hcju/isabelleps huggingface dataset:
bconfs_Vals: fixes P :: "'a prog"   and sh :: "sheap"   and vs :: "val list"   and b :: "bool" shows "(P,sh \<turnstile>\<^sub>b (map Val vs,b) \<surd>) = (\<not> b)"
connectedin_Union: fixes \<U> :: "'c set set"   and X :: "'c topology" assumes "\<And>S. S \<in> \<U> \<Longrightarrow> connectedin X S"   and "\<Inter> \<U> \<noteq> {}" shows "connectedin X (\<Union> \<U>)"
power_mult_cong: fixes x :: "'a"   and n :: "nat"   and a :: "'a"   and m :: "'a"   and y :: "'a"   and b :: "'a" assumes "[x ^ n = a] (mod m)"   and "[y ^ n = b] (mod m)" shows "[(x * y) ^ n = a * b] (mod m)"
ipPartioningAddSubset0: fixes ts :: "'b set set"   and a :: "'b set" assumes "disjoint ts" shows "ipPartition ts (addSubsetSet a ts)"
leg1_commutes: shows "trg.cod.leg1 \<cdot> trg.chine = trg.leg1"
sum: shows "(\<Sum>\<^sup>+ x. ennreal (f x)) = ennreal (weight_spmf p)"
ivt_increasing_component_on_1: fixes a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> 'b"   and k :: "'b"   and y :: "real" assumes "a \<le> b"   and "continuous_on {a..b} f"   and "f a \<bullet> k \<le> y"   and "y \<le> f b \<bullet> k" shows "\<exists>x\<in>{a..b}. f x \<bullet> k = y"
cut_and_update_imp_sc_completion: fixes s :: "('l, 'thread_id) locks \<times> (('thread_id \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('thread_id \<Rightarrow> 'w wait_set_status option) \<times> 'thread_id set"   and vs :: "'addr \<times> addr_loc \<Rightarrow> ('addr val \<times> bool) option" assumes "cut_and_update s vs" shows "sc_completion s vs"
bounded_iff: fixes b :: "'b"   and c :: "'b"   and a :: "'b" shows "(b \<sqinter> c \<le> a) = (b \<le> a \<and> c \<le> a)"
A2: fixes m :: "nat" shows "s m k \<in> carrier (residue_ring (p ^ k))"
frp_invar_def: fixes T1 :: "('d, 'f) tree_automaton_rec"   and T2 :: "('e, 'f) tree_automaton_rec" shows "frp_invar T1 T2 \<equiv> frp_invar_add (ta_rules T1) (ta_rules T2) \<inter> {s. frp_\<alpha> s \<in> dfs_invar (ta_initial T1 \<times> ta_initial T2) (f_succ (\<delta>_prod (ta_rules T1) (ta_rules T2)))}"
rec_prime_def: shows "rec_prime = Cn (Suc 0) rec_conj [Cn (Suc 0) rec_less [constn 1, recf.id (Suc 0) 0], rec_all (Cn 1 rec_minus [recf.id 1 0, constn 1]) (rec_all (Cn 2 rec_minus [recf.id 2 0, Cn 2 (constn 1) [recf.id 2 0]]) (Cn 3 rec_noteq [Cn 3 rec_mult [recf.id 3 1, recf.id 3 2], recf.id 3 0]))]"
g: shows "(\<Sum>i\<in>U. g i * i) = a"
simps_1: fixes \<theta>1 :: "('b + 'e, 'd) trm"   and \<theta>2 :: "('b + 'e, 'd) trm"   and \<sigma> :: "'e \<Rightarrow> ('b, 'd) trm" shows "FsubstFO (Geq \<theta>1 \<theta>2) \<sigma> = Geq (TsubstFO \<theta>1 \<sigma>) (TsubstFO \<theta>2 \<sigma>)"
less_le: fixes x :: "'a word"   and y :: "'a word" shows "(x <s y) = (x \<le>s y \<and> x \<noteq> y)"
maxsimpD_simplex: fixes x :: "'a set" assumes "chamber x" shows "x \<in> X"
R44_2: shows "pmf (sds R44) d = 1 - pmf (sds R12) a"
TC_mult: fixes x :: "V"   and y :: "V" shows "TC (x * y) = (SUP r\<in>elts (TC x). SUP u\<in>elts (TC y). set {x * u + r})"
