hcju/isabelleps huggingface dataset:
closest_point_le: fixes S :: "'b set"   and x :: "'b"   and a :: "'b" assumes "closed S"   and "x \<in> S" shows "dist a (closest_point S a) \<le> dist a x"
simps_10: fixes C :: "char list"   and F :: "char list"   and D :: "char list"   and e\<^sub>2 :: "'c exp" shows "ss_exp (C\<bullet>\<^sub>sF{D} := e\<^sub>2) = (case val_of e\<^sub>2 of None \<Rightarrow> ss_exp e\<^sub>2 | \<lfloor>v\<rfloor> \<Rightarrow> C\<bullet>\<^sub>sF{D} := e\<^sub>2)"
igFreshCls_fromMOD: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) Recursion.model" assumes "gFreshCls MOD" shows "igFreshCls (fromMOD MOD)"
hadd_succ_left: fixes y :: "hf"   and x :: "hf" assumes "Ord y" shows "succ x @+ y = succ (x @+ y)"
discounted_integrable: fixes N :: "'a measure"   and X :: "nat \<Rightarrow> 'a \<Rightarrow> real"   and n :: "nat"   and r :: "real" assumes "integrable N (X n)"   and "- 1 < r"   and "space N = space M" shows "integrable N (discounted_value r X n)"
length_list_update: fixes xs :: "'a list"   and i :: "nat"   and x :: "'a" shows "|xs[i := x]| = |xs|"
is_rbt_def: fixes less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and t :: "('a, 'b) rbt" shows "ord.is_rbt less t = (inv1 t \<and> inv2 t \<and> color_of t = B \<and> ord.rbt_sorted less t)"
Abs_dlist_inverse: fixes y :: "'d list" assumes "y \<in> {xs. distinct xs}" shows "list_of_dlist (Abs_dlist y) = y"
reducible\<^sub>dI: fixes p :: "'a poly" assumes "0 < degree p \<Longrightarrow> \<exists>q r. degree q < degree p \<and> degree r < degree p \<and> p = q * r" shows "\<not> irreducible\<^sub>d p"
of_weak_ranking_wf_1: fixes xs :: "'b set list"   and A :: "'b set" assumes "\<Union> (set xs) = A"   and "is_weak_ranking xs" shows "total_preorder_on A (of_weak_ranking xs)"
ideal_nsum_closed: fixes R :: "('c, 'd) Ring_scheme"   and I :: "'c set"   and n :: "nat"   and f :: "nat \<Rightarrow> 'c" assumes "Ring R"   and "ideal R I"   and "\<forall>j\<le>n. f j \<in> I" shows "\<Sigma>\<^sub>e R f n \<in> I"
vars\<^sub>s\<^sub>s\<^sub>t_Cons: fixes a :: "('d, 'c) stateful_strand_step"   and A :: "('d, 'c) stateful_strand_step list" shows "vars\<^sub>s\<^sub>s\<^sub>t (a # A) = vars\<^sub>s\<^sub>s\<^sub>t\<^sub>p a \<union> vars\<^sub>s\<^sub>s\<^sub>t A"
nvalidI: fixes \<Gamma> :: "'f \<Rightarrow> ('d, 'f, 'e) com option"   and c :: "('d, 'f, 'e) com"   and n :: "nat"   and P :: "'d set"   and F :: "'e set"   and Q :: "'d set"   and A :: "'d set" assumes "\<And>s t. \<lbrakk>\<Gamma>\<turnstile> \<langle>c,Normal s\<rangle> =n\<Rightarrow> t; s \<in> P; t \<notin> Fault ` F\<rbrakk> \<Longrightarrow> t \<in> Normal ` Q \<union> Abrupt ` A" shows "\<Gamma>\<Turnstile>n:\<^bsub>/F\<^esub> P c Q,A"
map_fun_def: fixes f :: "'r \<Rightarrow> 't"   and g :: "'u \<Rightarrow> 's"   and h :: "'t \<Rightarrow> 'u" shows "map_fun f g h = g \<circ> h \<circ> f"
gbinomial_Suc_rec: fixes x :: "'a"   and k :: "nat" shows "x gchoose Suc k = (x gchoose k) * ((x - of_nat k) / (of_nat k + (1::'a)))"
