hcju/isabelleps huggingface dataset:
elims_1: fixes x :: "('b, 'c) Combinators list"   and y :: "bool" assumes "noDenyAll1 x = y" obtains xs :: "('b, 'c) Combinators list" where "x = DenyAll # xs" and "y = noDenyAll xs"   | "x = []" and "y = noDenyAll []"   | vb :: "'b" and vc :: "'b" and va :: "('b, 'c) Combinators list" where "x = DenyAllFromTo vb__ vc__ # va__" and "y = noDenyAll (DenyAllFromTo vb__ vc__ # va__)"   | vb :: "'b" and vc :: "'b" and vd :: "'c" and va :: "('b, 'c) Combinators list" where "x = AllowPortFromTo vba__ vca__ vd # vaa__" and "y = noDenyAll (AllowPortFromTo vba__ vca__ vd # vaa__)"   | vb :: "('b, 'c) Combinators" and vc :: "('b, 'c) Combinators" and va :: "('b, 'c) Combinators list" where "x = vb \<oplus> vc # va" and "y = noDenyAll (vb \<oplus> vc # va)"
intro: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" assumes "l_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf known_ptr get_child_nodes get_child_nodes_locs" shows "l_new_element_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf known_ptr get_child_nodes get_child_nodes_locs"
assms: shows "snd (dispatch_instruction instr s) = False \<and> s' = snd (fst (dispatch_instruction instr s)) \<and> get_S (cpu_reg_val PSR s) = 0"
n_val_Pg: fixes v :: "'a \<Rightarrow> ant" assumes "valuation K v" shows "n_val K v (Pg K v) = 1"
identity_arrow_of_spans_axioms: shows "identity_arrow_of_spans (\<cdot>) (\<nu> \<star> \<pi>)"
step: fixes Sup :: "'b set \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and f :: "'b \<Rightarrow> 'b" assumes "class.ccpo Sup less_eq less"   and "x \<in> ccpo.iterates Sup less_eq f" shows "f x \<in> ccpo.iterates Sup less_eq f"
continuous_within_compose: fixes x :: "'b"   and s :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and g :: "'c \<Rightarrow> 'd" assumes "continuous (at x within s) f"   and "continuous (at (f x) within f ` s) g" shows "continuous (at x within s) (g \<circ> f)"
run_limit_two_connectedI: fixes E :: "('e \<times> 'e) set"   and r :: "nat \<Rightarrow> 'e"   and a :: "'e"   and b :: "'e" assumes "ipath E r"   and "a \<in> limit r"   and "b \<in> limit r" shows "(a, b) \<in> E\<^sup>+"
3: shows "LBINT x:{a<..}. (b + x) powr c = - ((b + a) powr (c + 1) / (c + 1))"
fls_times_base_factor_conv_shifted_times: fixes f :: "'a fls"   and g :: "'a fls" shows "fls_base_factor f * fls_base_factor g = fls_shift (fls_subdegree f + fls_subdegree g) (f * g)"
assms_2: shows "bdd_below {u n / real n |n. 0 < n}"
fieldE_1: fixes R :: "('c, 'd) ring_scheme" assumes "field R" shows "cring R"
assms_1: shows "A \<subseteq>\<^sub>\<circ> C"
rel_map_2: fixes Sa :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'c list"   and g :: "'e \<Rightarrow> 'd"   and y :: "'e list" shows "list_all2 Sa x (map g y) = list_all2 (\<lambda>x y. Sa x (g y)) x y"