assms: shows "\<bar>arctan x + arctan y\<bar> < pi / 2"
mtype_mbody_length: fixes CT :: "nat \<Rightarrow> classDef option"   and m :: "nat"   and C :: "nat"   and Cs :: "nat list"   and C0 :: "nat"   and xs :: "nat list"   and e :: "exp" assumes "mtype(CT,m,C) = Cs \<rightarrow> C0"   and "mbody(CT,m,C) = xs . e" shows "length xs = length Cs"
range_None: fixes A :: "'a set" assumes "local.MaxR.MaxR_opt A = None" shows "A \<inter> Field r = {}"
get_parent_locs_def: shows "get_parent_locs = (\<Union>ptr. get_child_nodes_locs ptr \<union> {preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t ptr RObject.nothing)})"
comp_eq_dest: fixes a :: "'i \<Rightarrow> 'h"   and b :: "'g \<Rightarrow> 'i"   and c :: "'j \<Rightarrow> 'h"   and d :: "'g \<Rightarrow> 'j"   and v :: "'g" assumes "a \<circ> b = c \<circ> d" shows "a (b v) = c (d v)"
filterlim_def: fixes f :: "'c \<Rightarrow> 'd"   and F2 :: "'d filter"   and F1 :: "'c filter" shows "filterlim f F2 F1 = (filtermap f F1 \<le> F2)"
ordLess_or_ordLeq: fixes r :: "('d \<times> 'd) set"   and r' :: "('e \<times> 'e) set" assumes "Well_order r"   and "Well_order r'" shows "r <o r' \<or> r' \<le>o r"
le_fin_ind': fixes ind :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and w :: "'a list"   and v :: "'a list" assumes "traces ind"   and "traces.le_fin ind [a] w"   and "traces.le_fin ind v w"   and "a \<notin> set v" shows "\<forall>a\<in>{a}. \<forall>b\<in>set v. ind a b"
iterate_rule_P: fixes S :: "'a"   and I :: "('Q, 'L) ta_rule set \<Rightarrow> '\<Sigma> \<Rightarrow> bool"   and \<sigma>0 :: "'\<Sigma>"   and f :: "('Q, 'L) ta_rule \<Rightarrow> '\<Sigma> \<Rightarrow> '\<Sigma>"   and P :: "'\<Sigma> \<Rightarrow> bool" assumes "invar S"   and "I (\<alpha> S) \<sigma>0"   and "\<And>x it \<sigma>. \<lbrakk>x \<in> it; it \<subseteq> \<alpha> S; I it \<sigma>\<rbrakk> \<Longrightarrow> I (it - {x}) (f x \<sigma>)"   and "\<And>\<sigma>. I {} \<sigma> \<Longrightarrow> P \<sigma>" shows "P (iteratei S (\<lambda>_. True) f \<sigma>0)"
min_elt_ex: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set"   and B :: "'a set" assumes "minimal_element P A"   and "B \<subseteq> A"   and "B \<noteq> {}" shows "\<exists>x. x \<in> B \<and> (\<forall>y\<in>A. P y x \<longrightarrow> y \<notin> B)"
lang_dual_lderiv_dual: fixes \<Sigma> :: "nat \<Rightarrow> 'a set"   and project :: "'a \<Rightarrow> 'a"   and embed :: "'a \<Rightarrow> 'a list"   and wf_atom :: "nat \<Rightarrow> 'b \<Rightarrow> bool"   and n :: "nat"   and r :: "'b rexp_dual"   and w :: "'a"   and lookup :: "'b \<Rightarrow> 'a \<Rightarrow> bool" assumes "Pi_Derivatives.embed \<Sigma> project embed"   and "alphabet.wf_dual wf_atom n r"   and "w \<in> \<Sigma> n" shows "project.lang_dual \<Sigma> project lookup n (embed.lderiv_dual lookup embed w r) = lQuot w (project.lang_dual \<Sigma> project lookup n r)"
