hcju/isabelleps huggingface dataset:
validCI_cont: fixes cid :: "'id"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat"   and \<gamma> :: "(nat \<Rightarrow> 'cmp) \<Rightarrow> nat \<Rightarrow> bool"   and t' :: "nat \<Rightarrow> 'cmp" assumes "\<exists>i. \<parallel>cid\<parallel>\<^bsub>t i\<^esub>"   and "\<not> (\<exists>i'\<ge>n. \<parallel>cid\<parallel>\<^bsub>t i'\<^esub>)"   and "\<gamma> (lnth (\<pi>\<^bsub>cid\<^esub>inf_llist t @\<^sub>l inf_llist t')) (\<^bsub>cid\<^esub>\<down>\<^bsub>t\<^esub>n)" shows "eval cid t t' n \<gamma>"
document_ptr_casts_commute: fixes ptr :: "(_) object_ptr"   and document_ptr :: "(_) document_ptr" shows "(cast ptr = Some document_ptr) = (cast document_ptr = ptr)"
typing_alt_simps_11: fixes \<Gamma> :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type"   and v :: "String.literal"   and \<tau> :: "'a OCL_Types.type option"   and init :: "'a expr"   and body :: "'a expr"   and \<sigma> :: "'a OCL_Types.type" shows "(\<Gamma> \<turnstile>\<^sub>E expr.Let v \<tau> init body : \<sigma>) = (\<exists>\<sigma>' \<tau>'. \<tau> = Some \<tau>' \<and> \<Gamma> \<turnstile>\<^sub>E init : \<sigma>' \<and> \<sigma>' \<le> \<tau>' \<and> \<Gamma>(v \<mapsto>\<^sub>f \<tau>') \<turnstile>\<^sub>E body : \<sigma>)"
rep_eq: fixes x :: "nat \<Rightarrow>\<^sub>0 nat"   and xa :: "'a" shows "mapping_of (monom x xa) = monomial xa x"
cobounded2: fixes a :: "'d"   and b :: "'d" shows "a \<sqinter> b \<le> b"
Caratheodory_theorem: fixes A :: "'a vec set" assumes "A \<subseteq> carrier_vec n" shows "cone A = \<Union> {finite_cone B |B. B \<subseteq> A \<and> lin_indpt B}"
assms_1: shows "ground\<^sub>l\<^sub>s C\<^sub>1' \<and> falsifies\<^sub>g (B @ [d]) C\<^sub>1'"
fminus_single_finsert: fixes A :: "'a fset"   and x :: "'a"   and B :: "'a fset" assumes "A |-| {|x|} |\<subseteq>| B" shows "A |\<subseteq>| finsert x B"
IH: fixes xs :: "bool list" assumes "n < harry_meas xs" shows "True \<in> set ((harry_step ^^ n) xs)"
crit_pair_cbelow_distinct_component: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c"   and d :: "'a \<Rightarrow> nat"   and m :: "nat"   and F :: "('t \<Rightarrow>\<^sub>0 'c) set" assumes "component_of_term (lt p) \<noteq> component_of_term (lt q)" shows "crit_pair_cbelow_on d m F p q"
left_adjoint_determines_right_up_to_iso: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and f :: "'b"   and g :: "'b"   and g' :: "'b" assumes "strict_bicategory V H \<a> \<i> src trg"   and "bicategory.adjoint_pair V H \<a> \<i> src trg f g"   and "bicategory.adjoint_pair V H \<a> \<i> src trg f g'" shows "category.isomorphic V g g'"
match_4: fixes g :: "'c"   and f :: "'c"   and h :: "'c" assumes "C.seq g f"   and "C.seq h g" shows "C.seq h (C g f)"
Re_winding_number_poly_part_circlepath: fixes p :: "complex poly"   and z0 :: "complex" assumes "0 < degree p" shows "((\<lambda>r. Re (winding_number (poly p \<circ> part_circlepath z0 r 0 pi) 0)) \<longlongrightarrow> real (degree p) / 2) at_top"
iT_Div_infinite_iff: fixes k :: "nat"   and I :: "nat set" assumes "0 < k" shows "infinite (I \<oslash> k) = infinite I"
pair_sigma_finite_def: fixes M1 :: "'d measure"   and M2 :: "'e measure" shows "pair_sigma_finite M1 M2 \<equiv> sigma_finite_measure M1 \<and> sigma_finite_measure M2"
