hcju/isabelleps huggingface dataset:
interior_diff: fixes S :: "'c set"   and T :: "'c set" shows "interior (S - T) = interior S - closure T"
S6RPCUnch: fixes p :: "PrIds"   and rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState" shows "\<turnstile> [RPCNext crCh rmCh rst p]_(r p) \<and> $S6 rmhist p \<longrightarrow> unchanged r p"
le_fin_transp: fixes ind :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and w\<^sub>1 :: "'a list"   and w\<^sub>2 :: "'a list"   and w\<^sub>3 :: "'a list" assumes "traces ind"   and "traces.le_fin ind w\<^sub>1 w\<^sub>2"   and "traces.le_fin ind w\<^sub>2 w\<^sub>3" shows "traces.le_fin ind w\<^sub>1 w\<^sub>3"
regular_cover_def: fixes I :: "'d set"   and \<psi> :: "'d \<Rightarrow> ('a, 'b) chart" shows "regular_cover I \<psi> = (countable I \<and> carrier = (\<Union>i\<in>I. domain (\<psi> i)) \<and> locally_finite_on carrier I (domain \<circ> \<psi>) \<and> (\<forall>i\<in>I. codomain (\<psi> i) = ball (0::'b) 3) \<and> carrier = (\<Union>i\<in>I. inv_chart (\<psi> i) ` ball (0::'b) 1))"
total_ordering_axioms: shows "total_ordering UNIV (\<le>) (<)"
poly_no_roots: fixes p :: "real poly" shows "(\<forall>x. poly p x \<noteq> 0) = (p \<noteq> 0 \<and> count_roots p = 0)"
distinct_getVars: fixes \<sigma>l :: "'tp list" shows "distinct (GE.getVars \<sigma>l)"
assms_2: shows "i && ~~ mask r2 = x && ~~ mask r2"
times_divide_eq_left: fixes divide :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and inverse :: "'e \<Rightarrow> 'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and one :: "'e"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and uminus :: "'e \<Rightarrow> 'e"   and b :: "'e"   and c :: "'e"   and a :: "'e" assumes "class.field divide inverse times one plus zero minus uminus" shows "times (divide b c) a = divide (times b a) c"
callee_invariant_extend_state_oracle2_const: fixes "oracle" :: "'a \<Rightarrow> 'c \<Rightarrow> ('d \<times> 'a) spmf"   and I :: "'b \<Rightarrow> bool" shows "callee_invariant oracle\<dagger> (\<lambda>(s, s'). I s')"
L_infinity_I_1: fixes f :: "'b \<Rightarrow> real"   and M :: "'b measure"   and C :: "real" assumes "f \<in> borel_measurable M"   and "AE x in M. \<bar>f x\<bar> \<le> C"   and "0 \<le> C" shows "f \<in> space\<^sub>N (\<LL> \<infinity> M)"
monad_state_optionT: fixes return :: "'b option \<Rightarrow> 'c"   and bind :: "'c \<Rightarrow> ('b option \<Rightarrow> 'c) \<Rightarrow> 'c"   and get :: "('d \<Rightarrow> 'c) \<Rightarrow> 'c"   and put :: "'d \<Rightarrow> 'c \<Rightarrow> 'c" assumes "monad_state return bind get put" shows "monad_state (return_option return) (bind_option return bind) (get_option get) (put_option put)"
hyps: fixes y :: "nat"   and x1 :: "'i list"   and x2 :: "'a list"   and y1 :: "'i list"   and y2 :: "'a list"   and m :: "nat" assumes "y < n"   and "list_all2 R x1 x2"   and "list_all2 R y1 y2" shows "poly_rel (karatsuba_main_i ops x1 y y1 m) (karatsuba_main x2 y y2 m)"
