hcju/isabelleps huggingface dataset:
INF_inf_const2: 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 :: "'c set"   and f :: "'c \<Rightarrow> 'b"   and x :: "'b" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "I \<noteq> {}" shows "Inf ((\<lambda>i. inf (f i) x) ` I) = inf (Inf (f ` I)) x"
T_append: fixes qs :: "'a list"   and as :: "'b list"   and step :: "'c \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'c"   and t :: "'c \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> nat"   and s :: "'c"   and q :: "'a"   and a :: "'b" assumes "length qs = length as" shows "On_Off.T step t s (qs @ [q]) (as @ [a]) = On_Off.T step t s qs as + t (On_Off.steps step s qs as) q a"
assms_1: shows "\<forall>op\<in>set ops. v \<notin> set (add_effects_of op) \<and> v \<notin> set (delete_effects_of op)"
rel_pmf_conj_2: fixes Q :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and P :: "bool"   and x :: "'c pmf"   and y :: "'d pmf" shows "rel_pmf (\<lambda>x y. Q x y \<and> P) x y = (P \<and> rel_pmf Q x y)"
qSwapAbs_preserves_qGoodAbs: fixes zs :: "'varSort"   and x :: "'var"   and y :: "'var"   and A :: "('e, 'f, 'varSort, 'var, 'g) qAbs" shows "qGoodAbs (A $[[x \<and> y]]_zs) = qGoodAbs A"
multLessRemdups: fixes a :: "Literal list"   and b :: "Literal list"   and M :: "(Literal \<times> bool) list" assumes "(a, b) \<in> multLess M" shows "(remdups a, remdups b) \<in> multLess M \<and> (remdups a, b) \<in> multLess M \<and> (a, remdups b) \<in> multLess M"
assms: shows "rel_pmf (R \<upharpoonleft> P \<otimes> Q) x y"
elims_2: fixes less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and proper_interval :: "'a option \<Rightarrow> 'a option \<Rightarrow> bool"   and x :: "'a option"   and xa :: "'a list"   and xb :: "'a list" assumes "proper_intrvl.proper_interval_Compl_set_aux less proper_interval x xa xb" obtains ao :: "'a option" and xc :: "'a" and xs :: "'a list" and y :: "'a" and ys :: "'a list" where "x = ao" and "xa = xc # xs" and "xb = y # ys" and "if less xc y then \<not> proper_interval ao (Some xc) \<and> proper_intrvl.proper_interval_Compl_set_aux less proper_interval (Some xc) xs (y # ys) else if less y xc then \<not> proper_interval ao (Some y) \<and> proper_intrvl.proper_interval_Compl_set_aux less proper_interval (Some y) (xc # xs) ys else \<not> proper_interval ao (Some xc) \<and> (ys = [] \<longrightarrow> xs \<noteq> [])"
finfun_upd_apply: fixes f :: "('b, 'a) finfun"   and a :: "'b"   and b :: "'a"   and a' :: "'b" shows "finfun_apply (finfun_update f a b) a' = (if a = a' then b else finfun_apply f a')"
wt_elim_cases_17: fixes E :: "env"   and dt :: "loc \<Rightarrow> ty option"   and D :: "qtname"   and blk :: "stmt"   and T :: "tys" assumes "E,dt\<Turnstile>In1l (Body D blk)\<Colon>T" obtains T :: "ty" where "Tb__ = Inl T" and "is_class (prg E) D" and "E,dt\<Turnstile>blk\<Colon>\<surd>" and "lcl E Result = Some T" and "is_type (prg E) T"