cint: fixes x :: "complex" assumes "x \<noteq> w"   and "cmod (x - w) < d" shows "(\<lambda>z. (f' z / (z - x) ^ k - f' z / (z - w) ^ k) / (x * of_nat k - w * of_nat k)) contour_integrable_on \<gamma>"
collapse_2: fixes step :: "('e, 'f) step" assumes "is_Yield step" shows "Yield (un_Yield1 step) (un_Yield2 step) = step"
SourcesS4_L2: shows "Sources level2 sS4 = {sS2}"
gr_implies_not_zero: fixes m :: "'q multiset"   and n :: "'q multiset" assumes "m \<subset># n" shows "n \<noteq> {#}"
ineq: shows "map of_int_hom.vec_hom fs ! i - gs2.gso i + rat_of_int c \<cdot>\<^sub>v map of_int_hom.vec_hom fs ! j - rat_of_int c \<cdot>\<^sub>v map of_int_hom.vec_hom fs ! j = map of_int_hom.vec_hom fs ! i - gs2.gso i"
brD: fixes c :: "'b"   and a :: "'c"   and \<alpha> :: "'b \<Rightarrow> 'c"   and I :: "'b \<Rightarrow> bool" assumes "(c, a) \<in> br \<alpha> I" shows "a = \<alpha> c \<and> I c"
is_processT3_SR: fixes s :: "'b event list"   and t :: "'b event list"   and X :: "'b event set"   and P :: "'b process" assumes "(s @ t, X) \<in> F P" shows "(s, {}) \<in> F P"
block_sizes: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and sizes :: "int set"   and bl :: "'b set" assumes "K_block_design point_set block_collection sizes"   and "bl \<in># block_collection" shows "int (card bl) \<in> sizes"
LeftDerivationIntro_LeftDerives1: fixes \<alpha> :: "symbol list"   and i :: "nat"   and r :: "symbol \<times> symbol list"   and ix :: "nat"   and D :: "(nat \<times> symbol \<times> symbol list) list"   and j :: "nat"   and \<gamma> :: "symbol list"   and a1 :: "symbol list"   and A :: "symbol"   and a2 :: "symbol list" assumes "LeftDerivationIntro \<alpha> i r ix D j \<gamma>"   and "splits_at \<alpha> i a1 A a2" shows "LeftDerives1 \<alpha> i r (a1 @ snd r @ a2)"
card_extensional_funcset_surj_on: fixes A :: "'c set"   and B :: "'d set" assumes "finite A"   and "finite B" shows "card {f \<in> A \<rightarrow>\<^sub>E B. f ` A = B} = fact (card B) * Stirling (card A) (card B)"
lunit_def: fixes a :: "'a" shows "MC.lunit a \<equiv> THE f. \<guillemotleft>f : MC.tensor MC.unity a \<rightarrow> a\<guillemotright> \<and> MC.tensor MC.unity f = MC.tensor \<iota> a \<cdot> MC.assoc' MC.unity MC.unity a"
star_one_def: shows "1 \<equiv> star_of (1::'b)"
Int_Diff: fixes A :: "'b set"   and B :: "'b set"   and C :: "'b set" shows "A \<inter> B - C = A \<inter> (B - C)"
IH: assumes "Var x \<sqsubseteq>\<^sub>s\<^sub>e\<^sub>t trms\<^sub>s\<^sub>s\<^sub>t A" shows "x \<in> vars\<^sub>s\<^sub>s\<^sub>t A"
id_take_nth_drop: fixes i :: "nat"   and xs :: "'a list" assumes "i < |xs|" shows "xs = take i xs @ xs ! i # drop (Suc i) xs"
rtrancl_0: fixes pair_of_term :: "'c \<Rightarrow> 'd \<times> 'e"   and term_of_pair :: "'d \<times> 'e \<Rightarrow> 'c"   and ord :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_strict :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_term :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ord_term_strict :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and F :: "('c \<Rightarrow>\<^sub>0 'f) set"   and x :: "'c \<Rightarrow>\<^sub>0 'f" assumes "ordered_term pair_of_term term_of_pair ord ord_strict ord_term ord_term_strict"   and "(ordered_term.red pair_of_term term_of_pair ord_term F)\<^sup>*\<^sup>* 0 x" shows "x = 0"
