hcju/isabelleps huggingface dataset:
simps: fixes "consider" :: "'a \<Rightarrow> bool"   and gpv :: "('b, 'a, 'c) gpv"   and n :: "enat" shows "interaction_bounded_by consider gpv n = (interaction_bound consider gpv \<le> n)"
cases_scheme: fixes r :: "'b gState_scheme" obtains vars :: "(String.literal, variable) assoc_list" and channels :: "channel list" and timeout :: "bool" and procs :: "pState list" and more :: "'b" where "r = \<lparr>gState.vars = vars, channels = channels, timeout = timeout, procs = procs, \<dots> = more\<rparr>"
assms_2: shows "A, n \<turnstile> ((\<^bold>\<not> p) # ps, a) # branch"
mult_right_dist_sup: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(x \<squnion> y) \<sqinter> z = x \<sqinter> z \<squnion> y \<sqinter> z"
VcompNml_Nml_Dom: fixes V :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and src :: "'a \<Rightarrow> 'a"   and trg :: "'a \<Rightarrow> 'a"   and t :: "'a local.term" assumes "bicategorical_language V src trg"   and "bicategorical_language.Nml V src trg t" shows "bicategorical_language.VcompNml V t (bicategorical_language.Dom V src trg t) = t"
r_leap_Suc_saturating: fixes t :: "nat"   and i :: "nat"   and x :: "nat" assumes "pdec1 (the (eval r_leap [t, i, x])) = 0" shows "eval r_leap [Suc t, i, x] = eval r_leap [t, i, x]"
OneElement_Card: fixes M :: "'d set"   and t :: "'d" assumes "finite M"   and "card M \<le> Suc 0"   and "t \<in> M" shows "M = {t}"
step_2: shows "h \<turnstile> get_dom_component child \<rightarrow>\<^sub>r c"
stut_fun1: fixes X :: "(nat \<Rightarrow> 'a) \<Rightarrow> 'b"   and f :: "'b \<Rightarrow> 'c" assumes "STUTINV X" shows "STUTINV f<X>"
assms_3: fixes as :: "real list"   and bs :: "real list"   and a :: "real" assumes "P as bs" shows "P (a # as) (a # bs)"
bvars\<^sub>s\<^sub>s\<^sub>t_cons_subset: fixes A :: "('b, 'a) stateful_strand_step list"   and a :: "('b, 'a) stateful_strand_step" shows "bvars\<^sub>s\<^sub>s\<^sub>t A \<subseteq> bvars\<^sub>s\<^sub>s\<^sub>t (a # A)"
ind: fixes as :: "nat"   and am :: "nat list"   and bs :: "nat"   and bm :: "nat list" assumes "abc_steps_l (as, am) ap stp = (bs, bm)" shows "abc_steps_l (as, am) ap (Suc stp) = abc_step_l (bs, bm) (abc_fetch bs ap)"
lfinite_prepend_induct: fixes xs :: "'b llist"   and P :: "'b llist \<Rightarrow> bool" assumes "lfinite xs"   and "P LNil"   and "\<And>xs. \<lbrakk>lfinite xs; \<And>zs. \<exists>ys. xs = prepend ys zs \<and> ys \<noteq> [] \<Longrightarrow> P zs\<rbrakk> \<Longrightarrow> P xs" shows "P xs"
adjunction_in_bicategory_axioms: shows "adjunction_in_bicategory (\<cdot>) (\<star>) \<a> \<i> src trg g f \<eta> \<epsilon>"
lfilter_eq_LCons: fixes P :: "'c \<Rightarrow> bool"   and xs :: "'c llist"   and x :: "'c"   and xs' :: "'c llist" assumes "lfilter P xs = LCons x xs'" shows "\<exists>xs''. xs' = lfilter P xs'' \<and> ldropWhile (Not \<circ> P) xs = LCons x xs''"
Rm_1: shows "integrable M (\<lambda>x. fm x * real_cond_exp M F g x)"
simps_8: fixes p :: "'a \<Rightarrow> bool"   and a :: "'a"   and g1 :: "'a \<Rightarrow> 'c"   and g21 :: "'a \<Rightarrow> 'b"   and q22 :: "'a \<Rightarrow> bool"   and g221 :: "'a \<Rightarrow> ('b, 'c) tllist"   and g222 :: "'a \<Rightarrow> 'a" assumes "\<not> p a" shows "\<not> is_TNil (corec_tllist p g1 g21 q22 g221 g222 a)"