1: shows "rep f \<in> f \<and> Arr (rep f) \<and> Can (rep f) \<and> Ide \<^bold>\<lfloor>rep f\<^bold>\<rfloor>"
domains_comp: fixes g :: "'a \<times> 'a"   and f :: "'a \<times> 'a" assumes "VV.seq g f" shows "VV.domains (VV.comp g f) = VV.domains f"
Red_F_Bot: fixes B :: "'f"   and N :: "'f set" assumes "B \<in> Bot"   and "N \<Turnstile> {B}" shows "N - Red_F N \<Turnstile> {B}"
sem_5bd3: fixes X :: "w \<Rightarrow> bool"   and Y :: "w \<Rightarrow> bool"   and Z :: "w \<Rightarrow> bool" shows "ob X Y \<and> X \<sqsubseteq> Z \<longrightarrow> ob Z ((\<sim>X) \<squnion> Y)"
snapshot_eq_filter: fixes PROB :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) set"   and s :: "('c, 'd) fmap" shows "snapshot PROB s = Set.filter (\<lambda>a. agree (fst a) s \<and> agree (snd a) s) PROB"
lcosI: fixes h :: "'a"   and H :: "'a set"   and x :: "'a" assumes "h \<in> H"   and "H \<subseteq> carrier G"   and "x \<in> carrier G" shows "x \<otimes> h \<in> x <# H"
Cons_4: shows "\<forall>c. c \<in> set (clause # Wl') \<longrightarrow> 0 \<le> c \<and> c < length (getF state)"
less_eq_rexp_def: fixes r :: "'a rexp"   and s :: "'a rexp" shows "r \<le> s \<equiv> local.le_rexp r s"
sp: shows "s \<notin> L1 \<union> L2 \<and> p \<notin> L1 \<union> L2 \<and> s \<noteq> p"
image_ring_carrier: fixes f :: "'e \<Rightarrow> 'd"   and R :: "('e, 'f) ring_scheme" shows "carrier (image_ring f R) = f ` carrier R"
circ_slide_1: fixes x :: "'a"   and y :: "'a" shows "x * (y * x)\<^sup>\<circ> = (x * y)\<^sup>\<circ> * x"
MapsToId: fixes X :: "'a" assumes "X \<in> obj" shows "id X \<approx>> id X"
unique_factorization_mI: fixes f :: "int poly"   and c :: "int"   and fs :: "int poly multiset" assumes "factorization_m f (c, fs)"   and "\<And>d gs. factorization_m f (d, gs) \<Longrightarrow> Mf (d, gs) = Mf (c, fs)" shows "unique_factorization_m f (c, fs)"
Abs_type': fixes \<Gamma> :: "binding list"   and x :: "vrs"   and s :: "trm"   and S :: "ty"   and U :: "ty"   and U' :: "ty" assumes "\<Gamma> \<turnstile> (\<lambda>x:S. s) : U \<rightarrow> U'"   and "x \<sharp> \<Gamma>" obtains S' :: "ty" where "\<Gamma>\<turnstile>U<:S" and "VarB x S # \<Gamma> \<turnstile> s : S'" and "\<Gamma>\<turnstile>S'<:U'"
mono_sync_FD_oneside: assumes "Q \<le> Q'" shows "(Q \<lbrakk>A\<rbrakk> P) \<le> (Q' \<lbrakk>A\<rbrakk> P)"
rel_pmf_return_pmf1: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'a"   and M :: "'b pmf" shows "rel_pmf R (return_pmf x) M = (\<forall>a\<in>set_pmf M. R x a)"
stable_rank_alt_def: fixes \<Sigma> :: "'c set"   and \<delta> :: "'d \<Rightarrow> 'c \<Rightarrow> 'd"   and q\<^sub>0 :: "'d"   and w :: "nat \<Rightarrow> 'c"   and x :: "nat"   and n :: "nat"   and j :: "nat" assumes "semi_mojmir \<Sigma> \<delta> q\<^sub>0 w" shows "(semi_mojmir_def.rank \<Sigma> \<delta> q\<^sub>0 w x n = Some j \<and> semi_mojmir.stable_rank \<Sigma> \<delta> q\<^sub>0 w x j) = (\<forall>m\<ge>n. semi_mojmir_def.rank \<Sigma> \<delta> q\<^sub>0 w x m = Some j)"