sup_left_commute: fixes sup :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and less_eq :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and less :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and x :: "'b::type"   and y :: "'b::type"   and z :: "'b::type" assumes "class.semilattice_sup (sup::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (less::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool)" shows "(sup::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (x::'b::type) (sup (y::'b::type) (z::'b::type)) = sup y (sup x z)"
card_Diff_insert: fixes A :: "'a set"   and a :: "'a"   and B :: "'a set" assumes "finite A"   and "a \<in> A"   and "a \<notin> B" shows "card (A - insert a B) = card (A - B) - 1"
globEA: fixes n :: "nat"   and c :: "'id"   and t :: "nat \<Rightarrow> cnf"   and t' :: "nat \<Rightarrow> 'cmp"   and \<gamma> :: "(nat \<Rightarrow> 'cmp) \<Rightarrow> nat \<Rightarrow> bool"   and n' :: "nat" assumes "\<exists>i\<ge>n. \<parallel>c\<parallel>\<^bsub>t i\<^esub>"   and "eval c t t' n (\<box>\<^sub>b\<gamma>)"   and "\<langle>c \<Leftarrow> t\<rangle>\<^bsub>n\<^esub> \<le> n'" shows "eval c t t' n' \<gamma>"
of_int_1_le_iff: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and z :: "int" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less_eq one (ring_1.of_int one plus zero minus z) = (1 \<le> z)"
distinct_21: fixes x2 :: "nat"   and x71 :: "nat"   and x72 :: "msg" shows "Number x2 \<noteq> Crypt x71 x72"
ordinal_plus_minus1: fixes x :: "ordinal"   and y :: "ordinal" shows "x + y - x = y"
ord_le_eq_trans: fixes a :: "'e"   and b :: "'e"   and c :: "'e" assumes "a \<le> b"   and "b = c" shows "a \<le> c"
psubst_preserves_wls: fixes s :: "'sort"   and X :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set option" assumes "wls s X"   and "wlsEnv rho" shows "wls s X #[rho]"
ex_k_mod: fixes e :: "nat"   and P :: "nat"   and Q :: "nat"   and d :: "nat" assumes "coprime e ((P - 1) * (Q - 1))"   and "P \<noteq> Q"   and "prime P"   and "prime Q"   and "d \<noteq> 0"   and "[e * d = 1] (mod P - 1)" shows "\<exists>k. e * d = 1 + k * (P - 1)"
dyn_control_dependence_path: fixes n :: "'node"   and n' :: "'node"   and as :: "'edge list" assumes "n controls n' via as" shows "n -as\<rightarrow>* n' \<and> as \<noteq> []"
h: fixes n :: "nat" shows "set_integrable lborel {0..1} (\<lambda>x. sum ((^) x) {..<n})"
zhmsetmset_diff: fixes x :: "zhmultiset"   and xa :: "zhmultiset" shows "zhmsetmset (x - xa) = zhmsetmset x - zhmsetmset xa"
simps_3: fixes Obj :: "'a set"   and Hom :: "'a \<Rightarrow> 'a \<Rightarrow> 'b set"   and Id :: "'a \<Rightarrow> 'b"   and Comp :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x21 :: "'c"   and x22 :: "'c"   and x23 :: "'d" assumes "concrete_category Obj Hom Id Comp" shows "concrete_category.MkArr x21 x22 x23 \<noteq> concrete_category.Null"
gFreshGOpRev_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd, 'e) Recursion.model_scheme" shows "gFreshGOpRev MOD \<equiv> \<forall>ys y delta inp' inp binp' binp. wlsInp delta inp' \<and> gWlsInp MOD delta inp \<and> wlsBinp delta binp' \<and> gWlsBinp MOD delta binp \<longrightarrow> gFresh MOD ys y (Op delta inp' binp') (gOp MOD delta inp' inp binp' binp) \<longrightarrow> gFreshInp MOD ys y inp' inp \<and> gFreshBinp MOD ys y binp' binp"