m_inverse: fixes i :: "nat" assumes "i \<in> N" shows "m' (m i) = i"
simps_2: fixes c :: "'b"   and s :: "('b \<times> 'b trm) list" shows "Const c \<lhd> s = Const c"
le_segment_segment: fixes a :: "'a"   and b :: "'a" assumes "a \<in> carrier D"   and "b \<in> carrier D" shows "a \<preceq> b = (segment (Iod D (segment D b)) a = segment D a)"
nbot: fixes x :: "('a, 'b, 'c) Resumption.resumption" assumes "x \<in> Y" shows "\<not> resumption_ord x ABORT"
bounded_linear_intro: fixes f :: "'d \<Rightarrow> 'e"   and K :: "real" assumes "\<And>x y. f (x + y) = f x + f y"   and "\<And>r x. f (r *\<^sub>R x) = r *\<^sub>R f x"   and "\<And>x. norm (f x) \<le> norm x * K" shows "bounded_linear f"
g_s: fixes n :: "nat" shows "(\<Sum>\<^sup>+ x\<in>Y n. ennreal (g n x)) = s n"
lossless_S1: fixes m1 :: "'msg1"   and out1 :: "'out1" shows "lossless_spmf (S1 m1 out1)"
listsetD_2: fixes ys :: "'b list"   and xs :: "'b set list"   and i :: "nat" assumes "ys \<in> listset xs"   and "i < length xs" shows "ys ! i \<in> xs ! i"
add2_Some: fixes x :: "'b option"   and y :: "'b"   and z :: "'b" shows "(total_partial.add2 x (Some y) = Some z) = (\<exists>x'. x = Some x' \<and> Sep_Eq.add2 x' y = Some z)"
simps_2: fixes v :: "name" shows "unconst (Free v) = None"
ipv4addr_of_nat_AND_mask8: fixes a :: "nat" shows "ipv4addr_of_nat a && mask 8 = ipv4addr_of_nat (a mod 256)"
induced_play_eq: fixes p :: "'b list"   and s1 :: "'b list \<Rightarrow> 'b"   and s2 :: "'b list \<Rightarrow> 'b" assumes "\<forall>p'. prefix p p' \<longrightarrow> augment_list s1 p' = augment_list s2 p'" shows "induced_play s1 p = induced_play s2 p"
card_le_PiE_subindex: fixes A :: "'a set"   and A' :: "'a set"   and B :: "'a \<Rightarrow> 'b set" assumes "A \<subseteq> A'"   and "Pi\<^sub>E A' B \<noteq> {}" shows "Pi\<^sub>E A B \<lesssim> Pi\<^sub>E A' B"
leq_zero_imp: fixes r :: "('d \<times> 'd) set"   and x :: "'d" assumes "wo_rel r"   and "(x, zero r) \<in> r" shows "x = zero r"
maskshift_eq_not_mask_generic: shows "mask l << LENGTH('i) - l = ~~ mask (LENGTH('i) - l)"
tappend_TNil: fixes b :: "'c"   and f :: "'c \<Rightarrow> ('a, 'b) tllist" shows "tappend (TNil b) f = f b"
rep_eq: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and x :: "rat" assumes "class.field_char_0 divide inverse times one plus zero minus uminus" shows "field_char_0.of_rat divide one plus zero minus x = divide (ring_1.of_int one plus zero minus (fst (Rep_Rat x))) (ring_1.of_int one plus zero minus (snd (Rep_Rat x)))"
multiplicity_same_power': fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and p :: "'a"   and n :: "nat" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize" shows "factorial_semiring.multiplicity times one p (power.power one times p n) = (if p = zero \<or> dvd.dvd times p one then 0 else n)"
subformulas\<^sub>\<nu>_list_distinct: fixes \<phi> :: "'a ltln" shows "distinct (subformulas\<^sub>\<nu>_list \<phi>)"
b5: fixes x :: "'b" shows "\<parallel>(A\<^sup>\<dagger>) x\<parallel>\<^sup>2 \<le> \<parallel>x\<parallel> * \<parallel>A ((A\<^sup>\<dagger>) x)\<parallel>"