map_iterator_genord_rule_insert_P: fixes it :: "('a \<Rightarrow> bool) \<Rightarrow> ('b \<times> 'c \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a"   and m :: "'b \<Rightarrow> 'c option"   and R :: "'b \<times> 'c \<Rightarrow> 'b \<times> 'c \<Rightarrow> bool"   and I :: "'b set \<Rightarrow> 'a \<Rightarrow> bool"   and \<sigma>0 :: "'a"   and c :: "'a \<Rightarrow> bool"   and f :: "'b \<times> 'c \<Rightarrow> 'a \<Rightarrow> 'a"   and P :: "'a \<Rightarrow> bool" assumes "map_iterator_genord it m R"   and "I {} \<sigma>0"   and "\<And>k v it \<sigma>. \<lbrakk>c \<sigma>; k \<in> dom m - it; m k = Some v; it \<subseteq> dom m; I it \<sigma>; \<forall>k' v'. k' \<in> dom m - it - {k} \<and> m k' = Some v' \<longrightarrow> R (k, v) (k', v'); \<forall>k' v'. k' \<in> it \<and> m k' = Some v' \<longrightarrow> R (k', v') (k, v)\<rbrakk> \<Longrightarrow> I (insert k it) (f (k, v) \<sigma>)"   and "\<And>\<sigma>. I (dom m) \<sigma> \<Longrightarrow> P \<sigma>"   and "\<And>\<sigma> it. \<lbrakk>it \<subseteq> dom m; it \<noteq> dom m; \<not> c \<sigma>; I it \<sigma>; \<forall>k v k' v'. k \<in> it \<and> m k = Some v \<and> k' \<notin> it \<and> m k' = Some v' \<longrightarrow> R (k, v) (k', v')\<rbrakk> \<Longrightarrow> P \<sigma>" shows "P (it c f \<sigma>0)"
a_2: shows "list_all is_Assignment (unlabel (transaction_selects T))"
simps_1: fixes P :: "'a \<Rightarrow> bool" shows "lookup [] P = None"
unique_rootD_5: fixes plr :: "int poly \<times> rat \<times> rat"   and y :: "real" assumes "\<exists>!x. root_cond plr x" shows "(real_of_1 plr = y) = root_cond plr y"
maximal_ideal_ideal: fixes mx :: "'a set" assumes "maximal_ideal R mx" shows "ideal R mx"
assms: shows "t \<sqsubseteq>\<^sub>s\<^sub>e\<^sub>t ik\<^sub>e\<^sub>s\<^sub>t A \<union> assignment_rhs\<^sub>e\<^sub>s\<^sub>t A"
val_is_vD: fixes ip :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option" assumes "ip \<in> kD rt"   and "the (flag rt ip) = val" shows "ip \<in> vD rt"
subst4Chain: fixes xvec :: "name list"   and Tvec :: "'a list"   and T :: "'c" assumes "length xvec = length Tvec"   and "distinct xvec"   and "xvec \<sharp>* Tvec" shows "xvec \<sharp>* substCond T xvec Tvec"
Sum_any_lookup_times_eq: fixes x :: "'b \<Rightarrow>\<^sub>0 'a"   and f :: "'b \<Rightarrow> 'a" shows "(\<Sum>k. lookup x k * f k) = (\<Sum>k\<in>keys x. lookup x k * f k)"
tendsto_add_zero: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'c" assumes "(f \<longlongrightarrow> (0::'c)) F"   and "(g \<longlongrightarrow> (0::'c)) F" shows "((\<lambda>x. f x + g x) \<longlongrightarrow> (0::'c)) F"
unwind_def: fixes istate :: "'a"   and step :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<times> 'a"   and \<phi> :: "('a, 'b, 'c) trans \<Rightarrow> bool"   and f :: "('a, 'b, 'c) trans \<Rightarrow> 'd"   and \<gamma> :: "('a, 'b, 'c) trans \<Rightarrow> bool"   and g :: "('a, 'b, 'c) trans \<Rightarrow> 'e"   and T :: "('a, 'b, 'c) trans \<Rightarrow> bool"   and \<Delta> :: "'a \<Rightarrow> 'd list \<Rightarrow> 'a \<Rightarrow> 'd list \<Rightarrow> bool" shows "BD_Security.unwind istate step \<phi> f \<gamma> g T \<Delta> \<equiv> \<forall>s vl s1 vl1. BD_Security.reachNT istate step T s \<and> IO_Automaton.reach istate step s1 \<and> \<Delta> s vl s1 vl1 \<longrightarrow> vl \<noteq> [] \<and> BD_Security.exit step \<phi> f T s (hd vl) \<or> BD_Security.iaction step \<phi> f \<gamma> \<Delta> s vl s1 vl1 \<or> (vl \<noteq> [] \<or> vl1 = []) \<and> BD_Security.reaction step \<phi> f \<gamma> g T \<Delta> s vl s1 vl1"