Lambda_ext: fixes s :: "ZF"   and f :: "ZF \<Rightarrow> ZF"   and t :: "ZF"   and g :: "ZF \<Rightarrow> ZF" shows "(Lambda s f = Lambda t g) = (s = t \<and> (\<forall>x. Elem x s \<longrightarrow> f x = g x))"
id_ideal_psub_sum: fixes I :: "'a set"   and a :: "'a" assumes "ideal R I"   and "a \<in> carrier R"   and "a \<notin> I" shows "I \<subset> I \<minusplus> R \<diamondsuit>\<^sub>p a"
reflexive: fixes Rel :: "(ccs \<times> ccs) set"   and P :: "ccs" assumes "Id \<subseteq> Rel" shows "P \<leadsto><Rel> P"
pnu_isotone: fixes f :: "'a \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'a" assumes "has_greatest_postfixpoint f"   and "has_greatest_postfixpoint g"   and "f \<le>\<le> g" shows "p\<nu> f \<le> p\<nu> g"
mem_hmsetmset_encodeD: fixes M :: "hmultiset"   and e :: "nat"   and n :: "nat" assumes "M \<in># hmsetmset (local.encode e n)" shows "\<exists>e'\<ge>e. M = local.encode 0 e'"
sum_distrib_left: fixes r :: "'a"   and f :: "'c \<Rightarrow> 'a"   and A :: "'c set" shows "r * sum f A = (\<Sum>n\<in>A. r * f n)"
cSup_eq_maximum: fixes z :: "'d multiset"   and X :: "'d multiset set" assumes "z \<in> X"   and "\<And>x. x \<in> X \<Longrightarrow> x \<subseteq># z" shows "Sup X = z"
nextl_Accessible_dfa: fixes q :: "hf"   and u :: "'a list" assumes "q \<in> accessible" shows "Accessible.nextl q u = nextl q u"
success_while_induct: fixes R :: "(heap \<times> heap) set"   and I :: "heap \<Rightarrow> bool"   and a :: "'b array"   and key :: "'b"   and i :: "nat ref"   and f :: "'c Heap"   and h :: "heap" assumes "wf R"   and "\<And>h. I h \<Longrightarrow> success (shiftr_p a key i) h"   and "\<And>h. \<lbrakk>I h; success (shiftr_p a key i) h; cond (shiftr_p a key i) h\<rbrakk> \<Longrightarrow> success f h"   and "I h"   and "\<And>h h' r. \<lbrakk>I h; success (shiftr_p a key i) h; cond (shiftr_p a key i) h; effect f h h' r\<rbrakk> \<Longrightarrow> (h', h) \<in> R \<and> I h'" shows "success (while (shiftr_p a key i) f) h"
assms_2: shows "length ts \<le> 4 * k + 1"
trg_def: fixes F :: "('a E.term, 'a) arr" shows "trg F \<equiv> if arr F then MkIde (E.Trg (Dom F)) else null"
obj_src: fixes \<mu> :: "'a" assumes "C.arr \<mu>" shows "C.obj (src\<^sub>C \<mu>)"
all_intersection_of: fixes P :: "'c set set \<Rightarrow> bool"   and Q :: "'c set \<Rightarrow> bool"   and R :: "'c set \<Rightarrow> bool" shows "(\<forall>S. (P intersection_of Q) S \<longrightarrow> R S) = (\<forall>T. P T \<and> T \<subseteq> Collect Q \<longrightarrow> R (\<Inter> T))"
distinct_pair_filter: fixes xs :: "('c \<times> 'd) list"   and a :: "'c"   and b :: "'d" assumes "distinct (map fst xs)"   and "(a, b) \<in> set xs" shows "map snd (filter (\<lambda>x. fst x = a) xs) = [b]"
ntsubst_preserves_free: fixes \<theta> :: "('h + 'i, 'j) trm"   and \<sigma> :: "'i \<Rightarrow> ('h, 'j) trm" assumes "dfree \<theta>"   and "\<And>i. dfree (\<sigma> i)" shows "dfree (TsubstFO \<theta> \<sigma>)"
