hcju/isabelleps huggingface dataset:
ss: fixes n :: "'b" assumes "n \<in> N" shows "sum f1 (B1 n) = b2 n"
do_store_action_def: fixes wact :: "('a, 'c, 'd) mem_store_action" shows "do_store_action wact = (\<lambda>s. case wact of mw_Mark r gc_mark \<Rightarrow> s\<lparr>heap := (heap s)(r := map_option (obj_mark_update (\<lambda>_. gc_mark)) (heap s r))\<rparr> | mw_Mutate r f new_r \<Rightarrow> s\<lparr>heap := (heap s)(r := map_option (\<lambda>obj. obj\<lparr>obj_fields := (obj_fields obj)(f := new_r)\<rparr>) (heap s r))\<rparr> | mw_Mutate_Payload r f pl \<Rightarrow> s\<lparr>heap := (heap s)(r := map_option (\<lambda>obj. obj\<lparr>obj_payload := (obj_payload obj)(f := pl)\<rparr>) (heap s r))\<rparr> | mw_fA gc_mark \<Rightarrow> s\<lparr>fA := gc_mark\<rparr> | mw_fM gc_mark \<Rightarrow> s\<lparr>fM := gc_mark\<rparr> | mw_Phase gc_phase \<Rightarrow> s\<lparr>phase := gc_phase\<rparr>)"
span_is_subset: fixes R :: "('b, 'c) ring_scheme"   and M :: "('b, 'd, 'e) module_scheme"   and S :: "'d set"   and N :: "'d set" assumes "module R M"   and "S \<subseteq> N"   and "LinearCombinations.submodule R N M" shows "LinearCombinations.module.span R M S \<subseteq> N"
ignore: fixes y :: "'b"   and s' :: "'s"   and s :: "'s"   and x :: "'a" assumes "(y, s') \<in> set_spmf (callee s x)"   and "I s"   and "\<not> consider x"   and "x \<in> outs_\<I> \<I>" shows "count s' \<le> count s"
foreachc_reflexive_transitive: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and c :: "'c \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> bool"   and S :: "'d set"   and a\<^sub>1 :: "'c"   and a\<^sub>2 :: "'c" assumes "reflp r"   and "transp r"   and "foreach\<^sub>C c f S a\<^sub>1 a\<^sub>2"   and "\<And>x a\<^sub>1 a\<^sub>2. \<lbrakk>x \<in> S; f x a\<^sub>1 a\<^sub>2\<rbrakk> \<Longrightarrow> r a\<^sub>1 a\<^sub>2" shows "r a\<^sub>1 a\<^sub>2"
assms: shows "FlexRayArch n nReturn nC nStore nGet"
lunit_simps_5: fixes f :: "'d" assumes "D.ide f" shows "D.cod \<l>\<^sub>D[f] = f"
funcset_mem: fixes f :: "'e \<Rightarrow> 'f"   and A :: "'e set"   and B :: "'f set"   and x :: "'e" assumes "f \<in> A \<rightarrow> B"   and "x \<in> A" shows "f x \<in> B"
step_sim: fixes a :: "'a"   and a' :: "'a"   and b :: "'b" assumes "(a, a') \<in> a.E"   and "(a, b) \<in> R" shows "\<exists>b'. (b, b') \<in> b.E \<and> (a', b') \<in> R"
oalist_inv_raw_update_by_fun_gr_pair: fixes xs :: "('a \<times> 'b) list"   and k :: "'a"   and f :: "'b \<Rightarrow> 'b" assumes "tc.oalist_inv_raw xs" shows "tc.oalist_inv_raw (update_by_fun_gr_pair_tc k f xs)"
evalOP_closed: fixes P :: "db"   and V :: "db" assumes "P \<Down> V"   and "closed P" shows "closed V"