homotopy_eqv_sing: fixes S :: "'a set"   and a :: "'b" shows "(S homotopy_eqv {a}) = (S \<noteq> {} \<and> contractible S)"
IH: assumes "list_all isState sl" shows "doo (local.getGMState (hd sl)) (map getGMUserCom (tl sl)) = local.getGMState (last sl)"
conjI3: fixes A :: "bool"   and B :: "bool"   and C :: "bool" assumes "A"   and "B"   and "C" shows "A \<and> B \<and> C"
relcomppE: fixes r :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and s :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and a :: "'d"   and c :: "'f" assumes "(r OO s) a c" obtains b :: "'e" where "r a b" and "s b c"
dominates_unsnoc': fixes g :: "'g"   and n :: "'node"   and m :: "'node"   and m' :: "'node"   and ms :: "'node list" assumes "invar g"   and "old.dominates g n m"   and "g \<turnstile> m'-ms\<rightarrow>m"   and "\<forall>x\<in>set (tl ms). x \<noteq> n" shows "old.dominates g n m'"
assms_2: assumes "P \<longmapsto> Rs"   and "a = b" shows "F a a"
le: shows "P \<le>\<^sub>L 1\<^sub>m d"
sbintrunc_n_0: fixes n :: "nat" shows "signed_take_bit n 0 = 0"
i_Exec_Stream_nth: fixes trans_fun :: "'b \<Rightarrow> 'a \<Rightarrow> 'a"   and input :: "nat \<Rightarrow> 'b"   and c :: "'a"   and n :: "nat" shows "i_Exec_Comp_Stream trans_fun input c n = f_Exec_Comp trans_fun (input \<Down> Suc n) c"
geodesic_segment_param_in_geodesic_spaces_1: fixes x :: "'a"   and y :: "'a" shows "geodesic_segment_param {x--y} x 0 = x"
pathVertices_fwd: fixes c :: "nat \<times> nat \<Rightarrow> 'b"   and u :: "nat"   and p :: "(nat \<times> nat) list"   and v :: "nat" assumes "Graph.isPath c u p v" shows "pathVertices u p = pathVertices_fwd u p"
induct: fixes P1 :: "tm \<Rightarrow> bool"   and P2 :: "tm list \<Rightarrow> bool"   and tm_list :: "tm list" assumes "\<And>x. P1 (Var x)"   and "\<And>x1 x2. P2 x2 \<Longrightarrow> P1 (Fn x1 x2)"   and "P2 []"   and "\<And>x1 x2. \<lbrakk>P1 x1; P2 x2\<rbrakk> \<Longrightarrow> P2 (x1 # x2)" shows "P2 tm_list"
simps_10: fixes \<phi> :: "'a ltln"   and \<psi> :: "'a ltln"   and \<nu> :: "'a set" shows "af_letter (\<phi> W\<^sub>n \<psi>) \<nu> = af_letter \<psi> \<nu> or\<^sub>n (af_letter \<phi> \<nu> and\<^sub>n \<phi> W\<^sub>n \<psi>)"
"1_1": fixes x :: "'a \<times> 'a list"   and xa :: "'a"   and y :: "'a list"   and finl :: "'a list" assumes "set todo \<noteq> {}"   and "x = (case todo of h # t \<Rightarrow> (h, t))"   and "(xa, y) = x"   and "\<exists>b\<in>fin. subsumes b xa"   and "fin = set finl" shows "set (mk_rtrancl_list_main subsumes r y finl) = mk_rtrancl_main y fin"
abs_eq: fixes xd :: "('d \<Rightarrow> ('d \<Rightarrow> 'e) \<Rightarrow> 'e) \<Rightarrow> 'f"   and xc :: "'f \<Rightarrow> 'g \<Rightarrow> 'h"   and xb :: "'h \<Rightarrow> 'i \<Rightarrow> 'c"   and xa :: "'i"   and x :: "'g" shows "ap_dual xd xc xb (B xa) (B x) = B (xb (xc (xd (\<lambda>x f. f x)) x) xa)"
nth_Cons: fixes x :: "'k"   and xs :: "'k list"   and n :: "nat" shows "(x # xs) ! n = (case n of 0 \<Rightarrow> x | Suc k \<Rightarrow> xs ! k)"
var_rename_inv_def: fixes n :: "nat"   and v :: "'f \<times> nat" shows "var_rename_inv n v \<equiv> Var (fst v, snd v - Suc n)"