bij_betw_mset_set: shows "bij_betw mset_set {A. finite A} {M. \<forall>x. count M x \<le> 1}"
diff_divide_distrib: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and inverse :: "'d \<Rightarrow> 'd"   and one :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and a :: "'d"   and b :: "'d"   and c :: "'d" assumes "class.division_ring divide inverse one times plus zero minus uminus" shows "divide (minus a b) c = minus (divide a c) (divide b c)"
assms_2: shows "formulaFalse F0 (elements (getM state))"
assoc_simps_3: fixes f :: "'a"   and g :: "'a"   and h :: "'a" assumes "ide f"   and "ide g"   and "ide h"   and "src f = trg g"   and "src g = trg h" shows "trg \<a>[f, g, h] = trg f"
indRelTEQ_to_TRel_4: fixes P :: "('procS, 'procT) Proc"   and Q :: "('procS, 'procT) Proc"   and TRel :: "('procT \<times> 'procT) set" assumes "P \<sim>\<lbrakk>\<cdot>\<rbrakk>T<TRel> Q" shows "\<forall>TP TQ. TP \<in>T P \<and> TQ \<in>T Q \<longrightarrow> (TP, TQ) \<in> (TRel \<union> _Collect (T1, T2) (\<exists>S. T1 = \<lbrakk>S\<rbrakk> \<and> T2 = \<lbrakk>S\<rbrakk>))\<^sup>+"
N_disjoint: shows "(h \<circ> list_of) -` {1 - i} \<inter> list.set ` AM \<inter> Pow N = {}"
eq_fin_le_fin_transp: fixes w\<^sub>1 :: "'item list"   and w\<^sub>2 :: "'item list"   and w\<^sub>3 :: "'item list" assumes "w\<^sub>1 =\<^sub>F w\<^sub>2"   and "w\<^sub>2 \<preceq>\<^sub>F w\<^sub>3" shows "w\<^sub>1 \<preceq>\<^sub>F w\<^sub>3"
bal_tree_def: fixes n :: "nat"   and t :: "'b tree" shows "Root_Balanced_Tree.bal_tree n t = Time_Monad.val (bal_tree_tm n t)"
IS_LEFT_UNIQUE_def: fixes R :: "('a \<times> 'b) set" shows "IS_LEFT_UNIQUE R \<equiv> single_valued (R\<inverse>)"
simps: fixes \<tau> :: "ty" shows "SRED \<tau> = _Collect k (\<forall>t\<in>RED \<tau>. SN ([t] \<star> k))"
Group_RSpan: fixes ms :: "'m list" assumes "set ms \<subseteq> M" shows "Group (RSpan ms)"
of_int_le_iff: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sgn :: "'d \<Rightarrow> 'd"   and abs :: "'d \<Rightarrow> 'd"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and w :: "int"   and z :: "int" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less_eq (ring_1.of_int one plus zero minus w) (ring_1.of_int one plus zero minus z) = (w \<le> z)"
fatom_of_nat_bij: shows "bij_betw fatom_of_nat UNIV ground_fatoms"
eq: shows "mbufnt_take f z (mbufn_add xss buf) nts = (z', buf', nts')"
PiE_empty_range: fixes i :: "'b"   and I :: "'b set"   and F :: "'b \<Rightarrow> 'c set" assumes "i \<in> I"   and "F i = {}" shows "Pi\<^sub>E I F = {}"
assms: shows "\<forall>op\<in>set \<psi>. op \<in> set (\<Psi>\<^sub>\<O>\<^sub>+)"
law_restrict_hom: fixes h :: "'b \<Rightarrow> 'c"   and A :: "('b, 'd) ring_scheme"   and B :: "('c, 'e) ring_scheme" shows "(h \<in> ring_hom A B) = (h \<in> ring_hom (law_restrict A) (law_restrict B))"