simps_15: fixes x31 :: "'b"   and x32 :: "('a, 'b) form list" shows "ff \<noteq> Modal x31 x32"
r_step_Mn_def: shows "r_step_Mn \<equiv> Cn 1 r_ifz [r_e2ls, Cn 1 r_prod_encode [Cn 1 r_cons [r_triple_encode (Cn 1 r_pdec22 [r_e2i]) (Cn 1 r_cons [Z, r_e2xs]) Z, Cn 1 r_cons [r_triple_encode r_e2i r_e2xs (Cn 1 r_singleton_encode [Z]), r_e2tail]], Z], Cn 1 r_ifeq [r_e2rv, r_const 1, Cn 1 r_prod_encode [r_e2tail, Cn 1 S [Cn 1 r_hd [r_e2ls]]], Cn 1 r_prod_encode [Cn 1 r_cons [r_triple_encode (Cn 1 r_pdec22 [r_e2i]) (Cn 1 r_cons [Cn 1 S [Cn 1 r_hd [r_e2ls]], r_e2xs]) Z, Cn 1 r_cons [r_triple_encode r_e2i r_e2xs (Cn 1 r_singleton_encode [Cn 1 S [Cn 1 r_hd [r_e2ls]]]), r_e2tail]], Z]]]"
P_def: shows "P = (\<lambda>n. {p \<in> {0<..n}. prime p})"
tendsto_exp_limit_at_top: fixes x :: "real" shows "((\<lambda>y. (1 + x / y) powr y) \<longlongrightarrow> exp x) at_top"
sup_inf_distrib1: fixes x :: "'b"   and y :: "'b"   and z :: "'b" shows "x \<squnion> y \<sqinter> z = (x \<squnion> y) \<sqinter> (x \<squnion> z)"
subdiv1_contr_axioms_def: fixes H :: "('c, 'd) pre_digraph"   and rev_H :: "'d \<Rightarrow> 'd"   and HM :: "'d pre_map" shows "subdiv1_contr_axioms H rev_H HM \<equiv> digraph_map H HM \<and> edge_rev HM = rev_H"
rep_eq: fixes x :: "mini_alg_unique" shows "mau_floor x = ma_floor (Rep_mini_alg_unique x)"
inter_aform_plane_refine_ex_typedef: fixes d :: "nat"   and xi :: "(real \<times> real pdevs) list"   and x :: "real list set"   and si :: "real list sctn"   and s :: "real list sctn"   and optns :: "nat" assumes "\<exists>Rep Abs. type_definition Rep Abs {0..<d}"   and "(xi, x) \<in> aforms_rel"   and "(si, s) \<in> \<langle>rl_rel\<rangle>sctn_rel"   and "length xi = d"   and "length (normal si) = d" shows "(nres_of (inter_aform_plane_lv (length xi) optns xi si), inter_sctn_specs d x s) \<in> \<langle>aforms_rel\<rangle>nres_rel"
ex: shows "\<exists>k<length vs. vs ! k \<noteq> 0"
carrier_vs_is_self: fixes N :: "'c set" shows "carrier (N.md N) = N"
coplanar_perm_20: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Coplanar Bet A B C D" shows "Tarski_neutral_dimensionless.Coplanar Bet D B A C"
less_fps_def: fixes f :: "('c, 'd) fps"   and g :: "('c, 'd) fps" shows "(f < g) = (f \<le> g \<and> f \<noteq> g)"
in_cong: fixes L :: "'a filter \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b) set"   and L' :: "'c filter \<Rightarrow> ('c \<Rightarrow> 'b) \<Rightarrow> ('c \<Rightarrow> 'b) set"   and Lr :: "'a filter \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> ('a \<Rightarrow> real) set"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and h :: "'a \<Rightarrow> 'b" assumes "landau_symbol L L' Lr"   and "\<forall>\<^sub>F x in F. f x = g x" shows "(f \<in> L F h) = (g \<in> L F h)"
card_Diff_subset: fixes B :: "'g set"   and A :: "'g set" assumes "finite B"   and "B \<subseteq> A" shows "card (A - B) = card A - card B"
