hcju/isabelleps huggingface dataset:
isPrefixElements: fixes a :: "('a \<times> bool) list"   and b :: "('a \<times> bool) list" assumes "isPrefix a b" shows "isPrefix (elements a) (elements b)"
checked_invariant_nsc: fixes nsc :: "('i \<times> 'a ns_constraint) list"   and J :: "'i set"   and s :: "'s" assumes "checked_nsc nsc J s" shows "invariant_nsc nsc J s"
alive_reach_alive: fixes x :: "Value"   and s :: "Store"   and l :: "Location" assumes "alive x s"   and "s\<turnstile> l reachable_from x" shows "alive (ref l) s"
moebius: fixes M :: "moebius"   and u :: "complex_homo"   and v :: "complex_homo"   and w :: "complex_homo" assumes "unit_disc_fix M"   and "u \<in> unit_disc"   and "v \<in> unit_disc"   and "w \<in> unit_disc"   and "P (moebius_pt M u) (moebius_pt M v) (moebius_pt M w)" shows "P u v w"
assms: shows "\<forall>u. u \<notin> S \<longrightarrow> D u = \<infinity>"
r_neg2: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier G"   and "y \<in> carrier G" shows "x \<oplus> (\<ominus> x \<oplus> y) = y"
Span_base_incl: fixes K :: "'a set"   and Us :: "'a list" assumes "subfield K R"   and "set Us \<subseteq> carrier R" shows "set Us \<subseteq> R.Span K Us"
h0: fixes x :: "real" shows "h (0, x) = g1 x"
as: shows "set as = {a \<in> set as. is_dvd a} \<union> {a \<in> set as. case a of Le x [] \<Rightarrow> False | Le x (k # xa) \<Rightarrow> 0 < k | _ \<Rightarrow> False} \<union> {a \<in> set as. case a of Le x [] \<Rightarrow> False | Le x (k # xa) \<Rightarrow> k < 0 | _ \<Rightarrow> False}"
GSubspace_is_Subspace: fixes G :: "'a set"   and smult :: "('b, 'a) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and V :: "'c set"   and U :: "'c set" assumes "FGModule G smult V"   and "RModule (aezfun_setspan G) smult U \<and> U \<subseteq> V" shows "RModule UNIV (aezfun_scalar_mult.fsmult smult) U \<and> U \<subseteq> V"
composites_are_arrows_1: shows "hseq \<mu> (\<nu> \<star> \<pi>)"
IH1: fixes \<nu> :: "(real, 'sz) vec \<times> (real, 'sz) vec"   and \<omega> :: "(real, 'sz) vec \<times> (real, 'sz) vec" assumes "Vagree \<nu> \<omega> (\<Union>y\<in>{y. Inl (Inr y) \<in> SIGO ODE1}. FVT (\<sigma> y))"   and "osafe ODE1" shows "ODE_sem (adjointFO I \<sigma> \<nu>) ODE1 = ODE_sem (adjointFO I \<sigma> \<omega>) ODE1"
assms_1: shows "Di \<in> set (subseqs (dbproj ln j D))"
entry_in_tree_keys: fixes k :: "'a"   and v :: "'b"   and t :: "('a, 'b) rbt" assumes "(k, v) \<in> set (entries t)" shows "k \<in> set (keys t)"
elimmeets: fixes meets :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and \<I> :: "'a \<Rightarrow> bool"   and p :: "'a"   and s :: "'a"   and r :: "'a"   and q :: "'a" assumes "class.interval meets \<I>" shows "(meets p s \<and> (\<exists>t. meets p t \<and> meets t s) \<and> (\<exists>t. meets r t \<and> meets t q)) = False"
fresh_Alts: fixes a :: "atom"   and e1 :: "exp"   and e2 :: "exp" shows "a \<sharp> Alts e1 e2 = (a \<sharp> e1 \<and> a \<sharp> e2)"
widen_refl: fixes P :: "'b prog"   and T :: "ty" shows "subtype P T T"
chine_hcomp_SPN_SPN: shows "Span.chine_hcomp (SPN g) (SPN h) = Maps.MkIde (src Tgh.p\<^sub>0)"