assms_6: fixes q :: "int poly"   and ri :: "root_info" assumes "(q, ri) \<in> set todo \<union> set old" shows "q \<noteq> 0"
fields: shows "r = \<lparr>xpos = xpos_, ypos = ypos_, \<dots> = more_\<rparr>"
simps_4: fixes v :: "ref"   and va :: "dag"   and vb :: "ref"   and vc :: "dag" shows "ordered (Node Tip v (Node va vb vc)) = (\<lambda>var. var (root (Node va vb vc)) < var v \<and> ordered (Node va vb vc) var)"
funion_commute: fixes A :: "'x fset"   and B :: "'x fset" shows "A |\<union>| B = B |\<union>| A"
NotEquivD: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>" assumes "\<not> [\<phi> \<^bold>\<equiv> \<psi> in v]" shows "[\<phi> in v] \<noteq> [\<psi> in v]"
Keys_minus_zero: fixes A :: "('e \<Rightarrow>\<^sub>0 'f) set" shows "Keys (A - {0}) = Keys A"
preserves_cod: fixes f :: "'a" assumes "arr f" shows "cod (L f) = L (cod f)"
add_sheaf_spec_extensional: fixes U :: "nat set set"   and a :: "nat set \<Rightarrow> (nat \<times> nat) set"   and b :: "nat set \<Rightarrow> (nat \<times> nat) set" shows "cring0.add_sheaf_spec U a b \<in> extensional U"
cong_mod_poly: fixes n :: "'c poly"   and a :: "'c poly" assumes "n \<noteq> 0" shows "[a mod n = a] (mod n)"
assms_1: shows "x \<in> Units (G\<lparr>carrier := H\<rparr>)"
AbstAtomicP_Mem: fixes v :: "tm"   and i :: "tm"   and x :: "tm"   and x' :: "tm"   and y :: "tm"   and y' :: "tm" shows "{AbstTermP v i x x', AbstTermP v i y y'} \<turnstile> AbstAtomicP v i (Q_Mem x y) (Q_Mem x' y')"
group_hom_axioms: shows "group_hom G (BijGroup X) act"
Refinement_basic: fixes Sc :: "('b, 'c) spec"   and R :: "('d \<times> 'b) set"   and Sa :: "('d, 'e) spec"   and pi :: "'c \<Rightarrow> 'e" assumes "init Sc \<subseteq> R `` init Sa"   and "{R} TS.trans Sa, TS.trans Sc {> R}"   and "obs_consistent R pi Sa Sc" shows "refines R pi Sa Sc"
fg: fixes x :: "'b" assumes "x \<in> span ((+) (- b) ` T)" shows "f (g x) = x"
simps_32: fixes vd :: "'a acom"   and ve :: "'a acom"   and v :: "bexp"   and va :: "'a acom"   and vb :: "'a acom"   and vc :: "'a" shows "(vd;; ve \<le> IF v THEN va ELSE vb {vc}) = False"
isLub_def: fixes R :: "'b set"   and S :: "'b set"   and x :: "'b" shows "isLub R S x = leastP (isUb R S) x"
elims_2: fixes x :: "nat"   and xa :: "cell list \<times> cell list" assumes "inv_end5_exit x xa" obtains xb :: "nat" and l :: "cell list" and r :: "cell list" where "x = xb" and "xa = (l, r)" and "0 < xb \<and> l = [] \<and> r = Bk # Oc \<up> xb @ Bk # Oc \<up> xb"
hb2: shows "ta_hb_consistent P (E @ list_of E') E''"
simps_4: fixes a :: "'b"   and b :: "'b" shows "New_max.merge_option (Some a, Some b) = Some a"
normalized_add: fixes xs :: "('c, 'd) bintree option list"   and x :: "('c, 'd) bintree option" assumes "normalized xs" shows "normalized (add x xs)"
cod_simp: fixes f :: "'c \<times> 'c" assumes "CopxC.arr f" shows "CopxC.cod f = (Cop.cod (fst f), C.cod (snd f))"