U'_def: shows "U' \<equiv> \<lambda>i x. indicat_real (space M) x * enn2real (U i x)"
powr_le1: fixes a :: "real"   and x :: "real" assumes "0 \<le> a"   and "0 \<le> x"   and "x \<le> 1" shows "x powr a \<le> 1"
times_ms_aux_MSLCons: fixes x :: "'a \<times> real"   and xs :: "('a \<times> real) msllist"   and y :: "'a \<times> real"   and ys :: "('a \<times> real) msllist" shows "times_ms_aux (MSLCons x xs) (MSLCons y ys) = MSLCons (fst x * fst y, snd x + snd y) (plus_ms_aux (scale_shift_ms_aux x ys) (times_ms_aux xs (MSLCons y ys)))"
exec_all_finalD: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> nat \<times> nat \<times> instr list \<times> (nat \<times> nat \<times> char list \<times> nat \<times> nat) list) list) list"   and x :: "nat option"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and \<sigma> :: "nat option \<times> (nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (val list \<times> val list \<times> char list \<times> char list \<times> nat) list" assumes "P \<turnstile> (x, h, []) -jvm\<rightarrow> \<sigma>" shows "\<sigma> = (x, h, [])"
assms_5: shows "Separation (remove_vertex v) v0 v1 S"
assms_5: shows "final_iteration M2 M1 \<Omega> V m i"
wf_matchF_aux_Cons3: fixes \<sigma> :: "(String.literal \<times> event_data list) trace"   and V :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and n :: "nat"   and R :: "event_data list set"   and I :: "\<I>"   and r :: "Formula.formula Regex.regex"   and t :: "nat"   and rels :: "event_data option list set list"   and rel :: "event_data option list set"   and aux :: "(nat \<times> event_data option list set list \<times> event_data option list set) list"   and ne :: "nat"   and i :: "nat" assumes "wf_matchF_aux \<sigma> V n R I r ((t, rels, rel) # aux) ne i" shows "qtable n (fv_regex r) (mem_restr R) (\<lambda>v. \<exists>j\<ge>ne. j < Suc (ne + length aux + i) \<and> mem (\<tau> \<sigma> j - \<tau> \<sigma> ne) I \<and> Regex.match (Formula.sat \<sigma> V (map the v)) r ne j) rel"
simps_2: fixes A :: "'b tree"   and a :: "'b" shows "splay_max \<langle>A, a, \<langle>\<rangle>\<rangle> = \<langle>A, a, \<langle>\<rangle>\<rangle>"
le_SUP_iff: 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 x :: "'b"   and f :: "'c \<Rightarrow> 'b"   and A :: "'c set" assumes "class.complete_linorder Inf Sup inf less_eq less sup bot top" shows "less_eq x (Sup (f ` A)) = (\<forall>y. less y x \<longrightarrow> (\<exists>i\<in>A. less y (f i)))"
cone_mkCone: fixes p0 :: "'a"   and p1 :: "'a" assumes "a0xa1.is_rendered_commutative_by p0 p1" shows "a0xa1.cone (local.dom p0) (a0xa1.mkCone p0 p1)"
encode_kind_5: fixes n :: "nat"   and f :: "recf" shows "encode_kind (encode (Mn n f)) = 5"