path_pd_ipd: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem"   and \<pi> :: "nat \<Rightarrow> 'a"   and k :: "nat"   and n :: "nat" assumes "IFC prob"   and "IFC_def.is_path prob \<pi>"   and "\<pi> k \<noteq> IFC_def.return prob"   and "\<pi> n \<noteq> \<pi> k"   and "IFC_def.is_pd prob (\<pi> n) (\<pi> k)"   and "k < n" obtains l :: "nat" where "k < l" and "l \<le> n" and "\<pi> l = IFC_def.ipd prob (\<pi> k)"
inverse_arrow_unique: fixes f :: "'b"   and g :: "'b"   and g' :: "'b" assumes "B.inverse_arrows f g"   and "B.inverse_arrows f g'" shows "g = g'"
smult_vec_def: fixes a :: "'c"   and v :: "'c Matrix.vec" shows "a \<cdot>\<^sub>v v \<equiv> Matrix.vec (dim_vec v) (\<lambda>i. a * v $v i)"
divide_strict_left_mono: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sgn :: "'c \<Rightarrow> 'c"   and abs :: "'c \<Rightarrow> 'c"   and b :: "'c"   and a :: "'c"   and c :: "'c" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less b a"   and "less zero c"   and "less zero (times a b)" shows "less (divide c a) (divide c b)"
seq_char: fixes g :: "'a \<times> 'a"   and f :: "'a \<times> 'a" shows "B.VV.seq g f = (B.VV.arr f \<and> B.VV.arr g \<and> B.VxV.seq g f)"
simps_2: fixes c :: "int"   and i :: "nat"   and P :: "pol" shows "Pc c \<langle>+\<rangle> Pinj i P = Pinj i (P \<langle>+\<rangle> Pc c)"
last_prog_hd_prog_last_prog: fixes p' :: "'b"   and sb :: "'b memref list"   and sb' :: "'b memref list"   and p :: "'b" assumes "last_prog p' (sb @ sb') = p"   and "hd_prog p' (sb @ sb') = p'" shows "last_prog (hd_prog p sb') sb = last_prog p' sb"
reg: shows "t = take i t @ take (j + 1 - i) (drop i t) @ drop (j + 1) t"
Units_one_closed: fixes G :: "('a, 'b) monoid_scheme" assumes "Group.monoid G" shows "\<one>\<^bsub>G\<^esub> \<in> Units G"
set_plus_imp_minus: fixes a :: "'c"   and b :: "'c"   and C :: "'c set" assumes "a \<in> b +o C" shows "a - b \<in> C"
left_idem: fixes a :: "'a::refinement_lattice"   and b :: "'a::refinement_lattice" shows "(a::'a::refinement_lattice) \<iinter> (a \<iinter> (b::'a::refinement_lattice)) = a \<iinter> b"
map2_val_pair_singleton_eq_update_by_fun_pair: fixes x :: "'o"   and xs :: "('a \<times> 'd) list"   and f :: "'a \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'd"   and g :: "('a \<times> 'd) list \<Rightarrow> ('a \<times> 'd) list"   and h :: "('a \<times> 'e) list \<Rightarrow> ('a \<times> 'd) list"   and k :: "'a"   and v :: "'e" assumes "oalist_inv_raw x xs"   and "\<And>k x. f k x (0::'e) = x"   and "\<And>zs. oalist_inv_raw x zs \<Longrightarrow> g zs = zs"   and "h [(k, v)] = map_val_pair (\<lambda>k. f k (0::'d)) [(k, v)]" shows "map2_val_pair x f g h xs [(k, v)] = update_by_fun_pair x k (\<lambda>x. f k x v) xs"
neg_mod_sign: fixes l :: "int"   and k :: "int" assumes "l < 0" shows "k mod l \<le> 0"
simps_2: fixes r1 :: "'b rexp"   and r2 :: "'b rexp" shows "simp (Times r1 r2) = simp_Times (simp r1) (simp r2)"
\<pi>_nonneg: fixes x :: "real" shows "0 \<le> \<pi> x"
emeasure_T': fixes X :: "('b \<times> 'a) stream set"   and I :: "('b \<times> 'a) pmf" assumes "X \<in> sets (stream_space (count_space UNIV))" shows "emeasure (T' I) X = \<integral>\<^sup>+ s. emeasure (T s) {\<omega> \<in> space (stream_space (count_space UNIV)). s ## \<omega> \<in> X} \<partial>measure_pmf I"
nn_integral_measure_spmf_conv_measure_pmf: fixes f :: "'h \<Rightarrow> ennreal"   and p :: "'h spmf" assumes "f \<in> borel_measurable (count_space UNIV)" shows "integral\<^sup>N (measure_spmf p) f = integral\<^sup>N (restrict_space (measure_pmf p) (range Some)) (f \<circ> the)"