closed_zero: shows "(0::'b) \<in> U\<^sub>M"
bind_fv2_changed: fixes v :: "variable \<times> typ"   and lev :: "nat"   and tm :: "Core.term" assumes "bind_fv2 v lev tm \<noteq> tm" shows "v \<in> fv tm"
name_calc_3: fixes a :: "name"   and b :: "name"   and c :: "name" shows "swap (a, b) c = (if a = c then b else if b = c then a else c)"
subset_proj_absorb: fixes vs1 :: "'a set"   and vs2 :: "'a set"   and PROB :: "(('a, 'b) fmap \<times> ('a, 'c) fmap) set" assumes "vs1 \<subseteq> vs2" shows "prob_proj (prob_proj PROB vs2) vs1 = prob_proj PROB vs1"
find_index_in_set: fixes x :: "'a"   and xs :: "'a list" assumes "x \<in> set xs" shows "xs ! find_index xs x = x"
row_def: fixes i :: "'d"   and A :: "(('b, 'c) Finite_Cartesian_Product.vec, 'd) Finite_Cartesian_Product.vec" shows "row i A = (\<chi>j. A $h i $h j)"
assms: shows "p \<subseteq> {s. f s \<in> q}"
stop_seq_rel: fixes c\<^sub>A :: "(Eg1.var, aexp, bexp) Stmt"   and c\<^sub>A_tail :: "(Eg1.var, aexp, bexp) Stmt"   and c\<^sub>C :: "(var\<^sub>C, aexp\<^sub>C, bexp\<^sub>C) Stmt"   and c\<^sub>C_tail :: "(var\<^sub>C, aexp\<^sub>C, bexp\<^sub>C) Stmt"   and mds\<^sub>A :: "Mode \<Rightarrow> Eg1.var set"   and mds\<^sub>C :: "Mode \<Rightarrow> var\<^sub>C set"   and mem\<^sub>A :: "Eg1.var \<Rightarrow> nat"   and mem\<^sub>C :: "var\<^sub>C \<Rightarrow> nat" assumes "c\<^sub>A = Stop ;; c\<^sub>A_tail"   and "c\<^sub>C = Stop ;; c\<^sub>C_tail"   and "mds\<^sub>A = mds\<^sub>A_of mds\<^sub>C"   and "mem\<^sub>A = mem\<^sub>A_of mem\<^sub>C"   and "RelS_Eg1Eg2p c\<^sub>A_tail mds\<^sub>A mem\<^sub>A c\<^sub>C_tail mds\<^sub>C mem\<^sub>C" shows "RelS_Eg1Eg2p c\<^sub>A mds\<^sub>A mem\<^sub>A c\<^sub>C mds\<^sub>C mem\<^sub>C"
not_ordLeq_iff_ordLess: fixes r :: "('f \<times> 'f) set"   and r' :: "('g \<times> 'g) set" assumes "Well_order r"   and "Well_order r'" shows "((r', r) \<notin> ordLeq) = (r <o r')"
assert_SBE_def: fixes e :: "'b \<Rightarrow> bool" shows "assert\<^sub>S\<^sub>B\<^sub>E e = (\<lambda>\<sigma>. if e \<sigma> then Some {((), \<sigma>)} else None)"
intro: fixes g :: "('c \<times> nat, 'd) rgraph_scheme"   and subs :: "(('c \<times> nat) \<times> 'c \<times> nat) set" assumes "sub_rel_of g subs"   and "wf_sub_rel subs" shows "wf_sub_rel_of g subs"
Base_foo_not_inherited_in_Ext: shows "\<not> tprg \<turnstile> Ext inherits (Base, mdecl Base_foo)"
1: shows "(r|\<sigma> @ \<tau>'|, r|\<sigma>| + r|\<tau>|) \<in> mul_eq r"
wp3EndMT: fixes p :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list"   and xs :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators"   and a :: "(int \<times> int \<times> protocol) set set"   and b :: "(int \<times> int \<times> protocol) set set"   and po :: "protocol \<times> int" assumes "wellformed_policy3Pr (p @ [xs])"   and "AllowPortFromTo a b po \<in> set p" shows "dom (Cp (AllowPortFromTo a b po)) \<inter> dom (Cp xs) = {}"
r_hd_prim: shows "prim_recfn 1 r_hd"