Gen_Shleg_altdef: fixes k :: "nat"   and x :: "real" assumes "k \<le> n" shows "local.Gen_Shleg k x = (\<Sum>i\<le>k. (- 1) ^ (k - i) * real (k choose i) * real_of_int (int (pochhammer (n - i + 1) i * pochhammer (n - k + i + 1) (k - i))) * x ^ (n - i) * (1 - x) ^ (n - k + i))"
prime_factorization_nat_2: fixes n :: "nat" assumes "n \<noteq> 0" shows "prod_list (prime_factorization_nat n) = n"
of_int_nonneg: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and z :: "int" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one"   and "0 \<le> z" shows "less_eq zero (ring_1.of_int one plus zero minus z)"
Simulation_Invariant_axioms: shows "Simulation_Invariant B A (\<lambda>x y. y \<sim> x) PB PA"
sum_incoming_alt: fixes g :: "nat \<times> nat \<Rightarrow> 'capacity" assumes "\<forall>e. (0::'capacity) \<le> g e \<and> g e \<le> cf e" shows "\<forall>v\<in>cf.V. sum g (cf.incoming v) = (\<Sum>u\<in>cf.V. g (u, v))"
cr_gen_ssa_cfg_wf_def: shows "cr_gen_ssa_cfg_wf \<equiv> \<lambda>x y. x = Rep_gen_ssa_cfg_wf y"
vangle_refl: fixes u :: "'b" assumes "u \<noteq> (0::'b)" shows "vangle u u = 0"
assms_2: shows "finite {i. X i \<noteq> topspace (T i)}"
outer_prod_dim: fixes v :: "'a vec"   and n :: "nat"   and w :: "'a vec"   and m :: "nat" assumes "v \<in> carrier_vec n"   and "w \<in> carrier_vec m" shows "outer_prod v w \<in> carrier_mat n m"
R_def: shows "R = Pascal_Property.inter (line F A) (line C D)"
assms: shows "P \<Rightarrow>\<langle>\<alpha>\<rangle> P'"
power_one_over: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a"   and n :: "nat" assumes "class.division_ring divide inverse one times plus zero minus uminus" shows "power.power one times (divide one a) n = divide one (power.power one times a n)"
is_rbt_fold_rbt_insert_impl: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and t :: "('b, 'c) rbt"   and t' :: "('b, 'c) rbt" assumes "class.linorder less_eq less"   and "ord.is_rbt less t" shows "ord.is_rbt less (RBT_Impl.fold (ord.rbt_insert less) t' t)"
cz: shows "cz = 1 - (cmod z')\<^sup>2"
filterlim_at_top_at_bot: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter" assumes "filterlim f at_top F"   and "filterlim f at_bot F"   and "F \<noteq> bot" shows "False"
CBlinfun_cases: fixes x :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b" obtains y :: "'a \<Rightarrow> 'b" where "x = CBlinfun y" and "y \<in> {f. bounded_clinear f}"
1: shows "\<forall>m. filtered_seq s (kth_res_equals k l) m k = l"
ty_of_htype_eq_Boolean: fixes hT :: "htype" shows "ty_of_htype hT \<noteq> Boolean"
HMA_I_def: shows "Mod_Type_Connect.HMA_I = (\<lambda>i a. i = to_nat a)"
kbo_std_axioms: shows "kbo_std ground_heads_var (>\<^sub>s) \<epsilon> 0 extf (\<lambda>_. \<infinity>) (\<lambda>_. \<infinity>) wt_sym"
Pi_pmf_pmf_of_set_Suc: fixes A :: "'b set" assumes "finite A" shows "Pi_pmf A 0 (\<lambda>_. geometric_pmf (1 / 2)) = pmf_of_set (Pow A) \<bind> (\<lambda>B. Pi_pmf B 0 (\<lambda>_. map_pmf Suc (geometric_pmf (1 / 2))))"
rtranclpD: fixes R :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and a :: "'g"   and b :: "'g" assumes "R\<^sup>*\<^sup>* a b" shows "a = b \<or> a \<noteq> b \<and> R\<^sup>+\<^sup>+ a b"