fst_clock_sim: fixes jkbp :: "'b \<Rightarrow> ('b, 'c, 'd) GC list"   and agents :: "'b odlist"   and t :: "'e Trace"   and envInit :: "'e list"   and envAction :: "'e \<Rightarrow> 'f list"   and envTrans :: "'f \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> 'e \<Rightarrow> 'e"   and envVal :: "'e \<Rightarrow> 'c \<Rightarrow> bool"   and envObs :: "'b \<Rightarrow> 'e \<Rightarrow> 'g" assumes "FiniteLinorderEnvironment jkbp agents"   and "t \<in> PreEnvironmentJView.jkbpC jkbp envInit envAction envTrans envVal (Environment.clock_jview envObs)" shows "fst (Environment.clock_sim jkbp envInit envAction envTrans envVal envObs t) = tLast ` PreEnvironmentJView.jkbpCn jkbp envInit envAction envTrans envVal (Environment.clock_jview envObs) (tLength t)"
simps_1: fixes t' :: "Syntax.term"   and x :: "Syntax.var" shows "Unit[t' / x] = Unit"
monom_term_car: fixes c :: "nat \<Rightarrow> int"   and x :: "nat \<Rightarrow> int"   and n :: "nat" assumes "c \<in> carrier Zp"   and "x \<in> carrier Zp" shows "c \<otimes> x [^] n \<in> carrier Zp"
binary_def: fixes a :: "'b"   and b :: "'b" shows "binary a b = (\<lambda>x. b)(0 := a)"
lta_trans: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and A1 :: "'p"   and B1 :: "'p"   and C1 :: "'p"   and A2 :: "'p"   and B2 :: "'p"   and C2 :: "'p" assumes "A B C LtA A1 B1 C1"   and "A1 B1 C1 LtA A2 B2 C2" shows "A B C LtA A2 B2 C2"
distinct_2: fixes x1 :: "real" shows "PInfty \<noteq> real_min_top.R x1"
plus_fun_def: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" shows "f + g = (\<lambda>x. f x + g x)"
discI_1: fixes sum :: "'a + 'b"   and x1 :: "'a" assumes "sum = Inl x1" shows "isl sum"
d1_2: fixes p :: "('a \<times> 'a set) set" assumes "p tagged_division_of cbox a b"   and "d1 fine p" shows "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - I) < e"
set_conv_nth: fixes xs :: "'b list" shows "set xs = {xs ! i |i. i < length xs}"
set_intros_2: fixes y :: "'e"   and x22 :: "'e list"   and x21 :: "'e" assumes "y \<in> set x22" shows "y \<in> set (Cons x21 x22)"
f'_in: shows "f' \<in> A' \<rightarrow> UNIV"
assms_2: shows "\<exists>ps. valid_graph.is_path (del_unEdge v e v' G) v ps v'"
simps_2: fixes i :: "nat"   and j :: "nat" shows "[i..<Suc j] = (if i \<le> j then [i..<j] @ [j] else [])"
eq_fin_length: fixes w\<^sub>1 :: "'action list"   and w\<^sub>2 :: "'action list" assumes "w\<^sub>1 =\<^sub>F w\<^sub>2" shows "length w\<^sub>1 = length w\<^sub>2"
lres_inverse: fixes x :: "'a"   and y :: "'a" shows "x / y * y \<le> x"
assms: shows "homogeneous\<^sub>l\<^sub>s\<^sub>t m \<A> Sec"
eq1: shows "spair p q = monom_mult ((1::'b) / punit.lc (rep_list p)) (l - punit.lt (rep_list p)) p + monom_mult (- (1::'b) / punit.lc (rep_list q)) (l - punit.lt (rep_list q)) q"
EASSUME_def: fixes \<Phi> :: "bool" shows "EASSUME \<Phi> \<equiv> if \<Phi> then ERETURN () else SUCCEED"