map_spmf_of_set_inj_on: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set" assumes "inj_on f A" shows "map_spmf f (spmf_of_set A) = spmf_of_set (f ` A)"
comp_group_morphisms: fixes \<eta> :: "'d \<Rightarrow> 'e"   and A :: "'d set"   and multA :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and oneA :: "'d"   and B :: "'e set"   and multB :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and oneB :: "'e"   and \<theta> :: "'e \<Rightarrow> 'f"   and C :: "'f set"   and multC :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and oneC :: "'f" assumes "group_homomorphism \<eta> A multA oneA B multB oneB"   and "group_homomorphism \<theta> B multB oneB C multC oneC" shows "group_homomorphism (\<theta> \<circ> \<eta> \<down> A) A multA oneA C multC oneC"
cinner_scaleC_right: fixes x :: "'a"   and r :: "complex"   and y :: "'a" shows "cinner x (r *\<^sub>C y) = r * cinner x y"
And_append_syntactic: fixes xs :: "'c ltl list"   and ys :: "'c ltl list" assumes "xs \<noteq> []" shows "And (xs @ ys) = And (And xs # ys)"
sort_Cons_append_swap: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and xs :: "'b list" assumes "class.linorder less_eq less" shows "linorder.sort_key less_eq (\<lambda>x. x) (x # xs) = linorder.sort_key less_eq (\<lambda>x. x) (xs @ [x])"
flow_solves_ode: fixes t0 :: "real"   and x0 :: "'a" assumes "t0 \<in> S"   and "x0 \<in> Y" shows "(Y.flow t0 x0 solves_ode g) (Y.existence_ivl t0 x0) Y"
lift_poly_fun_def: fixes pair_of_term :: "'a \<Rightarrow> 'b \<times> 'c"   and term_of_pair :: "'b \<times> 'c \<Rightarrow> 'a"   and f :: "('b \<Rightarrow>\<^sub>0 'd) \<Rightarrow> 'b \<Rightarrow>\<^sub>0 'd"   and p :: "'a \<Rightarrow>\<^sub>0 'd" assumes "term_powerprod pair_of_term term_of_pair" shows "term_powerprod.lift_poly_fun pair_of_term term_of_pair f p = term_powerprod.lift_poly_fun_2 pair_of_term term_of_pair (\<lambda>_. f) 0 p"
f'_v: shows "f' v \<noteq> (0::'a)"
surjective: fixes r :: "('b, 'c, 'd) ipurge_rec_scheme" shows "r = \<lparr>Pol = Pol r, Map = Map r, Doms = Doms r, List = List r, ListOp = ListOp r, Set = Set r, \<dots> = more r\<rparr>"
beukers_aux_integrable2: shows "set_integrable lborel D' (\<lambda>(z, x, y). P x * (x * y * z) ^ n * (1 - y) ^ n / (1 - (1 - x * y) * z) ^ (n + 1))"
is_least_prefixpoint_def: fixes f :: "'a \<Rightarrow> 'a"   and x :: "'a" shows "is_least_prefixpoint f x \<equiv> f x \<le> x \<and> (\<forall>y. f y \<le> y \<longrightarrow> x \<le> y)"
hyps_3: shows "\<exists>m. \<forall>n\<in>N. n < m"
SUP_empty: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and f :: "'c \<Rightarrow> 'b" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "Sup (f ` {}) = bot"
leadsTo_weaken: fixes F :: "'c program"   and A :: "'c set"   and A' :: "'c set"   and B :: "'c set"   and B' :: "'c set" assumes "F \<in> A \<longmapsto> A'"   and "B \<subseteq> A"   and "A' \<subseteq> B'" shows "F \<in> B \<longmapsto> B'"