silent_move_return_node: fixes S :: "'node SDG_node set"   and f :: "'edge \<Rightarrow> ('var, 'val, 'ret, 'pname) edge_kind"   and ms :: "'node list"   and s :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and a :: "'edge"   and ms' :: "'node list"   and s' :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" assumes "S,f \<turnstile> (ms,s) -a\<rightarrow>\<^sub>\<tau> (ms',s')" shows "Ball (set (tl ms')) return_node"
1: shows "\<forall>a\<in>A. finite (R `` {a})"
step_deterministic: fixes x :: "'state"   and y :: "'state"   and z :: "'state" assumes "step x y"   and "step x z" shows "y = z"
nths_map: fixes f :: "'b \<Rightarrow> 'a"   and xs :: "'b list"   and I :: "nat set" shows "nths (map f xs) I = map f (nths xs I)"
prefixes_hd: fixes t :: "nat"   and j :: "nat"   and b :: "nat" assumes "local.prefixes t j \<down>= b" shows "e_nth b 0 = j"
predict_stoch_proc_def: fixes F :: "nat \<Rightarrow> 'b measure"   and X :: "nat \<Rightarrow> 'b \<Rightarrow> 'd"   and N :: "'d measure" shows "predict_stoch_proc F X N = (X 0 \<in> F 0 \<rightarrow>\<^sub>M N \<and> (\<forall>n. X (Suc n) \<in> F n \<rightarrow>\<^sub>M N))"
bin_rep_def: fixes n :: "nat"   and m :: "nat" shows "bin_rep n m = butlast (bin_rep_aux n m)"
splitAt_ram3: fixes ram2 :: "'a"   and ram1 :: "'a"   and vs :: "'a list" assumes "ram2 \<notin> set (fst (splitAt ram1 vs))"   and "ram1 \<in> set vs"   and "ram2 \<in> set vs"   and "ram1 \<noteq> ram2" shows "ram2 \<in> set (snd (splitAt ram1 vs))"
not_self_fresh: fixes x :: "'b" shows "atom x \<sharp> x = False"
memb_imp_not_empty: fixes x :: "'c"   and S :: "'c set" assumes "x \<in> S" shows "S \<noteq> {}"
fA_tso_empty_locs_def: shows "fA_tso_empty_locs = - {mark_noop_mfence}"
C1: shows "card ({token_run x n |x n. True} - Collect sink) \<le> card (reach \<Sigma> \<delta> q\<^sub>0 - Collect sink)"
irrefl: fixes a :: "'t \<Rightarrow>\<^sub>0 'c" shows "\<not> a \<prec>\<^sub>p a"
MemInv_def: fixes mm :: "state \<Rightarrow> Locs \<Rightarrow> Vals"   and l :: "Locs" shows "MemInv mm l \<equiv> #l \<in> #MemLoc \<longrightarrow> mm!l \<in> #MemVal"
equivalence_mapE: fixes f :: "'a" assumes "equivalence_map f" obtains g :: "'a" where "quasi_inverses f g"
resumption_ord_outputD: fixes r :: "('f, 'g, 'h) Resumption.resumption"   and r' :: "('f, 'g, 'h) Resumption.resumption" assumes "resumption_ord r r'"   and "\<not> Resumption.resumption.is_Done r" shows "Resumption.resumption.output r = Resumption.resumption.output r'"
disc_23: fixes x31 :: "'b local.term"   and x32 :: "'b local.term" shows "is_Hcomp (x31 \<^bold>\<star> x32)"
get_shadow_root_new_document: fixes h :: "(_) heap"   and new_document_ptr :: "(_) document_ptr"   and h' :: "(_) heap"   and r :: "(_) heap \<Rightarrow> (_) heap \<Rightarrow> bool"   and ptr' :: "(_) element_ptr" assumes "h \<turnstile> new_document \<rightarrow>\<^sub>r new_document_ptr"   and "h \<turnstile> new_document \<rightarrow>\<^sub>h h'"   and "r \<in> get_shadow_root_locs ptr'" shows "r h h'"
singleton_conv: fixes a :: "'f" shows "{x. x = a} = {a}"