induct: fixes P1 :: "fml \<Rightarrow> bool"   and P2 :: "game \<Rightarrow> bool"   and game :: "game" assumes "\<And>x1 x2. P1 (Pred x1 x2)"   and "\<And>x1 x2. P1 (Geq x1 x2)"   and "\<And>x. P1 x \<Longrightarrow> P1 (! x)"   and "\<And>x1 x2. \<lbrakk>P1 x1; P1 x2\<rbrakk> \<Longrightarrow> P1 (x1 && x2)"   and "\<And>x1 x2. P1 x2 \<Longrightarrow> P1 (Exists x1 x2)"   and "\<And>x1 x2. \<lbrakk>P2 x1; P1 x2\<rbrakk> \<Longrightarrow> P1 (\<langle> x1 \<rangle> x2)"   and "\<And>x. P2 (Game x)"   and "\<And>x1 x2. P2 (x1 := x2)"   and "\<And>x. P1 x \<Longrightarrow> P2 (? x)"   and "\<And>x1 x2. \<lbrakk>P2 x1; P2 x2\<rbrakk> \<Longrightarrow> P2 (x1 \<union>\<union> x2)"   and "\<And>x1 x2. \<lbrakk>P2 x1; P2 x2\<rbrakk> \<Longrightarrow> P2 (x1 ;; x2)"   and "\<And>x. P2 x \<Longrightarrow> P2 x**"   and "\<And>x. P2 x \<Longrightarrow> P2 x^d"   and "\<And>x1 x2. P2 (ODE x1 x2)" shows "P2 game"
poly_mult_degree_eq: fixes K :: "'a set"   and p1 :: "'a list"   and p2 :: "'a list" assumes "subring K R"   and "polynomial K p1"   and "polynomial K p2" shows "degree (poly_mult p1 p2) = (if p1 = [] \<or> p2 = [] then 0 else degree p1 + degree p2)"
Class_equivalence: fixes a :: "'a"   and b :: "'a" assumes "a \<in> M"   and "b \<in> M" shows "(Class a = Class b) = ((a, b) \<in> E(\<eta>))"
suffixE: fixes xs :: "'b list"   and ys :: "'b list" assumes "suffix xs ys" obtains zs :: "'b list" where "ys = zs @ xs"
sinks_aux_subset: fixes U :: "'c set"   and I :: "('c \<times> 'c) set"   and D :: "'d \<Rightarrow> 'c"   and xs :: "'d list" shows "U \<subseteq> sinks_aux I D U xs"
Int_left_commute: fixes A :: "'e set"   and B :: "'e set"   and C :: "'e set" shows "A \<inter> (B \<inter> C) = B \<inter> (A \<inter> C)"
Invariant_func_integral_2: fixes f :: "'a \<Rightarrow> 'b" assumes "f \<in> borel_measurable Invariants" shows "AE x in M. f x = integral\<^sup>L M f /\<^sub>R Sigma_Algebra.measure M (space M)"
simps_2: fixes c :: "color"   and lt :: "('a, 'b) rbt"   and k :: "'a"   and v :: "'b"   and rt :: "('a, 'b) rbt" shows "bheight (Branch c lt k v rt) = (if c = B then Suc (bheight lt) else bheight lt)"
RedLAss: fixes E :: "char list \<Rightarrow> ty option"   and V :: "char list"   and T :: "ty"   and 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 v :: "val"   and v' :: "val"   and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and l :: "char list \<Rightarrow> val option" assumes "E V = \<lfloor>T\<rfloor>"   and "P \<turnstile> T casts v to v' " shows "P,E \<turnstile> \<langle>V:=Val v,(h, l)\<rangle> \<rightarrow> \<langle>Val v',(h, l(V \<mapsto> v'))\<rangle>"
targetInRel: shows "\<forall>T1 T2. (T1, T2) \<in> TRel \<longrightarrow> (TargetTerm T1, TargetTerm T2) \<in> Rel"
step_approx: fixes gc :: "(cmd, local_state, global_state) Gen_Scheduler.global_config option"   and prog :: "'a program_scheme"   and gc' :: "(cmd, local_state, global_state) Gen_Scheduler.global_config option" assumes "gc \<in> gco_approx prog"   and "(gc, gc') \<in> li'.s1.step" shows "gc' \<in> gco_approx prog"
arr_dom: fixes f :: "'a" assumes "S.arr f" shows "S.arr (S.dom f)"
C0: shows "0 < c\<^sup>2 + 3 * d\<^sup>2"