mat_multrow_gen_def: fixes mul :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and k :: "nat"   and a :: "'b"   and A :: "'b mat" shows "mat_multrow_gen mul k a A = mat (dim_row A) (dim_col A) (\<lambda>(i, j). if k = i then mul a (A $$ (i, j)) else A $$ (i, j))"
conforms_NormI: fixes x :: "abopt"   and s :: "st"   and G :: "prog"   and L :: "lname \<Rightarrow> ty option" assumes "(x, s)\<Colon>\<preceq>(G, L)" shows "Norm s\<Colon>\<preceq>(G, L)"
last_map: fixes xs :: "'a list"   and f :: "'a \<Rightarrow> 'b" assumes "xs \<noteq> []" shows "last (map f xs) = f (last xs)"
ideD_1: fixes a :: "('o, 'a) arr" assumes "ide a" shows "arr a"
fg_2: shows "g $ 0 * f $ 0 = (1::'a)"
treap_ins: fixes t :: "('a \<times> 'b) tree"   and k :: "'a"   and p :: "'b" assumes "treap t" shows "treap (ins k p t)"
smult_normalize_field_eq: fixes p :: "'a poly" shows "p = smult (lead_coeff p) (normalize p)"
assms_4: shows "Idem\<^bsub>L\<^esub> f"
StatesCarAudioSystem: shows "HAStates CarAudioSystem = {''SelectingPreviousTrack'', ''SelectingNextTrack'', ''Playing'', ''4'', ''3'', ''2'', ''1'', ''On'', ''Off'', ''CDEmpty'', ''CDFull'', ''ReadTracks'', ''CarAudioSystem''}"
Pi_cong: fixes A :: "'e set"   and f :: "'e \<Rightarrow> 'f"   and g :: "'e \<Rightarrow> 'f"   and B :: "'e \<Rightarrow> 'f set" assumes "\<And>w. w \<in> A \<Longrightarrow> f w = g w" shows "(f \<in> Pi A B) = (g \<in> Pi A B)"
\<phi>_\<psi>: fixes b :: "'c"   and a :: "'c"   and x :: "'s" assumes "C.ide b"   and "C.ide a"   and "x \<in> HomC.set (b, a)" shows "\<phi>C (b, a) (\<psi>C (b, a) x) = x"
finprod_one_eqI: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "\<And>x. x \<in> A \<Longrightarrow> f x = (0::'b)" shows "sum f A = (0::'b)"
Sup_inter_less_eq: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and A :: "'c set"   and B :: "'c set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "less_eq (Sup (A \<inter> B)) (inf (Sup A) (Sup B))"
fres_bot_bot: shows "\<bottom> \<leftarrow> \<bottom> = \<top>"
IH2_2: shows "factors_of_factor_tree (product_factor_tree p r) = factors_of_factor_tree r"
J: shows "P \<turnstile> (E, ws) weakly_justified_by J"
elims: fixes x :: "('b \<times> nat) tree"   and y :: "nat" assumes "size_wbt x = y" obtains "x = \<langle>\<rangle>" and "y = 0"   | uu_ :: "('b \<times> nat) tree" and uv_ :: "'b" and n :: "nat" and uw_ :: "('b \<times> nat) tree" where "x = \<langle>uu_, (uv_, n), uw_\<rangle>" and "y = n"
x1: shows "x = lc j \<cdot>\<^sub>v us ! j + (lc i \<cdot>\<^sub>v us ! i + M1)"
sig_red_singleD7: fixes sing_reg :: "'t \<Rightarrow> 't \<Rightarrow> bool"   and top_tail :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and p :: "'t \<Rightarrow>\<^sub>0 'b"   and q :: "'t \<Rightarrow>\<^sub>0 'b"   and f :: "'t \<Rightarrow>\<^sub>0 'b"   and t :: "'a" assumes "sig_red_single sing_reg top_tail p q f t" shows "top_tail (t + punit.lt (rep_list f)) (punit.lt (rep_list p))"
simulation_axioms_def: fixes as\<^sub>0 :: "'e"   and alstep :: "'f \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> bool"   and cs\<^sub>0 :: "'g"   and clstep :: "'f \<Rightarrow> 'g \<Rightarrow> 'g \<Rightarrow> bool"   and R :: "'e \<Rightarrow> 'g \<Rightarrow> bool" shows "simulation_axioms as\<^sub>0 alstep cs\<^sub>0 clstep R \<equiv> (\<forall>c. system.reachable cs\<^sub>0 clstep c \<longrightarrow> (\<exists>a. R a c \<and> system.reachable as\<^sub>0 alstep a)) \<and> (\<forall>cs. df_system.is_run cs\<^sub>0 clstep cs \<longrightarrow> (\<exists>as. (\<forall>i. R (as i) (cs i)) \<and> df_system.is_run as\<^sub>0 alstep as)) \<and> (\<forall>cs. df_system.is_fair_run cs\<^sub>0 clstep cs \<longrightarrow> (\<exists>as. (\<forall>i. R (as i) (cs i)) \<and> df_system.is_fair_run as\<^sub>0 alstep as))"