contour_integral_midpoint: fixes a :: "complex"   and b :: "complex"   and f :: "complex \<Rightarrow> complex" assumes "continuous_on (closed_segment a b) f" shows "contour_integral (linepath a b) f = contour_integral (linepath a (midpoint a b)) f + contour_integral (linepath (midpoint a b) b) f"
MPoly_inverse: fixes y :: "(nat \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" assumes "y \<in> UNIV" shows "mapping_of (MPoly y) = y"
pLp: shows "p ; 1 ; p\<^sup>T \<le> 1'"
Completion: fixes F :: "'c program"   and A :: "'c set"   and A' :: "'c set"   and C :: "'c set"   and B :: "'c set"   and B' :: "'c set" assumes "F \<in> A \<longmapsto>w A' \<union> C"   and "F \<in> A' Co A' \<union> C"   and "F \<in> B \<longmapsto>w B' \<union> C"   and "F \<in> B' Co B' \<union> C" shows "F \<in> A \<inter> B \<longmapsto>w A' \<inter> B' \<union> C"
inject_2: fixes x2 :: "nat"   and y2 :: "nat" shows "(Number x2 = Number y2) = (x2 = y2)"
circ_right_slide: fixes x :: "'a"   and y :: "'a" shows "x * (y * x)\<^sup>\<circ> \<le> (x * y)\<^sup>\<circ> * x"
inject: fixes x :: "'a \<Rightarrow> 'b"   and ya :: "'a \<Rightarrow> 'b" shows "(EnvT x = EnvT ya) = (x = ya)"
GFP_weak_unfold: fixes f :: "'a \<Rightarrow> 'a" assumes "Mono f"   and "f \<in> carrier L \<rightarrow> carrier L" shows "GFP f .= f (GFP f)"
fst_annotate_rlen_le: fixes k :: "nat"   and a :: "'b"   and l :: "'b list" assumes "(k, a) \<in> set (annotate_rlen l)" shows "k < length l"
assms_3: shows "\<GG> \<circ>\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<FF> = dghm_id \<AA>"
monofunI: fixes f :: "'c \<Rightarrow> 'd" assumes "\<And>x y. x \<sqsubseteq> y \<Longrightarrow> f x \<sqsubseteq> f y" shows "monofun f"
ord_Zp_geq: fixes x :: "nat \<Rightarrow> int"   and n :: "nat" assumes "x \<in> carrier Zp"   and "x n = 0"   and "x \<noteq> \<zero>" shows "int n \<le> ord_Zp x"
sel_1: fixes x :: "complex"   and y :: "complex" shows "Re (x * y) = Re x * Re y - Im x * Im y"
rep_eq: fixes x :: "('c, 'd) lifted_distrib_lattice_top"   and xa :: "('c, 'd) lifted_distrib_lattice_top" shows "(x < xa) = ((\<forall>f. Rep_lifted_distrib_lattice_top x f \<le> Rep_lifted_distrib_lattice_top xa f) \<and> \<not> (\<forall>f. Rep_lifted_distrib_lattice_top xa f \<le> Rep_lifted_distrib_lattice_top x f))"
simps_1: fixes k :: "nat"   and A :: "nat set" assumes "enat k < esize A" shows "nth_least_ext A k = nth_least A k"
parallel_subspace_aux: fixes A :: "'a set"   and B :: "'a set" assumes "subspace A"   and "subspace B"   and "affine_parallel A B" shows "B \<subseteq> A"
not_arr_Null: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "span_vertical_category C" shows "\<not> arrow_of_spans C \<lparr>Chn = partial_magma.null C, Dom = \<lparr>Leg0 = partial_magma.null C, Leg1 = partial_magma.null C\<rparr>, Cod = \<lparr>Leg0 = partial_magma.null C, Leg1 = partial_magma.null C\<rparr>\<rparr>"
e_length_0: fixes e :: "nat" assumes "e_length e = 0" shows "e = 0"