homeomorphic_Hausdorff_space: fixes X :: "'c topology"   and Y :: "'d topology" assumes "X homeomorphic_space Y" shows "Hausdorff_space X = Hausdorff_space Y"
simps_1: fixes x :: "nat" shows "FV (EVar x) = {x}"
initial_no_mon: fixes p :: "'p" assumes "initialproc fg p" shows "mon fg p = {}"
corthogonal_inv_def: fixes A :: "'c mat" shows "corthogonal_inv A = mat_of_rows (dim_row A) (map vec_inv (cols A))"
assms_4: shows "e = Binop_i t iop \<or> e = Relop_i t irop"
diagonalize_def: fixes A :: "(('a, 'c) Finite_Cartesian_Product.vec, 'b) Finite_Cartesian_Product.vec"   and bezout :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a \<times> 'a \<times> 'a" shows "local.diagonalize A bezout = (let (P, S, Q) = diagonalize_JNF (Mod_Type_Connect.from_hma\<^sub>m A) bezout in (Mod_Type_Connect.to_hma\<^sub>m P, Mod_Type_Connect.to_hma\<^sub>m S, Mod_Type_Connect.to_hma\<^sub>m Q))"
final_Accessible_dfa: shows "final Accessible_dfa = final M \<inter> accessible"
escape: fixes P :: "'b assn"   and G :: "'b triple set"   and c :: "com"   and Q :: "'b assn" assumes "\<forall>Z s. P Z s \<longrightarrow> G|-{\<lambda>Z s'. s' = s}. c .{\<lambda>Z'. Q Z}" shows "G|-{P}. c .{Q}"
path_component_mem_2: fixes S :: "'a set"   and x :: "'a"   and y :: "'a" assumes "path_component S x y" shows "y \<in> S"
compactinD: fixes X :: "'a topology"   and S :: "'a set"   and \<U> :: "'a set set" assumes "compactin X S"   and "\<And>U. U \<in> \<U> \<Longrightarrow> openin X U"   and "S \<subseteq> \<Union> \<U>" shows "\<exists>\<F>. finite \<F> \<and> \<F> \<subseteq> \<U> \<and> S \<subseteq> \<Union> \<F>"
intT_def: shows "intT \<equiv> ProblemIkTpartG_GEModel.intT tpD tpFD (\<lambda>\<sigma>. D) eintP"
rAS_set_def: shows "rAS_set \<equiv> {aseq. is_SK_IPC aseq \<or> is_SK_EV_WAIT aseq \<or> is_SK_EV_SIGNAL aseq} \<union> {[]}"
bot_entails_all: fixes B :: "'f"   and N1 :: "'f set" assumes "B \<in> Bot_F" shows "{B} \<Turnstile>\<inter>\<G> N1"
dist_Ps_upper_bound: fixes cf1 :: "('test, 'atom, 'choice) cmd \<times> 'state"   and cf2 :: "('test, 'atom, 'choice) cmd \<times> 'state"   and s :: "'state"   and n :: "nat"   and m :: "nat" assumes "proper (fst cf1)"   and "proper (fst cf2)"   and "fst cf1 \<approx>01 fst cf2"   and "snd cf1 \<approx> snd cf2" shows "dist (T.prob cf1 {bT \<in> space (T cf1). \<exists>n. qsend (cf1 ## bT) = enat n \<and> snd ((cf1 ## bT) !! n) \<approx> s}) (T.prob cf2 {bT \<in> space (T cf2). \<exists>n. qsend (cf2 ## bT) = enat n \<and> snd ((cf2 ## bT) !! n) \<approx> s}) \<le> T.prob cf1 {bT \<in> space (T cf1). \<not> discrCf ((cf1 ## bT) !! n)} + T.prob cf2 {bT \<in> space (T cf2). \<not> discrCf ((cf2 ## bT) !! m)}"