min_dnf_iff_prop_assignment_subset: fixes \<A> :: "'a ltln set"   and \<phi> :: "'a ltln" shows "\<A> \<Turnstile>\<^sub>P \<phi> = (\<exists>B. fset B \<subseteq> \<A> \<and> B \<in> min_dnf \<phi>)"
is_span: shows "C.span \<mu>\<nu>\<pi>.leg0 \<mu>\<nu>\<pi>.leg1"
arrI: fixes f :: "'a" assumes "right a f" shows "R.arr f"
lselect_lnth: fixes i :: "nat"   and w :: "'a llist"   and s :: "nat set" assumes "enat i < llength (lselect s w)" shows "lselect s w ?! i = w ?! nth_least s i"
preserves_hom: fixes f :: "'a"   and a :: "'a"   and b :: "'a" assumes "S.in_hom f a b" shows "S.in_hom (S.\<rr> f) (S.R a) (S.map b)"
splitAt_simpD: fixes ram :: "'c"   and a :: "'c"   and b :: "'c list" assumes "ram \<noteq> a" shows "snd (splitAt ram (a # b)) = snd (splitAt ram b)"
is_selection: fixes K :: "'a set"   and A :: "'a set"   and \<phi> :: "'a" assumes "K = Cn A"   and "K .\<bottom>. \<phi> \<noteq> {}" shows "\<gamma> K \<phi> \<subseteq> K .\<bottom>. \<phi>"
crename_Cut: fixes R :: "trm"   and a :: "coname"   and b :: "coname"   and c :: "coname"   and M :: "trm"   and x :: "name"   and N :: "trm" assumes "R[a\<turnstile>c>b] = Cut <c>.M x.N"   and "c \<sharp> (a, b, N, R)"   and "x \<sharp> (M, R)" shows "\<exists>M' N'. R = Cut <c>.M' x.N' \<and> M'[a\<turnstile>c>b] = M \<and> N'[a\<turnstile>c>b] = N \<and> c \<sharp> N' \<and> x \<sharp> M'"
sum_characters: fixes x :: "nat" assumes "x \<in> carrier G" shows "(\<Sum>\<chi>\<in>characters G. \<chi> x) = (if x = \<one>\<^bsub>G\<^esub> then of_nat (order G) else 0)"
preservation_of_barbs_in_barbed_encoding: fixes Rel :: "(('procS, 'procT) Proc \<times> ('procS, 'procT) Proc) set"   and P :: "('procS, 'procT) Proc"   and Q :: "('procS, 'procT) Proc"   and a :: "'barbs" assumes "rel_preserves_binary_pred Rel (\<lambda>a. hasBarb a (STCalWB SWB TWB))"   and "(P, Q) \<in> Rel"   and "(\<exists>S. S \<in>S P \<and> S\<down><SWB>a) \<or> (\<exists>T. T \<in>T P \<and> T\<down><TWB>a)" shows "(\<exists>S. S \<in>S Q \<and> S\<down><SWB>a) \<or> (\<exists>T. T \<in>T Q \<and> T\<down><TWB>a)"
Sink_idle: fixes s :: "'state" assumes "s \<in> Sink" shows "s \<in> \<delta> s"
mult_subdistr: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(x \<cdot> y) ; z \<le> x ; z"
simps_13: fixes v :: "'b acom"   and va :: "'b acom"   and vb :: "'b"   and vc :: "bexp"   and vd :: "'b acom"   and ve :: "'b" shows "(v;; va \<sqsubseteq> {vb} WHILE vc DO vd {ve}) = False"
Lim_def: shows "LIM \<equiv> {U. \<exists>\<psi> s. learn_lim \<psi> U s}"
a_set_tag_name_locs_def: fixes element_ptr :: "(_) element_ptr" shows "CD.a_set_tag_name_locs element_ptr \<equiv> all_args (put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t element_ptr tag_name_update)"
discI_1: fixes hd :: "('a, 'b) hd"   and x1 :: "'b" assumes "hd = Var x1" shows "is_Var hd"
gSubstGOp_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd, 'e) Recursion.model_scheme" shows "gSubstGOp MOD \<equiv> \<forall>ys y Y' Y delta inp' inp binp' binp. wls (asSort ys) Y' \<and> gWls MOD (asSort ys) Y \<and> wlsInp delta inp' \<and> gWlsInp MOD delta inp \<and> wlsBinp delta binp' \<and> gWlsBinp MOD delta binp \<longrightarrow> gSubst MOD ys Y' Y y (Op delta inp' binp') (gOp MOD delta inp' inp binp' binp) = gOp MOD delta (inp' %[Y' / y]_ys) (gSubstInp MOD ys Y' Y y inp' inp) (binp' %%[Y' / y]_ys) (gSubstBinp MOD ys Y' Y y binp' binp)"