list_rel_split_right_iff: fixes x :: "'a"   and xs :: "'a list"   and l :: "'b list"   and R :: "('a \<times> 'b) set" shows "((x # xs, l) \<in> \<langle>R\<rangle>list_rel) = (\<exists>y ys. l = y # ys \<and> (x, y) \<in> R \<and> (xs, ys) \<in> \<langle>R\<rangle>list_rel)"
hd_acc_lengths: fixes acc :: "nat"   and l :: "'a list"   and ls :: "'a list list" shows "hd (acc_lengths acc (l # ls)) = acc + length l"
rk_uniqueness: fixes A :: "Points"   and B :: "Points"   and C :: "Points"   and D :: "Points"   and M :: "Points"   and P :: "Points" assumes "rk {A, B} = 2"   and "rk {C, D} = 2"   and "rk {A, B, M} \<le> 2"   and "rk {C, D, M} \<le> 2"   and "rk {A, B, P} \<le> 2"   and "rk {C, D, P} \<le> 2"   and "3 \<le> rk {A, B, C, D}" shows "rk {M, P} = 1"
Abs_data_inverse: fixes y :: "'b list \<times> 'b dataspace" assumes "y \<in> data" shows "Rep_data (Abs_data y) = y"
assms_2: shows "s = C\<langle>t\<rangle>"
PNT2_imp_PNT3: assumes "(\<lambda>x. \<pi> x * ln (\<pi> x)) \<sim>[at_top] (\<lambda>x. x)" shows "(\<lambda>x. real (nth_prime x)) \<sim>[sequentially] (\<lambda>x. real x * ln (real x))"
odd_odd_even_nat': fixes a :: "nat"   and b :: "nat" assumes "odd a"   and "odd b" shows "even (((a * b)\<^sup>2 - 1) div 8) = even ((a\<^sup>2 - 1) div 8 + (b\<^sup>2 - 1) div 8)"
unary_rec_into_pr: fixes g :: "nat \<Rightarrow> nat"   and h :: "nat \<Rightarrow> nat" assumes "g \<in> PrimRec1"   and "h \<in> PrimRec1" shows "UnaryRecOp g h \<in> PrimRec1"
406: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "- (x \<sqinter> (y \<sqinter> z)) \<sqinter> - (x \<sqinter> y) = - (x \<sqinter> y)"
l3_obs_secrecy: shows "oreach l3 \<subseteq> l3_secrecy"
reducecoeffh: fixes t :: "MIR.num"   and g :: "int"   and bs :: "real list" assumes "dvdnumcoeff t g"   and "0 < g" shows "real_of_int g * Inum bs (reducecoeffh t g) = Inum bs t"
append_eq_as_proj_1: fixes as1 :: "'d list"   and as2 :: "'d list"   and as3 :: "'d list"   and f1 :: "'d \<Rightarrow> bool"   and f2 :: "'d \<Rightarrow> 'd"   and p :: "'d list" assumes "as1 @ as2 @ as3 = filter f1 (map f2 p)" shows "\<exists>p_1 p_2 p_3. p_1 @ p_2 @ p_3 = p \<and> as1 = filter f1 (map f2 p_1) \<and> as2 = filter f1 (map f2 p_2) \<and> as3 = filter f1 (map f2 p_3)"
simps_7: fixes v :: "'a" shows "qfree (Atom v) = True"
step_red: fixes c :: "nat \<times> cell list \<times> cell list"   and p :: "(action \<times> nat) list \<times> nat"   and n :: "nat" shows "steps c p (Suc n) = step (steps c p n) p"
Ss: shows "\<forall>y\<in>LVE \<union> LVQ. let Ss = SS y in \<forall>s1 s2. s1 = s2 on Ss \<longrightarrow> S s1 y = S s2 y"
lm064: fixes a :: "('a \<times> 'b set) set"   and G :: "'b set"   and N :: "'a set"   and bids :: "'a \<times> 'b set \<Rightarrow> 'c" assumes "a \<in> allAllocations N G" shows "sum (summedBidVector bids N G) a = sum (sum bids) (omega ` a)"
subsetD: fixes A :: "'x set"   and B :: "'x set"   and c :: "'x" assumes "A \<subseteq> B"   and "c \<in> A" shows "c \<in> B"
assms_5: shows "prv (imp \<phi>1 \<chi>1)"