EF: shows "is_sentence (\<alpha> @ \<delta>) \<and> is_sentence (\<beta> @ \<delta>') \<and> LeftDerivation (\<alpha> @ \<delta>) D (\<beta> @ \<delta>') \<and> i < length (\<alpha> @ \<delta>) \<and> j < length (\<beta> @ \<delta>') \<and> (\<alpha> @ \<delta>) ! i = (\<beta> @ \<delta>') ! j \<and> D = E @ derivation_shift F 0 (Suc j) \<and> LeftDerivation (take i (\<alpha> @ \<delta>)) E (take j (\<beta> @ \<delta>')) \<and> LeftDerivation (drop (Suc i) (\<alpha> @ \<delta>)) F (drop (Suc j) (\<beta> @ \<delta>'))"
normDH1_1: shows "poly_mod.Mp p D1 = D1"
simps_4: fixes f1 :: "'c"   and f2 :: "'d \<Rightarrow> 'd list \<Rightarrow> 'c" shows "(case [] of [] \<Rightarrow> f1 | x # xa \<Rightarrow> f2 x xa) = f1"
minimal_antichain_def: fixes A :: "'b set" shows "minimal_antichain A = {x \<in> A. \<not> (\<exists>y\<in>A. y < x)}"
forall_elim: fixes \<Theta> :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set"   and \<Gamma> :: "Core.term set"   and \<tau> :: "typ"   and B :: "Core.term"   and a :: "Core.term" assumes "\<Theta>,\<Gamma> \<turnstile> Ct STR ''Pure.all'' ((\<tau> \<rightarrow> propT) \<rightarrow> propT) $ Abs \<tau> B"   and "\<turnstile>\<^sub>\<tau> a : \<tau>"   and "wf_term (sig \<Theta>) a" shows "\<Theta>,\<Gamma> \<turnstile> subst_bv a B"
ereal_liminf_lim_add: fixes u :: "nat \<Rightarrow> ereal"   and a :: "ereal"   and v :: "nat \<Rightarrow> ereal" assumes "u \<longlonglongrightarrow> a"   and "\<bar>a\<bar> \<noteq> \<infinity>" shows "liminf (\<lambda>n. u n + v n) = a + liminf v"
mr_DA: shows "applied_rule_rev Cp x p = \<lfloor>DenyAll\<rfloor>"
P'''Trans: shows "P''' \<longmapsto> a[b] \<prec> P''"
eq_inf_concat_start: fixes w :: "'action list"   and w\<^sub>1 :: "'action stream"   and w\<^sub>2 :: "'action stream" shows "(w @- w\<^sub>1 =\<^sub>I w @- w\<^sub>2) = (w\<^sub>1 =\<^sub>I w\<^sub>2)"
upper_circle_rescale: fixes l :: "real"   and r :: "real" assumes "l < r" shows "upper_circle l r = (\<lambda>x. x * (complex_of_real r - complex_of_real l) + complex_of_real l) ` upper_circle_01"
slim_arcs_def: shows "S.slim_arcs \<equiv> \<Union>(uu_, p, uu_)\<in>S.slim_paths. set p"
intros_1: fixes c :: "('loc, 't) configuration"   and loc :: "'loc" shows "loc_imps_fw c loc loc (c_imp c loc) []"
that: fixes ct :: "'s \<Rightarrow> 's pmf" assumes "proper ct" shows "thesis"
"8_1": shows "f T2 \<subseteq> wfvarsoccs\<^sub>s\<^sub>s\<^sub>t T1 \<union> X"
univalent_comp_left_dist_inf: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "id x \<sqinter> x \<le> top" shows "x \<sqinter> (y \<sqinter> z) = x \<sqinter> y \<sqinter> (x \<sqinter> z)"
rQuot_lists: fixes c :: "'b"   and S :: "'b set" assumes "c \<in> S" shows "rQuot c (lists S) = lists S"
alpha: fixes a :: "name"   and x :: "'b"   and b :: "name"   and y :: "'b" shows "([a].x = [b].y) = (a = b \<and> x = y \<or> a \<noteq> b \<and> x = [(a, b)] \<bullet> y \<and> a \<sharp> y)"
hyps: assumes "k < n" shows "ket_zero_k k = Matrix.vec (2 ^ Suc k) (\<lambda>k. if k = 0 then 1 else 0)"
\<Z>_srng_vint: shows "\<Z>_srng \<alpha> \<SS>\<^sub>\<int>"
tag_Plus_def: fixes A :: "'a list set"   and B :: "'a list set" shows "tag_Plus A B \<equiv> \<lambda>x. (\<approx>A `` {x}, \<approx>B `` {x})"
