hcju/isabelleps huggingface dataset:
subst_typ_split_last: fixes xs :: "((variable \<times> String.literal set) \<times> typ) list"   and x :: "(variable \<times> String.literal set) \<times> typ"   and T :: "typ" assumes "distinct (map fst (xs @ [x]))"   and "\<And>y. y \<in> tvsT_Set (snd ` set (xs @ [x])) \<Longrightarrow> y \<notin> fst ` set (xs @ [x])" shows "subst_typ (xs @ [x]) T = subst_typ [x] (subst_typ xs T)"
CAND_NotL_elim: fixes x :: "name"   and a :: "coname"   and M :: "trm"   and B :: "ty" assumes "x:NotL <a>.M x \<in> \<parallel>(B)\<parallel>"   and "x:NotL <a>.M x \<notin> BINDINGn B (\<parallel><B>\<parallel>)" shows "\<exists>B'. B = NOT B' \<and> <a>:M \<in> \<parallel><B'>\<parallel>"
trans: fixes a :: "'b \<times> 'c"   and b :: "'b \<times> 'c"   and c :: "'b \<times> 'c" assumes "less_eq_prod a b"   and "less_eq_prod b c" shows "less_eq_prod a c"
intro: fixes return :: "'a \<Rightarrow> 'b"   and bind :: "'b \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b"   and sample :: "'c pmf \<Rightarrow> ('c \<Rightarrow> 'b) \<Rightarrow> 'b" assumes "monad return bind"   and "monad_prob_axioms bind sample" shows "monad_prob return bind sample"
derivs_Atom: fixes xs :: "'b list"   and as :: "'b" shows "derivs xs (Atom as) \<in> {Zero, One, Atom as}"
lockstep_to_plus_backward_simulation: fixes match :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and step2 :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and step1 :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and s1 :: "'a"   and s2 :: "'b"   and s2' :: "'b" assumes "\<And>s1 s2 s2'. \<lbrakk>match s1 s2; step2 s2 s2'\<rbrakk> \<Longrightarrow> \<exists>s1'. step1 s1 s1' \<and> match s1' s2'"   and "match s1 s2"   and "step2 s2 s2'" shows "\<exists>s1'. step1\<^sup>+\<^sup>+ s1 s1' \<and> match s1' s2'"
dpd4: fixes x :: "'a"   and y :: "'a" shows "d (x + y) = d x + d y"
simps_1: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and neutral :: "'b"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'b"   and u :: "nat"   and v :: "nat" assumes "class.linordered_ab_monoid_add plus less_eq less neutral" shows "linordered_ab_monoid_add.len plus m u v [] = m u v"
R_def: fixes m :: "int"   and f :: "'b mod_ring poly" assumes "poly_mod_type_irr m f" shows "poly_mod_type_irr.R f = \<lparr>carrier = poly_mod_type_irr.carrier_irr f, monoid.mult = poly_mod_type_irr.mult_irr f, one = 1, zero = 0, add = poly_mod_type_irr.plus_irr f\<rparr>"
liftAll_def: fixes phi :: "'e \<Rightarrow> bool"   and inp :: "'f \<Rightarrow> 'e option" shows "liftAll phi inp \<equiv> \<forall>i v. inp i = Some v \<longrightarrow> phi v"
list2policyR_Start: fixes p :: "int \<times> (int \<times> int) \<times> (int \<times> int) \<times> DummyContent"   and a :: "((int \<times> int) set set, int) Combinators"   and list :: "((int \<times> int) set set, int) Combinators list" assumes "p \<in> dom (C a)" shows "C (list2policyR (a # list)) p = C a p"