dimc: shows "dim_col (jordan_matrix n_as) = sum_list (map fst n_as)"
B: shows "ipurge_tr_rev_aux I D U (x # xs @ ys) = ipurge_tr_rev_aux I D V (x # xs)"
intro: fixes R :: "('a, 'b) ring_scheme" assumes "cring R"   and "domain_axioms R" shows "domain R"
subset_iff_less_eq_V: fixes B :: "V set"   and A :: "V set" assumes "small B" shows "(A \<subseteq> B) = (ZFC_in_HOL.set A \<le> ZFC_in_HOL.set B \<and> small A)"
assms_5: shows "lproject visible (llist_of_stream u) = lproject visible (llist_of_stream v)"
rm_set_lookup: fixes key :: "'b \<Rightarrow> 'c"   and ds :: "'b list"   and k :: "'c" shows "set (rm_set_lookup (elem_list_to_rm key ds) k) = {d \<in> set ds. key d = k}"
hom_lead_coeff: fixes p :: "'c poly poly" shows "lead_coeff (map_poly (pCons (0::'c)) p) = pCons (0::'c) (lead_coeff p)"
transp_less: fixes ko :: "'c nat_term_order" shows "transp (ko_ntm.lt ko)"
sorted_list_of_set_def: shows "sorted_list_of_set = folding.F insort []"
trms\<^sub>s\<^sub>t_update\<^sub>s\<^sub>t_eq: fixes x :: "('fun, 'var) strand_step"   and S :: "('fun, 'var) strand_step list"   and \<S> :: "('fun, 'var) strand_step list set" assumes "x # S \<in> \<S>" shows "\<Union> (trms\<^sub>s\<^sub>t ` update\<^sub>s\<^sub>t \<S> (x # S)) \<union> trms\<^sub>s\<^sub>t\<^sub>p x = \<Union> (trms\<^sub>s\<^sub>t ` \<S>)"
continuous_interval_vimage_Int: fixes a :: "real"   and b :: "real"   and g :: "real \<Rightarrow> real"   and c :: "real"   and d :: "real" assumes "continuous_on {a..b} g"   and "\<And>x y. \<lbrakk>a \<le> x; x \<le> y; y \<le> b\<rbrakk> \<Longrightarrow> g x \<le> g y"   and "a \<le> b"   and "c \<le> d"   and "{c..d} \<subseteq> {g a..g b}" obtains c' :: "real" and d' :: "real" where "{a..b} \<inter> g -` {c..d} = {c'..d'}" and "c' \<le> d'" and "g c' = c" and "g d' = d"
max2_greater: fixes a :: "'a"   and b :: "'a" assumes "a \<in> Field r"   and "b \<in> Field r" shows "(a, local.max2 a b) \<in> r \<and> (b, local.max2 a b) \<in> r"
simulation_silents2: fixes t :: "'t"   and s1 :: "'x1 \<times> 'm1"   and s2 :: "'x2 \<times> 'm2"   and s2' :: "'x2 \<times> 'm2" assumes "t \<turnstile> s1 \<approx> s2"   and "r2.silent_moves t s2 s2'" shows "\<exists>s1'. r1.silent_moves t s1 s1' \<and> t \<turnstile> s1' \<approx> s2'"
ternary_to_bool_None: fixes t :: "ternaryvalue" shows "(ternary_to_bool t = None) = (t = TernaryUnknown)"
pr_rec_last_scheme: fixes g :: "nat \<Rightarrow> nat"   and h :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat"   and f :: "nat \<Rightarrow> nat \<Rightarrow> nat" assumes "g \<in> PrimRec1"   and "h \<in> PrimRec3"   and "\<forall>x. f x 0 = g x"   and "\<forall>x y. f x (Suc y) = h x (f x y) y" shows "f \<in> PrimRec2"
M_def: shows "M = Max {dist (x (from_nat n)) (y (from_nat n)) |n. n \<le> N}"
frameIntAssociativity: fixes A\<^sub>F :: "name list"   and \<Psi> :: "'b"   and \<Psi>' :: "'b"   and \<Psi>'' :: "'b" shows "\<langle>A\<^sub>F, (\<Psi> \<otimes> \<Psi>') \<otimes> \<Psi>''\<rangle> \<simeq>\<^sub>F \<langle>A\<^sub>F, \<Psi> \<otimes> \<Psi>' \<otimes> \<Psi>''\<rangle>"
is_empty_sup: fixes A :: "'a pred"   and B :: "'a pred" shows "is_empty (A \<squnion> B) = (is_empty A \<and> is_empty B)"