subgroupE_2: fixes H :: "nat set" assumes "subgroup H H" shows "H \<noteq> {}"
weak_bisimilarity_implies_weak_equivalence_Act: fixes x :: "('idx, 'pred, 'act) formula"   and P :: "'state"   and Q :: "'state"   and \<alpha> :: "'act" assumes "\<And>P Q. P \<approx>\<cdot> Q \<Longrightarrow> P \<Turnstile> x = Q \<Turnstile> x"   and "P \<approx>\<cdot> Q"   and "P \<Turnstile> \<langle>\<langle>\<alpha>\<rangle>\<rangle>x" shows "Q \<Turnstile> \<langle>\<langle>\<alpha>\<rangle>\<rangle>x"
widest_spread_axis_def: fixes k :: "'a"   and K :: "'a set"   and ps :: "(real, 'a) vec set" shows "widest_spread_axis k K ps = (\<forall>k'\<in>K. spread k' ps \<le> spread k ps)"
ext_inject: fixes heap :: "heap"   and locals :: "locals"   and more :: "'a"   and heap' :: "heap"   and locals' :: "locals"   and more' :: "'a" shows "(\<lparr>heap = heap, locals = locals, \<dots> = more\<rparr> = \<lparr>heap = heap', locals = locals', \<dots> = more'\<rparr>) = (heap = heap' \<and> locals = locals' \<and> more = more')"
simps_1: fixes \<alpha> :: "'d \<Rightarrow> 'e set \<times> 'e set"   and w :: "'d list" shows "[] \<otimes>\<^bsub>\<alpha>\<^esub> w = {w}"
induced_map_unique: fixes w :: "'a"   and w' :: "'a" assumes "\<mu>.is_induced_map w"   and "\<mu>.is_induced_map w'" shows "w \<cong> w'"
resumption_ord_outputD: fixes r :: "('d, 'e, 'f) Resumption.resumption"   and r' :: "('d, 'e, 'f) Resumption.resumption" assumes "resumption_ord r r'"   and "\<not> Resumption.resumption.is_Done r" shows "Resumption.resumption.output r = Resumption.resumption.output r'"
simps_2: fixes C_info :: "(name, dt_def) fmap"   and cs :: "(Term_Class.term \<times> sterm) list"   and \<Gamma> :: "(name, value) fmap" shows "special_constants.vwelldefined' C_info (Vabs cs \<Gamma>) = (pred_fmap id (fmmap (special_constants.vwelldefined' C_info) \<Gamma>) \<and> list_all (\<lambda>(pat, t). consts t |\<subseteq>| fmdom \<Gamma> |\<union>| constructors.C C_info) cs \<and> fdisjnt (constructors.C C_info) (fmdom \<Gamma>))"
emsapss_decode_help5_def: fixes mHash :: "bit list"   and maskedDB :: "bit list"   and dbMask :: "bit list"   and H :: "bit list"   and emBits :: "nat" shows "emsapss_decode_help5 mHash maskedDB dbMask H emBits = emsapss_decode_help6 mHash (bvxor maskedDB dbMask) H emBits"
Un_eqpoll_cong: fixes A :: "'a set"   and C :: "'b set"   and B :: "'a set"   and D :: "'b set" assumes "A \<approx> C"   and "B \<approx> D"   and "disjnt A B"   and "disjnt C D" shows "A \<union> B \<approx> C \<union> D"
hom_decomp_exact_aux: fixes qs :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and k :: "nat"   and m :: "nat" assumes "valid_decomp X qs"   and "standard_decomp k qs"   and "exact_decomp m qs"   and "hom_decomp qs" shows "hom_decomp (exact_aux k m qs)"
axiom_instance: fixes \<phi> :: "\<o>"   and v :: "i" assumes "[[\<phi>]]" shows "[\<phi> in v]"
apply_power_order: fixes f :: "'a perm"   and a :: "'a" shows "(f ^ order f a) \<langle>$\<rangle> a = a"