inv_Mp_coeff: fixes f :: "int poly"   and n :: "nat" shows "coeff (inv_Mp f) n = inv_M (coeff f n)"
Predict_mk_regular1: fixes k :: "nat" shows "\<exists>P F. Predict k = mk_regular1 P F"
proots_count_uminus: fixes p :: "'b poly"   and s :: "'b set" shows "proots_count (- p) s = proots_count p s"
complex_homo_coords_eq_symp: shows "symp (\<approx>)"
npow_notin_prime: fixes P :: "'a set"   and x :: "'a" assumes "prime_ideal R P"   and "x \<in> carrier R"   and "x \<notin> P" shows "\<forall>n. x^\<^bsup>R n\<^esup> \<notin> P"
assms: fixes s :: "'b"   and a :: "'a"   and s' :: "'b" assumes "run_state m s = (a, s')" shows "P a"
arcsin_bounded: fixes y :: "real" assumes "- 1 \<le> y"   and "y \<le> 1" shows "- (pi / 2) \<le> arcsin y \<and> arcsin y \<le> pi / 2"
more_than_one_mset_mset_diff: fixes a :: "'b"   and M :: "'b multiset" assumes "a \<in># M - (a, {#})" shows "set_mset (M - (a, {#})) = set_mset M"
authTicket_crypt_authK: fixes A :: "agent"   and authK :: "nat"   and Ta :: "nat"   and evs :: "event list" assumes "Crypt (shrK Tgs) \<lbrace>Agent A, Agent Tgs, Key authK, Number Ta\<rbrace> \<in> parts (knows Spy evs)"   and "evs \<in> kerbIV_gets" shows "authK \<in> authKeys evs"
starts: fixes i :: "'b"   and j :: "'b" assumes "\<I> i"   and "\<I> j" shows "((i, j) \<in> s \<union> s\<inverse> \<union> e) = (BEGIN i = BEGIN j)"
one_closed: shows "\<zero>\<^bsub>K [X]\<^esub> \<in> carrier (K [X])"
intros_5: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and Exit :: "'b"   and Def :: "'b \<Rightarrow> 'c set"   and Use :: "'b \<Rightarrow> 'c set"   and ParamDefs :: "'b \<Rightarrow> 'c list"   and ParamUses :: "'b \<Rightarrow> 'c set list"   and n :: "'b SDG_node"   and ns :: "'b SDG_node list"   and n'' :: "'b SDG_node"   and p :: "'f"   and n' :: "'b SDG_node" assumes "SDG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit Def Use ParamDefs ParamUses"   and "SDG.intra_call_sum_SDG_path sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses n ns n''"   and "SDG.sum_SDG_edge sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses n'' None (Some (p, True)) False n'" shows "SDG.intra_call_sum_SDG_path sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses n (ns @ [n'']) n'"
DEFER_tag_def: fixes x :: "bool" shows "DEFER_tag x \<equiv> x"
Lfilled_exact_imp_Lfilled: fixes n :: "nat"   and lholed :: "Lholed"   and es :: "e list"   and LI :: "e list" assumes "Lfilled_exact n lholed es LI" shows "Lfilled n lholed es LI"
freeIn_substInfinitesimalLinear_fm: fixes var :: "nat"   and a :: "real mpoly"   and b :: "real mpoly"   and F :: "atom fm" assumes "var \<notin> vars a"   and "var \<notin> vars b" shows "freeIn var (substInfinitesimalLinear_fm var a b F)"
svP: fixes P :: "'s \<Rightarrow> real" assumes "sound P" shows "sound (v P)"
equivalent_symmetric: fixes S1 :: "'at Literal set set"   and S2 :: "'at Literal set set" assumes "equivalent S1 S2" shows "equivalent S2 S1"
starfun_eq: fixes f :: "'b \<Rightarrow> 'a"   and a :: "'b" shows "(*f* f) (star_of a) = star_of (f a)"