lossless_fast_uniform: shows "lossless_spmf local.fast_uniform = (0 < n)"
Lv: shows "(\<lambda>n. extended_Gromov_product_at basepoint (to_Gromov_completion (v n)) x) \<longlonglongrightarrow> \<infinity>"
SUP_upper: 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 i :: "'d"   and A :: "'d set"   and f :: "'d \<Rightarrow> 'b" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "i \<in> A" shows "less_eq (f i) (Sup (f ` A))"
induct: fixes P :: "nat \<Rightarrow> bool"   and a0 :: "nat" assumes "P 0"   and "P (Suc 0)"   and "\<And>v. P (Suc v) \<Longrightarrow> P (Suc (Suc v))" shows "P a0"
exhaust: fixes y :: "'b aexp" obtains x1 :: "value" where "y = L x1"   | x2 :: "'b" where "y = V x2"   | x31 :: "'b aexp" and x32 :: "'b aexp" where "y = Plus x31 x32"   | x41 :: "'b aexp" and x42 :: "'b aexp" where "y = Minus x41 x42"   | x51 :: "'b aexp" and x52 :: "'b aexp" where "y = Times x51 x52"
domA_append: fixes a :: "('d \<times> 'e) list"   and b :: "('d \<times> 'e) list" shows "domA (a @ b) = domA a \<union> domA b"
lmeasurable_iff_indicator_has_integral: fixes S :: "'a set"   and m :: "real" shows "(S \<in> lmeasurable \<and> m = Sigma_Algebra.measure lebesgue S) = (indicat_real S has_integral m) UNIV"
dT_He_eq_bot: fixes j :: "'a"   and d :: "'a"   and h :: "'a"   and g :: "'a" assumes "vector j"   and "regular j"   and "d * top \<le> - j"   and "forest_components h * j = j"   and "j \<noteq> bot" shows "(d[[forest_components h]]) * selected_edge h j g \<le> bot"
add_block_def: fixes b :: "'a set" shows "add_point_sys.add_block b \<equiv> \<B> + {#b#}"
idiv_imod_equality: fixes m :: "enat"   and n :: "enat"   and k :: "enat" shows "m div n * n + m mod n + k = m + k"
mult_coprime_aux: fixes f :: "nat \<Rightarrow> 'b"   and a :: "nat"   and b :: "nat" assumes "multiplicative_function f"   and "1 < a"   and "1 < b"   and "coprime a b" shows "f (a * b) = f a * f b"
carr: shows "{swap_cols_rows k l A, A, swaprows_mat n k l, swaprows_mat n k l} \<subseteq> carrier_mat n n"
id_comp: fixes g :: "'c \<Rightarrow> 'd" shows "id \<circ> g = g"
G: fixes t :: "real" assumes "t \<in> {0..dist x y}" shows "geodesic_segment_param G x t = g t"
dual_iff: fixes x :: "'a"   and y :: "'a" shows "(\<partial> x = y) = (x = \<partial> y)"
l_new_document_get_shadow_root_axioms_def: fixes get_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_new_document_get_shadow_root_axioms get_shadow_root_locs \<equiv> \<forall>h new_document_ptr h' r ptr'. h \<turnstile> new_document \<rightarrow>\<^sub>r new_document_ptr \<longrightarrow> h \<turnstile> new_document \<rightarrow>\<^sub>h h' \<longrightarrow> r \<in> get_shadow_root_locs ptr' \<longrightarrow> r h h'"
i: shows "\<forall>i<\<^bold>|w\<^bold>|. w ! i = r ! (i mod \<^bold>|r\<^bold>|)"
reduction_fresh: fixes t :: "trm"   and t' :: "trm"   and x :: "name" assumes "t \<mapsto> t'"   and "x \<sharp> t" shows "x \<sharp> t'"
frontier_subset_closed: fixes S :: "'a set" assumes "closed S" shows "frontier S \<subseteq> S"
guessed3_1: shows "zs' = zs1 @- x ## zs2"
exp_conj_unitary: fixes P :: "'a \<Rightarrow> real"   and Q :: "'a \<Rightarrow> real" assumes "unitary P"   and "unitary Q" shows "unitary (P && Q)"