fresh_star_eqvt_2: fixes pi :: "(name \<times> name) list"   and b :: "name list"   and x :: "'a" shows "pi \<bullet> b \<sharp>* x = (pi \<bullet> b) \<sharp>* (pi \<bullet> x)"
collapse: fixes taylor_model :: "taylor_model" shows "TaylorModel (tm_poly taylor_model) (tm_bound taylor_model) = taylor_model"
in_proj: fixes P :: "'c \<Rightarrow> bool"   and \<phi> :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c" assumes "equivP P \<phi>"   and "P x" shows "x \<in> proj \<phi> x"
inverse: fixes f :: "'c \<Rightarrow> 'a" shows "U1 (C1 f) = f"
distinct_15: fixes x3 :: "nat"   and x41 :: "typ"   and x42 :: "Core.term" shows "Bv x3 \<noteq> Abs x41 x42"
P_hits_A: shows "lset P \<inter> A \<noteq> {}"
trapped_backward_iff_rev_trapped_forward: fixes x :: "'a"   and K :: "'a set" shows "rev.trapped_backward x K = trapped_forward x K"
symmetric: fixes a :: "'a"   and b :: "'a" assumes "(a, b) \<in> Orbit_Relation" shows "(b, a) \<in> Orbit_Relation"
one_step: fixes p :: "nat"   and n :: "nat"   and base :: "nat \<Rightarrow> nat"   and upd :: "nat \<Rightarrow> nat"   and s :: "(nat \<Rightarrow> nat) set"   and a :: "nat \<Rightarrow> nat"   and j :: "nat"   and p' :: "nat" assumes "kuhn_simplex p n base upd s"   and "a \<in> s"   and "j < n"   and "\<And>a'. \<lbrakk>a' \<in> s; a' \<noteq> a\<rbrakk> \<Longrightarrow> a' j = p'" shows "a j \<noteq> p'"
bind_pure_returns_result_I: fixes f :: "('aq, 'ar, 'as) prog"   and h :: "'aq"   and x :: "'as"   and g :: "'as \<Rightarrow> ('aq, 'ar, 'at) prog"   and y :: "'at" assumes "pure f h"   and "h \<turnstile> f \<rightarrow>\<^sub>r x"   and "h \<turnstile> g x \<rightarrow>\<^sub>r y" shows "h \<turnstile> Heap_Error_Monad.bind f g \<rightarrow>\<^sub>r y"
ReturnNotReadWrite: fixes mm :: "state \<Rightarrow> Locs \<Rightarrow> Vals"   and ch :: "state \<Rightarrow> PrIds \<Rightarrow> (memOp, Vals) chan"   and rs :: "state \<Rightarrow> PrIds \<Rightarrow> Vals"   and p :: "PrIds" shows "\<turnstile> (\<forall>l. $MemInv mm l) \<and> MemReturn ch rs p \<longrightarrow> \<not> Read ch mm rs p \<and> (\<forall>l. \<not> Write ch mm rs p l)"
h_plus_outside: fixes \<Delta> :: "('a, 'b) network_scheme"   and g :: "'a \<times> 'a \<Rightarrow> ennreal"   and e :: "'a \<times> 'a"   and i :: "nat" assumes "flow_network \<Delta> g"   and "e \<notin> \<^bold>E\<^bsub>\<Delta>\<^esub>" shows "flow_network.h_plus \<Delta> g i e = 0"
stut_fun1: fixes X :: "(nat \<Rightarrow> 'b) \<Rightarrow> 'c"   and f :: "'c \<Rightarrow> 'd" assumes "STUTINV X" shows "STUTINV f<X>"
lsbis1_def: fixes B1 :: "'d set"   and B2 :: "'e set"   and s1 :: "'d \<Rightarrow> ('f, 'd, 'e) F1"   and s2 :: "'e \<Rightarrow> ('g, 'd, 'e) F2" shows "lsbis1 B1 B2 s1 s2 = \<Union> (fst ` {(R1, R2) |R1 R2. sbis B1 B2 s1 s2 R1 R2})"