Rmi_def: fixes s :: "bdd" shows "Rmi s \<equiv> {(a, b) |a b. Rmi_g a b s}"
is_min2_def: fixes x :: "'d \<times> 'e"   and xs :: "('d \<times> 'e) set" shows "is_min2 x xs \<equiv> x \<in> xs \<and> (\<forall>y\<in>xs. snd x \<le> snd y)"
word_atLeastLessThan_Suc_atLeastAtMost_union: fixes m :: "'b word"   and l :: "'b word"   and u :: "'b word" assumes "m \<noteq> - 1"   and "l \<le> m"   and "m \<le> u" shows "{l..m} \<union> {m + 1..u} = {l..u}"
assms_2: shows "i < 2 ^ (n + 1) \<and> j < 2 ^ (n + 1)"
goal_def: shows "goal = ((\<exists>I\<subseteq>{0..<m}. card (a ` I) = n \<and> lin_indpt (a ` I) \<and> b \<in> finite_cone (a ` I)) \<or> (\<exists>c I. I \<subseteq> {0..<m} \<and> c \<in> {normal_vector (a ` I), - normal_vector (a ` I)} \<and> Suc (card (a ` I)) = n \<and> lin_indpt (a ` I) \<and> (\<forall>i<m. (0::'a) \<le> c \<bullet> a i) \<and> c \<bullet> b < (0::'a)))"
cr_st_vec_def: shows "cr_st_vec \<equiv> \<lambda>x y. x = st_vec y"
assoc_ide_simp: fixes f :: "'a"   and g :: "'a"   and h :: "'a" assumes "B.ide f"   and "B.ide g"   and "B.ide h"   and "src f = trg g"   and "src g = trg h" shows "\<a>[f, g, h] = B.\<a>' h g f"
lbound: shows "wmax l1 (wneg u1) \<equiv>\<^sub>L \<bar>r\<bar>"
subadditive_content_division: fixes \<D> :: "'a set set"   and S :: "'a set"   and a :: "'a"   and b :: "'a" assumes "\<D> division_of S"   and "S \<subseteq> cbox a b" shows "sum content \<D> \<le> content (cbox a b)"
transformation_at_ide_is_transformation: fixes F :: "'j \<Rightarrow> ('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr"   and G :: "'j \<Rightarrow> ('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr"   and \<tau> :: "'j \<Rightarrow> ('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr"   and a :: "'a" assumes "natural_transformation (\<cdot>\<^sub>J) (\<cdot>\<^sub>[\<^sub>A\<^sub>,\<^sub>B\<^sub>]) F G \<tau>"   and "A.ide a" shows "natural_transformation (\<cdot>\<^sub>J) (\<cdot>\<^sub>B) (at a F) (at a G) (at a \<tau>)"
bis_coind: fixes Retr :: "('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set"   and theta :: "('a \<times> 'a) set" assumes "Bisim.mono Retr"   and "sym theta"   and "theta \<subseteq> Retr (theta \<union> bis Retr)" shows "theta \<subseteq> bis Retr"
STCal_steps_1: fixes S :: "'a"   and Source :: "'a processCalculus"   and Target :: "'b processCalculus"   and P' :: "('a, 'b) Proc" shows "SourceTerm S \<longmapsto>STCal Source Target* P' = (\<exists>S'. P' = SourceTerm S' \<and> S \<longmapsto>Source* S')"
assms: shows "x \<in> fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s (F \<cdot>\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s \<delta>)"
mat_mult_append: fixes A :: "'a mat"   and nr1 :: "nat"   and nc :: "nat"   and B :: "'a mat"   and nr2 :: "nat"   and v :: "'a vec" assumes "A \<in> carrier_mat nr1 nc"   and "B \<in> carrier_mat nr2 nc"   and "v \<in> carrier_vec nc" shows "(A @\<^sub>r B) *\<^sub>v v = A *\<^sub>v v @\<^sub>v B *\<^sub>v v"