relators: fixes S :: "'a set"   and P :: "('a \<times> bool) list set" assumes "GroupByPresentation S P" shows "order.greater_eq \<langle>\<Union>g\<in>FreeGroup S. lconjby g ` Abs_freeword ` P\<rangle> (Abs_freeword ` P)"
of_int_or_eq: fixes k :: "int"   and l :: "int" shows "of_int (k OR l) = of_int k OR of_int l"
isDERIV_approx: shows "isDERIV_approx prec x f vs"
set_less_aux_irrefl: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set" shows "\<not> ord.set_less_aux less_eq A A"
simps_2: fixes \<sigma> :: "'tp" shows "GE_parOf (Guard \<sigma>) = [\<sigma>]"
brn_cf2: fixes I :: "nat set"   and i :: "nat" assumes "I \<in> P"   and "i \<in> F I" shows "i < brn (fst cf2)"
insert_spec_set: fixes ref :: "'b"   and xs :: "'b list"   and oid :: "'b" assumes "ref \<in> set xs" shows "set (insert_spec xs (oid, Some ref)) = set xs \<union> {oid}"
flow_to_path_unfold: fixes x :: "'a"   and t :: "real"   and t' :: "real"   and s :: "real" shows "flow_to_path x t t' s = flow0 x ((1 - s) * t + s * t')"
elims_2: fixes G :: "'a graph"   and x :: "'a"   and xa :: "('a \<times> 'a) list"   and xb :: "'a" assumes "wf_graph G"   and "wf_graph.is_path G x xa xb" obtains v :: "'a" and v' :: "'a" where "x = vb__" and "xa = []" and "xb = v'a__" and "vb__ = v'a__ \<and> v'a__ \<in> nodes G"   | v :: "'a" and v1 :: "'a" and v2 :: "'a" and p :: "('a \<times> 'a) list" and v' :: "'a" where "x = v" and "xa = (v1, v2) # p" and "xb = v'" and "v = v1 \<and> (v1, v2) \<in> edges G \<and> wf_graph.is_path G v2 p v'"
CX3h_singular: fixes h :: "H1"   and X :: "X3 set" shows "inj_on X3d (CX3h h X)"
simpler_modify_def: fixes f :: "'b \<Rightarrow> 'b" shows "modify f = (\<lambda>s. (((), f s), False))"
type_wf\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a: fixes h :: "(_) heap" assumes "DocumentClass.type_wf h" shows "CharacterDataClass.type_wf h"
assms_1: shows "mset_ran a r' = mset_ran a' r'"
gbg_impl_rel_eext_internal_def: fixes Rm :: "('b \<times> 'd) set"   and Rv :: "('a \<times> 'c) set" shows "gbg_impl_rel_eext Rm Rv \<equiv> \<langle>Rm, \<langle>\<langle>Rv\<rangle>list_set_rel\<rangle>list_set_rel\<rangle>gen_gbg_impl_rel_eext"
authKlife_def: shows "authKlife \<equiv> SOME authKlife. 2 \<le> authKlife"
prefix_contains_msg: fixes es :: "('a \<times> ('a, 'b) operation) event list"   and i :: "nat"   and m :: "'a \<times> ('a, 'b) operation" assumes "es prefix of i"   and "m \<in> set (node_deliver_messages es)" shows "Deliver m \<in> set es"
simps_2: fixes \<gamma> :: "'b set \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "('b, 'a) flow_entry_match"   and as :: "('b, 'a) flow_entry_match list"   and p :: "'c" shows "OF_match_linear \<gamma> (a # as) p = (if \<gamma> (ofe_fields a) p then Action (ofe_action a) else OF_match_linear \<gamma> as p)"
that: assumes "d permutes S"   and "\<forall>x\<in>S. d x \<noteq> x" shows "thesis"
assms_2: shows "inj_on g (dom (f \<circ> g))"
triangle_paths_1: fixes \<E> :: "'b set"   and \<P> :: "'b set set"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "MinkowskiPrimitive \<E> \<P>"   and "MinkowskiPrimitive.kinematic_triangle \<E> \<P> a b c" shows "\<exists>Q. Q \<in> \<P> \<and> a \<in> Q \<and> b \<in> Q \<and> a \<noteq> b"
simps_2: fixes x :: "'c"   and y :: "'c"   and ys :: "'c list" shows "index_of x (y # ys) = (if x = y then 0 else index_of x ys + 1)"