oghoare_induct: fixes P2 :: "'b ann_com \<Rightarrow> 'b set \<Rightarrow> bool"   and P1 :: "'b set \<Rightarrow> 'b com \<Rightarrow> 'b set \<Rightarrow> bool"   and x1 :: "'b set"   and x2 :: "'b com"   and x3 :: "'b set" assumes "\<And>r f q. r \<subseteq> {s. f s \<in> q} \<Longrightarrow> P2 (AnnBasic r f) q"   and "\<And>c0 c1 q. \<lbrakk>\<turnstile> c0  pre c1; P2 c0 (pre c1); \<turnstile> c1  q; P2 c1 q\<rbrakk> \<Longrightarrow> P2 (AnnSeq c0 c1) q"   and "\<And>r b c1 q c2. \<lbrakk>r \<inter> b \<subseteq> pre c1; \<turnstile> c1  q; P2 c1 q; r \<inter> - b \<subseteq> pre c2; \<turnstile> c2  q; P2 c2 q\<rbrakk> \<Longrightarrow> P2 (AnnCond1 r b c1 c2) q"   and "\<And>r b c q. \<lbrakk>r \<inter> b \<subseteq> pre c; \<turnstile> c  q; P2 c q; r \<inter> - b \<subseteq> q\<rbrakk> \<Longrightarrow> P2 (AnnCond2 r b c) q"   and "\<And>r i b c q. \<lbrakk>r \<subseteq> i; i \<inter> b \<subseteq> pre c; \<turnstile> c  i; P2 c i; i \<inter> - b \<subseteq> q\<rbrakk> \<Longrightarrow> P2 (AnnWhile r b i c) q"   and "\<And>c r b q. \<lbrakk>atom_com c; \<parallel>- (r \<inter> b) c q; P1 (r \<inter> b) c q\<rbrakk> \<Longrightarrow> P2 (AnnAwait r b c) q"   and "\<And>c q q'. \<lbrakk>\<turnstile> c  q; P2 c q; q \<subseteq> q'\<rbrakk> \<Longrightarrow> P2 c q'"   and "\<And>Ts. \<lbrakk>\<forall>i<length Ts. \<exists>c q. Ts ! i = (Some c, q) \<and> \<turnstile> c  q \<and> P2 c q; interfree Ts\<rbrakk> \<Longrightarrow> P1 (\<Inter>i\<in>{i. i < length Ts}. pre (the (OG_Hoare.com (Ts ! i)))) (Parallel Ts) (\<Inter>i\<in>{i. i < length Ts}. OG_Hoare.post (Ts ! i))"   and "\<And>f q. P1 {s. f s \<in> q} (Basic f) q"   and "\<And>p c1 r c2 q. \<lbrakk>\<parallel>- p c1 r; P1 p c1 r; \<parallel>- r c2 q; P1 r c2 q\<rbrakk> \<Longrightarrow> P1 p (Seq c1 c2) q"   and "\<And>p b c1 q c2. \<lbrakk>\<parallel>- (p \<inter> b) c1 q; P1 (p \<inter> b) c1 q; \<parallel>- (p \<inter> - b) c2 q; P1 (p \<inter> - b) c2 q\<rbrakk> \<Longrightarrow> P1 p (Cond b c1 c2) q"   and "\<And>p b c i. \<lbrakk>\<parallel>- (p \<inter> b) c p; P1 (p \<inter> b) c p\<rbrakk> \<Longrightarrow> P1 p (While b i c) (p \<inter> - b)"   and "\<And>p' p c q q'. \<lbrakk>p' \<subseteq> p; \<parallel>- p c q; P1 p c q; q \<subseteq> q'\<rbrakk> \<Longrightarrow> P1 p' c q'" shows "\<parallel>- x1 x2 x3 \<longrightarrow> P1 x1 x2 x3"
interp_imp_interp: fixes C :: "'a literal multiset"   and D :: "'a literal multiset"   and D' :: "'a literal multiset" assumes "C \<le> D"   and "D \<le> D'"   and "interp D \<Turnstile> C" shows "interp D' \<Turnstile> C"
winning_region_in_V: fixes p :: "Player" shows "G'.winning_region p \<subseteq> V\<^bsub>G'\<^esub>"
simps_1: shows "Num.inc num.One = num.Bit0 num.One"
map_ident: fixes t :: "'a dnelist" shows "dnelist.map (\<lambda>x. x) t = t"
hiding_fronttickFree: fixes s :: "'b event list"   and A :: "'b set" assumes "front_tickFree s" shows "front_tickFree (trace_hide s (ev ` A))"
len: shows "\<forall>y\<in>set as. length y = x"