lconf_upd: fixes P :: "'a prog"   and h :: "heap"   and l :: "sfields"   and E :: "char list \<Rightarrow> ty option"   and v :: "val"   and T :: "ty"   and V :: "char list" assumes "P,h \<turnstile> l (:\<le>) E"   and "P,h \<turnstile> v :\<le> T"   and "E V = \<lfloor>T\<rfloor>" shows "P,h \<turnstile> l(V \<mapsto> v) (:\<le>) E"
is_nnf_nnf_id: fixes F :: "'b formula" assumes "is_nnf F" shows "nnf F = F"
enat_cocase_eSuc: fixes z :: "'c"   and s :: "enat \<Rightarrow> 'c"   and n :: "enat" shows "(case eSuc n of 0 \<Rightarrow> z | eSuc x \<Rightarrow> s x) = s n"
cases: fixes x :: "'at Literal" obtains A :: "'at" where "x = Pos Aa__"   | A :: "'at" where "x = Neg A"
intro: fixes x :: "'c \<Longrightarrow> 'b" assumes "\<And>\<sigma> v. get\<^bsub>x\<^esub> (put\<^bsub>x\<^esub> \<sigma> v) = v" shows "weak_lens x"
partitioningIps_equi: fixes ss :: "'a wordinterval list"   and ts :: "'a wordinterval list" shows "map wordinterval_to_set (partitioningIps ss ts) = partitioning1 (map wordinterval_to_set ss) (map wordinterval_to_set ts)"
component_guaranteesD: fixes F :: "'b program"   and X :: "'b program set"   and Y :: "'b program set"   and G :: "'b program"   and H :: "'b program" assumes "F \<in> X guarantees Y"   and "F \<squnion> G = H"   and "H \<in> X"   and "F ok G" shows "H \<in> Y"
compl_sup: fixes minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and uminus :: "'e \<Rightarrow> 'e"   and inf :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and sup :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and bot :: "'e"   and top :: "'e"   and x :: "'e"   and y :: "'e" assumes "class.boolean_algebra minus uminus inf less_eq less sup bot top" shows "uminus (sup x y) = inf (uminus x) (uminus y)"
denot_terminates: fixes v' :: "val"   and e :: "exp"   and \<rho>' :: "(nat \<times> val) list"   and \<rho> :: "(nat \<times> bval) list" assumes "v' \<in> E e \<rho>'"   and "good_env \<rho> \<rho>'" shows "\<exists>v. \<rho> \<turnstile> e \<Down> v \<and> v \<in> good v'"
bin_items_le: fixes k :: "nat"   and I :: "item set" shows "bin (items_le k I) k = bin I k"
inteq: shows "integral (cbox a b) (indicat_real S) = integral UNIV (indicat_real S)"
jumpNestingOk_def: fixes jmps :: "jump set"   and t :: "Term.term" shows "jumpNestingOk jmps t = (case t of In1 (Inl e) \<Rightarrow> True | In1 (Inr s) \<Rightarrow> jumpNestingOkS jmps s | _ \<Rightarrow> True)"
Neg': fixes a :: "'c"   and ps :: "('b, 'c) fm list"   and branch :: "(('b, 'c) fm list \<times> 'c) list"   and p :: "('b, 'c) fm"   and A :: "'c set"   and n :: "nat" assumes "(\<^bold>\<not> \<^bold>\<not> p) at a in' set ((ps, a) # branch)"   and "A, n \<turnstile> (p # ps, a) # branch" shows "A, n \<turnstile> (ps, a) # branch"
take_eq_val_take_tm: fixes n :: "nat"   and xs :: "'a list" shows "Time_Monad.val (take_tm n xs) = take n xs"
assms_2: fixes conv1 :: "('a, 'b, 'c, 'd) converter"   and conv2 :: "('e, 'f, 'g, 'h) converter" assumes "X conv1 conv2" shows "rel_fun A (rel_gpv'' (rel_prod B (\<lambda>conv1 conv2. X conv1 conv2 \<or> rel_converter A B C R conv1 conv2)) C R) (run_converter conv1) (run_converter conv2)"
swaprows_mat_def: fixes n :: "nat"   and k :: "nat"   and l :: "nat" shows "swaprows_mat n k l = mat n n (\<lambda>(i, j). if k = i \<and> l = j \<or> k = j \<and> l = i \<or> i = j \<and> i \<noteq> k \<and> i \<noteq> l then 1::'d else (0::'d))"