compress_interfaces_def: fixes ifces :: "iface negation_type list" shows "compress_interfaces ifces \<equiv> case compress_pos_interfaces (getPos ifces) of None \<Rightarrow> None | Some i \<Rightarrow> if \<exists>negated_ifce\<in>set (getNeg ifces). iface_subset i negated_ifce then None else if \<not> iface_is_wildcard i then Some ([i], []) else Some (if i = ifaceAny then [] else [i], getNeg ifces)"
cinner_simps_6: fixes x :: "'b"   and r :: "complex"   and y :: "'b" shows "x \<bullet>\<^sub>C r *\<^sub>C y = r * (x \<bullet>\<^sub>C y)"
addSubsetSet_def: fixes s :: "'b set"   and ts :: "'b set set" shows "addSubsetSet s ts = insert (s - \<Union> ts) ((\<inter>) s ` ts) \<union> (\<lambda>x. x - s) ` ts"
AssertionStatEq_def: fixes \<Psi> :: "'b"   and \<Psi>' :: "'b" shows "\<Psi> \<simeq> \<Psi>' \<equiv> AssertionStatImp \<Psi> \<Psi>' \<and> AssertionStatImp \<Psi>' \<Psi>"
ceq: shows "Consts (USubst.adjoint (dotsubstt \<theta>) I \<omega>) = Consts (repc I dotid (term_sem I \<theta> \<omega>))"
simps: fixes G :: "game"   and H :: "game" shows "ge_game (G, H) = (\<forall>x. if zin x (right_options G) then if zin x (left_options H) then \<not> (ge_game (H, x) \<or> ge_game (x, G)) else \<not> ge_game (H, x) else if zin x (left_options H) then \<not> ge_game (x, G) else True)"
grun_run: fixes A :: "('a, 'b) nba"   and w :: "'a stream"   and V :: "(nat \<times> 'b) set"   and k :: "nat"   and r :: "'b stream"   and p :: "'b" assumes "grun A w V (fromN (Suc k) ||| r) (k, p)" shows "run A (sdrop k w ||| r) p"
l: shows "revision_step r s\<^sub>1 s\<^sub>2"
get_owner_document_def: fixes ptr :: "(_) object_ptr" shows "get_owner_document\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M ptr = invoke local.CD.a_get_owner_document_tups ptr ()"
module_over_zeroring: fixes M :: "('e, 'a, 'f) Module_scheme" assumes "zeroring R"   and "R module M" shows "carrier M = {\<zero>\<^bsub>M\<^esub>}"
meth_C: shows "P \<turnstile> C sees M, b :  Ts\<rightarrow>T = (mxs, mxl\<^sub>0, ins, xt) in C"
afun_rel_intros_3: fixes A :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'d"   and y :: "'e" assumes "A x y" shows "rel_option A (Some x) (Some y)"
birr: shows "\<forall>b\<in>set bs. irreducible G b"
obj_simps_2: fixes a :: "'d" assumes "D.obj a" shows "src\<^sub>D a = a"
initial_def: fixes a :: "'s" shows "initial a \<equiv> ide a \<and> (\<forall>b. ide b \<longrightarrow> (\<exists>!f. \<guillemotleft>f : a \<rightarrow> b\<guillemotright>))"
prod_emb_trans: fixes J :: "'b set"   and K :: "'b set"   and L :: "'b set"   and M :: "'b \<Rightarrow> 'c measure"   and X :: "('b \<Rightarrow> 'c) set" assumes "J \<subseteq> K"   and "K \<subseteq> L" shows "prod_emb L M K (prod_emb K M J X) = prod_emb L M J X"
limit_cone_axioms: shows "D.has_as_pullback p0 p1"