cinner_add_left: fixes scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and scaleC :: "complex \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and cinner :: "'b \<Rightarrow> 'b \<Rightarrow> complex"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "class.complex_inner scaleR scaleC plus zero minus uminus dist norm sgn uniformity open cinner" shows "cinner (plus x y) z = cinner x z + cinner y z"
exp_base_independent: fixes K :: "'a set"   and x :: "'a" assumes "subfield K R"   and "x \<in> carrier R"   and "(algebraic over K) x" shows "local.independent K (exp_base x (degree (Irr K x)))"
assms_1: shows "inverse_transformations A B F G \<sigma> \<sigma>'"
MapsToMorDomCod_3: fixes C :: "('d, 'e, 'f) Category_scheme"   and f :: "'e"   and g :: "'e" assumes "Category C"   and "f \<approx>>\<^bsub>C\<^esub> g" shows "cod\<^bsub>C\<^esub> (f ;;\<^bsub>C\<^esub> g) = cod\<^bsub>C\<^esub> g"
rel_of_regexp_rel: fixes vs :: "('b \<times> 'b) set list"   and r :: "nat rexp" shows "rel_of_regexp vs r = rel ((!) vs) r"
le_bot: fixes a :: "'a" assumes "top \<le> a" shows "a = top"
id_fcomp: fixes g :: "'c \<Rightarrow> 'd" shows "fcomp id g = g"
gen_upper_bound_increasing: fixes a :: "real"   and x :: "real"   and fu :: "real \<Rightarrow> real"   and f :: "real \<Rightarrow> real"   and g :: "real \<Rightarrow> real" assumes "a \<le> x"   and "\<And>y. \<lbrakk>a \<le> y; y \<le> x\<rbrakk> \<Longrightarrow> ((\<lambda>x. fu x - f x) has_real_derivative g y) (at y)"   and "\<And>y. \<lbrakk>a \<le> y; y \<le> x\<rbrakk> \<Longrightarrow> 0 \<le> g y"   and "fu a = f a" shows "f x \<le> fu x"
valid_disj_PL: fixes p :: "'a"   and q :: "'a" shows "\<tturnstile> (p .\<or>. q) = ({.\<not> p} \<turnstile> q)"
distinct_77: fixes x61 :: "'d"   and x62 :: "'a \<Rightarrow> nat set"   and x63 :: "'a \<Rightarrow> 'b"   and x64 :: "('a, 'b, 'c, 'd) seqp"   and x10 :: "'c" shows "{x61}groupcast(x62, x63) . x64 \<noteq> call(x10)"
all_le: fixes i :: "nat" assumes "i \<in> {d. d dvd order (mult_of R)}" shows "card {a \<in> carrier (mult_of R). G.ord a = i} \<le> phi' i"
seq_pr_inc: fixes s :: "nat \<Rightarrow> nat \<Rightarrow> int"   and k :: "nat" assumes "s \<in> carrier Zp\<^bsup>\<omega>\<^esup>" shows "\<exists>l. \<forall>m. \<exists>n>m. kth_res_equals k l (s n)"
p_comp_pp: fixes x :: "'c \<times> 'c \<Rightarrow> 'd"   and y :: "'c \<times> 'c \<Rightarrow> 'd" shows "\<ominus> (x \<odot> \<ominus> \<ominus> y) = \<ominus> (x \<odot> y)"
Sup_level: fixes L :: "level set" shows "Sup L \<equiv> if Hi \<in> L then Hi else Lo"
p_def: shows "p \<equiv> poly_real_alg_1 plr"
word_eq_unatI: fixes v :: "'d word"   and w :: "'d word" assumes "unat v = unat w" shows "v = w"
lt_imp_ex_count_lt: fixes M :: "'a multiset"   and N :: "'a multiset" assumes "M < N" shows "\<exists>y. count M y < count N y"
extremum_unique: fixes a :: "'d" shows "(\<top> \<le> a) = (a = \<top>)"