same_cs_implies_same_resulting_cs: fixes c :: "('p, 's, 'm) configuration"   and i :: "nat"   and d :: "('p, 's, 'm) configuration"   and ev :: "('p, 's, 'm) event"   and c' :: "('p, 's, 'm) configuration"   and d' :: "('p, 's, 'm) configuration" assumes "cs c i = cs d i"   and "c \<turnstile> ev \<mapsto> c'"   and "d \<turnstile> ev \<mapsto> d'"   and "regular_event ev" shows "cs c' i = cs d' i"
FWdelay_bisimulation_lift_aux_def: fixes final1 :: "'a \<Rightarrow> bool"   and r1 :: "('d, 'b, 'a, 'c, 'e, 'f) semantics"   and final2 :: "'g \<Rightarrow> bool"   and r2 :: "('d, 'b, 'g, 'h, 'e, 'f) semantics"   and \<tau>move1 :: "('d, 'b, 'a, 'c, 'e, 'f) \<tau>moves"   and \<tau>move2 :: "('d, 'b, 'g, 'h, 'e, 'f) \<tau>moves" shows "FWdelay_bisimulation_lift_aux final1 r1 final2 r2 \<tau>move1 \<tau>move2 \<equiv> \<tau>multithreaded_wf final1 r1 \<tau>move1 \<and> \<tau>multithreaded_wf final2 r2 \<tau>move2"
edc_exp: fixes vs :: "nat list"   and vs' :: "nat list"   and \<Gamma> :: "nat \<Rightarrow> pdf_type"   and \<delta> :: "cexpr"   and e :: "expr"   and f :: "cexpr" assumes "(vs, vs', \<Gamma>, \<delta>) \<turnstile>\<^sub>c e \<Rightarrow> f" shows "(vs, vs', \<Gamma>, \<delta>) \<turnstile>\<^sub>c Exp $$ e \<Rightarrow> (IF\<^sub>c CReal 0 <\<^sub>c CVar 0 THEN (f \<circ>\<^sub>c ln\<^sub>c (CVar 0)) /\<^sub>c CVar 0 ELSE CReal 0)"
msgs_def: shows "msgs = keys \<times> {ms. set ms \<subseteq> messages \<and> length ms = n}"
simplicial_chain_def: fixes p :: "nat"   and S :: "('b \<Rightarrow> real) set"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real) \<Rightarrow>\<^sub>0 'c" shows "simplicial_chain p S c \<equiv> Poly_Mapping.keys c \<subseteq> Collect (simplicial_simplex p S)"
zdiff_int_split: fixes P :: "int \<Rightarrow> bool"   and x :: "nat"   and y :: "nat" shows "P (int (x - y)) = ((y \<le> x \<longrightarrow> P (int x - int y)) \<and> (x < y \<longrightarrow> P 0))"
mult_2: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and z :: "'a" assumes "class.semiring_numeral one times plus" shows "times (numeral.numeral one plus (num.Bit0 num.One)) z = plus z z"
fmapU_eq_bindU: fixes fmapU :: "(udom \<rightarrow> udom) \<rightarrow> udom\<cdot>'c \<rightarrow> udom\<cdot>'c"   and returnU :: "udom \<rightarrow> udom\<cdot>'c"   and bindU :: "udom\<cdot>'c \<rightarrow> (udom \<rightarrow> udom\<cdot>'c) \<rightarrow> udom\<cdot>'c"   and f :: "udom \<rightarrow> udom"   and xs :: "udom\<cdot>'c" assumes "class.monad fmapU returnU bindU" shows "fmapU\<cdot>f\<cdot>xs = bindU\<cdot>xs\<cdot>(\<Lambda> x. returnU\<cdot>(f\<cdot>x))"
as_needed_asses_submap_exec: fixes s1 :: "('a, 'b) fmap"   and s2 :: "('a, 'b) fmap"   and as :: "(('a, 'b) fmap \<times> ('a, 'b) fmap) list" assumes "s1 \<subseteq>\<^sub>f s2"   and "\<forall>a. ListMem a as \<longrightarrow> action_needed_asses a s2 \<subseteq>\<^sub>f action_needed_asses a s1" shows "exec_plan s1 as \<subseteq>\<^sub>f exec_plan s2 as"
euclidean_size_greater_0_iff: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and 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 modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and euclidean_size :: "'c \<Rightarrow> nat"   and b :: "'c" assumes "class.euclidean_semiring divide plus minus zero times one modulo euclidean_size" shows "(0 < euclidean_size b) = (b \<noteq> zero)"