cases: fixes \<alpha>n :: "'a \<Rightarrow> 'b list"   and defs :: "'a \<Rightarrow> 'b \<Rightarrow> 'c set"   and "uses" :: "'a \<Rightarrow> 'b \<Rightarrow> 'c set"   and phis :: "'a \<Rightarrow> 'b \<times> 'c \<Rightarrow> 'c list option"   and g :: "'a"   and a :: "'c" assumes "CFG_SSA_wf_base.liveVal \<alpha>n defs uses phis g a" obtains n :: "'b" and val :: "'c" where "a = val" and "n \<in> set (\<alpha>n g)" and "val \<in> uses g n"   | v :: "'c" and v' :: "'c" where "a = v'" and "CFG_SSA_wf_base.liveVal \<alpha>n defs uses phis g v" and "CFG_SSA_wf_base.phiArg \<alpha>n defs phis g v v'"
heun_remainder2_def: fixes p :: "real"   and x :: "real \<Rightarrow> 'f"   and f :: "real \<times> 'f \<Rightarrow> 'f"   and f'' :: "real \<times> 'f \<Rightarrow> (real \<times> 'f) \<Rightarrow>\<^sub>L (real \<times> 'f) \<Rightarrow>\<^sub>L 'e"   and t :: "real"   and h :: "real"   and s :: "real" shows "heun_remainder2 p x f f'' t h s = (h ^ 3 * p / 4) *\<^sub>R blinfun_apply (blinfun_apply (f'' (t + s * h * p, x t + (s * h * p) *\<^sub>R f (t, x t))) (1, f (t, x t))) (1, f (t, x t))"
distinct_45: fixes x3 :: "'a"   and x91 :: "nat"   and x92 :: "'a fm" shows "fm.Atom x3 \<noteq> ExN x91 x92"
prob_space_restrict_space: fixes S :: "'b set"   and M :: "'b measure" assumes "S \<in> sets M"   and "emeasure M S = 1" shows "prob_space (restrict_space M S)"
assms: shows "\<forall>x\<in>set xs. \<forall>y\<in>set (xs @ ys). P y x"
lemma_2_5_15: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "(a r\<rightarrow> b) * (b r\<rightarrow> c) \<le> a r\<rightarrow> c"
Un_Union_image: fixes A :: "'c \<Rightarrow> 'b set"   and B :: "'c \<Rightarrow> 'b set"   and C :: "'c set" shows "(\<Union>x\<in>C. A x \<union> B x) = \<Union> (A ` C) \<union> \<Union> (B ` C)"
fun_comp_eq_conv: fixes f :: "'e \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'e"   and fg :: "'c \<Rightarrow> 'd" shows "(f \<circ> g = fg) = (\<forall>x. f (g x) = fg x)"
vs: shows "vs \<equiv> \<lambda>\<phi>. getFrN (map snd txs) (map fst txs) [\<phi>] (length txs)"
mv_box_LE_def: shows "mv_box_LE \<equiv> inv_image Recursive.lex_triple mv_box_measure"
ih1: assumes "wt t \<ge>\<^sub>p wt s1" shows "vars s1 \<subseteq> vars t"
norm_rep: fixes f :: "'c local.term set" assumes "arr f" shows "\<^bold>\<parallel>rep f\<^bold>\<parallel> = rep f"
precedence_def: fixes prc :: "'a \<Rightarrow> 'a \<Rightarrow> bool \<times> bool"   and prl :: "'a \<Rightarrow> bool" shows "precedence prc prl \<equiv> ((\<forall>f. prc f f = (False, True)) \<and> SN {(f, g). fst (prc f g)} \<and> (\<forall>g f. prl g \<longrightarrow> snd (prc f g) = True)) \<and> (\<forall>f g. prl f \<longrightarrow> snd (prc f g) \<longrightarrow> prl g) \<and> (\<forall>f g s1 ns1 h s2 ns2 s ns. prc f g = (s1, ns1) \<longrightarrow> prc g h = (s2, ns2) \<longrightarrow> prc f h = (s, ns) \<longrightarrow> (ns1 \<and> ns2 \<longrightarrow> ns) \<and> (ns1 \<and> s2 \<longrightarrow> s) \<and> (s1 \<and> ns2 \<longrightarrow> s)) \<and> (\<forall>f g. fst (prc f g) \<longrightarrow> snd (prc f g))"
