'a set"   and z :: "'a" assumes "A \<subseteq> carrier L"   and "z \<in> carrier L"   and "\<And>x. x \<in> A \<Longrightarrow> z \<sqsubseteq> x" shows "z \<sqsubseteq> \<Sqinter>A"
phl_n_while: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "\<lbrace>!x \<cdot> !y\<rbrace>z\<lbrace>!x\<rbrace>" shows "\<lbrace>!x\<rbrace>(!y \<cdot> z)\<^sup>\<star> \<cdot> t y\<lbrace>!x \<cdot> t y\<rbrace>"
hyps: assumes "\<And>t'. Join t' \<in> set cas \<Longrightarrow> \<not> not_final_thread s t' \<and> t \<noteq> t'" shows "cond_action_oks s t cas"
is_inf_minimum: fixes x :: "'b"   and y :: "'b" shows "is_inf x y (minimum x y)"
comp_mapping_complement: fixes x :: "'a"   and y :: "'a" assumes "mapping x" shows "x * - y = - (x * y)"
in_set_spmf_iff_spmf: fixes x :: "'h"   and p :: "'h spmf" shows "(x \<in> set_spmf p) = (spmf p x \<noteq> 0)"
source: fixes S :: "'a"   and Enc :: "'a \<Rightarrow> 'b" shows "(SourceTerm S, SourceTerm S) \<in> encoding.indRelRPO Enc"
convex_halfspace_gt: fixes b :: "real"   and a :: "'c" shows "convex {x. b < a \<bullet> x}"
form_of_lit_is_nnf: fixes x :: "'b literal" shows "is_nnf (form_of_lit x)"
simps: fixes a1 :: "'a sequent list \<times> 'a sequent"   and a2 :: "'a form" shows "leftPrincipal a1 a2 = (\<exists>C F Fs Ps. a1 = (Ps, C) \<and> a2 = Compound F Fs \<and> C = ( \<LM> Compound F Fs  \<RM> \<Rightarrow>* Em))"