preserves_iso: fixes f :: "'d" assumes "D.iso f" shows "D.iso (src\<^sub>D f)"
simps_2: fixes p :: "real mpoly"   and i :: "nat"   and L :: "(real mpoly \<times> nat) list"   and R :: "(real mpoly \<times> nat) list" shows "chophelper ((p, i) # L) R = (if p = 0 then chophelper L (R @ [(p, i)]) else (R, (p, i) # L))"
mult_less_cancel_right1: fixes plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and uminus :: "'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and sgn :: "'e \<Rightarrow> 'e"   and abs :: "'e \<Rightarrow> 'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and one :: "'e"   and c :: "'e"   and b :: "'e" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less c (times b c) = ((less_eq zero c \<longrightarrow> less one b) \<and> (less_eq c zero \<longrightarrow> less b one))"
tl_lists: fixes us :: "'b list"   and A :: "'b set" assumes "us \<in> lists A" shows "tl us \<in> lists A"
phi_equiv_mapping_def: fixes p :: "'g \<Rightarrow> ('node \<times> 'val, 'val list) mapping"   and g :: "'g"   and nou\<^sub>1 :: "('val, 'a set) mapping"   and nou\<^sub>2 :: "('val, 'a set) mapping" shows "uninst_code.phi_equiv_mapping p g nou\<^sub>1 nou\<^sub>2 \<equiv> \<forall>v\<in>Mapping.keys (uninst_code.ssa.phidefNodes p g). (case Mapping.lookup nou\<^sub>1 v of None \<Rightarrow> {} | Some x \<Rightarrow> id x) = (case Mapping.lookup nou\<^sub>2 v of None \<Rightarrow> {} | Some x \<Rightarrow> id x)"
vector_matrix_left_distrib: fixes x :: "('a, 'c) vec"   and y :: "('a, 'c) vec"   and A :: "(('a, 'b) vec, 'c) vec" shows "(x + y) v* A = x v* A + y v* A"
converse_Image: fixes R :: "('c \<times> 'd) set"   and X :: "'c set" assumes "runiq R"   and "runiq (R\<inverse>)" shows "R\<inverse> `` R `` X \<subseteq> X"
preserves_seq: fixes g :: "'a"   and f :: "'a" assumes "B.seq g f" shows "B.seq (B.R g) (B.R f)"
unfold: shows "lrev_body_final = (\<Lambda> r v ys. Fixrec.run\<cdot>(match_lnil\<cdot>v\<cdot>(Fixrec.succeed\<cdot>ys) +++ match_lcons\<cdot>v\<cdot>(\<Lambda> x xs. Fixrec.succeed\<cdot>(r\<cdot>xs\<cdot>(x :@ ys)))))"
homeomorphism_apply1: fixes S :: "'c set"   and T :: "'d set"   and f :: "'c \<Rightarrow> 'd"   and g :: "'d \<Rightarrow> 'c"   and x :: "'c" assumes "homeomorphism S T f g"   and "x \<in> S" shows "g (f x) = x"
Ord_equality: fixes i :: "V" assumes "Ord i" shows "i = \<Squnion> (ZFC_in_HOL.succ ` elts i)"
project_Always_I: fixes F :: "'a program"   and G :: "'c program"   and A :: "'a set" assumes "extend h F \<squnion> G \<in> Always (extend_set h A)" shows "F \<squnion> project h (reachable (extend h F \<squnion> G)) G \<in> Always A"
simps_2: fixes x :: "'d \<times> 'f"   and xs :: "('d \<times> 'f) list"   and ys :: "('f \<times> 'e) list" shows "compose (x # xs) ys = (case map_of ys (snd x) of None \<Rightarrow> compose (delete (fst x) xs) ys | Some v \<Rightarrow> (fst x, v) # compose xs ys)"