lconf: shows "P,h \<turnstile> l (:\<le>) E"
simps_3: fixes uv :: "'d option" shows "zip_option None uv = None"
interchange: fixes \<nu> :: "'b"   and \<mu> :: "'b"   and \<tau> :: "'b"   and \<sigma> :: "'b" assumes "D.seq \<nu> \<mu>"   and "D.seq \<tau> \<sigma>" shows "\<nu> \<cdot>\<^sub>D \<mu> \<star>\<^sub>D \<tau> \<cdot>\<^sub>D \<sigma> = (\<nu> \<star>\<^sub>D \<tau>) \<cdot>\<^sub>D (\<mu> \<star>\<^sub>D \<sigma>)"
mod_eq_imp_diff_mod_eq_divisor: fixes a :: "nat"   and b :: "nat"   and m :: "nat" assumes "a < b"   and "0 < m"   and "a mod m = b mod m" shows "Suc ((b - Suc a) mod m) = m"
J_def: shows "J = (\<Sum>n. (- 1) ^ n *\<^sub>R v ^ n)"
image2pI: fixes R :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'d"   and y :: "'e"   and f :: "'d \<Rightarrow> 'f"   and g :: "'e \<Rightarrow> 'g" assumes "R x y" shows "BNF_Greatest_Fixpoint.image2p f g R (f x) (g y)"
proves_eq_mp_rule_better: fixes \<Theta> :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set"   and \<Gamma> :: "Core.term set"   and A :: "Core.term"   and B :: "Core.term" assumes "wf_theory \<Theta>"   and "\<Theta>,\<Gamma> \<turnstile> mk_eq A B"   and "\<Theta>,\<Gamma> \<turnstile> A"   and "finite \<Gamma>"   and "\<forall>A\<in>\<Gamma>. term_ok \<Theta> A"   and "\<forall>A\<in>\<Gamma>. typ_of A = Some propT" shows "\<Theta>,\<Gamma> \<turnstile> B"
pr_naturality_1: fixes g :: "'a"   and b :: "'a"   and d :: "'a"   and f :: "'a"   and a :: "'a"   and c :: "'a" assumes "arr g"   and "local.dom g = b"   and "cod g = d"   and "arr f"   and "local.dom f = a"   and "cod f = c" shows "prj0' c d \<cdot> ECC.prod f g = g \<cdot> prj0' a b"
empty: shows "(\<lambda>x. \<epsilon> (''2'' @ x)) \<cdot> (\<lambda>x. \<epsilon> (tl x)) = \<epsilon>"
map_cong: fixes x :: "'b llist"   and ya :: "'b llist"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "x = ya"   and "\<And>z. z \<in> lset ya \<Longrightarrow> f z = g z" shows "lmap f x = lmap g ya"
r_dettime_2: fixes x :: "nat"   and i :: "nat" assumes "\<not> determined i x" shows "dettime i x \<up>"
Blinfun_inverse: fixes y :: "'d \<Rightarrow> 'e" assumes "y \<in> {f. bounded_linear f}" shows "blinfun_apply (Blinfun y) = y"
\<Gamma>_def: fixes b :: "'a \<Rightarrow> bool"   and cd :: "('a \<times> 'a) set" shows "\<Gamma> b cd = (\<lambda>cw. {(s, t). if b s then (s, t) \<in> cd O cw else s = t})"
hseq_leg\<^sub>0: shows "hseq s\<^sub>0 \<mu>.chine"
monotoneD: fixes orda :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and ordb :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and f :: "'f \<Rightarrow> 'g"   and x :: "'f"   and y :: "'f" assumes "monotone orda ordb f"   and "orda x y" shows "ordb (f x) (f y)"
key: shows "\<forall>i n. N \<le> n \<longrightarrow> \<bar>(\<Sum>n<n. f n i) - (\<Sum>n. f n i)\<bar> < r"
one_add_one: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c" assumes "class.numeral plus" shows "plus one one = numeral.numeral one plus (num.Bit0 num.One)"