v11_vlepollE: fixes A :: "V"   and B :: "V" assumes "A \<lesssim>\<^sub>\<circ> B" obtains f :: "V" where "v11 f" and "\<D>\<^sub>\<circ> f = A" and "\<R>\<^sub>\<circ> f \<subseteq>\<^sub>\<circ> B"
UN_subset_iff: fixes A :: "'b \<Rightarrow> 'a set"   and I :: "'b set"   and B :: "'a set" shows "(\<Union> (A ` I) \<subseteq> B) = (\<forall>i\<in>I. A i \<subseteq> B)"
eval_elim_cases_14: fixes G :: "prog"   and s :: "st"   and va :: "Term.var"   and v :: "vals"   and s' :: "state" assumes "G\<turnstile>Norm s \<midarrow>In1l (Acc va)\<succ>\<rightarrow> (v, s')" obtains v :: "val" and f :: "val \<Rightarrow> state \<Rightarrow> state" where "vb__ = In1 v" and "G\<turnstile>Norm s \<midarrow>va=\<succ>(v, f)\<rightarrow> s'"
upt_conv_Nil: fixes j :: "nat"   and i :: "nat" assumes "j \<le> i" shows "[i..<j] = []"
simps_2: fixes h :: "'b \<times> bool"   and t :: "('b \<times> bool) list" shows "elements (h # t) = element h # elements t"
simps_2: fixes as :: "'b list"   and i :: "nat"   and j :: "nat"   and a :: "'b option"   and t :: "'b tree"   and u :: "'b tree" shows "well_valued0 as (Node i j a t u) = (0 < i \<and> j \<le> length as \<and> (a \<noteq> None \<longrightarrow> a = Some (SWA.sum as i j)) \<and> well_valued0 as t \<and> well_valued0 as u \<and> (u = Leaf \<or> val u \<noteq> None))"
intros_2: fixes sourcenode :: "'f \<Rightarrow> 'g"   and targetnode :: "'f \<Rightarrow> 'g"   and kind :: "'f \<Rightarrow> 'h edge_kind"   and valid_edge :: "'f \<Rightarrow> bool"   and Entry :: "'g"   and Def :: "'g \<Rightarrow> 'i set"   and Use :: "'g \<Rightarrow> 'i set"   and state_val :: "'h \<Rightarrow> 'i \<Rightarrow> 'j"   and Exit :: "'g"   and control_dependence :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and n :: "'g"   and V :: "'i"   and n' :: "'g" assumes "PDG sourcenode targetnode kind valid_edge Entry Def Use state_val Exit control_dependence"   and "CFG_wf.data_dependence sourcenode targetnode valid_edge Def Use n V n'" shows "PDG.PDG_edge sourcenode targetnode valid_edge Def Use control_dependence n (Some V) n'"
items_le_pointwise: fixes k :: "nat" shows "pointwise (items_le k)"
vrange_atD: fixes b :: "V" assumes "b \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> (\<dagger>\<^sub>D\<^sub>G\<^sub>.\<^sub>R\<^sub>e\<^sub>l \<alpha>\<lparr>ArrMap\<rparr>)" shows "\<exists>a\<in>elts (\<D>\<^sub>\<circ> (\<dagger>\<^sub>D\<^sub>G\<^sub>.\<^sub>R\<^sub>e\<^sub>l \<alpha>\<lparr>ArrMap\<rparr>)). \<dagger>\<^sub>D\<^sub>G\<^sub>.\<^sub>R\<^sub>e\<^sub>l \<alpha>\<lparr>ArrMap\<rparr>\<lparr>a\<rparr> = b"
getMinTree_cons: fixes y :: "('d, 'c) BinomialTree"   and x :: "('d, 'c) BinomialTree"   and xs :: "('d, 'c) BinomialTree list" shows "prio (getMinTree (y # x # xs)) \<le> prio (getMinTree (x # xs))"