SD_preorder: fixes q :: "'b pmf"   and p :: "'b pmf" shows "SD (R' i) q p = (p \<in> lotteries_on alts' \<and> q \<in> lotteries_on alts' \<and> (\<forall>x\<in>alts'. measure_pmf.prob q (preferred_alts (R' i) x) \<le> measure_pmf.prob p (preferred_alts (R' i) x)))"
nemp: shows "wordinterval_to_set b1 \<noteq> {}"
nxt_accessible: fixes q :: "hf"   and a :: "'a" assumes "q \<in> accessible" shows "nxt M q a \<in> accessible"
induct: fixes x :: "event list"   and P :: "event list \<Rightarrow> bool" assumes "x \<in> yahalom"   and "P []"   and "\<And>evsf X B. \<lbrakk>evsf \<in> yahalom; P evsf; X \<in> synth (analz (knows Spy evsf))\<rbrakk> \<Longrightarrow> P (Says Spy B X # evsf)"   and "\<And>evsr A B X. \<lbrakk>evsr \<in> yahalom; P evsr; Says A B X \<in> set evsr\<rbrakk> \<Longrightarrow> P (Gets B X # evsr)"   and "\<And>evs1 NA A B. \<lbrakk>evs1 \<in> yahalom; P evs1; Nonce NA \<notin> used evs1\<rbrakk> \<Longrightarrow> P (Says A B \<lbrace>Agent A, Nonce NA\<rbrace> # evs1)"   and "\<And>evs2 NB B A NA. \<lbrakk>evs2 \<in> yahalom; P evs2; Nonce NB \<notin> used evs2; Gets B \<lbrace>Agent A, Nonce NA\<rbrace> \<in> set evs2\<rbrakk> \<Longrightarrow> P (Says B Server \<lbrace>Agent B, Crypt (shrK B) \<lbrace>Agent A, Nonce NA, Nonce NB\<rbrace>\<rbrace> # evs2)"   and "\<And>evs3 KAB B A NA NB. \<lbrakk>evs3 \<in> yahalom; P evs3; Key KAB \<notin> used evs3; KAB \<in> symKeys; Gets Server \<lbrace>Agent B, Crypt (shrK B) \<lbrace>Agent A, Nonce NA, Nonce NB\<rbrace>\<rbrace> \<in> set evs3\<rbrakk> \<Longrightarrow> P (Says Server A \<lbrace>Crypt (shrK A) \<lbrace>Agent B, Key KAB, Nonce NA, Nonce NB\<rbrace>, Crypt (shrK B) \<lbrace>Agent A, Key KAB\<rbrace>\<rbrace> # evs3)"   and "\<And>evs4 A K B NA NB X. \<lbrakk>evs4 \<in> yahalom; P evs4; A \<noteq> Server; K \<in> symKeys; Gets A \<lbrace>Crypt (shrK A) \<lbrace>Agent B, Key K, Nonce NA, Nonce NB\<rbrace>, X\<rbrace> \<in> set evs4; Says A B \<lbrace>Agent A, Nonce NA\<rbrace> \<in> set evs4\<rbrakk> \<Longrightarrow> P (Says A B \<lbrace>X, Crypt K (Nonce NB)\<rbrace> # evs4)"   and "\<And>evso A B K NA NB X. \<lbrakk>evso \<in> yahalom; P evso; Says Server A \<lbrace>Crypt (shrK A) \<lbrace>Agent B, Key K, Nonce NA, Nonce NB\<rbrace>, X\<rbrace> \<in> set evso\<rbrakk> \<Longrightarrow> P (Notes Spy \<lbrace>Nonce NA, Nonce NB, Key K\<rbrace> # evso)" shows "P x"
a_get_owner_document\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def: fixes document_ptr :: "(_) document_ptr"   and uu :: "unit" shows "a_get_owner_document\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r document_ptr uu = Heap_Error_Monad.bind document_ptr_kinds_M (\<lambda>document_ptrs. if document_ptr \<in> set document_ptrs then return document_ptr else error SegmentationFault)"