step_2: fixes shadow_root :: "(_) shadow_root_ptr"   and host :: "(_) element_ptr" assumes "c = 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"   and "h \<turnstile> get_host shadow_root \<rightarrow>\<^sub>r host"   and "h \<turnstile> get_root_node_si (cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r host) \<rightarrow>\<^sub>r cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^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 child" shows "h \<turnstile> get_parent child \<rightarrow>\<^sub>r None"
PM_clearjunk0_cong: fixes xs :: "('a, 'b) fmap" shows "Pm_fmap (clearjunk0 xs) = Pm_fmap xs"
runit_hcomp_2: fixes f :: "'a"   and g :: "'a" assumes "ide f"   and "ide g"   and "src f = trg g" shows "\<r>\<^sup>-\<^sup>1[f \<star> g] = \<a>\<^sup>-\<^sup>1[f, g, src g] \<cdot> (f \<star> \<r>\<^sup>-\<^sup>1[g])"
good_imp_wf_ruleset: fixes rs :: "'a rule list"   and \<gamma> :: "('a \<Rightarrow> 'b \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'b \<Rightarrow> bool)"   and p :: "'b" assumes "good_ruleset rs" shows "wf_ruleset \<gamma> p rs"
rtrancl_path_Range: fixes R :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and x :: "'k"   and xs :: "'k list"   and y :: "'k"   and z :: "'k" assumes "rtrancl_path R x xs y"   and "z \<in> set xs" shows "Rangep R z"
simps_2: fixes l :: "'b tree"   and x :: "'b"   and r :: "'b tree"   and xs :: "'b list" shows "inorder2_tm \<langle>l, x, r\<rangle> xs =1 inorder2_tm r xs \<bind> (\<lambda>rs. inorder2_tm l (x # rs))"
bl2wc_simps_9: fixes list :: "cell list" shows "bl2wc (Oc # list) mod 2 = Suc 0"
dsg1: fixes x :: "'a" shows "x \<cdot> ar ar x = x"
mem_find_Some: fixes x :: "'a"   and xs :: "'a list"   and P :: "'a \<Rightarrow> bool" assumes "x \<in> set xs"   and "P x" shows "\<exists>x'. find P xs = Some x'"
length_map_idx: fixes f :: "'d \<Rightarrow> nat \<Rightarrow> 'c"   and xs :: "'d list"   and n :: "nat" shows "length (map_idx f xs n) = length xs"
semigroup_add_ow_axioms: shows "semigroup_add_ow U\<^sub>M (+\<^sub>M)"
lookup_nil: fixes e :: "('a, 'b, 'c) env" shows "lookup e [] = Some e"
path_component_in_topspace: fixes X :: "'c topology"   and x :: "'c"   and y :: "'c" assumes "path_component_of X x y" shows "x \<in> topspace X \<and> y \<in> topspace X"
wrapper_correctness: fixes Q_pos :: "(nat set \<times> 'b option list set) set"   and Q_neg :: "(nat set \<times> 'b option list set) set"   and n :: "nat"   and z :: "'b option list" assumes "1 \<le> card Q_pos"   and "\<forall>(A, X)\<in>Q_pos \<union> Q_neg. table n A X \<and> wf_set n A" shows "(z \<in> New_max_getIJ_wrapperGenericJoin Q_pos Q_neg) = (wf_tuple n (\<Union>(A, X)\<in>Q_pos. A) z \<and> (\<forall>(A, X)\<in>Q_pos. restrict A z \<in> X) \<and> (\<forall>(A, X)\<in>Q_neg. restrict A z \<notin> X))"
repU_leq: fixes w :: "32 Word.word"   and r :: "real"   and r' :: "real" assumes "w \<equiv>\<^sub>U r"   and "r' \<le> r" shows "w \<equiv>\<^sub>U r'"