face_cycle_succ_iso: fixes hom :: "('a, 'b, 'e, 'f) digraph_isomorphism"   and x :: "'f" assumes "digraph_isomorphism hom"   and "x \<in> iso_arcs hom ` arcs G" shows "pre_digraph_map.face_cycle_succ (map_iso hom) x = wrap_iso_arcs hom face_cycle_succ x"
finite_reach_af: fixes \<Sigma> :: "'b set set"   and \<phi> :: "'b ltl" shows "finite (reach \<Sigma> \<up>af (Abs \<phi>))"
FAss: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and e\<^sub>1 :: "expr"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and a :: "nat"   and Cs' :: "char list list"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and e\<^sub>2 :: "expr"   and v :: "val"   and h\<^sub>2 :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and l\<^sub>2 :: "char list \<Rightarrow> val option"   and D :: "char list"   and S :: "(char list list \<times> (char list \<Rightarrow> val option)) set"   and F :: "char list"   and T :: "ty"   and Cs :: "char list list"   and v' :: "val"   and Ds :: "char list list"   and fs :: "char list \<Rightarrow> val option"   and fs' :: "char list \<Rightarrow> val option"   and S' :: "(char list list \<times> (char list \<Rightarrow> val option)) set"   and h\<^sub>2' :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option" assumes "P,E \<turnstile> \<langle>e\<^sub>1,s\<^sub>0\<rangle> \<Rightarrow> \<langle>ref (a, Cs'),s\<^sub>1\<rangle>"   and "P,E \<turnstile> \<langle>e\<^sub>2,s\<^sub>1\<rangle> \<Rightarrow> \<langle>Val v,(h\<^sub>2, l\<^sub>2)\<rangle>"   and "h\<^sub>2 a = \<lfloor>(D, S)\<rfloor>"   and "P \<turnstile> last Cs' has least F:T via Cs"   and "P \<turnstile> T casts v to v' "   and "Ds = Cs' @\<^sub>p Cs"   and "(Ds, fs) \<in> S"   and "fs' = fs(F \<mapsto> v')"   and "S' = S - {(Ds, fs)} \<union> {(Ds, fs')}"   and "h\<^sub>2' = h\<^sub>2(a \<mapsto> (D, S'))" shows "P,E \<turnstile> \<langle>e\<^sub>1\<bullet>F{Cs} := e\<^sub>2,s\<^sub>0\<rangle> \<Rightarrow> \<langle>Val v',(h\<^sub>2', l\<^sub>2)\<rangle>"
min_lvar_not_in_bounds_None: fixes s :: "('i, 'a) state" shows "min_lvar_not_in_bounds s = None \<longrightarrow> (\<forall>x\<in>lvars (\<T> s). in_bounds x \<langle>\<V> s\<rangle> (\<B> s))"
set_of_graph_rulesD_2: fixes Rs :: "(('d, 'e) labeled_graph \<times> ('d, 'e) labeled_graph) set"   and R :: "('d, 'e) labeled_graph \<times> ('d, 'e) labeled_graph" assumes "set_of_graph_rules Rs"   and "R \<in> Rs" shows "finite_graph (snd R)"
of_rat_less_eq: fixes r :: "rat"   and s :: "rat" shows "(of_rat r \<le> of_rat s) = (r \<le> s)"
simps_1: fixes uu :: "'a simple_packet set" shows "rmshadow' [] uu = []"
zero_le: fixes plus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and zero :: "'g"   and less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and x :: "'g" assumes "class.canonically_ordered_monoid_add plus zero less_eq less" shows "less_eq zero x"
pick_inj_on_good: shows "inj_on pick (Collect good)"
assms: shows "t \<in>\<^sub>A frontier N - frontier M"