intro: fixes get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" assumes "\<And>ptr' shadow_root_ptr h h' r. \<lbrakk>ptr' \<noteq> cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^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 shadow_root_ptr; h \<turnstile> delete\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M shadow_root_ptr \<rightarrow>\<^sub>h h'; r \<in> get_child_nodes_locs ptr'\<rbrakk> \<Longrightarrow> r h h'" shows "l_delete_shadow_root_get_child_nodes get_child_nodes_locs"
inter: fixes B :: "'b set set"   and x :: "'b set"   and y :: "'b set" assumes "x open\<^bsub>topo B\<^esub>"   and "y open\<^bsub>topo B\<^esub>" shows "x \<inter> y open\<^bsub>topo B\<^esub>"
intros_4: fixes P :: "pi"   and Q :: "pi" shows "P \<oplus> Q \<equiv>\<^sub>s Q \<oplus> P"
bij_convert': fixes \<sigma> :: "complex \<Rightarrow> complex"   and X :: "complex multiset" assumes "\<sigma> permutes Roots" shows "bij_betw (\<lambda>f. f \<circ> (\<circ>) \<sigma>) (tuples X) (tuples X)"
preservation: shows "rel_weakly_preserves_barbs Rel (STCalWB SWB TWB)"
functor_axioms: shows "functor A C (K \<circ> G)"
natural_transformation_axioms: shows "natural_transformation J' C A'.map (D \<circ> F) (\<chi> \<circ> F)"
tms_cong_ax_def: shows "tms_cong_ax = Var X1 EQ Var X2 AND Var X3 EQ Var X4 IMP tms (Var X1) (Var X3) EQ tms (Var X2) (Var X4)"
respection_and_closures_2: fixes Rel :: "('a \<times> 'a) set"   and Pred :: "'a \<Rightarrow> bool" assumes "rel_respects_pred Rel Pred" shows "rel_respects_pred (symcl Rel) Pred"
homogeneous_homogenize_dehomogenize: fixes p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and x :: "'b" assumes "homogeneous p" obtains d :: "nat" where "d = poly_deg p - poly_deg (homogenize x (dehomogenize x p))" and "MPoly_Type_Class.punit.monom_mult (1::'c) (monomial d x) (homogenize x (dehomogenize x p)) = p"
"1_1": shows "\<forall>(t, t')\<in>set (map snd D). (fv t \<union> fv t') \<inter> set X = {}"
prog_instrs_append_Read\<^sub>s\<^sub>b: fixes sb :: "'b memref list"   and volatile :: "bool"   and a :: "nat"   and t :: "nat"   and v :: "nat" shows "prog_instrs (sb @ [Read\<^sub>s\<^sub>b volatile a t v]) = prog_instrs sb"
sum_less_eqI: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and y :: "'c" assumes "\<And>x. x \<in> A \<Longrightarrow> f x \<le> y" shows "sum f A \<le> y"
updSize: fixes h :: "('b \<times> 'a) list"   and a :: "'b"   and obj :: "'a"   and obj1 :: "'a" assumes "h\<down>a = Some obj" shows "|h[a\<mapsto>obj1]| = |h|"
SD_strict_singleton_left: fixes carrier :: "'b set"   and le :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and q :: "'b pmf" assumes "Order_Predicates.preorder_on carrier le"   and "x \<in> carrier"   and "q \<in> lotteries_on carrier" shows "return_pmf x \<prec>[SD le] q = ((\<forall>y\<in>set_pmf q. le x y) \<and> (\<exists>y\<in>set_pmf q. x \<prec>[le] y))"
sep_conj_sep_true_left: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool"   and h :: "'a" assumes "(P \<and>* Q) h" shows "((\<lambda>s. True) \<and>* Q) h"
rep_eq: fixes x :: "int"   and xa :: "int" shows "real_of_float (Float x xa) = real_of_int x * 2 powr real_of_int xa"