map_of_update_with_aux: fixes v :: "'b"   and k :: "'a"   and f :: "'b \<Rightarrow> 'b"   and ps :: "('a \<times> 'b) list" shows "map_of (AList.update_with_aux v k f ps) = map_of ps(k \<mapsto> case map_of ps k of None \<Rightarrow> f v | Some v \<Rightarrow> f v)"
zero_le_power: fixes a :: "'d"   and n :: "nat" assumes "(0::'d) \<le> a" shows "(0::'d) \<le> a ^ n"
components_are_iso: fixes a :: "'a" assumes "hom'.ide a" shows "hom'.iso (d\<^sub>0e\<^sub>0.\<phi> a)"
ord_spmf_reflI: fixes p :: "'a spmf"   and ord :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "\<And>x. x \<in> set_spmf p \<Longrightarrow> ord x x" shows "ord_spmf ord p p"
heap_is_wellformed_def: shows "heap_is_wellformed = (\<lambda>h. local.a_acyclic_heap h \<and> local.a_all_ptrs_in_heap h \<and> local.a_distinct_lists h \<and> local.a_owner_document_valid h)"
count_roots_below_def: fixes p :: "real poly"   and a :: "real" shows "count_roots_below p a = (if p = 0 then 0 else let ps = sturm_squarefree p in sign_changes_neg_inf ps - sign_changes ps a)"
integral_subset_le: fixes s :: "'a set"   and t :: "'a set"   and f :: "'a \<Rightarrow> real" assumes "s \<subseteq> t"   and "f integrable_on s"   and "f integrable_on t"   and "\<forall>x\<in>t. 0 \<le> f x" shows "integral s f \<le> integral t f"
Conj\<^sub>\<alpha>_def': fixes tset\<^sub>\<alpha> :: "('a, 'b, 'c, 'd) Tree\<^sub>\<alpha> set['a]" shows "Conj\<^sub>\<alpha> tset\<^sub>\<alpha> = abs_Tree\<^sub>\<alpha> (tConj (map_bset rep_Tree\<^sub>\<alpha> tset\<^sub>\<alpha>))"
left_uniqueD: fixes A :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'c"   and z :: "'d"   and y :: "'c" assumes "left_unique A"   and "A x z"   and "A y z" shows "x = y"
top_real_min_top_def: shows "top \<equiv> PInfty"
elims_3: fixes x :: "('d, 'e, 'varSort, 'var, 'f) qAbs" assumes "\<not> qGoodAbs x" obtains xs :: "'varSort" and x :: "'var" and X :: "('d, 'e, 'varSort, 'var, 'f) qTerm" where "xa__ = qAbs xs x X" and "\<not> qGood X"
is_map_def: fixes M :: "('c \<times> 'd) set" shows "is_map M = (\<forall>(a\<^sub>1, b\<^sub>1)\<in>M. \<forall>(a\<^sub>2, b\<^sub>2)\<in>M. a\<^sub>1 = a\<^sub>2 \<longrightarrow> b\<^sub>1 = b\<^sub>2)"
pedersen_asymp_def: fixes \<G> :: "nat \<Rightarrow> 'a cyclic_group" shows "pedersen_asymp \<G> \<equiv> \<forall>\<eta>. pedersen (\<G> \<eta>)"
meet_commute: fixes x :: "'c"   and y :: "'c" shows "x \<sqinter> y = y \<sqinter> x"
Cod_Diagonalize_rep: fixes f :: "'c local.term set" assumes "arr f" shows "Cod \<^bold>\<lfloor>rep f\<^bold>\<rfloor> = \<^bold>\<lfloor>rep (cod f)\<^bold>\<rfloor>"
lift_resumption_Done: fixes x :: "'d option" shows "lift_resumption (Resumption.resumption.Done x) = (case x of None \<Rightarrow> Fail | Some x' \<Rightarrow> Generative_Probabilistic_Value.Done x')"
Quotient3_rel: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and Abs :: "'c \<Rightarrow> 'd"   and Rep :: "'d \<Rightarrow> 'c"   and r :: "'c"   and s :: "'c" assumes "Quotient3 R Abs Rep" shows "(R r r \<and> R s s \<and> Abs r = Abs s) = R r s"