single_def: fixes x :: "'a" shows "single x = Pred ((=) x)"
integral_one_over_x_ln_x: fixes a :: "real"   and b :: "real" assumes "a \<le> b"   and "1 < a" shows "integral {a..b} (\<lambda>x. inverse (x * ln x)) = ln (ln b) - ln (ln a)"
eq: shows "matrix_sum d (\<lambda>k. f k + g k) n = matrix_sum d f n + matrix_sum d g n"
sel_3: fixes x1 :: "real"   and x2 :: "real"   and x3 :: "real"   and x4 :: "real"   and x5 :: "real"   and x6 :: "real"   and x7 :: "real"   and x8 :: "real" shows "Im2 (Octo x1 x2 x3 x4 x5 x6 x7 x8) = x3"
intros: fixes \<I> :: "('c, 'd) \<I>"   and res :: "('c, 'd) resource" assumes "\<And>q r res'. \<lbrakk>q \<in> outs_\<I> \<I>; (r, res') \<in> set_spmf (run_resource res q)\<rbrakk> \<Longrightarrow> r \<in> responses_\<I> \<I> q \<and> \<I> \<turnstile>res res' \<surd>" shows "\<I> \<turnstile>res res \<surd>"
stoch_proc_comp_proj_i_preimage: fixes i :: "nat"   and n :: "nat"   and X :: "nat \<Rightarrow> 'b \<Rightarrow> 'c"   and x :: "'b" assumes "i \<le> n" shows "X i -` {X i x} = (\<Union>z\<in>comp_proj_i X n i (X i x). proj_stoch_proc X n -` {z})"
handshake_invL_def: shows "gc.handshake_invL = (\<lambda>s. (atS gc gc.hs_noop_locs s \<longrightarrow> sys_hs_type s\<down> = ht_NOOP) \<and> (atS gc gc.hs_get_roots_locs s \<longrightarrow> sys_hs_type s\<down> = ht_GetRoots) \<and> (atS gc gc.hs_get_work_locs s \<longrightarrow> sys_hs_type s\<down> = ht_GetWork) \<and> (atS gc gc.hs_mut_in_muts_locs s \<longrightarrow> gc_mut s\<down> \<in> gc_muts s\<down>) \<and> (atS gc gc.hs_init_loop_locs s \<longrightarrow> (\<forall>x. x \<notin> gc_muts s\<down> \<longrightarrow> sys_hs_pending x s\<down> \<or> sys_ghost_hs_in_sync x s\<down>)) \<and> (atS gc gc.hs_init_loop_not_done_locs s \<longrightarrow> (\<forall>x. x \<in> gc_muts s\<down> \<longrightarrow> \<not> sys_hs_pending x s\<down> \<and> \<not> sys_ghost_hs_in_sync x s\<down>)) \<and> (atS gc gc.hs_init_loop_done_locs s \<longrightarrow> (sys_hs_pending (gc_mut s\<down>) s\<down> \<or> sys_ghost_hs_in_sync (gc_mut s\<down>) s\<down>) \<and> (\<forall>x. x \<in> gc_muts s\<down> \<and> x \<noteq> gc_mut s\<down> \<longrightarrow> \<not> sys_hs_pending x s\<down> \<and> \<not> sys_ghost_hs_in_sync x s\<down>)) \<and> (atS gc gc.hs_done_locs s \<longrightarrow> (\<forall>x. sys_hs_pending x s\<down> \<or> sys_ghost_hs_in_sync x s\<down>)) \<and> (atS gc gc.hs_done_loop_locs s \<longrightarrow> (\<forall>x. x \<notin> gc_muts s\<down> \<longrightarrow> \<not> sys_hs_pending x s\<down>)) \<and> (atS gc gc.hs_none_pending_locs s \<longrightarrow> (\<forall>x. \<not> sys_hs_pending x s\<down>)) \<and> (atS gc gc.hs_in_sync_locs s \<longrightarrow> (\<forall>x. sys_ghost_hs_in_sync x s\<down>)) \<and> (atS gc gc.hs_out_of_sync_locs s \<longrightarrow> (\<forall>x. \<not> sys_hs_pending x s\<down> \<and> \<not> sys_ghost_hs_in_sync x s\<down>)) \<and> (atS gc