and_eq_minus_1_iff: fixes 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 divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and drop_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and take_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and "and" :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and or :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and xor :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and mask :: "nat \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and not :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.ring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask uminus not" shows "(and a b = uminus one) = (a = uminus one \<and> b = uminus one)"
nres_rel_mono: fixes A :: "('c \<times> 'd) set"   and B :: "('c \<times> 'd) set" assumes "A \<subseteq> B" shows "\<langle>A\<rangle>nres_rel \<subseteq> \<langle>B\<rangle>nres_rel"
swap_preserves_wls3: fixes X :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and Y :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and xs :: "'varSort"   and x1 :: "'var"   and x2 :: "'var"   and ys :: "'varSort"   and y1 :: "'var"   and y2 :: "'var"   and s :: "'sort" assumes "good X"   and "good Y"   and "X #[x1 \<and> x2]_xs = Y #[y1 \<and> y2]_ys" shows "wls s X = wls s Y"
\<alpha>\<beta>: shows "\<alpha> * \<beta> = \<Union>\<^sub>\<circ> (VLambda \<beta> ((*) \<alpha>) `\<^sub>\<circ> \<beta>)"
assms: fixes x :: "'a" assumes "x \<le> c" shows "P x"
Mapping_RBT'_inverse: fixes y :: "('c, 'd) rbt" assumes "y \<in> {t. ord.is_rbt cless t \<or> ID ccompare = None}" shows "impl_of (Mapping_RBT' y) = y"
PiE_eq_empty_iff: fixes I :: "'d set"   and F :: "'d \<Rightarrow> 'e set" shows "(Pi\<^sub>E I F = {}) = (\<exists>i\<in>I. F i = {})"
hyps_2: assumes "y \<in> UNIV\<^sup>\<star>" shows "LList2.llast (l @@ a ## y) = LList2.llast (a ## y)"
bsok_simps_23: fixes e :: "'a expr1"   and C :: "String.literal"   and V :: "nat"   and e' :: "'a expr1"   and n :: "nat" shows "bsok (try e catch(C V) e') n = (bsok e n \<and> bsok e' (Suc n) \<and> V = n)"
actionI: fixes A :: "'a \<times> 'b \<Rightarrow> bool" assumes "\<And>s t. A (s, t)" shows "\<turnstile> A"
well_valued0_lchild_if_well_valued0: fixes as :: "'b list"   and t :: "'b tree" assumes "well_valued0 as t" shows "well_valued0 as (lchild t)"
elims_3: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and x :: "symbol list"   and xa :: "(nat \<times> symbol \<times> symbol list) list"   and xb :: "symbol list" assumes "CFG \<NN> \<TT> \<RR> \<SS>"   and "\<not> CFG.Derivation \<NN> \<TT> \<RR> x xa xb" obtains a :: "symbol list" and b :: "symbol list" where "x = aa__" and "xa = []" and "xb = ba__" and "aa__ \<noteq> ba__"   | a :: "symbol list" and d :: "nat \<times> symbol \<times> symbol list" and D :: "(nat \<times> symbol \<times> symbol list) list" and b :: "symbol list" where "x = a" and "xa = d # D" and "xb = b" and "\<nexists>x. CFG.Derives1 \<NN> \<TT> \<RR> a (fst d) (snd d) x \<and> CFG.Derivation \<NN> \<TT> \<RR> x D b"