seq_step_invariant_sterms_weaken: fixes A :: "('e \<times> ('e, 'f, 'g, 'h) seqp, 'f seq_action) automaton"   and I :: "'f seq_action \<Rightarrow> bool"   and \<Gamma> :: "'g \<Rightarrow> ('e, 'f, 'g, 'h) seqp"   and P :: "('e \<times> 'h) \<times> 'f seq_action \<times> 'e \<times> 'h \<Rightarrow> bool"   and l' :: "'h"   and q :: "('e, 'f, 'g, 'h) seqp"   and \<xi> :: "'e"   and p :: "('e, 'f, 'g, 'h) seqp"   and I' :: "'f seq_action \<Rightarrow> bool"   and p' :: "('e, 'f, 'g, 'h) seqp"   and a :: "'f seq_action"   and \<xi>' :: "'e" assumes "A \<TTurnstile>\<^sub>A (I \<rightarrow>) onll \<Gamma> P"   and "wellformed \<Gamma>"   and "automaton.trans A = seqp_sos \<Gamma>"   and "l' \<in> labels \<Gamma> q"   and "(\<xi>, p) \<in> reachable A I'"   and "((\<xi>, p'), a, \<xi>', q) \<in> automaton.trans A"   and "I' a"   and "p' \<in> sterms \<Gamma> p"   and "\<And>a. I' a \<Longrightarrow> I a" shows "\<forall>l\<in>labels \<Gamma> p'. P ((\<xi>, l), a, \<xi>', l')"
mod_nat_wlog: fixes b :: "nat"   and n :: "nat"   and P :: "nat \<Rightarrow> bool" assumes "0 < b"   and "\<And>k. \<lbrakk>k \<in> {0..<b}; n mod b = k\<rbrakk> \<Longrightarrow> P n" shows "P n"
wmsI: fixes A :: "(nat \<times> nat) multiset"   and B :: "(nat \<times> nat) multiset"   and Z :: "(nat \<times> nat) multiset" assumes "(set_mset A, set_mset B) \<in> max_strict \<or> A = {#} \<and> B = {#}" shows "(Z + A, Z + B) \<in> ms_weak"
to_hma_from_hma\<^sub>m: fixes v :: "(('b, 'c) Finite_Cartesian_Product.vec, 'd) Finite_Cartesian_Product.vec" shows "Mod_Type_Connect.to_hma\<^sub>m (Mod_Type_Connect.from_hma\<^sub>m v) = v"
stkOk_r_rewrite: fixes x :: "'b"   and xs :: "'b list"   and c :: "'b \<Rightarrow> bool"   and l :: "'b \<Rightarrow> 'b ref"   and r :: "'b \<Rightarrow> 'b ref"   and g :: "'b ref"   and iL :: "'b \<Rightarrow> 'b ref"   and iR :: "'b \<Rightarrow> 'b ref" assumes "x \<notin> set xs" shows "stkOk c l (r(x := g)) iL iR (Ref x) xs = stkOk c l r iL iR (Ref x) xs"
convergent_NSconvergent_iff: fixes X :: "nat \<Rightarrow> 'a" shows "convergent X = NSconvergent X"
Catch_decomp_star_Stuck: fixes \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and p\<^sub>1 :: "('d, 'e, 'f) com"   and p\<^sub>2 :: "('d, 'e, 'f) com"   and s :: "'d" assumes "\<Gamma> \<turnstile> (Catch p\<^sub>1 p\<^sub>2, Normal s) \<rightarrow>\<^sup>* (Skip, Stuck)" shows "\<Gamma> \<turnstile> (p\<^sub>1, Normal s) \<rightarrow>\<^sup>* (Skip, Stuck) \<or> (\<exists>s'. \<Gamma> \<turnstile> (p\<^sub>1, Normal s) \<rightarrow>\<^sup>* (Throw, Normal s') \<and> \<Gamma> \<turnstile> (p\<^sub>2, Normal s') \<rightarrow>\<^sup>* (Skip, Stuck))"