length_inorder: fixes t :: "'a tree" shows "length (inorder t) = size t"
seq_assoc: fixes seq :: "'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice"   and nil :: "'b::refinement_lattice"   and a :: "'b::refinement_lattice"   and b :: "'b::refinement_lattice"   and c :: "'b::refinement_lattice" assumes "sequential (seq::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) (nil::'b::refinement_lattice)" shows "(seq::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) (seq (a::'b::refinement_lattice) (b::'b::refinement_lattice)) (c::'b::refinement_lattice) = seq a (seq b c)"
assms: fixes \<sigma> :: "nat \<Rightarrow> 'a"   and \<tau> :: "nat \<Rightarrow> 'a" assumes "\<sigma> \<approx> \<tau>"   and "\<sigma> \<Turnstile>\<^sub>p \<phi>" shows "\<tau> \<Turnstile>\<^sub>p \<phi>"
simps: fixes \<sigma> :: "'b \<Rightarrow> 'b"   and P :: "'b hrel"   and \<rho> :: "'b \<Rightarrow> 'b"   and Q :: "'b hrel" shows "(\<sigma>, P) \<rightarrow>\<^sub>u (\<rho>, Q) = (\<langle>\<sigma>\<rangle>\<^sub>a ;; P \<sqsubseteq> \<langle>\<rho>\<rangle>\<^sub>a ;; Q)"
ftmx: shows "(\<lambda>x. f (t - x)) absolutely_integrable_on {- pi..pi}"
edom_AEstack: fixes Aexp :: "exp \<Rightarrow> Arity \<rightarrow> Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and as :: "Arity list"   and S :: "stack_elem list" assumes "EdomArityAnalysis Aexp" shows "edom (ArityAnalysis.AEstack Aexp as S) \<subseteq> fv S"
FHModule_addfunset: shows "FGModule H (\<currency>) (addfunset FH V)"
vconn_implies_hpar_3: fixes \<mu> :: "'a"   and f :: "'a"   and g :: "'a" assumes "\<guillemotleft>\<mu> : f \<Rightarrow>\<^sub>B g\<guillemotright>" shows "src\<^sub>B g = src\<^sub>B f"
ket_zero_dim: shows "ket_zero \<in> carrier_vec 2"
ncoplanar_perm_12: 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 "\<not> Tarski_neutral_dimensionless.Coplanar Bet A B C D" shows "\<not> Tarski_neutral_dimensionless.Coplanar Bet C A B D"
path_quotient_web: fixes \<Gamma> :: "('a, 'b) web_scheme"   and x :: "'a"   and p :: "'a list"   and y :: "'a"   and f :: "'a \<times> 'a \<Rightarrow> ennreal" assumes "path \<Gamma> x p y"   and "x \<notin> RF\<^sup>\<circ>\<^bsub>\<Gamma>\<^esub> (TER\<^bsub>\<Gamma>\<^esub> f)"   and "\<And>z. z \<in> set p \<Longrightarrow> z \<notin> RF\<^bsub>\<Gamma>\<^esub> (TER\<^bsub>\<Gamma>\<^esub> f)" shows "path (quotient_web \<Gamma> f) x p y"
inversion_number_between_cong_mset: fixes xs :: "'a list"   and xs' :: "'a list"   and ys :: "'a list"   and ys' :: "'a list" assumes "mset xs = mset xs'"   and "mset ys = mset ys'" shows "inversion_number_between xs ys = inversion_number_between xs' ys'"
finite_psupset_def: fixes S :: "'a set" shows "finite_psupset S \<equiv> {(Q', Q). Q \<subset> Q' \<and> Q' \<subseteq> S}"
mahler_measure_monic_ge_1: fixes p :: "complex poly" shows "1 \<le> mahler_measure_monic p"
grid_stepsize_nonneg: fixes t :: "nat \<Rightarrow> real"   and j :: "nat" assumes "grid t" shows "0 \<le> grid.stepsize t j"
c: shows "(i has_integral f (X + H) - (\<Sum>i<n. (1 / fact i) *\<^sub>R Df X i H)) {0..1}"