split_inside_simple_closed_curve: fixes c1 :: "real \<Rightarrow> complex"   and a :: "complex"   and b :: "complex"   and c2 :: "real \<Rightarrow> complex"   and c :: "real \<Rightarrow> complex" assumes "simple_path c1"   and "pathstart c1 = a"   and "pathfinish c1 = b"   and "simple_path c2"   and "pathstart c2 = a"   and "pathfinish c2 = b"   and "simple_path c"   and "pathstart c = a"   and "pathfinish c = b"   and "a \<noteq> b"   and "path_image c1 \<inter> path_image c2 = {a, b}"   and "path_image c1 \<inter> path_image c = {a, b}"   and "path_image c2 \<inter> path_image c = {a, b}"   and "path_image c \<inter> inside (path_image c1 \<union> path_image c2) \<noteq> {}" obtains "inside (path_image c1 \<union> path_image c) \<inter> inside (path_image c2 \<union> path_image c) = {}" and "inside (path_image c1 \<union> path_image c) \<union> inside (path_image c2 \<union> path_image c) \<union> (path_image c - {a, b}) = inside (path_image c1 \<union> path_image c2)"
lset_cases: fixes x :: "'c"   and xs :: "'c llist" assumes "x \<in> lset xs" obtains xs' :: "'c llist" where "xs = LCons x xs'__"   | x' :: "'c" and xs' :: "'c llist" where "xs = LCons x' xs'" and "x \<in> lset xs'"
that_1: fixes x :: "nat" assumes "Q = P"   and "n = m"   and "x = x" shows "thesis"
on_curve_def: fixes a :: "int"   and b :: "int"   and p :: "int point" shows "on_curve a b p = (case p of Infinity \<Rightarrow> True | Point x y \<Rightarrow> x \<in> carrier R \<and> y \<in> carrier R \<and> y [^] 2 = x [^] 3 \<oplus> a \<otimes> x \<oplus> b)"
A5: shows "\<lfloor>\<P> (\<lambda>x w. \<forall>xa. (\<forall>xb. (\<^bold>\<box>xb x) w = (\<^bold>\<box>(\<lambda>v. \<forall>x. (existsAt x \<^bold>\<rightarrow> xa x \<^bold>\<rightarrow> xb x) v)) w) \<longrightarrow> (\<^bold>\<box>mexistsActB xa) w)\<rfloor>"
c: shows "card {x \<in> A. Q x} < card {x \<in> A. P x}"
prob_space_distr: fixes f :: "'a cfg stream \<Rightarrow> 'b"   and s :: "'a cfg"   and M' :: "'b measure" assumes "MDP.MC.T.random_variable s M' f" shows "prob_space (distr (MDP.MC.T s) M' f)"
ex_sup: fixes le :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "dlat le" shows "\<exists>sup. dpo.is_sup le x y sup"
fin_def: fixes s :: "nat \<Rightarrow> 'b" shows "fin s \<equiv> \<exists>i. \<forall>j\<ge>i. s j = s i"
l11_13: 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"   and E :: "'a"   and F :: "'a"   and A' :: "'a"   and D' :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.CongA Bet Cong A B C D E F"   and "Bet A B A'"   and "A' \<noteq> B"   and "Bet D E D'"   and "D' \<noteq> E" shows "Tarski_neutral_dimensionless.CongA Bet Cong A' B C D' E F"
res_assoc_iff3: shows "(\<forall>x y z. x \<cdot> (y \<cdot> z) = x \<cdot> y \<cdot> z) = (\<forall>x y z. x \<rhd> y \<lhd> z = x \<rhd> (y \<lhd> z))"
base: fixes B :: "'a set"   and C :: "'a set" assumes "B \<subseteq> A"   and "f B = {}" shows "P C B []"
startsby_zero_power_iff: fixes a :: "'a fps"   and n :: "nat" shows "(fps_nth (a ^ n) 0 = (0::'a)) = (n \<noteq> 0 \<and> fps_nth a 0 = (0::'a))"
fresh_type_subst_fresh: fixes X :: "tyvrs"   and T' :: "ty"   and T :: "ty" assumes "X \<sharp> T'" shows "X \<sharp> T[X \<mapsto> T']\<^sub>\<tau>"