typeI_cube_line_integral_exists_boundary': assumes "\<forall>two_cube\<in>two_chain. typeI_twoCube two_cube"   and "\<forall>twoC\<in>two_chain. analytically_valid (cubeImage twoC) (\<lambda>x. F x \<bullet> i) j"   and "\<forall>two_cube\<in>two_chain. valid_two_cube two_cube" shows "\<forall>(k, \<gamma>)\<in>two_chain_vertical_boundary two_chain. line_integral_exists F {i} \<gamma>"
nonzero_norm_divide: fixes b :: "'a"   and a :: "'a" assumes "b \<noteq> (0::'a)" shows "norm (a / b) = norm a / norm b"
deletion: fixes ss :: "'w list" assumes "ss \<in> lists S"   and "\<not> S_reduced ss" shows "\<exists>a b as bs cs. ss = as @ [a] @ bs @ [b] @ cs \<and> sum_list ss = sum_list (as @ bs @ cs)"
measure_Un3: fixes A :: "'a set"   and M :: "'a measure"   and B :: "'a set" assumes "A \<in> fmeasurable M"   and "B \<in> fmeasurable M" shows "Sigma_Algebra.measure M (A \<union> B) = Sigma_Algebra.measure M A + Sigma_Algebra.measure M B - Sigma_Algebra.measure M (A \<inter> B)"
antisym: fixes b :: "'b word"   and a :: "'b word" assumes "b \<le>s a"   and "a \<le>s b" shows "a = b"
S_def: fixes vs :: "'a set"   and lss :: "('a, 'b) fmap list"   and PROB :: "(('a, 'b) fmap \<times> ('a, 'b) fmap) set"   and s :: "('a, 'b) fmap" shows "S vs lss PROB s \<equiv> wlp (\<lambda>x y. y \<in> state_successors (prob_proj PROB vs) x) (\<lambda>s. problem_plan_bound (snapshot PROB s)) max (\<lambda>x y. x + y + 1) s lss"
proj4_addpre: fixes v :: "nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set"   and pre :: "nat set" shows "\<pi>\<^sub>4 (addpre v pre) = \<pi>\<^sub>4 v"
distinct_remove_cycles_inv: fixes xs :: "'b list"   and ys :: "'b list"   and x :: "'b" assumes "distinct (xs @ ys)" shows "distinct (remove_cycles xs x ys)"
record_simps_3: fixes f :: "nat \<Rightarrow> nat"   and x :: "'a state" shows "ffi (update_clock f x) = ffi x"
max_MInf: fixes x :: "ereal" shows "max (- \<infinity>) x = x"
prems_12: shows "I = items_le k' (\<pi> k' {} (Scan T k (Gen (Prefixes p))))"
cbig_big_implies: fixes \<sigma>' :: "'state"   and coll :: "'coll"   and P :: "'prog"   and \<sigma> :: "'state" assumes "(\<sigma>', coll) \<in> cbig P \<sigma>" shows "\<sigma>' \<in> big P \<sigma>"
init_InvTempUpright: fixes c :: "('c, 'd) configuration" assumes "init_config c" shows "InvTempUpright c"
supp_Pred\<^sub>\<alpha>: fixes \<phi> :: "'f" shows "supp (Pred\<^sub>\<alpha> \<phi>) = supp \<phi>"
comp_fun_commute_on_def: fixes f :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and A :: "'e set" shows "comp_fun_commute_on f A \<equiv> (\<forall>y\<in>A. \<forall>x\<in>A. \<forall>z\<in>A. f y (f x z) = f x (f y z)) \<and> f \<in> A \<rightarrow> A \<rightarrow> A"
AE_measure_pmf_iff: fixes P :: "'e \<Rightarrow> bool"   and M :: "'e pmf" shows "almost_everywhere (measure_pmf M) P = (\<forall>y\<in>set_pmf M. P y)"
assms: shows "xs #\<^sub>\<circ> x \<subseteq>\<^sub>\<circ> ys"
RHS2: shows "Cring_Poly.compose R (Cring_Poly.compose R (monom P (f (deg R f)) (deg R f)) q) r = Cring_Poly.compose R (to_polynomial R a) r \<otimes>\<^bsub>P\<^esub> Cring_Poly.compose R (q [^]\<^bsub>P\<^esub> n) r"