natural_transformation_axioms: shows "natural_transformation J (\<cdot>\<^sub>A) \<chi>.A.map D \<chi>"
sub_non_empty_blocks: fixes A :: "'a set" assumes "A \<in># \<A>" shows "A \<noteq> {}"
assms: shows "theta \<subseteq> Sretr (theta \<union> Sbis)"
size_hps_link: fixes h1 :: "'b hp"   and h2 :: "'b hp" shows "size_hps (hps (link h1 h2)) = Pairing_Heap_List2_Analysis.size_hp h1 + Pairing_Heap_List2_Analysis.size_hp h2 - 1"
wfx_start: shows "ts_ok wfx (thr (h.start_state f P C M vs)) h.start_heap"
supp_property_res: fixes as :: "atom set"   and x :: "'a"   and p :: "perm"   and as' :: "atom set"   and x' :: "'a" assumes "(as, x) \<approx>res (=) supp p (as', x')" shows "p \<bullet> (supp x \<inter> as) = supp x' \<inter> as'"
assms_3: fixes k :: "'u"   and it :: "'u set"   and \<sigma> :: "'\<sigma>" assumes "k \<in> it"   and "\<forall>j\<in>it. k \<le> j"   and "\<forall>j\<in>\<alpha> m - it. j \<le> k"   and "it \<subseteq> \<alpha> m"   and "I it \<sigma>" shows "I (it - {k}) (f k \<sigma>)"
last_appendR: fixes ys :: "'b list"   and xs :: "'b list" assumes "ys \<noteq> \<epsilon>" shows "last (xs \<cdot> ys) = last ys"
w_values_vs_type_all_start_heap_obs: assumes "wf_syscls P" shows "w_values P (vs_type_all P) (map snd (lift_start_obs h.start_tid h.start_heap_obs)) = vs_type_all P"
one_point: fixes x :: "'a \<Longrightarrow> 'b"   and v :: "('a, 'b) uexpr"   and P :: "'b upred" assumes "mwb_lens x"   and "x \<sharp> v" shows "(\<exists> x \<bullet> P \<and> x =\<^sub>u v) = P\<lbrakk>v/x\<rbrakk>"
filt_equiv_borel_AE_eq: fixes F :: "'c \<Rightarrow> 'a measure"   and N :: "'a measure"   and f :: "'a \<Rightarrow> real"   and t :: "'c"   and g :: "'a \<Rightarrow> real" assumes "filt_equiv F M N"   and "f \<in> borel_measurable (F t)"   and "g \<in> borel_measurable (F t)"   and "AEeq M f g" shows "AEeq N f g"
P_fixed_point_of_P_conj: fixes P :: "'a set"   and p :: "nat" assumes "finite (carrier G)"   and "P \<in> subgroups_of_size p" shows "P \<in> group_action.fixed_points (G\<lparr>carrier := P\<rparr>) (conjugation_action p) (subgroups_of_size p)"
pseudonatural_equivalence_axioms: shows "pseudonatural_equivalence V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C F \<Phi>\<^sub>F F \<Phi>\<^sub>F F.map\<^sub>0 (\<lambda>f. V\<^sub>C (F.D.lunit' (F f)) (F.D.runit (F f)))"
analytic_zeta_derivdiff: obtains a :: "complex" where "(\<lambda>z. if z = 1 then a else (z - 1) * deriv zeta z - deriv zeta z / zeta z) analytic_on {s. 1 \<le> Re s}"
JJ: shows "JJ \<in> UNpart dl (\<lambda>n. F n ` P n)"
group_right_one: fixes inverse :: "'b \<Rightarrow> 'b"   and one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'b" assumes "class.group inverse one times" shows "times x one = x"
closedin_component: fixes C :: "'a set"   and s :: "'a set" assumes "C \<in> components s" shows "closedin (top_of_set s) C"
Red_F_\<G>_def: fixes N :: "'f set" shows "empty_ord.Red_F_\<G> N = {C. \<forall>D\<in>\<G>_F C. D \<in> Red_F_G (\<G>_Fset N)}"
eval_vars_det\<^sub>A: fixes e :: "'AExp"   and mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val" assumes "\<forall>x\<in>aexp_vars e. mem\<^sub>1 x = mem\<^sub>2 x" shows "ev\<^sub>A mem\<^sub>1 e = ev\<^sub>A mem\<^sub>2 e"