shift_read_follow: fixes m :: "bool"   and r :: "(bool \<times> 'b) rexp"   and x :: "'b" shows "shift m r x = read x (follow m r)"
assms_1: shows "eq_\<I>_gpv (=) \<I> gpv'' gpv'"
open_imp_ANR: fixes S :: "'a set" assumes "open S" shows "ANR S"
hoare_tvalid_def: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> bool"   and c :: "com"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> bool" shows "(\<Turnstile>\<^sub>t {P} c {Q}) = (\<forall>s. P s \<longrightarrow> (\<exists>t. (c, s) \<Rightarrow> t \<and> Q t))"
mssnth_mssalternate: fixes a :: "'a"   and b :: "'a"   and n :: "nat" shows "mssnth (mssalternate a b) n = (if even n then a else b)"
and_zero_eq: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bit :: "'c \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and drop_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and take_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and "and" :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and or :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and xor :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and mask :: "nat \<Rightarrow> 'c"   and a :: "'c" assumes "class.semiring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask" shows "and a zero = zero"
l_set_disconnected_nodes_get_shadow_root_def: fixes set_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_set_disconnected_nodes_get_shadow_root set_disconnected_nodes_locs get_shadow_root_locs \<equiv> \<forall>ptr h h' ptr'. \<forall>w\<in>set_disconnected_nodes_locs ptr. h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r\<in>get_shadow_root_locs ptr'. r h h')"
update_rightmost_eq_Leaf_iff: fixes f :: "'b \<Rightarrow> 'b"   and t :: "'b tree" shows "(update_rightmost f t = Leaf) = (t = Leaf)"
lang_INTERSECT: fixes n :: "nat"   and xs :: "'a atom rexp list" shows "lang n (INTERSECT xs) = (if xs = [] then lists ((set \<circ> \<sigma> \<Sigma>) n) else \<Inter> (lang n ` set xs))"
p: shows "smult c p = smult (c * d) (\<Prod>(a, i)\<in>set bs. a ^ Suc i)"
retraction_def: fixes g :: "'a" shows "retraction g \<equiv> \<exists>f. ide (g \<cdot> f)"
from_list_refine: fixes l :: "(('v \<times> 'v) \<times> nat) list" assumes "valid_wgraph_repr l" shows "invar (from_list l) \<and> \<alpha>g (from_list l) = graph_from_list (map fst l) \<and> \<alpha>w (from_list l) = weight_from_list l"
intros_1: fixes t :: "lam" shows " t \<longrightarrow>\<^sub>\<beta>\<^sup>* t"
map_S_rel_cong: fixes L1 :: "'b \<Rightarrow> 'd \<Rightarrow> bool"   and L2 :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and Co1 :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and Co2 :: "'i \<Rightarrow> 'j \<Rightarrow> bool"   and Contra1 :: "'k \<Rightarrow> 'l \<Rightarrow> bool"   and Contra2 :: "'m \<Rightarrow> 'n \<Rightarrow> bool"   and xf :: "('b, 'e, 'g, 'i, 'k, 'm, 'o) S"   and yf :: "('d, 'f, 'h, 'j, 'l, 'n, 'o) S"   and L1' :: "'p \<Rightarrow> 'q \<Rightarrow> bool"   and x :: "'b \<Rightarrow> 'p"   and y :: "'d \<Rightarrow> 'q"   and L2' :: "'r \<Rightarrow> 's \<Rightarrow> bool"   and xa :: "'e \<Rightarrow> 'r"   and ya :: "'f \<Rightarrow> 's"   and Co1' :: "'t \<Rightarrow> 'u \<Rightarrow> bool"   and xb :: "'g \<Rightarrow> 't"   and yb :: "'h \<Rightarrow> 'u"   and Co2' :: "'v \<Rightarrow> 'w \<Rightarrow> bool"   and xc :: "'i \<Rightarrow> 'v"   and yc :: "'j \<Rightarrow> 'w"   and Contra1' :: "'x \<Rightarrow> 'y \<Rightarrow> bool"   and xd :: "'x \<Rightarrow> 'k"   and yd :: "'y \<Rightarrow> 'l"   and Contra2' :: "'z \<Rightarrow> 'aa \<Rightarrow> bool"   and xe :: "'z \<Rightarrow> 'm"   and ye :: "'aa \<Rightarrow> 'n" assumes "rel_S L1 L2 Co1 Co2 Contra1 Contra2 xf yf"   and "\<And>xa ya. L1 xa ya \<Longrightarrow> L1' (x xa) (y ya)"   and "\<And>x y. L2 x y \<Longrightarrow> L2' (xa x) (ya y)"   and "\<And>x y. Co1 x y \<Longrightarrow> Co1' (xb x) (yb y)"   and "\<And>x y. Co2 x y \<Longrightarrow> Co2' (xc x) (yc y)"   and "\<And>x y. Contra1' x y \<Longrightarrow> Contra1 (xd x) (yd y)"   and "\<And>x y. Contra2' x y \<Longrightarrow> Contra2 (xe x) (ye y)" shows "rel_S L1' L2' Co1' Co2' Contra1' Contra2' (map_S x xa xb xc xd xe xf) (map_S y ya yb yc yd ye yf)"
root': shows "h' \<turnstile> get_root_node (cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r document_ptr) \<rightarrow>\<^sub>r cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r document_ptr"
subst_lit_in_negs_is_neg: fixes L :: "'a literal"   and \<sigma> :: "'s"   and AA :: "'a multiset" assumes "L \<cdot>l \<sigma> \<in># negs AA" shows "is_neg L"
add_mono: fixes m :: "nat"   and n :: "nat"   and ns :: "nat list" assumes "m \<le> n" shows "foldl (+) m ns \<le> foldl (+) n ns"
meet_lower2: fixes x :: "'b"   and y :: "'b" shows "x \<sqinter> y \<sqsubseteq> y"
uint_2p: fixes n :: "nat" assumes "0 < 2 ^ n" shows "uint (2 ^ n) = 2 ^ n"
ldistinct_LNil_code: shows "ldistinct LNil = True"
R: shows "cmod R \<le> cmod s / (2 * Re s) * a powr - Re s"
istep: fixes s :: "'a"   and s' :: "'a" assumes "C s"   and "I s"   and "B s = Some s'" shows "I s'"
is_increasing_comp: fixes f :: "nat \<Rightarrow> nat"   and g :: "nat \<Rightarrow> nat" assumes "is_increasing f"   and "is_increasing g" shows "is_increasing (f \<circ> g)"
simps_1: fixes v :: "'c"   and w :: "'c"   and xs :: "'c list" shows "H.first_edge_of_walk (v # w # xs) = (v, w)"
VCG_FRAME_def: fixes m :: "'b nres" shows "VCG_FRAME m \<equiv> m"
simps_3: fixes uu :: "'b" shows "(Some uu \<sqsubseteq> None) = False"
learn_lim_extra_wrt_goedel: fixes \<chi> :: "nat \<Rightarrow> nat \<Rightarrow> nat option"   and \<psi> :: "nat \<Rightarrow> nat \<Rightarrow> nat option"   and U :: "(nat \<Rightarrow> nat option) set"   and s :: "nat \<Rightarrow> nat option"   and extra :: "(nat \<Rightarrow> nat option) set \<Rightarrow> (nat \<Rightarrow> nat option) \<Rightarrow> nat \<Rightarrow> (nat \<Rightarrow> nat option) \<Rightarrow> bool" assumes "goedel_numbering \<chi>"   and "learn_lim \<psi> U s"   and "\<And>f n. f \<in> U \<Longrightarrow> extra U f n (\<psi> (the (s (f \<triangleright> n))))" shows "\<exists>t. learn_lim \<chi> U t \<and> (\<forall>f\<in>U. \<forall>n. extra U f n (\<chi> (the (t (f \<triangleright> n)))))"
select_memberE: fixes y :: "'a"   and f :: "'b \<Rightarrow> 'a option"   and S :: "'b set" assumes "y \<in> select f S" obtains x :: "'b" where "x \<in> S" and "f x = Some y"