simps_2: fixes a :: "'a \<times> ('b, 'c) fmap"   and as :: "('a \<times> ('b, 'c) fmap) list" shows "no_effectless_act (a # as) = (fmdom' (snd a) \<noteq> {} \<and> no_effectless_act as)"
Done_eq_ABORT: fixes a :: "'f option" shows "(Resumption.resumption.Done a = ABORT) = (a = None)"
simps_7: fixes x :: "nat"   and P :: "pol"   and Q :: "pol"   and y :: "nat"   and R :: "pol" shows "Pinj x P \<langle>*\<rangle> PX Q y R = (if x = 0 then P \<langle>*\<rangle> PX Q y R else if x = 1 then mkPX (Pinj x P \<langle>*\<rangle> Q) y (R \<langle>*\<rangle> P) else mkPX (Pinj x P \<langle>*\<rangle> Q) y (R \<langle>*\<rangle> Pinj (x - 1) P))"
ih: fixes ta :: "('s, 'v) tm"   and sa :: "('s, 'v) tm"   and i :: "nat" assumes "{#size ta, size sa#} < {#size t, size s#}"   and "ta >\<^sub>t sa" shows "subst_db i x ta >\<^sub>t subst_db i x sa"
P2_security: fixes x :: "nat"   and n :: "nat"   and y :: "nat" assumes "x < q n"   and "y < q n" shows "local.sim_non_det_def.perfect_sec_P2 n x y"
y: shows "y \<in> Collect (root_cond (p, l, r))"
assms_3: shows "pl.unique_factorization_m f (lead_coeff f, mset us)"
vsv_vrange_vsubset: fixes A :: "V" assumes "\<And>x. x \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> vrat_uminus \<Longrightarrow> -\<^sub>\<rat> x \<in>\<^sub>\<circ> A" shows "\<R>\<^sub>\<circ> vrat_uminus \<subseteq>\<^sub>\<circ> A"
Class_equals_Block: fixes S :: "'c set"   and P :: "'c set set" assumes "partition S P" shows "equivalence.Class S (partition.Equivalence P) = partition.Block S P"
pseudo_mod_hom: fixes p :: "'b poly"   and q :: "'b poly" shows "pseudo_mod (map_poly to_fract p) (map_poly to_fract q) = map_poly to_fract (pseudo_mod p q)"
permutation_swap_id: fixes a :: "'b"   and b :: "'b" shows "permutation (Fun.swap a b id)"
commutativity: fixes xs :: "'a list" assumes "op_history xs" shows "concurrent_ops_commute xs"
mismatchI: fixes u :: "'b list"   and v :: "'b list"   and i :: "nat" assumes "\<not> u \<bowtie> v"   and "i < min \<^bold>|u\<^bold>| \<^bold>|v\<^bold>|"   and "take i u = take i v"   and "u ! i \<noteq> v ! i" shows "mismatch_pair u v = (u ! i, v ! i)"
step: fixes x :: "'a node \<times> 'a node set" assumes "((\<forall>nm\<in>incoming (fst x). nm < name (fst x)) \<and> 0 < name (fst x) \<and> (\<forall>q\<in>snd x. name q < name (fst x) \<and> (\<forall>nm\<in>incoming q. nm < name (fst x)))) \<and> expand_name_ident (snd x)" shows "f x \<le> SPEC (\<lambda>a. (\<forall>q\<in>snd a. name q < fst a \<and> (\<forall>nm'\<in>incoming q. nm' < fst a)) \<and> ((name (fst x) \<le> fst a \<and> name ` snd x \<subseteq> name ` snd a) \<and> name ` snd a = name ` snd x \<union> name ` {nd \<in> snd a. name (fst x) \<le> name nd}) \<and> expand_name_ident (snd a))"
empty: fixes g :: "'l \<Rightarrow> 'k" shows "sum g {} = (0::'k)"