approx_mult1: fixes a :: "'a star"   and b :: "'a star"   and c :: "'a star" assumes "a \<approx> b"   and "c \<in> HFinite" shows "a * c \<approx> b * c"
wbt_insert: fixes t :: "('b \<times> nat) tree"   and x :: "'b" assumes "wbt t" shows "wbt (Weight_Balanced_Trees.insert x t)"
simps_28: fixes x2 :: "'f" shows "set1_step (Skip x2) = {}"
unique_expression4: fixes H :: "'a set"   and f :: "nat \<Rightarrow> 'a"   and n :: "nat"   and s :: "nat \<Rightarrow> 'b" assumes "free_generator R M H" shows "f \<in> {k. k \<le> n} \<rightarrow> H \<and> inj_on f {k. k \<le> n} \<and> s \<in> {k. k \<le> n} \<rightarrow> carrier R \<and> l_comb R M n s f \<noteq> \<zero> \<longrightarrow> (\<exists>m g t. g \<in> {k. k \<le> m} \<rightarrow> H \<and> inj_on g {k. k \<le> m} \<and> g ` {k. k \<le> m} \<subseteq> f ` {k. k \<le> n} \<and> t \<in> {k. k \<le> m} \<rightarrow> carrier R \<and> (\<forall>l\<in>{k. k \<le> m}. t l \<noteq> \<zero>\<^bsub>R\<^esub>) \<and> l_comb R M n s f = l_comb R M m t g)"
comp_distr: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(x + y) ; z = x ; z + y ; z"
elims: fixes x :: "complex \<times> complex \<times> complex \<times> complex"   and xa :: "complex \<times> complex"   and y :: "complex \<times> complex" assumes "x *\<^sub>m\<^sub>v xa = y" obtains a :: "complex" and b :: "complex" and c :: "complex" and d :: "complex" and xb :: "complex" and ya :: "complex" where "x = (a, b, c, d)" and "xa = (xb, ya)" and "y = (xb * a + ya * b, xb * c + ya * d)"
save_restore_instr_def: fixes instr :: "sparc_operation \<times> inst_operand list" shows "save_restore_instr instr \<equiv> let instr_name = fst instr; op_list = snd instr; rd = get_operand_w5 (op_list ! 3) in do psr_val \<leftarrow> gets (cpu_reg_val PSR); curr_win \<leftarrow> get_curr_win (); wim_val \<leftarrow> gets (cpu_reg_val WIM); if instr_name = ctrl_type SAVE then do new_cwp \<leftarrow> gets (\<lambda>s. word_of_int ((uint curr_win - 1) mod NWINDOWS)); if get_WIM_bit (unat new_cwp) wim_val \<noteq> 0 then do raise_trap window_overflow; return () od else do result \<leftarrow> gets (get_addr op_list); save_retore_sub1 result new_cwp rd od od else do new_cwp \<leftarrow> gets (\<lambda>s. word_of_int ((uint curr_win + 1) mod NWINDOWS)); if get_WIM_bit (unat new_cwp) wim_val \<noteq> 0 then do raise_trap window_underflow; return () od else do result \<leftarrow> gets (get_addr op_list); save_retore_sub1 result new_cwp rd od od od"
load_tmps_fresh: fixes i :: "nat"   and p :: "'a"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "'a memref list"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option" assumes "i < length ts\<^sub>s\<^sub>b"   and "ts\<^sub>s\<^sub>b ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>)" shows "load_tmps is \<inter> dom \<theta> = {}"