Astack_map_Dummy: fixes l :: "Vars.var list" shows "Astack (map Dummy l) = 0"
simps_203: fixes x141 :: "expr"   and x142 :: "expr"   and x101 :: "expr"   and x102 :: "char list option"   and x103 :: "char list"   and x104 :: "expr list" shows "while (x141) x142 \<noteq> Call x101 x102 x103 x104"
i_Exec_Comp_Stream_Acc_Output__Init__eq_NoMsg_iAll_start_event_conv: fixes k :: "nat"   and event :: "nat \<Rightarrow> bool"   and t0 :: "nat"   and t :: "nat"   and s :: "nat \<Rightarrow> 'a message_af"   and output_fun :: "'b \<Rightarrow> 'a message_af"   and trans_fun :: "'c message_af \<Rightarrow> 'b \<Rightarrow> 'b"   and input :: "nat \<Rightarrow> 'c message_af"   and c :: "'b" assumes "0 < k"   and "\<And>t. event t = ((t + k - Suc 0) mod k = 0)"   and "t0 = Suc (t * k)"   and "s = output_fun \<circ> i_Exec_Comp_Stream_Init trans_fun (input \<odot> k) c" shows "(i_Exec_Comp_Stream_Acc_Output k output_fun trans_fun input c t = NoMsg) = (s t0 = NoMsg \<and> (\<circle> t' t0 [0\<dots>]. s t1 = NoMsg. t1 \<U> t2 [0\<dots>] \<oplus> t'. event t2))"
deriv_diff: fixes f :: "'a \<Rightarrow> 'a"   and z :: "'a"   and g :: "'a \<Rightarrow> 'a" assumes "f field_differentiable at z"   and "g field_differentiable at z" shows "deriv (\<lambda>w. f w - g w) z = deriv f z - deriv g z"
bottom_in_below_in: fixes P :: "'a set"   and x :: "'a" assumes "has_bottom P"   and "x \<in> P" shows "bottom P \<in> P .\<^bold>\<le> x"
mem_Collect_eq: fixes a :: "'r"   and P :: "'r \<Rightarrow> bool" shows "(a \<in> Collect P) = P a"
rel_bset_bempty_1: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and X :: "'d set['e]" shows "rel_bset R bempty X = (X = bempty)"
\<Sigma>_protocol_def: shows "\<Sigma>_OR.\<Sigma>_protocol \<equiv> \<Sigma>_OR.completeness \<and> \<Sigma>_OR.special_soundness \<and> \<Sigma>_OR.HVZK"
sing_pow_hd_tl: fixes c :: "'b"   and w :: "'b list"   and a :: "'b" shows "c # w \<in> [a]* = (c = a \<and> w \<in> [a]*)"
ivl_disj_un_singleton_2: fixes u :: "'a" shows "{..<u} \<union> {u} = {..u}"
array_shl_def: fixes a :: "'b array"   and i :: "nat"   and k :: "nat" shows "array_shl a i k \<equiv> Array.len a \<bind> (\<lambda>l. safe_sblit a i (i - k) (l - i))"
U1_as_alpha: fixes P :: "('d, 'g) urel" shows "P ;; U1 = \<lceil>P\<rceil>\<^sub>1"
intra_proc_matched: fixes a :: "'edge"   and Q :: "('var \<Rightarrow> 'val option) \<times> 'ret \<Rightarrow> bool"   and r :: "'ret"   and p :: "'pname"   and fs :: "(('var \<Rightarrow> 'val option) \<Rightarrow> 'val option) list"   and a' :: "'edge" assumes "valid_edge a"   and "kind a = Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs"   and "a' \<in> get_return_edges a" shows "matched (CFG_node (targetnode a)) [CFG_node (targetnode a)] (CFG_node (sourcenode a'))"
invertible_qmpt_def: shows "I.invertible_qmpt \<equiv> bij Tinv \<and> inv Tinv \<in> M \<rightarrow>\<^sub>M M"
hmem_def: fixes a :: "hf"   and b :: "hf" shows "(a \<^bold>\<in> b) = (a \<in> hfset b)"