simps_2: fixes nodes :: "'b set"   and edges :: "('b \<times> 'b) set"   and more :: "'c" shows "edges \<lparr>nodes = nodes, edges = edges, \<dots> = more\<rparr> = edges"
h4: shows "satisfy_equation_R p (qs1 @ qs2) (get_subsets_R (snd (combine_systems_R p (qs1, calculate_data_R p qs1) (qs2, calculate_data_R p qs2)))) (get_signs_R (snd (combine_systems_R p (qs1, calculate_data_R p qs1) (qs2, calculate_data_R p qs2))))"
irreducible_normalized_divisors: 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 x :: "'a"   and y :: "'a" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize"   and "comm_semiring_1.irreducible times one zero x"   and "dvd.dvd times y x"   and "normalize y = y" shows "y = one \<or> y = normalize x"
inj_image_subset_iff: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and B :: "'b set" assumes "inj f" shows "(f ` A \<subseteq> f ` B) = (A \<subseteq> B)"
star_sum_unfold: fixes x :: "'a"   and y :: "'a" shows "x\<^sup>\<star> + x\<^sup>\<star> ; y ; (x + y)\<^sup>\<star> = (x + y)\<^sup>\<star>"
distributed_real_measurable: fixes N :: "'d measure"   and f :: "'d \<Rightarrow> real"   and M :: "'e measure"   and X :: "'e \<Rightarrow> 'd" assumes "\<And>x. x \<in> space N \<Longrightarrow> 0 \<le> f x"   and "distributed M N X (\<lambda>x. ennreal (f x))" shows "f \<in> borel_measurable N"
coalg_F2set2: fixes B1 :: "'d set"   and B2 :: "'e set"   and s1 :: "'d \<Rightarrow> ('f, 'd, 'e) F1"   and s2 :: "'e \<Rightarrow> ('f, 'd, 'e) F2"   and a :: "'e" assumes "coalg B1 B2 s1 s2"   and "a \<in> B2" shows "F2set2 (s2 a) \<subseteq> B1"
foldl_cons_def: fixes z :: "'a"   and s :: "'s" shows "local.foldl_cons z s = foldl f z (unstream g s)"
add_less_cancel_right: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and c :: "'a"   and b :: "'a" assumes "class.ordered_ab_semigroup_add_imp_le plus minus less_eq less" shows "less (plus a c) (plus b c) = less a b"
multiple_orig_sub_system: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and n :: "nat" assumes "incidence_system point_set block_collection"   and "0 < n" shows "sub_incidence_system point_set block_collection point_set (repeat_mset n block_collection)"
eq_def: fixes \<phi> :: "('state \<times> 'aprop set) stream list \<Rightarrow> bool"   and \<psi> :: "('state \<times> 'aprop set) stream list \<Rightarrow> bool" shows "eq \<phi> \<psi> \<equiv> con (imp \<phi> \<psi>) (imp \<psi> \<phi>)"
component_trans: fixes F :: "'b program"   and G :: "'b program"   and H :: "'b program" assumes "F \<le> G"   and "G \<le> H" shows "F \<le> H"
dual_ball: fixes A :: "'b set"   and P :: "'b dual \<Rightarrow> bool" shows "(\<forall>x\<in>A. P (dual x)) = (\<forall>x'\<in>dual ` A. P x')"
trans_less_nmultiset: fixes X :: "'a nmultiset"   and Y :: "'a nmultiset"   and Z :: "'a nmultiset" assumes "X < Y"   and "Y < Z" shows "X < Z"
ipurge_tr_ref_aux_failures: fixes P :: "'c process"   and I :: "('d \<times> 'd) set"   and D :: "'c \<Rightarrow> 'd"   and xs :: "'c list"   and X :: "'c set"   and U :: "'d set" assumes "secure P I D"   and "(xs, X) \<in> failures P" shows "(ipurge_tr_aux I D U xs, ipurge_ref_aux I D U xs X) \<in> failures P"
False: shows "even (order c p) \<noteq> sign_r_pos p c"
mset_zero_iff_right: fixes x :: "'c list" shows "({#} = mset x) = (x = [])"