gc.hp_Idle_locs s \<longrightarrow> sys_ghost_hs_phase s\<down> = hp_Idle) \<and> (atS gc gc.hp_IdleInit_locs s \<longrightarrow> sys_ghost_hs_phase s\<down> = hp_IdleInit) \<and> (atS gc gc.hp_InitMark_locs s \<longrightarrow> sys_ghost_hs_phase s\<down> = hp_InitMark) \<and> (atS gc gc.hp_IdleMarkSweep_locs s \<longrightarrow> sys_ghost_hs_phase s\<down> = hp_IdleMarkSweep) \<and> (atS gc gc.hp_Mark_locs s \<longrightarrow> sys_ghost_hs_phase s\<down> = hp_Mark))"
Inf_finite_5_def: fixes A :: "finite_5 set" shows "\<Sqinter> A = (if a\<^sub>1 \<in> A \<or> a\<^sub>4 \<in> A \<and> (a\<^sub>2 \<in> A \<or> a\<^sub>3 \<in> A) then a\<^sub>1 else if a\<^sub>2 \<in> A then a\<^sub>2 else if a\<^sub>3 \<in> A then a\<^sub>3 else if a\<^sub>4 \<in> A then a\<^sub>4 else a\<^sub>5)"
valid': shows "valid_msaux args (\<tau> \<sigma> ne) aux' auxlist'"
IH: assumes "pcp C"   and "S \<in> C" shows "pcp_seq C S n \<in> C"
rep_inverse: fixes x :: "('d, 'e) mapping" shows "mapping.Mapping (mapping.rep x) = x"
dim_row_matrix_A_R: fixes signs :: "rat list list"   and subsets :: "(nat list \<times> nat list) list" shows "dim_row (M_mat_R signs subsets) = length subsets"
assms_2: shows "time (Rep_run sub n c) = \<tau>"
continuous_at_eps_delta: fixes x :: "'c"   and f :: "'c \<Rightarrow> 'd" shows "isCont f x = (\<forall>e>0. \<exists>d>0. \<forall>x'. dist x' x < d \<longrightarrow> dist (f x') (f x) < e)"
elims: fixes x :: "nat"   and xa :: "atom list"   and y :: "real mpoly option" assumes "find_luckiest_eq x xa = y" obtains v :: "nat" where "x = v__" and "xa = []" and "y = None"   | v :: "nat" and p :: "real mpoly" and L :: "atom list" where "x = va__" and "xa = Eq p # La__" and "y = (if MPoly_Type.degree p va__ = 1 \<or> MPoly_Type.degree p va__ = 2 then let (a, b, c) = get_coeffs va__ p in case get_if_const a of None \<Rightarrow> find_luckiest_eq va__ La__ | Some a \<Rightarrow> case get_if_const b of None \<Rightarrow> find_luckiest_eq va__ La__ | Some b \<Rightarrow> case get_if_const c of None \<Rightarrow> find_luckiest_eq va__ La__ | Some c \<Rightarrow> if a \<noteq> 0 \<or> b \<noteq> 0 \<or> c \<noteq> 0 then Some p else find_luckiest_eq va__ La__ else find_luckiest_eq va__ La__)"   | v :: "nat" and va :: "real mpoly" and L :: "atom list" where "x = vb__" and "xa = Less vaa__ # Lb__" and "y = find_luckiest_eq vb__ Lb__"   | v :: "nat" and va :: "real mpoly" and L :: "atom list" where "x = vc__" and "xa = Leq vab__ # Lc__" and "y = find_luckiest_eq vc__ Lc__"   | v :: "nat" and va :: "real mpoly" and L :: "atom list" where "x = v" and "xa = Neq va # L" and "y = find_luckiest_eq v L"
pseudonatural_transformation_axioms: shows "pseudonatural_transformation (\<cdot>\<^sub>C) (\<star>\<^sub>C) \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C (\<cdot>\<^sub>D) (\<star>\<^sub>D) \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F G \<Phi>\<^sub>G \<tau>\<^sub>0 \<tau>\<^sub>1"