simps_2: fixes ev :: "event"   and evs :: "event list" shows "used (ev # evs) = (case ev of Says A B X \<Rightarrow> parts {X} \<union> used evs | Gets A X \<Rightarrow> used evs | Notes A X \<Rightarrow> parts {X} \<union> used evs)"
hyps_5: shows "(lookup_pair xs k = (0::'b)) = (k \<notin> fst ` set xs)"
intros: fixes p :: "'b \<times> 'c" shows "fst p \<in> Basic_BNFs.fsts p"
propa_dom_invariant_aux: fixes b :: "nat list"   and a :: "nat list list"   and ss :: "nat list list"   and w :: "nat list" assumes "propa f (step (hd b) (a ! hd b)) a (tl b) = (ss, w)"   and "b \<noteq> []"   and "\<forall>\<tau>\<in>set a. \<tau> \<in> A"   and "\<forall>p<n. Sorted_Less.sorted (rev (a ! p)) \<and> (a ! p \<noteq> rev [0..<n] \<longrightarrow> (\<forall>x\<in>set (a ! p). x < p)) \<and> (a ! p = rev [0..<n] \<longrightarrow> (\<exists>x\<in>set b. (x, p) \<in> g_E G \<and> x < p)) \<and> (p \<notin> set b \<longrightarrow> stable r step a p)"   and "Sorted_Less.sorted b"   and "length a = n"   and "\<forall>x\<in>set b. x < n" shows "(\<forall>\<tau>\<in>set ss. \<tau> \<in> A) \<and> (\<forall>p<n. Sorted_Less.sorted (rev (ss ! p)) \<and> (ss ! p \<noteq> rev [0..<n] \<longrightarrow> (\<forall>x\<in>set (ss ! p). x < p)) \<and> (ss ! p = rev [0..<n] \<longrightarrow> (\<exists>x\<in>set w. (x, p) \<in> g_E G \<and> x < p)) \<and> (p \<notin> set w \<longrightarrow> stable r step ss p)) \<and> Sorted_Less.sorted w \<and> length ss = n \<and> (\<forall>x\<in>set w. x < n)"
ket_plus_k_dim: fixes k :: "nat" assumes "k < n" shows "ket_plus_k k \<in> carrier_vec (2 ^ Suc k)"
simps_1: fixes x :: "complex"   and y :: "complex" shows "Re (x - y) = Re x - Re y"
inside_inside: fixes S :: "'b set"   and T :: "'b set" assumes "S \<subseteq> inside T" shows "inside S - T \<subseteq> inside T"
Nil: fixes \<O>s :: "nat set list"   and i :: "nat"   and \<theta> :: "nat \<Rightarrow> nat option"   and m :: "nat \<Rightarrow> nat"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<S> :: "nat \<Rightarrow> bool option" shows "\<O>s,i\<turnstile> ([], \<theta>, m, \<D>, \<O>, \<S>) \<surd>"
le_succE: fixes i :: "hf"   and j :: "hf" assumes "succ i \<le> succ j" shows "i \<le> j"
new_subc_put'_2: fixes c :: "'a"   and s :: "'a FOL_Fitting.term"   and l :: "'a FOL_Fitting.term list"   and u :: "'a FOL_Fitting.term"   and m :: "nat" assumes "new_term c s" shows "subc_list c s (l[u/m]) = subc_list c s (l[subc_term c s u/m])"
assms_2: shows "\<forall>b\<^sub>\<sigma>. local.etp.HCP_adv \<A> m2 b\<^sub>\<sigma> D \<le> HCP_ad"
assoc_bpp_def: fixes A :: "'e set"   and f :: "'e \<Rightarrow> 'e \<Rightarrow> 'e" shows "assoc_bpp A f = (\<forall>a\<in>addition_set f A. \<forall>b\<in>addition_set f A. \<forall>c\<in>addition_set f A. a \<^sub>f+ b \<^sub>f+ c = a \<^sub>f+ (b \<^sub>f+ c))"
disjoint_notin1: fixes A :: "'c set"   and B :: "'c set"   and x :: "'c" assumes "A \<inter> B = {}"   and "x \<in> B" shows "x \<notin> A"