HInv6_def: fixes s :: "state" shows "HInv6 s = ((chosen s \<noteq> NotAnInput \<longrightarrow> valueChosen s (chosen s)) \<and> (\<forall>p. outpt s p \<in> {chosen s, NotAnInput}))"
cobounded2: fixes a :: "'b"   and b :: "'b" shows "inf a b \<le> b"
zero_less_power_abs_iff: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and a :: "'a"   and n :: "nat" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less zero (power.power one times (abs a) n) = (a \<noteq> zero \<or> n = 0)"
abs_def: fixes R :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and Abs :: "'d set \<Rightarrow> 'e"   and Rep :: "'e \<Rightarrow> 'd set"   and x :: "'d" assumes "quot_type R Abs Rep" shows "quot_type.abs R Abs x = Abs (Collect (R x))"
simps: fixes a :: "('idx, 'pred, 'act) formula" shows "weak_formula a = ((\<exists>xset. a = Conj xset \<and> finite (supp xset) \<and> (\<forall>x. x \<in> set_bset xset \<longrightarrow> weak_formula x)) \<or> (\<exists>x. a = Formula.Not x \<and> weak_formula x) \<or> (\<exists>x \<alpha>. a = \<langle>\<langle>\<alpha>\<rangle>\<rangle>x \<and> weak_formula x) \<or> (\<exists>x \<phi>. a = \<langle>\<langle>\<tau>\<rangle>\<rangle>Conj (binsert (Pred \<phi>) (bsingleton x)) \<and> weak_formula x))"
carrier_integer_mod_group: fixes n :: "nat" shows "carrier (Z n) = (if n = 0 then UNIV else {0..<int n})"
H: shows "s\<^sub>0 \<Turnstile> u\<^sub>2 \<and> u\<^sub>2 \<turnstile> v\<^sub>2 \<and> (\<nexists>S. \<langle>n, PubKey S\<rangle> \<in> spied u\<^sub>2 \<and> Sec_PriK m \<in> S) \<and> (\<exists>S. \<langle>n, PubKey S\<rangle> \<in> spied v\<^sub>2 \<and> Sec_PriK m \<in> S)"
labeled_list_insert_eq_cases_2: fixes i :: "'a"   and d :: "'b"   and D :: "('a \<times> 'b) list" assumes "(i, d) \<in> set D" shows "List.insert d (unlabel D) = unlabel (List.insert (i, d) D)"
finite_degree_nonzero: fixes x :: "'b pdevs" shows "finite {i. pdevs_apply x i \<noteq> (0::'b)}"
add_edge_is_path: fixes G :: "('c, 'd) graph"   and x :: "'c"   and p :: "('c \<times> 'd \<times> 'c) list"   and y :: "'c"   and a :: "'c"   and b :: "'d"   and c :: "'c" assumes "valid_graph G"   and "is_path_undir G x p y" shows "is_path_undir (add_edge a b c G) x p y"
wf_atom_constT_imp_objT: fixes Q :: "variable \<Rightarrow> PDDL_STRIPS_Semantics.type option"   and a :: "PDDL_STRIPS_Semantics.term atom" assumes "wf_atom (ty_term Q constT) a" shows "wf_atom (ty_term Q objT) a"
ivp_unique_solution: fixes s :: "'a"   and U :: "'a \<Rightarrow> real set"   and t :: "real"   and Y\<^sub>1 :: "real \<Rightarrow> 'a"   and Y\<^sub>2 :: "real \<Rightarrow> 'a" assumes "s \<in> S"   and "is_interval (U s)"   and "U s \<subseteq> T"   and "t \<in> U s"   and "Y\<^sub>1 \<in> Sols f U S t\<^sub>0 s"   and "Y\<^sub>2 \<in> Sols f U S t\<^sub>0 s" shows "Y\<^sub>1 t = Y\<^sub>2 t"
IH_3: fixes a :: "atom"   and b :: "atom" shows "((a \<rightleftharpoons> b) \<bullet> t2 =\<^sub>\<alpha> t2) = (p \<bullet> (a \<rightleftharpoons> b) \<bullet> t2 =\<^sub>\<alpha> p \<bullet> t2)"
choose_iapath_def: fixes u :: "'a"   and v :: "'a" shows "choose_iapath u v \<equiv> let chosen_path = \<lambda>u v. SOME p. gen_iapath (verts3 (with_proj G)) u p v in if direct_arc (u, v) = (u, v) then chosen_path u v else rev_path (chosen_path v u)"