lm002: fixes x :: "'c"   and X :: "'d set" shows "{x} \<times> ({X} - {{}}) \<in> allocationsUniverse"
iSince_imp_iEx: fixes I :: "nat set"   and P :: "nat \<Rightarrow> bool"   and Q :: "nat \<Rightarrow> bool" assumes "P t1. t1 \<S> t2 I. Q t2" shows "\<diamond> t I. Q t"
dropWhile_eq_drop: fixes P :: "'c \<Rightarrow> bool"   and xs :: "'c list" shows "dropWhile P xs = drop |takeWhile P xs| xs"
1: shows "igSubstInpIPresIGWlsInpSTR (errMOD MOD) \<and> igSubstBinpIPresIGWlsBinpSTR (errMOD MOD)"
dnf0: fixes n :: "nat" assumes "2 \<le> n" shows "(deriv ^^ n) f 0 = 0"
ipset_prefix_match_m: fixes pfx :: "'b prefix_match"   and rg :: "'b word set" shows "fst (ipset_prefix_match pfx rg) = rg \<inter> prefix_to_wordset pfx"
w\<^sub>0_def: shows "Semantics.w\<^sub>0 \<equiv> dw"
assms_4: shows "prv (scnj (insert \<phi> F))"
sets_measure_of_eq: fixes \<mu> :: "'a set \<Rightarrow> ennreal" shows "sets (measure_of \<Omega> (sigma_sets \<Omega> E) \<mu>) = sigma_sets \<Omega> E"
allDefs_def: fixes defs :: "'b \<Rightarrow> 'c \<Rightarrow> 'a set"   and phis :: "'b \<Rightarrow> 'c \<times> 'a \<Rightarrow> 'a list option"   and g :: "'b"   and n :: "'c" shows "CFG_SSA_base.allDefs defs phis g n \<equiv> defs g n \<union> CFG_SSA_base.phiDefs phis g n"
bt_augment_v_union_2: fixes ts :: "('d, 'c) bintree list"   and r :: "('d, 'c) bintree"   and q :: "('c, 'd) pq" shows "set |bts_augment ts (bt_augment r q)| = set |bts_augment ts q| \<union> set |bt_augment r q|"
oracle_encrypt2_def: shows "oracle_encrypt2 = (\<lambda>(k_prf, k_upf) b D (msg1, msg0). case length msg1 = prf_clen \<and> length msg0 = prf_clen of True \<Rightarrow> spmf_of_set prf_domain \<bind> (\<lambda>x. spmf_of_set (nlists UNIV prf_clen) \<bind> (\<lambda>P. let p = case D x of None \<Rightarrow> P | Some r \<Rightarrow> r; c = p [\<oplus>] (if b then msg1 else msg0); t = upf_fun k_upf (x @ c) in return_spmf (Some (x, c, t), D(x \<mapsto> p)))) | False \<Rightarrow> return_spmf (None, D))"
a_b_exchange: fixes a :: "bool"   and b :: "bool" assumes "a"   and "a = b" shows "b"
\<tau>exec_methI: 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 h :: "'heap"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and xcp :: "'addr option"   and s' :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option" assumes "exec_meth ci P ins xt t h (stk, loc, pc, xcp) \<lbrace>\<rbrace> h s'"   and "xcp = None \<Longrightarrow> \<tau>instr P h stk (ins ! pc)" shows "\<tau>exec_meth ci P ins xt t h (stk, loc, pc, xcp) s'"
split_head_repeat_2: fixes list1 :: "cell list"   and j :: "nat"   and list2 :: "cell list" shows "(Bk # list1 = Oc \<up> j @ list2) = (j = 0 \<and> Bk # list1 = list2)"
poly_map_poly: fixes a :: "'a"   and p :: "'a poly poly"   and x :: "'a poly" shows "poly (map_poly (\<lambda>p. poly p a) p) (poly x a) = poly (poly p x) a"