ftb_to_option_def: fixes b :: "'d flowtable_behavior" shows "ftb_to_option b \<equiv> case b of Action a \<Rightarrow> Some a | NoAction \<Rightarrow> None"
assms_2: fixes n :: "nat" shows "f n \<in> borel_measurable M"
intros_2: fixes xs :: "'a list"   and y :: "'a" assumes "hb_consistent xs"   and "\<forall>x\<in>set xs. \<not> y \<prec> x" shows "hb_consistent (xs @ [y])"
subst_support_def: fixes \<theta> :: "'c \<Rightarrow> ('d, 'c) Term.term"   and \<tau> :: "'c \<Rightarrow> ('d, 'e) Term.term" shows "(\<theta> supports \<tau>) = (\<tau> = \<theta> \<circ>\<^sub>s \<tau>)"
star_sim_right: fixes p :: "'a"   and x :: "'a" assumes "test p" shows "p \<cdot> x = x \<cdot> p \<longrightarrow> p \<cdot> x\<^sup>\<star> = (p \<cdot> x)\<^sup>\<star> \<cdot> p"
DIAndConcl_def: fixes vid1 :: "'a"   and vid2 :: "'a"   and vid3 :: "'a"   and fid1 :: "'b"   and fid2 :: "'b"   and fid3 :: "'b"   and pid1 :: "'c"   and pid2 :: "'c"   and pid3 :: "'c"   and pid4 :: "'c" assumes "ids vid1 vid2 vid3 fid1 fid2 fid3 pid1 pid2 pid3 pid4" shows "ids.DIAndConcl vid1 pid1 pid2 pid3 pid4 = ([], [(Pc pid1 && Pc pid2) \<rightarrow> ([[$\<alpha> vid1]]Pc pid3 && Pc pid4) \<rightarrow> [[$\<alpha> vid1]]Pc pid1 && Pc pid2])"
Limit_vfinite_in_VsetI: fixes \<alpha> :: "V"   and A :: "V" assumes "Limit \<alpha>"   and "A \<subseteq>\<^sub>\<circ> Vset \<alpha>"   and "vfinite A" shows "A \<in>\<^sub>\<circ> Vset \<alpha>"
adjacent_appendI: fixes w :: "nat \<times> nat"   and ts :: "'b tree list"   and us :: "'b tree list" assumes "0 < fst w"   and "fst w \<le> snd w"   and "case us of [] \<Rightarrow> adjacent w ts | u # us' \<Rightarrow> adjacent (Suc (r u), snd w) ts \<and> adjacent (fst w, case ts of [] \<Rightarrow> snd w | a # list \<Rightarrow> r u) (u # us')" shows "adjacent w (ts @ us)"
simps_2: fixes k :: "nat"   and m :: "nat"   and qs :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list" shows "exact_aux k (Suc m) qs = exact_aux k m (local.shift k m qs)"
allDefs_in_allVars: fixes v :: "'val"   and g :: "'g"   and n :: "'node" assumes "v \<in> allDefs g n"   and "n \<in> set (\<alpha>n g)" shows "v \<in> allVars g"
parts_image_priEK: fixes C :: "agent"   and KK :: "nat set"   and evs :: "event list" assumes "Key (priEK C) \<in> parts (Key ` KK \<union> knows Spy evs)"   and "evs \<in> set_pur" shows "priEK C \<in> KK \<or> C \<in> bad"
initialproc_def: fixes fg :: "('f, 'g, 'h, 'i, 'j) flowgraph_rec_scheme"   and p :: "'g" shows "initialproc fg p \<equiv> p = main fg \<or> (\<exists>u v. (u, Spawn p, v) \<in> edges fg)"
update_vardict_inv: fixes ss :: "(integer \<times> edge list) iarray"   and proc :: "nat \<times> edgeIndex \<times> procArg list \<times> varDecl list"   and v :: "(String.literal, variable) assoc_list"   and k :: "String.literal"   and x :: "variable"   and x' :: "variable" assumes "vardict_inv ss proc v"   and "lm.lookup k v = Some x"   and "variable_inv x'" shows "vardict_inv ss proc (lm.update k x' v)"