semiring_norm_63: fixes a :: "'d" shows "(0::'d) * a = (0::'d)"
MGF_lemma1: fixes G :: "state triple set"   and c :: "com" assumes "state_not_singleton"   and "\<forall>pn\<in>dom body. G|-{=}.BODY pn.{->}"   and "WT c" shows "G|-{=}.c.{->}"
prob_non_zero: shows "\<exists>i\<in>I. 0 < prob (A i)"
subst_fm_graph_aux_def: shows "subst_fm_graph_aux \<equiv> lfp (\<lambda>p x1 x2. (\<exists>subst_fm_aux t u i x. x1 = (t IN u, i, x) \<and> x2 = subst i x t IN subst i x u) \<or> (\<exists>subst_fm_aux t u i x. x1 = (t EQ u, i, x) \<and> x2 = subst i x t EQ subst i x u) \<or> (\<exists>subst_fm_aux A B i x. x1 = (A OR B, i, x) \<and> x2 = ((\<lambda>uu_. uu_) \<circ> subst_fm_aux) (A, i, x) OR ((\<lambda>uu_. uu_) \<circ> subst_fm_aux) (B, i, x) \<and> p (A, i, x) (((\<lambda>uu_. uu_) \<circ> subst_fm_aux) (A, i, x)) \<and> p (B, i, x) (((\<lambda>uu_. uu_) \<circ> subst_fm_aux) (B, i, x))) \<or> (\<exists>subst_fm_aux A i x. x1 = (Neg A, i, x) \<and> x2 = Neg (((\<lambda>uu_. uu_) \<circ> subst_fm_aux) (A, i, x)) \<and> p (A, i, x) (((\<lambda>uu_. uu_) \<circ> subst_fm_aux) (A, i, x))) \<or> (\<exists>subst_fm_aux j i x A. x1 = (SyntaxN.Ex j A, i, x) \<and> x2 = SyntaxN.Ex j (((\<lambda>uu_. uu_) \<circ> subst_fm_aux) (A, i, x)) \<and> atom j \<sharp> (i, x) \<and> p (A, i, x) (((\<lambda>uu_. uu_) \<circ> subst_fm_aux) (A, i, x))))"
move_matrix_def: fixes A :: "'b matrix"   and y :: "int"   and x :: "int" shows "move_matrix A y x \<equiv> Abs_matrix (\<lambda>j i. if int j - y < 0 \<or> int i - x < 0 then 0::'b else Rep_matrix A (nat (int j - y)) (nat (int i - x)))"
K_mono: fixes x :: "'a set"   and y :: "'a set" assumes "x \<subseteq> y" shows "K x \<subseteq> K y"
weak_diamond_def: fixes r :: "('a \<times> 'a) set" shows "w\<diamond> r = (r\<inverse> O r - Id \<subseteq> r O r\<inverse>)"
poly_add_monom: fixes p :: "'a list"   and a :: "'a" assumes "set p \<subseteq> carrier R"   and "a \<in> carrier R - {\<zero>}" shows "poly_add (local.monom a (length p)) p = a # p"
I_def: shows "I \<equiv> 1\<^sub>m n"
field_sum_of_halves: fixes x :: "'b" shows "x / (2::'b) + x / (2::'b) = x"
stat_override_declclasses_relation: fixes new :: "'a"   and superNew :: "qtname"   and G :: "prog"   and old :: "qtname \<times> mdecl" assumes "G\<turnstile>declclass new\<prec>\<^sub>C1superNew"   and "G \<turnstile>Method old member_of superNew" shows "G\<turnstile>declclass new\<prec>\<^sub>C declclass old"
Gamma_bounds_1: fixes x :: "real" assumes "1 \<le> x" shows "sqrt (2 * pi / x) * (x / exp 1) powr x \<le> Gamma x"
\<pi>1_less: fixes k :: "nat" assumes "k < Suc i" shows "\<pi>1 !! k = \<pi> !! k"
skel_vsubst: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and ys :: "'varSort"   and y1 :: "'var"   and y2 :: "'var" assumes "good X" shows "skel (X #[y1 // y2]_ys) = skel X"
mpj_surjec: fixes H :: "'a set" assumes "submodule R M H" shows "surjec\<^bsub>M,M /\<^sub>m H\<^esub> mpj M H"