connect_eq_resource_cong: fixes \<I> :: "('i, 'j) \<I>"   and distinguisher :: "('i, 'j) distinguisher"   and res :: "('i, 'j) resource"   and res' :: "('i, 'j) resource" assumes "\<I> \<turnstile>g distinguisher \<surd>"   and "outs_\<I> \<I> \<turnstile>\<^sub>R res \<sim> res'"   and "\<I> \<turnstile>res res \<surd>" shows "connect distinguisher res = connect distinguisher res'"
sup_bot_right: fixes sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and bot :: "'a"   and x :: "'a" assumes "class.bounded_semilattice_sup_bot sup less_eq less bot" shows "sup x bot = x"
that: fixes k :: "nat" assumes "r \<^sup>@ k = u" shows "thesis"
random_scheduler_spec: fixes I :: "(('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set) set" assumes "\<alpha>.deterministic I" shows "scheduler_spec final r local.random_scheduler (\<lambda>_ _. True) thr_\<alpha> thr_invar ws_\<alpha> ws_invar is_\<alpha> is_invar I"
preserves_modes_mem_def: fixes \<R> :: "((((Eg1.var, aexp, bexp) Stmt \<times> (Mode \<Rightarrow> Eg1.var set)) \<times> (Eg1.var \<Rightarrow> nat)) \<times> ((var\<^sub>C, aexp\<^sub>C, bexp\<^sub>C) Stmt \<times> (Mode \<Rightarrow> var\<^sub>C set)) \<times> (var\<^sub>C \<Rightarrow> nat)) set" shows "preserves_modes_mem \<R> \<equiv> \<forall>c\<^sub>A mds\<^sub>A mem\<^sub>A c\<^sub>C mds\<^sub>C mem\<^sub>C. (\<langle>c\<^sub>A, mds\<^sub>A, mem\<^sub>A\<rangle>\<^sub>A, \<langle>c\<^sub>C, mds\<^sub>C, mem\<^sub>C\<rangle>\<^sub>C) \<in> \<R> \<longrightarrow> (\<forall>x\<^sub>A. mem\<^sub>A x\<^sub>A = mem\<^sub>C (Eg2_var\<^sub>C_of_Eg1 x\<^sub>A)) \<and> (\<forall>m. Eg2_var\<^sub>C_of_Eg1 ` mds\<^sub>A m = range Eg2_var\<^sub>C_of_Eg1 \<inter> mds\<^sub>C m)"
ord_rec_Seq_functional: fixes k :: "hf"   and T :: "hf"   and G :: "hf \<Rightarrow> hf"   and s :: "hf"   and y :: "hf"   and s' :: "hf"   and y' :: "hf" assumes "Ord k"   and "k \<noteq> 0"   and "ord_rec_Seq T G s k y"   and "ord_rec_Seq T G s' k y'" shows "y' = y"
has_integral_restrict_UNIV: fixes s :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and i :: "'d" shows "((\<lambda>x. if x \<in> s then f x else (0::'d)) has_integral i) UNIV = (f has_integral i) s"
strict_boundedE: fixes a :: "'b"   and b :: "'b"   and c :: "'b" assumes "a < min b c" obtains "a < b" and "a < c"
assms_2: shows "s1 = (x * ((y + (q - b)) mod q) + (q - r)) mod q"
assms_4: fixes exp :: "exp"   and var :: "Vars.var" assumes "P exp" shows "P (Lam [var]. exp)"
abs_eq: shows "I = Abs_pmf (distr M (count_space UNIV) (X 0))"
lens_plus_def: fixes X :: "'d \<Longrightarrow> 'f"   and Y :: "'e \<Longrightarrow> 'f" shows "X +\<^sub>L Y = \<lparr>lens_get = \<lambda>\<sigma>. (get\<^bsub>X\<^esub> \<sigma>, get\<^bsub>Y\<^esub> \<sigma>), lens_put = \<lambda>\<sigma> (u, v). put\<^bsub>X\<^esub> (put\<^bsub>Y\<^esub> \<sigma> v) u\<rparr>"
assms_3: shows "b \<in>\<^sub>\<circ> Vset \<alpha>"